On divergence-free stress fields and zero-energy stress functions in elasticity

• Novel identities for divergence free second-order tensors are proposed.
• A first-order stress function vector and a second-order stress function are introduced for generating self-equilibrated stress fields in 3D elasticity.
• The zero-energy modes of the proposed stress functions are derived.
• The structure of the zero-energy first-order stress functions is the same as that of the zero-energy displacements.

Applying two identities for divergence-free non-symmetric and symmetric second-order tensors, novel type of first- and second-order stress functions are proposed for three-dimensional elasticity problems. It is shown that self-equilibrated but non-symmetric 3D stress fields can be generated by one first-order stress function vector, whereas a self-equilibrated and symmetric 3D stress field can be generated by one Airy-type second-order stress function. Assuming linearly elastic materials, the zero-energy modes of the stress functions introduced are derived and investigated. It is pointed out that the structure of the zero-energy modes of the proposed first-order stress function vector is the same as that of the rigid-body displacements in the linear theory of elasticity.