Flow/damage surfaces for fiber-reinforced metals having different periodic microstructures

Flow/damage surfaces can be defined in terms of stress, inelastic strain rate, and internal variables using a thermodynamics framework. A macroscale definition relevant to thermodynamics and usable in an experimental program is employed to map out surfaces of constant inelastic power in various stress planes. The inelastic flow of a model silicon carbide/titanium composite system having rectangular, hexagonal, and square diagonal fiber packing arrays subjected to biaxial stresses is quantified by flow/damage surfaces that are determined numerically from micromechanics, using both finite element analysis and the generalized method of cells. Residual stresses from processing are explicitly included and damage in the form of fiber–matrix debonding under transverse tensile and/or shear loading is represented by a simple interface model. The influence of microstructural architecture is largest whenever fiber-matrix debonding is not an issue; for example in the presence of transverse compressive stresses. Additionally, as the fiber volume fraction increases, so does the effect of microstructural architecture. These results indicate that microstructural architecture needs to be accounted for in an accurate continuum model, thus complicating development of such a model. With regard to the micromechanics analysis, the overall inelastic flow predicted by the generalized method of cells is in excellent agreement with that predicted using a large number of displacement-based finite elements.     

Instability analysis of non-homogeneous materials under biaxial loading

In this paper, we formulate the instability problem of non-homogeneous materials under biaxial loading, in the sense that the mechanical parameters, and more specifically the strain hardening or softening and the strain-rate sensitivity, are spatially dependent functions. We first study the time behavior of the process and present the evolution of strain non-uniformities before the critical time, to show that it depends on the interplay between the spatial distribution of mechanical parameters and the initial non-uniformities. We next study the instability modes of a material exhibiting inhomogeneities along one direction by an effective instability analysis [Dudzinski, D., Molinari, A., 1991. Perturbation analysis of thermoviscoplastic instabilities in biaxial loading. International Journal of Solids and Structures 27 (5), 601–628] adapted to the non-homogeneous case. This method is based on a suitable non-uniform reference solution and allows to select the localization modes activated at different deformation levels for different “deformation paths” of the material zones.     

Experimental observations of evolving yield loci in biaxially strained AA5754-O

Experimental measurement of the plastic biaxial mechanical response for an aluminum alloy (AA5754-O) sheet metal is presented. Traditional methods of multiaxial sheet metal testing require the use of finite element analysis (FEA) or other assumptions of material response to determine the multiaxial true stress versus true strain behavior of the as-received sheet material. The method used here strives to produce less ambiguous measurements of data for a larger strain range than previously possible, through a combination of the Marciniak flat bottom ram test and an X-ray diffraction technique for stress measurement. The study is performed in conjunction with a study of the microstructural changes that occur during deformation, and these microstructural results are briefly mentioned in this work. Issues of calibration and applicability are discussed, and results are presented for uniaxial (U), plane strain (PS), and balanced biaxial (BB) extension. The results show repeatable behavior (within quantified uncertainties) for U to 20%, PS to almost 15%, and BB to above 20% in-plane strains. The results are first compared with three common yield locus models (von Mises’, Hill’48, and Hosford’79), and show some unexpected results in the shape change of the yield locus at high strain levels (>5% strain). These changes include the rotation of the locus toward the von Mises surface and elongation in the balanced biaxial direction. Comparison with a more complex yield locus model (Yld2000-2d with eight adjustable parameters) showed that the locus elongation in the biaxial direction could be fit well (for a specific level of work), but at the detriment of fit to the plane strain data. Artificially large plastic strain ratios would be needed to match both the biaxial and plane strain behavior even with this more complex model.     

混凝土动态双轴拉压破坏准则细观数值模拟研究

正常服役期内的混凝土结构往往处于复杂应力状态,并且不可避免地会受到偶发动力载荷作用.对于复杂载荷作用下的混凝土力学性能研究,破坏准则是基础.受试验设备等条件限制,现有的动态双轴拉压破坏准则形式复杂、缺乏更高应变率和侧应力比范围且尚未综合考虑应变率和侧应力比的耦合作用.为进一步提出适用范围更高且更准确的混凝土动态双轴拉压...     

Measurement of the shapes of foil bulge-test samples

Accurate measurements of important tensile properties of thin metal foils are often quite difficult to achieve in uniaxial tests because of sample-preparation difficulties and the tensile instability called necking. Consequently, hydraulic bulge tests have been introduced as a successful means of suppressing these problems through the use of a simplified specimen geometry and biaxial rather than uniaxial tensile-stress states. Considerable effort has been made by various investigators to relate such biaxial stress-strain and ductility data to uniaxial data, generally following the assumption that the bulge is shaped like a spherical cap. The present study evaluates this assumption for foils by measuring actual shapes with unprecedented accuracy and detail using the two-source holographic technique and a polynomial-spline computer analysis of the resulting interferograms. These measurements were made on nine specimens of 0.127-mm-thick annealed rolled copper foil which had been deformed into bulges of varying heights up to rupture. A comparison is made between the measured shapes and the spherical-cap shape generally assumed in the interpretation of bulge-test data. The spherical assumption gives results which are reasonably valid for the later stages of deformation. Indeed, the stress-strain curve obtained from bulge testing corresponds closely with the uniaxial tensile curves for this material. The strain at failure (i.e., elongation) was greater in the biaxial bulge test than in the uniaxial test but not nearly as great as the strain expected from a theoretical model proposed by Hill. However, all the specimens measured exhibited localized areas with larger radii of curvature. The presence of these “flats” may be associated with a mode of failure in the bulge test which corresponds to necking instability in the uniaxial test, and thereby account for the limited strain to failure.     

Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression

In this paper, employing the homogenization theory and the microscopic bifurcation condition established by the authors, we discuss which microscopic buckling mode grows in elastic honeycombs subject to in-plane biaxial compression. First, we focus on equi-biaxial compression, under which uniaxial, biaxial and flower-like modes may develop as a result of triple bifurcation. By forcing each of the three modes to develop, and by comparing the internal energies, we show that the flower-like mode grows steadily if macroscopic strain is controlled, while either the uniaxial or biaxial mode develops if macroscopic stress is controlled. Second, by analyzing several cases other than equi-biaxial compression, it is shown that a second bifurcation from either the uniaxial or biaxial mode to the flower-like mode, which is distorted, occurs under biaxial compression in a certain range of biaxial ratio under macroscopic strain control. Finally, the possibility of macroscopic instability under biaxial compression is discussed.     

On the deformation and fracture of porous materials

The constitutive behavior of porous materials (including the yield loci, the void growth rate, the macro stress-strain relation and the strain to localization instability) is examined based on the lower bound approach proposed by the present authors. These results are then compared with the experimental and the finite element results as well as those predicted by Gurson's equations. Emphasis is placed on approaching the real behavior from the upper and the lower bound analysis. Calculation is also made on the influence of void nucleation on the critical strain to instability and a modified strain-controlled nucleation criterion is proposed. Finally the instability and fracture of AISI4340 steel in plane strain tension is examined and comparison is made between theoretical and experimental results.     

Analysis of diffuse plastic stability in tubes and sheets

The diffuse plastic instability in tubes and sheets under biaxial stress conditions is examined by the use of perturbation methods. Very general constitutive relationships for material properties are used. This requires the inclusion of first order changes in the strain directions inside the patch and also treatment of the material anisotropy and strain rate sensitivity in addition to strain hardening. The inclusion of variations in strain direction is found to alter the form of the characteristic equation for stability from first order to second order but both roots are real for all cases investigated. The value of strain hardening at which the largest root becomes significantly positive is almost the same as that reached when changes in strain directors are ignored. However the strain hardening at which this root becomes formally zero can be very different. The former condition is considered to be of more practical importance than the latter. By this test the stability increases rapidly above a strain rate sensitivity of about 0.1.     

Cyclic multiaxial and shear finite deformation response of OFHC: Part I,experimental results

A series of experiments has been conducted on oxygen free high conductivity (OFHC) copper hollow cylinders under cyclic free-end torsion and biaxial tension–torsion at large strains. In addition, equations are developed to account for the finite rotation and strains in electrical resistance strain gages. In free-end cyclic torsion experiments with shear strain range equal to 23%, a significant strain in the axial direction is observed and it accumulates with a constant rate cycle by cycle. In the biaxial tension–torsion (multiaxial ratchetting) experiments, in which the primary (constant) axial stress is larger than the initial yield stress of the material, the loading conditions are varied to determine the influence of primary axial stress, cyclic shear strain range, pre-cyclic hardening and loading sequence on multiaxial ratchetting. Some important experimental features are high-lighted and recommended to help modeling efforts later.     

Photoplastic stress analysis considering rate effect

The possibilities of cellulose acetate as a photoplastic model material used in the uniaxial tension test were presented by the authors in Ref. 1. Empirical formulas were proposed for photoplastic stress analysis considering the effect of strain rate and temperature. In this paper, the stress-strain-optic laws derived from the uniaxial tension test were confirmed precisely under the biaxial stress field. Consequently, the biaxial stress-strain-optic laws of cellulose acetate can be represented by simply substituting the difference of the principal stress and the difference of the principal strain into the terms of the stress and the strain in the empirical formulas for the uniaxial stress state, respectively. Therefore, the empirical formulas considering the influence of strain rate were obtained for the photo-viscoelastoplastic plane-stress analysis during the usual static loading. This method was applied to an elastoplastic problem of a finite plate with a circular hole under uniaxial tension. The distribution of stress and strain, the stress-concentration factors in the elastoplastic region and the development of the plastic region were obtained experimentally taking the rate effect into account. As a result, the stress on a minimum section in a model increased as head speed increased. On the other hand, the distribution of strain, the development of the plastic region and the stress-concentration factors were almost never influenced by head speed.     

对膨胀壳体材料失稳的一种简化处理

提出一种简化的塑性失稳处理并被作为高应变率拉伸下材料破坏的粗估判据。这种准则假设局部化失稳在如下条件下发生:应变及应变率硬化率恰被几何及热软化率平衡。指出了经验本构方程及本征本构方程对膨胀断裂应变的影响。通过算例也说明了热软化壳体在内压作用下的破裂过程。     

Thermal softening induced plastic instability in rate-dependent materials

A linear perturbation analysis is performed for a class of rate-dependent materials, such as the Johnson-Cook model, in which the rate contribution to the stress can be separated from that of the plastic strain and temperature and in which the temperature rises adiabatically. The analysis is facilitated by perturbing both the rate of momentum equation and the momentum equation. An identical material stability/instability criterion is deduced from the characteristic spectral equations for one-dimensional deformation, one-dimensional shearing, and general three-dimensional field equations, and thus shows that the instability derived here is a material constitutive instability.The criteria indicate that the materials become unstable once the thermal softening overcomes the strain hardening, regardless of the strain rate. The strain rate enters the criteria through its effects on the accumulated temperature and the current stress. Based on the criterion, the three-dimensional instability surface is established in the space of plastic strain, plastic strain rate, and temperature. Instability surface is shown as a material property and independent of deformation histories or modes. Both necking and shear banding are simulated to validate the excellent predictive capability of the criterion.     

Lattice orientation effects on void growth and coalescence in fcc single crystals

Void growth and coalescence in fcc single crystals were studied using crystal plasticity under uniaxial and biaxial loading conditions and various orientations of the crystalline lattice. A 2D plane strain unit cell with one and two cylindrical voids was employed using three-dimensional 12 potentially active slip systems. The results were compared to five representative orientations of the tensile axis on the stereographic triangle. For uniaxial tension conditions, the void volume fraction increase under the applied load is strongly dependent on the crystallographic orientation with respect to the tensile axis. For some orientations of the tensile axis, such as [1 0 0] or [1 1 0], the voids exhibited a growth rate twice as fast compared with other orientations ([1 0 0], [2 1 1]). Void growth and coalescence simulations under uniaxial loading indicated that during deformation along some orientations with asymmetry of the slip systems, the voids experienced rotation and shape distortion, due mainly to lattice reorientation. Coalescence effects are shown to diminish the influence of lattice orientation on the void volume fraction increase, but noteworthy differences are still present. Under biaxial loading conditions, practically all differences in the void volume fraction for different orientations of the tensile axes during void growth vanish. These results lead to the conclusion that at microstructural length scales in regions under intense biaxiality/triaxiality conditions, such as crack tip or notched regions, the plastic anisotropy due to the initial lattice orientation has only a minor role in influencing the void growth rate. In such situations, void growth and coalescence are mainly determined by the stress triaxiality, the magnitude of accumulated strain, and the spatial localization of such plastic strains.     

Localization in elasto-plastic materials: Influence of the plasticity yield surface in biaxial loading conditions

The influence of the plasticity yield surface on the development of instabilities in plane plates in biaxial loading is analyzed in order to understand and simulate the localization pattern observed in an expanding hemisphere experiment. First, a criterion for the activation of slip bands is formulated in the form of a critical hardening coefficient: it is particularized to the Von Mises and Tresca surfaces. In the Von Mises case, the criterion gives a strongly negative hardening coefficient in biaxial loading conditions different from the ones of plane strain. In the Tresca case, the criterion is fulfilled for a perfectly plastic material in uniaxial and biaxial loading; besides, in equi-biaxial loading, two possible orientations for slip bands are exhibited; this can be understood, with a few approximations, by the existence of a vertex point on the Tresca yield surface which give additive degrees of freedom for the direction of the plastic strain rate. Second, the development of localization in the loading conditions met in an expanding hemisphere experiment is simulated using both plasticity yield surfaces; whereas the Von Mises simulation does not localize, the Tresca simulation exhibits a pattern composed of a network of shear bands of different orientations; this pattern is not far from the pattern observed experimentally.     

An investigation on microscopic and macroscopic stability phenomena of composite solids with periodic microstructure

An analysis of the effects of microscopic instabilities on the homogenized response of heterogeneous solids with periodic microstructure and incrementally linear constitutive law is here carried out. In order to investigate the possibility to obtain a conservative prediction of microscopic primary instability in terms of homogenized properties, novel macroscopic constitutive stability measures are introduced, corresponding to the positive definiteness of the homogenized moduli tensors relative to a class of conjugate stress–strain pairs.Numerical simulations, addressed to hyperelastic microstructural models representing cellular solids and reinforced composites, are worked out through the implementation of an innovative one-way coupled finite element formulation able to determine sequentially the principal equilibrium solution, the incremental equilibrium solutions providing homogenized moduli and the stability eigenvalue problem solution, for a given monotonic macrostrain path. Both uniaxial and equibiaxial loading conditions are considered.The exact microscopic stability region in the macrostrain space, obtained by taking into account microstructural details, is compared with the macroscopic stability regions determined by means of the introduced macroscopic constitutive measures. These results highlight how the conservativeness of the adopted macroscopic constitutive stability measure with respect to microscopic primary instability, strictly depends on the type of loading condition (tensile or compressive) and the kind of microstructure.     

Localization in elasto-plastic materials: Influence of an evolving yield surface in biaxial loading conditions

The influence of the plasticity yield surface – and of its evolution with plastic deformation – on the development of instabilities in metals is analyzed. Conditions for the activation of slip bands are taken as an instability criterion. They are exhibited in stress states identical to the ones encountered in a flat plate in biaxial tension. The classical bifurcation criterion is replaced by a criterion on the growth of a perturbation at a time scale comparable to the one of the homogeneous solution. This second criterion reveals less severe than the bifurcation one which is reached for the limit case of an infinite growth rate in the perturbation approach. The growth rate is a decreasing function of the biaxiality of the loading which is in agreement with previous studies. The possible destabilizing effect of texture evolution is also exhibited by using an evolving yield surface the curvature of which increases in the neighborhood of the homogeneous solution.     

An examination of dynamic fracture under biaxial-strain conditions

An experiment to investigate the influence of biaxial strain on the dynamic fracture of metals is outlined. A hollow cylindrical specimen is loaded on the inner diameter by a polyethylene-coated exploding wire which results in a circumferential spallation pattern. Prior to fracture, the spall surface undergoes biaxial deformation with a total hoop strain of the same order of magnitude as the total radial strain. This is in contrast to earlier studies in which fracture induced by stress waves was examined in experiments which are characterized by uniaxial-strain conditions. Comparing the results of the two configurations, it can be shown that the maximum principal-stress history required to cause fracture is the same in both cases, although the stresses normal to the fracture surface are quite different. High-speed photographs of the coated exploding wires and dynamic-flash X-rays of the specimens are included to demonstrate the axisymmetry of both the loading and the spall phenomenon.     

Large-deformation properties of wheat dough in uni- and biaxial extension. Part I. Flour dough

Rheological and fracture properties of optimally mixed flour doughs from three wheat cultivars which perform differently in cereal products were studied in uniaxial and biaxial extension. Doughs were also tested in small angle sinusoidal oscillation. In accordance with previously published results the linear region was found to be very small. The rheological properties at small deformations hardly depended on the cultivar. A higher water content of the dough resulted in a lower value for the storage modulus and a slightly higher value for tan . For both uniaxial and biaxial extension a more than proportional increase in stress was found with increasing strain, a phenomenon called strain hardening. In uniaxial extension (i) stresses at a certain strain were higher and (ii) the stress was less dependent on the strain rate than in biaxial extension. This indicates that in elongational flow orientational effects are of large importance for the mechanical properties of flour dough. This conclusion is consistent with published data on birefringence of stretched gluten. Fracture stress and strain increased with increasing deformation rate. The observed time-dependency of fracture properties can best be explained by inefficient transport of energy to the crack tip. Presumably, this is caused by energy dissipation due to inhomogeneous deformation because of friction between structural elements, e.g. between dispersed particles and the network. Differences in the rheological properties at large deformations between the cultivars were observed with respect to (i) stress, (ii) strain hardening, (iii) strain rate dependency of the stress, (iv) fracture properties and (v) the stress difference between uniaxial and biaxial extension.     

A strain localization analysis using a viscoplastic softening model for clay

Strain localization has become an attractive subject in geomechanics during the past decade. Shear bands are well known to develop in clay specimens during the straining process. Strain localization is closely related to plastic instability. In the present paper, a non-linear instability condition for the viscoplastic strain softening model during the creep process is firstly obtained. It is found that the proposed viscoplastic model is capable of describing plastic instability. Secondly, a two-dimensional linear instability analysis is performed and the preferred orientation for the growth of fluctuation and the instability condition are derived. It is worth noting that the two instability conditions are equivalent. Finally, the behavior of the clay is numerically analyzed in undrained plane-strain compression tests by the finite element method, considering a transport of pore water in the material at a quasi-static strain rate. The numerical results show that the model can predict strain localization phenomena, such as shear banding. From the numerical calculations, the effects of strain rate and permeability are discussed.     

The effect of microstructural representation on simulations of microplastic ratcheting

This paper assesses the sensitivity of cyclic plasticity to microstructure morphology by examining and comparing the microplastic ratcheting behavior of different idealized microstructures (square, hexagonal, tessellated, and digitized from experimental data). This analysis demonstrates the sensitivity of computational accuracy to the various approximations in microstructural representation. The methodology used to perform this study relies on a coupling between microstructural characterization, mechanical testing and numerical simulations to investigate the influence of the microstructure on the purely tensile uniaxial microplastic ratcheting behavior of pure nickel polycrystals. The morphology and deformation behavior of polycrystals were characterized using electron back-scatter diffraction (EBSD), while a finite element model (FEM) of crystal plasticity was used in a computational framework. The predicted cyclic behavior is compared to experimental results both at the macroscopic and microstructural scales. The stress–strain response is less sensitive to the details of the microstructural representation than might be expected with all representations displaying similar macroscopic constitutive response. However, the details of the plastic strain distribution at the microstructural scale and the related estimations of damage mechanics vary substantially from one microstructural representation to another.