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.
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