Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites

The aim of this work is to construct yield surfaces to describe initial yielding and characterize hardening behavior of a highly anisotropic material. A methodology for constructing yield surfaces for isotropic materials using axial–torsion loading is extended to highly anisotropic materials. The technique uses a sensitive definition of yielding based on permanent strain rather than offset strain, and enables multiple yield points and multiple yield surfaces to be conducted on a single specimen. A target value of 20 × 10⁻⁶ is used for Al₂O₃ fiber reinforced aluminum laminates having a fiber volume fraction of 0.55. Sixteen radial probes are used to define the yield locus in the axial–shear stress plane. Initial yield surfaces for [0₄], [90₄], and [0/90]₂ fibrous aluminum laminates are well described by ellipses in the axial–shear stress plane having aspect ratios of 10, 2.5, and 3.3, respectively. For reference, the aspect ratio of the Mises ellipse for an isotropic material is 1.73. Initial yield surfaces do not have a tension–compression asymmetry. Four overload profiles (plus, ex, hourglass, and zee) are applied to characterize hardening of a [0/90]₂ laminate by constructing 30 subsequent yield surfaces. Parameters to describe the center and axes of an ellipse are regressed to the yield points. The results clearly indicate that kinematic hardening dominates so that material state evolution can be described by tracking the center of the yield locus. For a nonproportional overload of (σ, τ) = (500, 70) MPa, the center of the yield locus translated to (σ, τ) = (430, 37) MPa and the ellipse major axis was only 110 MPa.     

近代海积软土结构性及弹-塑性模型研究

土的结构性数学模型是 2 1世纪土力学的核心问题。本文分析了近代海积软土的结构性以及其塑性体应变硬化和塑性偏应变硬化的机制 ,建立了弹塑性本构模型 ,模型采用了具有双硬化参数的屈服轨迹及含结构性参数 m的塑性应变方程。     

Necking of anisotropic micro-films with strain-gradient effects

Necking of stubby micro-films of aluminum is investigated numerically by considering tension of a specimen with an initial imperfection used to onset localisation. Plastic anisotropy is represented by two different yield criteria and strain-gradient effects are accounted for using the visco-plastic finite strain model. Furthermore, the model is extended to isotropic anisotropic hardening （evolving anisotropy）. For isotropic hardening plastic anisotropy affects the predicted overall nominal stress level, while the peak stress remains at an overall logarithmic strain corresponding to the hardening exponent. This holds true for both local and nonlocal materials. Anisotropic hardening delays the point of maximum overall nominal stress.     

Optimization of Digital Image Correlation for High-Resolution Strain Mapping of Ceramic Composites

Digital image correlation (DIC) is assessed as a tool for measuring strains with high spatial resolution in woven-fiber ceramic matrix composites. Using results of mechanical tests on aluminum alloy specimens in various geometric configurations, guidelines are provided for selecting DIC test parameters to maximize the extent of correlation and to minimize errors in displacements and strains. The latter error is shown to be exacerbated by the presence of strain gradients. In a case study, the resulting guidelines are applied to the measurement of strain fields in a SiC/SiC composite comprising 2-D woven fiber. Sub-fiber tow resolution of strain and low strain error are achieved. The fiber weave architecture is seen to exert a significant influence over strain heterogeneity within the composite. Moreover, strain concentrations at tow crossovers lead to the formation of macroscopic cracks in adjacent longitudinal tows. Such cracks initially grow stably, subject to increasing app lied stress, but ultimately lead to composite rupture. Once cracking is evident, the composite response is couched in terms of displacements, since the computed strains lack physical meaning in the vicinity of cracks. DIC is used to identify the locations of these cracks (via displacement discontinuities) and to measure the crack opening displacement profiles as a function of applied stress.     

Unit cell debonding analyses for arbitrary orientation of plastic anisotropy

Debonding of rigid inclusions embedded in the elastic–plastic aluminum alloy Al 2090-T3 is analyzed numerically using a unit cell model taking full account of finite strains. The cell is subjected to overall biaxial plane strain tension and periodical boundary conditions are applied to represent arbitrary orientations of plastic anisotropy. Plastic anisotropy is considered using two phenomenological anisotropic yield criteria, namely Hill [Proceedings of the Royal Society of London A 193 (1948) 281] and Barlat et al. [International Journal of Plasticity 7 (1991) 693]. For this material plastic anisotropy delays debonding compared to plastic isotropy except for the case of Hill’s yield function when the tensile directions coincided with the principal axes of anisotropy. For some inclinations of the principal axes of anisotropy relative to the tensile directions, the stress strain responses are identical but the deformation modes are mirror images of each other.     

Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure

The anisotropic plastic deformation behavior of extruded 5000 series aluminum alloy tubes, A5154-H112, of 76 mm outer diameter and 3.9 mm wall thickness is investigated, using a servo-controlled tension-internal pressure testing machine. This machine is capable of applying arbitrary stress or strain paths to a tubular specimen using an electrical, closed-loop control system. Detailed measurements were made of the initial yield locus, contours of plastic work for different levels of work-hardening, and the directions of the incremental plastic strain vectors for both linear and combined stress paths. It is found that the measured work contours constructed in the principal stress space are similar in shape, and that the directions of the incremental plastic strain vectors remain almost constant at constant stress ratios. The work-hardening behavior predicted using Hosford's or the Yld2000-2d yield functions under the assumption of isotropic hardening agrees closely with the observations for both linear and combined stress paths. The material is thus found to work-harden almost isotropically. Both yield functions are effective phenomenological plasticity models for predicting the anisotropic plastic deformation behavior of the material.     

Anisotropic yield functions with plastic-strain-induced anisotropy

In most anisotropic yield functions, the stress exponent, M, associated with the shape of the yield surface is usually independent of plastic-strain accumulation. This does not allow for different work-hardening characteristics under various strain states, as has been observed in aluminum alloys. Assuming coefficients characterizing anisotropy do not change with plastic deformation, the M value should vary with plastic strain, relaxing the isotropic hardening assumption. To verify this, plane-strain tests along with numerical analysis were carried out with 2008-T4 aluminium and 70/30 brass. The effective stress and effective plastic-strain curve assuming plane strain and plane stress was fit to the corresponding stress-strain data obtained in uniaxial tension. This was done by allowing M value to vary with effective plastic-strain. Hill's 1979 (case iv),Hosford's 1979 and Barlat's 1991 (6 component) yield functions were evaluated. Results showed that, with all the yield functions tested, the aluminum exhibited substantial variation of M value especially at larger strains while the brass showed minor change. Relevant numerical analysis indicated that, even though all the yield functions showed noticeable changes of M as strain increases in order for the plane-strain curve to match with the uniaxial curve, this variation of M will not affect much to the prediction with Hosford's and Barlat's yield functions, of which the typically valid M is much higher than that of Hill's. FEM simulation of plane-strain sheet forming with 2008-T4 aluminium alloy verified that implementation of varying M-value with Hill's yield function led to better agreement with experimental measurements, while the variation of M with Barlat's yield function exhibited little influence on the strain prediction.     

Theoretical considerations upon the MK model for limit strains prediction: The plane strain case,strain-rate effects,yield surface influence,and material heterogeneity

The paper presents a study of the Marciniak and Kuczynski (MK for short) model for the prediction of limit strains of orthotropic sheet metal under in-plane proportional biaxial stretching. In two particular cases analytical results can be obtained if the groove of the MK model is oriented along one of the in-plane symmetry axes. The first case is the plane strain loading mode. Necessary and sufficient conditions are derived for the MK-predicted plane strain limit strain to match exactly the experimentally measured limit strain. An example of material, the AA5182-O aluminum alloy, that does not satisfy these conditions is discussed. It is shown then that if a power-law strain rate sensitivity is included in the hardening law then the MK-model can match exactly any target plane strain limit strain. The second case is the non-hardening case for positive strain ratios. This case allows for an insight into the way the MK-predicted limit strains depend upon the yield function. Based on the theory developed for the plane strain case, material heterogeneity as a possible cause for unstable plastic flow is further discussed. It is shown that such heterogeneities can be modeled by perturbing the rate of deformation with an eigenstrain. This allows for an extension of the MK-model to sheets of uniform thickness.     

Microstructural effects on yield surface evolution in cubic metals using the viscoplastic ?-model

Polycrystalline yield surfaces of metals are a good way to characterize the anisotropy of plastic deformation. The evolution of these surfaces is impossible to accurately reproduce without taking into account the evolution of the material microstructure such as texture development. In this paper, a numerical computation of yield surfaces using the viscoplastic ?-model is proposed. Results concerning face-centered cubic metals subjected to a plane strain compression test are presented. The influence of several mechanical parameters (strain hardening, strain rate sensitivity coefficient and accumulated deformation) on subsequent yield surfaces evolution is studied. The analysis of the change in the shape and size of the yield surfaces shows that the results depend strongly on the parameter ? which controls the strength of the interactions in the polycrystal. In addition, the predictions are compared to the widely used viscoplastic self-consistent model as well as to experimental yield loci taken from the literature for various aluminum alloy sheets. A fairly good qualitative agreement between our ?-model results and the experimental ones is found. The probable links between the parameter ? and the microstructural features such as the stacking fault energy and the grain size of the polycrystal are also briefly discussed.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part II: A very high work hardening aluminum alloy (annealed 1100 Al)

Results are presented on the evolution of subsequent yield surfaces with finite deformation in a very high work hardening annealed 1100 aluminum alloy. In Part I [Khan, A.S., Kazmi, R., Stoughton, T., Pandey, A., 2009a. Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part 1: a very low work hardening aluminum alloy (Al-6061–T6511) 25, 1611–1625.] of this paper, similar results are presented for a very low work hardening aluminum alloy. Those results were very different from the present ones, and all the results were for proportional loading paths. The subsequent yield surfaces are determined in tension, free end torsion and combined tension–torsion proportional and non-proportional loading paths, using 10 με deviation from linearity definition of yield. Yield surfaces are also determined after linear, bi-linear, and non-linear unloading paths after finite deformation under tension, free end torsion, and combined tension–torsion loading. The initial yield surface is closer to the von-Mises surface and the subsequent yield surfaces show distortion, expansion, positive cross-effect, and “nose” in the loading direction. Additionally, the subsequent yield surfaces after non-proportional loading paths show shrinkage and compounded distortion. The yield surfaces after unloading depict strong anisotropy, positive cross-effect and exhibits different proportion of distortion in each loading conditions. The Young’s and shear modulus decrease with plastic deformation and this decrease is much less than those reported in the published literature.     

Stress integration method for a nonlinear kinematic/isotropic hardening model and its characterization based on polycrystal plasticity

Sheet metal forming processes generally involve non-proportional strain paths including springback, leading to the Bauschinger effect, transient hardening, and permanent softening behavior, that can be possibly modeled by kinematic hardening laws. In this work, a stress integration procedure based on the backward-Euler method was newly derived for a nonlinear combined isotropic/kinematic hardening model based on the two-yield’s surfaces approach. The backward-Euler method can be combined with general non-quadratic anisotropic yield functions and thus it can predict accurately the behavior of aluminum alloy sheets for sheet metal forming processes. In order to characterize the material coefficients, including the Bauschinger ratio for the kinematic hardening model, one element tension–compression simulations were newly tried based on a polycrystal plasticity approach, which compensates extensive tension and compression experiments. The developed model was applied for a springback prediction of the NUMISHEET’93 2D draw bend benchmark example.     

复杂加载下混凝土的弹塑性本构模型

混凝土材料在不同应力路径下或复杂加载条件下会表现出差异性显著的应力应变关系,在小幅循环加载条件下,其应力应变关系会表现出类似于弹性变形的滞回曲线.在不同应力水平下,混凝土的应力应变关系以及破坏特性都具有静水压力相关特点,即随着静水压力增大,各向异性强度特性弱化.此外,混凝土受压及受拉破坏机理不同,因而对应于混凝土硬化损伤亦有不同,即可分为受压硬化损伤,受拉硬化损伤及两者的混合硬化损伤类型.基于Hsieh模型,对该模型进行了三点改进.（1）针对小幅循环加载下混凝土无塑性变形的试验规律,而模型中在应力水平较低的循环加载条件下始终存在塑性变形的预测问题,采用在边界面模型框架下,设置了应力空间的弹性域,初始屈服面与后续临界状态屈服面几何相似的假定.（2）基于广义非线性强度准则将原模型采用变换应力方法将其推广为三维弹塑性本构模型,采用变换后模型可合理的考虑不同应力路径对于子午面以及偏平面上静水压力效应形成的影响,并避免了边界面应力点奇异问题.（3）分别对拉压两种加载损伤模式建议了相应的硬化参数表达式,可分别用于描述上述加载中产生的应变软化及强度退化行为.基于多种加载路径模拟表明:所建立的三维弹塑性本构模型可合理地用于描述混凝土的一般应力应变关系特性.      

应变率对SiC颗粒增强铝基复合材料拉伸性能的影响

本文利用岛津试验机和自行研制的冲击拉伸试验装置,对体积含量为10%的SiC颗粒增强铝基复合材料进行了准静态的拉伸试验、冲击拉伸试验和冲击拉伸加卸载试验,获得了复合材料在应变率为0.002s^-1-1000s^-1范围内从弹塑性变形直至断裂的完整应力应变曲线。试验结果表明,随着应变速率的提高,复合屈服应力,拉伸强度以及破坏应变均相应提高,具有明显的应变率强化效应和高速韧性现象;同时,由于冲击拉伸试验过程中热力耦合效应的影响,准静态加载下复合材料的应力指数与冲击拉伸加载下复合材料的应力指数相比降低了17.8%;在用冲击拉伸复元试验解耦出热力耦合效应的影响后,材料的静、动态等温应力应变曲线具有相同的应变硬化规律。最后,根据复合材料在不同应变率下的试验结果和Eshelby‘s等效夹杂理论,本文建立了一个计及应变率强化效应的弹塑性自洽模型,模型拟合结果与试验结果吻合得很好。     

A plane stress anisotropic plastic flow theory for orthotropic sheet metals

A phenomenological theory is presented for describing the anisotropic plastic flow of orthotropic polycrystalline aluminum sheet metals under plane stress. The theory uses a stress exponent, a rate-dependent effective flow strength function, and five anisotropic material functions to specify a flow potential, an associated flow rule of plastic strain rates, a flow rule of plastic spin, and an evolution law of isotropic hardening of a sheet metal. Each of the five anisotropic material functions may be represented by a truncated Fourier series based on the orthotropic symmetry of the sheet metal and their Fourier coefficients can be determined using experimental data obtained from uniaxial tension and equal biaxial tension tests. Depending on the number of uniaxial tension tests conducted, three models with various degrees of planar anisotropy are constructed based on the proposed plasticity theory for power-law strain hardening sheet metals. These models are applied successfully to describe the anisotropic plastic flow behavior of 10 commercial aluminum alloy sheet metals reported in the literature.     

A combined necking and shear localization analysis for aluminum sheets under biaxial stretching conditions

A combined necking and shear localization analysis is adopted to model the failures of two aluminum sheets, AA5754 and AA6111, under biaxial stretching conditions. The approach is based on the assumption that the reduction of thickness or the necking mode is modeled by a plane stress formulation and the final failure mode of shear localization is modeled by a generalized plane strain formulation. The sheet material is modeled by an elastic-viscoplastic constitutive relation that accounts for the potential surface curvature, material plastic anisotropy, material rate sensitivity, and the softening due to the nucleation, growth, and coalescence of microvoids. Specifically, the necking/shear failure of the aluminum sheets is modeled under uniaxial tension, plane strain tension and equal biaxial tension. The results based on the mechanics model presented in this paper are in agreement with those based on the forming limit diagrams (FLDs) and tensile tests. When the necking mode is suppressed, the failure strains are also determined under plane strain conditions. These failure strains can be used as guidances for estimation of the surface failure strains on the stretching sides of the aluminum sheets under plane strain bending conditions. The estimated surface failure strains are higher than the failure strains of the forming limit diagrams under plane strain stretching conditions. The results are consistent with experimental observations where the surface failure strains of the aluminum sheets increase significantly on the stretching sides of the sheets under bending conditions. The results also indicate that when a considerable amount of necking is observed for a sheet metal under stretching conditions, the surface failure strains on the stretching sides of the sheet metal under bending conditions can be significantly higher.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

Fundamental experiments in plasticity and creep of aluminum—Extension of previous results

Paper presents combined stress experiments in plasticity and creep of aluminum 1100-0. The purpose of these experiments was to determine the motion of the yield surface in tension-torsion space for three complicated prestressing paths, to investigate the validity of the normality hypothesis, to investigate the development of creep strains after prestressing, and finally to investigate the validity of the constant volume hypothesis.It is shown that the law of hardening proposed by the author previously [3,5,6] is valid, except possibly when the prestress path intersects the yield surface at a small angle. It is also shown that the normality hypothesis is valid. After prestressing the creep strain vector has in the beginning the same direction as the plastic strain vector but later its direction may change. Finally it is shown that at the level of permanent strains less than 1% the plastic strains follow the constant volume hypothesis but the creep strains do so only when they begin to appear.     

基于应力三轴度和罗德参数的6061和7075铝合金材料断裂失效分析

分别对6061铝合金和7075铝合金材料的缺口圆棒试件和凹槽平板试件进行准静态拉伸试验,并采用ABAQUS软件对拉伸过程进行数值模拟。模拟结果与试验测试结果吻合很好,验证了有限元模型的合理性和可靠性。通过有限元模拟,分别给出了不同试件的应力三轴度和罗德参数随等效塑性应变的变化曲线以及两种材料的失效轨迹,并对它们进行了分析讨论。结果表明:形状相同、材料性质不同的试件,应力三轴度的演化规律不同;材料的失效应变受应力三轴度和罗德参数的影响,并且不同性质的材料对罗德参数的敏感性不同。     

基于微纳米压入法提取无铅焊料合金弹塑性力学参数的研究

本文通过微纳米压入法结合数值模拟研究了无铅焊料合金SnAg3.5 的弹塑性力学性能，分别采用圆柱形压头及两种不同锥角压头对无铅焊料合金进行压入测试：基于圆柱形压头测试过程中接触面积恒定的特点得到了无铅焊料的弹性模量，进一步采用塑性应变梯度理论对两种锥角压头的测试结果予以修正并通过数值模拟反分析得到相应的特征应力值，同时基于压入特征塑性应变与压头锥角的关系式得到两种不同锥角压头下的特征应变值，在此基础上经求解方程组得到焊料合金的初始屈服应力与应变强化因子，进而得到了焊料合金的幂强化弹塑性本构关系．该方法剔除了压入尺度效应的影响并保证了所得本构关系的唯一性，给出了一种通过原位压入测试表征金属材料弹塑性力学性能的有效方法．     

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.