TECHNSCHE ECHANK, 7,, (07), 8-7 submitted: November 7, 0 ode Delamination Analsis of Asmmetrical Four Point Bend Laered Beams with Considering aterial Non-linearit V. Rizov An analtical stud is performed of mode delamination fracture in the asmmetrical Four Point Bend (FPB) beam configurations with considering non-linear material behavior. The beam mechanical behavior is described b using two non-linear constitutive models (ideall elastic-plastic model and model with power law stressstrain relation).the crack is located arbitraril along the beam height. Fracture is analzed b appling the - integral approach. B using the classical beam theor, analtical solutions of the -integral are obtained at characteristic levels of the external load. The solutions derived are compared with the strain energ release rate. The influence of material non-linearit and crack location along the beam height on the fracture behaviour is evaluated. The -integral solutions derived are ver useful for parametric investigations, since the simple formulae obtained capture the essentials of fracture in asmmetrical FPB beams that have non-linear material behaviour. ntroduction The integrit and load bearing capacit of laered beam structures depends upon their delamination fracture properties. Appearance and propagation of delamination cracks between laers significantl deteriorates the structure performance. For instance, cracking considerabl reduces the stiffness of beams and ma lead to catastrophic failure. Considerable efforts have been made in the analsis of delamination fracture in laered beam structures b man researchers (Guadette et al., 00; Guo et al., 00; Her and Su, 05; Hsueh et al., 009; iao et al., 998; Sørensen and acobsen, 00; Suo et al., 99; Szekrenes, 0; Szekrenes, 0; Tkacheva, 008; Yeung et al., 000). nfluence of residual stress (hgroscopic, thermal and piezoelectric induced stresses) on the strain energ release rate in laered beam specimens has been analzed b Guo et al., 00. ethods of linear-elastic fracture mechanics have been applied. Closed form analtical solutions for determination of the strain energ release rate have been derived b using the classical beam theor. The effects of geometrical and material parameters of the specimens on the delamination fracture behaviour have been evaluated. Delamination fracture behaviour of laminated composite beams have been investigated b appling the -integral approach b Sørensen and acobsen, 00. Determination of cohesive laws for different beam specimens has been disscussed. The effects of specimen shape on strength have been predicted b using the cohesive laws. Failure of adhesive joints has also been analized. icromechanisms of splitting of laminated composites have been studied both experimentall and theoreticall. The effects of various bridging mechanisms on the delamination fracture resistance of laminated composites have been evaluated b Suo et al., 99. A famil of stead-state mixed mode delamination beams has been considered. A complete solution has been obtained which has been used to construct R-curves for given model parameters. Double cantilever beams loaded b moments and b wedge forces have been compared in order to elucidate the significance of stead state fracture for understanding delamination R-curves. t has been recommended to use R-curves when studing localized damage response. 8
Dnamic delamination fracture in laered materials has been studied analticall b Tkacheva, 008. Linearelastic behaviour of the material has been assumed. Solutions have been obtained b using the classical beam theor. An equation for balance of energ has been used as a criterion for delamination crack growth. The sted-state delamination fracture behaviour of multilaered beams has been analzed assuming linear-elastic material behaviour b Hsueh et al., 009. The fracture has been studied in terems of the strain energ release rate b appling the classical beam theor. A closed form analtical solution has been derived for a delamination crack located arbitrar along the beam height. The solution can be used for multilaered beam configurations with an number of laeres. Besides, each laer can have individual thickness and modulus of elasticit. The delamination fracture toughness of a four-laer linear-elastic beam structure has been studied b Her and Su, 05. t has been assumed that a delamination crack is located between the second and the third laers. A formula for the strain energ release rate has been derived b using the classical beam theor. The fracture has been studied in a function of modulus of elasticit and thickness of the laers. t can be concluded that delamination fracture in laered beam configurations, including the FPB, has been analzed in terms of the strain energ release rate mainl assuming linear-elastic material behaviour. n realit, however, the beam structures ma have non-linear behaviour of material. Therefore, the purpose of present article is to perform an analtical stud of mode delamination fracture in the asmmetrical FPB beam configuration under static loading with taking into account the material non-linearit. Fracture is analzed b appling the -integral approach. Two non-linear constitutive models (ideall elastic-plastic model and elasticplastic model with power law stress-strain relation) are used to describe the mechanical response of beam considered. Closed form analtical solutions of the -integral are obtained b appling the conventional beam theor. The influence of crack location and material non-linearit on the fracture behaviour is discussed. Analsis of the Asmmetrical FPB Beam The FPB beam configuration considered is illustrated in Figure. The beam is in four point bending. The beam cross-section is a rectangle of width, b, and height, h. There is a delamination crack of length, a, located arbitraril along the beam height. The lower and upper crack arm thicknesses are h and h, respectivel (Figure ). t is assumed that h h. Figure. Geometr and loading of the asmmetrical FPB beam The fracture behaviour is analzed b using the -integral approach (Rice, 98; Rice et al., 97; Broek, 98): u cosα p Γ u + p x v ds x 0 x, () 9
Γ is a contour of integration going from one crack face to the other in the counter clockwise direction, u 0 is the strain energ densit, α is the angle between the outwards normal vector to the contour of integration and the crack direction, p and p are the components of the stress vector, u and v are the components of the x displacement vector with respect to the crack tip coordinate sstem x, and ds is a differential element along the contour Γ. Figure. Stress-strain diagram of the ideall elastic-plastic model ( u 0 and and complimentar strain energ densit, respectivel) pl * u 0 are the strain energ densit t should be specified that the present analsis is based on the small strain assumption (this assumption has been frequentl used in analses of delamination fracture in laered beam configurations (Hsueh et al., 009)). The -integral solution is derived b using integration contour, Γ, consisting of the cross-sections ahead and behind the crack tip (Figure ). Therefore, the -integral value is obtained b summation: + A + A B, () A, A and B are the -integral values in segments A, A and B, respectivel.. Analsis b Using the deall Elastic-plastic odel First, the mechanical response of the FPB beam is described b using the ideall elastic-plastic model (Figure ). At low magnitudes of the external load, the beam deforms linear-elasticall. The normal stresses, induced b bending are distributed linearl along the beam height. Thus, the components of the -integral in the segment A of the integration contour are written as p x σ, 0 z p, dz ds, cos α, () bh is the principal moment of inertia of the lower crack arm cross-section. The coordinate, z, originates from the lower crack arm cross-section centre and is directed downwards ( z varies in the interval h ; + h ). The cross-sectional bending moment in the lower crack arm,, that participates in (), is obtained b using the following formula (Rzhanitsn, 98) 0
h h + h, (4) Fl (5) is the bending moment in the crack tip beam cross-section (Figure ). The partial derivative, u, is written as x u σ ε x E E. () z E is the modulus of elasticit. The strain energ densit is obtained as u 0 σ E z E z. (7) E B substitution of (), () and (7) in (), we obtain the following solution of the -integral in segment A A. (8) The -integral solution in the segment A (Figure ) is found also b (8). For this purpose, and h are replaced with and h, respectivel A, (9) h h + h (0) is the cross-sectional bending moment in the upper crack arm (Rzhanitsn, 98). The -integral components in the segment B of the integration contour are written as (Figure ) p x σ z, 0 u x z, 0 E p, ds dz u z E, cos α, (), ()
( h) b is the principal moment of inertia of the un-cracked beam cross-section, is obtained b (5). The coordinate, z, originates from the beam cross-section centre and is directed downwards ( z varies in the interval [ h ; + h] ). After substitution of () and () in (), we obtain B F l 4Eb h. () The final solution is found b substitution of (8), (9) and () in () + t should be mentioned that at F l 4Eb h. (4) h h h equation (4) transforms into F l. (5) 4Eb h Equation (5) coincides with the formula for the strain energ release rate when the crack is located in the FPB beam mid-plane (Hutchinson and Suo, 99). When the external load magnitude increases, the ield stress limit arm, since h h. f will be attained first in the lower crack Figure. Elastic-plastic distribution of stresses and strains in cross-section A of the lower crack arm Two smmetric plastic zones will develop in the lower crack arm, while the rest of the beam will continue to deform linear-elasticall. The normal stresses and the longitudinal strain distribution in the segment, A, of the integration contour is shown in Figure. The -integral solution in A is obtained b integration along the elastic and plastic zones. A A el + A pl () B performing the necessar mathematical operations, we derive
( ) ep el A ep 8 + el ep el, (7) ep Q + Q + el + 4 ( Fl) (8) q Q (9) Q + Q q Q (0) p q + p + 4Fl + Fl q el + 4Fl + 08 el F l el Q () ( ) F l () el el + ( ) ( + 4Fl)( Fl + F l )+ f el + () bh el. (4) el The upper crack arm deforms linear-elasticall. Therefore, the -integral solution in the segment A of the integration contour can be found b (9). For this purpose, has to be replaced with e e A (5) e Fl ep. () The solution in segment B of the integration contour is obtained b (), since the un-cracked beam portion, x 0, deforms linear-elasticall (Figure ). The -integral final solution is found b combining of (), (), (7) and (5) ( ) ep el ep 8 + el ep + el Eb e F l h 4Eb h. (7) Solution (7) is valid if h < h /, because plastic collapse of the lower crack arm will occur before,lim h the onset of plastic deformation of the upper crack arm.
f h h,lim, plastic strains will develop also in the upper crack arm before the plastic collapse of lower crack arm. n this case, the -integral solution is found as ( ) ep el ep 8 + el ep + el ep ( el ) ep 8 + + el ep F l el 4Eb h bh el el, (8) f (9) Fl + el el el el ep (0) ( el + el ) Fl + el el el el el ep Fl. () ( el + el ) t should be specified that equation (8) is applicable when the external load magnitude F is in the boundaries between the elastic limit load and the plastic colapse load F for the lower crack arm, i.e. h ( h + h ) ( h + h ) F F pc pc. (). Analsis b Using the odel with Power Law Stress-strain Relation The delamination fracture is analzed also assuming that the mechanical behaviour of FPB beam can be described b using the constitutive model with power law stress-strain relation. The stress-strain equation is written as (Dowling, 007) H ε n σ, () σ and ε are the normal stresses and longitudinal strains, H and n are material constants. The fracture is analzed again b appling the -integral approach (). The -integral solution is ( n + )( n + ) n h n + + n + κ h H n + ( n + )( n + ) H n h n + + n + nκ h H n κ ( )( ) ( ) n ( ) + h h n + n + [ ] n + n +, (4) n + ( n ) + Fl n n + n + bh( h + h ) κ (5) 4
n + ( n ) n Fl + bh H κ. () The -integral solutions are compared with the strain energ release rate G determined with taking into account the material non-linearit b using the following equation * U a U G b a * b (7) * * U b and U a are the complimentar strain energies before and after the increase of crack, respectivel, a is a small increase of the crack length. The fact that the -integral solutions coincide with the strain energ release rate is an indication for consistenc of the non-linear delamination fracture analses performed in the present paper. Parametric nvestigations A parametric investigation is performed in order to evaluate the effect of non-linear behaviour of material on the mode delamination fracture in the asmmetrical FPB laered beam configuration. For this purpose, calculations of the -integral are carried-out b using formula (8). Figure 4. The -integral value in non-dimensional form plotted against F / F ratio (curve linear-elastic pc material behaviour, curve elastic-plastic material behaviour) The -integral value obtained is presented in non-dimensional form b using the formula N /( Eb). t is assumed that b0.0 m, h0.0 m, h 0. 009 m and l0.5 m. The -integral value in non-dimensional form is plotted against F / Fpc ratio in Fig. 4 ( F / F ratio varies in the boundaries determined b ()). The lower pc boundar of F / F ratio calculated b () at pc h 0. 009 and h 0. 0 is 0.755. n order to evaluate the effect of material non-linearit on the delamination fracture, the -integral value obtained assuming linear-elastic material behaviour b formula (4) is also plotted in non-dimensional form against F / F ratio in Figure 4 for comparison with the curve generated b non-linear solution. The curves in Figure 4 indicate that the material non-linearit leads to increase of the -integral value. Therefore, the non-linear behaviour of material has to be taken into account in fracture mechanics based safet design of structural members composed b laered materials. pc 5
The effect of crack location along the beam height on the elastic-plastic fracture behaviour is analzed too. For this purpose, the non-linear solution (4) is used. The -integral in non-dimensional form, /( Hb), is plotted against h / h ratio at n 0. 8 and F / 0. 95 in Figure 5. F pc Figure 5. The -integral value in non-dimensional form plotted against law stress-strain relation h / h ratio for the model with power t can be observed that the -integral value is maximum at h / h 0. 5, i.e. when the crack is located in the beam mid-plane (Figure 5). 4 Conclusions An analticall stud is conducted of mode delamination fracture in the asmmetrical FPB laered beams. The basic purpose is to analze the fracture behaviour with considering the material non-linearit when the crack is located arbitraril along the beam height. Fracture is investigated b appling the -integral approach. The FPB beam mechanical behaviour is described b using two non-linear material models (ideall elastic-plastic model and model with power law stress-strain relation). Closed form analtical solutions of the -integral are derived with the help of classical beam theor. The solutions are compared with the strain energ release rate derived with taking into account the material non-linearit. The effects of material non-linearit and crack location along the beam height on the fracture behaviour are analzed. t is found that the -integral value is maximal when the crack is in the beam mid-plane. The analtical solutions derived are ver convenient for parametric investigations, since the simple formulae capture the essentials of delamination fracture with considering the nonlinear material behaviour. The present stud contributes for the understanding of delamination fracture behaviour of laered beams that exhibit material non-linearit. Also, the analsis developed in the present paper can be applied for studing the mode strain energ release rate in the FPB adhesion test in which adherends are made of metalic materials with non-linear behaviour (for inctance, the mechanical behaviour of mild steel and aluminium can be reasonabl approximated b the ideall elastic-plastic model, the model with power law stressstrain relation can be applied for describing the mechanical behaviour of annealed copper (Denton, 9)). Acknowledgments The present stud was supported financiall b the Research and Design Centre (CNP) of the UACEG, Sofia (Contract BN 89/0).
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