Stress and strain mapping tensors and general work-conjugacy in large strain continuum mechanics

• A systematic procedure to develop large strain constitutive equations in arbitrary strain and stress measures.
• Mapping tensors are developed to transform any arbitrary work-conjugate pair of stress and strain measures into any other pair.
• A procedure is valid regardless of the type of constitutive equation and material symmetries.
• Mappings for constitutive tensors also available.
• Possible applications in large strain hyperelasticity, elastoplasticity and viscoelasticity.

In this paper we show that mapping tensors may be constructed to transform any arbitrary strain measure in any other strain measure. We present the mapping tensors for many usual strain measures in the Seth–Hill family and also for general, user-defined ones. These mapping tensors may also be used to transform their work-conjugate stress measures. These transformations are merely geometric transformations obtained from the deformation gradient and, hence, are valid regardless of any constitutive equation employed for the solid. Then, advantage of this fact may be taken in order to simplify the form of constitutive equations and their numerical implementation and thereafter, perform the proper geometric mappings to convert the results –stresses, strains and constitutive tangents– to usually employed measures and to user-selectable ones for input and output. We herein provide the necessary transformations. Examples are the transformation of small strains formulations and algorithms to large deformations using logarithmic strains.