Characterization of heterogeneous materials under shear loading at finite strain

This article presents a homogenization procedure to predict the effective shear response of heterogeneous materials at large deformation. Assuming local periodicity, heterogeneous microstructure is identified by a representative volume element that is subjected to an equivalent macroscopic deformation field. The energy balance and periodicity conditions are considered to relate macro and micro-stress fields. Based on the symmetrical planes of the microstructure and local periodicity, it is shown that the analysis of one-quarter of three-dimensional representative volume element is enough to evaluate the effective shear response at finite deformation. A computational method is subsequently developed to obtain the shear response of heterogeneous microstructures. The homogenization procedure is implemented to evaluate shear response of two specific heterogeneous materials, elastomeric composite and reinforced viscoelastic fluid. The performance is successfully verified by comparison of the deformation in the macroscopic level to the response of a homogenized cell.     

Discussion and calculation on welding residual longitudinal stress and plastic strain by finite element method

In recent years, some researchers have put forward the new viewpoint that the weld is merely formed during the cooling process, not concerned with the heating process. According to this view, it can be concluded that it is not the compressive but the tensile plastic strain that may remain in the weld. To analyze the formation mechanism of the longitudinal residual stress and plastic strain, finite element method (FEM) is employed in this paper to model the welding longitudinal residual stress and plastic strain. The calculation results show that both the residual compressive plastic strain and the tensile stress in the longitudinal direction can be found in the weld.     

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.     

On the influence of kinematic hardening on plastic anisotropy in the context of finite strain plasticity

The paper discusses the application of a newly developed material model for finite anisotropic plasticity to the simulation of earing formation in cylindrical cup drawing. The model incorporates Hill-type plastic anisotropy, nonlinear kinematic and nonlinear isotropic hardening. The constitutive framework is derived in the context of continuum thermodynamics and represents a multiplicative formulation of anisotropic elastoplasticity in the finite strain regime. Plastic anisotropy is described by means of second-order structure tensors which are used as additional tensor-valued arguments in the representation of the yield criterion and the plastic flow rule. The evolution equations are integrated by a form of the exponential map that fullfils plastic incompressibility and preserves the symmetry of the internal variables. The numerical examples investigate the influence of the hardening behaviour on an initially anisotropic yield criterion. In particular, the influence of using the kinematic hardening component of the model in addition to isotropic hardening in the earing simulations is examined. Comparisons with test data for aluminium and steel sheets display a good agreement between the finite element results and the experimental data.     

On the dynamic stress intensity factors in finite cracked bodies under harmonic loading

A method of computing stress intensity factors under harmonic loading is suggested. The method is based on the representation of the stress intensity factors in form of superposition of the modal stress intensity factors corresponding to the normalized free vibration modes with some weight coefficients. The motion equations are built with the help of the finite element method. The singularity of stresses in the crack tip is taken into account using special finite elements. To demonstrate the accuracy and possibilities of the method, the dependence of the stress intensity factors of the first and second kind on frequency in square plate with inclined central crack under harmonic extension-compression was calculated.     

Structure of the plastic zone at the crack tip and the endurance of steel during low-cycle loading

From microhardness, metallographic, and also layered and sight x-ray analyses, the mechanisms controlling changes in the phase composition, structure, and size of the plastic zone at the crack tip during low-cycle loading of steels 12Kh18N10T and Kh11N10M2T have been established. These steels had various initial structures due to directional changes in their strength and ductility. It was shown that with increase in the maximum uniform elongation, there is an increase in the amount of intense structural changes in the plastic zone, an increase in the number of load cycles to failure, and a decrease in the rate of stable crack growth. These mechanical effects can be explained by the positive influence of the martensitic transformation and of dislocation mobility on the energy intensity of failure activation in the plastic zone. In particular, dislocation mobility leads to a partial relaxation of microdistortion in the crystallographic lattice of the matrix phase.Translated from Problemy Prochnosti, No. 8, pp. 23–31, August, 1993.     

Thermal effect of plastic dissipation at the crack tip on the stress intensity factor under cyclic loading

Plastic dissipation at the crack tip under cyclic loading is responsible for the creation of an heterogeneous temperature field around the crack tip. A thermomechanical model is proposed in this paper for the theoretical problem of an infinite plate with a semi-infinite through crack under mode I cyclic loading both in plane stress or in plane strain condition. It is assumed that the heat source is located in the reverse cyclic plastic zone. The proposed analytical solution of the thermo-mechanical problem shows that the crack tip is under compression due to thermal stresses coming from the heterogeneous stress field around the crack tip. The effect of this stress field on the stress intensity factor (its maximum and its range) is calculated analytically for the infinite plate and by finite element analysis. The heat flux within the reverse cyclic plastic zone is the key parameter to quantify the effect of dissipation at the crack tip on the stress intensity factor.     

Study of the strain anisotropy of metals during nonmonotonic plastic deformation in a linear stress state

This article examines an experimental verification of the theory of anisotropic strain-hardening proposed by G. Backhaus. The tests were conducted under linearstress-state conditions in cyles of tension alternating with compression. The tests were conducted on cylindrical specimens of steels 45 and 30KhGSA by a method which makes it possible to realize such cycles on one specimen. It was established that the parameters which determine the Bauschinger effect and the hereditary effect of the strain history in the above-noted theory are the material characteristics determined from tension-compression tests. Functions which approximate these parameters are presented. In accordance with the theory, a new material characteristic is proposed: . This characteristic is the strain interval within which the stress state should be determined with allowance for strain anisotropy when there is a change in the loading trajectory. For strains in excess of the interval , the stress state should be determined from the theory of isotropic strain-hardening. The experimental data that is obtained shows that the Backhaus theory can be used to reliably evaluate the stress-strain state in tension-compression.Translated from Problemy Prochnosti, No. 12, pp. 53–56, December, 1990.     

On uniqueness criteria and minimum principles for crystalline solids at finite strain

Summary New uniqueness criteria and minimum principles for quasi-static, rate-type boundary value problems in crystalline solids are derived. The analysis is based upon a general constitutive theory of finite elastic-plastic deformation encompassing the distinct mechanisms of lattice straining and crystallographic slip. Locally sufficient uniqueness criteria and a related minimum principle permitting the independent variation of velocity and plastic shear rates are established for the case when the two dominant principal stresses are everywhere tensile.

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Eindeutigkeitskriterien und Minimalprinzipien für kristalline Festkörper endlicher Verzerrungen

Zusammenfassung Es werden neue Eindeutigkeitskriterien und Minimalprinzipien für quasistatische inkrementelle Randwertprobleme in kristallinen Festkörpern hergeleitet. Die Analyse beruht auf einer allgemeinen konstitutiven Theorie endlicher, elastisch-plastischer Verformungen, welche die getrennten Vorgänge von Gitterdehnung und Kristallgleitung einschließen. Örtlich hinreichende Eindeutigkeitskriterien und ein damit zusammenhängendes Minimalprinzip ermöglichen die unabhängige Variation von Verformungsgeschwindigkeit und plastischer Gleitgeschwindigkeit und werden aufgestellt für den Fall, daß die beiden dominierenden Hauptspannungen überall positiv sind.

     

Distributions of stress and plastic strain in notched tensile bars

The paper deals with a classical problem in fracture mechanics, namely the determination of stresses and plastic strains at the minimum cross-section of a notched tensile specimen. Ideas of Hill and Bridgman are combined to develop a new approximate method for solving this problem. The main aim of the paper is to introduce a damage variable in the analysis without any complicated numerical procedure. The material model proposed by Lemaitre is used to predict the evolution of the damage variable. However, any other model resulting in the incompressibility condition may be adopted with no difficulties. This revised version was published online in July 2006 with corrections to the Cover Date.     

Pseudo-elastic description of polymeric foams at finite deformation with stress softening and residual strain effects

Polymeric foams are typical materials for energy absorber in such areas as aircraft, car industry and in the field of electronic packaging. Besides the typical hyperelastic behaviour, non-linear stress–strain behaviour in large elastic deformation, polymeric foams may also exhibit some inelastic effects, like stress softening and residual strain. In this paper we first describe some experiment results that illustrate the stress softening in compressible expanded polypropylene (EPP) foams together with associated residual strain effects. Then, based on Ogden and Dorfmann’s results, a pseudo-elastic model is introduced to capture the stress softening and residual strain effects by including of two variables in the energy function. Numerical simulations of uniaxial-compression tests of two types of EPP foam are used to determine the material parameters of Ogden’s model, stress softening and residual strain effects. The numerical simulations indicate that the pseudo-elastic model provides reasonably accurate predictions of the inelastic behaviour of polymeric foam.     

On distribution of the plastic strain around the crack tip at the onset of ductile fracture

A very simple model is presented to explain the change in the plastic strain from the crack tip. It is based on a relaxation mechanism by dislocations.

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Zusammenfassung Ein sehr einfaches Modell wird eingebracht zur erklärung der Änderung in der plastischen Verbiegung von der Rißspitze. Es beruht auf einer Relaxationsmechanik durch Versetzungen.

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Résumé Un modèle très simple est présenté pour expliquer le changement dans le gauchissement plastique dès la pointe fente. C'est conformé à un méchanisme de relaxation par dislocations.

     

Change in plastic strain amplitude with regular and programmed soft loading of steels

A procedure is suggested for calculating the change in plastic strain amplitude with multicycle soft regular and programmed loading of cyclically weakening metals. The effect is studied of average stresses on the change in plastic strain amplitude with regular loading. An equation is obtained for calculating the coefficient of the basic cyclic strain curve with asymmetric loading. The dependences suggested are compared with experimental data.Translated from Problemy Prochnosti, No. 2, pp. 39–44, February, 1991.     

Quadrilateral elements for the solution of elasto‐plastic finite strain problems

In this paper two plane strain quadrilateral elements with two and four variables, are proposed. These elements are applied to the analysis of finite strain elasto‐plastic problems. The elements are based on the enhanced strain and B‐bar methodologies and possess a stabilizing term. The pressure and dilatation fields are assumed to be constant in each element's domain and the deformation gradient is enriched with additional variables, as in the enhanced strain methodology. The formulation is deduced from a four‐field functional, based on the imposition of two constraints: annulment of the enhanced part of the deformation gradient and the relation between the assumed dilatation and the deformation gradient determinant. The discretized form of equilibrium is presented, and the analytical linearization is deduced, to ensure the asymptotically quadratic rate of convergence in the Newton–Raphson method. The proposed formulation for the enhanced terms is carried out in the isoparametric domain and does not need the usually adopted procedure of evaluating the Jacobian matrix in the centre of the element. The elements are very effective for the particular class of problems analysed and do not present any locking or instability tendencies, as illustrated by various representative examples. Copyright © 2001 John Wiley & Sons, Ltd.     

Distributions of stress and plastic strain in circumferentially notched tension specimens

Distributions of stress and plastic strain at the minimum diameter in circumferentially notched bars with three different notch profiles are deduced from observation of the displacements of bands within the material and a modified Bridgman analysis. The strain distributions were found to be non-linear; the degree of non-linearity being dependent on the degree of straining and strongly dependent upon notch geometry, which cannot be described only by the ratio of the minimum radius to the profile radius.

For two of the three profiles studied the longitudinal stress distributions were found to be nearly uniform across the minimum diameter whilst the third was found to be extremely non-uniform. Qualitative confirmation of the strain distribution results were obtained by use of transversely orientated specimens.     

On singular stress at the crack tip of a thick plate under in-plane loading

By superposition of three asymptotic solutions, which collectively satisfy all the equilibrium and compatibility equations as well as boundary conditions, the three-dimensional crack tip stresses of a thick plate with a through-the-thickness cut are shown to be of square root singularity throughout the thickness. The near tip deformation is shown to be plane strain with stress intensity factor changing in the thickness direction. All components of the singular stresses vanish on plate surfaces. The profile of the stress intensity factor in the thickness direction cannot be determined by the asymptotic analysis alone.     

Instantaneous plastic strain associated with stress changes during the steady state creep of Al and Al-4.20 at% Mg alloy

The instantaneous strains resulting from stress changes during steady state creep of polycrystalline aluminium and an Al-4.2 at % Mg alloy in the temperature range 100 to 300° C, have been determined. Instantaneous plastic strains were found in both materials for stress increments and decrements. For polycrystalline aluminium the instantaneous plastic strain on a stress increment, (+), was considerably larger than the instananeous strain on a stress increment, (–), whereas for Al-4.2 at % Mg (+) was approximately equal to (–). Work hardening rates determined from (+) and (–) for polycrystalline aluminium vary from about one-tenth to one-half of Young's modulus and depend strongly on temperature and stress. The need to improve existing creep theories to include both climb (recovery) and glide components is suggested.On leave from Department of Engineering Materials, University of Windsor, Windsor, Ontario N9B 3P4, Canada.