Configurational balance laws for incompatibility in stress space

0 Department of Mechanical Engineering, University of California , Berkeley, CA 94720 , USA 1 Department of Mechanical and Aerospace Engineering, University of California , San Diego, La Jolla, CA 92093 , USA Conservation laws have been recently obtained by requiring that a positive definite functional of the stress gradient (the Euler-Lagrange equations of which are the Beltrami-Michell compatibility conditions) be invariant under certain transformations. Here these laws are extended to include body forces, thermal stresses and Kroner's incompatibility tensor as source terms in the configurational balance laws, which allows for the incompatibility in the volume to be measured from surface data. An example is presented. 1. Introduction In a recent paper, Li et al. (2005) obtained a class of conservation laws as a consequence of symmetries of a positive definite functional of the stress gradient in a variational principle of linear and isotropic elasticity. Such a variational principle appeared in a stress-based formulation of three-dimensional elasticity proposed by Pobedrja (1980). The equations of compatibility in terms of stresses (BeltramiMichell) are obtained as the EulerLagrange equations of this variational principle. Here, we consider the case when the right-hand side of compatibility equation (1.2) is non-zero due to the incompatibility tensor (Kroner 1958, 1981) and/or the gradient of body forces, and we modify the conservation laws to include them as source terms. These balance laws are identities, in the sense that they are satisfied identically if the equations of force equilibrium and compatibility are assumed to hold pointwise inside the domain under consideration. They prove to be of considerable importance if written in an integral form since the information regarding the volume content of the incompatibility and/or the body force gradient can be obtained from the value of the field variables (and their derivatives) on the surface. Possible applications would include, for example, the incompatibility arising from a continuous distribution of dislocations or thermal stress fields, which is a topic of current interest (e.g. Bako & Groma 2005; Fujimo et al. 2005). In the rest of this section, we briefly discuss the stress formulation of the traction boundary value problem of three-dimensional linear isotropic elasticity in the absence of body forces and incompatibility. The well-known BeltramiMichell boundary value problem with vanishing body forces is given in terms of the symmetric Cauchy stress sij(xk) by Gurtin (1972) (in Cartesian coordinates with indices ranging from 1 to 3), sij;kk C bskk;ij Z 0; c xk 2vU; where the Einsteins summation rule is used for repeated indices. The constant b is given in terms of the Poissons ratio n as bZ1/(1Cn). The traction force on the boundary vU of the domain U3E3 with an outward normal ni is denoted by pi. Pobedrja (1980) presented an alternative formulation which renders the traction problem to be a well-posed boundary value problem of threedimensional elasticity (see also Pobedrja & Holmatov 1982); the differential operator in the boundary value problem is self-adjoint and satisfies the Fredholm property (Kucher et al. 2004). For our purposes, we will only deal with the case of static, isotropic and linear elasticity and we make a special choice of the free parameters of Pobedrja so that the Poissons ratio appears as the only independent material parameter (Li et al. 2005). For a compatible strain field with vanishing body forces, Pobedrjas boundary value problem can be stated as sij;kk C bskk;ij C bsmn;mndij C asik;kj C sjk;ki Z 0; c xk 2vU; c xk 2vU; where the constant a is given by aZ(1Kn)/n(1Cn). Note that equilibrium is imposed only on the boundary. The term sij,j can be calculated on vU by obtaining its value at points in U which approaches vU. Pobedrja (1980) has shown that it is sufficient to satisfy the equilibrium condition over the boundary to ensure that equilibrium is satisfied in the domain. Kucher et al. (2004) have obtained the necessary and sufficient conditions for this formulation to be equivalent to the stress solution of the classical Naviers formulation in linear elasticity almost everywhere except on the boundary. Pobedrja & Radzhabov (1989) obtained a Greens function for Pobedrjas boundary value problem which has been extended to the case of anisotropy in Pobedrja (1994). 2. The BeltramiMichell equations from a variational principle and associated conservation laws The appendix contains a brief outline of Noethers theorem generalized to be applied to a positive definite tensor-valued functional. As a consequence of Noethers theorem ( Noether 1918), linearly independent combinations of the Lagrange expressions (defined as the left-hand side of EulerLagrange equations) become divergences (equation (A 13)). The resulting equations (A 13) are here applied to Pobedrjas functional (as defined in equation (2.3)) and thereupon conservation laws are obtained as a consequence of various symmetries of this functional by Noethers theorem. We will restrict attention to the functional which has an associated Lagrangian dependent only on the stress gradient, so that for a Lagrangian of the form LU(sij,k) with xi and sij being the independent and the dependent variables, respectively, equation (A 13) reduces to Here, the Lagrange expressions (denoted by Jij) involve only the second term in equation (A 10) In the above, 4ij and xijk represent the infinitesimal generators corresponding to xi and sij, respectively (equations (A 5) and (A 6) in appendix A). The variational principle for Pobedrjas stress formalism involves the following integral (Pobedrja 1980), LUsij;kdUK where cijZ(vLU/vsij,k)nk. The coordinate xi denotes the position in the domain. The Lagrangian LU is given as 1 a 2 sij;ksij;k C bskk;isij;j C 2 sik;ksij;j C sjk;ksji;i : The resulting EulerLagrange equations obtained as a consequence of the variation of the functional (2.3) with respect to stress (equations (A 10)(A 12)) coincide with equations (1.4)(1.6) and are indeed the BeltramiMichell compatibility equations of three-dimensional elasticity. The Lagrange expressions Jij are the left-hand side of equation (1.4) and vanish in the absence of an incompatibility and body force gradients. If we substitute JijZ0 in equation (2.1), we obtain a class of conservation laws from the corresponding symmetries of the problem (eqns (4.4)(4.14) of Li et al. (2005)), In the presence of an incompatibility and/or gradient of body forces, the BeltramiMichell equations have a non-zero right-hand side and therefore the Lagrange expressions Jij do not vanish. In 3, we evaluate them by using classical relations of incompatibility and equilibrium. A non-zero Jij will subsequently contribute as a source term in the right-hand side of equation (2.1). 3. The EulerLagrange equations in the presence of incompatibility and body forces The Lagrange expressions (...truncated)