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.     

Numerical modelling of shear bands by element bands

It is presently a concern and challenge to numerically model shear band localization. Many numerical methods have been developed to take into account the strain and displacement discontinuities across a shear band. In this paper, a contact band element method is proposed to model the shear band with a finite thickness under large shear deformation. The shear band elements, alternatively called contact band elements, are continuously updated based on their current configurations to prevent the large distortions of conventional finite elements and maintain realistic shear band configurations. The contact band element method, with a technique for the special shear band element, consists of the schemes to keep the shear band elements in good shapes, handle the band overlapping, kinking and separation problems. A few examples have shown that the contact band element method is a very efficient way to model the shear bands under large shear deformation. Copyright © 2002 John Wiley & Sons, Ltd.     

An extended finite element formulation for contact in multi‐material arbitrary Lagrangian–Eulerian calculations

Multi‐material Eulerian and arbitrary Lagrangian–Eulerian methods were originally developed for solving hypervelocity impact problems, but they are attractive for solving a broad range of problems having large deformations, the evolution of new free surfaces, and chemical reactions. The contact, separation, and slip between two surfaces have traditionally been addressed by the mixture theory, however the accuracy of this approach is severely limited. To improve the accuracy, an extended finite element formulation is developed and example calculations are presented. As a side benefit, the mixture theory is eliminated from the multi‐material formulation, eliminating the issues associated with the equilibration time between adjacent materials. By design, the new formulation is relatively simple to implement in existing multi‐material codes, parallelizes without difficulty, and has a low memory burden. Copyright © 2006 John Wiley & Sons, Ltd.     

Two-scale method for shear bands: thermal effects and variable bandwidth

A method for the analysis of shear bands using local partition of unity is developed in the framework of the extended finite element method (XFEM). Enrichments are introduced for both the displacement field and the thermal field. The shear band width is determined by minimizing the plastic work. A coupled finite strain thermo-elastoplastic constitutive law is used. The enrichment is injected into the mesh when the material law becomes unstable. The criterion based on a complete stability analysis for materials in the finite strain regime including heat conduction, strain hardening, strain rate hardening and thermal softening is presented. A mixed continuous quadrilateral element is employed. The method is applied to the Nesterenko experiments, which exhibit multiple propagating shear bands and other problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment

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 enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

In this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.     

A method for dynamic crack and shear band propagation with phantom nodes

A new method for modelling of arbitrary dynamic crack and shear band propagation is presented. We show that by a rearrangement of the extended finite element basis and the nodal degrees of freedom, the discontinuity can be described by superposed elements and phantom nodes. Cracks are treated by adding phantom nodes and superposing elements on the original mesh. Shear bands are treated by adding phantom degrees of freedom. The proposed method simplifies the treatment of element‐by‐element crack and shear band propagation in explicit methods. A quadrature method for 4‐node quadrilaterals is proposed based on a single quadrature point and hourglass control. The proposed method provides consistent history variables because it does not use a subdomain integration scheme for the discontinuous integrand. Numerical examples for dynamic crack and shear band propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

A combined space–time extended finite element method

The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX‐FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non‐polynomial part of the solution. This formulation allows a suitable form of the time‐stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T‐DGM), stability and accuracy properties of which can be derived from those of the TX‐FEM. Then Space and Time X‐FEM (STX‐FEM) are combined to obtain a unified space–time discretization. This combined STX‐FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.     

Space–time approach to numerical analysis of a string with a moving mass

Inertial loading of strings, beams and plates by mass travelling with near‐critical velocity has been a long debate. Typically, a moving mass is replaced by an equivalent force or an oscillator (with ‘rigid’ spring) that is in permanent contact with the structure. Such an approach leads to iterative solutions or imposition of artificial constraints. In both cases, rigid constraints result in serious computational problems. A direct mass matrix modification method frequently implemented in the finite element approach gave reasonable results only in the range of relatively low velocities. In this paper we present the space–time approach to the problem. The interaction of the moving mass/supporting structure is described in a local coordinate system of the space–time finite element domain. The resulting characteristic matrices include inertia, Coriolis and centrifugal forces. A simple modification of matrices in the discrete equations of motion allows us to gain accurate analysis of a wide range of velocities, up to the velocity of the wave speed. Numerical examples prove the simplicity and efficiency of the method. The presented approach can be easily implemented in the classic finite element algorithms. Copyright © 2008 John Wiley & Sons, Ltd.     

A cell‐based smoothed discrete shear gap method using triangular elements for static and free vibration analyses of Reissner–Mindlin plates

The cell‐based strain smoothing technique is combined with discrete shear gap method using three‐node triangular elements to give a so‐called cell‐based smoothed discrete shear gap method (CS‐DSG3) for static and free vibration analyses of Reissner–Mindlin plates. In the process of formulating the system stiffness matrix of the CS‐DSG3, each triangular element will be divided into three subtriangles, and in each subtriangle, the stabilized discrete shear gap method is used to compute the strains and to avoid the transverse shear locking. Then the strain smoothing technique on whole the triangular element is used to smooth the strains on these three subtriangles. The numerical examples demonstrated that the CS‐DSG3 is free of shear locking, passes the patch test, and shows four superior properties such as: (1) being a strong competitor to many existing three‐node triangular plate elements in the static analysis; (2) can give high accurate solutions for problems with skew geometries in the static analysis; (3) can give high accurate solutions in free vibration analysis; and (4) can provide accurately the values of high frequencies of plates by using only coarse meshes. Copyright © 2012 John Wiley & Sons, Ltd.     

On the evaluation of the method of finite spheres for the solution of Reissner–Mindlin plate problems using the numerical inf–sup test

In this paper we use the numerical inf–sup test to evaluate both displacement‐based and mixed discretization schemes for the solution of Reissner–Mindlin plate problems using the meshfree method of finite spheres. While an analytical proof of whether a discretization scheme passes the inf–sup condition is most desirable, such a proof is usually out of reach due to the complexity of the meshfree approximation spaces involved. The numerical inf–sup test (Int. J. Numer. Meth. Engng 1997; 40 :3639–3663), developed to test finite element discretization spaces, has therefore been adopted in this paper. Tests have been performed for both regular and irregular nodal configurations. While, like linear finite elements, pure displacement‐based approximation spaces with linear consistency do not pass the inf–sup test and exhibit shear locking, quadratic discretizations, unlike quadratic finite elements, pass the test. Pure displacement‐based and mixed approximation spaces that pass the numerical inf–sup test exhibit optimal or near optimal convergence behaviour. Copyright © 2007 John Wiley & Sons, Ltd.     

On nucleation and propagation of PLC bands in an Al–3Mg alloy

The processes of nucleation and propagation of Portevin–LeChatelier deformation bands during tensile deformation of the alloy AA5754 have been investigated by a special optical technique for local strain measurements, combining both a line-scan and a CMOS camera to record the changes of surface strain, applying digital image correlation. Examples for bands of types A and B are observed and discussed with respect to the micro-mechanisms of their nucleation and propagation, emphasizing local stress concentrations in addition to the well-known effects of dynamic strain aging and dislocation interactions for explaining the PLC behavior. Finally the nucleation and propagation of PLC bands are simulated by computations based on a simple constitutive model.     

The Effect of Steel Plates with Shear Studs for Weak Column–Strong Beam Connections in the Reinforced Concrete Structures under Earthquake Effect

Abstract: In this study, the effect of steel plates with shear studs used in the weak column–strong beam connections was investigated both experimentally and analytically. Five RC specimens were tested under cyclic loading in the experimental programme. Steel plates with shear studs in the column were used along the connection area. The specimens were derived from the exterior joint of a frame. As the main parameters, the width of shear stud heads and their spacing on the steel plates were changed to investigate their effects on the weak column. The test results confirmed that shear studs improved the strength and stiffness of the specimens. The control specimen collapsed because of joint failure, while the other four specimens collapsed because of bending at the bottom of the beam showing ductile behaviour. The comparison of experimental results with the analytical results obtained from ANSYS finite element programme showed similar outputs from the aspects of behaviour.     

Adaptive modelling of delamination initiation and propagation using an equivalent single‐layer shell approach

A major challenge for crash failure analysis of laminated composites is to find a modelling approach, which is both sufficiently accurate, for example, able to capture delaminations, and computationally efficient to allow full‐scale vehicle crash simulations. Addressing this challenge, we propose a methodology based on an equivalent single‐layer shell formulation which is adaptively through‐the‐thickness refined to capture initiating and propagating delaminations. To be specific, single shell elements through the laminate thickness are locally and adaptively enriched using the extended finite element method such that delaminations can be explicitly modelled without having to be represented by separate elements. Furthermore, the shell formulation is combined with a stress recovery technique which increases the accuracy of predicting delamination initiations. The paper focuses on the parameters associated with identifying, introducing and extending the enrichment areas; especially on the impact of these parameters on the resulting structural deformation behaviour. We conclude that the delamination enrichment must be large enough to allow the fracture process to be accurately resolved, and we propose a suitable approach to achieve this. The proposed methodology for adaptive delamination modelling shows potential for being computationally efficient, and thereby, it has the potential to enable efficient and accurate full vehicle crash simulations of laminated composites. Copyright © 2017 John Wiley & Sons, Ltd.     

A rheological interface model and its space–time finite element formulation for fluid–structure interaction

This contribution discusses extended physical interface models for fluid–structure interaction problems and investigates their phenomenological effects on the behavior of coupled systems by numerical simulation. Besides the various types of friction at the fluid–structure interface the most interesting phenomena are related to effects due to additional interface stiffness and damping. The paper introduces extended models at the fluid–structure interface on the basis of rheological devices (Hooke, Newton, Kelvin, Maxwell, Zener). The interface is decomposed into a Lagrangian layer for the solid‐like part and an Eulerian layer for the fluid‐like part. The mechanical model for fluid–structure interaction is based on the equations of rigid body dynamics for the structural part and the incompressible Navier–Stokes equations for viscous flow. The resulting weighted residual form uses the interface velocity and interface tractions in both layers in addition to the field variables for fluid and structure. The weak formulation of the whole coupled system is discretized using space–time finite elements with a discontinuous Galerkin method for time‐integration leading to a monolithic algebraic system. The deforming fluid domain is taken into account by deformable space–time finite elements and a pseudo‐structure approach for mesh motion. The sensitivity of coupled systems to modification of the interface model and its parameters is investigated by numerical simulation of flow induced vibrations of a spring supported fluid‐immersed cylinder. It is shown that the presented rheological interface model allows to influence flow‐induced vibrations. Copyright © 2010 John Wiley & Sons, Ltd.     

Contact with friction in multi‐material arbitrary Lagrangian‐Eulerian formulations using X‐FEM

A contact method with friction for the multi‐dimensional Lagrangian step in multi‐material arbitrary Lagrangian–Eulerian (ALE) formulations is presented. In our previous research, the extended finite element method (X‐FEM) was used to create independent fields (i.e. velocity, strain rate, force, mass, etc.) for each material in the problem to model contact without friction. The research presented here includes the extension to friction and improvements to the accuracy and robustness of our previous study. The accelerations of the multi‐material nodes are obtained by coupling the material force and mass fields as a function of the prescribed contact; similarly, the velocities of the multi‐material nodes are recalculated using the conservation of momentum when the prescribed contact requires it. The coupling procedures impose the same nodal velocity on the coupled materials in the direction normal to their interface during the time step update. As a result, the overlap of materials is prevented and unwanted separation does not occur. Three different types of contacts are treated: perfectly bonded, frictionless slip, and slip with friction. Example impact problems are solved and the numerical solutions are presented. Copyright © 2008 John Wiley & Sons, Ltd.