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