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.     

Finite elements with embedded strong discontinuities for the modeling of failure in solids

This paper presents new finite elements that incorporate strong discontinuities with linear interpolations of the displacement jumps for the modeling of failure in solids. The cases of interest are characterized by a localized cohesive law along a propagating discontinuity (e.g. a crack), with this propagation occurring in a general finite element mesh without remeshing. Plane problems are considered in the infinitesimal deformation range. The new elements are constructed by enhancing the strains of existing finite elements (including general displacement based, mixed, assumed and enhanced strain elements) with a series of strain modes that depend on the proper enhanced parameters local to the element. These strain modes are designed by identifying the strain fields to be captured exactly, including the rigid body motions of the two parts of a splitting element for a fully softened discontinuity, and the relative stretching of these parts for a linear tangential sliding of the discontinuity. This procedure accounts for the discrete kinematics of the underlying finite element and assures the lack of stress locking in general quadrilateral elements for linearly separating discontinuities, that is, spurious transfers of stresses through the discontinuity are avoided. The equations for the enhanced parameters are constructed by imposing the local equilibrium between the stresses in the bulk of the element and the tractions driving the aforementioned cohesive law, with the proper equilibrium operators to account for the linear kinematics of the discontinuity. Given the locality of all these considerations, the enhanced parameters can be eliminated by their static condensation at the element level, resulting in an efficient implementation of the resulting methods and involving minor modifications of an existing finite element code. A series of numerical tests and more general representative numerical simulations are presented to illustrate the performance of the new elements. Copyright © 2007 John Wiley & Sons, Ltd.     

Stress‐hybrid quadrilateral finite element with embedded strong discontinuity for failure analysis of plane stress solids

A formulation of a quadrilateral finite element with embedded strong discontinuity, suitable for the material failure numerical analysis of plane stress solids, is presented. The kinematics of standard finite element is enhanced by displacement jumps that vary linearly along the embedded discontinuity line. They are described by four kinematic parameters that are related to four element separation modes. The modes are designed for no stress transfer over the discontinuity line at its fully softened (opened) state. As for the material, the bulk of the element is assumed to be elastic, and the softening plasticity, in terms of discontinuity tractions and displacement jumps, is assumed along the discontinuity line. The bulk stresses are described by the optimal five‐parameter interpolation. The combination of stress interpolation and enhanced kinematics yields simple form of the element stiffness matrix. To achieve efficient implementation, the stiffness matrix is statically condensed for both the enhanced kinematic parameters and the stress parameters. In a set of numerical examples, the performance of the derived element is illustrated. Obtained results are compared with some other representative embedded discontinuity quadrilateral elements (displacement‐based and enhanced assumed strain based). It turns out that the element performs very well. Copyright © 2013 John Wiley & Sons, Ltd.     

3D modelling of strong discontinuities in elastoplastic solids: fixed and rotating localization formulations

This paper is concerned with the incorporation of displacement discontinuities into a continuum theory of elastoplasticity for the modelling of localization processes such as cracking in brittle materials. Based on the strong discontinuity approach (SDA) (Computational Mechanics 1993; 12: 277–296) mesh objective 2D and 3D finite element formulations are developed using linear and quadratic 2D elements as well as 8‐noded 3D elements. In the formulation of the finite‐element model proposed in the paper, the analogy with standard formulations is emphasized. The parameter defining the amplitude of the displacement jump within the finite element is condensed out at the material level without employing the standard static condensation technique. This approach results in linearized constitutive equations formally identical to continuum models. Therefore, the standard return mapping algorithm is used to solve the non‐linear equations. In analogy to concepts used in continuum smeared crack models, a rotating formulation of the SDA is proposed in addition to the standard concept of fixed discontinuities. It is shown that the rotating localization approach reduces locking effects observed in analyses based on fixed localization directions. The applicability of the proposed SDA finite‐element model as well as its numerical performance is investigated by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block subjected to a shear force. Copyright © 2003 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.     

On the strong discontinuity approach in finite deformation settings

Taking the strong discontinuity approach as a framework for modelling displacement discontinuities and strain localization phenomena, this work extends previous results in infinitesimal strain settings to finite deformation scenarios. By means of the strong discontinuity analysis, and taking isotropic damage models as target continuum (stress–strain) constitutive equation, projected discrete (tractions–displacement jumps) constitutive models are derived, together with the strong discontinuity conditions that restrict the stress states at the discontinuous regime. A variable bandwidth model, to automatically induce those strong discontinuity conditions, and a discontinuous bifurcation procedure, to determine the initiation and propagation of the discontinuity, are briefly sketched. The large strain counterpart of a non‐symmetric finite element with embedded discontinuities, frequently considered in the strong discontinuity approach for infinitesimal strains, is then presented. Finally, some numerical experiments display the theoretical issues, and emphasize the role of the large strain kinematics in the obtained results. Copyright © 2003 John Wiley & Sons, Ltd.     

A new method for modelling cohesive cracks using finite elements

A model which allows the introduction of displacements jumps to conventional finite elements is developed. The path of the discontinuity is completely independent of the mesh structure. Unlike so‐called ‘embedded discontinuity’ models, which are based on incompatible strain modes, there is no restriction on the type of underlying solid finite element that can be used and displacement jumps are continuous across element boundaries. Using finite element shape functions as partitions of unity, the displacement jump across a crack is represented by extra degrees of freedom at existing nodes. To model fracture in quasi‐brittle heterogeneous materials, a cohesive crack model is used. Numerical simulations illustrate the ability of the method to objectively simulate fracture with unstructured meshes. Copyright © 2001 John Wiley & Sons, Ltd.     

Enhanced mixed interpolation XFEM formulations for discontinuous Timoshenko beam and Mindlin‐Reissner plate

Shear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.     

Numerical simulation of crack problems using triangular finite elements with embedded interfaces

The aim of this paper is to propose numerical aspects for the modeling of discrete cracks in quasi-brittle materials using triangular finite elements with an embedded interface based on the formulation in [Computational Mechanics 27 (2001) 463]. The kinematics of the discontinuous displacement field and the variational formulation applied to a body with an internal discontinuity is given. The discontinuity is modeled by additional global degrees of freedom and the continuity of the displacement jumps across the element boundaries is enforced. To show the performance of the model, a single element test and two examples for mode-I dominated fracture, namely a tension test and a three-point bending beam, are discussed.