4 Puck s action plane fracture criteria

Transcription

1 4 Puk s ation plane frature riteria 4. Fiber frature riteria Fiber frature is primarily aused by a stressing σ whih ats parallel to the fibers. For (σ, σ, τ )-ombinations the use of a simple maximum stress formulation is reommendable. Suh a formulation was already proposed by Puk in 969 [Puk 969, Puk and Shneider 969]. It expresses the physial idea that fiber frature under multiaxial stresses in a UD-lamina ours when its stress parallel to the fibers σ is equal to or exeeds the stress neessary for a frature under uniaxial stress σ. From this hypothesis follows the simple FF-ondition σ = σ > t R σ ( R ) for 0 = for σ < 0 (Eq. 4) t If σ is a tensile stress, the strength R is used in (Eq. 4). If σ is a ompressive stress, R is used instead. Strengths are always defined as positive values. This is the reason for using the - sign at R in (Eq. 4). If σ reahes the frature stress σ fr (= R t or R ) the frature ondition is fulfilled. (Eq. 4) an easily be transformed to a frature riterion being formulated with the stress exposure f E,FF : f f σ = for σ 0 E, FF t R = σ R ) for σ < 0 E, FF ( (Eq. 5) For a preliminary analysis this simple frature riterion is suffiient. However, for a preise analysis a more sophistiated formulation an be

2 38 4 Puk s ation plane frature riteria useful whih takes into aount seondary effets, too. These effets are disussed in the following: A uniaxial σ -stress (or σ 3 -stress, respetively) leads due to a Poisson s effet to an additional miro-mehanial strain in fiber diretion ε. The effet is enlarged loally by the irumstane that the stress in the matrix is inhomogeneously distributed and effetively larger than the transverse stress σ (σ 3, respetively) on the lamina-level. This effet of stress and strain magnifiation an best be explained using the ommon model of serial springs. If a thin UD-lamina (whih onsists alternately of fibers and matrix) is stressed transverse to the fiber diretion, the load is on an intersetion without fibers arried by the matrix alone. On a neighboring intersetion, the load is arried by the fiber alone [Puk and Shneider 969]. From this simple model follows that the stress σ is the same in fiber and matrix and that the miro-mehanial strain is different due to the different Young s moduli. However, lose to the interfae between fiber and matrix there is additional biaxial strain in the matrix. This effet is taken into aount by the use of a magnifiation fator m σ,f. Puk proposes m σ,f =.3 for GFRP and m σ,f =. for CFRP 6 [Puk 996, Fisher 003]. Puk based his more sophistiated FF-ondition on the following frature hypothesis [Puk 996, Puk and Shürmann 998, Puk and Shürmann 00]: Fiber frature of a UD-lamina under ombined stresses will our when in the fibers the same stress σ f is reahed whih is ating in the fibers at an FF of the lamina aused by uniaxial tensile stress σ t or a uniaxial ompressive stress σ respetively. The starting point is the strain ε f of the fibers aused by the ombined stresses σ, σ, σ 3 : σ ν ε = σ + σ (Eq. 6) ( ) f f f mσ f 3 E f E f Using ν f /E f = ν f /E f, and ε f = ε, the stress σ f in the fibers ating in their longitudinal diretion an be alulated from σ f = E f ε+ ν f m σ f ( σ + σ3) (Eq. 7) 6 The value for CFRP is smaller beause the transverse Young s modulus of arbon fibers is lower than that of glass fibers.

3 4. Fiber frature riteria 39 From (Eq. 7) a new frature ondition an be derived. First ε and the fiber stress σ f are replaed by the elasti law of the UD-lamina and the frature resistane R f of the fibre, respetively: σ ν 3 E E ( ) ε = σ +σ (Eq. 8) R f E f e ; R = E e = with e = frature strain from uniaxial σ of both fiber and UD-lamina Based on this the FF-ondition for the UD-lamina is: E t, σ ν ( ) ν f mσ f σ +σ = 3 ± R E f t R for [...] > 0 R for... < 0 [ ] with (Eq. 9) (Eq. 0) This frature ondition (Eq. 0) is homogeneous of grade and an thus easily be transformed to a formulation for the stress exposure f E,FF : E fe,ff = t, σ ν ( ) ν f mσ f σ +σ 3 ± R E with f (Eq. ) t R for [...] 0 R for... < 0 [ ] Here, f E,FF is the FF-stress exposure of the UD-lamina, ν is the major Poisson s ratios of the UD-lamina, and ν f the major Poisson s ratio of the fiber. In general the definition of the Poisson s ratios is a well known soure of misunderstandings, beause there are different definitions used. In the US for instane, it is ommon standard to define the indies of ν the other way round (the first index for the diretion of the stress whih leads to the Poisson effet). In the above Equations E is the longitudinal modulus of the lamina parallel to the fibers and E f the longitudinal modulus of the fibers. m σf is a magnifiation fator for the transverse stress in the fibers (GFRP m σf,3 and for CFRP m σf,). Puk has found that in ase of plane (σ, σ, τ )- stress, the results of (Eq. 5) and (Eq. ) differ only by a few perent. However, the influene of transverse stresses on FF an beome important in the region of ombined σ <0 and σ 3 <0 of similar magnitude, where σ

4 40 4 Puk s ation plane frature riteria and σ 3 an exeed the transverse ompressive strength R by a fator of up to 3 or 4 [Kopp 000]). The maximum stress riteria (Eq. 5) and (Eq. ) presented above were put into question by Hart-Smith [Hart-Smith 998a, Hart-Smith 998b] who prolaimed that the frature stress σ fr is onsiderably redued in the presents of high transverse strains with a different algebrai sign ompared to the strain in fiber diretion ε. Hart-Smith based this on some experimental work [Hart-Smith 984]. However, this hypothesis has at least for monotonously inreasing loads been disproved by sophistiated experimental work [Fisher 003; Mannigel 007] whih approves the appropriateness of maximum stress riteria. A further aspet to be onsidered is the influene of shear stresses on the fiber frature. Whereas tensile stressing in fiber diretion σ t leads unquestionably to fiber rupture, the frature mode is not as obvious in the ase of ompressive stressing σ. In priniple, several forms of fiber frature are possible. The dominant failure mode is kinking of the fibers due to shear stresses in the fiber [Shürmann 004; Pinho et al 006]. Consequently, longitudinal shear stressing of the UD-lamina τ redues the ompressive frature strength σ fr. In [Puk 996] and [Fisher 003] pure empirial formulations are given to aount for this effet. Mannigel [Mannigel 007] quantified the effet for the first time by intensive experimental investigations. He found that for relatively small shear stresses there is no influene of τ on the fiber frature. If the shear stress surpasses a threshold and leads to mirodamage, the ompressive frature strength is signifiantly redued. Mannigel found a linear (degressive) influene of the shear stress on the ompressive frature strength above the threshold. A alibration an be made with the experimental results published in [Mannigel 007]. A further influene on the fiber frature whih is not taken into aount by the maximum stress riteria (Eq. 5) and (Eq. ) is the effet IFF has. Espeially under yli loading IFF redues the fiber frature stress σ fr onsiderably. However, the UD-lamina an in this ase no longer been used as an isolated model. Instead, IFF in adjaent lamina influenes FF [Knikrehm 000]. In this ase a more sophistiated analysis of the gradual failure proess is neessary. 4. Inter fiber frature (IFF) riteria 4.. Motivation The major motivation for the development of Puk s Inter Fiber Frature riteria is experimental evidene. Frature riteria are used to predit the

5 4. Inter fiber frature (IFF) riteria 4 Fig. 3. Frature limits for (σ, τ )-stress ombinations strength of a material for arbitrary states of multiaxial stress with the preliminary knowledge of only some easy to measure strength values. One an best ompare different riteria by taking experimental results that do exhibit some kind of unommon or speial material behavior. For arbon (squares) and glass fiber reinfored plastis (irles) Fig. 3 shows suh test results obtained in a large German researh projet [Cuntze et al. 997]. Some remarkable observations an be made here: The stresses σ t and τ interat. That means that when σ t and τ at at one frature ours before σ t = R t or τ = R respetively are reahed. Shear frature is impeded by moderate transverse ompressive stress σ. In other words: Higher shear stress τ an be sustained without frature, when τ and moderate σ at simultaneously. A third aspet beomes obvious when looking at Fig. 4: When σ and τ at simultaneously and the ratio σ /τ exeeds a ertain value, frature ours under an angle θ fp 0. The frature angle inreases with growing stress ratio σ /τ and reahes approximately ±54 for pure transverse ompressive stress (see Fig. 8). The real failure envelope is not symmetri to any vertial line. This is learly illustrated by the frature urve in Fig. 4 and annot be modeled by onventional global failure riteria. Besides, these riteria annot alulate the frature angle nor do they explain the effets observed and explained above. However, the understanding of these effets is ruial for the development of reasonable failure riteria. Of speial interest is in this ontext the analysis of the frature angle under pure transverse ompression σ. Frature ours on a plane inlined by approximately ±54 to the ation plane of σ. A simple sketh of the Mohr irle shows what stresses at on this frature plane (Fig. 5). This shall be derived in detail:

6 4 4 Puk s ation plane frature riteria Fig. 4. Frature urve (σ, τ ) On the ation plane of σ (θ = 0 ) pure transverse ompression (σ n = σ, τ nt = 0) is ating. On this plane no frature an our. On a perpendiular plane (θ = 90 ) there is no stress at all. On all other planes with 0 < θ < 90 ombinations of a transverse ompressive stress σ n (θ) and a transverse shear stress τ nt (θ) are ating. Under θ = 45 the maximum shear stress τ nt max = τ nt (45 ) is reahed. It has the same value as σ n (45 ) ating on the same plane and the magnitude is just half that of σ. Under 54 on the frature plane the shear stress τ nt (54 ) is just slightly smaller than τ nt max. However, the transverse ompressive stress σ nt (54 ) whih impedes frature is onsiderably smaller than the orresponding value un- Fig. 5. Frature under uniaxial ompressive stress σ

7 4. Inter fiber frature (IFF) riteria 43 der 45. This fat explains why the frature angle is greater than ±45. This is in aordane with the observation that higher shear stresses an be sustained in the presene of moderate transverse ompression. The analysis of the frature behavior whih will be presented here, works with the stresses on the ation plane and not with the lamina stresses σ, σ 3, τ 3, τ 3, τ. This is a fundamental fat and leads to the approah to formulate frature riteria using the stresses of the ation plane. These frature riteria will be presented and explained in the next setions. Before this, the frature hypotheses those riteria are based on, are disussed. Besides, the basi strengths and some inlination parameters needed for alibration are presented. 4.. Different IFF-frature modes The different kinds of IFF desribed above for the (σ, τ )-frature urve lead to the differentiation of three IFF-frature modes, namely Mode A, Mode B and Mode C defined as follows: Mode A: Transverse tensile stressing σ t or longitudinal shear stressing τ ause ating either alone or in ombination frature. In the ase of a D-state of stress with σ, σ, τ the raks run in thikness diretion and thus in the ommon ation plane of the lamina stresses σ and τ. The frature surfaes are separated from eah other due to the tensile stressing. This leads from a marosopi point of view to a degradation of both the Young s modulus E and the shear modulus G. Mode B: Frature is aused by longitudinal shear stressing τ. This frature ours on the ation plane of the external shear stress τ. In ontrast to Mode A, the transverse normal stressing σ whih ats on the frature plane simultaneously with τ is a ompressive stressing! Thus the rak does not open and the frature surfaes are pressed on eah other 7. Consequently the degradation of stiffness due to IFF Mode B is muh less signifiant than that due to IFF Mode A. An IFF Mode B ours as long as the ratio of the ompressive stress at frature and the transverse ompressive strength σ fr /R is smaller than roughly 0.4. Mode C: If the ratio of the ompressive normal stressing at frature and the transverse ompressive strength σ fr /R exeeds roughly 0.4, the ation plane of the external shear stress τ is no longer the frature plane. Instead frature ours on a plane inlined by an angle θ fp 0 to the ation plane of σ and τ. The frature angle θ fp inreases in the ase of a plane state of stress (σ, σ, τ ) from 0 at the threshold 7 Thermal tensile stresses in the matrix might lead to a separation of the frature surfaes nonetheless.

8 44 4 Puk s ation plane frature riteria between Mode B and Mode C to roughly 54 for pure transverse ompression (τ = 0). IFF Mode C implies the risk of delamination between the broken layer and adjaent layers. In a more general 3D state of stress a forth frature mode an our: Mode A*: On the frature plan ats a ombination of σ t, τ and τ Frature hypotheses Mohr s frature hypothesis Fiber frature (FF) is in the sense as it used here defined as the simultaneous breaking of thousands of filaments. FF of a UD-lamina is regarded as final failure of the affeted lamina. Inter fiber frature (IFF) on the other hand is defined as a marosopi rak whih runs parallel to the fibers and separates an isolated UD-layer into two piees. Usually, suh a marosopi separation is preeded by miro-mehanial damage of the matrix or the fiber/matrix-interfae, respetively. Fiber reinfored omposites (FRP) used in lightweight onstrution usually show brittle frature behavior onerning both FF and IFF. This means that frature ours suddenly on a ertain frature plane without any major plasti deformation. This harateristi material behavior is espeially distintive for σ t - and τ -stressing. In both ases brittle frature ours on the plane with the highest tensile stress (ompare Fig. 6). Paul [Paul 96] alled materials with suh a behavior intrinsially brittle. The ompressive strength of suh materials (here: R ) is muh higher than (usually twie the value of) the tensile strength (here: R t ). Uniaxial ompression leads to shearing on a plane whih is inlined to the diretion of the ompressive stressing (ompare Fig. 6). This frature is aused by the transverse shear stressing (here: τ ) ating on the inlined frature plane. The desribed brittle frature in FRP-strutures has been known for deades now. Nevertheless, most ommerial Finite-Element-programs still work with global strength riteria like that of Tsai/Wu [Tsai 99], whih are in priniple all based on the von Mises yielding riterion. Those global strength riteria an only be regarded as pure interpolating formulae. They are not based on a failure hypothesis and thus do not onsider the true material behavior. All stresses are put into one formula not regarding whether the single stress auses FF or IFF. Obviously, yielding riteria are not adequate for FRP-failure analysis. Beause of the brittle material behavior of FRP, failure analysis should instead be based on the ideas of Mohr [Mohr 900] and Coulomb

9 4. Inter fiber frature (IFF) riteria 45 [Coulomb 776]. Mohr s frature hypothesis for brittle materials is the following: The frature limit of a material is determined by the stresses on the frature plane. The ited original work refers to brittle isotropi material (for instane ast iron) Puk s frature hypotheses Most failure onditions in use are pure mathematial interpolation funtions with no physial basis. Puk, however, based his frature riteria on a reasonable understanding of the brittle frature, formulated in his frature hypotheses. He works with the stresses σ n, τ nt and τ n whih are ating on a ommon fiber parallel ation plane of a UD-lamina. The two shear stresses τ nt and τ n an be ombined to one resulting shear stress τ nψ (see (Eq. 4) and Fig. 8). Puk s frature hypotheses are the following:. Inter Fiber Frature on a plane parallel to the fibers is aused by the stresses σ n and τ nψ ating on the frature plane.. If σ n is a tensile stress it promotes frature together with the shear stress τ nψ or even alone for τ nψ = 0. In ontrast to that σ n impedes frature if it is a ompressive stress by raising the frature resistane of the frature plane against shear frature with inreasing ompressive stress σ n. Experimental experiene has shown, that transverse shear stressing τ auses a frature not in the ation plane of τ but in the plane of the tensile prinipal stress whih has the same magnitude as τ. Therefore, a rule has been formulated by Puk for the ase of ombined stresses σ, σ 3, τ 3 or σ II, σ III respetively: If the UD-lamina is just stressed in the plane of transverse isotropy (only stresses σ, σ 3, τ 3 ) frature ours either as tensile frature due to σ n as the highest normal stress or as shear frature due to τ nt impeded by a transverse ompressive stress σ n. Whih of the two kinds ours depends on the ratios of the stresses σ, σ 3, τ 3. This additional rule might not be valid for omposites with an unusually small ratio between transverse ompressive and transverse tensile strength R t /R <. Experimental results for thermoplasti FRP (TP-FRP) like PEEK (Polyetheretherketon) show however, that for those thermoplasti materials the rule is valid just as for thermoset FRP 8 [Kuhnel 008]. 8 These investigations show, too, that for PEEK the ratio R /R t is larger than,5.

10 46 4 Puk s ation plane frature riteria If there is a UD-material with an extraordinary low ratio R /R t it is expeted that the rule just mentioned is not fulfilled. τ -stressing would instead lead to a mixed mode frature of ombined σ t -stressing and τ - stressing Frature resistane of the ation plane In almost all failure riteria the used mathematial funtions are alibrated by using the maximum sustainable stresses for uniaxial tensile or ompressive stresses and pure shear stresses. The absolute values of these stresses are given by the so alled strengths R. For stress based frature riteria of UD-omposites the so alled basi strengths R t, R, R t, R, R, R are used for alibration. All these are given as positive values, also the ompressive strengths. This is a soure of misunderstanding and mistakes, beause in frature riteria sustainable stresses have to be used. As ompressive stresses have a negative sign, the orresponding maximum sustainable uniaxial stresses are for a UD-omposite -R and -R. The basi strengths orrespond to the frature limit under pure σ, σ, τ or τ stressing, respetively. These frature limits are measured without giving any attention to the kind of failure (failure by yielding, brittle frature, rushing et.). Thus, R is defined as the transverse shear stressing τ leading to failure in the absene of all other stresses. However, experimental experiene shows that in this ase in brittle material frature takes plae on a plane inlined by 45 to the ation plane of the applied τ (ompare Fig. 6). On this plane pure transverse tension σ t is ating. This is not onsidered when onventional failure riteria are used. The shear strength R is alulated by dividing the applied shear load by the area it has been applied on. In the same way as R, the transverse ompressive strength R orresponds to σ stressing leading to failure in the absene of all other stresses. However, it has been proved that by no means frature ours on the ation plane of the σ -stressing but on a plane inlined by some ±54 (see above). On this plane transverse shear stressing τ is dominant and responsible for IFF. But this experiene is also not onsidered when the transverse ompressive strength R has to be determined. R is also found by dividing the applied transverse ompressive failure load by the area of the ross-setion of the speimen.

11 4. Inter fiber frature (IFF) riteria 47 Fig. 6. Stressings of the UD-lamina ausing IFF and orresponding frature planes As long as a failure riterion is formulated with stresses defined in the natural oordinate system of the layer (x, x, x 3 ) the basi strengths are the usual alibration parameters. Puk s IFF-riteria, however, are formulated with the stresses of the ation plane (σ n, τ nt, τ n ). Consequently, the orresponding frature resistanes of the ation plane (defined as R A in order to distinguish them from the basi strengths R) are needed. Instead of employing the standard form of a failure ondition t t F σ, σ, σ, τ, τ, τ, R, R, R, R, R, R ) (Eq. ) ( = a IFF-frature ondition is formulated in the form A t A A ( θ ), τ ( θ ), τ ( θ ), R, R, R ) F σ. (Eq. 3) n ( fp nt fp n fp = Looking on (Eq. 3) it is important to remember that the stresses σ n (θ), τ nt (θ), τ n (θ) are the three stresses whih in the most general ase are ating simultaneously on one and the same ation plane 9. The ation plane whih is inlined by θ = θ fp is the frature plane beause on this ation plane the risk of frature is the highest one. 9 That σ n, τ nt, τ n have the same ation plane an be seen from the index n of all three stresses.

12 48 4 Puk s ation plane frature riteria If all three stresses σ n, τ nt, τ n at simultaneously on an ation plane, they represent a ombined stressing onsisting of a σ -stressing, a τ - stressing and a τ -stressing. In order to answer the question whether a ertain ombination of σ n, τ nt, τ n an be sustained by a fiber parallel ation plane a frature riterion has to be formulated mathematially. On the one hand it should ontain the stresses σ n, τ nt, τ n ating simultaneously on a ommon ation plane. On the other hand a frature riterion of this kind an only be alibrated by experimental values for the sustainable stress of the ation plane, when a σ -stressing is ating alone, when a τ -stressing is ating alone and when a τ -stressing is ating alone. This means that frature tests have to be performed with uniaxial σ, pure τ and pure τ. t If in suh a test with for instane a single tensile stressing σ the frature would our in the ation plane of the applied stressing this would be the sustainable tensile stressing of the ation plane. Puk has introdued the following expression: Frature resistane of the ation plane and the orresponding symbol R A. The definition of R A is the following: A frature resistane of the ation plane is the resistane (expressed in the dimension of a stress) by whih an ation plane resists its own frature aused by a single stressing (σ or τ or τ ) ating in the ation plane under onsideration. This definition shows how important it is for the frature experiments to hek whether or not the applied σ - or τ - or τ -stressing really leads to a frature in the ation plane of the applied stress. Some of the frature resistanes of the ation plane used in (Eq. 3) are idential to the basi strengths as will be shown in the following. However, it is important to understand that the underlying question is different. The question is for instane To whih value σ t has to be inreased in order to provoke failure on its ation plane? and not To whih value σ t has to be inreased in order to provoke frature (on any plane)? There are three single frature resistanes to be differentiated:

13 4. Inter fiber frature (IFF) riteria 49 A R t = Resistane of the ation plane against its frature due to transverse tensile stressing σ t ating in that plane. R A = Resistane of the ation plane against its frature due to transverse shear stressing τ ating in that plane. R A = Resistane of the ation plane against its frature due to longitudinal shear stressing τ ating in that plane. In the following some examples will be given to illustrate the meaning of ation plane frature resistanes. The first example is a pure longitudinal shear stressing τ aused by τ 0 in the absene of all other stresses. This will obviously our in a test with a pure longitudinal shear stressing realized by applying for instane a pure shear stress τ in absene of all other stresses. In this ase a shear stress τ n = τ osθ is ating on a plane inlined by the angle θ to the x -plane on whih τ is ating. Obviously, for all angles θ 0 the shear stress τ n is smaller than τ. Frature must therefore our on the ation plane of τ (θ fp = 0 ) and therefore R A = R (Eq. 4) This means that the ation plane frature resistane R A is idential to the longitudinal shear strength R. Transverse tension for instane σ t normally leads in the absene of other stresses to frature in its ation plane, too, and therefore: A t t R = R (Eq. 5) This, however, is not as evident as in the ase of pure longitudinal shear. t In the ase of transverse tension σ there is a ombination of transverse tension (σ t = σ os θ) and transverse shear (τ = 0.5 σ sinθ) on planes θ 0. If the resistane of the ation plane against τ shear frature would be onsiderably smaller than that against transverse tension a different frature angle θ fp 0 would our. The third relevant frature resistane R A annot as easily be determined as R and R t. When applying a pure τ 3 stress to a unidiretional layer in the absene of any other stress there are three planes on whih just one stressing is ating: θ = 0 : ation plane of τ 3 (τ -stressing) θ = 90 : ation plane of τ 3 (τ -stressing) θ = 45 : ation plane of the prinipal tensile stress (σ t -stressing) resulting from τ 3 and being of same magnitude as τ 3 and τ 3.

14 50 4 Puk s ation plane frature riteria Aording to Puk s hypotheses frature ours on the plane with the highest stress exposure. Frature tests on UD-speimens loaded by τ show frature under θ fp = 45. This leads to the lear onlusion that the frature resistane R A t against transverse tension is smaller than the frature resistane R A against transverse shear. In other words: With a transverse shear test not R A but instead the resistane against transverse tension R A t is measured. In fat there is no way to measure R A for intrinsially brittle fiber reinfored plastis diretly. However, the unidiretional ompression test offers an indiret way for the determination of R A as long as the hypothesis for pure transverse stressing (ompare hapter Frature hypotheses ) is valid. If just σ is applied, a frature angle of θ fp ±54 results. On this frature plane a ombination of transverse shear τ and transverse ompression σ is ating. Aording to frature hypothesis, a transverse ompressive stress adds an additional frature resistane to the intrinsial frature resistane R A. This additional frature resistane an be alulated by the used mathematial frature ondition. In this way the test result an be orreted in order to find R A. Thus, the alulated value of R A depends on the frature model used. Therefore this R A is perhaps not the real value of the material property R A but it an be used for the frature model by whih it has been alulated. At least it is guaranteed that by using this same frature model in a frature analysis for the speial ase of uniaxial ompression the result will be (σ ) fr = R with R being the real transverse ompressive strength. The example of longitudinal shear is well suited to explain again the differene between a frature resistane (of the ation plane) and a onventional basi strength. The stress τ is ating on the x -plane parallel to the fibers and the orresponding shear stress τ of the same magnitude is ating on the x -plane. This shear stress on the x -plane is orretly addressed as τ. However, there is just one strength parameter R and it does not ontain the information about the frature plane. This ould be either the x - or the x -plane or even another plane. The orresponding frature resistanes of the two planes under onsideration differ, however, onsiderably. Namely, the frature resistane (R A ) of the x -plane parallel to the fibers is muh smaller than the frature resistane R A of the x -plane perpendiular to the fibers. This phenomenon an easily be understood: On the x -plane the stressing τ provokes Inter Fiber Frature. The frature plane runs parallel to the fibers whereas a frature on the x -plane would be a Fiber Frature (shear off of the fibers)!

15 4. Inter fiber frature (IFF) riteria 5 Obviously it makes no sense to talk about a resistane of the ation plane against frature due to transverse ompressive stressing σ beause a ompressive stress annot produe a frature in its own ation plane. Summing up this, there are three ation plane frature resistanes to be differentiated: A R t = Resistane of the ation plane against its frature due to transverse tensile stressing σ t ating in that plane. For strutural UD-material under normal temperature and humidity onditions it an be expeted that R At = R t is valid, R A = Resistane of the ation plane against its frature due to transverse shear stressing τ ating in that plane. Attention has to be paid to the fat that R A R, R A = Resistane of the ation plane against its frature due to longitudinal shear stressing τ ating in that plane. For elementary reasons R A = R. Experiene has shown that most mehanial engineers are not familiar with the differene between strength and ation plane frature resistane. It ould be helpful to realize the different underlying question. Dealing with strengths the underlying question is To whih value an a ertain stressing (for instane uniaxial stressing σ t ) be inreased until failure ours somewhere (and somehow) in the evenly stressed volume of the test speimen? Dealing with ation plane frature resistanes the underlying question is: To whih value an a ertain stressing be inreased until a frature ours on the ation plane in whih the evenly applied stressing is ating? 4..5 Visualization of the stress/strength problem Preliminary remark: The ontent of the following hapter is not required to be read by somebody who is interested in the appliation only of the algorithms of Puk s laminate failure analysis. However, it is strongly reommended to have a look on the astonishing possibilities of the visualization beause they present a muh better understanding of the results of the mathematial treatment of the physial problems whih is explained in the subsequent hapters. But the reader being in a hurry may ontinue with the hapter Universal 3-D-formulation of the ation plane related IFF-riteria.

16 5 4 Puk s ation plane frature riteria Mohr s Cirles and Frature Envelopes Within a failure analysis independently of the kind of material the engineer has to omplete two tasks: First the stresses of the material aused by the loads have to be alulated and in a seond step these stresses have to be ompared to the failure limit. If the material fails by brittle frature, Mohr s irles are helpful for the first task and Mohr s envelope for the seond one. Beause of the fat that Puk s IFF hypotheses are based on the ideas of Mohr [Mohr 900] and Coulomb [Coulomb 776] one an partially use these lassial solutions. However, it must be reognized that the approahes of Mohr and Coulomb are valid only for marosopially isotropi materials. In the isotropi ase frature due to an arbitrary three dimensional state of stress an be treated as a D-problem. Any stress ombination leading to frature an be visualized by means of the well known Mohr s irle and Mohr s envelope. The reason for this opportunity of simplifiation is the isotropy and the validity of Mohr s frature hypothesis for brittle frature. Any 3D-stress ombination (σ, σ, σ 3, τ 3, τ 3, τ ) 0 to use the oordinates whih are established for the UD-lamina (ompare Fig. 6) an be transformed to an equivalent (σ a, σ b, σ ) stress state without any shear stresses on the ation planes of σ a, σ b, σ where σ a, σ b and σ are the so alled prinipal stresses. If σ b > σ a by σ b < σ a being the intermediate prinipal stress, frature depends aording to Mohr only on the major and minor prinipal stresses σ b and σ (Fig. 7). In the (σ, τ)-diagram Fig. 7 points on the irumferene of a irle with the oordinates σ and τ present the stresses σ and τ on setion planes through the material, the normal of whih is perpendiular to one of the three axes a, b or, that means perpendiular to one of the three prinipal stresses. It an be shown theoretially that the stresses σ and τ of all other setion planes are given by points loated in the hathed areas of Fig. 7. Therefore the highest shear and normal stresses ourring in the material under the given stresses depend only on the two extreme prinipal stresses. In Fig. 7 σ b and σ are hosen as the extreme prinipal stresses for demonstration. However, depending on the load ase it ould be any other pair of the three prinipal stresses σ a, σ b, σ too. The axes a, b, of the Cartesian oordinate system in whih the ating stress ombination an be presented for the isotropi material hanges its spatial orientation depending on the atual stress state (σ, σ, σ 3, τ 3, τ 3, τ ). That means that gen- 0 The additional orresponding shear stresses follow from τ 3 =τ 3, τ 3 =τ 3, τ =τ.

17 4. Inter fiber frature (IFF) riteria 53 Fig. 7. System of the three Mohr-irles for isotropi material erally the diretions a, b, in whih the prinipal stresses σ a, σ b, σ are ating do not oinide with any of the hosen axes x, x, x 3 of the original referene oordinate system for the atual stress ombination (σ, σ, σ 3, τ 3, τ 3, τ ), see Fig. 8. If there is isotropy there is no diffiulty in working with a oordinate system with axes a, b, of the prinipal stresses σ a, σ b, σ whih for different states of stress has a different spatial orientation and there is also no problem to work with an unlimited number of oordinate systems for the axes of the prinipal stresses. However, obviously suh versatility does not exist for transversely-isotropi materials like the UD-lamina where the fixed diretion of the fibers prevents suh a proedure. When dealing with transversely-isotropi material a hange of the diretion of oordinate axes Fig. 8. The stresses σ and τ given by the oordinates of a point on the largest irle in Fig. 7 are ating on a potential frature plane whih is parallel to σ a. The diretion of σ and τ is perpendiular to σ a

18 54 4 Puk s ation plane frature riteria is at least possible in the plane of transverse isotropy whih is perpendiular to the fixed axis parallel to the fibers Mohr s irle and Mohr s envelop for a transversely-isotropi UD-lamina under plane (σ, σ 3, τ 3 )-stress In a UD-lamina the stresses σ, σ 3, τ 3 whih are ating on ation planes being parallel to the fixed fiber diretion x are the only stresses to whih Mohr s irle an be applied effetively. Transverse isotropy allows to swith over from the (x, x 3 )-oordinate system to a (x n, x t ) oordinate system. In order to get familiar with the derivation and appliation of Mohr s irle, a plane (σ, σ 3, τ 3 ) stress state will be investigated now. The stresses ating on the plane whih is inlined by an angle θ to the x 3 -axis (ompare Fig. 33) an be determined by (Eq. ) and (Eq. ). Using the well known theorems for sin and os (Eq. ) and (Eq. ) gives the form σ n ( θ ) = ( σ + σ 3 ) + ( σ σ 3 ) osθ + τ 3 sin θ (Eq. 6) τ nt ( θ ) = ( σ σ 3) sin θ + τ 3 osθ (Eq. 7) These equations look already like the mathematial desription of a irle using the angle θ as a parameter. This an be onfirmed in the following way: Rearrangement of terms, squaring of (Eq. 6) and adding up of the squared form of (Eq. 7) finally leads to the following expression: ( ) ( ) [ ] ( ) n 3 nt( ) 3 3 σ θ σ +σ + τ θ = σ σ + τ (Eq. 8) (Eq. 8) onfirms the very welome opportunity for the visualization of stress transformation beause (Eq. 8) is the mathematial desription of a irle with the enter on the σ n -axis at 0.5(σ +σ 3 ) and the radius ( ) σ σ +τ 3 3. This irle is illustrated in Fig. 9. Figure 9 presents a graphial proedure by whih the stresses σ II and σ III an be determined if σ, σ 3 and τ 3 are known.

19 4. Inter fiber frature (IFF) riteria 55 Fig. 9. Mohr irle for {σ, σ 3, τ 3 } On planes haraterized by the angles ϕ and (ϕ+90 ) there is no shear stress τ nt. The orresponding normal stresses σ n (ϕ) and σ n (ϕ+90 ) are designated by σ II and σ III and are alled extreme normal stresses of the transversely-isotropi plane. In general they are not real prinipal stresses. This is explained later. It is important to realize that for instane an angle ϕ in the real material with a mathematially positive sense of rotation orresponds to an angle of ϕ with a mathematially negative sense of rotation in the Mohr-irle! The extreme normal stresses σ II and σ III of the transversely isotropi plane and the angle ϕ (ompare Fig. 9) an also very easily be found in the usual analytial way by setting τ nt = 0 in (Eq. 7). It is τ 3 ϕ = artan (Eq. 9) σ σ3 and with (Eq. 6) σ ΙΙ = ( σ + σ3 ) + ( σ σ3 ) os ϕ + τ3 sin ϕ (Eq. 30) σ ΙΙΙ = ( σ + σ3 ) ( σ σ3 ) os ϕ τ3 sinϕ. (Eq. 3) With known extreme normal stresses σ II and σ III the stresses σ n (Θ) and τ n (Θ) on planes inlined to the diretion of σ II by the angle Θ an be alu-

20 56 4 Puk s ation plane frature riteria lated with (Eq. 3) and (Eq. 33). These equations orrespond to (Eq. 6) and (Eq. 7). σ n ( Θ) = ( σ ΙΙ + σ ΙΙΙ ) + ( σ ΙΙ σ ΙΙΙ ) os Θ (Eq. 3) τ nt ( Θ) = ( σ ΙΙ σ ΙΙΙ ) sin Θ (Eq. 33) Mohr s irle as shown in Fig. 30 is based on (Eq. 3) and (Eq. 33). The oordinates σ n and τ n of a point P at a radian distane of Θ from σ II are the stresses ating on a plane inlined by Θ to the ation plane of σ II (ompare Fig. 30). The entre of the irle has a distane of 0.5(σ II +σ III ) = 0.5(σ +σ 3 ) from the point of origin. The radius of the irle is 0.5(σ II -σ III ). The term (σ II -σ III ) an be negative or positive. To make sure that the graphial onstrution is in line with (Eq. 3) and (Eq. 33) (positive radius), the sense of rotation and the starting point of the Θ-measurement has to be adapted in a way that σ n (Θ) and τ n (Θ) are orret by means of magnitude and algebrai sign, see sketh on the bottom of Fig. 30. All Mohr s irles for arbitrary states of stress have their enters on the σ n axis. Eah point on the irumferene of suh a irle with oordinates σ n and τ nt belongs to a ertain setion through the material with an angle Θ (in reality). The stress vetor belonging to suh a setion has its starting point in the origin of the (σ n, τ nt )-oordinate system and its tip touhes the point (σ n (Θ), τ nt (Θ)) on the irumferene of Mohr s irle. No vetor tip belonging to a vetor representing a sustainable state of stress an exeed a ertain frature limit of the material. Therefore Mohr Fig. 30. Mohr irle and derivation of stresses on inlined planes

21 4. Inter fiber frature (IFF) riteria 57 Fig. 3. Mohr s irles for speial stress states and Mohr s frature envelope has not only provided the helpful visualization of stresses ating on inlined ation planes by Mohr s irle but he has also tried to draw a frature limit of the material as a urve in the (σ n, τ nt )-diagram. This is alled Mohr s envelope, see Fig. 3. It envelops all sustainable stress ombinations. Obviously no Mohr s irle for sustainable stress an exeed this envelope; it an at the most touh it. It is important to realize that if the ontat point of a Mohr s irle with the Mohr-envelope is found, not only the ombined stresses σ n and τ nt on the frature plane but as well the frature angle θ fp an be found in the diagram, see for instane Fig. 3. For a UD-lamina the frature envelope is hosen by Puk as a parabola in the area of σ n < 0 and an ellipse in the area of σ n > 0. At σ n = 0 the urve might have a sharp bend or more mathematially spoken a disontinuity in terms of gradient. Aording to the frature hypotheses disussed in detail in the hapter Frature hypotheses, there is (as far as IFF is onerned) either tensile frature or shear frature in a UD-lamina stressed by (σ, σ 3, τ 3, 0, 0). Consequently, on the frature plane there is either a σ n > 0 (pure σ t ) stressing or a ombination of σ n < 0 and τ nt ((σ, τ )-stressing). A ombined (σ t, τ )-stressing or a pure τ -stressing on the frature plane are both not possible aording to the additional rule given with the two frature hypotheses (ompare hapter Frature hypotheses ). Under these onditions, all Mohr irles desribing states of stress whih lead to tensile frature touh the frature envelope at the same point,

22 58 4 Puk s ation plane frature riteria Fig. 3. (σ II, σ III ) frature urves resulting from Mohr s frature envelope namely T(R t, 0) (ompare Fig. 3). In a (σ II, σ III )-diagram the orresponding frature urve for tensile frature onsists of two straight lines, running at a distane of R t parallel to the σ II - and σ III -axes (ompare Fig. 3). This orresponds to Paul s [Paul 96] findings for intrinsially brittle materials. In the first quadrant of the (σ II, σ III )-diagram tensile frature ours as a onsequene of σ II (θ fp = 0 ) in the ase of σ II > σ III or as a onsequene of σ III (θ fp ± 90 ) in the ase of σ III > σ II. For the straight frature lines σ II = R t and σ III = R t Paul introdued the term tensile ut offs. In the nd and 4 th quadrant, either tensile or shear frature an our. What frature ours depends on the ratio σ II /σ III. In the 3 rd quadrant, where both σ II and σ III are ompressive stresses frature is always a shear frature. However, the frature plane is not the plane with the highest shear stressing (θ fp = ±45 with τ nt = 0.5 (σ II -σ III ) ; in Mohr s irle straight above and below the entre point at ±θ fp = ±90 ). Instead, the frature angle deviates by ±5 to ±0 from the plane with the highest shear stress. This beomes understandable when regarding the normal stress σ n ating simultaneously to τ nt. This ompressive normal stress is redued onsiderably when the angle deviates a bit from ±45, whereas the shear stress τ nt stays nearly the same (ompare Fig. 5). Knowing that ompressive normal stress impedes shear frature, this explains why the observed angle of shear frature is not ±45 but somewhat larger. This has been experimentally onfirmed with uniaxial ompression tests, see Fig. 8. Figure 3 shows that the frature envelope and the Mohr s irles have no points of ontat between the points T (R t, 0) and S (point of transition

23 4. Inter fiber frature (IFF) riteria 59 from tensile frature to shear frature). The dashed irle in Fig. 3 marks the irle of transition from tensile to shear frature. For this speial (σ II, σ III )-ompression/tension stress ombination there an either our a shear frature or a tensile frature. Also the irle for pure transverse shear shows ontat only in T, that means R = R t is to be expeted. The ourse of the frature envelope between the points T and S does not seem to be relevant for the states of stress treated so far. There are no ontat points on it. It is a so alled dead branh of the frature envelope. However, from the following paragraph it will beome lear, that the ourse of the (σ n, τ nt ) frature urve between T and S has an influene on the frature envelope for 3D-stressings (σ n, τ nt, τ n ), the so alled Master Frature Body (MFB). Suh a 3D-stressing is present as soon as there is longitudinal shear τ ϖ, that means τ and/or τ 3 too From Mohr s irle to Puk s osine-shaped ylinder The idea of Mohr s irles and Mohr s envelope is used now as the basis for a orresponding proedure for transversely-isotropi materials with the most general 3D-state of stress. Figure 33 shows all stresses and their orret designation of a UD-lamina. The stress σ leads to a fiber parallel stressing σ, whereas both the stress σ and the stress σ 3 ause a transverse stressing σ. The stress τ 3 = τ 3 leads to a transverse shear stressing τ and τ and τ 3 both lead to a longitudinal shear stressing τ. That means that obviously all three frature resistanes R At, R A, R A of the fiber parallel ation plane will now generally influene the IFF-proess. In order to get to visualization also of the most general stress situation one must try not to use more than three variables. For this reason one should ombine as muh stresses as possible in resulting stresses. The Fig. 33. Stresses of the UD-lamina and stresses on a possible (IFF)-frature plane whih is parallel to the fibers

24 60 4 Puk s ation plane frature riteria Fig. 34. Uniting τ and τ 3 to τ ϖ and τ and τ 3 to τ ϖ longitudinal shear stresses τ and τ 3 and their orresponding stresses τ and τ 3 are quite speial in this respet. Figure 34 shows that τ and τ 3 are in fat omponents of only one longitudinal shear stress τ ϖ. The two omponents are τ = τ ϖ osϖ and τ 3 = τ ϖ sinϖ. Adequate equations an also be written with the orresponding stresses τ, τ 3 and τ ϖ (this is the stress provoking IFF in its fiber parallel ation plane). Impliitly the fat that the longitudinal shear stresses an easily be united in one stress means: there does exist only one relevant longitudinal shear stress and this is τ ϖ. As long as only τ ω ats in the absene of any other stress, IFF will our in the plane in whih the resulting longitudinal shear stress τ ϖ is ating as soon as τ ϖ = R is reahed. The frature plane is orientated perpendiular to the diretion of τ ϖ as shown in Fig. 34. It is important to note that the stresses σ II and σ III are only true prinipal stresses like σ b and σ respetively, if there is no longitudinal shear τ ϖ. In this speial ase σ, σ II, σ III are real prinipal stresses. As soon as τ ϖ 0, there is longitudinal shear on the ation plane of σ II and/or σ III. The ation plane of true prinipal stresses is however free of any shear stresses. Obviously the ombined ation by whih the stresses σ II, σ III and τ ϖ ause an IFF will depend on the diretion of the shear stress τ ϖ ompared to the diretions of the normal stresses σ II and σ III. But due to the transverse isotropy of the UD-lamina (with respet to the x -axis) not the absolute

25 4. Inter fiber frature (IFF) riteria 6 Fig. 35. Combining σ, σ 3, τ 3 to σ II and σ III and τ and τ 3 to τ ϖ ; definition of angle δ values of the angles ϖ and ϕ are relevant for the risk of frature. Instead, the differene δ between ϖ and ϕ is relevant (see Fig. 35): δ = ω ϕ. (Eq. 34) Beause of the meaning of ω and φ (see Fig. 35 and (Eq. 9) δ an be written as: τ3 τ3 δ= artan artan τ σ σ 3. (Eq. 35) Longitudinal shear stress τ ϖ annot be eliminated like a transverse shear stress τ 3 by the introdution of stresses like σ II and σ III. Thus, in the general ase frature has to be alulated for a (σ, σ II, σ III, 0, τ ω ) state of stress. For the moment it is assumed that just as in the ase of isotropi material (with σ a being the intermediate prinipal stress) the fiber parallel stress σ has no influene on the frature angle Θ fp and no influene on the magnitude of the frature stresses σ II, σ III, τ ω at IFF this is at least orret as long as σ is onsiderably smaller than 50 % of the fiber frature strength R. In the partiular ase that the longitudinal shear τ ϖ ats alone (σ II = σ III = 0), frature ours in the ation plane of τ ϖ whih is turned by the angle differene δ = ϖ-ϕ with respet to the ation plane of σ II (ompare Fig. 35). In a further speial ase, τ ϖ ats on the frature plane of the tensile frature provoked by (σ II, σ III ) alone. In this ase, the longitudinal shear simply helps provoking frature without hanging the frature angle. The

26 6 4 Puk s ation plane frature riteria Fig. 36. Mohr s irle with longitudinal shear τ n representation for UD-lamina ommon presene of σ II, σ III and τ ϖ leads to a ombined (σ t, 0, τ ) mode of frature, that means a Mode A-frature. The tensile frature ours on the ation plane of σ II or that of σ III (depending on whih of both stresses is larger). If, however, the ation plane of τ ω is neither equal to that of a tensile σ II nor that of a tensile σ III, frature will sometimes our on a plane whih is not the ation plane of any of the 3 stresses σ II, σ III, τ ϖ. On the atual frature plane there will then be a (σ t, τ, τ ) stress ombination, alled a Mode A*-frature. This is a major differene to the frature behavior of brittle isotropi material, where a omparable stressing (a ombination of a tensile stress σ t and a shear stress τ) on the frature plane is not possible. It seems neessary to have a loser look on the stress distribution. In general, a (σ II, σ III, τ ϖ ) stress state leads to a stressing σ n (θ), τ nt (θ), τ n (θ) on all planes (parallel to the fiber diretion). The maximum of τ n with the magnitude of τ n = τ ϖ ours on a plane inlined by the angle δ to the ation plane of σ II, see Fig. 36. On planes inlined by ±90 to the plane of τ n = τ ϖ the longitudinal shear τ n is zero. In between these extreme inlinations of the fiber parallel ation plane τ n follows a osine funtion, ompare (Eq. 3). A rotation of the fiber parallel ation plane by ±90 (in the real material) orresponds to a rotation of ±80 in the Mohr s irle. This leads to an illustration of the state of stress as a Mohr s irle with a osine-halfwave on top of the Mohr s irle. The amplitude of the osine-funtion is τ n = τ ω (ompare Fig. 36). The maximum of τ n is with respet to the σ II -axis rotated by the angle δ in reality and δ in the Mohr s irle respetively.

27 4. Inter fiber frature (IFF) riteria 63 Fig. 37. Spatial vetor fan for ombined (σ, τ )-stressings: a) for σ < 0; b) for σ > 0 In addition to (Eq. 3) and (Eq. 33) there is now an additional equation for the longitudinal shear τ n in dependene of the angle Θ: ( Θ) = τ os( Θ δ ) = τ + τ os( Θ δ ) τ n ω 3 (Eq. 36) A stress vetor (σ n (Θ), τ nt (Θ), τ n (Θ)) in Fig. 36 reahes from the enter of origin to a point P on the osine-half-wave. The infinite number of tips of all stress vetors (ating on planes 90 (Θ-δ) +90 ) of a given (σ, σ 3, τ 3, τ 3, τ ) state of stress form together the osine-half-wave in Fig. 36. All these vetors form a spatial vetor fan, speial examples of whih are shown in Fig. 37. The frature limit for all stress vetors is now no longer an enveloping frature limit urve in the (σ n, τ nt )-plane, but a surfae in the (σ n, τ nt, τ n )- spae. This frature envelope omprises all osine-half-waves and orresponding stress vetors whih do not lead to frature. In order to differentiate this frature envelope from other well known envelopes for instane in the (σ, σ, τ )-spae, the envelope in the (σ n, τ nt, τ n )-spae is alled Master Frature Body (MFB) (ompare Fig. 38). The visualization of the stress state on any fiber parallel ation plane by the osine-half-waves and the visualization of the (σ n, τ nt, τ n )-frature limit by the Master Frature Body (MFB) are very helpful both for the interpretation of the results of a frature analysis of speial stress states and quite generally for a deeper understanding of the bakground and the results of Puk s riteria.

28 64 4 Puk s ation plane frature riteria Fig. 38. IFF-Master Frature Body [Lutz 006] It helps for instane to understand that on the surfae of the MFB at ertain plaes dead areas appear [Ritt 999]. One of these dead areas ontains the dead branh between the points S and T in Fig. 3. Stress points (σ n, τ nt, τ n ) in dead areas symbolize (σ n, τ nt, τ n )-stresses on the frature plane whih never our (The osine half wave is unable to get in ontat with these parts of the surfae of the MFB). All steps of the appliation of Puk s ation plane frature riteria an be visualized from Fig. 38. In order to answer for instane the question what stress ombinations are sustainable? for a given (σ, τ )-stress ombination, one steps into the (σ n, τ nt, τ n )-stress spae where the Master Frature Body (MFB) is given. Here, one has to find the point on the MFB-surfae where the osine shaped stress line touhes the MFB. What the algorithm of the frature analysis really performs is the following proedure: Every (σ n, τ nt, τ n ) stress vetor on any inlined setion is elongated (strethed) in its original diretion by the so alled streth fator f S = (f E ) the way that the tip of the vetor just gets in ontat with the MFB-surfae. The (σ n, τ nt, τ n )-vetor whih needs the minimal streth fator f Smin is the one whih in reality leads to IFF. For (σ, τ )-ombinations Puk has found an analytial solution. For the general (σ, σ, σ 3, τ 3, τ 3, τ )-stress ombinations fast numerial searh proedures have been developed. In Fig. 38 the osine line in the tensile region of σ n has a ontat

29 4. Inter fiber frature (IFF) riteria 65 point on the line τ nt = 0, whih means θ fp = 0. In the ompressive region the ontat point an be in an area where all 3 stresses σ n, τ nt, τ n exist. This means that θ fp 0 is possible. It is interesting to see in Fig. 38 how the frature urve on the (σ, τ )-plane is represented in the MFB. Surprisingly the stress σ n on the frature surfae is onstant (σ n = R A ) in the whole area of oblique fratures (θ fp 0 ), see part () to (d) of the orresponding frature urves Universal 3-D-formulation of the ation plane related IFF-riteria Preliminary remarks The ation plane related IFF-riteria are valid for arbitrary ombinations of the stresses σ, σ, σ 3, τ 3, τ 3, τ. The influene stresses parallel to the fibers (σ ) have on IFF is assumed to be negligible as long as σ is muh smaller than the fiber parallel strength R. Thus, that influene is negleted in a first approah and will be disussed later (ompare hapter Influene of stresses σ ating parallel to the fibers on IFF ). Puk s approah is based on the hypothesis that the stresses ating on a frature surfae provoke frature, this was first proposed by Hashin [Hashin980]. Consequently, Puk s riteria are formulated with the stresses σ n (θ), τ nt (θ), τ n (θ) (ompare Fig. 8) whih an easily be alulated from the stresses σ, σ 3, τ 3, τ 3, τ (see (Eq. ), (Eq. ), (Eq. 3)). The stresses σ n (θ), τ nt (θ), τ n (θ) depend on the angle θ of the plane under onsideration or in other words of their ation plane. The same is true for the risk of frature. On a plane with θ = θ the risk of frature might be higher than on a plane θ = θ. Eventually, when the lamina stresses are inreased to the frature limit the lamina will suffer IFF on a frature plane θ = θ fp. On this plane the risk of frature is highest and here the stresses σ n (θ), τ nt (θ), τ n (θ) reah first the frature limit. A measure for the likelihood of frature is the stress exposure f E (θ) on the ation plane. In fat, the plane with the highest stress exposure will be the frature plane (f E (θ) = [f E (θ)] max = f E θ=θfp ). If the stresses of the UDlamina (σ, σ 3, τ 3, τ 3, τ ) are multiplied by the smallest streth fator f Smin = /[f E (θ)] max the lamina will suffer IFF exlusively on that frature plane. All other planes remain intat. For the general 3D-state of stress, the frature plane has to be found numerially. That means that the stresses σ n (θ), τ nt (θ), τ n (θ) need to be

30 66 4 Puk s ation plane frature riteria Fig. 39. Searh for the frature plane alulated for all planes with angles 90 θ 90 and that for all these angles the stress exposure f E (θ) needs to be alulated, see Fig. 39. This is done by inserting the stresses into an IFF-riterion formulated with the stresses of the ation plane σ n (θ), τ nt (θ), τ n (θ). The proedure is usually done in -steps leading to 80 alulations. With modern omputers that is no problem. Nevertheless, it makes sense to redue the numerial effort as far as possible. Figure 40 illustrates the ourse of the angle dependent stress exposure f E (θ) as a funtion of the angle θ of the ation plane for a seleted state of stress (σ = σ = σ 3 = τ = 0; τ 3 = τ 3 ). As a result of the frature analysis one gets a frature angle θ fp = 58 and a stress exposure f E = 0.5. This orresponds to the global maximum of the bold urve in Fig. 40. If the stresses are multiplied by the streth fator f S = /f E =, for θ = 58 f E = is reahed in this ase. The thin urve in Fig. 40 illustrates the urve f E (θ) resulting from multiplying the stresses by (f S (θ fp ) = /f E (θ fp )). Originally in 993, Puk proposed the following IFF-ondition for ompressive stress σ n : R A τ ( ) nt θ fp p ( ) σn θ fp + R A τ ( ) n θ fp p ( ) σn θ fp =, forσ < n 0 (Eq. 37)

31 4. Inter fiber frature (IFF) riteria 67 Fig. 40. Stress exposure f E (θ) for a τ 3 /τ 3 -ombination With the inlination of the (σ n, τ nt )- and (σ n, τ n )-urves at the point σ n = 0: p τ nt = σ n σ =0 n (Eq. 38) p τ n = σ n σ = 0 n (Eq. 39) Here, the lassial quadrati additive approah was hosen for the interation of the two shear stresses τ nt and τ n. The two denominators fulfill Puk s seond hypothesis stating that a ompressive stress σ n (a stress with a negative value) adds an additional frature resistane p σ n (θ fp ) to R A and p σ n (θ fp ) to R A, respetively. At this point one should reall the orret meaning of this statement: It does not mean that adding an additional ompressive stress σ redues the IFF-stress exposure f E = [f E (θ)] max! It does only mean that the stress exposure on the ation plane f E (θ) under onsideration dereases, if σ n on this plane grows. This implies that on other planes with a different angle Even though that might be true in some ases.

32 68 4 Puk s ation plane frature riteria Fig. 4. Master Frature Body in the (σ n, τ nt, τ n )-spae θ where the same state of stress leads to different stresses σ n (θ), τ nt (θ), τ n (θ) the stress exposure might inrease. Another important aspet is the differene between a onventional frature body written in σ, σ 3, τ 3, τ 3, τ (and σ, if fiber frature is overed, too) on the one hand and the Master Frature Body (MFB) desribing whih stress-ombinations (σ n, τ nt, τ n ) lead to frature on their ation plane. Again, the differene an well be illustrated regarding the simple example of plane stress (σ 3 = τ 3 = 0). Figure 4 shows the frature urve for the (σ, τ )-stress ombination. The urve is losed indiating that any stress ratio σ /τ leads to frature, if the magnitude of the stresses is suffiiently high. Figure 4 shows shematially the orresponding Master Frature Body (MFB) in the (σ n, τ nt, τ n )-spae. Additionally, a ross setion as desribed by (Eq. 37) and an arbitrary longitudinal setion of the (MFB) are shown. Comparing the frature urve in the (σ, τ )-diagram and the longitudinal setion of the MFB ((σ n, τ nψ )-diagram) several misunderstandings and misinterpretations are possible as long as the subjet has not been fully understood. First thing to reognize is that any longitudinal setion of the MFB has no intersetion with the σ n -axis in the area of σ n < 0. At first glane that seems to be in ontrast both to ommon sense and the (σ, τ )- frature urve: Does transverse ompression not lead to frature?

33 4. Inter fiber frature (IFF) riteria 69 Of ourse, it does! But not on the ation plane of the transverse ompressive stress σ! A single transverse ompressive stress does never ever provoke frature on its ation plane. That is, why the MFB is open to the negative σ n -axis. For pure transverse ompression σ frature does our on a plane inlined by ±54. That means, that first the stresses σ n (θ = ±54 ) and τ nt (θ = ±54 ) have to be alulated. In the ase of pure (σ, τ )-loading (Eq. ) simplifies to σ n = σ os θ, for σ 3 = τ 3 = 0 (Eq. 40) τ = σ sinθos θ, for σ = τ = 0 nt 3 3 (Eq. 4) τ n = τ osθ, for τ 3 = 0 (Eq. 4) For pure transverse ompression the longitudinal shear stress τ and onsequently τ n are zero. Thus on the frature plane θ fp = ±54 just σ n and τ nt are ating. The magnitude of these stresses an be alulated with (Eq. 40) to (Eq. 4) and is also illustrated in Fig. 5. This simple example might help to understand the proedure the onept of ation plane related frature riteria implies. The riteria are formulated with the stresses of the ation plane. For all possible ation planes with inlination angles θ between 90 and +90 the stress exposure f E (θ) is alulated and the plane with the highest stress exposure determined. This plane is the frature plane (ompare Fig. 39 and Fig. 40). The frature ondition (Eq. 37) mirrors well the mehanis and the material behavior. However, it is not very well suited for the numerial searh of the frature plane as explained below. If a state of stress σ n (θ), τ nt (θ), τ n (θ) does not lead to frature on the plane examined, the expression on the left hand side of (Eq. 37) is smaller than. Frature will our on this plane if all three stresses are multiplied by the reiproal value of the stress exposure [/f E (θ)], the so alled streth fator f S. This magnifiation fator leading to the value on the left hand side of (Eq. 37) is unknown at first and must be alulated. However, if f S is introdued as a fator for σ n, τ nt, τ n in (Eq. 37), a 4 th grade equation for f E results. This equation annot easily be solved. Thus, (Eq. 37) seems to be unsuitable for the alulation of f E (θ). Geometrially (Eq. 37) desribes a frature body with elliptial ross setions for σ n = onst. (ompare Fig. 4). Of ourse the frature body an also mathematially be desribed by its longitudinal setions, the so alled

34 70 4 Puk s ation plane frature riteria ontour lines. By this means the numerial effort for the searh of the frature plane an onsiderably be redued. The simple reason for that is the following: The diretion of the stress vetor does not hange when it is strethed by the streth fator f S. This means that the ratios between the three stresses σ n (θ):τ nt (θ):τ n (θ) remain onstant. Geometrially, the extension of the stress vetor takes plae within a longitudinal setion haraterized by the ratio τ n (θ)/τ nt (θ) = tanψ (Fig. 4). This means, that f E (θ) an relative easily be alulated, if the frature body is mathematially defined by its longitudinal setions. In the following the mathematial derivation of the frature ondition is given. Here, of ourse, a distintion of the ases transverse tension (σ n > 0) and transverse ompression (σ n < 0) is neessary Frature ondition for tensile stress σ n The longitudinal setion of the MFB haraterized by τ nt = 0 (ψ = 90 ) is well known from experimental experiene. This speial setion is the same as the (σ, τ )-frature urve. Frature ours on the ommon ation plane of σ and τ, thus σ n (θ fp ) = σ and τ n (θ fp ) = τ. In the region of tensile stress σ n on the frature plane this frature urve an well be fitted with an elliptial funtion having a slightly negative inlination at σ n = 0, τ n = R and hitting the σ n -axis perpendiularly at σ n = R t, τ n = 0. Puk assumes that all other longitudinal setions (ψ = onst.) an be desribed with similar urves. The resultant shear stress for suh a setion is derived from the two stresses τ nt (θ) and τ n (θ) with ommon ation plane (ompare Fig. 8): τ n ψ nt + ( θ ) = τ ( θ ) τ ( θ ) (Eq. 43) n Now, the problem an be treated as -dimensional with the two stresses σ n (θ) (in the following abbreviated with σ n ) and τ nψ (θ) (abbreviated with τ nψ ) similar to that of the lassial frature envelope of Otto Mohr, ompare Fig. 3. Defining the frature resistane of the ation plane against a resultant shear stress τ nψ as R A ψ (see Fig. 4), the frature ondition formulated as an ellipti equation results to be: τ n R ψ A ψ + σ n R A t + σ n A t ( R ) =, for σ n 0 (Eq. 44)

35 4. Inter fiber frature (IFF) riteria 7 The onstants and an be determined by boundary onditions: For τ nψ = 0 the expression σ n = R A t must result. This leads to + (Eq. 45) = At the intersetion with the τ nψ -axis the frature urve takes the value R ψ A and the inlination shall be by definition τ nψ σ n σ n = 0 = p t ψ (Eq. 46) Impliit differentiation of (Eq. 44) at the point σ n = 0, τ nψ = R A ψ gives: R τ nψ + A A t ψ σ n R σ n = 0 = 0 (Eq. 47) That leads eventually to the frature ondition: t t At nψ p ψ n p ψ R n A A A ψ ψ ψ τ σ σ + + =, R R R R for σ 0 n At ( ) (Eq. 48) Frature ondition for ompressive stress σ n In priniple several mathematial funtions might be adequate to model the Master Frature Body in the zone of negative σ n. Up to a stress magnitude σ 0,4R (This is the region of θ fp = 0) the experimentally determined (σ, τ )-frature urve an well be modeled by a parabola. The mentioned experimental experiene and the fat that Otto Mohr himself has assumed that the inrease in sustainable shear stress aused by a superimposed ompressive stress σ n would grow less than linearly with σ n led Puk to a general paraboli approah for σ n < 0. A suitable parabola an be formulated as follows: τ n A R ψ ψ + σ n =, for σ < 0 (Eq. 49) n

36 7 4 Puk s ation plane frature riteria Here, the inlination at σ n = 0 shall be by definition: τ nψ σ n σ n = 0 = p ψ (Eq. 50) Differentiation of (Eq. 49) at σ n = 0, τ nψ = 0 gives with (Eq. 50): p = R ψ A ψ (Eq. 5) All this leads finally to the following frature ondition for ompressive stress σ n on the frature plane: τ nψ p ψ n, for n 0 A + σ σ A = < R ψ R ψ (Eq. 5) In (Eq. 48) and (Eq. 5) there are still the frature resistanes R A ψ and the inlination parameters p ψ as unknown parameters. Consequently, he next paragraphs deal with the definition of reasonable values for these parameters Desription of the ross setion of the MFB at σ n = 0, determination of R A ψ Figure 4 is helpful for the interpretation of the parameter R A ψ. In fat, the frature resistane R A ψ is at the point σ n = 0 the distane from the point of origin to the surfae of the MFB under the angle ψ. This angle ψ haraterizes the longitudinal setion and depends on the ratio of τ nt and τ n. For ψ = 0 (τ nψ = τ n0 = τ nt, τ n = 0) the frature resistane is R A and for ψ = 90 (τ nψ = τ n90 = τ n, τ nt = 0) it is R A. Considerations based on miro-mehanis suggest that these two frature resistanes (R A and R A ) are not equal but of very similar magnitude. Thus, the lassial ellipti approah for intermediate values R A ψ seems to be appropriate: τ R nψ 0 A ψ τ = R nt0 A τ + R n0 A = (Eq. 53)

37 4. Inter fiber frature (IFF) riteria 73 Here, the additional index 0 indiates that all stresses are taken at σ n = 0. With the orrelations τ nt0 = τ nψ0 os ψ and τ n0 = τ nψ0 sin ψ (Eq. 53) leads to: A R ψ osψ = A R sin ψ + A R = (Eq. 54) However, the normal stress σ n has no influene on ψ. For all values of σ n the following orrelations are valid: τ nt os ψ = (Eq. 55) τ nψ and τ n sin ψ = τ (Eq. 56) τ n A R nψ Thus, (Eq. 53) is generally valid for all σ n : ψ ψ τ nt = A R τ + R n A (Eq. 57) With (Eq. 57) the Master Frature Body is defined by a transverse setion (σ n = 0) and an infinite number of longitudinal setions ψ = onst. ((Eq. 48) and (Eq. 5)). The only parameter left to be defined is the inlination of the longitudinal setions at σ n = 0 (ompare Fig. 4). Guidelines onerning the hoie of values for the inlination parameters an be found in the next paragraphs. In the ase τ nt = τ n = 0 it does not matter whih value is taken for p ψ t /R A ψ sine this oeffiient an be (in the speial ase of no shear stress) eliminated from (Eq. 48). The frature ondition of the ation plane for negative σ n (Eq. 5) is not defined for τ nt = τ n = 0, beause there is no frature on a plane with pure transverse ompression. In this ase frature takes plae on another ation plane Rearrangement of the frature ondition for searhing the frature plane To use the ation plane related frature ondition ( ( θ ), τ ( θ ), τ ( θ )) = F σ (Eq. 58) n fp nt fp n fp

38 74 4 Puk s ation plane frature riteria first the frature angle θ fp needs to be determined. The frature plane (and with this the frature angle θ fp ) is haraterized as the ation plane with the maximum loal stress exposure: f E (θ fp ) = [f E (θ)] max with 90 θ 90. It is better to use the stress exposure and not the reiproal streth fator f S = /f E (θ) for the numerial searh of the frature plane, beause on planes being free of stress, [/f E (θ)] equals whereas f E (θ) equals 0 whih poses no numerial problems. Assuming that the stresses τ nψ (θ) and σ n (θ) in (Eq. 48) and (Eq. 5) do not yet ause frature, they must be multiplied by f S or divided by f E (θ) in order to lead to frature (on this very plane) and to fulfill the frature ondition. Keeping this in mind, it is obvious that the frature onditions (Eq. 48) and (Eq. 5) lead to quadrati equations for f E (θ), beause the stresses our just in the first and seond order in the frature onditions. The solution of these quadrati equations an be expliitly determined, see (Eq. ). The resulting expression for f E (θ) is homogeneous of first grade onerning the stresses. This means that the stress exposure f E (θ) grows linearly with the stresses. Thus the stress exposure is a diret measure of the risk of frature. In Fig. 4 all equations needed for the desribed numerial searh of the frature plane are summarized. Fig. 4. Equations needed for the numerial searh of the frature plane

39 4. Inter fiber frature (IFF) riteria 75 Fig. 43. Derivation of the relation between R A and R The equation for R A in Fig. 4 an be derived geometrially from Fig. 43. The oordinates of the point P in Fig. 43 are R σn = sin( θ fp 90 ) (Eq. 59) R τnt = os( θ fp 90 ) (Eq. 60) For these oordinates not only the equation of Mohr s irle for uniaxial ompression is valid, but also the equation for the paraboli envelope: nt ( ) A A τ = R p R σ (Eq. 6) n In point P irle and parabola have the same slope: A A p R R tan ( θ 90 ) = = p τ R p fp A nt From (Eq. 6) and (Eq. 6) follows: os R A ( θ fp ) R = σ n (Eq. 6) p = and (Eq. 63) + p ( + p ) (Eq. 64)

40 76 4 Puk s ation plane frature riteria Choie of inlination parameters p t ψ and p ψ The parameters p t and p (ψ = 90 ) an be dedued from the (σ, τ )- frature urve (ompare Fig. 4). This is possible due to the fat, that the frature angle θ fp is zero for both σ > 0 and σ < 0. Thus, the stresses on the frature plane are σ n = σ and τ n = τ. Those experimentally determined values for the parameters p t and p usually lie between 0.5 and 0.35 (Table ). Quite often experimental t results are best fit to (Eq. 48) and (Eq. 5) respetively, if p is hosen slightly higher than p. Generally speaking, values between 0. and 0.35 lead to reasonable frature urves (Puk et al. 00). Table shows typial values for unidiretional layers with thermosetting matrix. They represent mean values resulting from two major experimental projets [Cuntze et al. 997; Kopp000]. Corresponding validated data for p and p t (ψ = 0 ) do not exist, beause the (τ nt, σ n )-frature urve is experimentally non-aessible in the region σ n 0 [Puk 996, Ritt 999]. However, p an indiretly be drawn from transverse ompression frature tests reording the frature angle θ fp. Here the following equation holds: p = os θ fp (Eq. 65) As stated before, for both GFRP and CFRP one obtains under uniaxial transverse ompression frature angles slightly above 50 from experiments. In this ase values greater than p = 0. result from (Eq. 65). However, it should be reognized that the measurement of frature angles is very diffiult and that results satter onsiderably [Cuntze et al. 997; Kopp000]. Moreover, (Eq. 65) is very sensitive even to marginal hanges of the frature angle θ fp under pure transverse ompression. Thus, it annot be reommended to use (Eq. 65) for fitting p and it must be aepted for the moment that no experiments are available for the validated determination of p and p t. At first glane it seems logial to hoose p t and p idential to p t and p, respetively, beause both τ and τ are shear stressings in a plane parallel to the fibers. In fat, this approah is reasonable. However, from a miro-mehanial point of view τ - and τ -stressing lead to different stress distributions in the matrix. In the first ase the shear stressing is oriented parallel to the fibers, in the latter ase it is transverse. This has onsequenes for the miro-raking preeding IFF, too. Consequently the values for R and R A are slightly different for both ases. Likewise, for p and p similar but not neessarily equal values an be expeted. The

41 4. Inter fiber frature (IFF) riteria 77 same is valid for p t and p t. However, there is no reason for p = p t sine in the orresponding longitudinal ut through the master frature body σ n hanges its frature mode from impeding IFF-formation (σ n < 0) to promoting IFF-formation (σ n > 0). Evaluating all available information it is hereby reommended to use the set of parameters given in Table. Table. Inlination parameters for typial FRP [Cuntze et al. 997; Kopp 000; Puk et al. 00] t ϕ = 60 % p [ ] p [ ] t p [ ] GFRP/ to Epoxy 0.5 CFRP/ to Epoxy 0.30 p [ ] 0.0 to to 0.30 In order to keep the MFB at all points ontinuously differentiable an interpolation is neessary for all longitudinal setions ψ 0 and ψ 90. This is done as follows: p R t, t, t, ψ p = os ψ + sin A A A ψ R R p ψ (Eq. 66) with τ nt os ψ = τ + τ (Eq. 67) nt n sin ψ = os ψ (Eq. 68) Limits of validity of the reommended inlination parameters There is one last fator of relevane with regard to the hoie of the inlination parameters p t and p. This is due to the frature mehanism of brittle materials subjeted to a pure τ 3 -stress (Fig. 44). Intrinsially brittle materials subjeted to a pure τ 3 -stress always fail due to the prinipal normal stress σ n = σ II ourring on a frature plane whih is inlined towards the diretion of the τ 3 -stress by θ fp = 45 [Paul 96].

42 78 4 Puk s ation plane frature riteria Fig. 44. Frature due to pure τ 3 -stress The frature riterion will only predit this behavior and not a mixed mode frature due to simultaneously ating τ nt - and σ n t -stresses, if the following ondition is omplied [Cuntze et al. 997]: p t R R (Eq. 69) R R A t t A This has also been shown by numerial studies [Ritt 999]. In ase of p = p t = p as proposed above and applying the equation in Fig. 4 to replae R A one gets ( r + 4) + ( r 4) ( r + ) ( r + 4) p (Eq. 70) ( r + ) with r = R /R t. In aordane with Table usually p = 0.5 is hosen for GFRP and p = 0.3 for CFRP. Under terms of (Eq. 70) this does not ause any problems for CFRP as long as R /R t 3 holds, whih is a valid assumption for most CFRP having a thermosetting matrix. As long as R /R t.8 holds, p = 0.5 is appliable. For most but not for all GFRP having a thermosetting matrix this ratio will not be underut. For the remaining exeptional ases it makes sense to determine the value of p aording to (Eq. 70). However, in aordane with Table no values lower than p = 0. (orresponding to R /R t =.65) should be used sine otherwise unrealisti transverse ompression frature angles θ fp would be alulated from (Eq. 65).

43 4. Inter fiber frature (IFF) riteria 79 If in ase of R /R t <.65 the lower limit p = 0. is hosen one has to aept that the alulation proedure inorporating (Eq. 48) and (Eq. 5) will not lead to a frature angle of θ fp = 45 for a pure τ 3 -stress. In other words the frature riterion will indiate a mixed mode frature due to simultaneously ating shear stress τ nt - and tensile σ n -stress. However, all FRP known to the author have a ratio of R /R t >.65. This applies aording to latest measurements to arbon fiber reinfored PEEK (R = 70 N/mm ; R t = 90 N/mm ) and glass fiber reinfored Polyamid (R = 4 N/mm ; R t = 4 N/mm ), too. But on the other hand it an be expeted that for a material with an unusually low ratio R /R t a mixed mode frature and the orresponding frature angle θ fp 45 resulting from Puk s frature riteria will really our Analytial -D-formulation for plane states of stress In thin laminates there are often no onsiderable stresses present in thikness diretion. In this ase a D stress and strength analysis is suffiient and no numerial searh of the frature plane is required. The equations for the stresses on the ation plane σ n, τ nt and τ n (ompare (Eq. ), (Eq. ), (Eq. 3)) simplify to: ( θ ) = σ θ σ n os (Eq. 40) τ nt ( θ ) = σ sinθ osθ (Eq. 4) τ n ( θ ) = τ osθ (Eq. 4) This ase has been disussed before and used for the explanation of the different IFF frature modes (ompare hapter Different IFF-frature modes and Fig. 9). Figure 45 illustrates again the (σ, τ )-frature urve and the orrespondent lines on the surfae of the Master Frature Body (MFB). For deriving the analytial D-formulation of the IFF-riterion a separate inspetion of the three IFF-Modes is useful. If σ is a tensile stress, the frature ours on the plane perpendiular to the x -axis (θ fp = 0 ) and the frature mode is Mode A. The stresses on the frature plane are idential to the stresses σ, τ of the UD-lamina: σ n = σ, τ nt = 0 and τ n = τ. The IFF-riterion for σ n 0 (Mode A, ompare Fig. 4) an therefore be written with the stresses σ and τ : t t p, τ p f E IFF = σ + + σ, for σ 0 t R R R R (Eq. 7)

44 80 4 Puk s ation plane frature riteria Fig. 45. (σ, τ )-frature urve and its derivation from the Master-Frature-Body For σ n < 0 the IFF-riterion in Fig. 4 has the general form f τ ( θ ) τ ( θ ) nt fp n fp p ψ p ψ EIFF, θ σ θ σ θ σ n fp n fp for A A A n R R R ψ R ψ ( ) = + + ( ) + ( ), < 0 (Eq. 7)

45 4. Inter fiber frature (IFF) riteria 8 The angle θ fp of the frature plane is unknown in first plae. However, as long as the ratio of the transverse stress at frature and the transverse ompressive strength σ /R does not exeed a ritial value of roughly 0.4 the frature angle equals zero (θ fp = 0) just as for transverse tensile stress. The frature mode is Mode B and the frature ondition (Eq. 7) an also be written with the stresses σ and τ. In this ase the transverse shear stress is zero (τ nt = 0), the inlination parameter p ψ = p and the shear strength of the ation plane R A ψ = R (ompare Fig. 4). f A τ p p σ R EIFF, = + σ + σ, σ < 0 R R R τ τ, (Eq. 73) The fat that the transverse ompressive strength R does not our in (Eq. 73) reflets that IFF Mode B is a pure shear frature on the ation plane of τ. This shear frature is impeded by the ompressive stress σ ating simultaneously to τ on the frature plane. Higher shear stress τ an be sustained in the presene of a ompressive stress σ. This means that the frature resistane of the ation plane (θ = 0 ) is inreased by inreasing σ. If σ exeeds a ritial value, the frature resistane of the plane with θ = 0 is no longer the lowest of all ation planes. Consequently, frature takes plae on a different plane with θ 0. The ritial value for σ is R A, the frature resistane of the ation plane against transverse shear stressing. The point σ = R A marks the transition point to Mode C. The shear stress reahed at this point is τ,. Thus the point σ = R ; τ = R + p (Eq. 74) A, marks on the frature urve the transition from Mode B to Mode C. This will be explained in more detail in the following. To find an analytial solution in σ and τ for the frature ondition in Mode C is muh more diffiult than to formulate the (σ, τ )-frature onditions for Mode A and Mode B. In a first step the extremum problem d(f E )/dθ = 0 has to be solved. How the problem has been takled by Puk an be found in [Puk 996, Puk and Shürmann 998]. For mathematial simplifiation Puk oupled the two (usually independent) parameters p and p in the following way: p R A = R p (Eq. 75)

46 8 4 Puk s ation plane frature riteria Constraining the free hoie of parameters by this means does not onsiderably hange the results of frature analysis nor does it lead to ontraditions or a onfinement of the physial basis of Puk s riteria [Puk 996]. As a provisional result he found a surprising orrelation of the frature angle θ fp and the normal stress [σ n (θ fp )] fr at frature [Puk 996, Puk and Shürmann 998] A ( ) ( ) os σ θ = σ θ = n fp fr fp fr R (Eq. 76) That means that for the entire range of (σ, τ )-ombinations whih ause frature by Mode C the stress [σ n (θ fp )] fr on the frature plane is onstant and an be alulated from [σ n (θ fp )] fr = R A = R / [(+ p )] 0.4 R. That means that all frature points (σ, τ ) fr on the Master Frature Body are loated on the boundary of a ross setion of the MFB. This is the urve () (d) in Fig. 45. With these results the analytial solution in (σ, τ ) for the frature riterion of Mode C is τ σ R f E,IFF =, + ( + p ) R R σ R A σ < τ τ, for 0 σ (Eq. 77) Using (Eq. 77) in order to formulate os² θ fp = R A /(σ ) fr the following relationship for the frature angle θ fp is found A R τ os θ fp = + p R σ ( ) + (Eq. 78) This relation is very useful. For instane it allows to alulate the frature angle without having alulated the frature stress (σ ) fr first. It depends only on the ratio of τ /σ. Further it demonstrates that θ fp does not hange if R A and R are dereased by the same fator for instane by a weakening fator η w whih will be dealt with in the next Setion.

47 4.3 Extensions to the IFF-riteria Extensions to the IFF-riteria The following two hapters are in main parts idential to the orresponding hapters of the annex to VDI 04 Part 3 [VDI 006]. These hapters of the VDI 04 have been written by Alfred Puk and Günther Lutz and the use of this work is authorized by VDI Inlusion of stresses not ating on the frature plane in the ation-plane-related inter-fiber frature riteria The ation-plane-related strength riteria for inter-fiber frature (IFF) Fig. 4 and also (Eq. 7), (Eq. 73) and (Eq. 77) are based on Mohr s hypothesis. Aording to this, only the stresses σ n (θ fp ), τ nt (θ fp ) and τ n (θ fp ) ating on the parallel-to-fiber frature plane (angle of inlination θ fp ) are of deisive importane for frature stresses at IFF. All parallel-to-fiber setion planes are in priniple potential frature planes for IFF. Their angles of inlination θ (definition of θ in Fig. 8) vary between θ = 90 and θ = +90, where the setion planes with θ = 90 and θ = +90 are idential. The frature plane whih is to be expeted is determined by finding the setion plane with the highest stress exposure f E (θ) max dependent on the angle of intersetion (The stress exposure is defined in Fig. and Fig. ). The searh for the angle of the frature plane was analytially antiipated when equations (Eq. 7), (Eq. 73) and (Eq. 77) were set up. If the equations of Fig. 4 are used, the setion plane with the highest IFF stress exposure f E (θ max ) = f E (θ fp ) must be found by a numerial searh of the intersetionangle range from θ = 90 to θ = +90. If the frature plane has been identified in this way, the stresses (σ, σ 3, τ 3, τ 3, τ ) fr when the IFF ours may be obtained by dividing the omponents of the given stress (σ, σ 3, τ 3, τ 3, τ ) by the stress exposure f E (θ fp ) present on the frature plane. This therefore means that it does not matter whether the stresses σ n (θ), τ n (θ), τ nt (θ) ating on other setions with angles θ θ fp bring about relatively high or low IFF stress exposures. On aount of the gradual development of an IFF due to progressive formation of miro raks and for probabilisti reasons, it is, however, to be expeted that there are stress states where the Mohr approah gives results whih in dimensioning alulations would tend to be on the non-onservative side. This problem will be the subjet of hapter Aording to Mohr s hypothesis, even the parallel-to-fiber stress σ does not have any influene on the IFF sine it does not at on a parallel-to-fiber setion plane and thus does not at on a potential frature plane for IFF.

48 84 4 Puk s ation plane frature riteria Miromehanial aspets do mean that even in this ase a orretion will be needed. Chapter will deal with a orretion of this kind Inlusion of a parallel-to-fiber stress σ in the ation-plane IFF onditions Physial onsiderations Aording to Mohr s hypothesis, the stress σ does not have any influene on the IFF due to the fat that the ation plane of σ is perpendiular to the plane whih is ated on by the stresses σ n, τ n and τ nt whih are of deisive importane to the IFF. However, a series of effets appear to make it neessary to inlude a σ term in the IFF riteria whih redues IFF strength somewhat [Puk 996]. The most essential effet may be the following one. FF is taken to mean the frature of a very large number of elementary fibers whih in turn auses a lamina to loose some of its load-bearing apaity in the fiber diretion over a maro-region. Statistial laws, however, state that in the event of tensile stress σ some elementary fibers will already have ruptured before the FF limit of the UD lamina t at σ = R has been reahed. In the ase of ompressive stress σ it is possible that individual bundles of fibers ould already start kinking before total frature ours when σ = R. Oasional miro fiber fratures ause loal damage in the UD lamina whih takes the form of debonding at fiber matrix interfaes and of miro-fratures in the matrix material. They weaken the fiber matrix ohesion and thus also redue its resistane to IFF. Analytial treatment In order to inlude in IFF analysis the weakening effet resulting from σ in a way whih is appropriate to the physial irumstanes, the ationplane-frature resistanes R t, R A, R are multiplied by a degradation fator η w (w = weakening, =ˆ weakening due to σ ). A fator of this kind an be used not only in the tensile range but also in the ompressive range of σ. For the sake of simpliity it is assumed that the weakening fator η w has the same numerial value for all three ation plane-frature resistanes R t, R A, R. This assumption has the effet that the inlination of the frature plane given by the frature plane angle θ fp and the IFF frature mode onneted with it are not affeted by the weakening whih is now solely dependent on σ, sine σ does not depend on θ. Searhing the frature angle an be arried out in the same way as before negleting an influene of σ.

49 4.3 Extensions to the IFF-riteria 85 Reduing the frature resistanes results in an inreased stress exposure fator. Therefore the stress exposure fator f E when taking into aount the influene of σ beomes: f = (Eq. 79) E0 f E ηw As in [Puk 996, Puk and Shürmann 00] the weakening whih inreases progressively with σ is desribed by a frature urve in the form of a segment of an ellipse, Fig. 46. Previously η w had always to be alulated iteratively but now a losed solution has been found. The setion of the ellipse starts at s R and ends at the FF-limit namely at σ = R. Here, at the FF-limit the weakening fator η w reahes its minimum m. Both parameters s and m an be hosen independently between 0 and resulting in an exellent adaptability to experiments, espeially as the parameter pairs s and m an be set differently in the tension area and in the ompression area of σ (see Fig. 46). Under the given assumptions the following formula for the weakening fator η w has been derived: = ( a (a s ) + + s) ηw ( a) + fe with 0 s = and a =. fe (FF) m (Eq. 80) Fig. 46. Form of a (σ, σ )-frature urve with influene of σ on the IFF expressed as a funtion of parameters s and m

50 86 4 Puk s ation plane frature riteria f E (FF) is the FF-stress exposure fator, whih is alulated by using the FF-ondition σ = R given in (Eq. 4). The stress exposure fator f E0 is the IFF exposure without an influene of σ (The subsript 0 indiates that σ = 0). If weakening due to (m+p) effets is not taken into aount, then what should be used for f E0 are the stress exposure values f E alulated with the aid of (Eq. 7), (Eq. 73) and (Eq. 77) or the frature riteria from Fig. 4 with measured basi strengths and with the guide values given in Table, hapter for the inlination parameters. When (m+p) effets are in-luded, what should be used for f E0 is the value of f Em+p aording to t (Eq. 88) and using the orreted basi strengths R or, R or, R or, and the orreted parameter p or. The range of validity of the weakening fator η w is given by fe 0 m (Eq. 8) s fe (FF) Beyond this region there is either no damage or no IFF ours before the FF-limit is reahed. In the later ase it makes no sense to talk about an IFFstress exposure fator. Instead, there is just a FF-stress exposure fator. This is beause FF usually results in an extensive destrution of the fiber-matrixomposite. Thus there is no suh thing as an independent IFF anymore. So far there are no reliable experimentally determined values for the parameters s and m available [Kaiser Kuhnel Obst 004]. Thus assumed values must be hosen. It is reommended to use s = 0.5 and m = 0.5 both at σ > 0 and at σ < 0. With these values η w beomes η w = for 0.5. (Eq. 8) ( + 3) In priniple, one might doubt whether the weakening by η w ould influene the frature angle θ fp. Thus, the question whether there is suh an influene or not is disussed in the following. For doing so, the expression for η w doumented in (Eq. 80) needs to be examined further. Based on the assumption that IFF will our on the ation plane with a maximum of the stress exposure f E (inluding weakening by σ ), roots of d(f E (θ))/dθ have to be examined: f E ( θ ) ( a) + 0 fe ( θ) = = f ( ) E θ 0 η w ( a ( a s) + ) + s) (Eq. 83) fe ( θ) 0 with = f E(FF)

51 4.3 Extensions to the IFF-riteria 87 Here, f E(FF) does not depend of θ and is therefore not of interest for the searh for an extremum. ( E ( θ )) d f d d d df ( ( )) ( ( )) E = f 0 E = f E = 0 (Eq. 84) dθ d dθ d dθ This means, that the stress exposure f E inluding the weakening effet of σ has an extremum at the same point as the stress exposure f E0 alulated without weakening by σ. Now the question left to be answered is whether there are further extrema, namely a solution for d d ( ( ) ) f = 0. (Eq. 85) E The result for (Eq. 85) is =. (Eq. 86) a This means, that there is no real solution. There are no further extrema of the stress exposure apart from those whih are valid for the stress exposure without weakening by σ, too Inlusion of stresses σ () θ, τ () θ, τ () θ n nt n whih at on parallel-to-fiber planes but not on the frature plane Physial fundamentals The effets whih are examined will first be presented by taking the example of a (σ, τ 3 )-stress ombination, where σ is to be a tensile stress. Aording to Mohr s hypothesis, with (σ, τ 3 )-ombinations of this kind there either ours a σ -tensile frature on the ation plane of σ in other words, t at θ fp = 0 when σ = R or a τ -shear frature on the ation plane of τ 3 in other words, at θ fp = ±90 and this when τ 3 = R. Whih frature atually does our depends on where the (σ, τ 3 )-stress state vetor intersets the frature urve whih, aording to Mohr s hypothesis, onsists of the t two straight lines σ = R and τ 3 = R (frature urve for (σ, τ 3 ), see Fig. 47). If we had a uniaxial σ -stress state, then a stress σ n (θ) = σ os θ and τ nt (θ) = σ sinθ osθ would at on the setions adjaent to the frature plane at angles of inlination θ 0. We will now onsider a (σ, τ 3 )-vetor whih meets the straight line σ = R t lose to its intersetion with the straight line τ 3 = R. In this ase

52 88 4 Puk s ation plane frature riteria Fig. 47. Frature urves for three stress ombinations: (a) (σ, τ 3 ), (b) (σ, σ 3 ) and () (σ, τ ). The thin lines represent the starting urves for η m+p -orretion, alulated using the equations given in Fig. 4 with orreted basi strength values and orreted parameter p or aording to (Eq. 9): R t or = N/mm, R or = 04.8 N/mm, R or = 8.6 N/mm, p t = 0.35, p or = 0.45, p t = p = The thik lines represent the result urves of an η m+p -orretion aording to (Eq. 87), (Eq. 88), (Eq. 89) and (Eq. 90). the angle of the frature plane aording to Mohr will learly also be θ fp = 0. But if stresses σ and τ 3 are present simultaneously, in addition to σ n (θ) and τ nt (θ) a transverse/longitudinal shear stress τ n (θ) = τ 3 sinθ will our on setions at an angle θ 0. With this, as θ = 90 and θ = +90 is approahed and provided τ 3 /R is only a little less than σ /R t, there follows an IFF stress exposure f E (θ) whih is almost as high as that on the ation plane at θ fp = 0 (see standardized stress exposure f E (θ) in Fig. 48). For the ombination of σ with τ 3 whih we are onsidering, the IFF stress exposure over the whole range between θ = 90 and θ = +90 will not be muh lower than on the ation plane itself. Miromehanial studies show that an IFF does not simply happen suddenly for no partiular reason. One the IFF stress exposure f E (standardized to f Emax = ) exeeds a threshold value of about f E 0.5, instanes of

53 4.3 Extensions to the IFF-riteria 89 Fig. 48. Standardized stress exposure urves f E (θ) (parameters from Fig. 47); a) Left-hand side of diagram for different (σ, τ 3 )-stress ombinations; right-hand side for (σ, σ 3 )-stress ombinations. The omplete f E (θ)-urves for (σ, τ 3 )- and (σ, σ 3 )-stress ombinations are symmetrial to the f E axis at θ = 0. b) For different (σ, τ )-stress ombinations. The triangular area marked in the figure gives an impression of the magnitude of the referene sum S ref miro-damage to the fiber/matrix omposite will already our whih inrease progressively as stress inreases. In the ase of transverse/longitudinal shear stressing τ, these are the familiar 45 miro raks (hakles) whih are stopped at fibers and ause often tiny delaminations there (see Fig. 49). It is only when miro-fratures have exeeded a ertain magnitude that the IFF will suddenly our. In the ase of the (σ, τ 3 )-ombination under onsideration here, not only miro-fratures resulting from σ but also those resulting from τ 3 develop simultaneously the ase is different with a uniaxial σ stress. The latter type of miro-fratures due to τ 3 further weakens the fiber/matrix omposite as ompared with the situation with uniaxial σ - stress. The miro-fratures have a weakening effet on all setions with inlination angles between θ = 90 and θ = +90. They also redue the transverse tensile strength whih in the end is still available on the frature plane when IFF happens at θ fp = 0. It is therefore to be expeted that σ in ombination with a relatively high τ 3 stress will not reah at IFF that frature stress whih is obtained in the uniaxial transverse tension test with σ = R t.

54 90 4 Puk s ation plane frature riteria Fig. 49. Ourrene of matrix raks [Puk 99]; a) τ shear stress leads to raks under 45 (so alled hakles) whih are stopped at the fibers and are ause of parallel to fiber mini delaminations. b) With σ tensile stress raks are partiularly enouraged where there are flaws and an spread along the fibers without hindrane. ) σ stress results in raks ourring at the tip of broken individual fibers and running transverse with respet to the fiber until stopped at adjaent fibers. As a further example let us examine a biaxial (σ, σ 3 )-transverse tensile stress. In this ase a tensile frature ours at θ fp = 0 if σ > σ 3 or at θ fp = ±90 if σ 3 > σ. If σ = σ 3 = σ were the ase, the same stress would prevail at all setions at any angle θ due to the relation σ(θ) = σ os θ + σ 3 sin θ = σ and thus the intersetion-angle-dependent stress exposure f E (θ) would be equally high on every setion plane with θ between 90 and +90. In this speial individual ase Mohr s hypothesis predits that IFF will our simultaneously on all setion planes with angles of θ between 90 and +90. At what angle frature will atually take plae in a biaxial transverse tension test with two stresses of equal magnitude depends on ontingenies, e. g. on the distribution of flaws. We will now pass on to a (σ, σ 3 )-ombination where σ 3 = 0.95 σ. In this ase the stress exposure f E (θ) is nearly the same in all setions with angles θ between 90 and +90 (see again the standardized stress exposures f E(θ) in Fig. 48). For this reason a more massive instane of mirodamage is to be expeted before the IFF ours than would be the ase with uniaxial transverse tensile stress. A real fiber/matrix omposite in addition always ontains flaws for example, in the form of uring raks, flat air entrapments, or loal imperfetions in the bonding between fiber and matrix whih often over only a part of the irumferene of the fibers. Flaws of this kind have a sense of diretion. For example, a uring rak will have an espeially strengthreduing effet when a tensile stress σ n perpendiular to the rak plane ours. In the ase under onsideration, in whih σ 3 is only 5% less than

55 4.3 Extensions to the IFF-riteria 9 σ, the stress exposure f E (θ) is theoretially at its maximum at θ fp = 0. There is nevertheless a high probability that with a different angle of inlination θ of the setion plane, a partiularly serious flaw will trigger the IFF despite the somewhat lower theoretial stress exposure f E (θ) found there. This is a probabilisti effet. It makes predition of the angle of the frature plane unertain and auses the frature stresses at IFF to be somewhat lower than those alulated on the basis of Mohr s hypothesis. In both examples there is a mixture of the effets of mirodamage and of probabilisti effets; they annot be treated separately. Generalizing, these examples permit us to arrive at the following onlusion. It is to be expeted that the effets of mirodamage and probabilistis redue the magnitude of frature stresses when the IFF ours will rise the more parallel-to-fiber setions at different angles θ there are for whih relatively high values of the stress exposure fator f E (θ) an be alulated or, to put it another way, the rounder the stress exposure urve f E (θ) will be. A lear piture of whih situation of this kind exists with the stress state to be investigated an be obtained by examining the entire stress exposure urve f E (θ) alulated using Mohr s equations of Fig. 4 (see standardized stress exposures in Fig. 48). The stress ombination σ = σ 3 > 0 has been reognized as being an extreme ase where at any setion the value for f E (θ) is equally high. Analytial treatment Miromehanial failure analyses and mathematial methods from probabilistis annot be used in omponent design. A alulation method will therefore be presented below whereby it is possible, with minimal effort, to estimate the effets of miro-damage and probabilistis (in abbreviated form: (m+p) effets) with the aid of a phenomenologial approah on physial foundations and whih is appliable in engineering pratie. This method will be referred to as η m+p orretion. It is a slightly modified form of the approah whih appears in [Puk 996]. The η m+p -orretion is based on the following assumptions: The frature plane at IFF appears at the angle of intersetion θ fp for whih, aording to Mohr s hypothesis the maximum stress exposure f E (θ) max = f E (θ fp ) is alulated from the frature riteria in Fig. 4. The frature stresses at IFF are obtained by orreting the frature stresses alulated from f E (θ fp ) by Mohr s hypothesis to lower stresses using a orretion fator of η m+p <. (Correspondingly, the IFF stress exposures alulated by Mohr are orreted to higher values by dividing the Mohr stress exposures by η m+p ).

56 9 4 Puk s ation plane frature riteria Expressed formally (see also Fig. 47a): {} σ {} σ {} σ Mohr m p = fr ηm + p = ηm + p fe( θfp) {} σ + = (Eq. 87) f E m + p f fe = η ( θfp ). E m + p m+ p (Eq. 88) {σ} m+p = frature stress vetor taking (m+p) effets into aount, {σ} Mohr fr = frature stress vetor without taking (m+p) effets into aount, {σ} = vetor of the effetive stress state, η m+p = orretion fator of the frature stresses for taking (m+p) effets into aount, f Em+p = stress exposure of the frature plane taking (m+p) effets into aount, f E (θ fp ) = stress exposure of the frature plane without taking (m+p)- effets into aount. As will be explained later, {σ} fr Mohr and f E (θ fp ) have to be alulated by using orreted strengths R t or, R or, R or. The reason for this is the following. Experimentally determined strengths values are real values. This means that effets of miro-damage and probabilistis have already influened these values. The analytial proedure of the η m+p -orretion should therefore not a seond time degrade these values. In other words: When operating the η m+p -proedure and applying the η m+p -orretion to the ase of uniaxial transverse tension, to the ase of uniaxial transverse ompression and to pure longitudinal shear the results should be the experimental values for R t, R and R. This is ahieved in the following way: Before starting a frature analysis with η m+p -orretion the experimentally determined values R t, R and R are divided by η m+p belonging to uniaxial tensile stressing, uniaxial ompressive stressing and pure longitudinal shear stressing respetively. The appropriate fators η m+p for orreting R t, R and R are alulated from (Eq. 89) and (Eq. 90) by using the orreted strengths R t or, R or and R or. These are not known at the beginning. This is no problem as far as R or and R or are onerned, beause when normalizing f E to f E the hosen values for these strengths are anelled from the equation. This is different for uniaxial tension, where f E is dependent on R t or and R A or. In this ase a few iterations are neessary. Later on, perhaps also the sustainable strengths for uniaxial σ t and σ or pure τ are alulated in the ourse of a general frature analysis inluding

57 4.3 Extensions to the IFF-riteria 93 (m+p) effets. This is done now automatially by multiplying the orreted t strengths R or, R or and R or.with just those η m+p by whih R t, R and R had been divided before. Therefore, the results of these η m+p -orretions are just the experimentally determined strengths R t, R and R. The dependene of the (m+p) orretion fator η m+p on the roundness of the standardized stress exposure urve f E (θ) an be quantified by means of the following evaluation of the stress exposure urve f E (θ)with the aid of a summation formula. In the frature analysis of a general threedimensional stress state, during the neessary numerial searh for the frature plane angle the intersetion-angle-dependent IFF stress exposure f E (θ) from θ = 90 to θ = + 90 is alulated with the aid of the frature riteria in Fig. 4 in small angular inrements Δθ (in most ases with Δθ = ). The result is a large number of f E (θ)-values whih represent points of the stress exposure urve f E (θ). To ensure that all possible stress states are treated alike, the stress exposure urves are standardized in suh a way that at point θ = θ fp the stress exposure fator has a value of. The standardized stress exposure urve f E (θ) is then valid for just that magnitude of the stresses whih aording to the frature riteria in Fig. 4 would produe frature and this is true for any kind of stress ombination. It follows from the foregoing that setion planes with f E (θ)-values lose to are of muh greater importane than those with f E (θ) values whih are onsiderably smaller. For this reason, as already ours in [Puk 996], standardized f E (θ) values below a threshold f E thr = 0.5 are not taken into onsideration. On the basis of equation (0.5) in [Puk 996], the following sum is obtained with the standardized f E (θ) values: 89 ' ' ' ' ( E( ) E ), only for fe( θ) fethr + S f θ f Δθ 90 thr. (Eq. 89) (The limits 90 and +89 apply to the angular inrement Δθ = whih was used). This summing (with a zero line shifted by f E thr ) results in high f E (θ) values being weighted onsiderably more heavily than low values for example, a value f E (θ) = 0.9 is weighted four times higher than a value f E (θ) = 0.6. For the extreme ase whih we mentioned of σ = σ 3 > 0, when f E (θ) = 0.5 the maximum S value of S max = ( 0.5) 80 = 90 is obtained. If the summation formula (Eq. 89) is applied to the standardized stress exposure urves f E (θ) for the basi stressings, uniaxial transverse tension σ t, uniaxial transverse ompression σ and pure transverse/longitudinal shear τ (see Fig. 7), we obtain values for the orresponding sums S t,

58 94 4 Puk s ation plane frature riteria S, S whih will lie between 30 and 40. S = 39.4 is a materialindependent fixed value while S depends on the inlination parameter p and has a value of approximately S 36. For S t we find material dependent values with S t 33. Taking the value S whih orresponds to a given stress state and slightly modifying equation (0.6) in [Puk 996], we obtain the following for the (m+p)-orretion fator: η = Δ S S ref m+ p max (Eq. 90) Smax Sref for Δ max 0.5 with S max = 90 and S ref = 30. It is neessary to introdue a referene value S ref in order to obtain a reasonable sensitivity of η m+p to the differenes in the S values ourring with different stress states. Experiene shows that there is no S value of any stress state whih is less than the value S ref = 30. The variable Δ max is the relative differene between the frature stress vetor alulated aording to Mohr and that orreted by η m+p when σ = σ 3 > 0. Values for Δ max should lie between 0.5 and 0.5. Seletion of a value for Δ max results in a alibration of the η m+p -orretion fator. Appliations In ommon pratie of the design and dimensioning of fiber omposite omponents, frature riteria are used for alulating the IFF stress exposure (and FF stress exposure) at some loations of the omponent whih are regarded as ritial. In what follows, however, entire frature urves for IFF will be presented. Frature urves of this kind reveal in whih stress states η m+p -orretion is important and in whih it is not. In Fig. 47a it an be seen that with the (σ, τ 3 )-ombinations, η m+p -orretion is very important in the first quadrant. With the (σ, σ 3 )-ombinations (Fig. 47b), a very notieable influene also ours in the first quadrant of the (σ, σ 3 )-frature urve. In the ase where σ = σ 3, this yields the utilized alibration value Δ max = 0.5 used with η m+p -orretion. In the fourth (and seond) quadrants, on the other hand that is, where σ > 0 and σ 3 < 0 only a relatively minor effet of the η m+p -orretion is found. At σ 3 = σ the frature stress at IFF alulated with η m+p is even a little higher than that alulated on the basis of Mohr s hypothesis (σ = σ 3 = R t ). This results from the fat that for the pure transverse/transverse shearing stressing τ whih is here present the orresponding sum has the value S = 3.45, and this is somewhat less than S t = 3.45 for uniaxial transverse tension stressing σ t.

59 4.3 Extensions to the IFF-riteria 95 To date there has been a lak of redible experimental results for (σ, τ 3 )- and (σ, σ 3 )-stress ombinations. The situation is entirely different for the (σ, τ )-stress ombinations shown in Fig. 47. The (σ, τ )-frature urve is the only one whih has adequate experimental baking not only for CFRP but also for GFRP [Cuntze et al. 997]. An outstandingly good mathematial model is obtained with equations (Eq. 7), (Eq. 73) and (Eq. 77) or the frature riteria in Fig. 4 using inlination parameters from Table, hapter Before proeeding further, it is important to bear in mind the fat that the effets of η m+p -effets on IFF strength are basially already inluded in experimentally determined frature stresses, as also in the test results from whih the modeled (σ, τ )-frature urve (thik line in Fig. 47) is obtained. If one wished to generate mathematially a (σ, τ )-frature urve whih involved use of η m+p -orretion, then the experimentally obtained (σ, τ )- frature urve whih was modeled using the measured basi strengths R t, R, R and inlination parameters from Table is not the starting urve for the η m+p -orretion proedure but represents rather the result urve. If it is exlusively plane (σ, σ, τ )-stress states whih have to be investigated, then the question of the unknown starting urve for an η m+p - orretion is irrelevant sine the mathematially modeled, experimentally determined (σ, τ )-frature urve is being used diretly for frature analysis. On the other hand, use of η m+p -orretion is advisable in those ases where it is neessary, during dimensioning of FRP omponents, to also analyze load appliation zones in whih spatial stresses σ 3, τ 3, τ 3 of a similar magnitude to σ and τ our. To ensure that absolutely onsistent alulation results are obtained here, it is, of ourse, neessary that the same parameters be used uniformly in the frature riteria in Fig. 4 for all stress states whih our in other words, even for the (σ, σ, τ )-stress states prevailing in undisturbed areas. For this reason the parameters must be known whih are appropriate for the (σ, τ )-starting urve whih is itself as yet unknown. Sine in the ase of the (σ, τ )-stress ombinations the result urve is known and the starting urve is required, an inversion of the η m+p - orretion, as it were, is now neessary. The orreted basi strengths R t or, R or, R or for the starting urve are obtained by inverting the η m+p -orretion: the measured values R t, R, R are not multiplied but are instead now divided by the η m+p -value whih is assoiated with the orresponding basi stressing. (Comment: These orreted basi strengths of ourse apply not only to the (σ, τ )-stress state but of ourse also to any stress state.) Sine the sum values S t, S, S of the basi stressings are

60 96 4 Puk s ation plane frature riteria only a little higher than the referene value S ref = 30, the orretion they need remains less than 5%. In the range of high ompressive stress values σ with at the same time high values of τ, sum values up to around S 60 are however reahed in the viinity of the transition point between Mode B (θ fp = 0 ) and Mode C (θ fp 0 ) (see Fig. 4). This results in η m+p -orretions up to approximately %, see Fig. 48b and Fig. 47. The required starting urve of the (σ, τ )-stress ombination annot be obtained as a whole from the known result urve simply by inverting the η m+p -orretion. The reason for this lies in the omplex nature of the η m+p - orretion whih is only to be arried out numerially on a point by point basis and whih makes impossible a losed mathematial formulation of the required starting urve. What is needed, however, is a (σ, τ ) starting urve formulated using the equations in Fig. 4, sine only these equations happen to be available for the frature analysis of general threedimensional stress states. For this reason, an aeptable approximated starting urve whih is desribed using the equations in Fig. 4 must be found by seleting suitable parameters. What aeptable means is that when a η m+p -orretion of this starting urve has been arried out orretly, a (σ, τ )-frature urve is obtained as the result whih will deviate only to an aeptable extent from the known modeled experimental (σ, τ )- frature urve. By iterative seletion of different inlination parameters p or for the (σ, τ )-starting urve and reviewing to what extent the required aeptane is thereby ahieved, the following result is obtained. If a orreted inlination parameter p or = 0.45 is hosen for the required starting urve of the η m+p orretion when the alibration value Δ max = 0.5 is used, what is obtained as the result of a orretly performed η m+p -orretion is a (σ, τ )-frature urve whih omes very lose to the experimental urve modeled in the usual way using the equations in Fig. 4. (Modeling of the experimental urve was arried out using the basi strengths for CFRP R t = 48 N/mm, R = 00 N/mm, R = 79 N/mm [Soden et al. 998] and the inlination parameters p t = 0.35, p = 0.30, p t = p = 0.75 aording to [Puk Kopp Knops 00]). If, in response to one s own experimental results, one wishes to use a value different from the value Δ max = 0.5 used in the example and also, within the permitted limits [Puk Kopp Knops 00], to diverge from the guideline value p = 0.3 given in Table, you an proeed iteratively as desribed above to obtain a orreted parameter p or provided you have a omputer program whih an display (σ, τ )-frature urves. If this is

61 4.3 Extensions to the IFF-riteria 97 not the ase, the following extrapolation formula may be used within a restrited range of p and Δ max : p or Δ max = p p for 0.5 Δmax 0.5 and 0.5 p (Eq. 9) Comment It has been shown that depending on whether you work with or without η m+p -orretion, you will have to analyze the stress exposure urve f E (θ) by using different inlination parameters p or or p respetively whih may differ aording to (Eq. 9) by a fator of up to.6. If shear stresses τ and/or τ 3 our in the stress state under investigation, by alulating with and without η m+p -orretion you will therefore obtain angles of the frature plane θ fp whih differ somewhat. In the ase of (σ, τ 3 )-stress ombinations the differenes are from to 5. With (σ, τ )-stress ombinations similar differenes are found, provided the angles of the frature plane θ fp 30. Considerably greater differenes our in the viinity of the transition point from mode B to mode C (see Fig. 4). But here any statements about the angle of the frature plane whih atually ours will be tainted with a high degree of unertainty anyway: this is due to the probabilisti effets whih our and whih an be expeted due to the flat ourse of the standardized stress exposure urve. This has been shown in experiments as well [Kopp 000]. All in all, these deviations are not problemati in the frature analysis. L 4.3. Calulation of the streth fator f S of the loaddetermined stresses when residual stresses are present Basi onsiderations In the following the proedure to be used when alulating the streth fator f S L of the (hanging) load-determined lamina stresses σ L, σ L, τ L when (onstant) residual stresses σ r, σ r, τ r are simultaneously present will be explained for the ase of a (σ, σ, τ ) stress state. This proedure is based on Puk s ation-plane frature riteria for IFF and takes into onsideration the weakening influene of the parallel-to-fiber stress σ. The formulae required for the proedure are given.

62 98 4 Puk s ation plane frature riteria The frature onditions are the starting point for IFF when there is no influene from σ. These are (see equations (Eq. 7), (Eq. 73) and (Eq. 77)): IFF mode A f E0 t t p τ p σ t σ forσ = + + =, 0 R R R R IFF mode B f E0 A p p σ R σ σ, σ 0 τ τ τ = + + = < R R R IFF mode C f E0 A ( R + p R ) ( R ) τ σ = + = 4 σ τ τ for σ < 0 and 0 σ R A ( R ) (Eq. 7) (Eq. 73) (Eq. 9) (Eq. 9) is equivalent to (Eq. 77). The following oupling of inlination parameters was used in deriving equation (Eq. 9): p R A = R p For this reason (Eq. 93) A R needs to be alulated using equation (Eq. 94): R A R R = + p (Eq. 94) p R The frature onditions for IFF equations (Eq. 7), (Eq. 73) and (Eq. 77) ((Eq. 9) respetively) desribe in the (σ, σ, τ ) stress spae the irumferential surfae of a ylindrial frature body (see Fig. 50) whose onstant ross-setional ontour is the frature urve for (σ, τ ) stress ombinations shown in Fig. 45 whih has been ompleted symmetrially with respet to the σ axis. This body extends to infinity in both the positive and negative σ diretions; its real importane eases, however, no later than at the two fiber frature limits at σ = R t and σ = R. Any influene of σ on the IFF is taken into aount by introduing a weakening fator η w < (see above) in the ase of the strengths R t, R,

63 4.3 Extensions to the IFF-riteria 99 Fig. 50. Creation of the (σ, σ, τ ) frature body for IFF. a) (σ, σ, τ ) frature body for IFF without σ influene aording to (Eq. 7), (Eq. 73) and (Eq. 9). This frature body whih theoretially extends to infinity in both the positive and negative σ diretion is of pratial relevane only between the fiber frature limits at σ = R t and σ = ( R ). b) IFF frature body alulated with weakening σ influene using (Eq. 0), (Eq. 0) and (Eq. 03). ) Valid remaining frature body as intersetion of the bodies in aordane with a) and b) and following removal of the invalid parts R whih are of primary, deisive importane to the IFF. These strengths do not weaken until a threshold value of σ is passed. This value must be determined on the basis of experimental experiene and is given as a fration of the stress (σ ) fr = R or (σ ) fr = ( R ) leading to fiber frature. In t hapter an ellipse has been seleted for the ourse of the weakening fator η w as a funtion η w (σ ). In order to obtain a losed solution for f L S, and that in the form of the solution of a quadrati equation, it will be neessary to linearize the funtion for η w (see equation (Eq. 95) and Fig. 5). Weakening of the strengths R t, R, R inreases as FF stress exposure f E,FF inreases; in other words, the weakening fator η w beomes smaller as f E,FF inreases. Here the following linear equation is used: η (Eq. 95) ' w = ηw ηw f 0 E, FF η w,0 is a hypothetial value of η w when σ = 0, and η w is the slope of the straight line for η w aording to equation (Eq. 95). By definition, the stress exposure f E, FF is the ratio of the ating stress and the stress leading to frature (f. disussion). Aordingly, from the FF ondition equation (Eq. 5) we obtain: f σ EFF, = with ( σ) fr ( σ ) ( σ ) t = R for σ 0 fr = R for σ < 0 fr (Eq. 96)

64 00 4 Puk s ation plane frature riteria Fig. 5. Course of the weakening fator η w as a funtion of σ or as a funtion of f E,FF in the range R t σ R or 0 < f E,FF <.0 respetively. In order to approximate the ellipse with parameters s = m = 0.5 as reommended in hapter 4.3.., the values η w0 =.6 and η w = are hosen (S = M = 0.6). t The reason for using different signs with R and ( R ) lies in the international onvention whereby ompressive stresses are expressed as negative values while all strengths (even ompressive strengths) are expressed as positive values. Now, with the aid of the weakening fator η w = η w (σ ) whih we introdued in the ase of the three strengths R t, R, R a desription is obtained of that (onial) part of the frature body whose surfae equation depends not only on σ and τ but also on σ (Fig. 50b). To do so, it is neessary to selet numerial values for the two parameters η w0 and η w in the linear equation (Eq. 95) for the weakening fator η w. In hapter the starting point of the elliptial weakening was defined by the value s and the minimum value of the weakening fator at the FF limit by the value m. If the orresponding values for the linearized ourse of η w (f E,FF ) are designed by S and M this delivers the following relationships: S M η = w 0 S M η = w S (Eq. 97) (Eq. 98) In hapter the reommendation was made, should experimental results be laking, that the values s = 0.5 and m = 0.5 should be seleted when η w has an elliptial ourse. A good approximation to this ellipse may be obtained by using S = 0.6 and M = 0.6 for the straight line. On the basis

65 4.3 Extensions to the IFF-riteria 0 of (Eq. 97) this yields η w0 =.6 and on the basis of (Eq. 98) η w =.0. From this we obtain the ourse of the weakening fator η w shown in Fig. 5 as a funtion of σ or f E, FF. (In priniple even different pairs of values (S and M) an be used for the ranges σ > 0 and σ < 0.) By applying an equally high weakening fator η w to all three strengths R t, R, R whih have a deisive influene on IFF we obtain (σ, τ ) frature urves whih represent a geometrially similar redution by a fator η w < of the (σ, τ ) frature urve shown in Fig. 45 whih was ompleted symmetrially to the σ axis. As a onsequene, therefore, of the linear equation for η w (σ ), in the (σ, σ, τ ) stress spae we find a frature body for IFF whih has the shape of a double one (Fig. 50b). Its ross setions perpendiular to the σ axis are geometrially similar. The ylindrial frature body shown in Fig. 50a and the double-onial frature body interset eah other at σ = S R t and σ = S ( R ) in two setions whih are vertial with respet to the σ axis. The validity of the ylindrial frature body extends from the oordinates origin as far as these intersetion planes. The onial irumferential surfaes of the double-onial frature body for IFF as shown in Fig. 50b are valid in the t range between the intersetion planes and the planes at σ = R and σ = R whih are vertial to the σ axis and assigned to FF. Calulation of the desired streth fator f L S for the load-determined stresses σ L, σ L, τ L from the geometrial point of view requires determination of the point where the resulting stress vetor {σ} makes ontat with the valid range of the surfae of the (σ, σ, τ ) frature body for IFF (Fig. 50). The resulting stress vetor {σ} omes from the geometrial sum of the residual stress vetor {σ} r and the load stress vetor {σ} L elongated (strethed) by the fator f L S. The shape of the (σ, σ, τ ) frature body as shown in Fig. 50 and the requirement that the loaddetermined vetor {σ} L may only be strethed in its speified diretion means that there is only one possible point of ontat between the vetor and the surfae of the frature body. In most ases this point of ontat will be loated on the ylindrial or onial part of the irumferential surfae appliable to IFF. It an, however, also be loated on one of the two end faes whih are valid for FF Derivation of the formulae required for alulating f In the frature onditions for IFF equations (Eq. 7), (Eq. 73) and (Eq. 9) the same weakening fator η w (σ ) is used with the strengths L S

66 0 4 Puk s ation plane frature riteria R t, R, R. Using IFF stress exposure f E with weakening due to σ we therefore have as frature ondition with weakening: f E ( σ τ σ ) ( σ, τ) ( σ ) = f = (Eq. 99) E0,, ηw or also f ( σ, τ ) η ( σ ) E0 W = (Eq. 00) For the three IFF modes this therefore yields: For mode A R t t p t σ + τ + p σ = ηw σ R R ( ) (Eq. 0) For mode B τ + σ + σ = η σ R For mode C ( p ) p W ( ) ( R ) τ σ + = η 4( R p R ) R A + σ ( ) ( σ ) W (Eq. 0) (Eq. 03) ' σ with ηw = ηw η, 0 W σ where ( ) and ( ) t fr ( ) σ = R for σ > 0 σ = R for σ < 0. fr fr The validity ranges of frature onditions (Eq. 0) to (Eq. 03) are the same as those for equations (Eq. 7), (Eq. 73) and (Eq. 77) (see also Table 3). In the equations (Eq. 0) to (Eq. 03) the stresses σ and τ are the stresses at the frature limit. The strengths R t, R, R appearing in equations (Eq. 0) to (Eq. 03) are the unweakened strengths. For the stresses (σ, σ, τ ) ourring in

67 4.3 Extensions to the IFF-riteria 03 them we use the stress omponents orresponding to the ontat point of r L L σ = σ + f σ : the resultant frature vetor { } { } { } r L L S fr σ =σ + f σ, (Eq. 04) σ =σ + f σ, (Eq. 05) r L L S τ =τ + f τ. (Eq. 06) r L L S In this way we obtain equations whose only unknown is the desired L streth fator f S of the load-determined stresses. After neessary reformulation, whih in partiular inludes the removal of the root expressions, we L obtain for eah of the modes A, B, C a quadrati equation for f S to whih an be applied the known solution formula: ( 4 ) L fs = l q l. (Eq. 07) q Quantities q and l are the oeffiients of the quadrati terms or of the linear terms of f L S, and in are grouped the orresponding (onstant) terms L whih are independent of f S. Table shows quantities q, l and in eah ase for IFF modes A, B and C. S Proedure for alulating f and evaluation of results L S With this task the starting situation is generally the following. The omponent design proess ame up with a laminate whereby speifiation of fiber diretions and fiber quantities has been brought to a stage whereby the next step is to try to improve the laminate struture with regard to homogenizing the safety to IFF in the individual laminae. The load-determined stresses whih need to be taken into aount here in most ases relate to the maximum loading to be expeted when the omponent is in servie. With this maximum loading an addition safety margin with respet to IFF is to be provided and generally a larger safety margin with respet to FF. When onsiderable residual stresses are present, it is not possible on the basis of alulated stress exposure values f E to ome to any useful onlusions regarding these safety margins for the individual lamina, instead this is f of the load- entirely possible on the basis of the streth fators determined stresses. L S

68 04 4 Puk s ation plane frature riteria Before ommening with determining to be advisable to larify whether an IFF an even our in the lamina under onsideration or whether a FF will appear first before an IFF. L f S with regard to IFF, it appears From the simple FF ondition given in (Eq. 4) it follows that it is solely stress σ whih is responsible for the FF. At FF: f E, FF σ + f σ r L L σ S( FF) = = =. (Eq. 08) ( σ ) ( σ ) fr fr where (σ ) fr = R t for σ > 0 and (σ ) fr = R for σ < 0. If σ r and σ L have the same sign, σ = σ r +f S L(FF) σ L an also only have the same sign. If σ r and σ L have different signs, it an be assumed that σ r alone will not yet ause a FF. This is beause a FF due solely to σ r must not even be permitted and will have already been ruled out by the designer s predimensioning. Should a stress f S L(FF) σ L with a different sign appear in addition to σ r, then as it inreases it will initially work in opposition to the residual stress σ r. Not until after the disappearane of σ r will the resulting stress σ inrease with the same sign as σ L until the FF limit at σ = (σ ) fr is reahed and FF finally ours. Consequently the deision regarding the quantity (σ ) fr to be used in equation (Eq. 08) an be made as follows: ( ) fr t = R ( ) = L σ for σ > 0 L σ for σ < 0. R fr (Eq. 09) The streth fator f s(ff) L of the load-determined stresses with regard to FF is thus obtained from equations (Eq. 08) and (Eq. 09): ( σ ) σ =. (Eq. 0) r L fr f S ( FF ) σ L The three omponents σ L, σ L, τ L of the load-determined stress vetor are all inreased by the same streth fator. One the FF limit is reahed, the resulting stresses will therefore have these magnitudes: σ = σ + σ, (Eq. ) r L L f S ( FF ) σ = σ + σ, (Eq. ) r L L f S ( FF ) τ = τ + τ. (Eq. 3) r L L f S ( FF )

69 4.3 Extensions to the IFF-riteria 05 With the aid of these stresses a hek is made as to whether the ontat point of the resulting stress vetor (at σ = R t or σ = R ) lies inside or outside the ontour line of the frature body end fae as shown in Fig. 50. This ontour line is the intersetion line of the onial irumferential surfae for IFF and the FF plane. It is a redued (σ, τ ) frature urve for IFF whih is geometrially similar redued by the minimum weakening fator η w = M valid at the FF limit. With stresses σ and τ alulated at the FF limit on the basis of equations (Eq. ) and (Eq. 3) and η w = M, the IFF stress exposure with σ influene an be alulated from: f E = fe f 0 E0 η = M, (Eq. 4) W doing so with the equation for mode A, B or C whih is valid for the stress state under onsideration (see equations (Eq. 7), (Eq. 73) and (Eq. 9)). If the result f E = were obtained from equation (Eq. 4), the ontat point would be loated exatly on the IFF urve and simultaneously in the plane for FF. In theory this would therefore mean that both IFF and FF would our. If f E < is the result, the vetor passes through the FF plane without having previously touhed an IFF irumferential surfae in L other words, a f S value annot be alulated for IFF. Should the result be f E >, this means that the vetor must piere the L IFF irumferential surfae before it reahes the FF limit. In this ase f S is alulated for IFF using the proedure shown in the flow hart in Fig. 5. Instead of the sequene shown in this flow hart, the omputational operations and inquiries ould also be arried out in a different order. Whatever the ase, it ends up to a proedure whereby arbitrary assumptions are made first (for example, about whih IFF mode is to be expeted, A, B or C and (σ ) fr = R t or (σ ) fr = R as well as an assumption as to whether a σ influene is already beoming effetive or not. A deision is then made on the basis of the omputational results as to whether the assumptions made L were justified or not. If this does not appear to be the ase, f S is realulated using modified assumptions. The proedure in the flow hart (Fig. 5) is that in eah ase the investigation is arried out for a seleted mode A, B or C until the result is aepted or rejeted. The final test is to hek to see whether the alulated ontat point atually falls within the range of validity of the equations assoiated with the mode under onsideration. If the result for a mode passes in addition to all proeeding tests also this final one, this means that the result is the orret and only possible result, beause only one ontat point exists.

70 06 4 Puk s ation plane frature riteria Fig. 5. Cheks whih have to be arried out during alulation of the streth fator f S L for an IFF mode A, B or C in order to deide whether the weakening influene of σ needs to be onsidered for the IFF modes, as well as the final hek, whether the ontat point to the frature body surfae falls within the range of validity of the equations for the onsidered IFF-mode A, B or C.

71 4.3 Extensions to the IFF-riteria 07 Table. Terms of equation (Eq. 07) for the alulation of L f S IFF mode A B C ( σ ) fr ( ) ( ) ( ) ' ( w ) ( ) L t ' L L t t L L p η w σ σ p R ( ) t σ fr ( σ) fr τ σ η σ qa = + R R R R t ' L r t L r r L t t p η ' w σ σ p σ σ σσ p R l A = + ηw η 0 w + + R ( σ ) R ( σ ) t fr fr ( R ) R ' L r r L ηw σ ' σ ττ + η w η 0 w + ( σ) ( σ ) R fr fr q r ( ) ' r σ t ( R ) ( σ ) r ( ) t t t r r p R σ τ p σ ' σ A = η w η 0 w + + η w η 0 w R R R ( σ ) fr fr B L ( ) ' L ( w ) ( ) ' L L p ηw σ σ R ( σ) fr ( σ) fr τ η σ = R p η σ σ p σ l = + + r L ' L r L r ττ w ' σ B η w η 0 w R R ( σ ) R ( σ) fr fr ' L r η wσ ' σ + η w η 0 w ( σ) ( σ) fr fr ( τ r ) r r r ' σ p σ ' σ B = η w η 0 w + ηw η 0 w R ( σ ) R ( σ) fr fr ( ) ( + ) ' ( ) w ( ) R ( σ) L L τ σ L L η σσ A fr r L ' r L ' L r L L r ττ ηw σσ ηw σσ η wσ 0 σ σ C = + + qc = + 4 R p R R l A ( ) ( ) ( ) R p R R σ R σ R + fr fr ( R ) r r r ( τ ' r r ) ηw σ ( σ ) 0 ηw σσ A ( R ) R ( ) R ( ) p R R σ + fr = C A Note: Beause of the parameter oupling p R = p R aording to A (Eq. 93) R needs to be alulated using (Eq. 94): R R A R = p p + R t σ = R for σ > 0 ( ) ( ) fr σ = R fr for σ < 0 The relations given in Table are also suitable for the alulation of without influene of σ by setting η w =.0 and η = 0 0 w L f S

72 08 4 Puk s ation plane frature riteria Table 3. Range of validity of the equations for IFF modes A, B, C IFF mode Range of validity σ + σ f S A r L L A 0 B C σ σ σ + f σ r L L A r L L S B + fs B σ < 0and 0 r L L τ + fs B τ τ r L L r L L S C + fs C σ < 0and 0 r L L A σ + fs C σ R τ + f τ R τ with τ = R p + ( aording to (Eq. 94)) p aording to (Eq. 93) and A R With the oordinates σ, σ, τ of this ontat point on the frature body surfae, the value f E = for the IFF stress exposure f E must result from equation (Eq. 99) when applying the relevant equation (Eq. 0), (Eq. 0) or (Eq. 03). This an be used for heking purposes. L If a value is alulated for f S whih is <, this indiates that the IFF limit will have already been exeeded with the existing load stresses L L L σ, σ, τ used in the alulation! 4.4 Visualization of frature bodies The visualization of frature onditions for FRP has been treated by Kopp [Kopp and Mihali 999]. This visualization an very muh help to see and omprehend the harateristis of different frature onditions. For the prinipal understanding it is important to reall the remarks on the oordinate systems in the hapter 3. The visualization of frature onditions for isotropi material is not diffiult at all. In this ase the 6 stress omponents σ, σ, σ 3, τ 3, τ 3, τ are substituted by 3 prinipal normal stresses σ IP, σ IIP, σ IIIP. For fibrous transversely isotropi materials suh a hange of the oordinate systems is not possible, as a rotation is only allowed in the so alled transversely isotropi (x, x 3 )-plane. The x -axis must remain parallel to the fiber diretion, beause only for these natural diretions the basi strength parameters of the UD-layer are known.

73 4.4 Visualization of frature bodies 09 The following investigations will only onsider stresses having a diret influene on IFF. The fiber parallel stress σ is disregarded, sine IFF ours in a fiber parallel plane [, 3] and with that the frature plane is parallel to σ. The orresponding situation for an isotropi material is that σ = σ IP is neither the minimal nor the maximal prinipal normal stress. Aording to Mohr only the extreme values of the prinipal stresses σ IIP and σ IIIP do have an influene on the frature proess; for this reason frature ours parallel to the diretion of σ IP similar to the IFF in an UD-lamina. For human imagination it is neessary to restrit frature bodies to three dimensions. Therefore it seems reasonable in a first step to transform the (σ, σ 3, τ 3, τ 3, τ )-stress state in onsideration of the transverse isotropy into a (σ II, σ III, 0, τ III, τ II )-stress state (Fig. 0). By no means σ II and σ III are real prinipal stresses σ IIP and σ IIIP, beause also the shear stresses τ III, τ II our on their ation planes (Fig. 0). Using the following transformation rule one is enabled to substitute any stress omponent appearing in a frature ondition in the way wanted: σ os ϕ σ 3 sin ϕ τ 3 = sinϕ osϕ τ 3 0 τ 0 sin ϕ os ϕ sinϕ osϕ 0 0 sinϕ osϕ sinϕ osϕ os ϕ sin ϕ osϕ sinϕ 0 σ II 0 σ III 0 0 sinϕ τ III osϕ τ II (Eq. 5) with τ 3 ϕ = artan σ σ 3 The next step is to replae the shear stresses τ III and τ II by the resulting longitudinal shear stress τ ω as shown in (Fig. 53). The Figure larifies that τ III and τ II are nothing else but the omponents of a longitudinal shear stress τ ω ating on a plane perpendiular to the x -diretion [Puk 997]: τ = τ + τ = τ + τ = τ ω III II III II ω (Eq. 6) The diretion of τ ω is defined by the angle δ: τ III τ δ = artan = artan τ τ II III II (Eq. 7) δ is alled the differene angle or deviation angle beause its value is a measure for the deviation of the ation planes of τ ω and σ II. Stress states

74 0 4 Puk s ation plane frature riteria Fig. 53. Resulting shear stress τ ω and deviation angle δ without transversal stresses σ II and σ III will lead to an IFF in the fiber parallel plane the resultant shear stress τ ω is ating in. In this speial ase the frature angle is θ fp = ω = artan τ 3 /τ and frature takes plae when τ ω has reahed the longitudinal shear strength R. One an imagine that in general the deviation angle δ between the ation planes of τ ω and σ II is of main importane for the frature proess. It is obvious that in dependene of δ the interation between the transverse extreme stresses σ II, σ III and the resulting shear stress τ ω is more or less distint. As an example one an regard a (σ II > 0, 0, τ ω, δ)-stress state with δ = 0 on the one hand and δ = 90 on the other hand. In ontrast to the omplete interation of σ II and τ ω for δ = 0 there is no interation at all for δ = 90 and frature does not take plae as long as either σ II reahes the transverse tension strength R t or τ ω reahes R. It is inevitable to onsider δ as a parameter for the visualization of frature onditions. From an aademi point of view this leads to an infinite number of frature bodies in the (σ II, σ III, τ ω )-stress spae, as any small hange of δ might modify the frature body slightly. For the omparison of different frature onditions it is absolutely suffiient to make a Δδ =,5 grading. All frature onditions visualized in this hapter were alulated with the same set of basi strength values published in [Cuntze et al. 997]. If a riterion is omplemented by additional material oeffiients the suggestions of the aompanying developer were fulfilled.

75 4.4 Visualization of frature bodies Tsai,Wu-riterion A lassial representative of a global strength riterion is the well-known riterion of Tsai and Wu (Tsai and Wu 97). It is a pure interpolation polynomial dedued from the basi strength values of the UD-layer, not regarding the fat that some strength values are dominated by fiber strength and others by matrix or interfae strength. The result is a smooth frature body without any edges. Most ertainly the frature body does not hange its shape if the influene of σ is negleted. Aording to expetations the Tsai,Wu-riterion takes on the shape of an ellipsoid in the (σ II, σ III, τ ω )-stress spae, too (Fig. 54). There is no dependene on the deviation angle δ. As σ II and σ III are treated absolutely equally in the invariant formulation of the riterion the frature body is moreover symmetrial with regard to the (σ II = σ III )-plane. Therefore stress states having a transverse normal stress and a longitudinal shear stress ating on the same plane are treated equally to those having their ation planes perpendiular to eah other. Another remarkable aspet of this frature body is that it is losed for biaxial ompressive stresses σ II σ III. From a maro mehanial point of view whih is the basis of all strength riteria investigated here the shear frature due to transverse ompression is not possible for σ II = σ III as the maximum transverse shear Fig. 54. Frature body of the Tsai,Wu-riterion

76 4 Puk s ation plane frature riteria Fig. 55. Frature body of the Hashin-riterion stress τ max = ½ (σ II σ III ) always remains zero. However beause of miromehanial inhomogeneity a damage limit for biaxial transverse ompressive stresses must be expeted ([Kopp et al. 997]). After exeeding this damage limit the miro-raks in the matrix will redue the original mehanial properties notieably but a real material separation does not happen Hashin-riterion Usually the term Hashin-riterion for IFF is related to two invariant frature onditions published in (Hashin 980), although in the same publiation an ation plane related approah is disussed too, whih gave Puk the impulse to develop his ation plane related IFF-riterion (Puk 99). As Hashin was aware of the importane of the sign of the normal stress σ n on the frature plane, his invariant formulations differentiate between the ases σ n 0 and σ n < 0. Unfortunately the frature plane is not known a priori and annot be found by using invariant formulations. Therefore Hashin had to define a more pratial ondition to delimit his frature onditions from eah other. He fixed the boundary of the setions for σ n 0 and for σ n < 0 in the plane σ III = σ II. However he simultaneously pointed out the severe physial ontraditions, whih result from this rather arbitrary boundary. The orresponding frature body is omposed of a paraboloid with elliptial ross setions for σ n < 0 and a rosswise oriented ellipsoid for σ n 0. Both are symmetri to the (σ II = σ III )-plane and interset eah other in the

77 4.4 Visualization of frature bodies 3 (σ III = σ II )-plane in an elliptial borderline. The frature urve for τ ω = 0 and σ n 0 ontradits to the theory of Paul (Paul 96), whih assumes no interations of the prinipal stresses in this area, but Hashin already takes into aount that a real frature due to biaxial transverse ompression is not possible. Aordingly the frature body is not losed for σ II σ III. Again it has to be emphasized that the Hashin frature body is independent of the deviation angle δ Puk riterion Figure 56 displays the frature bodies of Puk s IFF-riterion in the (σ II, σ III, τ ω )-stress spae in dependene on δ. Of ourse all frature bodies show the same base in the (σ II, σ III )-plane, whih has exept of the ompression/ompression domain the shape of Paul s frature envelope for intrinsially brittle isotropi material (Paul 96). Comparable to Hashin s approah in the third quadrant frature due to biaxial transverse ompression is not possible. Partiularly interesting is the shape of the frature bodies for δ = 0 and δ = 90. Here the omponents of the frature surfae, whih belong to the different frature modes marked in the figure, touh eah other in sharp edges. (A frature mode is defined by the ombination of stressings ating on the frature plane (Puk 997)). It is remarkable that Puk s frature Fig. 56. Frature bodies of the ation plane related riterion by Puk