On effective stress rate in Non-Linear Continuum Mechanics

Continuum Mech. Thermodyn. (2026) 38:34 https://doi.org/10.1007/s00161-026-01449-5 O R I G I NA L A RT I C L E Marzia Sara Vaccaro · Daniele Ussorio · Raffaele Barretta On effective stress rate in Non-Linear Continuum Mechanics Received: 19 December 2025 / Accepted: 2 January 2026 / Published online: 1 April 2026 © The Author(s) 2026 Abstract Structures undergoing large displacements and deformations are investigated in a differential geometric framework by a 4-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-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-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-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. Keywords Rate elasticity · Nonlinear continuum mechanics · Stress rate · Covariant hypo-elasticity 1 Introduction Over last centuries Non-Linear Continuum Mechanics has been one of the most debated topics in the scientific community, with still open challenging issues to be tackled. Among several approaches to address nonlinear structural problems [1–3], two main methodologies can be identified. As discussed in [4], the analysis of strain, introduced in the XIX century by Cauchy and Green, laid the foundation for the finite elasticity theory. In the context of finite deformations and purely elastic behaviours, logarithmic strain measure, also known as true strain [5], leads to results in accordance with experimental observances for a wide range of materials. M. S. Vaccaro (B) · D. Ussorio · R. Barretta Department of Structures for Engineering and Architecture, University of Naples Federico II, Via Claudio 21, 80125 Naples, Italy E-mail: D. Ussorio E-mail: R. Barretta E-mail: 34 Page 2 of 10 M.S. Vaccaro et al. Nevertheless, as also pointed out in [6], such a theory attempts to relate a state variable, id est a variable based at an event on the trajectory of the material body, to an entity pertaining to a pair of events: respectively, stress and strain. Other issues related to the finite elasticity theory arise when non-elastic effects come into play. Indeed, as discussed in [7], ultra-elastic behaviors [8,9] need investigation by means of refined models. In light of these considerations, a theory that could extend the basic ideas of linear elasticity to structures undergoing large displacements and deformations sounded attractive, paving the way to rate theories in NonLinear Continuum Mechanics. As stated in [10], one-dimensional viscosity was investigated in terms of rate in [11] for hygrosteric materials, a class of materials including both solids and fluids. A three-dimensional extension of the latter relations was given in [12] as a generalization of the classic theory of viscous fluids. Further investigations were carried out in [13,14] in the field of nonlinear solid mechanics providing the preliminary aspects of hypo-elasticity. The concept of hypo-elasticity was explicitly introduced for the first time in [7] as a constitutive law relating the rate of stress to the rate of deformation, with the intention, as recalled in [15], of finding a new concept of elastic behavior, mutually exclusive with the theory of finite deformations. Indeed, as stated in [16], the variables involved in the law of hypo-elasticity are defined in the current configuration of the body. In the work provided in [17], two common rates of stress are recalled: the Zaremba- Jaumann rate and the Truesdell rate. The same rates were also investigated in [18] in a more rigorous geometrical approach set forth in [19]. A further acknowledged corotational rate of stress was introduced in [20], known as GreenNaghdi stress rate. It is possible to show (see [5]) how these three mentioned rates can lead to completely different results when integrated in time, yielding ambiguity to the discussion. Further investigation of the topic was also carried out in [21] and [22]. Besides the several definitions of stress rates existing in literature, in [15] and [23] the non-conservativeness of the hypo-elastic model proposed in [7] is highlighted, making it not suitable to describe elastic materials and leaving room to a wide spread of finite elasticity models. In recent decades, issues and controversies arising from the application of the mentioned nonlinear theories have been discussed (see e.g. [24]). In order to rigorously and consistently address Non-Linear Continuum Mechanics problems, new insights were provided in [25–30]. Notably, an innovative rate elasticity theory based on a geometric covariance paradigm was proposed in [31], providing definitive answers to controversial questions. The present work aims at addressing nonlinear structural problems by providing a theoretically consistent framework and a computationally convenient methodology based on the original notion of effective stress rate. It is shown that such a definition naturally arises from the variational scheme of rate equilibrium. Constitutive laws are formulated as instantaneous incremental responses to a finite set of tensor state variables and to their time convective rates along the motion. Notably, the integrable and conservative covariant hypo-elasticity model contributed in [31] is exploited. Effectiveness of the proposed methodology is tested by inspecting geometrically nonlinear problems of elastic trusses. A computationally effective approach based on a straightforward incremental solution procedure is developed and put into operation. The plan is the following. In Section 2 the Rate Virtual Power Principle (RVPP) is derived for a threedimensional c (...truncated)