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

Linear Elastic Materialsmi@se.edu.cn

Linear Elastic Materials

Outline·Introduction（引言）·Linear elasticmodel（线弹性本构）·Matrixrepresentation（线性本构的矩阵表示）·Symmetryof stiffness/compliancetensor（刚度/柔度张量的对称性）·Linearelasticanisotropic models（线性各向异性模型）·Linearelasticorthotropic model（线性正交对称模型）·Linearelasticcubicmodel（线性立方对称模型）·Linearelastic isotropicmodel（线性各向同性本构）·Smallstretchesbutlargerotations（小应变大转动）2

Outline • Introduction（引言） • Linear elastic model（线弹性本构） • Matrix representation（线性本构的矩阵表示） • Symmetry of stiffness/compliance tensor（刚度/柔度张量 的对称性） • Linear elastic anisotropic models（线性各向异性模型） • Linear elastic orthotropic model（线性正交对称模型） • Linear elastic cubic model（线性立方对称模型） • Linear elastic isotropic model（线性各向同性本构） • Small stretches but large rotations（小应变大转动） 2

Introduction: Relations that characterize the mechanical behavior ofmaterials. Perhaps one of the most challenging fields in mechanics,due to the endless variety of materials and loadings The mechanical behavior of solids is normally defined byconstitutive stress-strain relationso= f(c,c,t,T,..)Linear elastic model (Hooke's law)Elastic-plastic modelVisco-elastic modelVisco-plastic model3

Introduction • Relations that characterize the mechanical behavior of materials • Perhaps one of the most challenging fields in mechanics, due to the endless variety of materials and loadings • The mechanical behavior of solids is normally defined by constitutive stress-strain relations 3 • Linear elastic model (Hooke’s law) • Elastic-plastic model • Visco-elastic model • Visco-plastic model σ  f t ε, , , , ε T 

Introduction: Neglect strain rate, time and loading history dependency: Set aside thermal, electrical, pore-pressure, and otherloadsInclude only mechanical loadsAssume linear stress-strain relationship Defined as materials that recover original configurationwhen mechanical loads are removedAgree well with experimental tests of metals0,=ESteelCast IronAluminumA

• Neglect strain rate, time and loading history dependency • Set aside thermal, electrical, pore-pressure, and other loads • Include only mechanical loads • Assume linear stress-strain relationship • Defined as materials that recover original configuration when mechanical loads are removed • Agree well with experimental tests of metals Introduction   x x  E 4

Linear Elastic Model. Hooke's law in 1D:C=E. In 3D, one might generalize this in tensor form as=C:& O, =Cjku;&=S: j=SijkOl Most generally, C has 81 independent components: Thanks to the (minor) symmetry propertiesO,=Oj =Cjh=G = =Cjk =jikl,Ai: The number of independent components is reduced to 36.5

• Hooke’s law in 1D: Linear Elastic Model • In 3D, one might generalize this in tensor form as : ; : σ     C ε     ij ijkl kl ij ijkl kl C S ε S σ • Most generally, C has 81 independent components. • Thanks to the (minor) symmetry properties ,     ij ji kl kl kl lk ij ij       C C C C ij ji kl lk • The number of independent components is reduced to 36. 5

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

Matrix RepresentationO, =CjkEh= Cj1 +Cj33633 +20i12G12+2CCi13Gj3 +2C+22822j23823(C)Ci133Ci112Ci113Cl123a1Ci12261C 222C 23C212C213C 22238202C s33Cs312Cs313C s3230383C212Cr213Cv2232812T122613Cr313C1323Symm.132823C 2323T23 21 elastic constants9

11 11 22 22 33 33 12 12 13 13 23 23 1 1111 1122 1133 1112 1113 1123 2 2222 2233 2212 2213 2223 3 3333 3312 3313 3323 12 1212 1213 1223 13 23 2 2 2 Symm. 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                                           1 2 3 12 1313 1323 13 2323 23 2 2 2 C C C                   Matrix Representation • 21 elastic constants 9

Linear Elastic Anisotropic Models: Differences in material properties along differentdirectionsMaterials like wood, crystalline minerals, fiber-reinforced composites have such behaviorTypical WoodBody-CenteredHexagonalFiberReinforcedStructureCrystalCompositeCubic Crystal. Note particular material symmetries indicated by arrows10

Body-Centered Cubic Crystal Fiber Reinforced Composite Hexagonal Crystal Typical Wood Structure • Note particular material symmetries indicated by arrows. Linear Elastic Anisotropic Models • Differences in material properties along different directions. • Materials like wood, crystalline minerals, fiber￾reinforced composites have such behavior. 10