A full tangent stiffness field-boundary-element formulation for geometric and material non-linear problems of solid mechanics

The field-boundary-element method naturally admits the solution algorithm in the incompressible regimes of fully developed plastic flow. This is not the case with the generally popular finite-element method, without further modifications to the method such as reduced integration or a mixed method for treating the dilatational deformation. The analyses by the field-boundary-element method for geometric and material non-linear problems are generally carried out by an incremental algorithm, where the velocities (or displacement increments) on the boundary are treated as the primary variables and an initial strain iteration method is commonly used to obtain the state of equilibrium. For problems such as buckling and diffused tensile necking, involving very large strains, such a solution scheme may not be able to capture the bifurcation phenomena, or the convergence will be unacceptably slow when the post-bifurcation behaviour needs to be analysed. To avoid this predicament, a full tangent stiffness field-boundary-element formulation which takes the initial stress–velocity gradient (displacement gradient) coupling terms accurately into account is presented in this paper. Here, the velocity field both inside and on the boundary are treated as primary variables. The large strain plasticity constitutive equation employed is based on an endochronic model of combined isotropic/kinematic hardening plasticity using the concepts of material director triad and the associated plastic spin. A generalized mid-point radial return algorithm is presented for determining the objective increments of stress from the computed velocity gradients. Numerical results are presented for problems of diffuse necking, involving very large strains and plastic instability, in initially perfect elastic–plastic plates under tension. These results demonstrate the clear superiority of the full tangent stiffness algorithm over the initial strain algorithm, in the context of the integral equation formulations for large strain plasticity.     

Bounding-surface plasticity for non-linear analysis of space structures

This paper presents a method for the non-linear analysis of space structures subjected to static and cyclic loading. A bounding-surface kinematic hardening plasticity model is used to simulate the hardening and hysteritic material behaviour. The model is used in conjunction with the lumped plasticity assumption coupled with the concept of a yield surface in force space. A hardening coefficient matrix which is a function of the plastic strain and the elastic stiffness matrix is introduced while the vectorial nature of the material memory parameters is maintained. This provides a smooth transition from the elastic to the plastic regime which simulates the hysteresis loops quite accurately. An updated Lagrangian formulation is used together with a predictor/corrector solution method. Several examples are presented to demonstrate the accuracy and efficiency of the method.     

DEVELOPMENT OF A CONSTITUTIVE MODEL FOR BIAXIAL LOW-CYCLE FATIGUE

Abstract— A version of the endochronic theory of plasticity for the modelling of nonproportional cyclic loading has been developed. To describe an additional hardening of a material a new engineering method for defining a nonproportionality parameter is proposed for a wide class of cyclic strain paths with a prescribed maximum range of plastic or total strains. This parameter makes it possible to establish an unambiguous linear dependence between the cyclic strain path shape and the stress level in a stabilized state. Conjugation conditions have been formulated to describe complex histories of nonproportional cyclic loading. The results of the modelling are shown to be in fair agreement with the experimental data.     

Nonlinear response in glass fibre non-crimp fabric reinforced vinylester composites

This paper addresses the nonlinear stress-strain response in glass fibre non-crimp fabric reinforced vinylester composite laminates subjected to in-plane tensile loading. The nonlinearity is shown to be a combination of brittle and plastic failure. It is argued that the shift from plastic to brittle behaviour in the vinylester is caused by the state of stress triaxiality caused by the interaction between fibre and vinylester. A model combining damage and plasticity is calibrated and evaluated using data from extensive experimental testing. The onset of damage is predicted using the Puck failure criterion, and the evolution of damage is calibrated from the observed softening in plies loaded in transverse tension. Shear loading beyond linear elastic response is observed to result in irreversible strains. A yield criterion is implemented for shear deformation. A strain hardening law is fitted to the stress-strain response observed in shear loaded plies. Experimental results from a selection of laminates with different layups are used to verify the numerical models. A complete set of model parameters for predicting elastic behaviour, strength and post failure softening is presented for glass fibre non-crimped fabric reinforced vinylester. The predicted behaviour from using these model parameters are shown to be in good agreement with experimental results.     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

Numerical modelling of ductile damage evolution in tensile and bending tests of timber structures

The characteristic values for strength and stiffness of all sorts of timber products are based on the assumption of a linear relation between stress and strain prior to failure and consequently verification of the load-bearing capacity of individual members in a construction is also based on a similar linear relation. Such an approach is very conservative and ill suited for performance-based design, which requires a full analysis of the structure with the possibility of moment and/or stress redistribution within parts of the structure. The development of material models that encompass the complex behaviour of wood is therefore necessary. The present work presents a model formulated within the frameworks of plasticity and continuum damage mechanics (CDM). It applies the classical flow theory of plasticity to formulate ductile failure of wood in compression and damage mechanics for the brittle failure modes. It takes into account the orthotropic elastic behaviour, the plastic anisotropic isotropic hardening, the isotropic ductile damage, and the large plastic deformations. The model was used to predict the initiation and growth of ductile damage in tensile and bending tests on different timbers types. Good agreement was found between the predictions of the model and the experimental results.     

A high-cycle fatigue life model for variable amplitude multiaxial loading

A high‐cycle fatigue life model for structures subjected to variable amplitude multiaxial loading is presented in this paper. It treats any kind of repeated blocks of variable amplitude multiaxial loading without using a cycle counting method. This model based on a mesoscopic approach is characterized by the following features: (i) the choice of a damage factor related to the accumulated mesoscopic plastic strain per stabilised cycle; (ii) the use of a mesoscopic mechanical behaviour taking into account the fatigue mechanisms such as plasticity and void growth. This behaviour is a von Mises elastoplastic model with linear kinematic hardening and hydrostatic stress dependent yield stress. The fatigue life model has six parameters identified with one SN curve and two fatigue limits. In‐phase and out‐of‐phase experimental tests from the literature are simulated. The predicted fatigue lives are compared to experimental ones.     

Werkstoffmechanische Beanspruchungsanalyse gekerbter Strukturen bei synchronen Belastungskombinationen

Elastic‐Plastic analysis of notched structures under synchronous cyclic loading This paper focuses on analysing the elastic‐plastic stress‐strain behaviour at the failure‐critical location of notched components and structures under multiaxial synchronous cyclic loading. For this, various load configurations are investigated numerically consisting of constant and cyclic load components with constant and variable amplitudes. The von Mises yield criterion and the kinematic hardening rule of Prager and Ziegler describe the elastic‐plastic material property. The finite element software of ABAQUS is used to solve the boundary element problem. A parametrical study is carried out and numerical results are presented to show the effects of load amplitude, mean load and spectrum shape on the local stress‐strain paths.     

Asymptotic deformation and stress fields at the tip of a sharp notch in an elastic-plastic material

The singular elastic-plastic stress, strain and the displacement fields at the tip of a sharp notch for both plane stress and plane strain conditions are investigated analytically. The material is assumed to be governed by the deformation theory of plasticity with linear strain hardening characteristic. Since the elastic strain is retained in the analysis, the final strain and displacement fields can be separated into the elastic and the plastic parts. In the case with zero notch angle, the results reduce to the classical crack problem. The relationship of the amplitude of the near crack tip elastic-plastic field to the elastic far field is obtained. Both mode I and mode II cases are investigated. The mixed mode case is also discussed.     

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.     

Hybrid-stress models for elastic–plastic analysis by the initial-stress approach

Two alternative numerical methods are presented, based on the assumed-stress hybrid finite element model and the initial-stress approach, for the elastic–plastic small-deflection analysis of structures under static loading. The use of the initial-stress approach results in a set of simultaneous linear algebraic incremental equations to be solved at each loading step, with the elastic stiffness matrix remaining unchanged throughout the loading process and the effects of plasticity included as equivalent element loads. The derivation of these alternative methods differs in the assigning of the incremental stress which satisfies equilibrium; in one case it is the actual stress increment while in the other it is a fictitious stress increment. An equilibrium imbalance correction is included in each of these methods to prevent drifting of the solution during the incremental process. Example solutions are presented which demonstrate the accuracy of the two methods and permit comparisons of the relative efficiencies of the two methods.     

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.     

A robust kinematic hardening rule for cyclic plasticity with ratchetting effects Part II. Application to nonproportional loading cases

Summary Performance of the proposed kinematic hardening rule is examined using several examples of cyclic plasticity phenomena observed in experiments. Results obtained and compared with experimental observations on various loading histories are presented. With the memory effects added to the model, impressive results are obtained without using an anisotropic yield model. Drifting of the yield surface occurs during the numerical computation of the plastic response due to nonproportional loading paths. The drift due to the finite increments of stress or strain is corrected using a simple and efficient method proposed in this paper. The new kinematic hardening rule proposed for the limit surface as being related directly to the yield surface kinematic hardening rule ensures nesting using the blended rule discussed in the part presenting the theoretical formulation [14].     

J-ESTIMATION FOR SEMI-ELLIPTICAL SURFACE CRACKS IN WIDE PLATES UNDER DIRECT TENSION

Large marine structures can be subjected to extensive localized damage, with strains reaching 10  times the yield strain. Small defects might propagate, and accurate defect assessment is required for safe operation. To simulate this problem, J -integrals have been computed for semielliptical cracks in wide steel plates under tension. Three-dimensional (3D) elastic–plastic finite element analysis was used to model shallow crack geometries with 0.2 ≤ a / c ≤ 0.57 and 0.05 ≤ a / t ≤ 0.15. The material responses were linear elastic followed either by power hardening, or perfect plasticity and power hardening. It was found that, in contrast to previous studies on single edge notch geometries, the material law does not have a major influence on the J –strain behaviour. Results obtained from the 3D analyses form the basis for the development of a J -based estimation scheme.     

Theoretical investigation of elastoplastic notch fields under triaxial stress constraint

In this paper, an exact elastic-plastic solution has been obtained based on the J ₂-deformation theory of plasticity for a plate having a circular hole under biaxial tension and triaxial stress constraint in linear elastic strain-hardening materials. The theoretical solution shows that a linear elastic solution of the equivalent strain can be used to linear elastic-power hardening plastic situation just by a simple variable replacement. Then a strain equivalent rule (SER) is proposed to predict the elastoplastic notch fields by use of the elastic solution. Validations against theoretical analyses and finite element calculation for various combinations of material properties, triaxial stress constraints, load levels show that the SER can be used to predict stress-strain distributions in the whole plastic zone effectively and conveniently.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part II: 3‐D non‐linear behaviour

In a companion paper, the effects of approximations in the flexural‐torsional stability analysis of beams was studied, and it was shown that a second‐order rotation matrix was sufficiently accurate for a flexural‐torsional stability analysis. However, the second‐order rotation matrix is not necessarily accurate in formulating finite element model for a 3‐D non‐linear analysis of thin‐walled beams of open cross‐section. The approximations in the second‐order rotation matrix may introduce ‘self‐straining’ due to superimposed rigid‐body motions, which may lead to physically incorrect predictions of the 3‐D non‐linear behaviour of beams. In a 3‐D non‐linear elastic–plastic analysis, numerical integration over the cross‐section is usually used to check the yield criterion and to calculate the stress increments, the stress resultants, the elastic–plastic stress–strain matrix and the tangent modulus matrix. A scheme of the arrangement of sampling points over the cross‐section that is not consistent with the strain distributions may lead to incorrect predictions of the 3‐D non‐linear elastic–plastic behaviour of beams. This paper investigates the effects of approximations on the 3‐D non‐linear analysis of beams. It is found that a finite element model for 3‐D non‐linear analysis based on the second‐order rotation matrix leads to over‐stiff predictions of the flexural‐torsional buckling and postbuckling response and to an overestimate of the maximum load‐carrying capacities of beams in some cases. To perform a correct 3‐D non‐linear analysis of beams, an accurate model of the rotations must be used. A scheme of the arrangement of sampling points over the cross‐section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3‐D non‐linear elastic–plastic behaviour of beams. Copyright © 2001 John Wiley & Sons, Ltd.     

Nonlinear subincremental method for determination of elastic–plastic–creep behaviour

The frequently used subincremental method has so far been based on a linear interpolation of the total strain path within each main step. This method has proven successful when elastic–plastic behaviour and secondary creep is involved. The present paper proposes a nonlinear subincremental method applicable to general elastic–plastic–creep behaviour including problems with a highly nonlinear total strain path caused by the occurrence of creep hardening. This nonlinear method degenerates to the linear-approach for elastic–plastic behaviour and when secondary creep is present. It is also linear during step loadings and it becomes increasingly more nonlinear, the more creep hardening deformations dominate the behaviour. A wide range of structures are analysed and the results from both subincremental methods are compared; the nonlinear strategy increases the accuracy by a factor of typically 3–5 without affecting the computer time. Moreover, the implementation of the nonlinear method is extremely simple. The optimum number of substeps in each main step is found to be around 5. For such a choice, the advantage of using the subincremental method as compared to the more conventional solution technique, where only one type of time step is used, is a significant reduction in computer time without, in practice, affecting the accuracy.     

Finite element analysis of a cyclically loaded notched bar using constitutive models based on the cyclic nonhardening region

It is shown that simple and elaborate constitutive models based on the cyclic nonhardening region are implemented without difficulty in existing finite element methods. The cyclic nonhardening region, which is defined in plastic strain space, enables us to describe an important feature of cyclic plasticity, i. e., dependence of cyclic hardening of materials on the cyclic strain and stress ranges. This dependence is essential to structural analysis of cyclic plasticity, because structures have, in general, nonuniform distributions of the cyclic strain and stress ranges, resulting in nonuniform development of cyclic hardening. An axisymmetric notched bar subjected to axial cyclic loading is analyzed as an example, and the results by the simple and elaborate models are compared.This paper was presented at International Conference on Computational Mechanics, May 25–29, 1986, Tokyo     

Effect of silicon carbide particulate on cyclic plastic strain response characteristics and fracture of aluminium alloy composites

The cyclic stress-response characteristics of powder-metallurgy-processed high-purity aluminium alloy 2124 discontinuously reinforced with varying volume fractions of silicon carbide particulates were studied over a range of plastic strains. The specimens were cycled using tension/compression loading under total strain control. The composite material, in the heat-treated condition, displayed cyclic hardening at all cyclic strain amplitudes and for different volume fractions of the ceramic reinforcement in the aluminium alloy matrix. The degree of hardening was observed to be greater at the higher cyclic strain amplitudes than at corresponding lower strain amplitudes. Micromechanisms controlling the hardening response during cyclic straining are highlighted and rationale for the observed hardening behaviour is attributed to concurrent and competing influences of an increase in dislocation-dislocation interaction, dislocation multiplication and dislocation-particle interactions, and is a mechanical effect. The kinetics of the cyclic fracture process of the composite alloy is discussed in light of composite microstructural effects, plastic strain amplitude and concomitant response stress.