An efficient augmented finite element method for arbitrary cracking and crack interaction in solids

This paper presents an augmentation method that enables bilinear finite elements to efficiently and accurately account for arbitrary, multiple intra‐elemental discontinuities in 2D solids. The augmented finite element method (A‐FEM) employs four internal nodes to account for the crack displacements due to an intra‐elemental weak or strong discontinuity, and it permits repeated elemental augmentation to include multiple interactive cracks. It thus enables a unified treatment of damage evolution from a weak discontinuity to a strong discontinuity, and to multiple interactive cohesive cracks, all within a single bilinear element that employs standard external nodal DoFs only. A novel elemental condensation procedure has been developed to solve the internal nodal DoFs as functions of the external nodal DoFs for any irreversible, piece‐wise linear cohesive laws. It leads to a fully condensed elemental equilibrium equation with mathematical exactness, eliminating the need for nonlinear equilibrium iterations at elemental level. The new A‐FEM's high‐fidelity simulation capabilities to interactive cohesive crack formation and propagation in homogeneous, and heterogeneous solids have been demonstrated through several challenging numerical examples. It is shown that the proposed A‐FEM, empowered by the new elemental condensation procedure, is numerically very efficient, accurate, and robust. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical simulation of dynamic fracture using finite elements with embedded discontinuities

This paper presents the extension of some finite elements with embedded strong discontinuities to the fully transient range with the focus on dynamic fracture. Cracks and shear bands are modeled in this setting as discontinuities of the displacement field, the so-called strong discontinuities, propagating through the continuum. These discontinuities are embedded into the finite elements through the proper enhancement of the discrete strain field of the element. General elements, like displacement or assumed strain based elements, can be considered in this framework, capturing sharply the kinematics of the discontinuity for all these cases. The local character of the enhancement (local in the sense of defined at the element level, independently for each element) allows the static condensation of the different local parameters considered in the definition of the displacement jumps. All these features lead to an efficient formulation for the modeling of fracture in solids, very easily incorporated in an existing general finite element code due to its modularity. We investigate in this paper the use of this finite element formulation for the special challenges that the dynamic range leads to. Specifically, we consider the modeling of failure mode transitions in ductile materials and crack branching in brittle solids. To illustrate the performance of the proposed formulation, we present a series of numerical simulations of these cases with detailed comparisons with experimental and other numerical results reported in the literature. We conclude that these finite element methods handle well these dynamic problems, still maintaining the aforementioned features of computational efficiency and modularity.     

Phase‐field modeling of ductile fracture at finite strains: A robust variational‐based numerical implementation of a gradient‐extended theory by micromorphic regularization

This work provides a robust variational‐based numerical implementation of a phase field model of ductile fracture in elastic–plastic solids undergoing large strains. This covers a computationally efficient micromorphic regularization of the coupled gradient plasticity‐damage formulation. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. It has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture‐specific computational structures such as finite element design of crack discontinuities or intricate crack‐tracking algorithms. The proposed gradient‐extended plasticity‐damage formulation includes two independent length scales that regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones or vice versa and guarantees on the computational side a mesh objectivity in post‐critical ranges. The proposed setting is rooted in a canonical variational principle. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient‐extended internal variables that enter plastic yield functions and fracture threshold functions. With this viewpoint on the generalized internal variables at hand, the thermodynamic formulation is outlined for gradient‐extended dissipative solids with generalized internal variables that are passive in nature. It is specified for a conceptual model of von Mises‐type elasto‐plasticity at finite strains coupled with fracture. The canonical theory proposed is shown to be governed by a rate‐type minimization principle, which fully determines the coupled multi‐field evolution problem. This is exploited on the numerical side by a fully symmetric monolithic finite element implementation. An important aspect of this work is the regularization towards a micromorphic gradient plasticity‐damage setting by taking into account additional internal variable fields linked to the original ones by penalty terms. This enhances the robustness of the finite element implementation, in particular, on the side of gradient plasticity. The performance of the formulation is demonstrated by means of some representative examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Three‐dimensional finite elements with embedded strong discontinuities to model material failure in the infinitesimal range

This paper presents new three‐dimensional finite elements with embedded strong discontinuities in the small strain infinitesimal range. The goal is to model localized surfaces of failure in solids, such as cracks at fracture, through enhancements of the finite elements that capture the propagating discontinuities of the displacement field in the element interiors. In this way, such surfaces of discontinuity can be sharply resolved in general meshes not necessarily related to the detailed geometry of the surface, unknown a priori. An important issue is also the consideration of general finite element formulations in the developments (e.g., basic displacement‐based, mixed or enhanced assumed strain finite element formulations), as needed to optimally resolve the continuum problem in the bulk. The actual modeling of the discontinuity effects, including the incorporation of the cohesive law defining the discontinuity constitutive response, is carried out at the element level with the proper enhancement of the discrete strain field of the element. The added elemental degrees of freedom approximate the displacement jumps associated with the discontinuity and are defined independently from element to element, thus allowing their static condensation at the element level without affecting the global mechanical problem in terms of the number and topology of the global degrees of freedom. In fact, this global‐local structure of the finite element methods developed in this work arises naturally from a multi‐scale characterization of these localized solutions, with the discontinuities understood to appear in the small scales, thus leading directly to these computationally efficient numerical methods for their numerical resolution, easily incorporated to an existing finite element code. The focus in this paper is on the development of finite elements incorporating a linear interpolation of the displacement jumps in the general three‐dimensional setting. These interpolations are shown to be necessary for hexahedral elements to avoid the so‐called stress locking that occurs with simpler constant approximations of the jumps (namely, a spurious transfer of stresses across the discontinuity not allowing its full release and, hence, resulting in an overstiff or locked numerical solution). The design of the new finite elements is accomplished in this work by a direct identification of the separation modes to be incorporated in the discrete strain field of the element, rather than from an assumed discontinuous interpolation of the displacements, assuring with this approach their locking‐free response by design. An additional issue addressed in the paper is the geometric characterization and propagation of the discontinuity surfaces in the general three‐dimensional setting of interest here. The paper includes a series of numerical simulations illustrating and evaluating the properties of the new finite elements. Copyright © 2012 John Wiley & Sons, Ltd.     

Strong discontinuities in antiplane/torsional problems of computational failure mechanics

This paper considers the analysis of localized failures and fracture of solids under antiplane conditions. We consider the longitudinal cracking of shafts in torsion, with the crack propagating through the cross section, besides pure antiplane problems (that is, with loading perpendicular to the plane of analysis). The main goal is the theoretical characterization and the numerical resolution of strong discontinuities in this setting, that is, discontinuities of the antiplane displacement field modeling the cracks. A multi-scale framework is considered, by which the discontinuities are treated locally in the (global) antiplane mechanical boundary value problem of interest, incorporating effectively the contributions of the discontinuities to the failure of the solid. We can identify among these contributions, besides the change in the stiffness of the solid or structural member, the localized energy dissipation associated to the cohesive law governing the physical response of the discontinuity surfaces. A main outcome of this approach is the development of new finite elements with embedded discontinuities for the antiplane problem that capture these solutions, and physical effects, locally at the element level. This local structure allows the static condensation at the element level of the degrees of freedom considered in the approximation of the antiplane displacement jumps along the discontinuity. In this way, the new elements result not only in a cost efficient computational tool of analysis for these problems, but also in a technique that can be easily incorporated in an existing finite element code, while resolving objectively those physical dissipative effects along the localized surfaces of failure. We develop, in particular, quadrilateral finite elements with the embedded discontinuities exhibiting constant and linear approximations of the displacement jumps, showing the superior performance of the latter given the stress locking associated with quadrilateral elements with constant jumps only. This limitation manifests itself in spurious transfers of stresses across the discontinuities, leading to severe oscillations in the stress field and an overall excessively stiff solution of the problem. These features are illustrated with several numerical examples, including convergence tests and validations with analytical results existing in the literature, showing in the process the treatment of characteristic situations like snap-backs, commonly encountered in the modeling of these structural members at failure.     

弹塑性杆非线性断裂问题的新型增强有限元法

针对一维弹塑性材料杆件的非线性断裂,该文提出了一种新型弹塑性增强有限元。该单元采用von Mises屈服准则和线性等向硬化模型描述开裂前的材料弹塑性变形,而结合利用内聚力关系来描述随后的裂纹萌生和非线性断裂过程。引入内部节点来描述单元内由于裂纹引起的位移不连续,通过单元凝聚获得含裂纹单元的刚度矩阵,并对数值稳定性问题进行了分析。通过与基于材料力学方法推导得到的解析解的对比验证了该新型弹塑性增强有限单元法在列式上的正确性和在数值上的高效性与精确性。     

Cohesive surface model for fracture based on a two-scale formulation: computational implementation aspects

The paper describes the computational aspects and numerical implementation of a two-scale cohesive surface methodology developed for analyzing fracture in heterogeneous materials with complex micro-structures. This approach can be categorized as a semi-concurrent model using the representative volume element concept. A variational multi-scale formulation of the methodology has been previously presented by the authors. Subsequently, the formulation has been generalized and improved in two aspects: (i) cohesive surfaces have been introduced at both scales of analysis, they are modeled with a strong discontinuity kinematics (new equations describing the insertion of the macro-scale strains, into the micro-scale and the posterior homogenization procedure have been considered); (ii) the computational procedure and numerical implementation have been adapted for this formulation. The first point has been presented elsewhere, and it is summarized here. Instead, the main objective of this paper is to address a rather detailed presentation of the second point. Finite element techniques for modeling cohesive surfaces at both scales of analysis (FE\(^2\) approach) are described: (i) finite elements with embedded strong discontinuities are used for the macro-scale simulation, and (ii) continuum-type finite elements with high aspect ratios, mimicking cohesive surfaces, are adopted for simulating the failure mechanisms at the micro-scale. The methodology is validated through numerical simulation of a quasi-brittle concrete fracture problem. The proposed multi-scale model is capable of unveiling the mechanisms that lead from the material degradation phenomenon at the meso-structural level to the activation and propagation of cohesive surfaces at the structural scale.     

A marching cubes based failure surface propagation concept for three‐dimensional finite elements with non‐planar embedded strong discontinuities of higher‐order kinematics

This paper presents an advanced failure surface propagation concept based on the marching cubes algorithm initially proposed in the field of computer graphics and applies it to the embedded finite element method. When modeling three‐dimensional (3D) solids at failure, the propagation of the failure surface representing a crack or shear band should not exhibit a strong sensitivity to the details of the finite element discretization. This results in the need for a propagation of the discrete failure zone through the individual finite elements, which is possible for finite elements with embedded strong discontinuities. Whereas for two‐dimensional calculations the failure zone propagation location is easily predicted by the maximal principal stress direction, more advanced strategies are needed to achieve a smooth failure surface in 3D simulations. An example for such method is the global tracking algorithm, which predicts the crack path by a scalar level set function computed on the basis of the solution of a simplified heat conduction like problem. Its prediction may though lead to various scenarios on how the failure surface may propagate through the individual finite elements. In particular, for a hexahedral eight‐node finite element, 256 such cases exist. To capture all those possibilities, the marching cubes algorithm is combined with the global tracking algorithm and the finite elements with embedded strong discontinuities in this work. In addition, because many of the possible cases result in non‐planar failure surfaces within a single finite element and because the local quantities used to describe the kinematics of the embedded strong discontinuities are physically meaningful in a strict sense only for planar failure surfaces, a remedy for such scenarios is proposed. Various 3D failure propagation simulations outline the performance of the proposed concept. Copyright © 2013 John Wiley & Sons, Ltd.     

On enrichment functions in the extended finite element method

This paper presents mathematical derivation of enrichment functions in the extended finite element method for numerical modeling of strong and weak discontinuities. The proposed approach consists in combining the level set method with characteristic functions as well as domain decomposition and reproduction technique. We start with the simple case of a triangular linear element cut by one interface across which displacement field suffers a jump. The main steps towards the derivation of enrichment functions are as follows: (1) extension of the subfields separated by the interface to the whole element domain and definition of complementary nodal variables; (2) construction of characteristic functions for describing the geometry and physical field; (3) determination of the sets of basic nodal variables; (4) domain decompositions according to Step 3 and then reproduction of the physical field in terms of characteristic functions and nodal variables; and (5) comparison of the piecewise interpolations formulated at Steps 3 and 4 with the standard extended finite element method form, which yields enrichment functions. In this process, the physical meanings of both the basic and complementary nodal variables are clarified, which helps to impose Dirichlet boundary conditions. Enrichment functions for weak discontinuities are constructed from deeper insights into the structure of the functions for strong discontinuities. Relationships between the two classes of functions are naturally established. Improvements upon basic enrichment functions for weak discontinuities are performed so as to achieve satisfactory convergence and accuracy. From numerical viewpoints, a simple and efficient treatment on the issue of blending elements is also proposed with implementation details. For validation purposes, applications of the derived functions to heterogeneous problems with imperfect interfaces are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Variable-node elements for non-matching meshes by means of MLS (moving least-square) scheme

A new class of finite elements is described for dealing with non-matching meshes, for which the existing finite elements are hardly efficient. The approach is to employ the moving least-square (MLS) scheme to devise a class of elements with an arbitrary number of nodal points on the parental domain. This approach generally leads to elements with the rational shape functions, which significantly extends the function space of the conventional finite element method. With a special choice of the nodal points and the base functions, the method results in useful elements with the polynomial shape functions for which the C¹ continuity breaks down across the boundaries between the subdomains comprising one element. The present scheme possesses an extremely high potential for applications which deal with various problems with discontinuities, such as material inhomogeneity, crack propagation, phase transition and contact mechanics. The effectiveness of the new elements for handling the discontinuities due to non-matching interfaces is demonstrated using appropriate examples. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical implementation of imperfect interfaces

One approach to characterizing interfacial stiffness is to introduce imperfect interfaces that allow displacement discontinuities whose magnitudes depend on interfacial traction and on properties of the interface or interphase region. This work implemented such imperfect interfaces into both finite element analysis and the material point method. The finite element approach defined imperfect interface elements that are compatible with static, linear finite element analysis. The material point method interfaces extended prior contact methods to include interfaces with arbitrary traction-displacement laws. The numerical methods were validated by comparison to new or existing stress transfer models for composites with imperfect interfaces. Some possible experiments for measuring the imperfect interface parameters needed for modeling are discussed.     

A finite element method with mesh‐separation‐based approximation technique and its application in modeling crack propagation with adaptive mesh refinement

This paper presents a FEM with mesh‐separation‐based approximation technique that separates a standard element into three geometrically independent elements. A dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation. The novel technique makes it very easy for mesh generation of problems with complex or solution‐dependent, varying geometry. It offers a flexible way to construct displacement approximations and provides a unified framework for the FEM to enjoy some of the key advantages of the Hansbo and Hansbo method, the meshfree methods, the semi‐analytical FEMs, and the smoothed FEM. For problems with evolving discontinuities, the method enables the devising of an efficient crack‐tip adaptive mesh refinement strategy to improve the accuracy of crack‐tip fields. Both the discontinuities due to intra‐element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level. The effectiveness and robustness of the present method are benchmarked with several numerical examples. The numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test, and it is essential to apply local adaptive mesh refinement for low fracture energy problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Computational and experimental study of high-speed impact of metallic Taylor cylinders

High-speed impact of metallic Taylor cylinders is investigated computationally and experimentally. On the computational side, a modular explicit finite element hydrocode based on updated Lagrangian formulation is developed. A non-classical contour integration is employed to calculate the nodal forces in the constant strain axisymmetric triangular elements. Cell and nodal averaging of volumetric strain formulations are implemented on different mesh architectures to reduce the incompressibility constraints and eliminate volumetric locking. On the experimental side, a gas gun is designed and manufactured, and Taylor impact tests of cylinders made of several metallic materials are performed. Computational predictions of the deformed profiles of Taylor cylinders and experimentally determined deformed profiles are compared for verification purposes and to infer conclusions on the effect of yield strength, strain hardening and strain rate on the material response. The article also compares the performance of different plastic flow stress models that are incorporated into the hydrocode with the experimental results and results provided by previously reported simulations and tests.     

Calculation of stress intensity factors based on mode I crack opening integral parameters

We present stress intensity factor assessment using nodal displacements of the crack surfaces determined by the finite element method for cracked bodies. The equation is solved by expanding the crack opening displacement in the Chebyshev function, where crack front asymptotic behavior corresponds to the regulations of the linear elastic fracture mechanics. Results of the stress intensity factor calculations are obtained for test problems with analytical solution. Crack opening displacements are defined with the help of the 3D SPACE software package designed to model mixed variational formulation of the finite element method for displacements and strains of the thermoelastic boundary value problems. Translated from Problemy Prochnosti, No. 6, pp. 122–127, November–December, 2008.     

The finite element absolute nodal coordinate formulation incorporated with surface stress effect to model elastic bending nanowires in large deformation

Surface stress was incorporated into the finite element absolute nodal coordinate formulation in order to model elastic bending of nanowires in large deformation. The absolute nodal coordinate formulation is a numerical method to model bending structures in large deformation. The generalized Young-Laplace equation was employed to model the surface stress effect on bending nanowires. Effects from surface stress and large deformation on static bending nanowires are presented and discussed. The results calculated with the absolute nodal coordinate formulation incorporated with surface stress show that the surface stress effect makes the bending nanowires behave like softer or stiffer materials depending on the boundary condition. The surface stress effect diminishes as the dimensions of the bending structures increase beyond the nanoscale. The developed algorithm is consistent with the classical absolute nodal coordinate formulation at the macroscale.     

The hybrid-Trefftz finite element model and its application to plate bending

This paper presents a new hybrid element approach and applies it to plate bending. In contrast to more conventional models, the formulation is based on displacement fields which fulfil a priori the non-homogeneous Lagrange equation (Trefftz method). The interelement continuity is enforced by using a stationary principle together with an independent interelement displacement. The final unknowns are the nodal displacements and the elements may be implemented without any difficulty in finite element libraries of standard finite element programs. The formulation only calls for integration along the element boundaries which enables arbitrary polygonal or even curve-sided elements to be generated. Where relevant, known local solutions in the vicinity of a singularity or stress concentration may be used as an optional expansion basis to obtain, for example, particular singular corner elements, elements presenting circular holes, etc. Thus a high degree of accuracy may be achieved without a troublesome mesh refinement. Another important advantage of the formulation is the possibility of generating by a single element subroutine a large number of various elements (triangles, quadrilaterals, etc.), presenting an increasing degree of accuracy. The paper summarizes the results of numerical studies and shows the excellent accuracy and efficiency of the new elements. The conclusions present some ideas concerning the adaptive version of the new elements, extension to nonlinear problems and some other developments.     

An Alternative Positional FEM Formulation for Geometrically Non-linear Analysis of Shells: Curved Triangular Isoparametric Elements

This work presents an alternative finite element shell formulation based on non-conventional nodal parameters. The considered parameters are nodal positions (not displacements) and generalized vector components that comprise both director cossines and shell thickness variation at the same time. Although any objective strain measure could be adopted to develop the proposed formulation, non-linear engineering strain is chosen in order to take advantage of well know linear engineering stress–strain relations and to complement the easy geometrical appeal of positional formulation. The resulting formulation presents six degrees of freedom by each node and considers constant thickness variation. Consequently, the formulation fulfills a three dimensional compatible mapping and requires a relaxed three-dimensional constitutive relation to avoid thickness locking. Curved triangular elements with cubic approximation are adopted following a very simple notation. Several numerical simulations illustrate and confirm the accuracy and applicability of the proposed formulation.