东南大学：《弹性力学》课程教学课件（英文讲稿）05 Constitutive Relations

Constitutive Relations

Constitutive Relations

Outline.Constitutive Laws.Strain Energy.Linear Constitutive Relations? Anisotropy and Stiffness Tensor· Isotropic Hooke's Law: Physical Meaning of Elastic Moduli? Simple Engineering Tests: Relationship among Elastic Constants· Hooke's Law in Curvilinear CoordinatesThermoelasticConstitutiveRelations. Typical Values of Elastic Constants. Inhomogeneity2

Outline • Constitutive Laws • Strain Energy • Linear Constitutive Relations • Anisotropy and Stiffness Tensor • Isotropic Hooke’s Law • Physical Meaning of Elastic Moduli • Simple Engineering Tests • Relationship among Elastic Constants • Hooke’s Law in Curvilinear Coordinates • Thermoelastic Constitutive Relations • Typical Values of Elastic Constants • Inhomogeneity 2

Constitutive Laws: Relations that characterize the mechanical properties ofmaterials: Perhaps one of the most challenging fields in mechanics,due to the endless variety of materials and loadingsThe mechanical behavior of solids is normally defined byconstitutive stress-strain relations.Generallyo = f(ε,ε,t,T,.3

Constitutive Laws • Relations that characterize the mechanical properties of materials • Perhaps one of the most challenging fields in mechanics, due to the endless variety of materials and loadings σ f ε ε   , , , ,  t T   • The mechanical behavior of solids is normally defined by constitutive stress-strain relations • Generally 3

Perfect (Linear) Elastic Solids Neglect strain rate, time and loading history dependencySet aside thermal, electric, pore-pressure, and other loads Include only mechanical loadsAssume linear stress-strain relationshipDefined as materials that recover original configurationwhen mechanical loads are removedAgree well with experimental tests=E8SteelCastlronAluminum

• Neglect strain rate, time and loading history dependency • Set aside thermal, electric, 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 Perfect (Linear) Elastic Solids • Agree well with experimental tests   x x  E 4

Strain Energy (Stress-Deformation Work) In physical terms, stress represents the variation of strainenergy with respect to strain changea"U(0)aU (0)U (s,)=U (0)+5kl211aokCGmmU(c,)=U(0)+buCu +Cklmmuemn +O(c3aU(6) =bu +(+Ckl)u +0(2)d5

                2 3 0 0 0 ij kl kl mn kl kl mn U U U U O                  • In physical terms, stress represents the variation of strain energy with respect to strain change Strain Energy (Stress-Deformation Work)             3 2 0 ij kl kl klmn kl mn ij ij ij ijkl klij kl ij U U b c O U b c c O                      5

Stiffness of Perfect Elastic SolidsIsothermal and perfect elasticity results irCiki+Cklj)ok~Cuikkl Symmetry propertyO,=Oj=Cjk iik-Skl=Sik =jki = Cjiki +Cklj =CijklkliiThere remain 21 independent elastic components6

Stiffness of Perfect Elastic Solids • Symmetry property    ij ijkl klij kl ijkl kl    c c C  • Isothermal and perfect elasticity results in   ij ji kl kl    C C ij ji kl lk ij ij ji kl kl kl lk ij ij ijkl ijkl klij kl klij C C C C C c c C C               • There remain 21 independent elastic components 6

Generalized Hooke's Law in Matrix Form0,= CujuCul=Cj1161 + Cj222 + Cyj33633 + 2Cj12612 + 2Cyj1313 + 2Cuj23623(Ci1C133C112C113Cμ122Ci123af81C 2223C 233C 2212C 222C2213622C 333C 3323C3312C331363832812C/212C213C1223T12C1313Ci13232813Symm.1132623C2323T2323: 21 elastic constants7

11 11 22 22 33 33 12 12 13 13 23 23 1 1111 1122 1133 1112 1113 1123 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                        1           Generalized Hooke’s Law in Matrix Form 2 2222 2233 2212 2213 2223 3 3333 3312 3313 3323 12 1212 1213 1223 13 23 Symm. C C C C C C C C C C C C                         2 3 12 1313 1323 13 2323 23 2 2 2 C C C              • 21 elastic constants 7

Anisotropy: Differences in material properties along differentdirections Materials like wood, crystalline minerals, fiber-reinforced composites have such behaviorTypical WoodBody-CenteredHexagonalFiberReinforcedStructureCrystalCubic CrystalComposite: Note particular material symmetries indicated by arrows8

Anisotropy • Differences in material properties along different directions. • Materials like wood, crystalline minerals, fiber￾reinforced composites have such behavior. Body-Centered Cubic Crystal Fiber Reinforced Composite Hexagonal Crystal Typical Wood Structure • Note particular material symmetries indicated by arrows. 8

Monotropic Materials· Symmetric about one planeEamineCl1,Ci13,C223,C2223,C3,C33,C,Ci2Cl13=QimQ,Q1.OspCmmop(m= l;n =l;o= l;p=3)= C(l13 = QuQiQuQ,Ci113 = -Ci13 = Cium3=0= 0|C2223= 01113=0：3323=02312= 0C1312=03313CilllCi122Cμ133Ci112a61C222C2233C22126262001C3312C 33336303001Qi2812T12Ci21200-12813T13SymmC1313322823T232323.13 different elastic constants9

Monotropic Materials • Symmetric about one plane 1113 1 1 1 3   1113 11 11 11 33 1113 1113 1113 1123 2213 2223 1113 1123 2213 2223 3313 3323 1213 122 1 3 11 3 1 Examine ; 1; 1; 3 0 0 0 0 0 0 : , , , , , , 0 , C Q Q Q Q C m n o p m n o p mnop C Q Q Q Q C C C C C C C C C C C C C C C C C C                                C  0 1 0 0 0 1 0 0 0 1 Qij                 3313 3323 2 13 2 C C C  0 0   0 111 1312 1111 1122 1133 1112 1 2 3 12 13 23 3 0 C C  C C C C                            C1123 C C C 2222 2233 2212 C2213 C2223 C C 3333 3312 C3 13 3 C3323 C1212 C12 31 C1223 1 2 3 12 13 1313 1323 23 2323 2 2 Symm. 2 C C C                               • 13 different elastic constants 9

Orthotropic Materials: Symmetric about two orthogonal planesExamine:Cl12,C2212,C3312,C/323Cl12=Q1mQinQ1.Q2pCmmop(m=l;n=l;o=1,p=2)= C(l12 = QQuQQ22Ci112 = -Ci112 = Ci11=0C22122313.=0=01112CullCi122C1133g81C222C22330262C3333Q3632812T12C121226130T130SymmC131328230T23-1Qu23230-1·9 different elastic constants10

Orthotropic Materials • Symmetric about two orthogonal planes 1112 1 1 1 2   1112 11 11 11 22 1112 1112 111 1112 2212 3312 2313 1112 2212 33 2 Examine: , , , 12 1323 1; 1; 1; 2 0 0 0 0 m n o p mnop C C C C C Q Q Q Q C m n o p C Q Q Q Q C C C C C C C C C C                           C C C 1 0 0 0 1 0 0 1 Qij                  1111 1122 11 1 2 3 12 13 23 C C C                            33 C1112 C1113 C1123 C C 2222 2233 C2 12 2 C2213 C2223 C3333 C33 21 C3313 C3323 C1212 C12 31 C1223 Symm. C1313 C1323 1 2 3 12 13 23 2323 2 2 2 C                                 10 • 9 different elastic constants