On conservation and balance laws in stress space

We study conservation and balance laws of linear three‐dimensional elasticity in terms of stress function tensors of first and second order. Such conservation and balance laws are sometimes called dual ones. The dual Eshelby stress tensor, angular momentum tensor and scaling flux of elasticity in stress space are derived within the framework of Noether's theorem on variational principles. Certain conservation and balance (or broken conservation) laws of translational, rotational, and dilatational symmetries including inhomogeneities, elastic anisotropy, dislocations, and incompatibilities are found. The conserved and non‐conserved dual J, L, and M integrals are derived and discussed. Also we give the conservation and balance laws corresponding to the gauge symmetries and the addition of solutions. From the addition of solutions we derive reciprocity theorems for the stress functions of first and second order. Explicit formulae for the configurational forces, moments, and work terms in terms of stress function tensors are formulated. The authors study conservation and balance laws of linear three‐dimensional elasticity in terms of stress function tensors of first and second order. Such conservation and balance laws are sometimes called dual ones. The dual Eshelby stress tensor, angular momentum tensor, and scaling flux of elasticity in stress space are derived within the framework of Noether's theorem on variational principles. Certain conservation and balance (or broken conservation) laws of translational, rotational, and dilatational symmetries including inhomogeneities, elastic anisotropy, dislocations, and incompatibilities are found. The conserved and non‐conserved dual J, L, and M integrals are derived and discussed. Also they give the conservation and balance laws corresponding to the gauge symmetries and the addition of solutions. From the addition of solutions they derive reciprocity theorems for the stress functions of first and second order. Explicit formulae for the configurational forces, moments, and work terms in terms of stress function tensors are formulated.