An extension of 3-D procedure to large strain analysis of shells

A numerical stress integration procedure for general 3-D large strain problems in inelasticity, based on the total formulation and the governing parameter method (GPM), is extended to shell analysis. The multiplicative decomposition of the deformation gradient is adopted with the evaluation of the deformation gradient practically in the same way as in a general 3-D material deformation. The calculated trial elastic logarithmic strains are transformed to the local shell Cartesian coordinate system and the stress integration is performed according to the GPM developed for small strain conditions. The consistent tangent matrix is calculated as in case of small strain deformation and then transformed to the global coordinate system.A specific step in the proposed procedure is the updating of the left elastic Green–Lagrangian deformation tensor. Namely, after the stresses are computed, the principal elastic strains and the principal vectors corresponding to the stresses at the end of time step are determined. In this way the shell conditions are taken into account appropriately for the next step.Some details are given for the stress integration in case of thermoplastic and creep material model.Numerical examples include bulging of plate (plastic, thermoplastic, and creep models for metal) and necking of a thin sheet. Comparison of solutions with those available in the literature, and with solutions using other type of finite elements, demonstrates applicability, efficiency and accuracy of the proposed procedure.     

Improvements in the membrane behaviour of the three node rotation-free BST shell triangle using an assumed strain approach

In this paper an assumed strain approach is presented in order to improve the membrane behaviour of a thin shell triangular element. The so called Basic Shell Triangle (BST) has three nodes with only translational degrees of freedom and is based on a Total Lagrangian Formulation. As in the original BST element the curvatures are computed resorting to the surrounding elements (patch of four elements). Membrane strains are now also computed from the same patch of elements which leads to a non-conforming membrane behaviour. Despite this non-conformity the element passes the patch test. Large strain plasticity is considered using a logarithmic strain–stress pair. A plane stress behaviour with an additive decomposition of elastic and plastic strains is assumed. A hyperplastic law is considered for the elastic part while for the plastic part an anisotropic quadratic (Hill) yield function with non-linear isotropic hardening is adopted. The element, termed EBST, has been implemented in an explicit (hydro-)code adequate to simulate sheet-stamping processes and in an implicit static/dynamic code. Several examples are given showing the good performance of the enhanced rotation-free shell triangle.     

On plastic incompressibility within time-adaptive finite elements combined with projection techniques

This article treats the interpretation of quasi-static finite elements applied to constitutive equations of evolutionary-type as a solution scheme to solve globally differential-algebraic equations. This concept is applied to finite strain viscoplasticity based on a model with non-linear kinematic hardening under the assumption of plastic incompressibility. The model is based on multiple multiplicative decomposition both for the deformation gradient into an elastic and an inelastic part as well as for the inelastic part into a kinematic hardening (energy storage) and a dissipative part. Both intermediate configurations are described by inelastic right Cauchy-Green tensors satisfying inelastic incompressibility in the theoretical context. The attention in view of the numerical treatment within finite elements is focused on diagonally implicit Runge-Kutta methods which destroy the assumption of plastic incompressibility during the time-integration due to an additive structure of the integration step. In combination with a Multilevel-Newton algorithm these algorithms embed the classical strain-driven radial-return method. To this end, a concept of geometric numerical integration is applied, where the plastic incompressibility condition is taken into account as an additional side-condition. Since the literature states large integration errors if the side-condition is not taken into account, a particular focus lies on the application of a time-adaptive procedure. Accordingly, the article investigates (i) the algorithmic treatment of kinematic hardening within time-adaptive finite elements, (ii) the influence of the Perzyna-type viscoplasticity approach in view of an order reduction phenomenon, and (iii) the influence of taking into account the exact fulfillment of plastic incompressibility using a projection method having the advantage of simple implementation.     

Topology optimization based on finite strain plasticity

In this paper infinitesimal elasto-plastic based topology optimization is extended to finite strains. The employed model is based on rate-independent isotropic hardening plasticity and to separate the elastic deformation from the plastic deformation, use is made of the multiplicative split of the deformation gradient. The mechanical balance laws are solved using an implicit total Lagrangian formulation. The optimization problem is solved using the method of moving asymptotes and the sensitivity required to form convex separable approximations is derived using a path-dependent adjoint strategy. The optimization problem is regularized using a PDE-type filter. A simple boundary value problem where the plastic work is maximized is used to demonstrate the capability of the presented model. The numerical examples reveal that finite strain plasticity successfully can be combined with topology optimization.     

A universal integration algorithm for rate-dependent elastoplasticity

An algorithm is developed for integrating rate-dependent constitutive equations of elstoplasticity including isotropic and kinematic hardening, as well as thermal softening and non-coaxiality of the plastic strain rate and the driving stress. The method is unconditionally stable and accurate for large time steps and all possible ranges of rate-dependency. Under a constant loading rate the algorithm gives exact results at arbitrary step sizes for rate-independent materials without hardening, and in proportional loading for rate-independency with hardening, and linear viscosity without hardening. The present method is an extension of a recently proposed integration algorithm for stiff equations to domains of high rate-sensitivity like, for example, in power-law creep. The algorithm employs a plastic predictor-elastic corrector scheme, which, in general, requires less numerical effort in the return mapping process than the assumption of an elastic predictor. Numerical examples underline the efficiency of this integration algorithm in comparison to gradient techniques and an extended radial return method for rate-dependent plasticity.     

Operator split methods for the numerical solution of the elastoplastic dynamic problem

The elastoplastic dynamic problem is first formulated in a form that facilitates the application of product formula techniques. The additive decomposition of the dynamic equations into elastic and plastic parts is taken as a basis for the definition of product algorithms that exploit such decomposition. In the context of a finite element discretization, these product algorithms entail, for every time step, the solution of an elastic problem followed by the application of plastic algorithms that operate on the stresses and internal variables at the integration points and bring in the plastic constitutive relations. Suitable plastic algorithms are discussed for the cases of perfect and hardening plasticity and viscoplasticity. The proposed formalism does not depend on any notion of smoothness of the yield surface and is applicable to arbitrary convex elastic regions, with or without corners. The stability properties of the product algorithm are identical to those of the elastic algorithm used whereas the computational expense is practically equal to that of an elastic problem.     

An implicit incrementally objective formulation for the solution of elastoplastic problems at finite strain by the F.E.M.

The mechanical formulation presented in this paper is based on an incremental updated Lagrange procedure using the principle of virtual work at the end of each load increment and an implicit incremental flow rule obtained by an approximate time integration of the objective rate constitutive equations. The approximate time integration is carried out along a particular path in the deformation and rotation space. This path ensures the incremental objectivity and minimizes the equivalent strain over the increment among all the possible paths, consequently avoiding an artificial increase of the plastic equivalent strain during the interpolation. The mechanical formulation presented leads to a fixed set of nonlinear equations, whose unknowns are the nodal displacements of the structure. A numerical algorithm based on a quasi Newton-Raphson method is then proposed to solve this system. The separation of the mechanical formulation from the resolution algorithm ensures the path independence. Numerical tests are carried out for a material obeying an isotropic with work-hardening von Mises criterion and associated flow rule. Single element tests show that this approach gives a very accurate solution even when the strain increment reaches twenty times the elastic strain up to yield. A structural test on a beam measures the influence of the incremental objectivity on the displacements, the equivalent plastic strain and the stresses.     

Hybrid strain based three node flat triangular shell elements—I. Nonlinear theory and incremental formulation

Aspects and theories of nonlinear analysis of structures, with special emphasis on structures that are discretized by the finite element method, are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Accordingly, the incremental second Piola-Kirchhoff stress and the incremental Washizu strain are selected as the incremental stress and strain measures. Various schemes for the transformation of the second Piola-Kirchhoff stress to Cauchy stress are included. Two versions of linear and nonlinear element stiffness and mass matrices are considered. These are the director and simplified versions. Variable thickness of the shell is considered so as to account for the ‘thinning effect’ due to large strain. Material nonlinearity studied in this paper is of elasto-plastic type with isotropic strain hardening. Cases in which small elastic but large plastic strain condition applies are considered and the J₂ flow theory of plasticity, in conjunction with Ilyushin's yield criterion, is employed. To simplify the derivation of (small displacement) stiffness matrix and to facilitate the derivation of explicit expressions for the element matrices, the non-layered approach has been applied.     

Nonlinear inelastic uniform torsion of composite bars by BEM

In this paper the elastic–plastic uniform torsion analysis of composite cylindrical bars of arbitrary cross-section consisting of materials in contact, each of which can surround a finite number of inclusions, taking into account the effect of geometric nonlinearity is presented employing the boundary element method. The stress–strain relationships for the materials are assumed to be elastic–plastic–strain hardening. The incremental torque–rotation relationship is computed based on the finite displacement (finite rotation) theory, that is the transverse displacement components are expressed so as to be valid for large rotations and the longitudinal normal strain includes the second-order geometric nonlinear term often described as the “Wagner strain”. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross-section’s torsional rigidity is evaluated exactly without using the so-called Saint Venant’s torsional constant. The torsional rigidity of the cross-section is evaluated directly employing the primary warping function of the cross-section depending on both its shape and the progress of the plastic region. A boundary value problem with respect to the aforementioned function is formulated and solved employing a BEM approach. The influence of the second Piola–Kirchhoff normal stress component to the plastic/elastic moment ratio in uniform inelastic torsion is demonstrated.     

Variational inequality in classical plasticity. Applications to Armstrong–Frederick elasto-plastic model

In this paper the quasi-static initial and boundary value problem for an elasto-plastic mixed hardening material is reformulated within the constitutive framework of small strains. The plastic factor plays the basic role in describing the rate independent evolution equations for the plastic strain and hardening variables. The plastic factor is equivalently represented as the solution of an appropriate local inequality involving the yield function. The main idea was to introduce the variational inequality at any time $t$ to be solved for the velocity field and the complementary plastic factor. There is the plastic factor in a strain-driven process. The solution procedure proposed here to solve the initial and boundary value problem is based on the solutions of the variational inequality at time $t$, coupled with an update algorithm in order to evaluate the current state of the material for an incremental deformation process. This time the return mapping algorithm is avoided as the values of the plastic factor and the velocity are known at time $t.$ As we developed a procedure to simultaneously solve the equilibrium equation coupled with the rate-independent evolution equations, no necessity to compute the algorithmic elasto-plastic tangent moduli occurs. The numerical simulations are done for the mixed hardening elasto-plastic model involving Armstrong–Frederick kinematic hardening. To validate the proposed numerical algorithms, we compare the solutions based on the variational inequality and those based on return mapping algorithm, computed for the same Prager kinematic hardening law.     

Finite element cyclic thermoplasticity analysis by the method of subvolumes

The subject of this paper is the development of an analytical tool capable of economically evaluating the cyclic plasticity which occurs in areas of strain concentration resulting from the combination of both mechanical and thermal stresses. The techniques developed are capable of handling large excursions in temperatures with the associated variations in material properties, including plasticity. The techniques are capable of reproducing real cyclic material behavior including Bauschinger effect, cross-hardening and memory.These analytical techniques have been implemented in a time-sharing finite element computer program. Cyclic plasticity has been introduced into this program using incremental loading and an iterative solution technique. The plasticity theory involved makes use of the von Mises yield criterion and the Prandtl-Reuss flow rule. The major portion of the developmental work in this effort was expended in the establishment of a temperature variable hardening rule and its finite element implementation. The plane stress, constant strain triangle is the finite element used in this work.The incremental plasticity solution is obtained by iteratively revising the right-hand side of the system of finite element equations by the addition of a vector of plastic pseudo forces. The method of subvolumes is used to generate the vector of plastic pseudo forces such that real material cyclic plasticity behavior is mathematically reproduced.The effects of the plastic deformations are introduced into the system of finite element equations by considering them as load terms in much the same way as thermal expansions are usually treated. The nonlinear solution is then attained through solution of a series of elastic problems and by variation of the plastic load terms until the requirements of compatibility, equilibrium and the specified nonlinear stress-strain relations are all met within a given tolerance.     

Finite element cyclic thermoplasticity analysis by the method of subvolumes

The subject of this paper is the development of an analytical tool capable of economically evaluating the cyclic plasticity which occurs in areas of strain concentration resulting from the combination of both mechanical and thermal stresses. The techniques developed are capable of handling large excursions in temperatures with the associated variations in material properties, including plasticity. The techniques are capable of reproducing real cyclic material behavior including Bauschinger effect, cross-hardening and memory.These analytical techniques have been implemented in a time-sharing finite element computer program. Cyclic plasticity has been introduced into this program using incremental loading and an interative technique. The plasticity theory involved makes use of the von Mises yield criterion and the Prandtl-Reuss flow rule. The major portion of the developmental work in this effort was expended in the establishment of a temperature variable hardening rule and its finite element implementation. The plane stress, constant strain triangle is the finite element used in this work.The incremental plasticity solution is obtained by interatively revising and right-hand side of the system of finite element equations by the addition of a vector of plastic pseudo forces. The method of subvolumes is used to generate the vector of plastic pseudo forces such that real material cyclic plasticity behavior is mathematically reproduced.The effects of the plastic deformations are introduced into the system of finite element equations by considering them as load terms in much the same way as thermal expansions are usually treated. The nonlinear solution is then attained through solution of a series of elastic problems and by variation of the plastic load terms until the requirements of compatibility, equilibrium and the specified non-linear stress-strain relations are all met within a given tolerance.     

Simulation of strength difference for adhesive materials in finite deformation elasto-plasticity

The experimental investigation of certain adhesive materials reveals elastic strains, plastic strains and hardening, respectively. Additionally a pronounced strength difference between tension, torsion or combined loading is observed. The purpose of this work is the simulation of these phenomena in the framework of large strain elasto-plasticity. To this end a yield function dependent on the first and second basic invariants of the Cauchy stress tensor is introduced. Furthermore, a plastic potential with the same mathematical structure is used to formulate the evolution equations under the assumption of small elastic strains. Upon considering thermodynamic consistency of the model equations some restrictions on the material parameters are derived. Furthermore numerical aspects are addressed concerning the integration of the constitutive relations and the finite element equilibrium iteration. In the numerical examples, firstly, we compare simulated and experimental results exhibiting the yield strength difference between tension and torsion for the adhesive material Betamate 1496. A second example investigates the deformation evolution of a compact tension specimen with an adhesive zone.     

Incremental formulation in nonlinear mechanics and large strain elasto-plasticity — Natural approach. Part 1

The paper discusses numerical solution techniques of problems in continuum mechanics in the presence of finite elastic as well as plastic strain components. The paper is based on the natural finite element formulation of large deformations. The proposed model requires the determination of an intermediate (stress-free) configuration at each step of the loading process. Certain approximations are adopted and comparisons are made to assess the difference between the natural approach with an intermediate reference configuration and conventional models for large displacements and small strains. Some numerical illustrations are given.     

Finite element elastic-plastic-creep analysis of two-dimensional continuum with temperature dependent material properties

A technique is presented for performing finite element elastic-plastic-creep analysis of two-dimensional continuum composed of material with temperature dependent elastic, plastic, and creep properties. The plastic analysis utilizes the Prandtl-Reuss flow equations assuming isotropic material properties and linear strain-hardening. A power creep flow law formulated by Odquist is used to determine the steady state creep strain rate. The plastic and creep flow laws are employed to derive a ‘softened’ plastic-creep stress-strain matrix. These modified stress-strain relations are then used to formulate the element stiffness matrix in the usual manner. The differences in the elastic, plastic, and creep properties of the material due to the temperature change during the increment result in the formation of pseudo stresses, which in turn lead to load terms that appear on the right hand side of the equilibrium equations. The load terms resulting from these pseudo stresses not only keep the solution on the temperature dependent stress-strain curve of the material, but also correct for the elastic ‘overshoot’ that occurs when an element changes from an elastic to a plastic state. The effect of large displacements is included by the formulation of the geometric stiffness matrix for each element being used in the computer code. With this procedure it becomes economically feasible to perform elastic-plastic-creep stress analysis of two-dimensional continuum subjected to transient thermal and mechanical loadings. Several examples of both elastic-plastic and creep analyses are presented, and the finite element solutions are compared to either other theoretical solutions or experiment.     

Nonlinear finite element analysis of shell structures using the semi-loof element

The Semi-Loof Shell element originally developed by Irons [2] for linear elastic analysis of thin shell structures is formulated to include large deflection and plastic deformation effects. In this paper the details of the finite element formulation of the problem using total Lagrangian coordinate systems are presented and different element matrices are given. For plastic materials following the Prandtl-Reuss flow rule with isotropic strain hardening a multi-layer approach using a subincremental technique is employed. Numerical results on the performance of the element for a variety of applications are presented. These computer studies include complete load-deflection curves into the post-buckling range and comparisons are made with other existing results. Current experience with the element indicates that it is a reliable and competitive element for nonlinear analysis of shells of general geometry.     

Large deformation elastic-plastic buckling analysis of spherical caps with initial imperfections

Large deformation elastic-plastic buckling loads are obtained for axisymmetric spherical caps with initial imperfections. The problem formulation is based on equilibrium equations in which the plastic deformation is taken as an effective plastic load. Both perfectly plastic and strain hardening behavior are considered. Strain hardening is represented by the Prager-Ziegler kinematic hardening theory, so that the Bauschinger effect is accounted for. Solutions of elastic-plastic circular plates and spherical caps are in good agreement with previous results. For the spherical cap it was determined that both initial imperfection and plastic deformation have the same effect of reducing buckling capacity; as the magnitude of the imperfection increases, the influence of plastic deformation becomes less important. It is also found that the geometric parameter λ, which is used as an important factor in elastic response, becomes meaningless for the elastic-plastic buckling analysis of spherical caps.     

An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model

Pure elastic damage models or pure elastic plastic constitutive laws are not totally satisfactory to describe the behaviour of concrete. They indeed fail to reproduce the unloading slopes during cyclic loading which define experimentally the value of the damage in the material. When coupled effects are considered, in particular in hydro-mechanical problems, the capability of numerical models to reproduce the unloading behaviour is essential, because an accurate value of the damage, which controls the material permeability, is needed. In the context of very large size calculations that are needed for 3D massive structures heavily reinforced and pre-stressed (such as containment vessels), constitutive relations ought also to be as simple as possible. Here an elastic plastic damage formulation is proposed to circumvent the disadvantages of pure plastic and pure damage approaches. It is based on an isotropic damage model combined with a hardening yield plastic surface in order to reach a compromise as far as simplicity is concerned. Three elementary tests are first considered for validation. A tension test, a cyclic compression test and triaxial tests illustrate the improvements achieved by the coupled law compared to a simple damage model (plastic strains, change of volumetric behaviour, decrease in the elastic slope under hydrostatic pressures). Finally, one structural application is also considered: a concrete column wrapped in a steel tube.     

Determination of the growth strain of LPCVD polysilicon

This work presents a semi-empirical procedure for determining the through-the-thickness variation of the eigenstrain (eigenstrain is a generic term for any inelastic strain, including plastic strain, free thermal expansion, phase transformation, etc.) that develops during the growth of thin polysilicon films formed using low-pressure chemical vapor deposition (LPCVD). This variation is assumed to depend on the polysilicon microstructure and deposition conditions, but not on the characteristics of the (single crystal silicon) substrate. The procedure involves the use of an elastic laminated plate model to determine the eigenstrain distribution that predicts the experimentally measured substrate curvatures. In comparison to the "shaving method" presented by A.Ni et al., which relies on incremental etching of a single specimen, an alternative experimental procedure is followed to measure the substrate curvatures of a series of different thickness films. While being significantly more time-consuming, the alternative procedure was expected to lead to improved predictions of the eigenstrain distribution, as it avoids the nonuniform film thicknesses produced by the etching procedure. However, a comparison of the curvature histories measured using the two approaches demonstrates that, as long as sufficiently small increments are used in the shaving method, then the improvement is insignificant. This suggests that the plasma etching does not alter the polysilicon's intrinsic growth strain, and that the etch rate nonuniformities across the substrate are small. The eigenstrain distributions could be used, in conjunction with structural mechanics models, to design multilayered polysilicon devices with prescribed curvatures.