Rate-independent Plasticity Research Articles

Determining plastic slips in rate-independent crystal plasticity models through machine learning algorithms

Dislocation slip-based crystal plasticity models have been a great success in connecting the fundamental physics with the macroscopic deformation of crystalline materials. Pioneered by Taylor in his work on “plastic strain in metals” (Taylor, 1938), and further advanced by Bishop and Hill (1951a, 1951b), the Taylor–Bishop–Hill theory laid the foundation of today’s constitutive models on crystal plasticity. An intriguing part of those modeling is to determine the active slip systems—which system to be involved in and how much it contributes to the deformation. In this paper, we developed a machine learning-based algorithm to determine accurately and efficiently the active slip systems in crystal plasticity constitutive models. Applications to the common three polycrystalline metals, face-centered cubic (FCC) copper, body-centered cubic (BCC) α-iron, and hexagonal close-packed (HCP) AZ31B, demonstrate that even a simple neural network could give rise to accurate and efficient results in comparing with traditional routines. There seems to be plenty of space for further reducing the computation time and hence scaling up the simulating samples.

Influence of specimen width on the elongation-to-fracture in cyclic-bending-under-tension of commercially pure titanium sheets

In recent publications, we have shown that repeated bending and unbending during tension can significantly increase the percent elongation-to-fracture (ETF) of metallic sheets. A custom-built machine for cyclic bending under tension (CBT) testing was designed and used to test several sheet metals. In particular, the improvements in ETF using CBT were over 3× relative to simple tension (ST) for commercially pure titanium (cp-Ti) sheets. The present paper evaluates the influence of specimen width on the achieved ETF in CBT of cp-Ti sheets. To facilitate the study, the CBT machine was furnished with wide grips to enable processing of sheets in addition to testing narrow strips. The ETF was observed to reduce with the sheet width. To rationalize the observations, the strain rate-independent plasticity theory of von Mises (J2) with isotropic expansion of yield surfaces of cp-Ti was used in finite elements (FE) to perform a set of simulations. To facilitate the FE simulations of the CBT process stretching the sheets of cp-Ti far beyond the point of maximum uniform strain in ST, the post-necking hardening behavior was inferred along RD, TD, and 45 sheet directions using an established methodology combining ST tests, CBT tests, and modeling material behavior during CBT. Simulated geometries and mechanical fields were found to be in good agreement with corresponding measurements for every specimen width. While predicted axial strains were similar for every width, the strain path shifted from uniaxial tension for the narrow specimen toward plane-strain tension for the wider specimens. As such, as plane-strain tension was approached via progressively wider specimens, the width strains reduced, while the thinning strains increased. The increase in the thinning strain was established as the primary reason for the reduced ETF with increasing sheet width. Experimental and FE simulation results along with the insights into the influence of specimen width on the ETF in CBT of cp-Ti are presented and discussed.

Basic properties of ferromagnetic vector hysteresis play models and their validation by finite element method

PurposeMagnetic hysteresis holds significant technical and physical importance in the design of electromagnetic components. Despite extensive research in this area, modeling magnetic hysteresis remains a challenging task that is yet to be fully resolved. The purpose of this paper is to study vector hysteresis play models for anisotropic ferromagnetic materials in a physical, thermodynamical approach.Design/methodology/approachIn this work, hysteresis play models are implemented to interpret magnetic properties, drawing upon classical rate-independent plasticity principles derived from continuum mechanics theory. By conducting qualitative and quantitative verification and validation, various aspects of ferromagnetic vector hysteresis were thoroughly examined. By directly incorporating the hysteresis play models into the primal formulations using fixed point method, the proposed model is validated with measurements in a finite element (FE) environments.FindingsThe proposed vector hysteresis play model is verified with fundamental properties of hysteresis effects. Numerical analysis is performed in an FE environment. Measured data from a rotational single sheet tester (RSST) are validated to the simulated results.Originality/valueThe results of this work demonstrates that the essential properties of the hysteresis effects by electrical steel sheets can be represented by the proposed vector hysteresis play models. By incorporation of hysteresis play models into the weak formulations of the magnetostatic problem in the h-based magnetic scalar potential form, magnetic properties of electrical steel sheets can be locally analyzed and represented.

Elastoplastic peridynamic formulation for materials with isotropic and kinematic hardening

We present an ordinary state-based peridynamic model in 2D and 3D consistent with rate-independent J2 plasticity with associated flow rule. The new contribution is the capability of the elastoplastic law to describe isotropic, kinematic and mixed hardening. The hardening formulations follow those available in the literature for classical elastoplasticity. The comparison between the results obtained with the peridynamic model and those obtained with a commercial FEM software shows that the two approaches are in good agreement. The extent of the plastic regions and von Mises stress computed with the new model for 2D and 3D examples match well those obtained with FEM-based solutions using ANSYS.

Rate-independent gradient-enhanced crystal plasticity theory — Robust algorithmic formulations based on incremental energy minimization

Numerically robust algorithmic formulations suitable for rate-independent crystal plasticity are presented. They cover classic local models as well as gradient-enhanced theories in which the gradients of the plastic slips are incorporated by means of the micromorphic approach. The elaborated algorithmic formulations rely on the underlying variational structure of (associative) crystal plasticity. To be more precise and in line with so-called variational constitutive updates or incremental energy minimization principles, an incrementally defined energy derived from the underlying time-continuous constitutive model represents the starting point of the novel numerically robust algorithmic formulations. This incrementally defined potential allows to compute all variables jointly as minimizers of this energy. While such discrete variational constitutive updates are not new in general, they are considered here in order to employ powerful techniques from non-linear constrained optimization theory in order to compute robustly the aforementioned minimizers. The analyzed prototype models are based on (1) nonlinear complementarity problem (NCP) functions as well as on (2) the augmented Lagrangian formulation. Numerical experiments show the numerical robustness of the resulting algorithmic formulations. Furthermore, it is shown that the novel algorithmic ideas can also be integrated into classic, non-variational, return-mapping schemes.

A numerical study on the visco-plastic regularization of a rate-independent strain gradient crystal plasticity formulation

A common practice in computational investigations of rate-independent plasticity is to approximate the rate-independent behavior by a visco-plastic regularization. As the fidelity of the approximation increases, the numerical solution of the non-linear problem becomes challenging and it can eventually lead to divergence; thus, there is a numerical limit to the regularization parameter. This limit may be exacerbated by complex material models and critical values of material parameters. Due to these constraints, the regularization may be rendered insufficient and the artificially introduced rate-dependency may lead to a behavior that can be mistakenly attributed to the material model and that we thus identify as spurious in a rate-independent context. To study these spurious effects and their onset, here the problem of an infinite strip under shear loading is numerically solved using a visco-plastic regularized gradient crystal plasticity formulation. The required accuracy of the rate-dependent approximation is found to vary with respect to the type of loading. Furthermore, a set of sigmoid functions used for the regularization is investigated and a subset is shown to deliver improved approximation of the rate-independent case.

A Small-Deformation Rate-Independent Continuous-Flow Model for Elasto-Plastic Frames Allowing Rapid Fatigue Predictions in Metallic Structures

Fatigue analysis in metallic frame structures can be challenging due to associated computational costs; if localized plasticity is involved, then the approach of three-dimensional (3D) continuum plasticity models for direct computation of stresses will be infeasible for the analysis of cyclic loading that would need to be modeled in medium- to high-cycle fatigue and vibratory fatigue applications. This difficulty is particularly accentuated in architected structures, for which high-resolution 3D finite element analysis (FEA) would be prohibitively expensive. In this work, we propose an alternative approach based on the use of novel elasto-plastic frame model with continuous flow (i.e. no sharp yield function) for modeling 3D frame and lattice structures. Rather than splitting the strains (as is done in classical plasticity) we split the deformation measures, extension, curvature and twist, into elastic and plastic components and postulate a rate type evolution rule for the plastic variables in terms of the stress resultants (axial force, bending moment, and torque). The combination of structural models together with the use of elasto-plastic operator split to solve the resulting boundary value problem allows for much faster determination of localized plasticity than continuum models can provide. The use of a continuous transition from elastic to rate-independent plasticity (as opposed to an abrupt change with classical plasticity models) allows us to capture localized microplasticity and determine resulting fatigue progression using a cycle-count-free, plastic work-based approach, formulated in terms of the curvatures and resultants. We demonstrate that (a) the model is able able to reproduce the response of 3D FEA with very few elements and (b) the model has the ability to rapidly predict the fatigue life under variable amplitude combined loading with relatively few frame elements.

A rate-independent crystal plasticity algorithm based on the interior point method

A robust rate-independent crystal plasticity algorithm is proposed where the amount of plastic slip required to satisfy the multi-surface yield conditions are determined using a flow rule that stems from the maximization of plastic dissipation. The stress update procedure, which is based on the backward Euler time integration, is treated as an optimization problem and the interior point method is utilized in obtaining the stress. As opposed to common return mapping algorithms, such as the closest point projection method, the yield conditions are not violated during the iterative update of the slip amounts which helps to avoid convergence issues for both the local consistency and the global equilibrium solutions. An efficient implementation of this method for large deformations is given together with an algorithmic tangent modulus that makes it attractive to be used in both Finite Element simulations as well as analytical homogenization algorithms.

Numerically efficient and robust interior-point algorithm for finite strain rate-independent crystal plasticity

The challenges of classic rate-independent crystal plasticity models regarding the determination of active slip systems and computation of solutions due to linear dependent slip systems are still a field of active research. In that regard, interior-point methods are promising approaches because a logarithmic barrier term naturally avoids the combinatorial active set search, and these methods are applicable even when the gradient of inequality constraints is linearly dependent. In this paper, a new interior-point algorithm suitable for rate-independent crystal plasticity at finite strains is developed. For that purpose, the postulate of maximum dissipation is recast as a constrained optimization problem whose optimality conditions are derived by utilizing interior-point approaches. This optimization problem is discretized in time by means of an exponential time integration of the flow rule, and the time-discrete equations are solved by a Newton scheme. The focus of this paper is the efficiency and robustness of the implementation of this approach. In particular, the effect of different line search procedures, favorable residual forms, preconditioning methods, barrier parameter updates and the consistent algorithmic tangent modulus are discussed and further elaborated. In addition, aspects regarding uniqueness and well-posedness of the algorithm are analyzed in a geometrically linearized setting. Efficiency and robustness of the developed algorithm are shown by numerical examples of an FCC crystal including finite element simulations without stabilizing hardening or rate-dependent effects. Furthermore, the results are compared to the well-established augmented Lagrangian formulation.

The strength of a constrained lithium layer

A constrained compression test is developed to replicate the mechanical state of a lithium filament within a solid state battery. Lithium microspheres are compressed between parallel quartz plates into a pancake shape of thickness on the order of 15 µm. Full adhesion with no slip exists between the lithium and platens, and the attendant mechanical constraint implies that the average pressure on the pancake-shaped specimens increases with increasing aspect ratio of radius to height. In addition to mechanical constraint, a thickness-dependant size effect is observed whereby the apparent flow strength of the lithium increases from 0.7 MPa in the bulk to 2.0 MPa at a thickness of 15 µm. The lithium deforms in a power-law creeping manner at room temperature, and to simplify interpretation of the results, the relative velocity of the loading platens is adjusted to ensure that the true compressive strain rate is held fixed at 10−3 s−1. Additional measurements of lithium flow strength are obtained by subjecting the pancake-shaped specimens to simple shear. The size effect under shear loading is comparable to that in constrained compression. The observed size effect for lithium is consistent with that reported in the literature for lithium in indentation tests and in single pillar compression tests. Finally, the size effect of lithium in the power law creep regime is compared with that for rate-independent plasticity (for copper).

“Implicit” vs “Explicit” gradient plasticity models: Do they always remove mesh dependence in softening materials?

Finite element quasi-static solutions of rate-independent softening plasticity models are known to depend on the mesh size (i.e., are unreliable) due to loss of ellipticity of the governing partial differential equations, which allows for the development of non-smooth solutions, such as shear bands of zero thickness. A “regularization” scheme that has been proposed is the use of “non-local” (Bažant et al., 1984) or “strain-gradient” theories (Aifantis, 1984). We examine in detail the families of “explicit” and “implicit” gradient plasticity models and show that use of such theories does not always eliminate mesh dependence of finite element solutions in rate-independent softening materials. Depending on the value of the non-local hardening modulus, the mathematical problem may still lose ellipticity and allow for the development of solutions with discontinuous spatial velocity gradients, which are responsible for the mesh dependence of numerical solutions. As an example, a rate-independent “implicit” gradient version of the von Mises yield condition with the associated flow rule is implemented in ABAQUS, and the formation of shear bands in plane strain tension is analyzed numerically; it is shown that, unless the material parameters in the non-local model are such that ellipticity is retained, the finite element solutions depend on the mesh size. The example of isotropic porous metal plasticity with a hardening matrix material is considered (e.g., the Gurson (1977) model); it is shown that regularization is achieved if an implicit non-local model is used, in which the local porosity f is replaced by its non-local (neighborhood average) counterpart f̄ in the yield condition. In all non-local plasticity models, dimensional consistency requires the introduction of one or more “material lengths” ℓi, which multiply the additional terms with the higher-order spatial derivatives introduced by the non-local models. The boundary layers associated with the singular perturbation problem when the material lengths ℓi→0 are examined. A linear stability analysis is carried out to study the stability of homogeneous problems under small harmonic perturbations; it is shown that the stability range of the non-local models is always larger than that of the corresponding local models.

Energy approach to the selection of deformation pattern and active slip systems in single crystals

The recently introduced quasi-extremal energy principle for incremental non-potential problems in rate-independent plasticity is applied to select the deformation pattern and active slip systems in single crystals. The standard crystal plasticity framework with a non-symmetric slip-system interaction matrix at finite deformation is used. The incremental work criterion for the formation of deformation bands is combined with the quasi-extremal energy principle for determining the active slip systems and slip increments in the bands. In this way, the incremental energy minimization approach has been extended to the non-potential problem of deformation banding in metal single crystals. It is shown that fulfilment of the mathematical criterion for incipient deformation banding in a homogeneous crystal in the multiple-slip case under certain conditions requires non-positive determinant of the hardening moduli matrix. Numerical examples of energetically preferable patterns of deformation bands are presented for Cu and Ni single crystals.

The elastic threshold for strain-gradient plasticity, and comparison of theoretical results with experiments

This work continues an earlier study, aimed at determining the threshold of elastic behaviour for specimens at the microscale. Given the size-dependent nature of the response the theory makes use of a model of rate-independent strain-gradient plasticity to establish conditions for lower and upper bounds to the threshold. The model is based on a family of yield or dissipation functions. In the earlier work bounds were established for one member of the family, in which the dissipation depends on the root mean square of the magnitude of plastic strain and its gradient. In this work bounds are derived based on a second alternative, in which the dependence is linear. These two alternatives are most relevant in determining length scales through fits to experimental results. The second aim of this work is to carry out comparisons between the bounds obtained theoretically, with experimental data on initial yielding for torsion tests. The correlations are in some cases reasonable, while in others there are discrepancies. Generally they illustrate the complexities – for example in accurate experimental determination of the elastic threshold, and of the length scale – in predicting aspects of behaviour at the microscale.

Physically enhanced training for modeling rate-independent plasticity with feedforward neural networks

In recent years, a lot of progress has been made in the field of material modeling with artificial neural networks (ANNs). However, the following drawbacks persist to this day: ANNs need a large amount of data for the training process. This is not realistic, if real world experiments are intended to be used as data basis. Additionally, the application of ANN material models in finite element (FE) calculations is challenging because local material instabilities can lead to divergence within the solution algorithm. In this paper, we extend the approach of constrained neural network training from [28] to elasto-plastic material behavior, modeled by an incrementally defined feedforward neural network. Purely stress and strain dependent equality and inequality constraints are introduced, including material stability, stationarity, normalization, symmetry and the prevention of energy production. In the Appendices, we provide a comprehensive framework on how to implement these constraints in a gradient based optimization algorithm. We show, that ANN material models with training enhanced by physical constraints leads to a broader capture of the material behavior that underlies the given training data. This is especially the case, if a limited amount of data is available, which is important for a practical application. Furthermore, we show that these ANN models are superior to classically trained ANNs in FE computations when it comes to convergence behavior, stability, and physical interpretation of the results.

On the numerical homogenization of real polycrystalline microstructures

AbstractThe FE2 method, cf. [1], a direct micro‐macro homogenization approach, has become a standard procedure for scale‐transition applications. Therein, the modeling of a micro‐heterogeneous material described by a representative volume element (RVE) based on realistic microstructures can give rise to a barely unmanageable computational effort. Alternatively, statistically similar RVEs (SSRVEs) can be used, which are constructed based on morphological information of the real microstructure and lead to a reduction of computational cost, see [2]. In their construction, a least‐square functional is used to minimize the deviation of statistical properties, such as volume fraction, spectral density and lineal‐path function, of the SSRVE and the real microstructure. The application of SSRVEs has been shown to lead to an adequate representation of the mechanical behavior of the real microstructure. The first part of the talk will give an overview on the construction of SSRVEs and present examples of multiscale analyses using the FE2 approach with simplified microstructures in different engineering applications of steel material.The paper also focusses on the details of the microstructure and discusses crystal plasticity models, see e.g. [3], in order to account for the material anisotropy induced by the texture of the crystalline structure of steel. It is well known that for rate independent single crystal plasticity, the ambiguity of the choice of active slip systems and linear dependency of slip criteria may cause instabilities in the algorithm. Classical perturbation methods are often used to solve the problem as well as rate dependent algorithms which model the rate independent case as the limit of vanishing viscosity. However, this leads to stiff constitutive equations and thus requires small time increments. In [4], an alternative approach has been proposed recently which is based on handling the constrained optimization problem in the framework of infeasible primal‐dual interior point methods (IPDIPM). We modify the original constrained optimization problem using slack variables in order to stabilize the algorithm and allow for temporary violation of the constraints. Numerical examples are presented for crystalline structures with face centered cubic properties.

Response of 2D and 3D crystal plasticity models subjected to plane strain condition

The plane strain assumption is generally applied in crystal plasticity finite element (CPFE) simulations in a 2D space to characterize the macroscopic material response considering microstructural features. However, the reliability and accuracy of 2D approximations need to be addressed. In this paper, crystal plasticity finite element simulations of 2D and 3D RVEs are performed with local and averaged plane strain assumptions in Abaqus/Standard. Plane strain postulation is implemented via plane strain elements in 2D and zero average thickness strain in 3D. Irregularly shaped RVEs are generated using the open-source software library Voro++. A conforming mesh is rendered to assign periodic boundary conditions on geometrically periodic RVEs. Periodic boundary condition (PBC) is applied using a prescribed macroscopic deformation gradient tensor. A rate-independent finite strain crystal plasticity model is employed as the user-defined material behavior in finite element simulations. A discrepancy is observed between macroscopic flow curves of 2D and 3D RVEs. The comparison was made for three cases of latent hardening in the crystal plasticity model. In all cases, 3D flow curves exceed 2D results. The results indicate that the deviation is caused by out-of-plane slip activation in 3D simulations, which proves to be an additional hardening source.

A thermodynamically consistent lumped-element model for magnetic shape memory components

A novel lumped-element modeling approach for magnetic shape memory alloys is presented. Building on concepts borrowed from rate-independent plasticity, the model describes the magnetic and magneto-mechanical behavior of a magnetic shape memory component subjected to a particular load case in a thermodynamically consistent way. The approach remedies the common issues of existing models regarding the representation of inner hysteresis loops and small-signal behavior. The model is parametrized in terms of a small number of parameters, which can be determined from single variant magnetic curves and mechanical first order reversal curves at constant magnetic input. The results indicate that the model predicts the magnetic and magneto-mechanical behavior with sufficient accuracy, which makes it appropriate for system design.

Non-iterative stress integration method for anisotropic materials

A non-iterative stress update method is proposed to accelerate finite element simulations while retaining the numerical robustness and accuracy in rate-independent plasticity. The stress tensor is directly integrated on the basis of the elastoplastic constitutive law without a recursive root-finding process such as the Newton-Raphson method. The iterative return mapping process for stress integration is avoided by first introducing an equation that guarantees the positive-definite nature of the effective plastic strain increment. Furthermore, the uncertainty of the yield condition fulfillment in a non-iterative stress update method is completely resolved by employing a stress projection technique. The numerical robustness, accuracy, and computational efficiency are comprehensively assessed through a wide range of strain increments in both implicit and explicit finite element analyses. Consequently, the computation time is remarkably reduced by more or less 50% with high fidelity in a variety of finite element simulation examples.

A novel multiscale computational methodology for numerical material testing based on finite element polycrystal model

In this paper, a novel computational procedure for numerical material testing method based on a strain rate-independent crystal plasticity model and an optimization scheme is presented. In order to simulate deformation of materials that exhibit mechanical anisotropy with high accuracy, it is necessary to apply material models that include many material parameters. Therefore, if difficult-to-perform material tests such as biaxial tests can be replaced by numerical material tests, the cost and time required for experiments can be significantly reduced. The proposed method was developed using a crystal plasticity-based finite element model, known as finite element polycrystal model (FEPM), and an optimization scheme based on a genetic algorithm (GA) for microscopic parameters. FEPM is a model that performs analysis while determining the activity of all slip systems without using the strain rate-dependent constitutive law that is commonly used and is suitable for numerical material testing because it has less physical ambiguity. In the optimization process, deformation anisotropy is related to the grain orientation expressed by the Euler angle, and yield stress anisotropy is related to the work-hardening coefficients of the slip system. In addition, by generating pseudo-anisotropic crystal orientation by numerical rolling, the need for crystal orientation observation is eliminated. In this paper, after giving a description of the proposed method, experimental verifications on aluminum alloys to prove the effectiveness of the proposed method are presented. Data will be available on reasonable request.

Uniformly strained anisotropic elastoplastic rods: Determination of elastoplastic constitutive relations and yield surface in terms of rod’s variables

This work presents a formulation for the elastoplastic deformation of the rod’s cross-section wherein the rod material can obey an arbitrary three-dimensional stored energy density model together with a rate-independent arbitrary phenomenological plasticity model. In order to do so, we impose a uniform strain field along the rod’s arc-length which reduces the three-dimensional nonlinear elastoplastic rod problem to just its cross-section. To solve this elastoplastic cross-sectional deformation problem, a three-dimensional elastoplastic moduli tensor is first derived along with a return map algorithm for the update of relevant internal variables at every point in the cross-section. A finite element formulation is then presented to solve the aforementioned cross-sectional problem. The elastoplastic stress resultants (internal contact force and moment in the rod’s cross-section) and elastoplastic stiffnesses (such as bending, twisting, extensional and shearing) of an elastically isotropic but plastically transversely-isotropic rod are then obtained numerically along with the warped state of the plastified cross-section. Several deformations (increment/decrement in combinations of extension, torsion, bending and shearing) are studied to observe the evolution of plasticity at each quadrature point in the rod’s cross-section. We observe that off-axis anisotropy during plasticity leads to unusual out-of-plane warping even during torsion of circular cross-sections. It also leads to coupling between various modes such as between bending and twisting. We also demonstrate that the rod’s various elastoplastic stiffnesses are not only dependent on the current strain state of the rod but also on the current direction of strain rate. Finally, we integrate the plastic work over the rod’s cross-section and use the criterion of limiting plastic work to obtain discrete points on the rod’s yield surface in terms of stress resultants. The obtained yield surface turns out to be different from the one obtained for purely isotropic material model.