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A Convex Modular Modelling (CMM) framework for developing thermodynamically consistent constitutive models
Abstract
This paper presents a theoretical framework termed the convex modular modelling (CMM) framework, which provides a convenient and expedient approach for constructing thermodynamically consistent constitutive models. This paper demonstrates how the CMM framework can be used to build increasingly complex constitutive models by mixing and matching re-usable components from a library of convex base functions in a systematic manner. It also describes the use of the modified LogSumExp (MLSE) function as a general and smooth approximation to the pointwise maximum function for any yield function (e.g. the Mohr-Coulomb/Tresca yield function). The MLSE function is then used to develop several new yield functions such as a convex and smooth approximation of the Matsuoka-Nakai yield function, a generalised polygonal yield function and a ‘Reuleaux triangle’-shaped yield function. As CMM is simple to use, it potentially offers a more accessible path for constitutive modellers to take advantage of the hyperplasticity framework to develop robust constitutive models.
... Thus, Eq. (18) is guaranteed to be globally convex as long as 0 ≤ w ≤ 1. Further details are provided in Suryasentana and Houlsby (2022). Through an analysis of the variation of the skirt and base failure envelopes with respect to ...
Winkler modelling offers a flexible and computationally efficient framework for estimating suction caisson capacity. However, there is a limited understanding of the local soil resistance acting on caissons at capacity under combined six degrees-of-freedom (6DoF) loading, which is essential for accurately estimating caisson failure envelopes. Furthermore, existing simplified design models for caissons cannot assess capacity under non-planar lateral and moment loading, which is common in offshore wind applications. To address these limitations, this paper presents a comprehensive three-dimensional (3D) finite element analysis (FEA) study, which investigates the local soil resistance acting on the caisson at capacity in undrained clay under combined 6DoF loading. The paper introduces the concept of 'soil reaction failure envelopes' to characterise the interactions between soil reactions at capacity. Closed-form formulations are derived to approximate these soil reaction failure envelopes. An elastoplastic Winkler model is then developed, incorporating linear elastic perfectly plastic soil reactions based on these formulations. The results demonstrate that the Winkler model can provide efficient and reasonably accurate estimations of caisson capacity under combined 6DoF loading, even for irregular soil profiles that pose much uncertainty and challenges to existing macro-element models.
... In computational framework it has resulted in numerous studies and also many approaximate expressions for multi-surface yield surfaces with corners have been presented, see e.g. [1,2,12,13,14]. Similar problem arises in continuum damage mechanics, especially when it is formulated in terms of the maximum principal stress, like in the classic Rankine failure criterion, which is linear in the principal stress space but non-linear in the global stress space [3]. Approximation of such yield/failure surfaces with a continuous function written in terms of the stress invariants offers clear advantages since it does not involve solution of the principal stresses and coordinate transformations. ...
In this paper, a recursion formula is given for the integer power of a second-order tensor in 3D Euclidean space. It can be used in constitutive modelling for approximating failure or yield surfaces with corners, and it is demonstrated for the case of Rankine failure criterion. Removing corners provides clear advantages in computational plasticity. We discuss the consequences of the approximation errors for failure analyses of brittle and quasi-brittlematerials.
Failure envelope formulations are typically employed to assess the ultimate capacity of foundations under combined loading and for incorporation in macro-element models. However, the complex interaction between each load component, especially for six degree of freedom (6DoF) loading, means that determining satisfactory formulations is often a complex process. Previous researchers have identified this difficulty as an obstacle to the adoption of the failure envelope approach in foundation engineering applications. To address this issue, the paper describes a systematic procedure for deriving globally convex failure envelope formulations; the procedure is applied to a circular surface foundation, bearing on undrained clay, in 6DoF load space. The formulations are shown to closely represent the failure load combinations determined from finite element analyses for a variety of loading conditions, including non-planar horizontal-moment loading. An example macro-element model based on the proposed formulation is described; the macro-element model provides a close representation of the foundation behaviour determined from a separate finite element analysis. A key aspect of the paper is that it demonstrates an automated process to determine well-behaved failure envelope formulations. The automated nature of the process has considerable advantages over the manual procedures that have previously been employed to determine failure envelope formulations.
The failure envelope approach is widely used to assess the ultimate capacity of shallow foundations for combined loading, and to develop foundation macro-element models. Failure envelopes are typically determined by fitting appropriate functions to a set of discrete failure load data, determined either experimentally or numerically. However, current procedures to formulate failure envelopes tend to be ad hoc, and the resulting failure envelopes may not have the desirable features of being convex and well-behaved for the entire domain of interest. This paper describes a new systematic framework to determine failure envelopes – based on the use of sum of squares convex polynomials – that are guaranteed to be convex and well-behaved. The framework is demonstrated by applying it to three data sets for failure load combinations (vertical load, horizontal load and moment) for shallow foundations on clay. An example foundation macro-element model based on the proposed framework is also described.
This paper explains why the critical state of sand is non-unique when expressed in terms of stress and void ratio only. For this purpose, a thermodynamically consistent, micromechanically inspired constitutive modelling framework with competing grain crushing and dilation is developed. While grain crushing is described through the theory of breakage mechanics, dilation is modelled in a novel way by acknowledging its negative contribution to the overall positive rate of dissipation. The competition between dilation and grain crushing underpinned by this framework yields a unique critical state in a space of stress, void ratio and breakage, in agreement with recent experiments. As an example, a simple constitutive model with only five mechanical parameters is proposed, which not only predicts the critical state but also quantitatively connects the full constitutive behaviour to key index properties related to grading- and breakage-dependent minimum and maximum densities.
We describe a non-linear anisotropic hyperelastic model appropriate for geomaterials, deriving the full stress-strain response from strain energy or complementary energy functions. Specific forms of the functions are chosen so that the stiffness and compliance matrices have the appropriate minor symmetries. The model employs two material parameters to describe basic volumetric and shear response, one to express nonlinearity of stiffness as a function of mean stress, and two more (together with the directions of the principal axes of anisotropy) to express the degree of anisotropy. The model is modular, so that non-linearity and anisotropy can be included separately or in combination. For specific parameter settings it reduces to simpler cases such as linear isotropic elasticity. Because the model employs hyperelasticity, thermodynamic acceptability is ensured and all appropriate cross-coupling terms are included between the shear and volumetric behaviour.
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, i.e. 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 rigorously formulated by integrating some of the main relations which emerge from experimental results. The model satisfies all theoretical requirements and also possesses features which 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 new yield function recently introduced by Lagioia and Panteghini (2016), herein referred to as the Generalised Classical (GC) yield function, combines a series of criteria commonly used in geotechnical analysis into a single equation, including those of Tresca, Mohr-Coulomb and Matsuoka-Nakai. This makes for efficient implementation of multiple criteria into finite element software, and in this paper two key improvements are made to further enhance the usefulness of the GC yield function. The first is the development of a new expression for the shape parameter γ, corresponding to the so-called ‘Inner Mohr-Coulomb’ option, which ensures that a true inner rounding of the hexagonal Mohr-Coulomb deviatoric section is always obtained. The second is the introduction of a hyperbolic rounding to eliminate a discontinuity which can occur at the tip in the meridional section of the GC yield surface. The resulting yield surface is at least C2 continuous everywhere, provided a rounded criterion is selected, and can thus be used in consistent tangent finite element formulations. The results of finite element analyses carried out for two benchmark problems (a thick cylinder and a rigid strip footing) demonstrate the benefits of the rounding techniques in the new yield surface. Comparisons are made with the original yield surface and also the Hyperbolic Rounded Mohr-Coulomb (HRMC) yield surface originally developed by Abbo and Sloan (1995).
This paper proposes a new constitutive model for geotechnical materials that consists two basic constitutive functions, the free energy function and the dissipation rate function, within the framework of hyperplastic theory. This free energy function is capable of describing the pressure-dependent elastic behavior of soils. The new constructed dissipation rate function accounts for the frictional mechanism of energy dissipation. Based on this dissipation rate function, the non-associated flow rule can be obtained. Furthermore, the convexity of the yield surface that is derived from the dissipation rate function is proved. Predictions of the behavior of a soil sample using this new constitutive model agree well with triaxial test data under drained and undrained conditions.
Yield and plastic potential surfaces are often affected by problems related to con-vexity. One such problem is encountered when the yield surface that bounds the elastic domain is itself convex; however, convexity is lost when the surface expands to pass through stress points outside the current elastic domain. In this paper, a technique is proposed, which effectively corrects this problem by providing linear homothetic expansion with respect to the centre of the yield surface. A very compact implicit integration scheme is also presented, which is of general applicability for isotropic constitutive models, provided that their yield and plastic potential functions are based on a separate mathematical definition of the meridional and deviatoric sections and that stress invariants are adopted as mechanical quantities. The elastic predictor-plastic corrector algorithm is based on the solution of a system of 2 equations in 2 unknowns only. This further reduces to a single equation and unknown in the case of yield and plastic potential surfaces with a linear meridional section. The effectiveness of the proposed convexification technique and the efficiency and stability of the integration scheme are investigated by running numerical analyses of a notoriously demanding boundary value problem.
We present a theoretical model to describe the response of a one dimensional mechanical system under cyclic loading. Specifically, the model addresses the non-linear response on loading, hysteretic behaviour on unloading and reloading, and the phenomenon of ratcheting under very many cycles. The methods developed are formulated within the hyperplasticity framework. The model can be expressed in the form of general incremental relationships, can therefore be applied without modification directly to any loading history, and can be readily implemented within a time-stepping numerical code. A rigorous procedure is described to accelerate the ratcheting process, so that the effects of very large numbers of cycles can be analysed through a reduced number of cycles. A generalisation from unidirectional to multidirectional loading is described, together with a tensorial form for application to material modelling. The original motivation was for the application to design of piles under lateral loading, and an example of this application is provided. However, the model is equally applicable to many other problems involving unidirectional or bi-directional cyclic loading in which the system exhibits a similar character of hysteretic behaviour, with ratcheting under large numbers of cycles.