Stress and Failure Analysis of Laminated Composite Structures

Transcription

1 PDHnline Curse M37 (6 PDH) Stress and Failure Analysis f aminated Cmpsite Structures Instructr: Jhn J. Engblm, Ph.D., PE PDH Online PDH Center 57 Meadw Estates Drive Fairfax, VA Phne & Fax: An Apprved Cntinuing Educatin Prvider

2 PDH Curse M37 ABE OF CONENS. Intrductin 4. Material Definitins 5. Istrpic Material Behavir 5. Anistrpic Material Behavir 5.3 Orthtrpic Material Behavir 6 3. Hke s aw fr Orthtrpic Materials 6 4. Restrictins n Elastic Cnstants 4 5. Stress-Strain Relatins fr Generally Orthtrpic amina 6 6. Biaxial Strength heries fr Orthtrpic amina 3 6. Separable Strength (Failure) heries Maximum Stress hery Maximum Strain hery Hashin uadratic hery Chang uadratic hery Generalized Strength (Failure) heries sai-hill hery sai-wu ensr hery Anther Example Cmparing Failure heries Failure Envelpes (Generalized heries) fr 49 Biaxial Stress State 6.5 Effect f Shear Stress Directin n amina Strength Analysis f aminated (Multi-ayered) Cmpsites Specifying Stress and Strain Variatin in a aminate Relating Resultant Frces and Mments t 59 Strain and Curvature Jhn J. Engblm Page f 9

3 PDH Curse M Including Hygrthermal Effects in aminate Analysis Cnstructin and Prperties f Varius aminates Symmetric aminates Unidirectinal, Crss-Ply and Angle-Ply aminates uasiistrpic aminates Sme Examples f aminate Analysis w-ply [45/] aminate Subjected t 74 Applied ads 7.5. w-ply [45/] aminate Subjected t 78 hermal ad Only w-ply [45/] aminate Subjected t 8 Applied and hermal ads uasiistrpic [/45/-45/9] S aminate 84 Subjected t Applied ads 8. Summary References 87 Jhn J. Engblm Page 3 f 9

4 PDH Curse M37 INRODUCION his curse fcuses n presenting a well established cmputatinal methd fr calculating stresses/strains in reinfrced laminated cmpsite structures. he basis fr the presented cmputatinal methd is ften referred t as classical laminatin thery. A clear understanding f this apprach is supprted by the develpment f the fundamental mechanics f an rthtrpic lamina (ply). Varius failure theries are presented each requiring that stresses/strains be quantified n a ply-by-ply basis in rder t make failure predictins. Bth applied lads and hygrthermal (thermal and misture) effects are treated in the cmputatinal prcedure. Stress and failure predictins are an imprtant part f the prcess required in the design f laminated cmpsite structures. he learning bjectives fr this curse are as fllws:. Understanding the differences between istrpic, rthtrpic and anistrpic material behavir. Having knwledge f the material cnstants required t define Hke s law fr an rthtrpic lamina (ply) 3. Understanding the restrictins n the material cnstants required in evaluating experimental data 4. nwing the difference between reference and natural (material) crdinates fr an rthtrpic lamina 5. Being familiar with the stress-strain relatins in reference and natural crdinates fr an rthtrpic lamina 6. Understanding the crdinate transfrmatins used in transfrming stresses and/r strains frm ne crdinate system t anther 7. nwing generally the types f tests perfrmed t determine the stiffness and strength prperties f an rthtrpic lamina 8. Having knwledge f a number f biaxial strength (failure) theries used in the design f laminated cmpsite structures 9. Understanding which in-plane strength quantities are needed, as a minimum, in applying varius failure theries. nwing the difference between separable and generalized failure theries. Understanding that the maximum stress and maximum strain failure theries make similar predictins except under certain material behavir. Appreciating under what cnditins the Chang failure criteria reduces t the Hashin failure criteria 3. nwing the basis fr the fact that the sai-wu failure criteria is mre general than the sai-hill failure criteria 4. Being familiar with the effect f the directin f shear stress n lamina strength 5. Understanding the laminate rientatin cde used t define stacking sequence 6. Being familiar with a number f special laminate cnstructins designed t eliminate undesirable cmpsite material behavir 7. Understanding the cmputatinal prcedure fr determining the stresses/strains in a laminated cmpsite subject t applied lads and/r hygrthermal effects 8. Having knwledge f the limitatins f classical laminatin thery. Jhn J. Engblm Page 4 f 9

5 PDH Curse M37 It is imprtant t nte a limitatin n the cmputatinal methdlgy presented in this curse. Stress predictins frm classical laminatin thery are quite accurate in lcatins away frm bundaries, e.g., free edges, edge f a hle r cutut, etc., f the laminate. hus at distances equal t the laminate plate(shell) thickness r greater, the cmputatinal methd presented herein is accurate and useful in the preliminary design f laminated cmpsite structures. he basis fr this limitatin is that laminatin thery assumes a generalized state f plane stress which is reasnably accurate away frm bundaries. Alng bundaries, the state f stress becmes three-dimensinal with the pssibility that interlaminar shear and/r interlaminar nrmal stresses can becme significant. Deviatin f laminatin thery alng laminate bundaries is ften referred t as a bundary-layer phenmenn. Cmputatin f stresses alng laminate bundaries is generally accmplished thrugh the applicatin f finite difference, finite element r bundary element methd cmputer prgrams and is beynd the scpe f the methdlgy presented in this curse.. MAERIA DEFINIIONS A lamina r ply can be thught f as a single layer within a cmpsite laminate and is cmprised f a matrix material and reinfrcing fibers. When the fibers are lng the layer is referred t as a cntinuus-fiber-reinfrced cmpsite and the matrix serves primarily t bind the fibers tgether. Alternatively layers with shrt fibers are dented as discntinuus-fiber-reinfrced cmpsites. amina are quite thin, i.e., generally n the rder f mm r.5 in. thick. amina can have unidirectinal r multi-directinal fiber reinfrcement. herefre a number f lamina bnded tgether frm a laminate. Mst laminated cmpsites used in structural applicatins are in fact multilayered. aminates have identical cnstituent materials in each ply; therwise the term hybrid laminate is used fr laminates cmprised f plies with different cnstituent materials. Fiber reinfrced cmpsites are hetergeneus but fr purpses f design analysis are typically assumed t be macrscpically hmgeneus. hus fr the cmputatinal methdlgy presented in this curse, rthtrpic lamina (plies) are treated as hmgenus with directinally dependent prperties. Orthtrpic material behavir falls smewhere between that f istrpic and anistrpic materials.. Istrpic Material Behavir Fr istrpic materials defrmatin behavir is independent f directin. hus nrmal stresses prduce nrmal strains nly and shear stresses prduce shear strains nly, as depicted in the figure belw. Jhn J. Engblm Page 5 f 9

6 PDH Curse M37 Figure.. Extensinal and Shear Defrmatin, Istrpic Material. Anistrpic Material Behavir In the case f anistrpic materials, defrmatin behavir is dependent n directin. hus, uniaxial tensin prduces bth extensinal and shear cmpnents f defrmatin. ikewise, pure shear lads als prduce extensinal and shear defrmatin. Anistrpic material behavir is depicted in the simple sketch belw. Figure.. Extensinal and Shear Defrmatin, Anistrpic Material.3 Orthtrpic Material Behavir In the case f rthtrpic materials defrmatin is, in general, directin dependent. An exceptin ccurs when lads are applied in natural (material) crdinates. hese are by definitin crdinates in the plane f the lamina, wherein the lngitudinal crdinate is aligned with the fiber reinfrcement and the transverse crdinate is aligned nrmal t the fiber reinfrcement. ngitudinal and transverse directins are material axes f symmetry in a unidirectinally reinfrced cmpsite. When lads are applied in these natural crdinates the material respnse is similar t that f istrpic materials, i.e., nrmal stresses prduce nrmal strains nly and shear stresses prduce shear strains nly as shwn belw. Jhn J. Engblm Page 6 f 9

7 PDH Curse M37 Figure.3. Extensinal and Shear Defrmatin, Orthtrpic Material, (aded Alng Material Crdinates) Here the lngitudinal and transverse axes are labeled as and, respectively. Unidirectinally reinfrced cmpsites are ften referred t as specially rthtrpic. Furthermre, unidirectinally reinfrced laminas are istrpic in the ut-f-plane (nrmal t the plane f the lamina) directin. 3. HOOE S AW FOR ORHOROPIC MAERIAS Generalized Hke s law has the tensrial frm (3.) ij E ijkl kl where stresses are related t strains thrugh the elastic cnstants E ijkl. In the matrix frm f the cnstitutive equatins, we have { } [ E]{} (3.) 9x 9x9 9x Here, the stress and strain tensrs are f rder 9x and there are 9x9 r a ttal f 8 elastic cnstants in the stiffness matrix [E]. It will be shwn that these 8 elastic cnstants reduce t cnstants even withut any axes f symmetry. With independent elastic cnstants we have an anistrpic material. he stress tensr ntatin is sketched in figure 3. belw. Jhn J. Engblm Page 7 f 9

8 PDH Curse M37 Figure 3.. Stress ensr Ntatin Cnsider ging thrugh this reductin in the number f elastic cnstants. First, cnsider that we have symmetry in the strains, i.e., j i ij ji It is therefre easily shwn that E ijkl E ijlk We als have symmetry in the stress tensr, i.e., and therefre, ij ji E ijkl E jikl And thus the tw symmetries reduce the elastic cnstants frm 8 t 36. We have { } [ E]{} (3.3) 6x 6x6 6x Here we have a ttal f 36 elastic cnstants. Nw cnsider the strain-energy density functin defined as a functin f the strains as with the prperty ( ) U U (3.4) ij ij U / (3.5) ij Jhn J. Engblm Page 8 f 9

9 PDH Curse M37 he simple -D analgy f (3.5) relates t the fact that the area under the stress-strain curve is equal t the strain energy density. In the -D case we have U Substituting the -D frm f Hke s law, i.e., E int the abve gives U E hus we have simply U / E fr a simple uniaxial state f stress. Getting back t the 3-D case, we substitute the stress-strain relatins (3.) int (3.5) U / (3.6) ij E ijkl kl aking the derivative again we have Interchanging indices gives kl ( U ij ) E ijkl / / (3.7) ij ( U kl ) E klij / / (3.8) Since the rder f differentiatin is immaterial we have herefre ij ( U / ) / ( U ) / / kl kl ij E ijkl E klij Since ij and kl are interchangeable, we nw have cnstants fr an anistrpic material. In the matrix frm f the previusly written cnstitutive equatins (3.3), the stiffness matrix [ E ] is therefre a symmetric matrix. We have n(n)/ independent cnstitutive terms in [E], where n6. Jhn J. Engblm Page 9 f 9

10 PDH Curse M37 It is mre cnvenient t write Hke s law in matrix frm as { } [ ]{} (3.9) Where [] is symmetric as befre, s that the ff-diagnal stiffness terms are defined as ij ji. If we think f the and axes as crdinates in the plane f the lamina, where the axis aligns with the fiber reinfrcement, the axis is transverse t the fibers and the 3 axis is then nrmal t the (, ) plane, these axes are sketched belw. Figure 3.. Natural (Material) Crdinates f Unidirectinally Reinfrced amina he ply (lamina) depicted abve shws nly ne fiber thrugh the ply thickness. his is atypical as there are nrmally several fibers thrugh the thickness f a typical ply. Nte that in develping all f the frmulatin presented herein, the (, ) axes are interchangeable with the (, ) axes and aligns with the 3 axis. A mre typical crss sectin f a cmpsite taken frm a single ply is shwn in the phtgraph (Figure 3.3) belw. Jhn J. Engblm Page f 9

11 PDH Curse M37 Figure 3.3. ypical Fiber Distributin in Unidirectinally Reinfrced amina (Ply) An imprtant factr in determining the stiffness and strength prperties f cmpsite materials is the relative prprtin f matrix t reinfrcing materials. hese prprtins can be quantified as either weight fractins r vlume fractins. he vlume fiber fractin is defined as V v / v (3.) f f c Here v f is the vlume f fibers and vc is the assciated vlume f cmpsite. Similarly, the weight fiber fractin is given as W w / w (3.) f f c where w f is weight f fibers and w c is the assciated weight f cmpsite. he stress tensr cntains the terms,, 3, τ, τ 3, τ 3 and the strain tensr cntains the terms,, 3, γ, γ 3, γ 3, respectively. Here, γ ij are the engineering shear strains. Nte the relatinship γ ij ij (3.) where ij are the tensrial shear strains. If we assume that we have ne plane f material symmetry, 3, i.e., the, plane, then the cnstitutive equatins can be written in matrix frm as Jhn J. Engblm Page f 9

12 PDH Curse M τ 3 τ 3 τ γ 3 γ 3 66 γ (3.3) hus n nrmal (extensinal) stresses are prduced by the ut-f-plane ( γ 3,γ 3 ) shear stresses. Due t symmetry in the ij cnstitutive terms, we have reduced the number f independent material cnstants frm t 3. As nted the crdinates,, and 3 align with the material (natural) crdinates, we have aligning with the fiber directin, is transverse t the fiber directin and 3 is nrmal t the plane f the lamina. his crdinate alignment results in the (, 3 ) plane becming an additinal plane f symmetry. In these natural crdinates stresses, and 3 d nt prduce in-plane shear strainγ, and ut-f-plane shear stresses τ 3 and τ 3 becme decupled. In these natural crdinates, the cnstitutive equatins reduce t 3 τ 3 τ 3 τ γ 3 γ 3 66 γ (3.4) We nw have reduced the number f independent material cnstants t 9 fr the 3D case f an rthtrpic material, i.e., utilizing material (natural) crdinates fr the lamina. If we cnsider the special case f a D rthtrpic material and cntinue t use the material (natural) crdinates, the cnstitutive equatins simply reduce t τ 66 γ (3.5) In this case, there are nly 4 independent material cnstants. his particular material case can be described as a specially rthtrpic lamina. Inversin f the cnstitutive matrix [] gives the strains as a functin f stresses. In matrix frm we have Jhn J. Engblm Page f 9

13 PDH Curse M37 {} [ S]{ } (3.6) where [S] [] -, and [S] is dented the cmpliance matrix. In expanded matrix frm this becmes S S γ S S S 66 τ (3.7) Fr the specially rthtrpic lamina where the reference axes cincide with the material axes f symmetry, the engineering cnstants can be defined in mre familiar terms. he, axes becme the - axes and the 3 axis becmes the axis as shwn is the sketch belw (Figure 3.4). he - axes are in the plane f the lamina, where the lngitudinal axis is directed alng the fibers and the transverse axis is directed perpendicular t the fiber reinfrcement. he axis is nrmal t the plane f the lamina (ften referred t as the thrugh-the-thickness directin). Figure 3.4. ngitudinal, ransverse and hrugh-he-hickness Axes, Unidirectinally Reinfrced amina he engineering cnstants are defined as E Elastic mdulus in lngitudinal (alng the fibers) directin E Elastic mdulus transverse t the fiber directin ν Majr Pissn s rati (transverse strain prduced by lngitudinal stress) ν Minr Pissn s rati (lngitudinal strain prduced by transverse stress) he cmpliance terms in [S] can be defined in terms f these engineering cnstants as Jhn J. Engblm Page 3 f 9

14 PDH Curse M37 S E S E (3.8) S S ν E 66 G ν E hus in matrix frm we have E γ E γ γ E E G τ (3.9) Since [] [S] - the cnstitutive (stiffness) terms can be defined by inverting [ S ] ν E ν ν E ν ν E ν ν ν E ν ν (3.) 66 G As an example, these engineering cnstants fr Carbn/Epxy AS/H35 are given as E 38 GPa, E 8.96 GPa, G 7. GPa and ν.3 We knw that we have symmetry such that ij ji and S ij S ji. herefre, Jhn J. Engblm Page 4 f 9

15 PDH Curse M37 ν E ν (3.) E hus if the majr Pissn s rati ν is knwn, then the Minr Pissn s rati is determined frm E ν E ν (3.) and nly 4 independent material cnstants E, E, G, and ν are needed t specify the behavir f a specially rthtrpic lamina. 4. RESRICIONS ON EASIC CONSANS Mre experimental measurements are needed t characterize the behavir f an rthtrpic material relative t an istrpic material. Fr the varius materials we have 3D Orthtrpic, need 9 independent cnstants D Orthtrpic, need 4 independent cnstants Istrpic, need nly independent cnstants Fr an istrpic material, we have a relatinship between ung s mdulus, shear mdulus and Pissn s rati, i.e., G E / ( ν ). hus we need nly tw f the three material cnstants t determine the third. A unidirectinal fiber cmpsite can be cnsidered t be transversely istrpic. Cnsider the crdinate system where is nrmal t the (lamina) plane. he material cnstants are related as belw E E G G ν ν and E G ( ν ) herefre in this case we have 5 independent cnstants (,,ν, ν ). E E, G and Cnstraints fr istrpic materials are that E, G, and are all psitive, where is the Bulk mdulus. Als Jhn J. Engblm Page 5 f 9

16 PDH Curse M37 Remember that and ν.5 E G ; thus ν fr G > ( ν ) E ; thus 3( ν ) ν fr > Similar cnstraints exist fr rthtrpic materials and are defined as fllws S ii > and ii > Which is the same as E, E, E3, G, G3, G3 > r equivalently E, E, E, G, G, G > all essentially the same cnstraints. he fllwing cnstraints are als required ( ν ( ν ( ν ν ν ν ) > ) > ) > (4.) hese cnstraints are required because, e.g., ( ν ν ) >. Since we have the previusly shwn relatinship E ν E ν (3.) E We can cmbine (3.) with the first equatin in (4.) t give a cnstraint f the frm Jhn J. Engblm Page 6 f 9

17 PDH Curse M37 ν E < E / similarly we have the additinal cnstraints / / E E E E ν < ; ν < E ; ν < E ; ν E < ; ν E / / E < E / he preceding cnstraints can be used t great benefit t evaluate experimental data. Fr example, tensile testing bth in the lngitudinal () and transverse () directins gives E, ν frm lngitudinal lading and E, ν frm the transverse lading. A check n the validity f the data requires that the cnstraint equatins ν E < E / be satisfied. ν E < E / and 5. SRESS-SRAIN REAIONS FOR GENERA ORHOROPIC AMINA Cnsider laminated cmpsite structures that are cnstructed by stacking a number f unidirectinal lamina (plies) in a specified rientatin sequence. hus the principal material (natural) crdinates f each lamina can be riented at a different angle with respect t a cmmn reference crdinate system. he behavir f each lamina can be described by the previusly derived stress-strain relatins in terms f the material (natural) axes. Fr the purpse f analyzing laminated cmpsite structures, it is necessary t refer the stress-strain relatins t a cnvenient reference crdinate system. hus we need t derive the stiffness and cmpliance matrices fr an rthtrpic lamina in terms f arbitrary axes. A lamina referred t arbitrary axes is called a generally rthtrpic lamina. he principal material - axes f each rthtrpic lamina are riented at an angle θ with respect t a cmmn set f reference - axes, as sketched belw. Jhn J. Engblm Page 7 f 9

18 PDH Curse M37 Figure 5.. Orthtrpic amina with Oriented Fiber Reinfrcement he fllwing transfrmatin relatins can be derived frm equilibrium relating stresses and strains in - crdinates t - crdinates. and τ γ [ ] τ [ ] γ (5.) (5.) Here, the transfrmatin matrix [ ] is defined as c s sc s c sc sc ( ) sc c s (5.3) where c cs(θ ) and s sin(θ ). Inversin gives the relatin between stresses and strains in material - crdinates t thse in - crdinates. We have Jhn J. Engblm Page 8 f 9

19 PDH Curse M37 Jhn J. Engblm Page 9 f 9 [ ] τ τ (5.4) and [ ] γ γ (5.5) and the inverted transfrmatin matrix ] [ is defined as ( ) s c sc sc sc c s sc s c (5.6) We have the stress-strain relatinships fr generally rthtrpic laminas in natural (material) crdinates. It is useful t have these relatinships defined in the reference crdinates as well. In rder t derive the relatinship between strain and stress in crdinates, first substitute (5.) int (3.9) giving [ ][ ] S τ γ (5.7) We can intrduce a useful transfrmatin between tensrial and engineering shear strains as belw [ ] R γ γ (5.8) where [ ] R (5.9)

20 PDH Curse M37 Jhn J. Engblm Page f 9 Substituting (5.8) int (5.7) gives [ ] [ ][ ] S R τ γ (5.) Nw substituting (5.) int the left hand side f (5.) yields [ ][ ] [ ][ ] S R τ γ (5.) he transfrmatin matrix [R] can als be used t define the transfrmatin [ ] R γ γ (5.) where [ ] R (5.3) Substituting (5.) int the left hand side f (5.) gives [ ][ ][ ] [ ][ ] S R R τ γ (5.4) Simply rearranging the matrix relatinship abve gives

21 PDH Curse M37 γ [ R][ ] [ R] [ S][ ] τ (5.5) It is easily shwn that the transpse f [ ] can be defined as [ ] [ R][ ] [ R] (5.6) Substituting (5.6) int (5.5) yields the simplified strain-stress relatinship in reference - crdinates. γ [ S ] τ (5.7) where [ S ] [ ] [ S][ ] (5.8) Nte that the cmpliance matrix [ S ] is fully ppulated and is herein represented as S S S6 S S S S 6 (5.9) S6 S 6 S 66 [ ] Relating stress t strain in the reference crdinates fllws by inverting (5.7) τ [ S ] γ (5.) r simply τ [ ] γ (5.) where [ ] [ S ] [ ] [ S] [ ] (5.) Jhn J. Engblm Page f 9

22 PDH Curse M37 As with the cmpliance matrix in the reference - crdinate system, the stiffness matrix [ ] is fully ppulated and can be written as 6 6 (5.3) [ ] he stiffness terms in [ ] are related t the 4 independent terms in [ ] as given belw. 4 4 c s ( 66 ) s c 4 4 s c ( 66 ) s c 4 ( 466 ) s c ( c s 4 66 ( 66 ) s c 66 ( s c 4 ) 4 ) (5.4) ( c s cs ) ( 66 ) 3 3 ( ) cs ( ) Similarly, the cmpliance terms in [ ] written belw. c 3 s S are related t the 4 independent terms in [ ] S as S 4 4 Sc S s ( S S 66 ) s c 4 4 Ss S c ( S S66 ) s c S 4 S ( S S S ) c s S ( c s S (S S 4S S ) c s S ( c 66 s ) ) (5.5) S 6 3 S S ) S66 ) c s (S S S 66 ( cs 3 S 6 3 ( S S S ) cs (S S S ) c 66 As an example f calculating stresses cnsider the lamina shwn belw 66 3 s Jhn J. Engblm Page f 9

23 PDH Curse M37 Figure 5.. Stresses and Applied t Angled Ply, Fiber Orientatin θ 6 Assume that we knw the stress values in - crdinates and that the lamina is a typical E-glass epxy cmpsite material. Stresses have values and MPa(.9psi) 4MPa(5.8psi) he E-glass epxy prperties are given as V.45 (vlume fiber fractin) f 3 ρ.8g / cm (density) E 38.6GPa(5.6MPsi) E 8.7GPa(.MPsi) G 4.4GPa(.6MPsi) ν.6 Nte that the fibers are rientated at 6 t the crdinate axis. We determine the stresses in natural ( ) crdinates by applying equatin (5.) Jhn J. Engblm Page 3 f 9

24 PDH Curse M37 τ MPa he strains in natural crdinates are easily btained frm equatin (3.9) as given belw.59e 6.736E γ 6.736E.9E 35E E E 8.66E 6 9 μ Failure theries fr rthtrpic lamina are generally defined in terms f the natural (material) crdinates and therefre it is essential in designing laminated cmpsite structures t be able t apply the crdinate transfrmatins as just demnstrated. It can be shwn that strains in - crdinates are related t strains in natural crdinates thrugh the transfrmatin γ [ ] γ (5.6) hus in the present example, the strains in - crdinates are given as.5.75 γ μ 6. BIAIA SRENGH HEORIES FOR ORHOROPIC AMINA Fr failure criteria t have validity, they must be able t predict the strength f materials under multi-axial lading cnditins based n data btained frm a set f simplified lading tests. Failure criteria fr istrpic materials are written in terms f principal stresses in cmbinatin with ultimate tensile, cmpressive and shear strengths. hus applying failure theries in the design f istrpic materials requires that these three strength quantities be knwn. he situatin is cnsiderably mre cmplex in the case f rthtrpic materials. Fr these engineered materials, bth strength as well as stiffness (cnstitutive) prperties are directin dependent. Fr design purpses, the failure theries are generally based n five in-plane strength quantities defined in natural (material) crdinates. hese strength quantities are herein defined as Jhn J. Engblm Page 4 f 9

25 PDH Curse M37 ngitudinal tensile strength (in the directin f fiber reinfrcement) U ransverse tensile strength (nrmal t the directin f fiber reinfrcement) τ Shear strength in the plane f the lamina ngitudinal cmpressive strength U ransverse cmpressive strength One f the failure theries presented later includes the transverse (ut-f-plane) shear strength τ in the frmulatin even fr the -D biaxial stress state cnsidered here. U here is als the pssibility f utilizing an additinal strength quantity based n experiments invlving the applicatin f a biaxial state f stress. he lngitudinal and transverse stiffness and strength prperties can be btained thrugh uniaxial testing f unidirectinally reinfrced cmpsite specimens. hese tests invlve lading specimens alng natural (material) crdinates. Uniaxial tensin testing serves t determine the lngitudinal and transverse mduli E and E, tensile strength values and U, as well as Pissn s ratis ν andν. Uniaxial cmpressin tests are mre difficult t perfrm than uniaxial tensin tests because the test must be designed t prevent ut f plane buckling and als t prevent edge damage. Hwever, varius test methds d exist t vercme these difficulties. hus uniaxial cmpressin testing is used t btain the cmpressive strength values and U. In-plane shear stiffness G and shear strength τ values can be btained frm a number f different types f tests, including trsin tube [], rail shear [], Isipescu [3,4], Arcan [5], ff-axis specimen [6] and ± 45 specimen [7]. As nted in [7], the ± 45 specimen des nt require any specialized fixtures and is therefre used ften t determine the in-plane shear stressstrain respnse f cmpsite materials. he relevant test methd is ASM D358/D358M-94() Standard est Methd fr In-Plane Shear Respnse f Plymer Matrix Cmpsite Materials by ensile est f ± 45 aminate. his standard test methd is based n measuring the uniaxial stress-strain respnse f a ± 45 laminate which is symmetrically laminated abut the mid-plane. Obtaining shear stress/strain data using the ff-axis specimen requires that blique end tabs be used in rder t achieve a hmgeneus strain field ver the entire specimen [8-]. Since failure theries fr cmpsite materials invlve strengths in material - crdinates, design calculatins require transfrmatin f the stress field frm sme - crdinate system t - crdinates. Failure criteria used in the design f cmpsite materials are thus written in terms f stresses in material crdinates rather than in terms f principal stresses, as is the case fr istrpic materials. Jhn J. Engblm Page 5 f 9

26 PDH Curse M37 It may be bvius but is useful t pint ut that a uniaxial stress applied in any ff-axis directin, i.e., nt alng a material axis, prduces a multiaxial stress state in - crdinates. herefre, an apprpriate failure thery must be used even fr this simple lading cnditin. Failure theries fr rthtrpic materials can be represented as theretical failure envelpes in stress space. hese failure envelpes are similar t yield surface envelpes used t represent the terminatin f linear elastic behavir fr istrpic materials. A number f strength (failure) theries, widely used in the design f fiber reinfrced cmpsite structures, will nw be presented. hese appraches can be brken int separable theries, i.e., thse that can identify the mde f failure, and thse that are mre generalized in that they identify a failure limit but d nt separate ut r identify any particular failure mde. An estimatin f the use f varius failure criteria by peple wrking in the cmpsites design field has been reprted, see Paris []. his estimatin rated the relative utilizatin f the varius criteria as fllws: maximum strain 3% use; maximum stress 3% use; sai-hill 8% use; sai-wu 3% use; and all thers 9% use. he maximum strain and maximum stress failure theries are herein dented as separable failure theries, whereas the sai-hill and sai-wu failure theries are dented as generalized failure theries. w ther failure theries t be presented herein, which are included in all thers regarding their utilizatin by designers, are dented the Hashin failure thery and the Chang failure thery. Each f these failure theries are defined as separable failure theries. It is interesting t nte that in a review f research papers the majrity f researchers base their prpsals n variatins f Hashin s criteria []. 6. Separable Strength (Failure) heries 6.. Maximum Stress hery In this thery the ntin is that failure ccurs if any f the stresses in the natural (material) crdinates exceeds the crrespnding allwable stress. In rder t avid failure, the fllwing inequalities must be satisfied < < U (6.) τ < τ When the nrmal stresses are cmpressive, and U are replaced with the allwable cmpressive stresses as belw < < U (6.) Jhn J. Engblm Page 6 f 9

27 PDH Curse M37 Nte that in this failure criterin there is assumed t be n interactin between the axial and shear mdes f failure. his ver simplificatin can lead t an ver predictin f allwable strength. As an example f applying this failure thery, cnsider the E-glass epxy material f the previus example. he strength prperties are given as U U τ 6MPa(54.Psi) 6MPa(88.5Psi) 3MPa(4.5Psi) 8MPa(7.Psi) 7MPa(.45Psi) Cnsider an rthtrpic lamina subjected t a stress making an angle θ with the lngitudinal fiber directin as illustrated in the sketch belw. Figure 6.. Unidirectinally aded amina with Offset Angle θ he applied stress is transfrmed t material crdinates using equatin (5.), we have cs θ [ ] sin θ (6.3) τ sinθ csθ Jhn J. Engblm Page 7 f 9

28 PDH Curse M37 Cmbining (6.3) with the maximum stress criteria represented in (6.) and (6.) gives the fllwing inequalities, nrmalized by. < cs θ U < (6.4) sin θ < τ sinθ csθ When the applied stress is cmpressive, the first tw f these inequalities becme < cs U < sin θ θ (6.5) Fr any particular value f θ, the inequality giving the lwest value f strength is the apprpriate failure predictin. he ff-axis strength predictins using the maximum stress criteria are pltted belw fr values f θ ranging frm t 9. he strength results are pltted in terms f nrmalized stress /.. Nrmalized Stress ensin Cmpressin 5 9 Off-Axis Angle (Degrees) Figure 6.. Nrmalized Stress / Related t Off-Axis Angleθ, Maximum Stress Failure hery Jhn J. Engblm Page 8 f 9

29 PDH Curse M37 At small values f θ the lad is parallel r nearly parallel with the lngitudinal fiber directin. he difference in tensile and cmpressive strengths at these lw angles is attributable t different failure mdes in tensin and cmpressin fr this particular cmpsite material. Failure in tensin is characterized by fiber fracture while failure in cmpressin is characterized by fiber micr-buckling. his result wuld nt be the case fr all cmpsite materials and certainly wuld nt be expected fr istrpic materials. he difference in tensile and cmpressive strengths at large angles f θ is again attributable t differences in tensile and cmpressive failure mdes in the transverse () directin. Relatively lw tensile strength in the transverse directin f a lamina (ply) is typical as the matrix material fractures with multiple cracks frming parallel t the fiber reinfrcement. his effect is minimized in cmpsite structures by stacking plies at varying angles t achieve quasi-istrpic behavir. 6.. Maximum Strain hery his failure criterin states that failure ccurs when strains in any f the natural (material) axes exceeds the crrespnding allwable strain. hus the fllwing inequalities must be satisfied t avid failure < < U (6.6) γ < γ If the nrmal strains are cmpressive, then and U are replaced by the allwable cmpressive strains as belw < < U (6.7) Again cnsider an rthtrpic lamina subjected t a stress making an angle θ with the lngitudinal fiber directin (see Figure 6.). Substituting values fr the stresses in material crdinates int the cmpliance equatins (3.9) yields the fllwing. E γ E γ γ E E G cs θ sin θ x sinθ csθ (6.8) Jhn J. Engblm Page 9 f 9

30 PDH Curse M37 Carrying ut the matrix multiplicatin and cmbining with the maximum strain criteria gives the fllwing inequalities. < E (cs θ ν sin θ ) E U < (6.9) (sin θ ν cs θ ) < G γ sinθ csθ If we assume that the material behavir is linear elastic t failure, these inequalities can be simplified by substituting E E (6.) U U τ G γ hus, in this example, the maximum strain criteria given in (6.9) reduces t < (cs θ ν sin θ ) U < (6.) (sin θ ν cs θ ) < τ sinθ csθ When the applied stress is cmpressive, the first f these tw inequalities are mdified by replacing the tensile strength values with their crrespnding cmpressive strength values. he third inequality in (6.) remains unchanged as it invlves the limit n shear strain which is unaffected by whether r nt the lading is tensile r cmpressive. he maximum strain criteria fr cmpressive lads becmes Jhn J. Engblm Page 3 f 9

31 PDH Curse M37 < < U (cs (sin θ ν θ ν sin θ ) cs θ ) (6.) Cmparing the maximum strain criteria t the maximum stress criteria, we see that the criteria lk identical except fr the Pissn s rati terms. herefre the differences in the failure predictins f these tw theries are minimal. It shuld be nted, hwever, that if the cmpsite material des nt behave linearly elastic t failure then the predictins can be quite different. Cnsidering the same E-glass epxy lamina, again fr any particular value f θ, the inequality giving the lwest value f strength is the apprpriate failure predictin. he ff-axis strength predictins using the maximum strain criteria are pltted belw fr values f θ ranging frm t 9. he strength results are again pltted in terms f nrmalized stress /. Nrmalized Stress ensin Cmpressin 9 Off-Axis Angle (Degrees) Figure 6.3. Nrmalized Stress / Related t Off-Axis Angleθ, Maximum Strain Failure hery he results in this case are virtually identical t thse btained using the maximum stress criteria Hashin uadratic hery Jhn J. Engblm Page 3 f 9

32 PDH Curse M37 As a third example f a separable failure criteria, cnsider the quadratic strength thery as develped by Hashin []. In this criteria, there is cupling between extensinal and shear mdes f failure. It is nt uncmmn in applying Hashin s failure thery t replace the transverse (ut-fplane) shear strength τ U with the in-plane shear strength value τ. his assumptin mdifies Hashin s cmpressive matrix failure predictin. his is t sme extent due t the difficulty in experimentally quantifying the transverse shear strength. Als there is sme questin as t the lgic f including an ut-f-plane strength term in a tw dimensinal plane stress frmulatin. In any event, there is a certain cmpensatin f errrs in replacing τ U with τ in Hashin s -D frmulatin []. Hashin based his frmulatin n lgical reasning rather than micrmechanics. his criteria has been successfully applied t prgressive failure analysis f varying laminate ply lay-ups by using in-situ unidirectinal strengths [3]. Use f in-situ strengths prvides a methd t accunt fr the cnstraining interactins between plies. he gverning equatins are listed belw fr a biaxial state f stress. Fiber Mde (ensin) τ τ < (6.3) Fiber Mde (Cmpressin) < (6.4) (same as maximum stress criteria) Matrix Mde (ensin) U τ τ < (6.5) Matrix Mde (Cmpressin) τ τ U τ τ U U U < (6.6) Again cnsider an rthtrpic lamina subjected t a stress making an angle θ with the lngitudinal fiber directin (see Figure 6.). Assuming that τ U τ and using the same E-glass epxy prperties, the inequality giving the lwest value f strength is the apprpriate failure predictin. Jhn J. Engblm Page 3 f 9

33 PDH Curse M37 Substituting stresses in material ( ) crdinates frm (6.3) int (6.3) gives cs 4 θ sin θ cs θ τ (6.7) Rearranging yields the nrmalized stress fr the tensile fiber failure mde as (6.8) 4 cs θ sin θ cs θ τ Fr the cmpressive fiber failure mde, we have the equivalent f the maximum stress criteria. his cnstraint is written as cs θ (6.9) Substituting the stresses int (6.5) gives the criteria fr tensile matrix failure as U sin 4 θ sin θ cs θ τ (6.) Slving fr the nrmalized stress gives (6.) 4 sin θ sin θ cs θ τ U Finally, fr the cmpressive matrix failure mde in this example we have frm (6.6) 4τ sin 4 U θ sin θ cs θ sin 4 θ τ τ U (6.) As can be seen, (6.) is a quadratic equatin which can be slved fr. Again nte that the inequality giving the lwest value f strength is the apprpriate failure predictin. Results are pltted belw fr the E-glass epxy lamina. he Hashin quadratic criteria is cmpared t results previusly btained using the maximum stress criteria. It can be bserved that the failure predictins are in clse agreement fr applied cmpressive stresses, hwever the maximum stress thery ver predicts strength in this example when the applied stresses are tensile in nature. Jhn J. Engblm Page 33 f 9

34 PDH Curse M37 Nrmalzed Stress Max Stress- Max Stress-C Hashin-C Hashin Off-Axis Angle (Degrees) Figure 6.4. Nrmalized Stress / Related t Off-Axis Angleθ, Hashin uadratic vs. Maximum Stress Failure heries here is evidence that when a cmpsite is subjected t a cmbined, τ lading, it becmes strnger when is cmpressive. his implies that the in-plane shear stress τ at failure crrespnding t is appreciably greater than the shear stress τ at failure crrespnding t [4]. Sun et al. [5] prpsed an empirical mdificatin t the failure criteria prpsed by Hashin in 973 [6] fr matrix cmpressin failure t accunt fr the beneficial rle that cmpressive has n matrix shear strength. his mdificatin is written as: Matrix Mde (Cmpressin) τ U τ η < (6.3) In this expressin, η is an experimentally determined cnstant and can be thught f as an internal material frictin parameter. he denminatr in the shear stress term is effectively an in-plane shear strength term that increases with the transverse cmpressive stress. his mdificatin t Hashin s criteria fr cmpressive matrix failure is nt pursued further here due t the added cmplexity required t experimentally determine Jhn J. Engblm Page 34 f 9

35 PDH Curse M37 the frictin parameterη. Fr additinal insight int this particular mdificatin t Hashin s criteria and int ther alternative criteria requiring mre extensive experimentatin see [,3,4] Chang uadratic hery As a furth and final example f a separable failure criteria, cnsider the quadratic thery as develped by Chang et al. [7-8]. Actually the Chang criteria presented here evlves frm the references cited and is the versin used in the finite element based cmputer cde MSC Dytran, see []. his criteria is a mdificatin t Hashin s criteria and therefre cuples the extensinal and shear mdes f failure. he gverning equatins are listed belw fr the biaxial state f plane stress. Fiber Mde (ensin) < (6.4) Matrix Mde (ensin) U < (6.5) Matrix Mde (Cmpressin) τ τ U U < (6.6) In these expressins, the quantity takes the frm τ τ 3 αg τ 3 αg τ (6.7) Here α is an experimentally defined cefficient used t represent the nnlinear in-plane shear strain-stress behavir as represented belw. τ ατ (6.8) γ G 3 Jhn J. Engblm Page 35 f 9

36 PDH Curse M37 Observe that fr α these failure criteria reduce t Hashin s criteria except that the inplane shear strength τ replaces the transverse (ut-f-plane) shear strengthτ U. Furthermre, fr shear dminated failures where τ is the dminant stress and τ τ the Chang criteria again reduces t the Hashin criteria. As befre cnsider an rthtrpic lamina subjected t a stress making an angle θ with the lngitudinal fiber directin (see Figure 6.). Using the same E-glass epxy prperties, the inequality giving the lwest value f strength prvides the apprpriate failure predictin. Substituting stresses in material ( ) crdinates frm (6.3) int (6.4) gives cs 4 θ sin θ cs θ ( ) θ τ (6.9) Fr (θ ) we have the fllwing. 3 αg sin cs θ θ ( θ ) (6.3) 3 αg τ Rearranging (6.9) yields the nrmalized stress fr the tensile fiber failure mde as (6.3) 4 cs θ sin θ cs θ ( θ ) τ Substituting the stresses int (6.5) gives the criteria fr tensile matrix failure as U sin 4 θ sin θ cs θ ( ) θ τ (6.3) and slving fr the nrmalized stress gives (6.33) 4 sin θ sin θ cs θ ( θ ) U τ Finally, fr the cmpressive matrix failure mde in this example we have frm (6.6) Jhn J. Engblm Page 36 f 9

37 PDH Curse M37 4τ sin 4 U θ sin ( ) 4 θ θ τ U (6.34) Clearly (6.34) is a quadratic equatin which can be slved fr. Again nte that the inequality giving the lwest value f strength is the apprpriate failure predictin. Results are pltted belw fr the Chang quadratic criteria and are cmpared t the results previusly btained using the Hashin criteria. he cefficient α is based n a least squares fit t experimental data btained fr E-glass epxy [7]. Nte that cmpressive fiber failure is nt cnsidered by the Chang failure criteria. hus fr small values f θ (less than 6 in this example), the Chang criteria makes n valid predictin and the limiting failure curve fr cmpressive lading is simply cut ff fr small values f θ. In this particular example, the Chang and Hashin criteria are in clse agreement. Hwever, it shuld be nted that while all f the failure criteria under cnsideratin can be implemented in a material nnlinear analysis, nnlinear material behavir is explicit in the Chang criteria due t the representatin f shear behavir in (6.8). hus the results btained in this example with the Chang criteria are versimplified because the results are based simply n a linear analysis using classical laminatin thery.. Nrmalized Stress Hashin-C Hashin- Chang- Chang-C 9 Off-Axis Angle (Degrees) Figure 6.5. Nrmalized Stress / Related t Off-Axis Angleθ, Hashin uadratic vs. Chang uadratic Failure heries Jhn J. Engblm Page 37 f 9

38 PDH Curse M Generalized Strength (Failure) heries 6.. sai-hill hery A failure thery fr anistrpic materials was prpsed by Hill [9]. he thery as prpsed is actually a yield criteria but in the cntext f cmpsite materials the yield strengths are treated as limits n linear elastic behavir. herefre Hill s yield strengths are treated herein as failure strengths. Hill s yield criteria is an extensin f the well knwn and much applied vn Mises yield criteria fr istrpic materials. he vn Mises criteria is related t distrtinal strain energy and nt t dilatatin (change in vlume). In the case f rthtrpic materials distrtinal and dilatatinal effects can nt be separated, thus this thery as applied t cmpsite materials is nt a distrtinal energy thery. he failure strength parameters in Hill s thery were first related t the failure strengths f an rthtrpic lamina by sai []. hus this failure thery fr rthtrpic lamina is referred t as the sai-hill thery. It is als referred t as the maximum wrk thery. Experimental supprt fr this thery has been demnstrated by several authrs, e.g., []. Hill s criteria fr yielding f anistrpic materials has the frm ( G H ) τ ( F H ) Mτ Nτ ( F G) < H G F (6.35) he failure strength parameters can be related t the usual failure strengths by cnsidering the separate applicatin f simple stress states. Cnsider first that τ acts alne. Based n the criteria in (6.35) this gives (6.36) N τ If acts alne we have G H When acts alne, criteria (6.35) gives F H U and if acts alne Jhn J. Engblm Page 38 f 9

39 PDH Curse M37 F G U Cmbining the abve three equatins prvides definitin f three strength parameters. hese parameters are given as H U U G U U (6.37) F U U Fr the biaxial (plane) stress state f interest we can assume that the thrugh-thethickness f the lamina stresses are essentially zer. his gives τ τ (6.38) If we cnsider the crss sectin f a typical lamina (ply) as depicted in the sketch belw Figure 6.6. Crss Sectin f Unidirectinal amina With Fibers in Directin and simply cnsider the gemetrical symmetry, it is cncluded that (6.39) U U Substituting (6.38) and (6.39) int (6.37) gives Jhn J. Engblm Page 39 f 9

40 PDH Curse M37 H F G U (6.4) Rearranging the strength parameters in (6.4) yields and G H (6.4) F H (6.4) U Substituting the strength parameters int (6.35) gives the sai-hill failure thery fr the case f biaxial (plane) stress. Failure is initiated when the inequality belw is vilated. U τ τ < (6.43) When nrmal stresses are cmpressive, the tensile strengths are replaced with cmpressive strengths. It is interesting t see that the sai-hill thery reduces t the vn Mises thery fr istrpic materials by making the fllwing substitutins τ U (6.44) where and are the principal stresses fr the istrpic material and the yield strength. Fr an istrpic material, (6.43) then reduces t the vn Mises yield criteria as belw < (6.45) he sai-hill failure thery given in (6.43) prvides a single functin t predict strength. Again cnsider the same example f an E-glass epxy (angled ply) lamina with stress applied (see Figure 6.). Substituting the stresses in natural (material) crdinates int (6.43) in this example yields the fllwing fr the case f tensile lading Jhn J. Engblm Page 4 f 9

41 PDH Curse M37 cs (6.46) τ 4 4 θ cs θ sin θ sin θ < U A similar expressin is btained fr the case f cmpressive lading cs 4 θ τ cs θ sin θ U sin 4 θ < (6.47) Fr pltting purpses, these equatins can be written in the general frm < f,, τ,, ) (6.48) ( U U he ff-axis strength predictins using the sai-hill criteria are cmpared t the maximum stress criteria fr values f θ ranging frm t 9. he strength results are again pltted in terms f nrmalized stress /.. Nrmalized Stress sai-hill- sai-hill-c Max Stress- Max Stress-C Off-Axis Angle (Degrees) Figure 6.7. Nrmalized Stress / Related t Off-Axis Angleθ, sai-hill vs. Max. Stress Failure heries he sai-hill thery predicts lwer strengths than thse predicted by the maximum stress thery and has been shwn t be in better agreement with experimental data than thse results btained using either the maximum stress r maximum strain thery [9]. One reasn fr the better agreement with experiments is the fact that there is cnsiderable Jhn J. Engblm Page 4 f 9

42 PDH Curse M37 interactin between the failure strengths (, U, τ ) in the sai-hill criteria. his interactin des nt exist fr either the maximum stress r maximum strain criteria, i.e., in the latter tw theries, axial, transverse and shear failures are assumed t ccur independently. In this example f applying t an angle ply with θ ranging frm t 9, the sai- Hill and Hashin quadratic strength theries are in clse agreement when the applied stress state is tensile, as shwn in Figure 6.8 belw. his is primarily because these strength theries each exhibit cupling between axial and shear defrmatin under a tensile stress state. Fr a cmpressive stress state, the Hashin criteria is mre similar t the maximum stress criteria, particularly fr lw values fθ.. Nrmalized Stress sai-hill- sai-hill-c Hashin-C Hashin Off-Axis Angle (Degrees) Figure 6.8. Nrmalized Stress / Related t Off-Axis Angleθ, sai-hill vs. Hashin uadratic Failure heries 6.. sai-wu ensr hery A way t theretically imprve the crrelatin between thery and experiment fr strength theries is t increase the number f terms, particularly with respect t terms relating t the interactin between stresses in tw directins. sai and Wu [] accmplished this bjective in their tensr strength thery fr cmpsites. hey pstulated a failure surface in stress space f the frm F F ; i,j,6 (6.49) i i ij i j Jhn J. Engblm Page 4 f 9

43 PDH Curse M37 wherein F i and F ij are strength tensrs f the secnd and furth rank. he usual cntracted stress ntatin is used, i.e., 4 τ, 5 τ and 6 τ. Fr the case f an rthtrpic lamina under plane stress cnditins, (6.49) reduces t the frm F F F τ F F F τ F (6.5) 6 66 he linear strength cnstants serve t represent different strengths in tensin and cmpressin. uadratic strength cnstants prvide the representatin f an ellipsid in stress space. he F term is the basis fr representing the interactin between the nrmal stresses in material crdinates. he ability t represent the interactin between and prvides mre generality than achieved with the sai-hill thery. Of curse, mre experimental data is required in that sme tests are needed with the applicatin f either biaxial stresses r an ff-axis uniaxial stress. All f the strength cnstants in equatin (6.5), except fr the interactin term F, can be defined n the basis f simple uniaxial r pure shear testing. Nte that all f the strength quantities, including and U, are treated as psitive quantities in the fllwing equatins. First cnsider the case where the nly nnzer stress is. ading the uniaxial specimen t failure gives F F (tensin) F F (cmpressin) hen slving fr the strength cnstants yields F F (6.5) Similarly, applying the nly nnzer stress t a uniaxial test specimen until failure gives F F (tensin) U U F F (cmpressin) U U Jhn J. Engblm Page 43 f 9

44 PDH Curse M37 Slving fr the strength cnstants F F U U U U (6.5) Applying pure shear τ in material crdinates gives the fllwing F (because sign f shear nt imprtant in crdinates) 6 F τ 66 r F 66 (6.53) τ he remaining interactive term F can be determined based n the perfrmance f biaxial stress tests. Fr example, cnsider the biaxial stress state and ther stresses zer. Here, is the biaxial stress required t prduce failure in the specimen. Substituting int (6.5) gives ( F F ) ( F F F ) Slving this equatin fr the interactive term gives where and F C C ( C C ) (6.54) U U U U hus the interactive F term depends n the engineering strengths in the and directins as well as n the biaxial tensile failure strength. Nte that ff-axis uniaxial tests culd be used as an alternative t determining the interactive F term. Jhn J. Engblm Page 44 f 9