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2.1.5 Stress-Strain Relation for Anisotropic Materials

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2.1.5 Stress-Strain Relation for Anisotropic Materials ,/... + -:\ 2.1.5Stress-strain Relation for AnisotropicMaterials The generalizedHook's law relatingthe stressesto strainscan be writtenin the form U2, 13,30-32]: o,=C,,€, (i,i =1,2,.........,6) Where: o, - Stress('ompttnents C,, - Elasticity matrix e,-Straincomponents The material properties (elasticity matrix, Cr) has thirty-six constants,stress-strain relations in the materialcoordinates are: [30] O1 LI CT6 a, l O2 03_ a, -t ....(2-6) T23 1/ )1 T32 1/1l 'Vt) Tt2 C16 c66 I The relations in equation (3-6) are referred to as characterizing anisotropicmaterials since there are no planes of symmetry for the materialproperties. An alternativename for suchanisotropic material is a triclinic material. If there is one plane of material property symmetry,the stress- strainrelations reduce to: -'+1-/ ) o1 c| ct2 ct3 0 0 crc €l o2 c12 c22 c23 0 0 c26 a: ct3 C23 ci3 0 0 c36 o-\ o3 ....(2-7) 723 000cqqc450 y)1 T3l 000c4sc5s0 Y3l T1a ct6 c26 c36 0 0 c66 1/t) Where the plane of symmetryis z:0. Such a material is termed monoclinicwhich containstfirfen independentelastic constants. If there are two brthogonalplanes of material property symmetry for a material,symmetry will existrelative to a third mutuallyorthogonal plane. The stress-strainrelations in coordinatesaligned with principal material directionswhich are parallel to the intersectionsof the three orthogonalplanes of materialsymmetry, are'. Ol ctl ct2ctl000 tl O2 Ct2 c22c23000 a./ o3 ct3 c23c33000 * o-\ ....(2-8) T23 0 00c4400 y)1 yll 731 0 000css0 712 0 0000cr'e 1/1) 'lt"' -' , :t i' ,/ Suchmaterial is calledorthotropic material.,There is no interaction betweennormal StreSSeS ot r 02, 03) andanisotropic materials. There are only nine independentconstants in the elasticitymatrix, lr2l. If at every point of a material,there is one plane in which the mechanicalproperties are equal in all directions,then the material is namedtransversely isotro pic, 12,l2l ti),t For I -2 planeof isotropy,the stress-strainrelationshave only five -.t I independentconstants which are: 61 ctl ct2 c13 0 0 0 el o2 ct2 c22 c23 0 0 0 e2 ct3 c23 c33 0 0 0 e3 o3 ....(2-e) T23 0 00 C44 0 0 .,,/,/ \ 0 00 0 L)) 0 T3l 13r C\t - Ct2 0 00 0 0 Tt2 r12 If there are infinite numbers of planes of material property symmetry,then the foregoing relations simplify to the isotropic material casewith only two independentconstants in the elasticitymatrix. Stress- strainrelations become: c| c12 c)3 0 0 0 Cl Ol ct2 c)2 c2l 0 0 0 a.z o2 ct3 c23 0 0 0 - c\t c12 o-) O3 00 0 0 0 ....(2-10) 'l/ 2 )1 L 2J c|-cn 0 0 0 Y31 731 z cr - ct2 a/t) Tt2 0 0 The elasticity matrix Ciifor an orthotropicmaterial in terms of engineeringconstants is obtainedas follows: (7-vr.v.r)*E,, 'll - F vr, ! vpr/",t)*E, - L [ F '?, (v,r*vtzvzz)*Er, l-r r f 1l- v,.,ur,)* L'rz I . rt F (7-v,rvr1)*Er, 13n (v2r+v1u' ,)+ E-,, z) r Cu: Gzs,Css: Gt3,Coa: Gtz F: I-V12V2t - Vt3V3t - ISZIZS - l121t2jl j1 - 121V1jl j2 LaminatedComposite Material Laminated compositeis made by the consolidationof prepreg sheet,with individual plies having fiber orientations,here orthotropic symmetry with nine independentelastic constants,has orthotropic symmetry Analogous to orthorhombicsymmetry: there are three mutually perpendicularaxes of symmetry,and the elastic constantsalong these threeaxes and different.For the cambricsystem the elasticmatrix is: - - - - F: I-V 12Vzt Y tSVSt lS:VzS | 1212jVj1 l2fl1jV j2 2.6The Stress-strainRelation for An Isotropic . Material of Orthorhombic Symmetry(in plasticrange) i ' 'nl; i i r-evy-Misesequations for an isotropicplastic material arel2Sl: d",, = o i1d), drr, :---:--dr.r, -dr., -drr.--:- -dr=., -d"r;-=cll" -.,, .,.,.. (2-11) O.rx Oy 6:: O ,, 6;x O :.v By analogwith Levy-Misesequations .Hill 125,331hasproposed the following relationsfor the tensor componentsof plastic strainsfor anisotropicmaterial related to the principleaxis of anisotropy , ,"t.,. - - .l d o, = dtrlH (or' du') * G(o" o=')l dr, = dblr@r,-o..)-r H(a',-o,")] ......(2-r2) ,1.-= il,lc(o,, - o,,)-r G(o,,- t,' )] d*, = d)"Lor. d,, = d)"Mo^ doy = cl)"No', It is to be noted that in this casethe principal axesof the plastic strain incrementonly coincide with the axes of anisotropywhen the principal axesof stresscoincide with the latter.It is alsoto be notedthat the proportionalityfactor d,t is dimensionallydifferent from that in the (2- 1I ) Levy-Misesequations for an isotropicplastic material. 2-7 Yield and Failure Criteria for Anisotropic Material The source of anisotropy in amaterial can be any of the following12,25,34): a- A single crystalcan have differentproperties in different directions due to its inherentcrystal symmetry. b- A cold-rolledsheet, tube, or wire of a metal or alloy can show a very high degree of preferred orientation of grains, polymers are also frequently processedby drawing, extrusion, or injection moulding techniques.Such techniquesimport a high degreeof anisotropyto the polymer. c- In a lined fiber-reinforcedcomposite, the properties Yary very markedlywith the functionof the angle0 of the fiber direction. A very simpleyield criterionfor anisotropicmaterial is the critical resolvedshear stress Law of Schmid.This law statesthat yield occurs whenthe resolvedshear stress in the slip planereaches a criticalvalue for a tensilestress (o) making anglescr and B with the slip directionand the normalto the slip planerespectively .This gives the criticalresolved shear stressrc : Tc: OCOSCICOSB Althoughthis law is extensivelyused in metalplasticity it is of restrictedapplication in polymers.For polymersa generalizationof Vcln Misesyield criterionby Hills hasproved more useful. Hills yield criterion is appliedto anisotropicmaterials which possessthree mutually orthogonalplanes of symmetryat everypoint i.e.the materialpossesses at testorthorhombic symmetry. It will alsoapply in a simplifiedform to solidspossessing transverse isotropy. .
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