Institute of Fundamental Technological Research

Superparamagnetic iron oxide nanoparticles/hydroxyapatite (SPION/HAp), Magnetic nanoparticles, Apatite ore, Adsorption, Congo red, Wastewater treatment

hydroxyapatite, myo-inositol, 5-fluorouracil, anticancer treatment, SKOV-3, calendula extract, Langmuir trough

Nanostructural hydroxyapatite, Apatite ore, Wastewater treatment, Heavy metal ions removal, Novel eco-friendly adsorbent, Adsorption

Presuming that the incremental free energy is invariant under a change of the Lagrangean finite strain measure and/or the reference configuration Hill’s transformation rules for the basic quantities occurring in mechanics of elastic–plastic solids are recasted in general 3D situation. On this background the invariant incremental plastic work is defined. The basic connections between Hill–Rice theoretical framework and Eckart–Mandel approach, involving the mobile stress-free configuration, are discussed both in generalized coordinates and in the tensorial notation. To this end the selected fundamentals of solid mechanics including the work-conjugacy are recalled. The structure of the updated Lagrangean plastic increment of the total strain is exhibited accounting for the deformation and stress effects due to possible damage and pressure sensitivity of a solid. Special simple approximate relations are derived for the situations when non-dilatational elastic strains are small. The merits of using the logarithmic elastic strain as a state variable are also discussed.

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).