Ductile damage analysis of elasto‐plastic shells at large inelastic strains

The objective of this contribution is to model ductile damage phenomena under consideration of large inelastic strains, to couple the corresponding constitutive law with a multi‐layer shell kinematics and to give finally an adequate finite element formulation. An elastic–plastic constitutive law is formulated by using a spatial hyperelasto‐plastic formulation based on the multiplicative decomposition of the deformation gradient. To include isotropic ductile damage the continuum damage model of Rousselier is modified so as to consider large strains and additionally extended by various void nucleation and macro‐crack criteria. In order to achieve numerical efficiency, elastic strains are supposed to be sufficiently small providing a numerical effective integration based on the backward Euler rule. Finite element formulation is enriched by means of the enhanced strain concept. Thus the well‐known deficiencies due to incompressible deformations and the inclusion of transverse strains are avoided. Several examples are given to demonstrate the performance of the algorithms developed concerning large inelastic strains of shells and ductile damage phenomena. Copyright © 2000 John Wiley & Sons, Ltd.     

High rate material behaviour at hot forming conditions

Modern processes of hot forming use very high strain rates and large, mostly incrementally applied strains. For the simulation of such forming processes relevant material data are needed, which have to be recorded under accordant forming conditions. This places extraordinary demands on the experimental technique, because high temperatures, high strain rates and large strains have to be implemented simultaneously. In the following contribution such an aligned experimental technique is introduced. In order to apply a very large range of strain rate, compression tests are performed using different technical equipment up to a drop tower and Hopkinson pressure bar. To reach large plastic strains, a hot torsion test was developed, which allowed true plastic shear strains up to 10 at strain rates in the order of 100 1/s at hot forming temperatures.     

Macroscopic model with anisotropy based on micro–macro information

Physical experiments can characterize the elastic response of granular materials in terms of macroscopic state variables, namely volume (packing) fraction and stress, while the microstructure is not accessible and thus neglected. Here, by means of numerical simulations, we analyze dense, frictionless granular assemblies with the final goal to relate the elastic moduli to the fabric state, i.e., to microstructural averaged contact network features as contact number density and anisotropy. The particle samples are first isotropically compressed and then quasi-statically sheared under constant volume (undrained conditions). From various static, relaxed configurations at different shear strains, infinitesimal strain steps are applied to “measure” the effective elastic response; we quantify the strain needed so that no contact and structure rearrangements, i.e. plasticity, happen. Because of the anisotropy induced by shear, volumetric and deviatoric stresses and strains are cross-coupled via a single anisotropy modulus, which is proportional to the product of deviatoric fabric and bulk modulus (i.e., the isotropic fabric). Interestingly, the shear modulus of the material depends also on the actual deviatoric stress state, along with the contact configuration anisotropy. Finally, a constitutive model based on incremental evolution equations for stress and fabric is introduced. By using the previously measured dependence of the stiffness tensor (elastic moduli) on the microstructure, the theory is able to predict with good agreement the evolution of pressure, shear stress and deviatoric fabric (anisotropy) for an independent undrained cyclic shear test, including the response to reversal of strain.     

An analysis of a new class of integration algorithms for elastoplastic constitutive relations

An accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors, is carried out in this paper. For simplicity, attention is confined to infinitesimal deformations. The integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations. General rate-independent and rate-dependent behaviour, with plastic hardening or softening, associated or non-associated flow rules and nonlinear elastic response can be efficiently treated within the present framework. Isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding.     

Strain Hardening of Initially Isotropic Metals under Loading along Slightly Curved Trajectories

By the example of martensitic steel we study regularities of strain hardening under loading along two-link broken paths corresponding to slightly curved strain paths. It is shown that the loading surface separating the domains of elastic and elastoplastic strains (yield surface) is displaced in the direction of a vector connecting the surface center with the loading path image point, while the shape of its frontal part remains unchanged. The yield surface center displacement versus the intensity of accumulated plastic strains is described by a curve invariant to the loading path.     

Boundary element method for shear deformable plates with combined geometric and material nonlinearities

In this paper a new boundary element formulation for shear deformable plate theory with combined geometric and material nonlinearities is presented. The material is assumed to undergo large deflection with small strains. The von Mises criteria is used to evaluate the plastic zone and an elastic perfectly plastic material behaviour is assumed. An initial stress formulation is used to formulate the boundary integral equations. The domain integrals involving geometrical and material nonlinear terms are evaluated using a cell discretization technique. A total incremental method is applied to solve the nonlinear boundary integral equations. Numerical examples are presented to demonstrate the validity and the accuracy of the proposed formulation.     

Formulation and numerical treatment of incompressibility constraints in large strain elastic–plastic analysis

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate‐independent finite strain analysis of solids undergoing large elastic‐isochoric plastic deformations. The formulation relies on the introduction of a mixed‐variant metric deformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure which can be additively decomposed into the exact isochoric (deviatoric) and volumetric (spheric) strain measures. This fact may be seen as the basic idea in the formulation of appropriate mixed finite elements which guarantee the accurate computation of isochoric strains. The mixed‐variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response whereas the plastic material behavior is assumed to be governed by a generalized J₂ yield criterion and rate‐independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on higher‐order Padé approximations. To be able to take into account the plastic incompressibility constraint a modified mixed variational principle is considered which leads to a quasi‐displacement finite element procedure. Finally, the numerical solution of finite strain elastic‐plastic problems is presented to demonstrate the efficiency and the accuracy of the algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

A notch strain calculation of a notched specimen under axial-torsion loadings

In this paper, a notch analysis model is presented for the numerical prediction of multiaxial strains of a notched 1070 steel specimen under combined axial and torsion loadings. The proposed model is based on the notion of a structural yield surface and uses a small-strain cyclic plasticity model to describe stress–strain relations. A notch load–strain curve is calculated with Neuber’s rule and incremental nonlinear finite element analysis. The presented model is applied to simulate the notch root deformations of a circumferentially notched specimen under cyclic tension–compression–torsion loading histories. The model predictions are evaluated with strain measurements at the notch root of the specimen in a comprehensive set of cyclic tests. The computed strain loops were in accord with experimental data and matched qualitatively with measured shear–axial strain histories irrespective of loading path of the test. In proportional balanced torsion-axial loading, the nonlinear shear strain–axial strain loops were calculated properly. The modeling errors were determined to be a function of the loading path shape, and compared to shear strains, axial strain predictions were more accurate.     

F.E.M. ANALYSIS OF THE EFFECT OF ASPECT RATIO IN PLANE STRAIN COMPRESSION OF A PRESHAPED STRAIN HARDENING MATERIAL BETWEEN INCLINED TOOLS.

Metal forming problems generally involve large plastic deformations. in cold forming most materials have a nonlinear strain hardening behaviour. F.E.M. solutions of. such problems require elastic plastic analysis and an incremental approach. While a number of such approaches exist they require fairly small step size and hence a large computational effort

Madnaik, Maiti and Chaturvedi (1) had proposed a modification to the tangent modulus method of Yamada et al(2) and showed that this method is able to solve the problem of plane strain compression with considerably less computational effort and reasonable accuracy; and presented results to show the stress distributions at various levels of reductions and for different aspect ratios(3)

This paper uses the same procedure termed the Modified Elastic Plastic Incremental Analysis Procedure (MEPIAP) for obtaining solution to another problem i.e. Plane Strain Compression of Preshaped Material between Inclined Tools.     

An analytical solution for a low velocity impact between a rigid sphere and a transversely isotropic strain-hardening plate supported by a rigid substrate

An analytical solution for a low velocity impact between a thin transversely isotropic plate made of a strain-hardening material supported by a rigid substrate and a rigid sphere is presented. One of the novelties of this work is employing a linear strain-hardening model for investigating the indentation phenomenon in the plastic zone, rather than the traditional perfectly plastic model. Another novelty of this work is employing the homotopy perturbation method to derive analytical solutions for the highly nonlinear governing equations of contact. Since it is very important to accurately predict the contact force and its time history, the three stages of the indentation process, i.e., (1) the elastic indentation, (2) the plastic indentation, and (3) the elastic unloading stages, are investigated in detail. Comparison of the present results with results obtained from the iterative numerical time integration method confirms the accuracy and efficiency of the present solution.     

On the mechanics of large inelastic deformations: kinematics and constitutive modeling

Summary In this paper we examine the complexities associated with the kinematics of finite elastoplastic deformations and other issues related to the development of constitutive equations. The decomposition of the total strain and strain rate tensors into elastic and plastic constituents is investigated by considering both a multiplicative decomposition of the deformation gradient and an additive decomposition of the deformation vector field. Physically based definitions for the elastic and plastic strain rate tensors are given and compared with other values found in the literature. Constitutive equations for the plastic flow are derived by considering both a phenomenological-energy approach and a physically motivatedmesomechanical approach based on the double-slip idealization. It is shown that by resorting to the mechanics of the double slip, specific relations for the plastic stretching and plastic spin can be rigorously derived, taking into account the effect of noncoaxiality and material rotation. Finally, the implication of such effects to large deformations is examined in connection with the localization phenomenon.     

The application of optical fibres as witness devices for the detection of plastic strain and cracking

Low cost optical fibres have recently become readily available for telecommunications purposes. Silica fibres are characterised by high elastic strains to failure. The feasibility of using these fibres for structural integrity monitoring particularly for offshore structures is investigated. The basis of the technique is that a fibre may be bonded to a critical part of a structure and provides an optical path which will be broken if the fibre fails due to plastic strain or crack opening in the critical area.
Groups of fibres which have been given predetermined fracture strains by surface etching were encapsulated in special packs. These packs were bonded to steel and concrete tensile specimens. Strain transfer occurred successfully between the specimens and individual fibres. The distribution of strain to fibre fracture appeared to be uniform along the fibre. The use of several fibres with a range of fracture strains caused fibres to break progressively with increasing strain. For applications to offshore structures it has been found possible to use water-repellent adhesives which can be applied and cured in sea water and suffer no deterioration.
The advantages of this system include versatility, relatively low cost, adaptability to continuous monitoring and the possibility of being fitted retrospectively and refitted after repair operations.     

Irreversible deformation of metal matrix composites: A study via the mechanism-based cohesive zone model

An elastic–plastic interface model at finite deformations is utilized to predict the irreversible deformation of metal matrix composites (MMCs) under the transverse loading and unloading conditions. The associated benefit of the cohesive model is to provide a physical insight on the main irreversible deformation mechanisms, i.e., the geometrically nonlinear, localized plastic deformation and damage induced debonding, at the interface of MMCs. The extensive parametric study is conducted using this cohesive model to investigate the effects of the cohesive parameters on the stress–strain response of MMCs under transverse loading. Further, the ductile mechanism of the matrix is considered to characterize the competition between the plastic flow of the matrix and the inelastic interface induced irreversible deformation. Moreover, the predictions using the cohesive model are compared with those available experimental data in the literature to demonstrate the inelastic behaviors, including the interfacial plasticity and damage induced debonding, as well as the plastic flow of the matrix. The numerical results of the stress–strain responses for both loading and unloading conditions show good agreements with those obtained by the experiment. The deformation and failure modes of MMCs predicted by the model are also consistent with the observations of the experiment.     

复合材料多层板壳在非线性弹性失稳时的本构关系

在研究复合材料的多层、夹层和加筋板壳的非线性弹性和塑性失稳问题时,需要用到失稳前的应力与应变之间和失稳时的应力增量与应变增量之间的非线性关系。复合材料在纤维断裂和基体开裂以前的物理非线性,主要是由基体引起的。本文对非线性弹性和处于主动塑性变形阶段的各向同性基体,采用小弹塑性变形理论,导得了应力与应变之间的各向同性非线性关系和应力增量与应变增量之间的各向异性非线性关系。假定纤维是线性弹性的。用复合材料的复合定律,求得了单向增强纤维复合材料层片的应力与应变之间和应力增量与应变增量之间的非线性关系。     

On the incremental plastic work and related aspects of invariance – Part I

Summary Taking for granted that the free energy function is invariant under a change of a finite strain measure and/or the reference configuration, Hill's transformation rules for selected fundamental constitutive quantities (such as tangent elastic modulus, plastic increments of total strain and work conjugate stress, the work of work-conjugate stress, the work expended in the plastic part of incremental strain etc.) are derived in a manner different from that of Hill. On this background distinguished by Hill [6] subtle aspects of invariance in mechanics of elastic plastic solids are discussed. It is shown that the plastic part of the increment of elastic strain energy (when taken with reverse sign) defines the true invariant incremental plastic work which in general is not equal to the work expended in the plastic part of the strain increment. It plays the role of a potential for the plastic part of the increment of work-conjugate stress. This fundamental fact has not found proper account in the literature. The analytical interrelations between two apparently different theoretical frameworks, Hill-Rice (fixed reference configuration) and Eckart-Mandel (mobile unloaded configuration) are discussed showing their equivalence. Since the transformation rules are complex in the general 3D case, the first part of the paper illustrates instructively the discussed aspects in a 1D situation (simple tension or simple extension).     

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

This paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.     

Design sensitivity analysis and optimization of non‐linear transient dynamics. Part I—sizing design

A continuum‐based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non‐linear structural systems with elastic–plastic material and large deformation. The methodology is aimed for applications in non‐linear dynamic problems, such as crashworthiness design. The first‐order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct differentiation method and updated Lagrangian formulation are used since they are more appropriate for the path‐dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central difference method and the finite element method are used to discretize the temporal and spatial domains, respectively. The Hughes–Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate‐based incrementally objective stress integration scheme are used to handle the finite strain and rotation. An elastic–plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the finite difference results. Design optimization is carried out using the design sensitivity information. Copyright © 2000 John Wiley & Sons, Ltd.     

Experimental and finite element simulation methods for rate-dependent metal forming processes

An experimental procedure and a finite element simulation method for rate-dependent metal forming processes are developed. The development includes the formulation of a tangential stiffness matrix for an axisymmetric solid finite element with four node, eight degree of freedom, quadrilateral cross-section. The formulation includes the effects of elasticity, viscoplasticity, temperature, strain rate and large strains. The solution procedure is based on a Newton-Raphson incremental-iterative method which solves the non-linear equilibrium equations and gives temperatures and incremental stresses and strains. Three examples are studied. In example 1, finite element simulation for the upsetting of a cylindrical workpiece between two perfectly rough dies is performed and the results are compared with alternative finite element solutions. In examples 2 and 3, both experimental and finite element studies are performed for the upsetting of a cylindrical billet and the forging of a ball, respectively. Annealed aluminium 1100 workpieces are used in both examples. For the finite element analysis, uniaxial compression tests are first performed to provide the material properties. The tests generate elastic moduli and two sets of stress-strain curves (quasi-static and constant strain rate), which are used to establish a rate-dependent material model for input. For both examples 2 and 3, comparisons between the experimental and finite element simulation results for the forming force vs. die displacement relations and also for the deformed configurations show good agreement. The versatility of finite element methods allows for displaying detailed knowledge of the metal forming process, such as the distributions of temperature rise, yield stress, effective stress, plastic strain, plastic strain rate, forming forces and deformed configurations, etc. at any instance during the forming process.     

Stress based prediction of formability and failure in incremental sheet forming

A strain-based forming limit criterion is widely used in sheet-metal forming industry to predict necking. However, this criterion is usually valid when the strain path is linear throughout the deformation process [1]. Strain path in incremental sheet forming is often found to be severely nonlinear throughout the deformation history. Therefore, the practice of using a strain-based forming limit criterion often leads to erroneous assessments of formability and failure prediction. On the other hands, stress-based forming limit is insensitive against any changes in the strain path and hence it is first used to model the necking limit in incremental sheet forming. The stress-based forming limit is also combined with the fracture limit based on maximum shear stress criterion to show necking and fracture together. A derivation for a general mapping method from strain-based FLC to stress-based FLC using a non-quadratic yield function has been made. Simulation model is evaluated for a single point incremental forming using AA 6022-T43, and checked the accuracy against experiments. By using the path-independent necking and fracture limits, it is able to explain the deformation mechanism successfully in incremental sheet forming. The proposed model has given a good scientific basis for the development of ISF under nonlinear strain path and its usability over conventional sheet forming process as well.