Micropolar hyperelasticity: constitutive model, consistent linearization and simulation of 3D scale effects

This study describes a computational framework for three-dimensional finite strain and finite curvature micropolar hyperelasticity. The model is based on the non-linear kinematic setting and features an appropriate hyperelastic material law which is derived within the thermodynamically consistent framework. The material tangent operator is obtained by consistent linearization. An implicit finite element method with a Newton-Raphson procedure is employed for the computation of the nodal displacements and rotations. A number of numerical examples is presented. The results demonstrate (i) that the methodology is capable of capturing 3D length scale effects in finite deformation, (ii) that it is robust and computationally efficient and (iii) that the proposed micropolar element tangent renders asymptotically quadratic convergence of the Newton-Raphson procedure. It is shown that the classical Neo-Hooke type material behaviour is recovered as a special case within the proposed micropolar setting.     

Multisurface plasticity for Cosserat materials: Plate element implementation and validation

The macroscopic behavior of materials is affected by their inner micro‐structure. Elementary considerations based on the arrangement, and the physical and mechanical features of the micro‐structure may lead to the formulation of elastoplastic constitutive laws, involving hardening/softening mechanisms and non‐associative properties. In order to model the non‐linear behavior of micro‐structured materials, the classical theory of time‐independent multisurface plasticity is herein extended to Cosserat continua. The account for plastic relative strains and curvatures is made by means of a robust quadratic‐convergent projection algorithm, specifically formulated for non‐associative and hardening/softening plasticity. Some important limitations of the classical implementation of the algorithm for multisurface plasticity prevent its application for any plastic surfaces and loading conditions. These limitations are addressed in this paper, and a robust solution strategy based on the singular value decomposition technique is proposed. The projection algorithm is then implemented into a finite element formulation for Cosserat continua. A specific finite element is considered, developed for micropolar plates. The element is validated through illustrative examples and applications, showing able performance. Copyright © 2016 John Wiley & Sons, Ltd.     

Implicit integration and consistent tangent modulus of a time‐dependent non‐unified constitutive model

This paper describes the implicit integration and consistent tangent modulus of an inelastic constitutive model with transient and steady strain rates, both of which are time‐ and temperature‐dependent; the transient rate is influenced by the evolution of back stress decomposed into parts, while the steady rate depends only on applied stress and temperature. Such a non‐unified model is useful for high‐temperature structural analysis and is practical owing to the ease in determining material constants. The implicit integration is shown to result in two scalar‐valued coupled equations, and the consistent tangent modulus is derived in a quite versatile form by introducing a set of fourth‐rank constitutive parameters into the discretized evolution rule of back stress. The constitutive model is, then, implemented in a finite element program and applied to a lead‐free solder joint analysis. It is demonstrated that the implicit integration is very accurate if the multilinear kinematic hardening model of Ohno and Wang is employed, and that the consistent tangent modulus certainly affords quadratic convergence to the Newton–Raphson iteration in solving nodal force equilibrium equations. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite deformation frictional mortar contact using a semi‐smooth Newton method with consistent linearization

A two‐dimensional, finite deformation frictional contact formulation with Coulomb's law is presented. The approach considers multibody contact and is based on a mortar formulation. The enforcement of contact constraints is realized with dual Lagrange multipliers. These alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase. Friction kinematic variables are formulated in an objective way and enter non‐smooth complementarity functions for expressing the contact constraints. An active set strategy is derived by applying a semi‐smooth Newton method, which treats contact nonlinearities, material and geometrical nonlinearities in one single iterative scheme. By further carrying out a consistent linearization for both normal and frictional contact forces and constraints, a robust and highly efficient algorithm for linear and higher‐order (quadratic) interpolation is achieved. Efficiency of the proposed method and quality of results are demonstrated in several examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Two‐scale shear band evolution by local partition of unity

We introduce a methodology to model shear band evolution in the quasi‐static regime using the extended finite element method. We enrich the finite element polynomial displacement field with a fine scale function, which models the high displacement gradient in the shear band. For this purpose we use a local partition of unity and a parameterized displacement enrichment based on closed form solutions for one‐dimensional shear bands. A stabilized consistent penalty method is used to circumvent locking in the regularized elasto‐viscoplastic plane‐strain regime and to guarantee element stability. The loss of stability of the boundary value problem is used as an indicator of shear band initiation point and direction. Shear band development examples are shown, illustrating the capabilities of the method to track shear band evolution and strains as high as 1000%. Copyright © 2005 John Wiley & Sons, Ltd.     

A thermo‐hydro‐mechanical model for multiphase geomaterials in dynamics with application to strain localization simulation

In this paper, the non‐isothermal elasto‐plastic behaviour of multiphase geomaterials in dynamics is investigated with a thermo‐hydro‐mechanical model of porous media. The supporting mathematical model is based on averaging procedures within the hybrid mixture theory. A computationally efficient reduced formulation of the macroscopic balance equations that neglects the relative acceleration of the fluids, and the convective terms is adopted. The modified effective stress state is limited by the Drucker–Prager yield surface. Small strains and dynamic loading conditions are assumed. The standard Galerkin procedure of the finite element method is applied to discretize the governing equations in space, while the generalized Newmark scheme is used for the time discretization. The final non‐linear set of equations is solved by the Newton method with a monolithic approach. Coupled dynamic analyses of strain localization in globally undrained samples of dense and medium dense sands are presented as examples. Vapour pressure below the saturation water pressure (cavitation) develops at localization in case of dense sands, as experimentally observed. A numerical study of the regularization properties of the finite element model is shown and discussed. A non‐isothermal case of incipient strain localization induced by temperature increase where evaporation takes place is also analysed. Copyright © 2016 John Wiley & Sons, Ltd.     

A 3D anisotropic elastoplastic‐damage model using discontinuous displacement fields

This paper is concerned with the development of constitutive equations for finite element formulations based on discontinuous displacement fields. For this purpose, an elastoplastic continuum model (stress–strain relation) as well as an anisotropic damage model (stress–strain relation) are projected onto a surface leading to traction separation laws. The coupling of both continuum models and, subsequently, the derivation of the corresponding constitutive interface law are described in detail. For a simple calibration of the proposed model, the fracture energy resulting from the coupled elastoplastic‐damage traction separation law is computed. By this, the softening evolution is linearly dependent on the fracture energy. The second part of the present paper deals with the numerical implementation. Based on a local and incompatible additive split of the displacement field into a continuous and a discontinuous part, the parameters specifying the jump of the displacement field are condensed out at the material level without employing the standard static condensation technique. To reduce locking effects, a rotating localization zone formulation is applied. The applicability and the performance of the proposed numerical implementation is investigated by means of a re‐analysis of a two‐dimensional L‐shaped slab as well as by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block. Copyright © 2004 John Wiley & Sons, Ltd.     

A 3D SMA constitutive model in the framework of finite strain

The aim of the present paper is to propose a phenomenological thermodynamically consistent 3D model for shape memory alloys (SMA) in the finite strain range. In particular, a model able to predict the main features of SMA materials, such as the superelastic and the shape‐memory effects, is proposed. The model is based on the assumption of the local multiplicative split of the deformation gradient into an elastic and a phase transformation part. The governing state and evolutive equations are written in the undeformed configuration. The material parameters of the model are characterized by a clear physical meaning so that they can be determined by simple experimental tests. The finite deformation SMA model is also reformulated in the framework of small strain, linearizing the strain and stress measures in order to obtain a consistent constitutive model preserving the nonlinear material response. A robust algorithm is adopted in order to integrate the nonlinear evolutive equations; 2D and 3D finite elements are implemented in a numerical code considering finite and small deformations. Some numerical applications are carried out showing the performances of the proposed model and the developed numerical procedure to describe the superelastic and the shape‐memory effects of SMA devices. Comparisons of different results obtained by the small and finite strain formulations are reported. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical integration of a pressure‐dependent,non‐linear kinematic hardening constitutive model for large strain cyclic plasticity of metals

A Gurson‐based constitutive model is presented, which includes non‐linear mixed isotropic–kinematic hardening and creep, and allows the analysis of problems involving arbitrarily large plastic strains. This model was developed with the main objective of allowing, on the basis of a single set of material parameters, the numerical simulation of all the main features of cold metal forming processes, which usually imply severe loading–unloading cycles with very large plastic strains, difficult to be correctly reproduced numerically. A suitable integration scheme of the rate equations is described and implemented into a finite element code. The results obtained are compared with some reference experimental ones; an application of the model for the simulation of wire drawing processes is also presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient numerical methods for the large‐scale,parallel solution of elastoplastic contact problems

Quasi‐static elastoplastic contact problems are ubiquitous in many industrial processes and other contexts, and their numerical simulation is consequently of great interest in accurately describing and optimizing production processes. The key component in these simulations is the solution of a single load step of a time iteration. From a mathematical perspective, the problems to be solved in each time step are characterized by the difficulties of variational inequalities for both the plastic behavior and the contact problem. Computationally, they also often lead to very large problems. In this paper, we present and evaluate a complete set of methods that are (1) designed to work well together and (2) allow for the efficient solution of such problems. In particular, we use adaptive finite element meshes with linear and quadratic elements, a Newton linearization of the plasticity, active set methods for the contact problem, and multigrid‐preconditioned linear solvers. Through a sequence of numerical experiments, we show the performance of these methods. This includes highly accurate solutions of a three‐dimensional benchmark problem and scaling our methods in parallel to 1024 cores and more than a billion unknowns. Copyright © 2015 John Wiley & Sons, Ltd.     

On the robustness and efficiency of integration algorithms for a 3D finite strain phenomenological SMA constitutive model

Most devices based on shape memory alloys experience large rotations and moderate or finite strains. This motivates the development of finite‐strain constitutive models together with the appropriate computational counterparts. To this end, in the present paper a three‐dimensional finite‐strain phenomenological constitutive model is investigated and a robust and efficient integration algorithm is proposed. Properly defining the variables, extensively used regularization schemes are avoided and a nucleation–completion criterion is defined. Moreover, introducing a logarithmic mapping, a new form of time‐discrete equations is proposed. The solution algorithm as well as a suitable initial guess for the resultant nonlinear equations are also deeply discussed. Extensive numerical tests are performed to show robustness as well as efficiency of the proposed integration algorithm. Implementation of the integration algorithm within a user‐defined subroutine UMAT in the commercial nonlinear finite element software ABAQUS/Standard makes also possible the solution of a variety of boundary value problems. The obtained results show the efficiency and robustness of the proposed approach and confirm the improved efficiency (in terms of solution CPU time) when a nucleation–completion criterion is used instead of regularization schemes, as well as when a logarithmic mapping is used for the time‐discrete evolution equation instead of an exponential mapping. Copyright © 2010 John Wiley & Sons, Ltd.     

A return‐mapping algorithm for plastic‐damage models: 3‐D and plane stress formulation

Three‐dimensional and plane stress formulations of the return‐mapping algorithm for a class of plastic‐damage models are derived using the spectral decomposition form of the stress. An efficient plane stress computation scheme based on the spectral return‐mapping algorithm is developed. The consistent algorithmic tangent stiffness for the present algorithm is formulated. The validation and performance of the present return‐mapping algorithm is demonstrated by numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.     

A novel 3D mixed finite‐element model for statics of angle‐ply laminates

Analysis of angle‐ply laminates becomes critical and computationally involved because of the presence of extension–shear coupling. A refined three‐dimensional, mixed, 18‐node finite element (FE) model has been developed to analyse angle‐ply laminates under static loading. The minimum potential energy principle has been used for the development of the mixed FE model, where the transverse stress components (τ_xz, τ_yz and σ_z, where z is the thickness direction) have been incorporated as the nodal degrees of freedom, in addition to the three displacement fields. Further, continuity of transverse stress fields through the thickness of the plate and layerwise continuity of displacement fields have been enforced in the formulation. Because all the constitutive and the compatibility conditions have been ensured within the continuum, the present formulation is unique amongst the family of mixed FE models. Results have been obtained for various angle‐ply laminates and compared with analytical and finite‐element solutions, which have been found to be in good agreement with them. Some new results on angle‐ply with clamped–clamped support condition have also been presented to serve as benchmark results. Copyright © 2003 John Wiley & Sons, Ltd.     

Control of analyses with isoparametric elements in both 2‐D and 3‐D

This paper presents an extension of the computation method for the error in the constitutive relation to include isoparametric elements. Developed over the past few years at ENS Cachan's laboratory, this technique is based on the strict building of admissible displacement and stress fields. This building process, validated for several types of 2‐D and 3‐D straight finite elements (triangles and quadrilaterals, tetrahedra and hexahedra), cannot be extended to isoparametric elements. For such elements, the method consists of seeking an approximation of the statically admissible stress field by solving a high‐degree finite element problem on each element. This technique, as implemented in our error computation code, which is associated both with a method of computing optimal sizes and with meshers able to respect a size map, allows us to optimize 2‐D and 3‐D meshes. Examples demonstrate the capabilities of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

Non‐planar 3D crack growth by the extended finite element and level sets—Part I: Mechanical model

A methodology for solving three‐dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended finite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The branch functions include asymptotic near‐tip fields that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be defined by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three‐dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems. Copyright © 2002 John Wiley & Sons, Ltd.     

A new multi‐scale dispersive gradient elasticity model with micro‐inertia: Formulation and ‐finite element implementation

Motivated by nano‐scale experimental evidence on the dispersion characteristics of materials with a lattice structure, a new multi‐scale gradient elasticity model is developed. In the framework of gradient elasticity, the simultaneous presence of acceleration and strain gradients has been denoted as dynamic consistency. This model represents an extension of an earlier dynamically consistent model with an additional micro‐inertia contribution to improve the dispersion behaviour. The model can therefore be seen as an enhanced dynamic extension of the Aifantis' 1992 strain‐gradient theory for statics obtained by including two acceleration gradients in addition to the strain gradient. Compared with the previous dynamically consistent model, the additional micro‐inertia term is found to improve the prediction of wave dispersion significantly and, more importantly, requires no extra computational cost. The fourth‐order equations are rewritten in two sets of symmetric second‐order equations so that ‐continuity is sufficient in the finite element implementation. Two sets of unknowns are identified as the microstructural and macrostructural displacements, thus highlighting the multi‐scale nature of the present formulation. The associated energy functionals and variationally consistent boundary conditions are presented, after which the finite element equations are derived. Considerable improvements over previous gradient models are observed as confirmed by two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Detonation gas model for combined finite‐discrete element simulation of fracture and fragmentation

The equation of state for expansion of detonation gas together with a model for gas flow through fracturing solid is proposed and implemented into the combined finite‐discrete element code. The equation of state proposed enables gas pressure to be obtained in a closed form for both reversible and irreversible adiabatic expansion, while the gas flow model proposed considers only 1D compressible flow through cracks, hence avoiding full 2D or 3D gas flow through the fracturing solid. When coupled with finite‐discrete element algorithms for solid fracture and fragmentation, the model proposed enables gas pressure to be predicted and energy balance to be preserved. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element implementation of non‐linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control

Implicit stress integration algorithms have been demonstrated to provide a robust formulation for finite element analyses in computational mechanics, but are difficult and impractical to apply to increasingly complex non‐linear constitutive laws. This paper discusses the performance of fully explicit local and global algorithms with automatic error control used to integrate general non‐linear constitutive laws into a non‐linear finite element computer code. The local explicit stress integration procedure falls under the category of return mapping algorithm with standard operator split and does not require the determination of initial yield or the use of any form of stress adjustment to prevent drift from the yield surface. The global equations are solved using an explicit load stepping with automatic error control algorithm in which the convergence criterion is used to compute automatically the coarse load increment size. The proposed numerical procedure is illustrated here through the implementation of a set of elastoplastic constitutive relations including isotropic and kinematic hardening as well as small strain hysteretic non‐linearity. A series of numerical simulations confirm the robustness, accuracy and efficiency of the algorithms at the local and global level. Published in 2001 by John Wiley & Sons, Ltd.     

Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities

Gmsh is an open‐source 3‐D finite element grid generator with a build‐in CAD engine and post‐processor. Its design goal is to provide a fast, light and user‐friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright © 2009 John Wiley & Sons, Ltd.