An improved monolithic Newton-Raphson scheme for solving plastic flow with nonlinear flow laws

<p>Brittle-plastic flows where the yield strength is a decreasing, non-linear function of plastic strain are thought to be commonplace in the Earth, and responsible for some of its most catastrophic events. Recent work [1] has highlighted again the computational benefit of an iterative Newton-Raphson scheme that contains a linearization of the plastic flow problem that is consistent with its time discretization. However, such a consistent linearization requires a nested set of iterations to converge on a yield strength if it is governed by a law that is non-linear in strain (or strain rate).</p><p>Eckert and co-authors [2] have shown that the construction of a consistent linearization can be avoided altogether, including these inner iterations, though at the considerable cost of including the full plastic strain tensor as an objective variable alongside the displacement vector. The resulting system is therefore larger, but as it can be expressed directly, posesses the quality that it may be linearized automatically, cheaply, and accurately by finite-differencing the non-linear residual with respect to the solution variables. Their algorithm naturally incorporates predictor and corrector polynomials that are second-order accurate in time, contrasting with traditional methods that are often derived using a Backward Euler time integrator. We present a modification to this algorithm that suppresses the cost of operating it significantly by replacing the symmetric second-order plastic strain tensor with a single effective plastic strain scalar objective variable, cutting the number of unknowns by 40% (2D) and 55% (3D) This makes it computationally more on par with existing schemes that employ a consistent tangent modulus.</p><p>We demonstrate this improved algorithm with test cases of non-linear strain softening laws relevant to Earth scientists, that include regularization by both Kelvin visco-plasticity [3] and non-local measures of effective plastic strain [4]. In addition, we analyse performance of this scheme with respect to existing algorithms.</p><p><em>References</em><br>[1] Duretz et al. (2018). “The benefits of using a consistent tangent operator for viscoelastoplastic computations in geodynamics.” <em>Geochemistry, Geophysics, Geosystems</em>, 19, 4904–4924.</p><p>[2] Eckert et al. (2004). “A BDF2 integration method with step size control for elasto-plasticity.” <em>Computational Mechanics</em> 34.5, 377–386.</p><p>[3] Duretz et al. (2019). “Finite Thickness of Shear Bands in Frictional Viscoplasticity and Implications for Lithosphere Dynamics.” <em>Geochemistry, Geophysics, Geosystems</em>, 20, 5598–5616.</p><p>[4] Engelen et al. (2003). “Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour.” <em>International Journal of Plasticity</em><br>19.4, 403–433.</p>

The Influence of Equivalent Contact Area Computation in 3D Extended Node to Surface Contact Elements

This paper extends the frictionless penalty-based node to contact formulation with area regularization to a 3D framework. Based on our previous work [1] focused on axisymmetric modeling, two computational methods are also considered for the determination of the slave node area. The first method, named as the geometrical approach, is based on a force equivalence system, while the second one, named as the consistent approach, is derived from a more sophisticated scheme elaborated upon the virtual work principle. Then, the extended contact elements are derived for the contact formulations with geometrical and consistent area regularization and a consistent linearization is provided accordingly, which guarantees a quadratic rate of convergence of the global Newton Raphson iterative procedure. Finally, two numerical examples assess the performance of both contact formulations with area regularization and demonstrates the robustness and the efficiency of the node to surface contact formulation with consistent area regularization in reproducing a constant contact pressure distribution across the interface between a deformable body and a analytically-defined rigid body, irrespective of the mesh. Our findings will certainly encourage further developments towards the design of a penaltybased node to surface contact algorithm passing the contact patch test, as was already done successfully in 2D contact problems [2].

Consistent crystal plasticity kinematics and linearization for the implicit finite element method

Purpose – The purpose of this paper is to describe several novel techniques for implementing a crystal plasticity (CP) material model in a large deformation, implicit finite element framework. Design/methodology/approach – Starting from the key kinematic assumptions of CP, the presentation develops the necessary CP correction terms to several common objective stress rates and the consistent linearization of the stress update algorithm. Connections to models for slip system hardening are isolated from these processes. Findings – A kinematically consistent implementation is found to require a correction to the stress update to include plastic vorticity developed by slip deformation in polycrystals. A simpler, more direct form for the algorithmic tangent is described. Several numerical examples demonstrate the capabilities and computational efficiency of the formulation. Research limitations/implications – The implementation assumes isotropic slip system hardening. With simple modifications, the described approach extends readily to anisotropic coupled or uncoupled hardening functions. Practical implications – The modular formulation and implementation support streamlined development of new models for slip system hardening without modifications of the stress update and algorithmic tangent computations. This implementation is available in the open-source code WARP3D. Originality/value – In the process of developing the CP formulation, this work realized the need for corrections to the Green-Naghdi and Jaumann objective stress rates to account properly for non-zero plastic vorticity. The paper describes fully the consistent linearization of the stress update algorithm and details a new scheme to implement the model with improved efficiency.

The Influence of Equivalent Contact Area Computation in Extended Node to Surface Contact Elements

This article aims at extending the node to surface formulation for contact problems withan area regularization as proposed by [1]. For that purpose, two methods are proposed to computethe equivalent contact area attributed to each slave node. The first method, which is based on a geo-metrical approach through force equivalence, is an original extension of the one proposed in [1] fortwo-dimensional contact problems, i.e. plane stress and plane strain state, to the axisymmetric mod-elling context. The second method relies on an energy consistent way obtained through the virtualwork principle and the same expression for the equivalent contact area as the one originally cited in[2] is then recovered. First, the node to surface strategy with area regularization is introduced and theaforementioned methods for the equivalent contact area are presented in detail and compared. After-wards a consistent linearization technique is applied to achieve a quadratic convergence rate in theNewton Raphson iterative procedure used to solve the non-linear equilibrium equations of the under-lying finite element model. Finally, two axisymmetric numerical examples are provided in order tocompare the aforementioned equivalent contact area evaluations and to demonstrate the performanceand the robustness of the consistent approach especially in the neighbourhood the revolution axis.

A SIMPLIFIED MESHLESS METHODS FOR BRITTLE FRACTURE OF CONCRETE

A simplified meshless methods for brittle fracture and nonlinear material is presented. In this method, the crack is modeled by a set of discrete crack segments crossing the entire domain of influence of the meshless shape functions. The key advantage of this method is its simplicity since no representation of the crack topology is needed. A nonlocal stress tensor around the crack tip is used as fracture criterion. A neo-Hooke material in the bulk material is used and a cohesive zone model is employed once discrete cracks occur. We also present consistent linearization of the cohesive zone model. The method is applied to fracture modeling in concrete that is accompanied by excessive cracking and therefore methods that represent the crack path have major drawbacks. We demonstrate the accuracy of the proposed method for complex problems involving mode-I and mixed mode failure.