16.21 Techniques of Structural Analysis and sig Spring 2003 Unit #5-Constitutive Equations Constitutive Equations For elastic materials 0j=0i(e)=DU If the relation is linear. Generalized Hookes Law In this expression: Cijkl fourth-order tensor of material properties or Elastic moduli(How many material constants? ) Making use of the symmetry of the stress tensor. Proof by(generalizable) example C 21=012→C2kkl=C12kEk� � 16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations Constitutive Equations For elastic materials: ∂U� σij = σij (�) = ρ (1) ∂�ij If the relation is linear: σij = Cijkl�kl , Generalized Hooke’s Law (2) In this expression: Cijkl fourth-order tensor of material properties or Elastic moduli (How many material constants?). Making use of the symmetry of the stress tensor: σij = σji ⇒ Cjikl = Cijkl (3) Proof by (generalizable) example: σ21 = C21kl�kl, σ12 = C12kl�kl σ21 = σ12 ⇒ C21kl�kl = C12kl�kl C21kl − C12kl �kl = 0 ⇒ C21kl = C12kl 1