东南大学：《固体力学基础》课程教学课件（英文讲稿）04 Linear Elastic Materials

Matrix RepresentationO, =Cjkeu= Cy111 +C,12812 +201313 +2082 +(3633 +2(ij2323213[CiC(122Cl133Ci113Cl123Ci11201S110C 23CC211QD622Q 22C2212C2222C2213C2223CC 3Cs312C 313C323C333C3311C332233C v22C123C(212Ci213C(223Ci2112612T12CC(33Ci312C1313CCi3112613Ti3C1323C1322C2311C 232C 233CC23132823C 2323T23C2312: 36 elastic constants6

11 11 22 22 33 33 12 12 13 13 23 23 11 1111 1122 1133 1112 1113 1123 22 2211 2222 2233 2212 2213 2223 33 3311 3322 3333 3312 3313 3323 12 1 13 23 2 2 2 ij ijkl kl ij ij ij ij ij ij C C C C C C C C C C C C C C C C C C C C C C C C C C                                           11 22 33 211 1222 1233 1212 1213 1223 12 1311 1322 1333 1312 1313 1323 13 2311 2322 2333 2312 2313 2323 23 2 2 2 C C C C C C C C C C C C C C C C C                   Matrix Representation • 36 elastic constants 6

Major Symmetry of Stiffness/Compliance Tensor. Assume an increment of uu→u+SuF.u. The work increment done by FSW = FSu = αAS(l)=Vod = 8U? Strain energy density:aU.aSSU。 = SU/V = SW/V =oSc;U。=U.()a8: Generalize to 3DSW =F.udv +T.SudS= JJ, Fou,dV +T,Su,dsm FSu,dV +n,o,du,ds7

• Assume an increment of u Major Symmetry of Stiffness/Compliance Tensor u u u   • The work increment done by F        W F u A l V U       • Strain energy density: t t V t i i S j ji i S i i V i i V S T u dS n u W dV dS F u dV F u dV dS                       F u T u • Generalize to 3D     0 0 0 0 0 0 ; , , U U U V W V U U U                    7

Major Symmetry of Stiffness/Compliance Tensor: Applying the divergence theorem on the surface integral:0oiSu,+OjioxjaasuSW = JdvF,Su,dV=lFSu. +uaxxagTFSu, +O,dvSc,+So,axOnFSodvo.So.dV=SU5Ox? Major symmetry property=U。=[o,08, =JCuksud8, =CSU=0,06jaU.=Cjki = Cklj0808l8

• Applying the divergence theorem on the surface integral: Major Symmetry of Stiffness/Compliance Tensor     ji i i i ji i i i i ji V V j j j ji i i ij V j ji i j ij ij u W F u u dV F u u dV x x x F u dV x F x                                                                    i ij ij ij ij V V       u dV dV U                      0 0 2 0 1 2 ij ij ij ij ijkl kl ij ijkl ij kl ijkl ij i kl k j l j i kl U U C C U C C C                          • Major symmetry property 8