Modification of the Normality Rule

In the theory of elasticity an elastic potential (strain-energy function W) is assumed, from which the constitutive equations can be derived by using the relation σ_ij = ∂W/ε_ij, where σ_ij and ε_ij are the components of appropriately defined stress and strain tensors, respectively. In the isotropic special case, when the elastic constitutive equation can be represented as an isotropic tensor function 1$${{\sigma }_{ij}}={{\phi }_{0}}{{\delta }_{ij}}+{{\phi }_{1}}{{\varepsilon }_{ij}}+{{\phi }_{2}}\varepsilon _{ij}^{\left( 2 \right)},$$ the elastic potential is a scalar-valued function only of the strain tensor and can be represented in the form W =S₁, S₂, S₃), where S₁, S₂, S₃ are the basic invariants of the strain tensor (finite or infinitesimal strain tensor). In (1) it has been shown in detail that the scalar coefficients in (1) can be expressed through the elastic potential: 2a,b,c$${{\phi }_{0}}\equiv \partial W/\partial {{S}_{1}},{{\phi }_{1}}\equiv 2\partial W/\partial {{S}_{2}},{{\phi }_{2}}\equiv 3\partial W/\partial {{S}_{3}}.$$ Eliminating the elastic potential, one can find the following sufficient and necessary conditions 3$$2\partial {{\phi }_{0}}/\partial {{\phi }_{2}}=\partial {{\phi }_{1}}/\partial {{S}_{1}},3\partial {{\phi }_{1}}/\partial {{S}_{3}}=2\partial {{\phi }_{2}}/\partial {{S}_{2}},3\partial {{\phi }_{_{0}}}/\partial {{S}_{3}}=\partial {{\phi }_{2}}/\partial {{S}_{1}},$$ i.e., the elastic potential is “compatible” with the tensor function theory (1), if the conditions (3) have been fulfiled (1).