The difficult challenge of modelling the non‐linear elastic behaviour of soils within a theoretically sound framework

Abstract

Summary Hypo‐elastic relations are often adopted to simulate the recoverable non‐linear behaviour of soils within elasto‐plastic constitutive models. In reality, they are unable to reproduce the elastic, ie, recoverable, response of materials; hence, they introduce severe inconsistencies in models based on the decomposition of the total strain tensor into its recoverable and permanent parts. Hyper‐elasticity should then be used. However, existing models developed within this framework do not satisfy a number of fundamental theoretical requirements. A new hyper‐elastic model is proposed, which is rigourously formulated by integrating some of the main relations which emerge from experimental results. The model satisfies all theoretical requirements and also possesses features that are fundamental for its numerical integration. The model can be considered as the correct hyper‐elastic version of the classical hypo‐elastic constitutive relation adopted in models based on the Critical State framework, such as the Modified Cam‐Clay, with a constant Poisson's ratio.

A Convex Modular Modelling (CMM) framework for developing thermodynamically consistent constitutive models

“…The free energy function proposed by Houlsby et al (2005) may be used as its convexity has been confirmed by Lagioia and Panteghini (2019), who investigates the convexity of several existing hyperelastic free energy functions for soils. Lagioia and Panteghini (2019)…”

A micropolar isotropic plasticity formulation for non‐associated flow rule and softening featuring multiple classical yield criteria

“…von Mises and Drucker‐Prager) and strain energy potentials (e.g. Houlsby, 28 Lagioia and Panteghini 29 ), often dropping one or two of the three invariants, resulting in incomplete formulations. For example the third invariant is dropped in the Drucker–Prager criterion, thus loosing the dependency on the relative magnitude of the intermediate principal stress.…”

A micropolar isotropic plasticity formulation for non-associated flow rule and softening featuring multiple classical yield criteria Part I -- Theory

“…Let σ and μ indicate the stress and couple-stress tensors conjugated in the meaning of Hill [6] to the strain γ and wryness χ tensors introduced in Eqs. ( 8) and (9). The rate of the internal work of deformation over the volume Ω is Ẇint =…”

“…von Mises and Drucker-Prager) and strain energy potentials (e.g. Houlsby [7], Lagioia and Panteghini [9]), often dropping one or two of the three invariants, resulting in incomplete formulations. For example the third invariant is dropped in the Drucker-Prager criterion, thus loosing the dependency on the relative magnitude of the intermediate principal stress σ II with respect to σ I and σ III , where σ I ≥ σ II ≥ σ III .…”