Elastoplastic Problems Research Articles

A GPU-based framework for finite element analysis of elastoplastic problems

A GPU-based framework for finite element analysis of elastoplastic problems

A robust stress update algorithm for elastoplastic models without analytical derivation of the consistent tangent operator and loading/unloading estimation

AbstractA robust and concise implicit stress integration algorithm of elastoplastic models is presented. It does not require the loading/unloading estimation and analytical derivation operation for the stress update. First, the elastoplastic stress update problem is recast into an unconstrained minimization problem by utilizing the smooth function to bypass the loading/unloading estimation. Then, the object problem is solved by the line search method instead of the Newton method for better convergence. The consistent tangent operator is evaluated by the complex step derivative approximation without the subtraction cancellation error, which provides the quadratic convergence rate of global iteration. The rationality of the numerical consistent tangent operator is validated by the one obtained by the analytical derivation. A recently developed non‐orthogonal elastoplastic (NEP) clay model is implemented using the new algorithm. The algorithm is confirmed through comparing the numerical solution and the analytical one for a cavity expansion problem. The algorithm performance is assessed based on a series of geotechnical boundary value problems. It is found that the new algorithm is more robust than the one employed by ABAQUS. The source code of the model implementation can be downloaded from https://github.com/zhouxin615.

A second-order reduced homogenization for nonlinear structures with periodic configurations in cylindrical coordinates

A high-efficiency second-order reduced homogenization (SORH) method is established for the inelastic problem of periodic heterogeneous structures in cylindrical coordinate. The composite structures investigated in this paper are periodical in radial, axial and circumferential directions. The new second-order linear/nonlinear local solutions at microscale are given by introducing various multiscale auxiliary functions. And, the nonlinear high-order homogenization solutions are obtained at macroscale using a modified asymptotic expansion method. The significant novelties of the work are (i) an effective reduced model on the basis of transformation field analysis given to solve the nonlinear multiscale problems in cylindrical coordinates with less computational amount and (ii) a new second-order nonlinear multiscale algorithm established using asymptotic expansion technique for simulating the heterogeneous cylindrical structure. Finally, by three typical numerical examples including nonlinear elastic and elasto-plastic periodic problems, the validity and accuracy of the proposed algorithms are verified.

A globally convergent smoothing Newton method for the second order cone complementarity approach of elastoplasticity problems

A new algorithm for solving small-deformation elastoplasticity problems as second order cone complementarity problems (SOCCPs) is presented. The SOCCP is derived from the Hellinger–Reissner variational principle, and then solved with a smoothing Newton method that is modified from the globally convergent Qi-Sun-Zhou method. The method is compared with two well-established methods: the return-mapping method and the conic programming (CP) approach with primal–dual interior-point method. Numerical results indicate that the proposed method shares common characteristics of CP class methods and offers better convergence efficiency in some cases. Moreover, it is shown that the proposed method can be imbedded in the framework of conventional finite element codes with only a few modifications, which makes it easy-to-implement and adaptive to large-scale parallel computing.

A deep learning energy-based method for classical elastoplasticity

The deep energy method (DEM) has been used to solve the elastic deformation of structures with linear elasticity, hyperelasticity, and strain-gradient elasticity material models based on the principle of minimum potential energy. In this work, we extend DEM to elastoplasticity problems involving path dependence and irreversibility. A loss function inspired by the discrete variational formulation of plasticity is proposed. The radial return algorithm is coupled with DEM to update the plastic internal state variables without violating the Kuhn–Tucker consistency conditions. Finite element shape functions and their gradients are used to approximate the spatial gradients of the DEM-predicted displacements, and Gauss quadrature is used to integrate the loss function. Four numerical examples are presented to demonstrate the use of the framework, such as generating stress–strain curves in cyclic loading, material heterogeneity, performance comparison with other physics-informed methods, and simulation/inference on unstructured meshes. In all cases, the DEM solution shows decent accuracy compared to the reference solution obtained from the finite element method. The current DEM model marks the first time that energy-based physics-informed neural networks are extended to plasticity, and offers promising potential to effectively solve elastoplasticity problems from scratch using deep neural networks.

Arbitrary polygon mesh for elastic and elastoplastic analysis of solids using smoothed finite element method

This work extends the smoothed finite elements method (S-FEM) to the use of arbitrarily complicated meshes, including nonconvex polygonal meshes for the first time. A novel formulation for arbitrarily polygonal elements based on the S-FEM is presented for analyses of elastic and elastoplastic problems. This enable the use of polygonal elements that are of conventional quadrilateral, convex polygonal, concave polygon elements and n-sides concave polygonal elements. This is achieved by making use of the exceptional unique features of S-FEM in terms of stability, accommodation to mesh distortion, and free from differentiation in strain field computation. The ear clipping technique is used to automatically generate smoothing domains without any dummy nodes. The stability in using such a complicated mesh is ensured by the gradient smoothing technique in S-FEM, and no additional stability control measure is required. In addition, domain integration is converted to boundary integral along sub-triangular domain, which requires only the shape function values at the field nodes. This avoiding coordinate mapping, and hence it is a Jacobian free formulation. The von Mises elastoplastic model is implemented in our S-FEM by decomposing the tangent stiffness matrix into three parts: elastic stiffness matrix, strain–displacement matrix and the derivative of the stress–strain matrix. The first two matrices are calculated only once at the initial stage, and only the derivative of the stress–strain matrix is updated in each iteration for capture the nonlinearity. The numerical results show that the proposed method can handle the general non-convex elements effectively with good accuracy and robustness. It offers an effective tool in dealing with polycrystalline metallic materials with crystal grains of arbitrary shapes.

A neural network-based data-driven local modeling of spotwelded plates under impact

Solving large structural problems with multiple complex localized behaviors is extremely challenging. To address this difficulty, both intrusive and non-intrusive Domain Decomposition Methods (DDM) have been developed in the past, where the refined model (local) is solved separately in its own space and time scales. In this work, the Finite Element Method (FEM) at the local scale is replaced with a data-driven Reduced Order Model (ROM) to further decrease computational time. The reduced model aims to create a low-cost, accurate and efficient mapping from interface velocities to interface forces and enable the prediction of their time evolution. The present work proposes a modeling technique based on the Physics-Guided Architecture of Neural Networks (PGANNs), which incorporates physical variables other than input/output variables into the neural network architecture. We develop this approach on a 2D plate with a hole as well as a 3D case with spot-welded plates undergoing fast deformation, representing nonlinear elastoplasticity problems. Neural networks are trained using simulation data generated by explicit dynamic FEM solvers. The PGANN results are in good agreement with the FEM solutions for both test cases, including those in the training dataset as well as the unseen dataset, given the loading type is present in the training set.

Topology optimization for elasto-plastic problems based on exact volume constraint method

Topology optimization for elasto-plastic problems based on exact volume constraint method

Explicit phase-field material point method with the convected particle domain interpolation for impact/contact fracture in elastoplastic geomaterials

A novel explicit phase-field material point method with the convected particle domain interpolation (ePF-CPDI) is proposed for solving large deformation dynamic impact/contact fracture problems in elastoplastic geomaterials. In this method, we derive an explicit rate-dependent phase-field fracture model relying on the microforce balance law and the second law of thermodynamics. A coupled explicit phase-field plasticity model is then developed to describe dynamic elastoplastic fracture responses of geomaterials. Here the explicit time integration strategy and the staggered solution scheme are utilized in this work to solve the coupled-field governing equations based on the material point method. To eliminate the numerical noises caused by material points crossing cell boundaries, the convected particle domain interpolation technique is adopted to improve the computational accuracy in large deformation simulation. Furthermore, the proposed approach combining with the particle to particle contact algorithm is extended to deal with more complicated high-velocity impact and multi-body contact elastoplastic fracture problems. Numerical experiments are employed to validate the high accuracy and excellent capability of the proposed method and discuss the influences of main parameters on the phase-field fracture modeling.

Convergence acceleration by Padé approximants in the ANM algorithm for large strain elasto-plasticity

In this work, we show how to introduce improved vectorial Padé approximants in the Asymptotic Numerical Method (ANM) for solving elasto-plasticity problems in finite transformation. In this way, we have proposed an algorithm, based on Taylor series which has shown its efficiency for computing elasto-plastic structures in large deformations. We show, in this paper, how to accelerate the convergence of ANM using a new class of vectorial Padé approximants instead of the Taylor series in the case of finite elasto-plasticity. The efficiency and robustness of the proposed algorithm are tested on 2 D examples. These examples show that some vectorial Padé approximants allow to reduce by 30% the number of tangent matrices. • The paper shows how to introduce improved vectorial Padé approximants in the Asymptotic Numerical Method (ANM) for solving elasto-plasticity problems in finite transformation. • Work shows how to accelerate the convergence of ANM using new Padé approximants. • Two examples show that some vectorial Padé approximants allow to reduce by 30% the number of tangent matrices.

LOCAL INDENTATION AS METHOD OF REDUCING FATIGUE CRACK GROWTH RATE

The possibility of a significant decrease in the growth rate of a fatigue crack is studied. The goal can be attained by creating a local field of residual stresses near the crack tip which arises due to the indentation of a spherical indenter. A methodology and program algorithm (in ANSYS) have been developed for numerical simulation of the problem in a three-dimensional formulation of the process of fatigue crack growth in the field of residual stresses. Considering the prospects of developing the technique with regard to the use of dynamic indentation, the ANSYS Explicit STR solver fully integrated into the calculation module was used in the macro. Proceeding from the solution of the elastoplastic problem the program provides determination of the fields of residual stresses (RS) at the first stage of the calculation and numerical simulation of the fatigue crack growth on the second stage. The effect of the indentation parameters (magnitude of the force applied to the indenter and the location of the indentation point, conditions of fixing) on the crack growth rate is considered. Methods for a significant reduction in the crack growth rate based on multiple preliminary indentation are substantiated. Using the developed program, we managed to solve a series of problems regarding the effect of different types of loading and fastenings on the growth rate of a fatigue crack in a plate with a through crack. The use of one-sided indentation of thin-walled objects with cracks, which are installed on the support surface, greatly simplifies the practical application of the technique for creating a residual stress field in the vicinity of the crack tip. It is shown that with multiple indentation along the crack propagation line, the fatigue crack growth rate is significantly lower than that when using a single indenter.

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.

A peridynamic approach to solving general discrete dislocation dynamics problems in plasticity and fracture: Part II. Applications

In Part II of this work we extend the method introduced in Part I to consider dislocation dynamic evolution through dislocations nucleation, glide, pile-up, and annihilation. The SP DDD-PD scheme is employed to investigate uniaxial tension in a single crystal and a polycrystal and verify its accuracy. The model is then used to simulate elastoplastic fracture by considering interactions between dislocations and crack growth. For Mode I elastoplastic fracture in a single crystal, we observe that the crack path is “attracted” towards regions of high density of gliding dislocations, leading to an undulating crack paths, as observed in experiments but never replicated by continuum-level computational models before. Tests on different sample sizes show how the proximity of constraints to the crack tip can lead to plastic hardening. Ductile-to-brittle transition happens naturally in this model when the crack, under Mode I displacement-controlled loading, approaches a free edge. A new way to calibrate the critical bond strain based on the material toughness or fracture energy is proposed. The present SP DDD-PD scheme can be used to investigate complicated elastoplastic fracture problems in which the interaction between dislocation motion and damage is critical.

Fast solution of elasto-plastic problems by reduced order finite element method with manifold learning

Solution of elasto-plastic problems is often computationally intensive. The most effective way to accelerate the computation is to reduce the number of the degree of freedom of the model. Considering the fact that constitutive relations of solid materials are very diverse and usually quite complicated, the paper investigates a model order reduction technique which can be embedded into existing finite element frameworks and reuse their constitutive models. The semi-intrusive technique can thus be used to solve nontrivial problems. Two typical models, i.e., the perforated plate under tensile load and the pressure vessel head under cyclic load, are tested to evaluate the accuracy and the efficiency of the reduced order model. It is shown that a speedup of around two orders of magnitude can be achieved by the reduced order model while maintaining the same level of accuracy. Moreover, the requirements for processor and memory of the computer are also significantly reduced.

Anisotropic multi-element polynomial chaos expansion for high-dimensional non-linear structural problems

This work presents an hp-adaptive variant of multi-element polynomial chaos expansion (ME-gPCE), referred to as anisotropic multi-element polynomial chaos expansion (AME-gPCE). The main advantage of the proposed framework is that the basis functions of the local gPCE are selected adaptively within each local element. The p-adaptivity allows the order of the local uni-variate polynomials to be increased only for the dimensions associated with highly non-linear behaviour of the response surface, while the h-adaptivity allows the mesh refinement to be applied only where the local response surface behaves non-smoothly. Thus, the computational cost can be significantly reduced compared with standard methods, which opens the door to high-dimensional problems. The efficiency, accuracy and convergence of AME-gPCE are studied numerically and compared with the traditional ME-gPCE for non-linear structural analysis. It outperforms the traditional ME-gPCE for the elasto-plastic problem, considering the material proprieties as random variable and random fields.

Simultaneous identification of multi-parameter for power hardening elastoplastic problems in three-dimensional geometries

PurposeThe purpose of this study is to simultaneously determine the constitutive parameters and boundary conditions by solving inverse mechanical problems of power hardening elastoplastic materials in three-dimensional geometries.Design/methodology/approachThe power hardening elastoplastic problem is solved by the complex variable finite element method in software ABAQUS, based on a three-dimensional complex stress element using user-defined element subroutine. The complex-variable-differentiation method is introduced and used to accurately calculate the sensitivity coefficients in the multiple parameters identification method, and the Levenberg–Marquardt algorithm is applied to carry out the inversion.FindingsNumerical results indicate that the complex variable finite element method has good performance for solving elastoplastic problems of three-dimensional geometries. The inversion method is effective and accurate for simultaneously identifying multi-parameters of power hardening elastoplastic problems in three-dimensional geometries, which could be employed for solving inverse elastoplastic problems in engineering applications.Originality/valueThe constitutive parameters and boundary conditions are simultaneously identified for power hardening elastoplastic problems in three-dimensional geometries, which is much challenging in practical applications. The numerical results show that the inversion method has high accuracy, good stability, and fast convergence speed.

The extended peridynamic model for elastoplastic and/or fracture problems

AbstractThe strain‐based implementation method for the extended peridynamic model (XPDM) resolves the limitation of standard models where only a fixed Poisson's ratio can be achieved. In this contribution, the XPDM formulation is extended to include bond breakage and/or plasticity mechanisms. The elastoplastic and bond breakage algorithms are elaborated. To capture the fracture process, a shear mechanism is now incorporated to the bond breakage response, in addition to the standard stretching failure mode. It is shown that the shear mechanism is required to accurately reproduce mixed mode fracture behavior observed experimentally. To demonstrate the predictive ability of the strain‐based XPDM, a wide range of quasi‐static and dynamic loading conditions, for both brittle and elasto‐plastic materials, is considered against experimental results or practical engineering scenarios.

Revisiting the Persson Theory of Elastoplastic Contact: A Simpler Closed-Form Solution and a Rigorous Proof of Boundary Conditions

Persson’s theory of contact is extensively used in the study of the purely normal interaction between a nominally flat rough surface and a rigid flat. In the literature, Persson’s theory was successfully applied to the elastoplastic contact problem with a scale-independent hardness H. However, it yields a closed-form solution, \(P(p, \xi )\), in terms of an infinite sum of sines. In this study, \(P(p, \xi )\) is found to have a simpler form which is a superposition of three Gaussian functions. A rigorous proof of the boundary condition \(P(p=0, \xi )=P(p=H, \xi ) = 0\) is given based on the new solution.

Torsional deformation of an infinite elastic solid weakened by a penny-shaped crack in the presence of surface elasticity and Dugdale plastic zone

We consider the torsional deformation of an infinite three-dimensional isotropic elastic solid weakened by a penny-shaped crack whose boundary is enhanced by the incorporation of surface elasticity. Furthermore, we assume the presence of a Dugdale-type plastic zone around the crack front. The ensuing elastoplastic problem gives rise to a non-standard mixed boundary value problem which is then reduced to a hypersingular integral equation upon application of the Hankel transform. The hypersingular integral equation is subsequently solved numerically using orthogonal expansions revealing the effects of surface elasticity on the size of the plastic zone and crack-face torsional displacements. Our results demonstrate the effect of surface elasticity on the width of the plastic zone thus providing a significant shielding effect while increasing the load-bearing capacity of the solid in the vicinity of the defect. Our conclusions will be of particular interest when the presence of surface elasticity along a crack boundary is used in enhanced continuum-based models to study fracture at micro- and nano-scales, for example, in the prediction of fracture and failure in micro- and nano-structures.

Three-Dimensional Boundary Element Strategy for Stress Sensitivity of Fractional-Order Thermo-Elastoplastic Ultrasonic Wave Propagation Problems of Anisotropic Fiber-Reinforced Polymer Composite Material.

A new three-dimensional (3D) boundary element method (BEM) strategy was developed to solve fractional-order thermo-elastoplastic ultrasonic wave propagation problems based on the meshless moving least squares (MLS) method. The temperature problem domain was divided into a number of circular sub-domains. Each node was the center of the circular sub-domain surrounding it. The Laplace transform method was used to solve the temperature problem. A unit test function was used in the local weak-form formulation to generate the local boundary integral equations, and the inverse Laplace transformation method was used to find the transient temperature solutions. Then, the three-dimensional elastoplastic problems could be solved using the boundary element method (BEM). Initial stress and strain formulations are adopted, and their distributions are interpolated using boundary integral equations. The effects of the fractional-order parameter and anisotropy are investigated. The proposed method’s validity and performance are demonstrated for a two-dimensional problem with excellent agreement with other experimental and numerical results.