On effective stress rate in Non-Linear Continuum Mechanics

Abstract Structures undergoing large displacements and deformations are investigated in a differential geometric framework by a $$\,4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> -D Euclid spacetime approach. The developed variational scheme of rate equilibrium leads to the original notion of effective stress rate, which proves useful for addressing applicative problems in Non-Linear Continuum Mechanics. Notably, the rate virtual power principle is exploited according to the presented paradigm and the coordinate-free expression of the effective stress rate for $$\,3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> -D Cauchy continua is contributed. Constitutive relations are formulated as instantaneous incremental responses to a finite set of tensor state variables and to their time convective rates along the motion, namely the (natural) stress state per unit mass, its convective rate and the elastic stretching. Geometrically nonlinear structural problems are addressed by resorting to an integrable and conservative covariant hypo-elasticity model. Specifically, the simplest linear hypo-elastic constitutive law is considered. Particularization to the $$\,1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> -D case of elastic trusses is provided and two variants of the mentioned constitutive relation are examined. A straightforward incremental solution procedure is implemented to solve the nonlinear structural problem of a representative case-study and comparisons with standard finite elasticity strategies are carried out. It is shown that outcomes of conventional methodologies can be recovered for $$\,1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> -D purely elastic structures by implementing the proposed computational strategy. This denouement paves the way for a theoretically and computationally promising formulation of nonlinear elasto-visco-plastic problems, circumventing long-known and debated issues that affect approaches to finite deformations.