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>

A fourth-order gauge-invariant gradient plasticity model for polycrystals based on Kröner’s incompatibility tensor

In this paper we derive a novel fourth-order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kröner’s incompatibility tensor [Formula: see text], where [Formula: see text] is the symmetric plastic strain tensor. Here, gauge-invariance denotes invariance under diffeomorphic reparametrizations of the reference configuration, suitably adapted to the geometrically linear setting. The model features a defect energy contribution that is quadratic in the tensor [Formula: see text] and it contains isotropic hardening based on the rate of the plastic strain tensor [Formula: see text]. We motivate the new model by introducing a novel rotational invariance requirement in gradient plasticity, which we call micro-randomness, suitable for the description of polycrystalline aggregates on a mesoscopic scale and not coinciding with classical isotropy requirements. This new condition effectively reduces the increments of the non-symmetric plastic distortion [Formula: see text] to their symmetric counterpart [Formula: see text]. In the polycrystalline case, this condition is a statement about insensitivity to arbitrary superposed grain rotations. We formulate a mathematical existence result for a suitably regularized non-gauge-invariant model. The regularized model is rather invariant under reparametrizations of the reference configuration including infinitesimal conformal mappings.

Учет влияния полей остаточных деформаций в современных физико-механических технологиях обработки конструкционных материалов

A strict determination of the mechanisms of redistribution of previously accumulated irreversible strains as a result of additional elastic shock actions on the material is given for a nonlinear gradient model of large elastic–plastic strain. It is shown that this redistribution is limited by rigid transport and rotation of the plastic strain tensor. Formulas for a change in the initial components of the plastic strain tensor in elastic waves are derived. It is shown that the preliminary plastic field affects the dynamics of further reversible strain as one of the factors of formation of the initial quasi-static elastic field, which cannot be obtained in a purely elastic process.

Reaction Stresses among the Grains during Tensile Deformation of Polycrystalline Metals

A new model describing the reaction stresses during plastic deformation of metals is proposed in which the reaction stresses among the grains and their accumulation are calculated. The model could overcome the shortages of the Sachs and Taylor deformation model. According to the model, the plastic strain tensor induced by activation of slip systems will produce certain elastic reaction stress in the surrounding matrix, which influences the choice of further activation of slip systems as well as the orientation evolution. The model gives more attention to both of the stress and strain compatibility among the grains. The simulation on the tensile deformation of pure copper indicates that the model could exhibit the main characteristics of the real deformation process.

Continuum Foundations of Endochronic Plasticity

In this paper a number of issues are addressed. The proper measure of intrinsic time is shown to be the norm of the increment of the plastic strain tensor, if the memory path is a line in the cojoint stress-strain space and elasticity at reversal points is stipulated. The convexity of the kernel function suffices to show that a periodic plastic strain history will give rise to a periodic stress history—as is observed. Standard tests are then established for the unequivocal experimental determination of the kernel and hardening function, valid for use in general three-dimensional histories.

On the Plastic and Viscoplastic Constitutive Equations—Part I: Rules Developed With Internal Variable Concept

The description of monotonic and cyclic behavior of material is possible by generalizing the internal stress concept by means of a set of internal variables. In this paper the classical isotropic and kinematic hardening rules are briefly discussed, using present plastic strain tensor and cumulated plastic strain as hardening variables. Some additional internal variables are then proposed, giving rise to many possibilities. What is called the “nonlinear kinematic hardening” leads to a natural description of the nonlinear plastic behavior under cyclic loading, but is connected to other concepts such as the Mroz’s model, limited to only two surfaces, and similarities with other approaches are pointed out in the context of a generalization of this rule to viscoplasticity.