Elasto-plastic constitutive relations of unidirectional fiber composites for cyclic loadings

The clasto-plastic constitutive behaviors of continuous fiber reinforced composites under cyclic loadings are studied by the micromechanics method in which the equal-strain model is used in the fiber direction, the equal-stress model in the other directions. It is supposed that fiber is linearly elastic and matrix is clastic-viscoplastic. The constitutive equations of the matrix are described by Bodner-Partom's unified constitutive theory. Boron/Aluminum composite, as an example, is investigated in detail for an understanding of the stress-strain relations and initial yield behaviors of metal matrix composites. Present results are compared with the experimental data.The project was supported by the Chinese National Natural Science Foundation.     

Influence of the loading path on the strength of fiber-reinforced composites subjected to transverse compression and shear

The influence of the loading path on the failure locus of a composite lamina subjected to transverse compression and out-of-plane shear is analyzed through computational micromechanics. This is carried out using the finite element simulation of a representative volume element of the microstructure, which takes into account explicitly fiber and matrix spatial distribution within the lamina. In addition, the actual failure mechanisms (plastic deformation of the matrix and interface decohesion) are included in the simulations through the corresponding constitutive models. Two different interface strength values were chosen to explore the limiting cases of composites with strong or weak interfaces. It was found that failure locus was independent of the loading path for the three cases analyzed (pseudo-radial, compression followed by shear and shear followed by compression) in the composites with strong and weak interfaces. This result was attributed to the fact that the dominant failure mechanism in each material was the same in transverse compression and in shear. Failure is also controlled by the same mechanisms under a combination of both stresses and the failure locus depended mainly on the magnitude of the stresses that trigger fracture rather than in the loading path to reach the critical condition.     

Mechanisms of stress transfer and interface integrity in carbon/epoxy composites under compression loading. Part II: Numerical approach

The finite element method is used to get an insight into the micromechanics of the compressive behaviour of carbon fibre composites. First the developed model is validated with existing experimental data and good agreement between predictions and experiments was found. Then the FE model is used to derive the complete stress field in the fibre and the matrix in the vicinity of a fibre fracture location. It was found that the perturbation of the stress field occurs mainly in the direction transverse to the fibre axis and this could explain the failure modes observed in composites tested in compression. Finally, a parametric study was performed on the effect of matrix modulus and matrix yield stress on the compressive fragmentation process.     

Theoretical and experimental considerations in representing macroscale flow/damage surfaces for metal matrix composites

This report shows how robust, multiaxial, constitutive models for advanced materials can be formulated by using micromechanics to address theoretical and experimental issues. An analytical micromechanical model that includes viscoplastic matrix response, as well as fiber-matrix debonding, is used to predict the multiaxial response of metal matrix composites in terms of macro flow/damage surfaces at room and elevated temperatures. Macro flow/damage surfaces (i.e. debonding envelopes, matrix threshold surfaces, macro ‘yield’ surfaces, surfaces of constant inelastic strain rate, and surfaces of constant dissipation rate) are determined for a silicon carbide/ titanium composite in three stress spaces. The flow/damage surfaces are shown to have their centers offset from the origin by residual stresses and their shape altered by debonding. The normality condition is shown to be reasonably well satisfied for macro surfaces of constant dissipation rate in the presence of fiber-matrix debonding. These results indicate which types of flow/damage surfaces should be characterized and what loadings must be applied to obtain the most meaningful experimental data for guiding theoretical model development and verification.     

颗粒增强复合材料的屈服极限

基于渐近均匀化方法,导出了颗粒增强复合材料的屈服准则,给出了初始屈服应力的解析表达式。因为颗粒增强体的弹性模量远比弹塑性基体的弹性模量高,这个模量差在增强体和基体中产生了装配刚度。解局部问题可以得到该装配刚度。从屈服应力的表达式可以看出,增强体和基体两者的平均装配刚度和剪切模量之比决定了屈服应力的提高程度。给出了两个数值算例。采用菱形十二面体单胞求解了局部问题。取单胞形状为菱形十二面体的优点是增强体的体积比可以高达74%。     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part-I: A very low work hardening aluminum alloy (Al6061-T6511)

In the present study, the initial and subsequent yield surfaces in Al 6061-T6511, based on 10 με deviation from linearity definition of yield, are presented. The subsequent yield surfaces are determined during tension, free end torsion, and combined tension–torsion proportional loading paths after reaching different levels of strains. The yield surfaces are also obtained after linear, bi-linear and non-linear unloading paths after finite plastic deformation. The initial yield surface is very close to the von-Mises yield surface and the subsequent yield surfaces undergo translation and distortion. In the case of this low work hardening material, the size of the yield surfaces is smaller and negative cross-effect is observed with finite plastic deformation. The subsequent yield have a usual “nose” in the loading direction and flattened shape in the reverse loading direction; the observed nose is more dominant in the case of tension and combined tension–torsion loading than in torsional loading. The size of the yield surfaces after unloading is smaller than the initial yield surface but larger than subsequent yield surfaces obtained during prior loading, show much smaller cross-effect, and the shape of these yield surfaces depends strongly on the loading and unloading paths. Elastic constants (Young’s and shear moduli) are also measured within each subsequent yield surfaces. Evolution of these constants with finite deformation is also presented. The decrease of the two moduli is found to be much smaller than reported earlier in tension by Cleveland and Ghosh [Cleveland, R.M., Ghosh, A.K., 2002. Inelastic effects on springback in metals. Int. J. Plast. 18, 769–785]. Part-II and III [(Khan et al., 2009a) and (Khan et al., 2009b)] of the papers will include experimental results on annealed 1100 Al (a very high work hardening material) and on both Al alloys (Al6061-T6511 and annealed 1100 Al) in tension- tension stress space, respectively. The results for both cases are quite different than the ones that are presented in this paper.     

非理想界面对三相复合材料应力场的影响

对非理想界面的三相复合材料,提出了计算弹性应力场的微观力学模型,在适当的简化假设下,对带界相的颗粒增强和纤维增强复合材料,得到了应力场的计算公式。以剪切载荷为例给出了数值例子。给出的数值结果表明非理想界面对三相复合材料应力场的影响。     

基于等效特征应变原理的复合材料有效模量细观力学分析

基于等效特征应变原理,提出了一种新的复合材料有效模量细观力学分析方法。首先,在等效特征应变原理基础上提出平均等效特征应变原理,它可用于解决有限体下任意形状(无论是凸或凹形)的单个夹杂或多个夹杂的弹性变形问题。其次,将平均等效特征应变原理与细观力学直接均匀法相结合,来分析确定复合材料的有效模量。最后利用复合材料纤维与基体的力学性能参数及纤维的体分比,借助MATLAB编程方法,预测其有效模量。通过将理论预测值与已有的的试验值、其它理论预测值进行对比,验证了新分析方法的合理性和分析精度。     

Yield surfaces of various periodic metal honeycombs at intermediate relative density

Initial yield surfaces are derived for several periodic metal honeycomb cell structures with sufficiently high relative density that failure occurs by plastic yielding. Both in-plane stress states (normal stresses perpendicular to cell axes, with in-plane shear) and triaxial stress states with one principal stress direction along the cell axes are considered. Beam/column and plate/shell yield criteria are adopted to address general in-plane loading and 3D triaxial loading, respectively, accounting for combined cell wall stretching and bending as appropriate. Cell wall behavior is assumed to be elastic-perfectly plastic. The initial yield surfaces for different periodic cell structures are systematically compared. Some issues related to the initial yield surfaces of various honeycombs are discussed, including dependencies on relative density and in-plane and out-of-plane applied stresses, as well as the influence of joints between cell walls.     

Crystallographic orientation and size dependence of tension-compression asymmetry in molybdenum nano-pillars

Uniaxial tension and compression experiments on [0 0 1] and [0 1 1] oriented molybdenum nano-pillars exhibit tension-compression asymmetry, a difference in attained stresses in compression vs. tension, which is found to depend on crystallographic orientation and sample size. We find that (1) flow stresses become higher at smaller diameters in both orientations and both loading directions, (2) compressive flow stresses are higher than tensile ones in [0 0 1] orientation, and visa versa in [0 1 1] orientation, and (3) this tension-compression asymmetry is in itself size dependent. We attribute these phenomena to the dependence of twinning vs. antitwinning deformation on loading direction, to the non-planarity of screw dislocation cores in Mo crystals, and to the possibly lesser role of screw dislocations in governing nano-scale plasticity compared with bulk Mo.     

Modeling the effects of material non-linearity using moving window micromechanics

In the analysis of materials with random heterogeneous microstructure the assumption is often made that material behavior can be represented by homogenized or effective properties. While this assumption yields accurate results for the bulk behavior of composite materials, it ignores the effects of the random microstructure. The spatial variations in these microstructures can focus, initiate and propagate localized non-linear behavior, subsequent damage and failure. In previous work a computational method, moving window micromechanics (MW), was used to capture microstructural detail and characterize the variability of the local and global elastic response. Digital images of material microstructure described the microstructure and a local micromechanical analysis was used to generate spatially varying material property fields. The strengths of this approach are that the material property fields can be consistently developed from digital images of real microstructures, they are easy to import into finite element models (FE) using regular grids, and their statistical characterizations can provide the basis for simulations further characterizing stochastic response. In this work, the moving window micromechanics technique was used to generate material property fields characterizing the non-linear behavior of random materials under plastic yielding; specifically yield stress and hardening slope, post yield. The complete set of material property fields were input into FE models of uniaxial loading. Global stress strain curves from the FE–MW model were compared to a more traditional micromechanics model, the generalized method of cells. Local plastic strain and local stress fields were produced which correlate well to the microstructure. The FE–MW method qualitatively captures the inelastic behavior, based on a non-linear flow rule, of the sample continuous fiber composites in transverse uniaxial loading.     

Mechanisms of stress transfer and interface integrity in carbon/epoxy composites under compression loading: Part I: Experimental investigation

Raman spectroscopy is used to get an insight into the microstructural aspects of the compressional behavior of carbon fiber composites. This is done by a comparative assessment of the stress transfer efficiency in tension and compression in single-fiber discontinuous model geometries. It was found that axial stress is transferred in the fiber through the generation of shear stresses at the interface for both tension and compression loading. Experimental evidence is presented to verify that the values of the maximum interfacial shear stress that the system sustains is a function of the applied strain and independent of the type of loading. However, compressive failure is quite different as fiber fragments remain in contact, thus can still bear load.     

On thermoelastostatics of composites with nonlocal properties of constituents II. Estimation of effective material and field parameters

One considers a linear thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities. It is assumed that the stress–strain constitutive relations of constituents are described by the nonlocal integral operators, whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity. The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. In a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities, one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties (local or nonlocal) of heterogeneities. In such a case, the Levin’s (1967) formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart. A solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method. The operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and “ellipsoidal symmetry” assumption. Quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented.     

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.     

室温下镁合金板循环塑性随动强化本构模型研究

本文在具有各向异性屈服强度和拉压不对称的CPB06屈服准则的基础上,建立了基于随动强化的循环塑性本构模型．通过引入滑移、孪晶以及去孪等不同变形模式下的背应力演化方程,对室温下镁合金板材异常循环硬化行为进行了模拟．选取了AZ31B-O和AZ31B两种镁合金板材,通过拉伸-压缩-拉伸(T-C-T)和压缩-拉伸(C-T)等不同加载路径下的部分实验曲线确定模型的参数,采用三次插值多项式建立了背应力参数与上一变形模式中累积的等效塑性应变（即预应变）之间的函数关系．使用本模型对剩下的实验曲线进行了预测,发现预测结果与实验结果有良好的一致性,说明了当前模型的正确性．     

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     

Axisymmetric micromechanics of elasticity-perfectly plastic fibrous composites under uniaxial tension loading

The uniaxial response of a continuous fiber elastic-perfectly plastic composite is modelled herein as a two-element composite cylinder. An axisymmetric analytical micromechanics solution is obtained for the rate-independent elastic-plastic response of the two-element composite cylinder subjected to tensile loading in the fiber direction for the case wherein the core fiber is assumed to be a transversely isotropic elastic-plastic material obeying Tsai-Hill's yield criterion, with yielding simulating fiber failure. The matrix is assumed to be an isotropic elastic-plastic material obeying Tresca's yield criterion. It is found that there are three different circumstances that depend on the fiber and matrix properties: (1) fiber yield, followed by matrix yielding; (2) complete matrix yield, followed by fiber yielding; and (3) partial matrix yield, followed by fiber yielding, followed by complete matrix yield. The order in which these phenomena occur is shown to have a pronounced effect on the predicted uniaxial effective composite response.     

压应力对压剪裂纹扩展的影响研究

本文针对压剪裂纹的启裂及扩展问题,研究了脆性材料中裂纹面压应力变化对其扩展的影响规律.借助双轴加载试验机可自由调节两个轴位移和力的优势,设计了一种单边对称双裂纹压剪试样.试验中,施加裂纹面压应力至不同的预设值后,使剪应力单调增大直至裂纹启裂及扩展,得到不同预设压应力下压剪裂纹启裂及扩展规律. 随着预设压应力增大,启裂角增大,裂纹扩展路径与初始裂纹的偏角也增大.但随着压应力增大,启裂角增大至一定值后趋于稳定,实验发现,依据裂纹是否闭合,压应力对压剪裂纹扩展的作用大致可分为两个阶段:闭合前,压应力对裂纹启裂载荷及启裂角、扩展路径均有影响,预设压应力增大,裂纹启裂载荷增大、启裂角增大,扩展路径愈来愈偏离初始裂纹方向;闭合后,压应力虽然增大,启裂角和临界压剪应力比始终恒定,压应力对临界剪力和扩展路径存在一定影响.研究表明,裂纹启裂角与启裂时的压剪应力比存在一定的对应关系.启裂时的压剪应力比增大,启裂角增大,启裂时的压剪应力比恒定,启裂角不变.