Article

An approach for providing quasi-convexity to yield functions and a generalized implicit integration scheme for isotropic constitutive models based on 2 unknowns

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Abstract

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.

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... More recently a different approach has been presented by Panteghini and Lagioia,6 which is based on the use of invariants as the main variables during the iterations. This considerably improves the efficiency of the backward Euler scheme since a single equation in a single unknown needs to be solved to integrate the constitutive relationship when yield and plastic potential surfaces are linear in the meridional section. ...
... The de Souza Neto et al. 8 method can be used to deal with singularities in the yield and plastic potential surfaces, which results in an extremely stable and fast integration algorithm. In fact, as reported by Panteghini and Lagioia,6,9 numerical analyses of a shallow footing problem, with no lateral surcharge and no effective cohesion in the failure criterion can be carried out without any difficulty, requiring very few iterations for each load increment, provided that an associated flow is adopted. It should be noted that the analysis of that boundary value problem is usually considered not feasible and researches apply a small conhesion and/or lateral load to avoid crushing. ...
... This further reduces to a single equation in a single unknown when the meridional section of the yield function is linear, as assumed in this paper, thus achieving the same results as for the Cauchy medium. 6 To achieve quadratic convergence in the structural Newton, nine consistent tangent operators are required. Moreover, as in Panteghini and Lagioia, 6 the proposed integration scheme allows for an exact treatment of the cusp of yield and plastic potential functions. ...
Article
A Finite Element (FE) procedure based on a fully implicit backward Euler pre-dictor/corrector scheme for the Cosserat continuum is here presented. The integration algorithm is suitable for yield and plastic potential surfaces with general shape in the deviatoric plane. The key element of the integration scheme is the spectral decomposition of the stress tensor, which is achieved, despite the lack of symmetry, because of the mathematical structure of the yield function and the set of invariants chosen as independent variables. It is also shown that the choice of invariants enables considerable mathematical simplifications, which result in the reduction of the system of equations and unknowns of the elasto-plastic problem from 19 to 1, and to rigorously handle the discontinuity at the apex of the surfaces. The algorithm has been implemented in a proprietary FE programme, and used for the constitutive model recently proposed by the same authors in this journal for the Cosserat continuum, which allows to set various classical failure criteria as yield and plastic potential surfaces. Numerical analyses have been conducted to simulate a biaxial compression test and a shallow strip footing resting on a Tresca, Mohr-Coulomb, Matsuoka-Nakai and Lade-Duncan soil. The benefits of the Cosserat continuum over the Cauchy/Maxwell medium are discussed considering mesh refinement, non-associated flow and softening behaviour.
... However, Panteghini and Lagioia 20,21 have shown that the hypothesis of isotropy can be further exploited, resulting in an integration scheme, which is based on the solution of a single equation in a single unknown (with linear meridional sections). This not only brings advantages in terms of a considerable reduction of the run-time of the numerical analyses but also improves the stability of the algorithm. ...
... An elasto-plastic isotropic constitutive model is here formulated for the Cosserat continuum, which incorporates the features required to extend the integration approach of Panteghini and Lagioia. 21 The model features associated/nonassociated plasticity, hardening/softening and the possibility of selecting yield surfaces with different deviatoric sections. This requires an extension of definition of the equivalent von Mises stress and of the Lode's angle to the Cosserat continuum. ...
... In fact, in general, the eigenvalues of a non-symmetric tensor˜may be imaginary, hence impeding its spectral decomposition (and its computation as a function of invariants). With the definition of the Lode's angle proposed in this section, all the features of the integration proposed by Panteghini and Lagioia 20,21 for the Cauchy continuum can be extended to the Cosserat medium. respectively, where the( ⋅) is used to distinguish the terms of the two functions. ...
Article
The Cosserat continuum is very effective in regularizing the ill-posed governing equations of the Cauchy/Maxwell continuum. An elasto-plastic constitutive model for the linear formulation of the Cosserat continuum is here presented, which features non-associated flow, hardening/softening behaviour and multiple yield and plastic potential surfaces, whilst linear hyper-elasticity is adopted to reproduce the recoverable response. For the definition of the yield and plastic potential functions, the equivalent von Mises stress is formulated using energy considerations and the theory of representations, the latter being also used to retrieve an expression for the Lode's angle. The recent Generalized classical criterion is then used to define the yield and plastic potential functions so that Lode's angle-dependent deviatoric sections can be used.
... More recently a different approach has been presented by Panteghini and Lagioia [10] [11] which is based on the use of invariants as the main variables during the iterations. This considerably improves the efficiency of the backward Euler scheme since a single equation in a single unknown needs to be solved to integrate the constitutive relationship. ...
... The De Souza Neto et al. [3] method can be used to deal with singularities in the yield and plastic potential surfaces, which results in an extremely stable integration algorithm. In fact, as reported by Panteghini and Lagioia [10] [11] numerical analyses of a shallow footing problem, with no lateral surcharge and no effective cohesion in the failure criterion can be carried out without any difficulty, provided that an associated flow is adopted. The analysis of that boundary value problem is usually considered not feasible. ...
... In this article the Panteghini and Lagioia [11] scheme is extended to the Cosserat continuum. As shown in what follows the formulation is laborious, as it requires the evaluation of an elevated number of derivatives and of fourth order tensors. ...
Preprint
A Finite Element procedure based on a full implicit backward Euler predictor/corrector scheme for the Cosserat continuum is here presented. Since this is based on invariants of the stress and couple stress tensors and on the spectral decomposition of the former, considerable benefits are achieved. The integration requires the solution of a single equation in a single unknown, which is a considerable improvement as compared to the system of seven or four equations required by other approaches available in the literature for the Cauchy medium. The scheme also allows for a very efficient treatment of the singularity which affects the apex of most of the existing yield and plastic potential surfaces. Moreover, no complications arise when some of the principal stresses coincide. The algorithm has been implemented in a proprietary Finite Element program, and used for the constitutive model proposed in part I of this paper. Numerical analyses have been conducted to simulate a biaxial compression test and a shallow strip footing resting on a Tresca, Mohr-Coulomb, Matsuoka-Nakai and Lade-Duncan soil. The benefits of the Cosserat continuum over the Cauchy/Maxwell medium are discussed considering mesh refinement, non-associated flow and softening behaviour.
... with the initial yield surface size controlled by p c0 [67]. B p represents the slope of the ε p ν − lnp ′ curve [65] and can be related to the conventional MCC parameters by (detailed in the Appendix): ...
... Linear elasticity was considered in alignment with the work by Lagioia and Nova [65]. The MCC model developed by Panteghini and Lagioia [67,68], which was adopted in the study, has advantages in convergence and integration efficiency by using fewer unknowns and improved yield functions with full convexity and homothety. For the elliptical yield surface used in the present study, the new expression is given by: ...
... More details about the constitutive model can be found in Refs. [67,69]. ...
Article
Subsea pipelines buried in the seabed may undergo large lateral displacement under environmental, operational, and accidental loads at different interaction rates and hence different drainage conditions. The undrained shear strength is commonly used in practice to assess the pipe-soil interaction assuming a sufficiently high displacement rate. This approach neglects consolidation effects and the rate-dependent response of the soil and may significantly underestimates the lateral resistance for a pipeline moving slowly relative to the ground. In this study, a coupled large deformation finite element (LDFE) framework is developed via a remeshing and interpolation technique with small strain (RITSS). A Modified Cam-Clay (MCC) model with efficient numerical integration is used. The proposed coupled LDFE framework is verified against selected physical model tests. Effects of the interaction rate and hence drainage condition on the p-y curve, excess pore pressure generation and dissipation, and failure mechanisms are discussed. An empirical relationship between the ultimate resistance and the normalized velocity of the pipe (denoting the drainage condition) is proposed, which may be applied for the integrity and safety analysis of buried pipes in landslide or fault-crossing regions.
... Tamagnini et al. (2002) and Borja et al. (2003) showed that for isotropic materials the number of unknowns can be reduced to four only. However, as shown by Panteghini & Lagioia (2018), the adoption of an uncoupled formulation based on two distinct functions for the meridional and the deviatoric sections, makes it possible to work with two unknowns only, which further reduce to one in the case of a linear meridional section. ...
... In this paper full convexity will be used to indicate that f is a quasi-convex function, so that any level set f ¼ f 0 is convex (for a more formal definition see e.g. Panteghini & Lagioia (2018)). Simple convexity or, for brevity, convexity will indicate that only the zero level set of f ¼ 0 is convex (i.e. the yield curve itself) while convexity is lost for all or some values ...
... The effect of lack of full convexity is particularly severe when a backwards Euler implicit integration is used, as during the iterations the plastic normals are evaluated at stresses characterised by a concave plastic potential. This is discussed by Panteghini & Lagioia (2018), who have concluded that when the yield function is quasi-convex, convergence is achieved even for unrealistically large strain increments. However, with simple convexity convergence is retained only when very small strain increments are applied. ...
Article
This paper presents a new yield function, defined in terms of stress invariants and suitable for isotropic geomaterials. It is a generalization of that of the Modified Cam-Clay model and as such it retains all the mathematical advantages of the original formulation which are particularly convenient for the numerical integration of the constitutive law. In addition the proposed function is capable of providing a wide range of shapes and it is therefore suitable for defining both the yield and the plastic potential surfaces. As compared to the original MCC ellipse, one additional parameter is introduced for defining the shape of the meridional section, which conveniently controls also the relative position of the Normal Compression and Critical State lines. In the deviatoric plane the function not only provides the exact shape of classical failure criteria, such as von Mises, Drucker-Prager, Matsuoka-Nakai, Lade-Duncan, Tresca and Mohr-Coulomb, but it is also capable of rounding the hexagons of the last two criteria with a continuity of class at least C2 required for achieving a quadratic convergence of the integration scheme. The new function has an unrestricted domain of definition, expands/shrinks homothetically with respect both to the origin of the stress space and to its centre and is characterized by convexity for any level set. The last two important features were obtained by applying the convexification technique proposed by Panteghini and Lagioia (2017).
... One may use the convexification technique to overcome this problem by developing an alternative formulation of the yield function which resembles and preserves the non-elliptical shape of the yield surface at all values; for example, Stupkiewicz et al. [18] used such an approach to extend the formulation of a previously proposed yield surface [13]. Moreover, Panteghini & Lagioia [19] corrected the undesired elastic domain of a distorted MCC-type yield surface with a single shape-parameter via a convexification technique [20], to develop a fully convex and unique yield function expression. However, in both of these works, the analytical expression for the yield function is complex and the approach they pursue may not be practical or convenient for all yield surfaces, as imaginary solutions may arise. ...
... Eq. (20) demonstrates the formulation of the yield surface that is divided into two separate functions defined in the meridian, h(p), and deviatoric, z((q−βp), θ ), planes. Note that q−βp is the radius of the yield surface, at a shear level of β, in the deviatoric plane. ...
Full-text available
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This paper presents a new Modified Cam Clay (MCC) type yield function, that is designed for robust and efficient use with implicit stress integration algorithms. The proposed yield function attains non-elliptical (e.g. tear and bullet) shapes, as well as the typical elliptical shape of the MCC model. Like that of MCC, and unlike most other yield functions with non-elliptical shapes available in literature, it is non-singular and unique throughout stress space. The experimental yielding stresses of a wide range of geomaterials have been accurately simulated using the yield surface. The yield function can be used in constitutive models based on classical elasto-plasticity theory.
... The latter is essential to provide good convergence characteristics in implicit numerical integration schemes. 64 Even though the limit of the elastic domain Y is a convex surface, it is crucial to conserve the convexity when the surface expands beyond stress states outside the current plastically admissible domain E. ...
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A complex elastoplastic model requires a robust integration procedure of the evolution equations. The performance of the finite element solution is directly affected by the convergence characteristics of the state‐update procedure. Thereby, this study proposes a comprehensive numerical integration scheme to deal with generic multi‐surface plasticity models. This algorithm is based on the backward Euler method aiming at accuracy and stability, and on the Newton‐Raphson method to solve the unconstrained optimization problem. In this scenario, a line search strategy is adopted to improve the convergence characteristics of the algorithm. The golden section method, an exact line search, is considered. Also, a substepping scheme is implemented to provide additional robustness to the state‐update procedure. Therefore, this work contributes to computational plasticity presenting an adaptive substep size scheme and a consistent tangent modulus according to the substepping technique. Finally, some numerical problems are evaluated using the proposed algorithm. Single‐surface and novel multi‐surface plasticity models are employed in these analyses. The results testify how the line search and substepping strategies can improve the robustness of the nonlinear analysis.
... Following Panteghini and Lagioia [15] considerable advantages are achieved if a slightly different structure is adopted ...
Preprint
The Cosserat continuum is used in this paper to regularize the ill-posed governing equations of the Cauchy/Maxwell continuum. Most available constitutive models adopt yield and plastic potential surfaces with a circular deviatoric section. This is a too crude an approximation which hinders the application of the Cosserat continuum into practice, particularly in the geotechnical domain. An elasto-plastic constitutive model for the linear formulation of the Cosserat continuum is here presented, which features non-associated flow and hardening/softening behaviour, whilst linear hyper-elasticity is adopted to reproduce the recoverable response. For the formulation of the yield and plastic potential functions, a definition of the equivalent von Mises stress is used which is based on Hencky's interpretation of the von Mises criterion and also on the theory of representations. The dependency on the Lode's angle of both the yield and plastic potential functions is introduced through the adoption of a recently proposed Generalized classical criterion, which rigorously defines most of the classical yield and failure criteria.
... The importance of convexity of the yield function has been largely overlooked, especially in geotechnics, over the past few decades. This particular issue was raised by Panteghini & Lagioia (2014, 2018a, 2018b, who reported numerical problems (e.g. lack of convergence in boundary value problems) when using implicit integration algorithms with non-convex yield functions. ...
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... An extensive literature review performed by Franchi et al 52 on yield functions revealed the extent of the ambiguity and confusion regarding the requirements of convexity. However, the discussion on convexity has been more recently reopened (Gluege and Bucci 53 ), and remedies have been proposed to provide convexity to non-convex functions (Panteghini and Lagioia, 54, 55 Lagioia et al, 56 and Lagioia and Panteghini 57 ). ...
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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.
... When documented, convexity-related issues might be remedied by resorting to recent convexification techniques. 74,75 The discussion offered in this work also aimed to discourage simplistic use of viscoplasticity as a mere numerical expedient against mesh dependence in strain-localisation problems. Conversely, the viscoplastic framework was reappraised as a physically sound approach to sand modelling, easy to extend to nonlocality whenever also characteristic length effects are relevant. ...
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... We have further extended this approach to three-invariants surfaces with linear meridional sections [4,3] and a framework for general isotropic surfaces was formulated in [6]. For the GC criterion, the integration reduces to a single nonlinear equation in terms of the Lode's angle. ...
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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.
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This paper presents a new yield function, defined in terms of stress invariants and suitable for isotropic geomaterials. It is a generalization of that of the Modified Cam-Clay model and as such it retains all the mathematical advantages of the original formulation which are particularly convenient for the numerical integration of the constitutive law. In addition the proposed function is capable of providing a wide range of shapes and it is therefore suitable for defining both the yield and the plastic potential surfaces. As compared to the original MCC ellipse, one additional parameter is introduced for defining the shape of the meridional section, which conveniently controls also the relative position of the Normal Compression and Critical State lines. In the deviatoric plane the function not only provides the exact shape of classical failure criteria, such as von Mises, Drucker-Prager, Matsuoka-Nakai, Lade-Duncan, Tresca and Mohr-Coulomb, but it is also capable of rounding the hexagons of the last two criteria with a continuity of class at least C2 required for achieving a quadratic convergence of the integration scheme. The new function has an unrestricted domain of definition, expands/shrinks homothetically with respect both to the origin of the stress space and to its centre and is characterized by convexity for any level set. The last two important features were obtained by applying the convexification technique proposed by Panteghini and Lagioia (2017).
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This paper presents an approach to use the method of characteristics in plane strain problems with failure criteria other than the Mohr–Coulomb and Tresca. Although the method is of general validity, an instance of application is presented for the evaluation of the vertical plastic collapse load of a rigid shallow strip footing resting on a purely frictional material with no lateral surcharge. A comparison with results from finite-element analyses confirms the correctness of the approach.
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: The principal im is to generalize h one- im ion l con itutive equ tions for r te-sensi ive plastic materials to general sta s of s ress. T HE DYNAMIC L YIELD CONDITIONS FOR LASTIC, VISCO-PLASTIC MATERIALS ARE DISCUS ED A D NEW R L XATION FUNCTIONS ARE INTRODUCED. S OLUTION OF THE RELAXATION EQUATIONS FOR SUCH MATERIALS ARE GIVEN. (Author)
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This paper presents a reformulation of the original Matsuoka–Nakai criterion for overcoming the limitations which make its use in a stress point algorithm problematic. In fact, its graphical representation in the principal stress space is not convex as it comprises more branches, plotting also in negative octants, and it does not increase monotonically as the distance of the stress point from the failure surface rises. The proposed mathematical reformulation plots as a single, convex surface, which entirely lies in the positive octant of the stress space and evaluates to a quantity which monotonically increases as the stress point moves away from the failure surface. It is an exact reproduction, and not an approximated one, of the only significant branch of the original criterion. It is also suitable for shaping in the deviatoric plane the yield and plastic potential surfaces of complex constitutive models. A very efficient numerical algorithm for the implicit integration of the proposed formulation is presented, which enables the evaluation of the stress at the end of each increment by solving a single scalar equation, both for associated and non-associated plasticity. The algorithm can be easily adapted for other smooth surfaces with linear meridian section. Finally, a close expression of the consistent Jacobian matrix is given for achieving quadratic convergence in the external structural newton loop. It is shown that all this results in extremely fast solutions of boundary value problems. Copyright © 2013 John Wiley & Sons, Ltd.
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In any elasto-plastic constitutive model there are three main ingredients, namely a yield surface, a plastic potential surface and a hardening/ softening rule. In this paper a versatile mathematical expression is presented which can be used to describe the yield and plastic potential surfaces. The expression is defined completely by a maximum of only four parameters. These parameters can easily be obtained from observable soil behaviour in simple triaxial tests. A major advantage of the expression is that by suitable adjustment of the parameters a wide range of surface shapes can be achieved. For example, it is possible to reproduce the so called “bullet shape” typical of the plastic potential used in the original Cam clay model and the “tear shape” yield surfaces employed in the more recent models. In fact the expression is capable of accurately reproducing the shapes of many of the yield and plastic potential surfaces currently in use. The expression is also shown to be in good agreement with experimental data.
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The authors expand the formerly proposed concept of three mobilized planes (compounded mobilized planes) among the three principal stress axes into a new postulate that a stress plane called ″spatial mobilized plane″ occurs in the three-dimensional stress space. Then they propose to verify, with various test data, the fact that stress-strain relationships of soil under three different principal stresses can be expressed by interpreting the rlationships with respect to this plane. They also propose a new yield condition (failure criterion) of soil that soil yields when the shear-normal stress ratio on this plane has reached a fixed value.
Article
This paper proposes a new isotropic yield and failure criterion for geomaterials under general stress conditions. The new criterion is a function of the three stress invariants and the paper shows the well known criteria by Matsuoka & Nakai (1974) and Lade & Duncan (1975) are included in the new formulation. In order to include different features characterising soil and rock behaviour, the obtained function is successively modified to account for tensile strength and curvature in meridian planes. The influence of each parameter is discussed and finally the comparison between the new criterion and some experimental data on soils and rocks is shown. Finally, the domain of the new criterion is analysed in order to show if the yield function takes negative values also outside the shown elastic domain.
Article
Presents a constitutive model for describing the stress-strain behaviour of partially saturated soils. The model is formulated within the framework of hardening plasticity using two independent sets of stress variables: the excess of total stress over air pressure and the suction. The model is able to represent, in a consistent and unified manner, many of the fundamental features of the behaviour of partially saturated soils which had been treated separately by previously proposed models. On reaching saturation, the model becomes a conventional critical state model. -from Authors
Article
The ability of the 15-noded displacement based finite element to assess accurately collapse loads of soil structures is examined for high friction angles and non-associated flow rules. Solutions are presented for strip and circular footings, for the trap door problem and for the cone penetration test. The last two problems are introduced in order to demonstrate that the method can be used to generate realistic solutions for problems that cannot be solved analytically. The footing problems are treated in order to assess the accuracy and stability of the numerical scheme. The accuracy is shown to be very high, but stability problems occur when non-associated flow rules are applied. In soil mechanics, this constitutes a fundamental problem since this type of instability is of a physical nature rather than of a numerical nature. L'article examine dans le contexte de grands angles de frottement et de règles d'écoulement non-associées l'utilité d'un système basé, sur les elements finis sur le deplacement de 15 noeuds pour évaluer de façon correcte les charges de ruption des sols. Des solutions du problème sont présentées pour les semelle niantes et circulaires, pour le problème des trappes et l'essai de pénétration statique. Ces deux derniers problèmes sont présentés afin de démontrer que la méthode peut s'employer pour obtenir des solutions réalistes de problèmes qui ne peuvent pas se résoudre de façon analytique. Les problèmes des semelles sont traités afin de pouvoir évaluer la précision et la stabilité du système numérique. On voit que la précision est très satisfaisante, mais des problèmes de stabilité se présentent lorsque les règles d'écoulement non-associées sont appliquées. Dans la mécanique du sol ceci représente un problème fondamental, puisque ce type d'instabilité est de nature physique plutôt que numérique.
Article
The essential feature of the observed behaviour is the occurrence of a destructuration phase, which marks the transition from rock-like to soil-like behaviour. During this phase the state of stress remains constant, while strains increase steadily. Three phases can be distinguished: an initial elastic, a destructuration phase and a hardening or softening phase which ends on an ultimate state locus which is linear in the p′-q plane. The observed behaviour is more and more ductile for increasing confining pressures. In the softening regime the specimen is unstable. It is shown that by means of a mathematical model based on the theory of strain-hardening plasticity it is possible to describe mathematically the overall behaviour of the calcarenite in various types of triaxial compression test. -from Authors
Article
Analytical solutions are derived for a three-invariant Cam clay model subjected to proportional and circular drained loading histories. The solutions are presented for a specific volume, and volumetric and generalized shear strains. In the case of a proportional loading only straight effective stress paths are considered while in the case of a circular loading the maximum possible change in Lode's angle is /3 due to plastic isotropy. Additionally, a concept of deviatoric stiffness is devised and an analytical expression for the generalized hardening modulus is derived. Qualitative and quantitative analyses are carried out in the form of direct comparisons between analytical solutions for drained and undrained loading histories thus offering an improved understanding of the three-invariant model.
Article
A large proportion of the constitutive models currently employed in Geomechanics are based on the theory of plasticity. Owing to the limitations of the data obtained from conventional testing equipment, arbitrary assumptions are often made about the behaviour of the material in generalized stress space. In this paper it is shown that the Lode angle of the stress state at failure in plane strain deformation is dependent on the shape adopted for the plastic potential. The influence of the shape of the plastic potential in the deviatoric plane on the predicted behaviour of two boundary value problems both prior to and at failure is then considered. In both cases a form of the Modified Cam Clay model is employed to describe the soil behaviour, and numerical predictions are obtained using a finite element computer code. For drained situations it is shown that the Lode angle of the stress state at failure has a dominating influence on the predicted behaviour. However, for undrained cases the effects are not so important as long as the correct undrained strength at failure is enforced.
Article
When using numerical methods in soil mechanics, one often needs to define a yield surface in three-dimensional principal-stress space. A special class of yield surfaces, given by J = (p+a)α(1−β sin 3ν)n, where ν is the Lode angle, is considered from the point of view of convexity and agreement with experimental data. Some recently proposed yield functions which belong to this class are compared. It is shown that the model with n = −0.229 is optimal as regards convexity, and can give reasonable agreement with the data.
Article
The relations between the inequality expressing Drucker's postulate, the convexity of the admissible stress domain (yield surface) and the quasi-convexity of the yield function are examined. A formal, analytical proof is given of two theorems: the first one, which states the convexity of the yield domain on the assumption of Drucker's postulate; the second one which follows the inverse path. A link role is played by the concept of quasi-convexity of the yield function.
Article
We investigate the performance of a numerical algorithm for the integration of isotropically hardening three-in-variant elastoplastic constitutive models with convex yield surfaces. The algorithm is based on a spectral representation of stresses and strains for infinitesimal and finite deformation plasticity, and a return mapping in principal stress di-rections. Smooth three-invariant representations of the Mohr–Coulomb model, such as the Lade–Duncan and Matsu-oka–Nakai models, are implemented within the framework of the proposed algorithm. Among the specific features incorporated into the formulation are the hardening/softening responses and the tapering of the yield surfaces toward the hydrostatic axis with increasing confining pressure. Isoerror maps are generated to study the local accuracy of the numerical integration algorithm. Finally, a boundary-value problem involving loading of a strip foundation on a soil is analyzed with and without finite deformation effects to investigate the performance of the integration algorithm in a full-scale non-linear finite element simulation. Ó 2002 Elsevier Science B.V. All rights reserved.
Article
An extended version of the classical Generalized Backward Euler (GBE) algorithm is proposed for the numerical integration of a three-invariant isotropic-hardening elastoplastic model for cemented soils or weak rocks undergoing mechanical and non-mechanical degradation processes. The restriction to isotropy allows to formulate the return mapping algorithm in the space of principal elastic strains. In this way, an efficient and robust integration scheme is developed which can be applied to relatively complex yield surface and plastic potential functions. Moreover, the proposed algorithm can be linearized in closed form, thus allowing for quadratic convergence in the global Newton iteration. A series of numerical experiments are performed to illustrate the accuracy and convergence properties of the algorithm. Selected results from a finite element analysis of a circular footing on a soft rock layer undergoing chemical weathering are then presented to illustrate the algorithm performance at the boundary value problem level. Copyright © 2002 John Wiley & Sons, Ltd.
Article
The purpose of the present review article is twofold:•recall elementary notions as well as the main ingredients and assumptions of developing macroscopic inelastic constitutive equations, mainly for metals and low strain cyclic conditions. The explicit models considered have been essentially developed by the author and co-workers, along the past 30 years;•summarize and discuss a certain number of alternative theoretical frameworks, with some comparisons made with the previous ones, including more recent developments that offer potential new capabilities.
Article
A sufficient condition is established for uniqueness of the boundary-value problem set by given velocities on a part of the surface of a body and given nominal traction-rates on the remainder. No restriction is placed on changes in geometry either in the boundary-value problem itself or in the postulated material properties, which are a generalization of those conventionally assumed for workhardening elastic-plastic solids. The solution, when it is unique, is characterized by an extremum principle.Stability under dead loading is also examined, and a criterion is proposed. This has a formal resemblance to the uniqueness criterion, but differs from it in a significant respect when the body is partly plastic, so that stability and uniqueness need not be parallel properties.Finally, the present, theorems are compared with those previously proved for rigid-plastic solids (Hill 1957a, b. d).
Article
Various modes of failure in geomaterials have been observed in practice. Different criteria have been proposed to analyse these failures. In particular, Hill's Condition of Stability and diffuse modes of failure are considered in this paper in a dual framework: continuum mechanics and discrete mechanics. With the assumption of continuous media, experiments have shown that q constant loading paths (q characterizes the second stress invariant. For axisymmetric conditions, q is equal to: q=σ1−σ3, where σ1 is the axial stress and σ3 the lateral stress) can exhibit non-localized failure modes and are analyzed by the second order work criterion. With the assumption of discrete media, grain avalanches are considered, and spatial and temporal correlations between bursts of kinetic energy and peaks of negative values of second order work are demonstrated from discrete computations. It is concluded that the second order work criterion (under its dual form: continuous and discrete) can be a proper tool to analyse diffuse modes of failure in geomaterials.
Article
A new yield/damage function is proposed for modelling the inelastic behaviour of a broad class of pressure-sensitive, frictional, ductile and brittle-cohesive materials. The yield function allows the possibility of describing a transition between the shape of a yield surface typical of a class of materials to that typical of another class of materials. This is a fundamental key to model the behaviour of materials which become cohesive during hardening (so that the shape of the yield surface evolves from that typical of a granular material to that typical of a dense material), or which decrease cohesion due to damage accumulation. The proposed yield function is shown to agree with a variety of experimental data relative to soil, concrete, rock, metallic and composite powders, metallic foams, porous metals, and polymers. The yield function represents a single, convex and smooth surface in stress space approaching as limit situations well-known criteria and the extreme limits of convexity in the deviatoric plane. The yield function is therefore a generalization of several criteria, including von Mises, Drucker–Prager, Tresca, modified Tresca, Coulomb–Mohr, modified Cam-clay, and––concerning the deviatoric section––Rankine and Ottosen. Convexity of the function is proved by developing two general propositions relating convexity of the yield surface to convexity of the corresponding function. These propositions are general and therefore may be employed to generate other convex yield functions.
Deformation of Soft Clay
  • J B Burland
Burland JB. Deformation of Soft Clay. Ph.D.: University of Cambridge; 1967.
A note on sand liquefaction and soil stability
  • R Nova
Nova R. A note on sand liquefaction and soil stability. In: Desai CS, Krempl E, Frantziskonis G, Saadatmanesh H, eds. Constitutive Laws for Engineering Materials: Theory and Applications. Tucson, Arizona: ASME Press; 1991:153-156.
The behaviour of shallow foundations on structured soils
  • Lagioia
Lagioia R, Potts DM. The behaviour of shallow foundations on structured soils. Rivista Italiana di Geotecnica. 1999;4: 52-64.
How to cite this article: Panteghini A, Lagioia R. An approach for providing quasi-convexity to yield functions and a generalized implicit integration scheme for isotropic constitutive models based on 2 unknowns
  • De Borst
  • R Vermeer
De Borst R, Vermeer PA. Possibilities and limitations of finite elements for limit analysis. Géotechnique. 1984;34(2):199-210. How to cite this article: Panteghini A, Lagioia R. An approach for providing quasi-convexity to yield functions and a generalized implicit integration scheme for isotropic constitutive models based on 2 unknowns. Int J Numer Anal Methods Geomech. 2018;1-27. https://doi.org/10.1002/nag.2767