Showing papers in "European Journal of Mechanics A-solids in 1995"

Discontinuous and continuous damage evolution in Ogden-type large-strain elastic materials

Abstract: The paper presents an application of continuum damage mechanics to large-strain Ogden-type elastic materials. A Eulerian setting of elasticity with isotropic damage-caused softening effect is discussed exclusively in terms of the Finger tensor as a strain measure. The local damage accumulation is related to two phenomenological variables, the maximum value and the arclength of effective free energy attained during a typical loading process. The frame is specified for a rigorously decoupled volumetric-isochoric response based on a multiplicative split of the Finger tensor into spherical and unimodular parts. In this context, the damage is related to the isochoric part of the deformation only, which is assumed to be governed by an Ogden-type effective free-energy function formulated in terms of the eigenvalues of the unimodular part of the Finger tensor. A constitutive algorithm for the computation of the nominal stresses and nominal tangent moduli for plane problems is developed. Finally, some fundamental finite-element simulations for rubber-like materials are presented.

An improved Gurson-type model for hardenable ductile metals

Abstract: The aim of this paper is to improve the modelling of strain hardening within the framework of the famous Gurson model for porous ductile metals. Indeed, although the original derivation of this model, for an ideal-plastic matrix, was based on a micromechanical analysis of some representative volume element, namely a hollow rigid-plastic sphere loaded axisymmetrically, the extension to the case of a hardenable matrix was of purely phenomenological and macroscopic nature, and this entailed a number of drawbacks. The phenomenological model was incompatible with the classical, exact solution to the problem of a hollow rigid-hardenable sphere loaded hydrostatically; also, the prediction that for any loading path corresponding to a fixed triaxiality, the curve representing porosity as a function of equivalent strain depended only on the initial porosity and the triaxiality but not on the hardening exponent, was incorrect. A new model solving these difficulties, based on an approximate analysis of a hollow rigid-hardenable sphere subjected to some axisymmetric loading, is proposed. Two types of hardening are considered: isotropic, as in Gurson's original model, and kinematic, as in Mear and Hutchinson's variant of Gurson's model. Comparisons with some finite element simulations evidence the improvements brought.

Homogenization for granular materials

Abstract: The homogenization techniques allow the macroscopic constitutive equations of a given material to be derived from its microscopic behaviour. The case of granular material is difficult because the microscopic level must be described in discrete terms (contact forces and relative displacements). The present paper is devoted to a general discussion of this problem and to a presentation of different models for granular homogenization. Since we shall primarily be concerned with methodological issues and with comparison to the standard microcontinuous case, attention will be focussed to the simplest case of micro-elasticity assuming a linear contact law

Ultimate bearing capacity of shallow foundations under inclined and eccentric loads. I: Purely cohesive soil

Abstract: The problem of determining the bearing capacity of a strip footing resting on the surface of a homogeneous half space and subjected to an inclined, eccentric load is solved within the framework of the yield design theory, assuming the soil to be purely cohesive according to Tresca's strength criterion. The soil foundation interface is also cohesive, in terms of the homologous strength criterion. Both the static and kinematic approaches of the yield design theory are used to derive lower and upper bounds for the ultimate bearing capacity. For an inclined centered load, an almost exact solution is derved; for increasing values of the load eccentricities, the ultimate bearing capacity can only be bracketed but neverthless determined within a sufficient degree of accuracy from an engineering standpoint. In a companion paper, the same problem is solved for a soil and an interface with a tension cut-off. These two solutions represent extreme conditons for the tensile resistance of the purely cohesive soil and could be used to bracket the variations of the bearing capacity as a function of the soil tensile strength

Effect of inertia on the necking behaviour of ring specimens under rapid radial expansion

Abstract: Dynamic necking is analysed numerically for plane strain specimens under rapid deformation. Inertia and finite strain effects are accounted for, with the material described by a simple isotropic hardening elastic-plastic constitutive model, so that thermal softening and material strain-rate sensitivity are neglected. Plane strain tensile test specimens are analysed to confirm a size dependence of final failure found previously for round bars under rapid extension. The imperfection-sensitivity of this size dependence is discussed. Main focus is on necking in thin'ring specimens, thus modelling experiments in which electromagnetic loading was employed to expand ring specimens very rapidly. For perfect rings under axisymmetric loading there is no wave propagation around the circumference; but small initial imperfections change that, and a large number of necks around the circumference are predicted for various imperfections

Ultimate bearing capacity of shallow foundations under inclined and eccentric loads. II: Purely cohesive soil without tensile strength

Abstract: The problem of determining the bearing capacity of a strip footing resting on the surface of a homogeneous half space and subjected to an inclined, eccentric load, is solved within the framework of the yield design theory assuming that the soil is purely cohesive without tensile strength according to Tresca's strength criterion with a tension cut-off. The soil foundation interface is also purely cohesive, in terms of the homologous strength criterion with a tension cut-off. As in a companion paper [Salencon & Pecker, 1995], both the static and the kinematic approaches of the yield design theory are used. New stress fields are constructed, in order to comply with the condition of a tension cut-off within the soil medium, and new lower bounds are determined as substitues to those given in the companion paper. Velocity fields taking advantage of the tension cut-off contribution in the expression of the maximum resisting work are also implemented, giving new lower bounds. As may be expected from common sense and from the general results of the theory, it appears that the tension cutt-off condition within the soil medium results in lower values of the bearing capacity of the foundation, and that the gravity forces acting in the soil mass have a stabilizing effect

A macroscopic model of the swelling phenomenon of a saturated clay

Abstract: A non-linear poroelastic constitutive law is proposed for modelling the swelling phenomenon which occurs when a saturated clay is put into contact with a salt solution which is not in thermodynamic equilibrium with the pore fluid. The porous medium is composed of three phases : the solid skeleton, the free water in the connected porosity and the electrolyte in solution. The coefficients of the macroscopic constitutive law which take the electric interaction between the electrolyte and the negatively charged clay particles into account are determined theoretically from the results of the electric double layer theory.

Large displacement frictional contact: a continuum framework for finite element discretization

Abstract: The papers develops a continuum formulation of large displacement frictional contact of deformable bodies, which is suitable for finite element discretization. It is based on the construction of a particular type of convected coordinate system: an approach which is largely influenced by existing computational procedures. Objective measures of nearness of contact and relative tangential slip, which form the basis for a derivation of a virtual work principle, are given. It is pointed out that the virtual work of contact is objective only after the strong form of action and reaction is assumed satisfied. This imposes a rather restrictive requirement which has to be satisfied by any possible constitutive law. However, the combination of the classical laws of Coulomb and Signorini are admissible since these laws imply that contact force measures are different from zero only in exact physical contact. By combining these laws with the virtual work principle a mixed variational inequality formulation of the problem is given. Finally, the linearization of this formulation is prepared for by developing linearizations of the strain-like measures of contact interaction

Ratchetting behaviour of 304 stainless steel at 650oC under multiaxially stain-controlled and uniaxially/multiaxially stress-controlled conditions

Abstract: Three types of ratchetting tests were conducted using thin-walled tubular specimens of 304 stainless steel at 650 o C: (1) multiaxially strain-controlled ratchetting under combined tension and cyclic torsional straining; (2) uniaxially controlled ratchetting under cyclic tension-compression with a mean stress, with or without peak-stress hold; and (3) mutliaxially stress-controlled ratchetting under combined torsion and cyclic tension-compression with a mean stress, with or without peak-stress hold. The numerical simulation of the stress/strain responses during the ratchetting was carried out using two types of constitutive equations: a classical plasticity-creep superposition model and a unified model. The effects of the strain (or stress) rate, the stress ratio, a peak-stress hold and a primary-stress change on the ratchetting behaviour are investigated by comparing the experimental results with the corresponding simulated results. The correlation between the uniaxial and multiaxial behaviour; and between the strain-controlled and the stress-controlled ratchetting are discussed

Constitutive modelling of the stored energy of cold work under cyclic loading

Abstract: A treatment of the Bodner-Partom elastic-viscoplastic constitutive equations which includes the consideration of the stored energy of cold work (SECW) from a thermodynamic point of view is developed. It is assumed that the SECW can be expressed entirely as a function of the isotropic and directional hardening variables of the constitutive theory. Simulations are performed corresponding to available test data of Halford on changes in SECW of annealed copper during cyclic loading. These simulations indicate a fairly good agreement wiht the original data

Experimental studies of yield surfaces of aluminium alloy and low carbon steel under complex biaxial loadings

Abstract: In order to develop an efficient computer code for the numerical prediction and simulation of forming processes, a constitutive law which adequately accounts for induced anisotropy is required. Cordebois and Boucher propose [Boucher, 1991], [Boucher, 1994] a new method of determining the plastic behaviour using a model which differs completely from classical ones. The model formulation consists of describing the yield surface evolution in terms of velocity. The yield function is not directly known but is obtained through the integration of a differential law starting from an initial locus. The incremental evolution law uses three strain hardening functions. The development of this theory is based upon a hypothesis connected with experimental observations and multiaxial tests are required in order to obtain a multidimensional representation of the yield surface. Then, in order to observe the phenomena during more complex loading paths, to identify the three hardening functions and to validate the model, we realize bidimensional tests of tension-compression-torsion on thin-walled tubes in aluminium alloy and low carbon steel. The aim is to investigate the evolution of the yield surface for large plastic strain. The yield surfaces are obtained using an automatic procedure in which several yield points can be determined from a single specimen. The test sequence begins with a prescribed prestress or prestrain. Then, points on the yield surface are obtained on radial stress paths starting from an elastic loading state. Plastic yielding is detected from an offset equal to 4.10 −5 . The tensile load and twisting torque are applied using a MTS servo-controlled hydraulic machine piloted by a DN 3500 Apollo workstation. Strains are measured by mean of three strain gages attached to the specimen. Our numerical simulations compute a stress loading incrementally. Some numerical results show yield after proportional loading, monotonic plastic loading and intermittent plastic loading [Cayla, 1990]. We shall make the assumption that a monotonic plastic loading is prevalent if, at any instant, the plastic strain velocity is not nul. This means that no unloading process, in terms of plasticity, is occurring. When unloading does occur, we shall speak of intermittent plastic loading

On the effective mechanical behavior of weakly inhomogeneous nonlinear materials

Abstract: This paper is concerned with the prediction of the effective properties of nonlinear composite materials with local constitutive behavior controlled by a convex potential w depending on a small parameter t. The case where t serves to characterize the amplitude of the local variations in the properties of the composites is studied in detail. Expressions are generated for each of the first four terms in a perturbation series expansion for the effective potential of the composites in the contrast t; their derivation being reduced to the solution of standard linear-elasticity problems for homogeneous anisotropic media with body force distributions determined by the relevant polarization tensors. The first two terms in such an expansion are found to add up to the volume average of the potentials of the N phases. For the case where the microstructure of the composite is periodic, explicit expressions are derived for the second- and third-order terms, in terms of Fourier series involving the characteristic functions of the constituent phases in the unit cell. For the case of random microstructures, the expressions generated for the second- and third-order terms depend on up to two- and three-point correlation functions, respectively. When the microstructure is further assumed to be statistically homogeneous and isotropic, the second-order term can be shown to depend only on the volume fractions and properties of the phases. The results are specialized to composites with N isotropic phases, and explicit expressions are generated for statistically isotropic and transversely isotropic fiber-reinforced and laminated materials. It is found that the results for the statistically isotropic composites have explicit dependence on the determinant of the applied strain, as expected from general symmetry considerations. In addition, it is found that the perturbation expansions diverge in the limit of rigid/perfectly plastic behavior for the constituent phases, under certain special types of loading conditions, which may be associated with the possible development of shear bands in the composites

Finite element analysis of the stress-strain behavior of single-crystal tubes

Abstract: In this paper, we investigate the anisotropic macroscopic behavior of single-crystal superalloy subjected to torsion loading. Two viscoplastic models are studied. One is a crystallographic approach based on Schmid law, the second one is phenomenologically built up. Both models are implemented into finite element codes. Their predictive ability is compared through the results delivered by the F. E. calculations of cylindrical specimens subjected to torsion. Results are also compared to the experiments

Shape sensitivity in mixed Dirichlet-Neumann boundary-value problems and associated class of path-independent integrals

Abstract: Considering an arbitrary behavioural functional specified for a mixed Dirichlet-Neumann boundary-value problem, its first variation with respect to variation of domain shape is derived using the material derivative concept and adjoint approach. Next the invariance of this functional is proved for the case of translation, rotation and scale change of an arbitrary homogeneous and isotropic domain of a problem. The associated path-independent integrals are derived and it is shown that the sensitivity analysis with respect to translation, rotation or expansion of defects can be performed by using these integrals

The crack tip field in a rubber sheet

Abstract: The distribution of stress, strain and displacement near the crack-tip in a rubber like sheet is investigated. The constitutive relation of the material is taken from [Gao, 1990], which is valid for finite and infinite strain. The analytical results show that when the crack tip is approached the thickness of the sheet shrinks to zero as R t (R is the distance to the crack-tip before deformation, t is a positive exponent). The crack tip field is composed of two kinds of sectors, i.e. two shrink sectors and one expansion sector. The stress and strain singularities are obtained analytically and verified numerically.

On localization in ductile-brittle materials under compressive loadings

Abstract: This paper focuses on the onset of strain localization for pure compression in concrete like materials. An elastoplastic damage model allowing for different types of coupling between damage and elasticity and different forms of permanent strain is formulated. The consequences of the several different factors on the strain localization predictions are investigated: alternative couplings, form of the yield function of the inelastic potential. The analytical results for the onset of localization predicted with this class of model are compared with the experimental results obtained during compression tests on concrete specimens. The critical hardening modulus and orientation of the localized band at localization are discussed in detail

Mechanical characterization of heterogeneous materials during setting

Abstract: The vibroscope, a non destructive vibratory test at low frequency (100 Hz-I kHz) is presented in this article. It was developed in order to study the rheological evolutions of crude cellular concrete during setting. This highly heterogeneous material changes from a bubbly fluid to a porous solid in four hours. The action of the vibroscope consists of sending short impulses of compressional waves into the material at regular intervals. The recorded signals allow the calculation of the celerity and the damping coefficients as well as the ratio of normal stresses in two orthogonal directions. From celerity measurements, the transition between fluid and solid state of the cellular crude paste is determined. The three quantities measured are analyzed and are then related to the mechanical properties of the material by an inverse analysis. The latter is based on the assumption that the paste behaves as a suspension of gas bubbles in a viscoelastic medium. From this analysis, the evolution with time of the values of the rheological parameters can be calculated. Finally it is shown that the vibroscope can be adapted to the study of other heterogeneous materials during setting.

Elastic-viscoplastic buckling of circular cylindrical shells under axial compression

Abstract: The time-dependent behaviour of materials has an influence on buckling and collapse of shell structures even for small deformation rates and at room temperature. This is investigated by an accurate incremental finite element analysis of inelastic buckling of axially compressed elastic-viscoplastic circular cylindrical shells. Buckling in an axisymmetric mode and a non-axisymmetric mode is analysed. The effect of a localized buckling pattern is investigated. Special interest is devoted to the maximum support load and on the energy-absorption of the viscoplastic structure. The deformation rate analysed is such that inertia effects are negligible. A small strain theory and a moderate rotation shell theory are used. Cases with significant rate hardening and deformation rate sensitivity are presented, as well as cases with little rate hardening.

On the poroplastic behaviour of porous rocks

Abstract: The general formulation of poroplastic models for porous rocks is firstly presented. The concepts of plastic porosity and plastic effective stress tensor are introduced. The parameters defining poroplastic coupling have been determined from appropriate laboratory tests. A particular poroplastic model for a porous chalk is developed and its validity is verified for undrained hydrostatic and triaxial tests.

A method for predicting linear viscoelastic mechanical behavior of composites, a comparison with other methods and experimental validation

Abstract: In this paper is firstly presented a survey of methods of determining the mechanical viscoelastic behavior of unidirectional continuous fiber composites with an organic matrix. We mention in the first part the composite cylinder assemblage (CCA) of Hashin-Rosen, complemented by Hermans and by the results of Christensen and Lo, and the semi-analytic method of Mori-Tanaka. We propose several numerical formulations for composite materials with periodical microstructures as well as the effective modulus approach presented in [Nguyen & Pastor, 1993]. In the second part, the matrix Poisson's ratio v is valued as a temperature-dependent complex number in order to forecast the complex transverse Young's modulus for comparison with results with real values of v. In each case the analytic and numerical predictions are compared with experimental results obtained on a viscoelasticimeter.

Strain localization in single crystals : bifurcation analysis, effects of boundaries and interfaces

Abstract: The bifurcation theory in elastoplasticity at large strain has been extensively used in the past to model the development of shear bands in ductile single crystals. In contrast we perform here a bifurcation analysis for single crystals undergoing symmetric multiple slip at small strain with a view to modelling slip band formation during incipient plasticity. In that case strain softening is required for such bifurcation modes to occur. The actual three-dimensional geometry of f.c.c. single crystals is taken into account and slip configurations are considered for which 1, 2, 4, 6 or 8 slip systems are simultaneously activated. Three-dimensional finite element calculations are carried out to induce the occurrence of some of the predicted bifurcation modes in a one-element thick single crystal plate in tension. Viscoplasticity is used to reduce the mesh dependence of the results. To trigger localization geometrical imperfections are introduced as usually done in the literature but we also introduce material flaws, such as slightly lower local yield stress, because it is thought to be physically more relevant. A bifurcation analysis at boundaries and interfaces is then performed in the case of single slip. It is proved firstly that no deviation occurs when a slip band regarded as a single slip bifurcation mode reaches a boundary and secondly that equilibrium conditions allow the crossing of an interface between two crystals as soon as localization conditions are fulfilled on each side. Finite element simulations of the crossing of a grain boundary separating two misoriented single crystals are presented. Finally the results of some finite element calculations indicate that a slip band initiated in a domain of a single crystal with a softening behaviour can propagate through neighbouring hardening zones. This provides a better understanding of the mechanisms associated with slip band initiation and propagation.

Scale dependent constitutive relations-information from wavelet analysis and application to localization problems

Abstract: This study explores wavelet analysis as a tool to include the influence of scale on material behavior. The wavelet representation of the failure state of a one-dimensional problem provides the relevant scale as a very important parameter in the material law. The consequences are paramount since the local scale involved, and its influence on material response, is decided from the global deformation field, rather then being defined as a fixed quantity independent of deformation. The scale dependent constitutive relations are examined in detail with respect to localization problems in the context of viscoplasticity.

A straightforward numerical method for evaluating the ultimate loads of structures

Abstract: In order to meet increasingly compelling safety requirements, engineers should employ rigorous and efficient design methods for properly assessing the ultimate behaviour of structures. Such is the main purpose of the numerical method presented in this paper, which is aimed at producing reliable estimates of their ultimate loads. Based upon the kinematic approach of the yield design theory, this method consists of restricting the related minimization problem to velocity fields generated by means of a finite element discretization of the structure. It leads to a search for the minimum of a convex function which is carried out by a specific numerical algorithm. Whereas the method can be directly implemented for «plane stress» problems where the strength criterion of the constituent material remains bounded in every direction of stress space, specific developments are presented which make its application to «plane strain» and three dimensional problems possible. This is achieved in the case of Tresca or von Mises materials for which the relevant velocity fields are generated through a «stream function». Several examples of the application of this method, such as a bearing capacity problem are given, where the numerical estimates, which are rigorous upperbounds for the ultimate loads, compare favourably with available exact solutions, thus illustrating the suitability and performance of the proposed method

An analysis of coupling between plane cracks and contacts

Abstract: A numerical analysis is presented to discover if the change in compliance of a half-plane, due to the presence of flaw, has any significant effect on the pressure distribution beneath a rigid, flat-ended indenter sliding over the surface of the half-plane close to the defect. It is shown that, when the indenter is constrained to remain vertical, the change in pressure distribution may be of sufficient magnitude to alter the stress intensity factors at the crack tip compared with the standard uncoupled approximation. However, this effect is only significant when the coefficient of friction between the indenter and the half-plane takes values towards the upper end of those experienced in practice, and when the geometry of the problem lies within a very small range. In most practical problems, therefore, the influence of coupling, as defined above, may be neglected

Interactions between cracks and circular holes in two-dimensional linear elastic media

Abstract: For a better understanding of the behaviour of cracked materials, we propose a numerical tool for linear elastic two-dimensional cracked media. The number of singularities induced by cracks indicates that the pseudo-tractions technique can be used to study brittle materials such as rocks. The developments presented in this paper extend this technique to finite media containing cracks and circular cavities, including the crack closure effect, and take into account the friction that may exist between the crack faces. The accuracy of the technique is demonstrated by comparison between the results of this pseudo-tractions technique and results found in the literature for some simple geometries. The effects of the parameters such as interactions, porosity and friction coefficient introduced at the mesoscopic scale are illustrated by some examples which consider the elastic domain of a cracked medium and the propagation of cracks under compression. The validity of the approximation and the limits of the proposed model are also discussed

Beam buckling as a coexistence of phases phenomenon

Abstract: A Bifurcation analysis for the equilibrium path of an axially loaded pinned-pinned beam with a non-convex stress strain law is presented. Adopting global stability criteria and Maxwell's law of equal area, Van der Waals [1895], the equilibrium configurations of a beam exhibiting the coexistence of phases phenomenon are studied. These states are similar to the elastic-plastic buckling configurations.

Fatigue Lifetime Estimation Based on Rainflow Counted Data Using the Local Strain Approach

Abstract: In the automotive industry both the loca l strain approach and rainflow counting are well known and approved tools in the numerical estimation of the lifetime of a new developed part especially in the automotive industry. This paper is devoted to the combination of both tools and a new algorithm is given that takes advantage of the inner structure of the most used damage parameters.

Acceleration waves in finitely deformed elastic-plastic solids

Abstract: Conditions under which acceleration waves may propagate in elastic-plastic solids undergoing finite deformations, are studied. The constitutive theory employed incorporates models corresponding to yield surfaces with vertices. Three types of acceleration wave are considered, according to whether the regions ahead and behind the wavefront are loading plastically or unloading elastically. For the case of smooth yield surfaces it is shown that wave speeds for elastic-plastic materials are bounded on either side by corresponding elastic wave speeds; these are what are generally known as Mandel's inequalities. Principal waves are investigated for the case of smooth and nonsmooth yield surfaces, and for an elastically isotropic but otherwise arbitrary material. The propagation condition for transverse waves is independent of plastic behaviour, while the condition for longitudinal waves incorporates the plastic state of the material in a simple way. Finally, the wave speeds corresponding to longitudinal waves propagating in a material which is in a state of uniaxial stress, are considered for the case of the von Mises and Tresca yield conditions, and in the case of the Tresca condition, for an isotropic hardening law which incorporates coupling between two distinct active yield surfaces. A simple bound on the coupling constant dictates which of the two wave speeds (von Mises or Tresca) is larger.

The effect of friction on multibody dynamics

Abstract: This paper discusses the nature of frictional effects on multibody dynamics and the influence of the effects on the numerical analysis of systems with friction. The joints in a multibody system are modelled as nonideal (friction-affected) constraints, and a general and easy manipulation method of the mathematical modelling of the problem in absolute variables is proposed. The governing equations arising are highly nonlinear differential-algebraic equations (DAEs). Two approaches to the numerical solution of the equations are discussed : using the DAE solvers directly and introducing a symbolic-numerical procedure of converting the DAEs into a smaller set of equations. Some variant formulations of the governing equations concerning the two approaches are demonstrated and discussed. The theoretical considerations are illustrated by a simple example.

Optimization methods in (structural) engineering

Abstract: The aim of this article is to present some general concepts about optimization and methods of application pertinent to engineering practice. In spite of more general applications, for the sake of clearness, attention will be addressed to the optimization of structures, with the aim of reflecting on the general features of the problems of optimal design. Making reference to new trends in structural optimization, analytical and numerical methods will be presented and discussed and their use will be explained through some sample applications