Discretization and numerical realization of contact problems for elastic‐perfectly plastic bodies. PART I – discretization, limit analysis

Sysala, Stanislav; Haslinger, Jaroslav; Hlaváček, Ivan; Čermák, Martin

doi:10.1002/zamm.201300112

Cited by 15 publications

(40 citation statements)

References 17 publications

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“…Then ζ α → ζ lim and from Lemma 5.8 in [32] we know that σ α → σ lim in S h as α → +∞, where σ lim is the solution to (P) * h,ζ lim . Consequently,…”

Section: Variational Characterization Of the Function ϑmentioning

confidence: 84%

“…On the other hand, is Lipschitz continuous in S h , therefore differentiable in the sense of Clark (see [10]). One of the elements of the generalized gradient ∂ (e) of at e ∈ S h is the vector function o : S h → L(S h , S h ) defined element-wise by (see [32])…”

Section: Modified Semi-smooth Newton Methods With Dampingmentioning

confidence: 99%

“…The dual and primal formulation of the discretized problem for the von Mises criterion with the load ζ L, ζ ≥ 0, reads as follows (see [32]):…”

Section: Setting Of the Problemmentioning

confidence: 99%

“…as follows from the definition of in [32] and the fact that 0 ∈ P h . Summing (2.18) and (2.20), we obtain…”

Section: Variational Characterization Of the Function ϑmentioning

confidence: 99%

“…If α is used then a constrained minimization problem for the stored energy function on the intersection of the set of kinematically admissible displacements with the hyperplane L(v) = α is solved to evaluate ϑ(α). It is worth noticing that the limit loading parameter ζ lim is an upper bound for the limit loading parameter of the continuous setting (see [32,Remark 5.3]). …”

Section: Introductionmentioning

confidence: 99%

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Discretization and numerical realization of contact problems for elastic‐perfectly plastic bodies. PART II – numerical realization, limit analysis

et al. 2015

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The paper deals with numerical realization of discretized, frictionless static contact problems for elastic-perfectly plastic materials and the computational limit analysis. Two numerical methods based on the variational formulation in terms of stresses are analyzed: the semi-smooth Newton method with damping and the alternating direction method of multipliers. These methods are used for tracking the loadings path to determine the discretized limit loading parameter and for solving elastic-perfectly plastic problems.

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“…Then ζ α → ζ lim and from Lemma 5.8 in [32] we know that σ α → σ lim in S h as α → +∞, where σ lim is the solution to (P) * h,ζ lim . Consequently,…”

Section: Variational Characterization Of the Function ϑmentioning

confidence: 84%

Section: Modified Semi-smooth Newton Methods With Dampingmentioning

confidence: 99%

“…The dual and primal formulation of the discretized problem for the von Mises criterion with the load ζ L, ζ ≥ 0, reads as follows (see [32]):…”

Section: Setting Of the Problemmentioning

confidence: 99%

“…as follows from the definition of in [32] and the fact that 0 ∈ P h . Summing (2.18) and (2.20), we obtain…”

Section: Variational Characterization Of the Function ϑmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Discretization and numerical realization of contact problems for elastic‐perfectly plastic bodies. PART II – numerical realization, limit analysis

et al. 2015

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Optimization and variational principles for the shear strength reduction method

Sysala

Hrubešová

Michalec

et al. 2021

Num Anal Meth Geomechanics

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In this paper, a modified shear strength reduction method (MSSR) and its optimization variant (OPT‐MSSR) are suggested. The idea of MSSR is to approximate the standard shear strength reduction to be more stable and rigorous from the numerical point of view. The MSSR method consists of a simplified associated elasto‐plastic model completed by the strength reduction depending on the dilatancy angle. Three Davis' modifications suggested by Tschuchnigg et al. (2015) are interpreted as special cases of MSSR and their factors of safety are compared. The OPT‐MSSR method is derived from MSSR on the basis of rigid plastic assumption, similarly as in limit analysis. Using the variational approach, the duality between the static and kinematic principles of OPT‐MSSR is shown. The numerical solution of OPT‐MSRR is obtained by performing a regularization method in combination with the finite element method, mesh adaptivity and a damped Newton method. In‐house codes (Matlab) are used for the implementation of this solution concept. Finally, two slope stability problems are considered, one of which follows from analysis of a real slope. The softwares packages Plaxis and Comsol Multiphysics are used for comparison of the results.

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Subdifferential‐based implicit return‐mapping operators in Mohr‐Coulomb plasticity

2017

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Key wordsInfinitesimal plasticity, Mohr-Coulomb yield surface, implicit return-mapping scheme, consistent tangent operator, semismooth Newton method, incremental limit analysis, slope stability. MSC (2010) 35Q74, 74C05, 74S05, 90C25The paper is devoted to constitutive solution, limit load analysis and Newton-like methods in elastoplastic problems containing the Mohr-Coulomb yield criterion. Within the constitutive problem, we introduce a self-contained derivation of the implicit return-mapping solution scheme using a recent subdifferential-based treatment. Unlike conventional techniques based on Koiter's rules, the presented scheme a priori detects a position of the unknown stress tensor on the yield surface even if the constitutive solution cannot be found in a closed form. This eliminates blind guesswork from the scheme and enables to analyze properties of the constitutive operator. It also simplifies the construction of the consistent tangent operator, which is important for the semismooth Newton method when applied to the incremental boundary-value elastoplastic problem. The incremental problem in Mohr-Coulomb plasticity is combined with limit load analysis. Beside a conventional direct method of incremental limit analysis, a recent indirect one is introduced and its advantages are described. The paper contains 2D and 3D numerical experiments on slope stability with publicly available Matlab implementations. / www.zamm-journal.org 1503 respectively. It is worth mentioning that the solution schemes depend mainly on the formulation of the plastic flow rule, its discretization and eventual other approximations. Let us now briefly discuss these aspects.The plastic flow rule is usually formulated by using the so-called Koiter rule in engineering practice. This rule was introduced for associative models with multisurface yield criteria in [23] and consequently extended to nonassociative models, see, e.g., [15]. It consists of several formulas that depend on a position of the unknown stress tensor on the yield surface. These formulas have a different number of plastic multipliers. Within the Mohr-Coulomb pyramid, one plastic multiplier is used for the smooth portions, two multipliers at the edge points, and six multipliers at the apex. Therefore, the resulting solution schemes are different for each Koiter's formula. However, only one of them usually leads to the correct stress tensor. Moreover, handling different numbers of plastic multipliers is not suitable for analyzing the stress-strain operator even if the solution can be found in a closed form. If the elastoplastic model contains a convex plastic potential as the Mohr-Coulomb one, then it is possible to replace the Koiter rule with a subdifferential of the potential (see, e.g., [17]). Such a formulation is independent of the unknown stress position, contains just one plastic multiplier, and thus it is more convenient for mathematical analysis of the constitutive operators. In [34], it was shown that this formulation is also convenient for solution of some con...

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Discretization and numerical realization of contact problems for elastic‐perfectly plastic bodies. PART I – discretization, limit analysis

Cited by 15 publications

References 17 publications

Discretization and numerical realization of contact problems for elastic‐perfectly plastic bodies. PART II – numerical realization, limit analysis

Discretization and numerical realization of contact problems for elastic‐perfectly plastic bodies. PART II – numerical realization, limit analysis

Optimization and variational principles for the shear strength reduction method

Subdifferential‐based implicit return‐mapping operators in Mohr‐Coulomb plasticity

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