Yield surfaces and principle of superposition: Revisit through incrementally non-linear constitutive relations

Darve, Félix; Flavigny, É.; Meghachou, Mourad

doi:10.1016/s0749-6419(95)00037-2

Cited by 90 publications

(61 citation statements)

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“…In the first section, the limit of the bifurcation domain plotted in the 3D stress space is depicted for the incrementally nonlinear and incrementally piecewise linear constitutive relations from Darve et al [25]. An analytical equation of instability cones is given and it is presented in the 3D principal stress space for both constitutive models.…”

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confidence: 99%

Bifurcation modeling in geomaterials: From the second‐order work criterion to spectral analyses

Prunier

Laouafa

Lignon

et al. 2008

Num Anal Meth Geomechanics

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SUMMARYThe present paper investigates bifurcation analysis based on the second-order work criterion, in the framework of rate-independent constitutive models and rate-independent boundary-value problems. The approach applies mainly to nonassociated materials such as soils, rocks, and concretes. The bifurcation analysis usually performed at the material point level is extended to quasi-static boundary-value problems, by considering the stiffness matrix arising from finite element discretization. Lyapunov's definition of stability (Annales de la faculté des sciences de Toulouse 1907; 9:203-274), as well as definitions of bifurcation criteria (Rice's localization criterion (Theoretical and Applied Mechanics. Fourteenth IUTAM Congress, Amsterdam, 1976; 207-220) and the plasticity limit criterion are revived in order to clarify the application field of the second-order work criterion and to contrast these criteria. The first part of this paper analyses the second-order work criterion at the material point level. The bifurcation domain is presented in the 3D stress space as well as 3D cones of unstable loading directions for an incrementally nonlinear constitutive model. The relevance of this criterion, when the nonlinear constitutive model is expressed in the classical form (d = Mdε) or in the dual form (dε = N d ), is discussed. In the second part, the analysis is extended to the boundary-value problems in quasi-static conditions. Nonlinear finite element computations are performed and the global tangent stiffness matrix is analyzed. For several examples, the eigenvector associated with the first vanishing eigenvalue of the symmetrical part of the stiffness matrix gives an accurate estimation of the failure mode in the homogeneous and nonhomogeneous boundary-value problem.

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confidence: 99%

Bifurcation modeling in geomaterials: From the second‐order work criterion to spectral analyses

Prunier

Laouafa

Lignon

et al. 2008

Num Anal Meth Geomechanics

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“…That is why the above discussion has to be made in a given tensorial zone. 2 Moreover, it is necessary to verify that solutions belong geometrically to the tensorial zone considered and to cut them (i.e. to keep only the part of the cone include in the tensorial zone for example).…”

Section: Basic Conceptsmentioning

confidence: 99%

“…Numerical results, are displayed with the constitutive models of Darve [2] and with the discrete element method.…”

Section: Illustrationmentioning

confidence: 99%

Discrete Numerical Analysis of Failure Modes in Granular Materials

Sibille

Prunier

Nicot

et al. 2011

Particle-Based Methods

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To cite this version:Luc Discrete Numerical Analysis of Failure Modes in Granular MaterialsAbstract The question of failure for geomaterials, and more generally for nonassociative materials, is revisited through the second-order work criterion defining, for such media, a whole domain of bifurcation included in the plastic limit surface. In a first theoretical part of the chapter, relations between the vanishing of the second-order work, the existence of limit states and the occurrence of failures characterized by a transition from a quasi-static pre-failure regime to a dynamic post-failure regime, are presented and illustrated from discrete element computations. Then boundaries of the bifurcation domain and cones of unstable loading directions are given in fully three-dimensional loading conditions for a phenomenological incrementally non-linear relation, and in axisymmetric loading conditions for a numerical discrete element model. Finally, conditions for the triggering and the development of failure inside the bifurcation domain are described and emphasized from direct simulations with the discrete element method for proportional stress loading paths.

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“…Despite a discrete nature of granular materials, their mechanical behaviour can be reasonably described by principles of continuum mechanics using an elastoplastic [30][31][32] and hypoplastic [33][34][35][36] approach. Non-polar hypoplastic constitutive laws formulated at the Karlsruhe University [35][36][37][38][39] describe the evolution of effective stress tensor depending on the current void ratio, stress state and rate of deformation by isotropic linear and non-linear tensorial functions according to the representation theorem by Wang [40].…”

Section: Micro-polar Hypoplastic Modelmentioning

confidence: 99%

Deterministic and statistical size effect during shearing of granular layer within a micro‐polar hypoplasticity

Tejchman

Górski

2007

Num Anal Meth Geomechanics

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SUMMARYThe paper deals with numerical investigations of a deterministic and statistical size effect in granular bodies during quasi-static shearing of an infinite layer under plane strain conditions, free dilatancy and constant pressure. For a simulation of the mechanical behaviour of a cohesionless granular material during a monotonous deformation path, a micro-polar hypoplastic constitutive relation was used which takes into account particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. The proposed model captures the essential mechanical features of granular bodies in a wide range of densities and pressures with a single set of constants. In the paper, a deterministic and statistical size effect is analysed. The deterministic calculations were carried out with an uniform distribution of the initial void ratio for four different heights of the granular layer: 5, 50, 500 and 2000 mm. To investigate the statistical size effect, the Monte Carlo method was applied. The random distribution of the initial void ratio was assumed to be spatially correlated. Truncated Gaussian random fields were generated in a granular layer using an original conditional rejection method. The sufficient number of samples was determined by analysing the convergence of the outcomes. In order to reduce the number of realizations without losing the accuracy of the calculations, stratified and Latin hypercube methods were applied. A parametric analysis of these methods was also presented. Some general conclusions were formulated.

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Yield surfaces and principle of superposition: Revisit through incrementally non-linear constitutive relations

Cited by 90 publications

References 1 publication

Bifurcation modeling in geomaterials: From the second‐order work criterion to spectral analyses

Bifurcation modeling in geomaterials: From the second‐order work criterion to spectral analyses

Discrete Numerical Analysis of Failure Modes in Granular Materials

Deterministic and statistical size effect during shearing of granular layer within a micro‐polar hypoplasticity

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