International Journal for Numerical Methods in Engineering

The application of the mixed finite element method to two‐dimensional elastic contact problems is investigated. Since in the mixed method, both displacements and stresses are retained as variables, it is found that all the contact conditions—displacement as well as stress—can be approximated directly. A significant novelty is that some of the displacement variables are treated as natural boundary conditions in the contact region. In cases where the contact region is independent of the applied loading, an iterative procedure is used to establish the sliding and adhering portions of the contact region. In cases where the contact region is a function of the applied loading, for example progressive contact, an incremental formulation is employed to describe the discretized contact stages before invoking the former iterations. Several numerical examples are presented and the results are compared with those from the conventional potential energy or displacement finite element method.

A return mapping algorithm is presented for the numerical time integration of the constitutive equations for elastoplasticity with isotropic yield surfaces, constructed from all three invariants of the stress tensor. Based on the first‐order backward Euler difference formula (BDF), the governing equations for the stresses are solved in the space of the invariants and the discretized persistence parameter. The stresses are recovered afterwards. The solution concept is applied to a pressure‐independent yield function, expressed in terms of the second and third invariant of the stress tensor. The numerical performance of the method is demonstrated with two examples.

A method for modelling the growth of multiple cracks in linear elastic media is presented. Both homogeneous and inhomogeneous materials are considered. The method uses the extended finite element method for arbitrary discontinuities and does not require remeshing as the cracks grow; the method also treats the junction of cracks. The crack geometries are arbitrary with respect to the mesh and are described by vector level sets. The overall response of the structure is obtained until complete failure. A stability analysis of competitive cracks tips is performed. The method is applied to bodies in plane strain or plane stress and to unit cells with 2–10 growing cracks (although the method does not limit the number of cracks). It is shown to be efficient and accurate for crack coalescence and percolation problems. Copyright © 2004 John Wiley & Sons, Ltd.

We have developed a new crack tip element for the phantom‐node method. In this method, a crack tip can be placed inside an element. Therefore, cracks can propagate almost independent of the finite element mesh. We developed two different formulations for the three‐node triangular element and four‐node quadrilateral element, respectively. Although this method is well suited for the one‐point quadrature scheme, it can be used with other general quadrature schemes. We provide some numerical examples for some static and dynamic problems. Copyright © 2007 John Wiley & Sons, Ltd.

A methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial differential equation loses hyperbolicity. The approach is limited to rate‐independent materials, so that the transition occurs on a set of measure zero. The discrete discontinuity is treated by the extended finite element method (XFEM) whereby arbitrary discontinuities can be incorporated in the model without remeshing. Loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model. A new method was developed for the case when the discontinuity ends within an element; it facilitates the modelling of crack tips that occur within an element in a dynamic setting. The method is applied to several dynamic crack growth problems including the branching of cracks. Copyright © 2003 John Wiley & Sons, Ltd.

An extended finite element method scheme for a static cohesive crack is developed with a new formulation for elements containing crack tips. This method can treat arbitrary cracks independent of the mesh and crack growth without remeshing. All cracked elements are enriched by the sign function so that no blending of the local partition of unity is required. This method is able to treat the entire crack with only one type of enrichment function, including the elements containing the crack tip. This scheme is applied to linear 3‐node triangular elements and quadratic 6‐node triangular elements. To ensure smooth crack closing of the cohesive crack, the stress projection normal to the crack tip is imposed to be equal to the material strength. The equilibrium equation and the traction condition are solved by the Newton–Raphson method to obtain the nodal displacements and the external load simultaneously. The results obtained by the new extended finite element method are compared to reference solutions and show excellent agreement. Copyright © 2003 John Wiley & Sons, Ltd.

One possibility of formulating the tinite element method is founded on the principle of virtual displacement, in which we want to include a rate‐independent elastoplastic constitutive model based on the assumption of a yield surface. The constitutive equations result from the assumptions of small deformations, Hooke's law for the elastic domain, the normality rule for the evolution of plastic strains, the von Mises yield condition and a special kind of kinematic hardening due to Armstrong and Frederick,¹ in which linear kinematic hardening is generalized with a saturation term. We show that it is not generally recommendable to propose large load steps. To this end, we investigate the influences of the non‐linear kinematic hardening model on the stress computation and the resulting consistent elastoplastic tangent operator. The main topics of this paper are: (1) development of a problem‐optimized backward Euler method with regard to the kinematic hardening model, (2) study of the influence of the saturation term on the numerical accuracy through isoerror maps and (3) computation of the consistent elastoplastic tangent operator.

An unconditionally stable algorithm for plane stress elastoplasticity is developed, based upon the notion of elastic predictor‐return mapping (plastic corrector). Enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation. Consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm. Use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of Newton methods. The accuracy of the algorithm is assessed by means of iso‐error maps. The excellent performance of the algorithm for large time steps is illustrated in numerical experiments.

We describe a methodology for solving the constitutive problem and evaluating the consistent tangent operator for isotropic elasto/visco‐plastic models whose yield function incorporates the third stress invariant . The developments presented are based upon original results, proved in the paper, concerning the derivatives of eigenvalues and eigenprojectors of symmetric second‐order tensors with respect to the tensor itself and upon an original algebra of fourth‐order tensors obtained as second derivatives of isotropic scalar functions of a symmetric tensor argument . The analysis, initially referred to the small‐strain case, is then extended to a formulation for the large deformation regime; for both cases we provide a derivation of the consistent tangent tensor which shows the analogy between the two formulations and the close relationship with the tangent tensors of the Lagrangian description of large‐strain elastoplasticity. Copyright © 2004 John Wiley & Sons, Ltd.