Elastic-plastic behavior of fibrous composites

The elastic-plastic behavior of fibrous composites is explored with self-consistent models. The overall and local yield surfaces, instantaneous moduli, thermal coefficients, plastic strains and thermal microstresses are calculated for selected material systems, using R. Hill's (1965) model, and a modified scheme. Axisymmetric mechanical loads and uniform thermal changes are considered and the extension to shear loads is discussed. A limited comparison of the calculated microstresses is made with available experimental results.     

Fourier-based incremental homogenisation of coupled unilateral damage–plasticity model for masonry structures

In structural analysis of large masonry structures, nondemanding computation effort, numerical stability and simplified model assembly and meshing often have a higher priority over precise details of local stress or strain responses. This paper presents the development of a Fourier-based incremental homogenisation technique, where the macro–micro transformations of mechanical variables are derived by incremental variational problems to minimise the potential energy in representative volume elements (RVEs) with respect to local fluctuating displacement fields expanded in Fourier series. In addition to the proposed homogenisation technique, a unilateral damage–plasticity constitutive model for mortar joints in the RVE is developed within the framework of thermomechanics, which accounts for the stiffness and strength degradation (or recovery) due to the transverse crack opening/closing in the mortar joints. The numerical solution for the homogenisation problem and the performances of the proposed coupled-damage plastic mode and Fourier-based homogenisation scheme verified by detailed case studies are presented. It has been shown that the computational effort of the analysis with the proposed modelling technique can be considerably reduced by more than 20% as compared with that of the discrete modelling technique.     

A scheme of automatic stress-updating on yield surfaces for a class of elastoplastic models

For a large class of elastoplastic models, the present paper proposes an integration scheme which updates stress points on yield surfaces automatically. Associated plastic flow models with the rules of kinematic, isotropic, and distortional hardening/softening are contained in the model class. The underlying structure of the class of elastoplastic models is exposed and utilized to develop the return-free integration scheme that keeps the stress points on the yield surfaces without any extra enforcement. The return-free capability of the integration scheme is underpinned by the theory of Lie group and Lie algebra. Numerical demonstrations show that the stress points are updated on the yield surfaces automatically and exactly within the machinery round-off error.     

Experimental determination of strain-induced anisotropy during nonproportional straining of an A1/Mg alloy at room temperature

A servohydraulic, computer controlled MTSn axial-torsion testing machine with a bi-axial clip-on extensometer is used to test thin-walled tubes of an A1/Mg alloy under strain control. A plastic offset strain of 10⁻⁴ determines the yield surfaces. Straining and yield surface probing is governed by a computer program which also controls digital data acquisition. Yield surfaces in stress and in strain space as well as the axial and shear stress-strain diagrams can be reconstructed from the digitally recorded data. The specimens were subjected to a strain path in the form of a regular 16-sided polygon which was followed on some specimens by a square path. The total inelastic strain path length can exceed 15% while the equivalent strain excursion is less than 2%. It is shown that yield surfaces measured on specimens withclose initial stress-strain diagrams are very consistent and that yield surface probing has an insignificant effect on subsequent yield surfaces. Yield surfaces are shown to translate, changein shape and size and to exhibit a cross effect. A post processor which includes a least square smoothing routine calculates the area and the centroid of each yield surface. The size increase is initially rapid but the rate of increase decreases as a saturation is approached. After strining for less than 1% in a fixed direction a characteristic yield surface shape is established. Yield surfaces obtained at the same point in strain space with identical prestrain direction of at least 1% but with increased amounts of accumulated plastic strain have the same shape but show an increase in size. The yield surfaces differ in shape and size when the same strain point is reached from different directions. The centroid of the yield surface in stress space moves almost in a circular path for a polygonal strain path. All stress space yield surfaces contain the origin but this is not the case for the surfaces in strain space.     

A parametric description of orthotropic plasticity in metal sheets

A polar-coordinate representation of the yield surface in principal stress space is utilized to formulate constitutive equations for plane-stress plasticity of orthotropic sheets. The yield function and the associated flow rule are analysed by taking account of the orientation of the principal stress axes, and conditions for internal consistency of the model are derived. An orthotropic yield criterion is proposed, which is devised as an extension of a previous isotropic yield function involving the second and third invariants of the deviatoric stress tensor. Comparisons with micro-macro computations and experimental measurements of yield surfaces are discussed.     

Thermo-mechanical modelling of the contact between rough surfaces using homogenisation technique

In this study, thermo-mechanical behaviour of contacting rough surfaces has been modelled. Firstly, a numerical microscopic contact model that considers the properties of engineering surfaces has been developed. Geometrical characteristics of rough surfaces are deduced using the standard procedure for roughness and waviness parameter determination according to the so-called “motif” procedure. Secondly, an equivalent macroscopic contact model using a homogenisation technique has been presented. The interfacial behaviour of this model has been governed by the curves deduced from the microscopic model. The transition from microscopic to macroscopic scale was also validated.     

3D homogenised strength criterion for masonry: Application to drystone retaining walls

A 3D strength criterion for masonry is constructed based on yield design theory. Yield design homogenisation provides a rigorous theoretical framework to determine the yield strength properties of a periodic medium, based on the properties of its constituent materials. First, theoretical basis of 2D homogenisation of periodic media, and more particularly its application in the framework of yield design, will be retrieved. Then, 2D principles are extended to exhibit a 3D domain of running-bond masonry. This criterion is finally used to assess the stability of a drystone retaining wall loaded by an axle load, and theoretical results are compared to experimental data. Perspectives on this work are given as a conclusion.     

A micro-mechanical model for the homogenisation of masonry

Masonry is a composite material made of units (brick, blocks, etc.) and mortar. For periodic arrangements of the units, the homogenisation techniques represent a powerful tool for structural analysis. The main problem pending is the errors introduced in the homogenisation process when large difference in stiffness are expected for the two components. This issue is obvious in the case of non-linear analysis, where the tangent stiffness of one component or the tangent stiffness of the two components tends to zero with increasing inelastic behaviour.The paper itself does not concentrate on the issue of non-linear homogenisation. But as the accuracy of the model is assessed for an increasing ratio between the stiffness of the two components, the benefits of adopting the proposed method for non-linear analysis are demonstrated. Therefore, the proposed model represents a major step in the application of homogenisation techniques for masonry structures.The micro-mechanical model presented has been derived from the actual deformations of the basic cell and includes additional internal deformation modes, with regard to the standard two-step homogenisation procedure. These mechanisms, which result from the staggered alignment of the units in the composite, are of capital importance for the global response. For the proposed model, it is shown that, up to a stiffness ratio of one thousand, the maximum error in the calculation of the homogenised Young's moduli is lower than five percent. It is also shown that the anisotropic failure surface obtained from the homogenised model seems to represent well experimental results available in the literature.     

A stress point algorithm for an elastoplastic model in unsaturated soils

Two stress fields, combination of total stresses, liquid pressure and gas pressure have to be considered to explain the deformational behaviour of unsaturated media. Elastoplastic models developed for these materials consider generally two yield surfaces, each one associated to a stress field, and whose intersection produces a corner in the space of generalised stress components. In this paper, a stress point algorithm is proposed to cope with the integration of such constitutive laws, which can be seen as non smooth multisurface plastic models in the space of the two stress fields. The basic model developed by Alonso et al. (Alonso, E.E., Gens, A., 1990. A constitutive model for partially saturated soils. Géotechnique 40 (3), 405–430), which will be used to test the algorithm, is first described. Generalised stress and strain variables are then defined. Implementation of the return mapping algorithm, based on an implicit integration scheme, is presented with special attention devoted to the problem of mixed control imposed by the F.E. formulation generally used to analyse the hydromechanical behaviour of unsaturated media. Validation results on distinct generalised stress paths are given at the end.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part III: Yield surface in tension–tension stress space (Al 6061–T 6511 and annealed 1100 Al)

Subsequent yield surfaces for aluminum alloys are determined for three proportional loading paths (i.e., axial, hoop, and combined hoop and axial stress) using 10 με deviation from linearity as the definition of yield. This paper is in continuation with Parts I and II of the author’s previous papers on subsequent yield surfaces under tension–torsion (σ11–√3σ₁₂) stress space using similar small offset definition of yield. In the current paper comprehensive experimental results on subsequent yield surfaces under tension–tension (σ₁₁–σ₂₂) stress space are presented. Comparison of subsequent yield surfaces under (σ₁₁–√3σ₁₂) stress space, investigated in the earlier papers, clearly indicated distinctive hardening behavior under various loading paths. However, subsequent yield surfaces for Al 6061–T 6511 (a low work hardening alloy) showed contraction and negative cross-effect with finite deformation as compared to the annealed 1100 Al (a high work hardening alloy) where expansion and positive cross-effect was observed.     

A model for the through-thickness elastic–plastic behaviour of paper

An elastic–plastic material model for the out-of-plane mechanical behaviour of paper is presented. This model enables simulation the elastic–plastic behaviour under high compressive loads in the through-thickness direction (ZD). Paper does not exhibit a sharp transition from elastic to elastic–plastic behaviour. This makes it advantageous to define critical stress states based on failure stresses rather than yield stresses. Moreover, the failure stress in out-of-plane shear is strongly affected by previous plastic through-thickness compression. To cover these two features, a model based on the idea of a bounding surface that grows in size with plastic compression is proposed. Here, both the bounding and the yield surfaces are suggested as parabolas in stress space. While the bounding surface is open for compressive loads, the yield surface is bordered by the maximum applied through-thickness compression.     

Prediction of tension–compression cycles in multiphase steel using a modified incremental mean-field model

A micromechanical model is proposed for multiphase metals consisting of a ductile matrix reinforced by hard, equiaxed inclusions. The model belongs to a class of incremental mean-field theories of the first order and is suitable for general, non-monotonic loading paths. This paper specifically addresses the definition of the effective, instantaneous shear modulus of the isotropic comparison materials which intervene in the solution of the linearized, homogenisation problem. Time integration of the composite response is calculated using a Newton–Raphson scheme and a consistent tangent operator is derived. Predictions of the phase stresses developed under various loading modes and for a wide range of volume fractions of inclusions are assessed by a comparison with finite element simulations of periodic unit cells. It is argued that the latter predictions depend dramatically on the topological arrangement of the inclusions. Considering a cubic ordering of inclusions, it is possible to reproduce FE predictions with the mean-field model on the condition that the comparison materials are stiffer than the real phases during the first few percent of plastic strain. The mean-field model provides an accurate prediction of the average stress developed within individual phases.     

A multi-dimensional composite model for plastic continua under polyaxial loading condition

By replacing a medium with reinforcing components oriented and distributed uniformly in a multi-dimensional space, a constitutive model is constructed. The components are extended/compressed compatibly with the strain and the resultant of load exerted on them to balance the stress. Their load-elongation relation can be determined from a conventional material test. Each component undergoes different elongation history depending on its own orientation during deformation, so that the model can simulate elasto-plastic behavior of materials under polyaxial loading conditions. The incremental constitutive matrix has been derived for application in numerical analysis and a yield criterion is also introduced. A few subsequent yield surfaces have been predicted and compared with experiments.     

On loading,yield and quasi-yield hypersurfaces in plasticity theory

In the context of the author's previously published “simple” theory of plasticity[1] in which no loading or yield surfaces are assumed to exist, it is shown that (a) loading surfaces must exist for a plastic material as a result of Caratheodory's theorem on Pfaffian forms, and that (b) a yield hypersurface in state space may be defined as the boundary of the region in which no loading surfaces exist (the elastic region) if this region has a positive volume, otherwise this region degenerates into the quasi-yield hypersurface. The significance of loading and yield (or quasi-yield) hypersurfaces is further explored for one-component loadings, with particular attention to the Bauschinger effect and kinematic hardening.     

A theoretical investigation of the influence of dislocation sheets on evolution of yield surfaces in single-phase B.C.C. polycrystals

Accurate and reliable predictions of yield surfaces and their evolution with deformation require a better physical representation of the important sources of anisotropy in the material. Until recently, the most physical approach employed in the current literature has been the use of polycrystalline deformation models, where it is assumed that crystallographic texture is the main contributor to the overall anisotropy. However, recent studies have revealed that the grain-scale mesostructural features (e.g. cell-block boundaries) may have a large impact on the anisotropic stress-strain behaviour, as evidenced during strain-path change tests (e.g. cross effect, Bauschinger effect).In previous papers, the authors formulated an extension of the Taylor-type crystal plasticity model by incorporating some details of the grain-scale mesostructural features. The main purpose of this paper is to study the evolution of yield surfaces in single-phase b.c.c. polycrystals during deformation and strain-path changes using this extended crystal plasticity model. It is demonstrated that the contribution of the grain-scale substructure in these metals on yield loci is comparable in magnitude to the effects caused by the differences in texture. Furthermore, it is shown that the shape of yield loci cannot be predicted accurately by the traditional polycrystalline deformation model with equal slip hardening. The trends predicted by the extended crystal plasticity model are in much better agreement with the experimental evidence reported in the literature than those represented in classical treatments by isotropic and kinematic hardening.     

On the experimental determination of yield surfaces and some results of annealed 304 stainless steel

Factors affecting the experimental determination of yield surfaces are discussed. They include the elastic moduli and the zero offset strain, the strain domain used to determine the yield stress, the probing path, and the strain rate of probing. To obtain yield surfaces consistently, it is necessary to account for these factors. The initial and subsequent yield surfaces of annealed AISI type 304 stainless steel have been experimentally determined in the axial-torsional stress space. Three loading paths have been studied. They are a pure axial path, a pure torsional path, and a proportional axial-torsional path. Each path includes loading, unloading, reloading, and the cyclically steady state.     

Resonant waves in elastic structured media: Dynamic homogenisation versus Green’s functions

We address an important issue of dynamic homogenisation in vector elasticity for a doubly periodic mass-spring elastic lattice. The notion of logarithmically growing resonant waves is used in the analysis of star-shaped wave forms induced by an oscillating point force. We note that the dispersion surfaces for Floquet–Bloch waves in the elastic lattice may contain critical points of the saddle type. Based on the local quadratic approximations of a dispersion surface, where the radian frequency is considered as a function of wave vector components, we deduce properties of a transient asymptotic solution associated with the contribution of the point source to the wave form. The notion of local Green’s functions is used to describe localised wave forms corresponding to the resonant frequency. The special feature of the problem is that, at the same resonant frequency, the Taylor quadratic approximations for different groups of the critical points on the dispersion surfaces (and hence different Floquet–Bloch vectors) are different. Thus, it is shown that for the vector case of micro-structured elastic medium there is no uniformly defined dynamic homogenisation procedure for a given resonant frequency. Instead, the continuous approximation of the wave field can be obtained through the asymptotic analysis of the lattice Green’s functions, presented in this paper.     

Localized viscoelastoplastic strain in mesovolume of heterogeneous medium under different loading types

The localized viscoelastoplastic strain in the mesovolume of heterogeneous media under quasi-static and dynamic loading is investigated. The generalized Bingham–Shwedov model is used; it consists of a combination of Dragon–Mroz's for elastoplasticity and Maxwell's model of viscoelasticity. Any variational finite-difference scheme for solving the quasi-static problem of elastoplastic yielding of a heterogeneous solid can be taken into account. A modified Lagrange's variance equation for analyzing the stress–strain state can be described by the non-symmetric stress tensor. Approximation of spatial derivatives is made by using the twofold partition of spatial domain in tetrahedronal or three-angular (in two-dimensional space) unit cell of mesh-work. Finite difference for deformation is made use of in two or three space dimensions and time. Results for heterogeneous medium with complex form and large number of interior surfaces are obtained for quasi-static and dynamics problems.