An isogeometric analysis approach to gradient‐dependent plasticity

Gradient‐dependent plasticity can be used to achieve mesh‐objective results upon loss of well‐posedness of the initial/boundary value problem because of the introduction of strain softening, non‐associated flow, and geometric nonlinearity. A prominent class of gradient plasticity models considers a dependence of the yield strength on the Laplacian of the hardening parameter, usually an invariant of the plastic strain tensor. This inclusion causes the consistency condition to become a partial differential equation, in addition to the momentum balance. At the internal moving boundary, one has to impose appropriate boundary conditions on the hardening parameter or, equivalently, on the plastic multiplier. This internal boundary condition can be enforced without tracking the elastic‐plastic boundary by requiring ‐continuity with respect to the plastic multiplier. In this contribution, this continuity has been achieved by using nonuniform rational B‐splines as shape functions both for the plastic multiplier and for the displacements. One advantage of this isogeometric analysis approach is that the displacements can be interpolated one order higher, making it consistent with the interpolation of the plastic multiplier. This is different from previous approaches, which have been exploited. The regularising effect of gradient plasticity is shown for 1‐ and 2‐dimensional boundary value problems.     

Approximate J estimates for axial part-through surface-cracked pipes

This paper provides net‐section limit pressures and a reference stress based J estimation method for pipes with constant depth, internal axial surface cracks under internal pressure. Based on systematic small strain finite element (FE) limit analyses using elastic perfectly plastic materials, net‐section limit pressures are firstly determined, and based on FE results, a closed‐form limit pressure solution is proposed. Furthermore, based on the proposed limit pressure solution, a method to estimate elastic–plastic J is proposed based on the reference stress approach. When the reference stress is defined by the proposed (global) limit pressure, estimated J values based on the reference stress approach are overall slightly lower than FE results, implying that the method is non‐conservative. By re‐defining the reference using optimised reference loads, resulting J estimates agree well with FE results.     

Gradient-dependent plasticity: Formulation and algorithmic aspects

A plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure (hardening parameter), but also on the Laplacian thereof. The consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity. To properly solve the set of non-linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements. For appropriate boundary conditions this formulation can also be derived from a variational principle. Accordingly, the theory is complete. The addition of gradient terms becomes significant when modelling strain-softening solids. Classical models then result in loss of ellipticity of the governing set of partial differential equations. The addition of the gradient terms preserves ellipticity after the strain-softening regime has been entered. As a result, pathological mesh dependence as obtained in finite element computations with conventional continuum models is no longer encountered. This is demonstrated by some numerical simulations.     

Non—Local Analysis of Forming Limits of Ductile Material Considering Void Growth

The current study performed a finite element analysis of the strain localization behavior of a voided ductile material using a non-local plasticity formulation in which the yield strength depends on both an equivalent plastic strain measurement (hardening parameter) and Laplacian equivalent. The introduction of gradient terms to the yield function was found to play an important role in simulating the strain localization behavior of the voided ductile material. The effect of the mesh size and characteristic length on the strain localization were also investigated. An FEM simulation based on the proposed non-local plasticity revealed that the load-strain curves of the voided ductile material subjected to plane strain tension converged to one curve, regardless of the mesh size. In addition, the results using non-local plasticity also exhibited that the dependence of the deformation behavior of the material on the mesh size was much less sensitive than that with classical local plasticity and could be successfully eliminated through the introduction of a large value for the characteristic length.     

A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems

This paper presents a novel numerical method for effectively simulating the singular stress field for mode-I fracture problems based on the edge-based smoothed finite element method (ES-FEM). Using the unique feature of the ES-FEM formulation, we need only the assumed displacement values (not the derivatives) on the boundary of the smoothing domains, and hence a new technique to construct singular shape functions is devised for the crack tip elements. Some examples have demonstrated that results of the present singular ES-FEM in terms of strain energy, displacement and J-integral are much more accurate than the finite element method using the same mesh.     

Convergence of curved shell elements based on assumed covariant strain interpolations

A C⁰ 9-node shell element based on assumed interpolations of covariant strain components defined with respect to the element natural co-ordinate system has recently been proposed. In this formulation, the covariant strains are obtained directly from the Cartesian strains by tensor transformation without any need to compute laminar co-ordinate based strains. In the present work, the interpolated covariant strains used in this element are analysed to determine their satisfaction of the basic requirements for successful strain interpolation. These basic requirements are stated as invariance to rigid body motions and ability to represent constant and linear strain states. In the finite element formulation, the weak form of momentum balance is expressed in terms of covariant strains and contravariant stresses. The corresponding elasticity tensor is a function of the components of the metric tensor associated with the element natural co-ordinate system. The invariance properties of the metric tensor in the context of the finite element approximation are also discussed.     

Enhanced error estimator based on a nearly equilibrated moving least squares recovery technique for FEM and XFEM

In this paper a new technique aimed to obtain accurate estimates of the error in energy norm using a moving least squares (MLS) recovery-based procedure is presented. In the techniques based on the superconvergent patch recovery (SPR) the continuity of the recovered field is provided by the shape functions of the underlying mesh. We explore the capabilities of a recovery technique based on an MLS fitting, more flexible than SPR techniques as it directly provides continuous interpolated fields without relying on any FE mesh, to obtain estimates of the error in energy norm as an alternative to SPR. In the enhanced MLS proposed in the paper, boundary equilibrium is enforced using a nearest point approach that modifies the MLS functional. Lagrange multipliers are used to impose a nearly exact satisfaction of the internal equilibrium equation. The numerical results indicate the high accuracy of the proposed error.     

An implicit BEM formulation for gradient plasticity and localization phenomena

The aim of this paper is to discuss a boundary element formulation for non‐linear structural problems involving localization phenomena. In order to overcome the well‐known mesh dependency observed in local plasticity, a gradient plasticity model is used. An implicit boundary element formulation is proposed and the underlying consistent tangent operator defined. This formulation is based on the classical displacement and strain integral representations combined with an integral representation of the plastic multiplier. First numerical examples are presented to illustrate the application of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

Multi-grid solutions to the elastic plastic torsion problem in multiply connected domains

The elastic plastic torsion problem for an elastic, perfectly plastic cylinder with multiply connected cross section twisted around its longitudinal axis is formulated as an obstacle problem for an associated stress potential, the obstacle being defined in terms of a generalized distance function. Based upon the reformulation of the obstacle problem as an equivalent linear complementarity problem, the latter is discretized by means of finite difference techniques, and a monotonically convergent iterative scheme for its numerical solution is developed. At each step of the iteration the solution of a reduced system of discrete Poisson equations is required which is done by applying multi-grid techniques with respect to a hierarchy of grid-point sets. Combined with a suitably chosen nested iteration process this results in a computationally very efficient algorithm for the approximate solution of the elastic plastic torsion problem.     

A new discretization method and its application to solve incompressible Navier–Stokes equations

 A new numerical method is presented in this paper. This method directly solves partial differential equations in the Cartesian coordinate system. It can be easily applied to solve irregular domain problems without introducing the coordinate transformation technique. The concept of the present method is different from the conventional discretization methods. Unlike the conventional numerical methods where the discrete form of the differential equation only involves mesh points inside the solution domain, the new discretization method reduces the differential equation into a discrete form which may involve some points outside the solution domain. The functional values at these points are computed by the approximate form of the solution along a vertical or horizontal line. This process is called extrapolation. The form of the solution along a line can be approximated by Lagrange interpolated polynomial using all the points on the line or by low order polynomial using 3 local points. In this paper, the proposed new discretization method is first validated by its application to solve sample linear and nonlinear differential equations. It is demonstrated that the present method can easily treat different solution domains without any additional programming work. Then the method is applied to simulate incompressible flows in a smooth expansion channel by solving Navier–Stokes equations. The numerical results obtained by the new discretization method agree very well with available data in the literature. All the numerical examples showed that the present method is very efficient, which is suitable for solving irregular domain problems. Received 19 July 2000     

A simple triangular element for thick and thin plate and shell analysis

A new plate triangle based on Reissner–Mindlin plate theory is proposed. The element has a standard linear deflection field and an incompatible linear rotation field expressed in terms of the mid-side rotations. Locking is avoided by introducing an assumed linear shear strain field based on the tangential shear strains at the mid-sides. The element is free of spurious modes, satisfies the patch test and behaves correctly for thick and thin plate and shell situations. The element degenerates in an explicit manner to a simple discrete Kirchhoff form.     

An extended Galerkin weak form and a point interpolation method with continuous strain field and superconvergence using triangular mesh

A point interpolation method (PIM) with continuous strain field (PIM-CS) is developed for mechanics problems using triangular background mesh, in which PIM shape functions are used to construct both displacement and strain fields. The strain field constructed is continuous in the entire problem domain, which is achieved by simple linear interpolations using locally smoothed strains around the nodes and points required for the interpolation. A general parameterized functional with a real adjustable parameter α are then proposed for establishing PIM-CS models of special property. We prove theoretically that the PIM-CS has an excellent bound property: strain energy obtained using PIM-CS lies in between those of the compatible FEM and NS-PIM models of the same mesh. Techniques and procedures are then presented to compute the upper and lower bound solutions using the PIM-CS. It is discovered that an extended Galerkin (x-Galerkin) model, as special case resulted from the extended parameterized functional with α = 1, is outstanding in terms of both performance and efficiency. Intensive numerical studies show that upper and lower bound solutions can always be obtained, there exist α values at which the solutions of PIM-CS are of superconvergence, and the x-Galerkin model is capable of producing superconvergent solutions of ultra accuracy that is about 10 times that of the FEM using the same mesh.     

Active macro‐zone approach for incremental elastoplastic‐contact analysis

The symmetric boundary element method, based on the Galerkin hypotheses, has found an application in the nonlinear analysis of plasticity and in contact‐detachment problems, but both dealt with separately. In this paper, we want to treat these complex phenomena together as a linear complementarity problem. A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem‐elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) between contiguous substructures have to be introduced, in order to attain the solving equation system governing the elastoplastic‐contact/detachment problem. The elastoplasticity is solved by incremental analysis, called for active macro‐zones, and uses the well‐known concept of self‐equilibrium stress field here shown in a discrete form through the introduction of the influence matrix (self‐stress matrix). The solution of the frictionless contact/detachment problem was performed using a strategy based on the consistent formulation of the classical Signorini equations rewritten in discrete form by utilizing boundary nodal quantities as check elements in the zones of potential contact or detachment. Copyright © 2013 John Wiley & Sons, Ltd.     

Verhalten gekerbter Biegeproben aus der Aluminiumlegierung AA7075 im Kurzzeitermüdungsbereich

Prediction on Fatigue Life of Notched Specimens under Cyclic Bending Loading Pulsating 3P‐bending fatigue tests are conducted on edge‐notched specimens of AA7075. Measurements of electrical potential drop across notches were used to determine the number of cycles up to crack initiation. Cyclic material data determined from strain–controlled constant amplitude loading are use in FE‐analyses to the determination time functions of the local stresses and strains at the notch root using non‐linear material model according to Chaboche and Lemaitre. Using these FE computations, the fatigue life is predicted by the equivalent strain approach of the “ASME Boiler and Pressure Vessel Code” and compared with the results of the plastic strain energy approach. It is found that both approaches lead to relatively good predictions.     

Non‐linear version of stabilized conforming nodal integration for Galerkin mesh‐free methods

A stabilized conforming (SC) nodal integration, which meets the integration constraint in the Galerkin mesh‐free approximation, is generalized for non‐linear problems. Using a Lagrangian discretization, the integration constraints for SC nodal integration are imposed in the undeformed configuration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed configuration. The proposed method is independent to the path dependency of the materials. An assumed strain method is employed to formulate the discrete equilibrium equations, and the smoothed deformation gradient serves as the stabilization mechanism in the nodally integrated variational equation. Eigenvalue analysis demonstrated that the proposed strain smoothing provides a stabilization to the nodally integrated discrete equations. By employing Lagrangian shape functions, the computation of smoothed gradient matrix for deformation gradient is only necessary in the initial stage, and it can be stored and reused in the subsequent load steps. A significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison with the mesh‐free solution using Gauss integration. The performance of the proposed method is shown to be quite robust in dealing with non‐uniform discretization. Copyright © 2002 John Wiley & Sons, Ltd.     

Discrete element methods for the plastic analysis of structures subjected to cyclic loading

Methods for the analysis of complex, highly redundant structures subjected to intermittent loads causing biaxial membrane stress and stress reversal into the plastic range are presented. The Bauschinger effect in multi-axial stress is taken into account by the use of Ziegler's modification of Pragers kinematic hardening theory. The implementation of this plasticity theory in the discrete element methods involves the application of the loading in small increments. A linear relationship between increments of plastic strain and of stress, arising out of the theory, is used in conjunction with a linear matrix equation that governs the elastic behaviour of the structure. In the latter equation, plastic strains are interpreted as initial strains. A solution to the linear matrix equation, expressed in terms either of stress or of total strain, may be obtained by utilizing one of two alternative procedures. The methods are capable of treating materials which exhibit elastic–plastic behaviour involving ideal plasticity, linear or non-linear strain hardening, or limited strain hardening. Application is made to several representative structures. Comparison of some of the results with existing test data for both monotonic and reversed loading shows good correlation.     

Finite element plastic loads for circumferential cracked pipe bends under in-plane bending

This paper proposes plastic loads (limit load and twice-elastic-slope (TES) plastic load) for pipe bends with circumferential through-wall and part-through surface cracks under in-plane bending, based on three-dimensional FE limit analyses. The material is assumed to be elastic-perfectly plastic, and both the geometrically linear (small strain) and nonlinear (large geometry change) effects are considered. Regarding a crack location, both extrados and intrados cracks are considered. Based on the FE results, closed-form approximations of limit and TES plastic loads are proposed for practical applications, and compared with corresponding solutions for straight pipes.