Finite strain analysis of simple shear using recent anisotropic yield criteria

The finite strain response of a rectangular block subjected to constrained simple shearing deformations is considered in order to evaluate the predictive capability of some recently proposed anisotropic yield functions. It is shown that, in the presence of plane anisotropy, the prediction of realistic second order normal stresses cannot be expected since every different initial orientation of the material axes relative to the loading axes results in a different response due to the rotation of the material axes with shear. Parametric studies are performed in order to determine possible limits on the material constants so that the predicted normal stresses remain second order with respect to the shear stress itself. Our numerical results indicate that, in particular, the commonly employed range of one parameter associated with grain related anisotropy renders results which no longer imply predicting a proper second order effect but rather introduces errors of the first order. The results suggest that the modelling of anisotropy with such phenomenological anisotropic yield functions should be limited to near-quadratic yield surfaces for applications involving stress states outside the biaxial tensile stress range.     

A Ring-Shaped Plastic Zone at an Elliptic Crack under Triaxial Axisymmetric Loading

The paper addresses a three-dimensional problem for an elliptic crack with a ring plastic zone under uni- and triaxial loading at infinity. The normal stresses in the plastic zone are found from the conditions that the stresses are constrained and the plane strain is local and from the yield criterion for the given material. The size of the elliptic ring is calculated by Rice's variational formula. It is shown that the constraint ratio for plastic strains under triaxial loading may be greater than that under loads close to hydrostatic tension. The contour of the plastic zone is confocal to the initial crack     

Mechanical Properties of two novel planar lattice structures

Two novel statically indeterminate planar lattice materials are designed: a new Kagome cell (N-Kagome) and a statically indeterminate square cell (SI-square). Their in-plane mechanical properties, such as stiffness, yielding, buckling and collapse mechanisms are investigated by analytical methods. The analytical stiffness is also verified by means of finite element (FE) simulations. In the case of uniaxial loading, effective modulus, yield strength, buckling strength and critical relative density are compared for various lattice structures. At a critical relative density, the collapse mode will change from buckling to yielding. Elastic buckling under macroscopic shear loading is found to have significant influence on failure of lattice structures, especially at low relative densities. Comparison of the analytical bulk and shear moduli with the Hashin–Shtrikman bounds indicates that the mechanical properties of the SI-square honeycomb are relatively close to being optimal. It is found that compared with the other existing stretching-dominated 2D lattice structures, the N-Kagome cell possesses the largest continuous cavities for fixed relative densities and wall thicknesses, which is convenient for oil storage, disposal of heat exchanger, battery deploying and for other functions. And the initial yield strength of the N-Kagome cell is slightly lower than that of the Kagome cell. The SI-square cell has similar high stiffness and strength as the mixed cell while its buckling resistance is about twice than that of the mixed cell.     

Effect of stress-state and spacing on voids in a shear-field

Unit cell model analyses are carried out for a material with a periodic array of voids, subject to shear loading. Thus the focus is on ductile fracture in conditions of low stress triaxiality. It has been shown recently that voids in shear are flattened out to micro-cracks, which rotate and elongate until interaction with neighboring micro-cracks gives coalescence, so that the failure mechanism is very different from that under tensile loading. In the present studies the plane strain unit cell has fully periodic boundary conditions, so that any combination of the stress components in the overall average stress state can be prescribed. This also allows for studies of the effect of different initial void spacing in the two in-plane coordinate directions. The stress states considered are essentially simple shear, with various levels of tensile stresses or compressive stresses superposed, i.e. low positive stress triaxiality or even negative stress triaxiality. For high aspect ratio unit cells a clear localization band is found inside the cell, which actually represents several parallel bands, due to periodicity. In the materials represented by a low aspect ratio unit cell localization would also occur after that the maximum shear stress has been passed, but this is not shown when periodicity is enforced. The effect of superposed tensile or compressive stresses is found to be bigger for high aspect ratio unit cells than for low aspect ratios.     

Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals

A Taylor-like polycrystal model is adopted here to investigate the plastic behavior of body centered cubic (b.c.c.) sheet metals under plane-strain compression and the subsequent in-plane biaxial stretching conditions. The <111> pencil glide system is chosen for the slip mechanism for b.c.c. sheet metals. The {110} <111> and {112} <111> slip systems are also considered. Plane-strain compression is used to simulate the cold rolling processes of a low-carbon steel sheet. Based on the polycrystal model, pole figures for the sheet metal after plane-strain compression are obtained and compared with the corresponding experimental results. Also, the simulated plane-strain stress—strain relations are compared with the corresponding experimental results. For the sheet metal subjected to the subsequent in-plane biaxial stretching and shear, plastic potential surfaces are determined at a given small amount of plastic work. With the assumption of the equivalence of the plastic potential and the yield function with normality flow, the yield surfaces based on the simulations for the sheet metal are compared with those based on several phenomenological planar anisotropic yield criteria. The effects of the slip system and the magnitude of plastic work on the shape and size of the yield surfaces are shown. The plastic anisotropy of the sheet metal is investigated in terms of the uniaxial yield stresses in different planar orientations and the corresponding values of the anisotropy parameter R, defined as the ratio of the width plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial tensile loading. The uniaxial yield stresses and the values of R at different planar orientations from the polycrystal model can be fitted well by a yield function recently proposed by Barlat et al. (1997b).     

Plasticity of formable all-metal sandwich sheets: Virtual experiments and constitutive modeling

The mechanical behavior of a metallic sandwich sheet material composed of two flat face sheets and two bi-directionally corrugated core layers is analyzed in detail. The manufacturing of the sandwich material is simulated to obtain a detailed unit cell model which accounts for the non-uniform thickness distribution and residual stresses associated with the stamping of the core layers. Virtual experiments are performed by subjecting the unit cell model to various combinations of bi-axial in-plane loading including the special cases of uniaxial tension, uniaxial compression, equi-biaxial tension and shear. The results demonstrate that the core structure’s contribution to the in-plane load carrying capacity of the sandwich sheet material is similar to that of the face sheets. The numerical results are also used to identify the effective yield surface and hardening response of both the core layer and the face sheets. An anisotropic yield function with linear pressure dependency is proposed to approximate the equal-plastic work surfaces for the core structure and face sheets. Furthermore, a new two-surface model with non-linear interpolation based on plastic work density is presented to describe the observed combined isotropic-distortional hardening of the core structure.     

PLASTIC ZONE OF SEMI-INFINITE CRACK IN PLANAR KAGOME AND TRIANGULAR LATTICES

The fracture investigations of the planar lattices made of ductile cell walls are currently limited to bending-dominated hexagonal honeycomb. In this paper, the plastic zones of stretching-dominated lattices, including Kagome and triangular lattices, are estimated by analyzing their effective yield loci. The normalized in-plane yield loci of these two lattices are almost identical convex curves enclosed by 4 straight lines, which is almost independent of the relative density but is highly sensitive to the principal stress directions. Therefore, the plastic zones around the crack tip of Kagome and triangular are estimated to be quite different to those of the continuum solid and also hexagonal lattice. The plastic zones predictions by convex yield surfaces of both lattices are validated by FE calculations, although the shear lag region caused by non-local bending effect in the Kagome lattice enlarges the plastic zone in cases of small ratio of rp/l.     

Effect of sample size on the response of DEM samples with a realistic grading

This paper shows that for DEM simulations of triaxial tests using samples with a grading that is repre- sentative of a real soil, the sample size significantly influences the observed material response. Four DEM samples with identical initial states were produced： three cylindrical samples bounded by rigid wails and one bounded by a cubical periodic cell, When subjected to triaxial loading, the samples with rigid boundaries were more dilative, stiffer and reached a higher peak stress ratio than the sample enclosed by periodic boundaries. For the rigid-wall samples, dilatancy increased and stiffness decreased with increasing sample size, The periodic sample was effectively homogeneous, The void ratio increased and the contact density decreased close to the rigid walls, This heterogeneity reduced with increasing sample size. The positions of the critical state lines （CSLs） of the overall response in e-log p＇ space were sensitive to the sample size, although no difference was observed between their slopes. The critical states of the interior regions of the rigid-wall-bounded samples approached that of the homogeneous periodic sample with increasing sample size. The ultimate strength of the material at the critical state is independent of sample size.     

Shakedown theory for elastic plastic kinematic hardening bodies

Shakedown static and kinematic theorems for elastic–plastic (generally nonlinear) kinematic hardening solids are derived in classical (path-independence) spirit with new constructions. The generally plastic-deformation-history-dependent hardening curve is assumed to be limited by the initial yield stress and ultimate yield strength, and to obey a positive hysteresis postulate for closed plastic cycles, but else can be arbitrary and unspecified. The theorems reveal that the shakedown of structures is not affected by the particular form of the hardening curve, but just by the initial and ultimate yield stresses. While the ultimate yield strength is clearly defined macroscopically and attached to the incremental collapse mode with unbounded plastic deformations, the initial yield stress, which is responsible for the bounded cyclic plasticity collapse mode, should not be taken as the convenient one at a fixed amount of plastic deformation (0.2%), but is suggested to be taken as low as the fatigue limit to preserve the classical load-history-independence spirit of the shakedown theorems. Otherwise, for our pragmatic application purpose, it may be given empirical values between the low fatigue limit and high ultimate yield stresses according to particular loading processes considered, which may range anywhere between the high-cycle and low-cycle ones. The theorems appear as simple as those of Melan and Koiter for perfect plasticity but applied to the much larger class of more realistic kinematic hardening materials.     

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.     

A plasticity model for pressure-dependent anisotropic cellular solids

The initial and subsequent yield surfaces for an anisotropic and pressure-dependent 2D stochastic cellular material, which represents solid foams, are investigated under biaxial loading using finite element analysis. Scalar measures of stress and strain, namely characteristic stress and characteristic strain, are used to describe the constitutive response of cellular material along various stress paths. The coupling between loading path and strain hardening is then investigated in characteristic stress–strain domain. The nature of the flow rule that best describes the plastic flow of cellular solid is also investigated. An incremental plasticity framework is proposed to describe the pressure-dependent plastic flow of 2D stochastic cellular solids. The proposed plasticity framework adopts the anisotropic and pressure-dependent yield function recently introduced by Alkhader and Vural [Alkhader M., Vural M., 2009a. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids 57(5), 871–890]. It has been shown that the proposed yield function can be simply calibrated using elastic constants and flow stresses under uniaixal loading. Comparison of stress fields predicted by continuum plasticity model to the ones obtained from FE analysis shows good agreement for the range of loading paths and strains investigated.     

An efficient constitutive model for room-temperature,low-rate plasticity of annealed Mg AZ31B sheet

Metals and alloys with hexagonal close packed (HCP) crystal structures can undergo twinning in addition to dislocation slip when loaded mechanically. The complexity of the plastic response and the limited extent of twinning are impediments to their room-temperature formability and thus their widespread adoption. In order to exploit the unusual deformation characteristics of twinning sheet materials in designing novel forming operations, a practical plane stress material model for finite element implementation was sought. Such a model, TWINLAW, has been constructed based on three phenomenological deformation modes for Mg AZ31B: S (slip), T (twinning), and U (untwinning). The modes correspond to three testing regimes: initial in-plane tension (from the annealed state), initial in-plane compression, and in-plane tension following compression, respectively. A von Mises yield surface with initial non-zero back stress was employed to account for plastic yielding asymmetry, with evolution according to a novel isotropic and nonlinear kinematic hardening model. Texture and its evolution were represented throughout deformation using a weighted discrete probability density function of c-axis orientations. The orientation of c-axes evolves with twinning or untwinning using explicit rules incorporated in the model.     

Approximate analysis of progressive deformation in honeycomb structures subjected to in-plane loading

Progressive deformation of honeycomb structures subjected to in-plane loading was approximately analyzed by using the collapse modes of hexagonal unit cells. The collapse modes were categorized as freely compressive, restricted compressive, and shear. Moreover, there were five characteristic deformation patterns, namely deformation bands. Average stresses of the collapsing honeycomb models were evaluated in terms of the plastic collapse stress per hinge and total number of hinges of progressively arising deformation bands. The displacements of the models were obtained by multiplying the displacement per cell with the number of collapsed cells. The present method was used to analyze progressive deformation of typical honeycomb structures. The validity of the stress–displacement relations derived for some structures was confirmed by comparing them with finite element method (FEM) results. Our method is much simpler than FEM but just as effective.     

用均匀化理论分析蜂窝结构的等效弹性参数

在线弹性范围内,根据均匀化理论,并结合有限元方法推导出适用于二维周期性结构的均匀化的有限元格式（Homo FEM）,计算出不同相对密度下的规则蜂窝结构的等效弹性模量Ee和泊松比νe.同时,利用蜂窝结构的代表胞元模型,用常规的有限元方法（FEM）模拟计算出相应的等效弹性参数.最后将两种数值计算结果与己有的理论公式进行了比较和分析讨论.结果表明:在考察的相对密度全场范围内（0～0.4）,HOmO FEM得到的蜂窝结构的 Ee和νe 与 FEM使用平面实体单元模拟计算得到的结果一致吻合,反映出 Homo FEM数值方法的客观准确性和可行性.而 Gibson公式和 W-K得到的等效弹性模量值 Ee只是在较小相对密度的情况下（小于0.15）与数值计算结果吻合.当结构的相对密度较大时,必须考虑胞棱附近区域由应力集中导致的复杂的应力和应变分布的影响.     

A Plane-Strain Formulation of Elastic-Plastic Constitutive Laws

ABSTRACT

Constitutive laws for elastic-plastic materials are derived by eliminating the transverse stress component on the basis of the plane-strain constraint. This leads to a fictitious hardening and temperature dependence of the loading function. For standard elastic-plastic materials the resulting laws are associated; however, the plastic strain state is represented by equivalent plastic-strain measures, which also account for transverse yielding. The new constitutive laws, together with the standard reduced form of the equilibrium and compatibility equations, permit the formulation of the plane-strain elastic-plastic analysis problem in terms of the in-plane stress components only. In the case of perfectly plastic materials, the subsequent plane-strain yield surfaces are contained within a domain bounded by a limit surface which represents the yield condition normally adopted in plane-strain limit analysis.     

Multi-mechanism anisotropic model for granular materials

By representing the assembly by a simplified column model, a constitutive theory, referred to as sliding–rolling theory, was recently developed for a two-dimensional assembly of rods subjected to biaxial loading, and then extended to a three-dimensional assembly of spheres subjected to triaxial (equibiaxial) loading. The sliding–rolling theory provides a framework for developing a phenomenological constitutive law for granular materials, which is the objective of the present work. The sliding–rolling theory provides information concerning yield and flow directions during radial and non-radial loading. In addition, the theory provides information on the role of fabric anisotropy on the stress–strain behavior and critical state shear strength. In the present paper, a multi-axial phenomenological model is developed within the sliding–rolling framework by utilizing the concepts of critical state, classical elasto-plasticity and bounding surface. The resulting theory involves two yield surfaces and falls within the definition of the multi-mechanism models. Computational issues concerning the solution uniqueness for stress states at the corner of yield surfaces are addressed. The effect of initial and induced fabric anisotropy on the constitutive behavior is incorporated. It is shown that the model is capable of simulating the effect of anisotropy, and the behavior of loose and dense sands under drained and undrained loading.     

Dynamic crushing and energy absorption of regular,irregular and functionally graded cellular structures

The in-plane dynamic crushing of two dimensional honeycombs with both regular hexagonal and irregular arrangements was investigated using detailed finite element models. The energy absorption of honeycombs made of a linear elastic-perfectly plastic material with constant and functionally graded density were estimated up to large crushing strains. Our numerical simulations showed three distinct crushing modes for honeycombs with a constant relative density: quasi-static, transition and dynamic. Moreover, irregular cellular structures showed to have energy absorption similar to their counterpart regular honeycombs of same relative density and mass. To study the dynamic crushing of functionally graded cellular structures, a density gradient in the direction of crushing was introduced in the computational models by a gradual change of the cell wall thickness. Decreasing the relative density in the direction of crushing was shown to enhance the energy absorption of honeycombs at early stages of crushing. The study provides new insight into the behavior of engineered and biological cellular materials, and could be used to develop novel energy absorbent structures.     

An experimental investigation of elastic-plastic behavior of a fibrous boron-aluminum composite: II. Fiber-dominated mode

In the companion paper, part I, initial and subsequent yield surfaces, and plastic strains in a unidirectionally reinforced boron-6061-O aluminum composite were found experimentally under loading conditions which activate matrix-dominated deformation. In the present paper, we present experimental results related to the fiber-dominated behavior of the B/AI composite. Both monotonic and cyclic loading programs were applied in the axial tension stress-longitudinal shear stress plane. The results indicate that the shape of the experimentally observed initial and subsequent yield surfaces conforms with the predictions of the bimodal theory. The agreement, however, does not extend to the observed and predicted direction and magnitude of plastic strains.     

Quasi-static crush behavior of aluminum honeycomb specimens under compression dominant combined loads

The influence of shear stress on the quasi-static crush behavior of aluminum honeycomb specimens under compression dominant combined loads is investigated by experiments. A test fixture was designed such that dominant compressive and shear loads with respect to the strongest material symmetry direction can be controlled and applied independently. Honeycomb specimens were also designed such that the secondary non-uniform stresses due to the stress-free boundary can be minimized. The experimental results indicate that the normal crush strengths under combined compressive and shear loads are lower than that under pure compressive loads. A phenomenological yield criterion for specimens with different in-plane orientation angles is proposed based on the experimental normal crush and shear strengths under combined loads. The experimental results suggest non-normality plastic flow based on the yield criterion. The non-normality flow behavior becomes more pronounced as the in-plane orientation angle increases. The experimental results also indicate that the energy absorption rate depends upon the ratio of the shear stress to the compressive stress and the in-plane orientation angle. In addition, specimens crushed under combined loads show inclined stacking patterns of folds due to the asymmetric location of horizontal plastic hinge lines and the rupture of aluminum cell walls along the adhesive lines. These experimental observations are useful to develop microscopic plasticity models of aluminum honeycombs under compression dominant combined loads.     

A non-quadratic yield function for polymeric foams

The yield behavior of a closed cell polymeric foam is investigated under multiaxial loadings. A phenomenological yield function is developed to characterize the initial yield behavior of the closed cell polymeric foam under a full range of loading conditions. The principal stresses of a relative stress tensor and the second invariant of the deviatoric stress tensor are the main parameters in the yield function. The yield function is a linear combination of non-quadratic functions of the relative principal stresses and the second invariant of the deviatoric stress tensor. The convexity of the yield surface based on the non-quadratic yield function is proved. The non-quadratic yield function is shown to well characterize the yield behavior of a closed cell polymeric foam in [Deshpande, V.S., Fleck, N.A., 2001. Multi-axial yield behavior of polymer foams. Acta Materialia 49, 1859–1866] under a full range of loading conditions. Finally, a comparison of different phenomenological yield functions to characterize the yield behavior of the foam is presented.