On the incremental plastic work and related aspects of invariance – Part I

Summary Taking for granted that the free energy function is invariant under a change of a finite strain measure and/or the reference configuration, Hill's transformation rules for selected fundamental constitutive quantities (such as tangent elastic modulus, plastic increments of total strain and work conjugate stress, the work of work-conjugate stress, the work expended in the plastic part of incremental strain etc.) are derived in a manner different from that of Hill. On this background distinguished by Hill [6] subtle aspects of invariance in mechanics of elastic plastic solids are discussed. It is shown that the plastic part of the increment of elastic strain energy (when taken with reverse sign) defines the true invariant incremental plastic work which in general is not equal to the work expended in the plastic part of the strain increment. It plays the role of a potential for the plastic part of the increment of work-conjugate stress. This fundamental fact has not found proper account in the literature. The analytical interrelations between two apparently different theoretical frameworks, Hill-Rice (fixed reference configuration) and Eckart-Mandel (mobile unloaded configuration) are discussed showing their equivalence. Since the transformation rules are complex in the general 3D case, the first part of the paper illustrates instructively the discussed aspects in a 1D situation (simple tension or simple extension).