Effects of strain path shapes on non-proportional cyclic plasticity

The title problem was discussed to facilitate the formulation of constitutive models of cyclic plasticity under general states of loading. A series of plastic strain controlled cyclic tests was performed by applying combined axial force and torque to thin-walled tubular specimens of Type 316 stainless steel at room temperature. These tests consist of cyclic loading along uniaxial, torsional, cruciform, stellate in eight directions, square and circular plastic strain paths with a constant amplitude of equivalent plastic strain.The results of these tests showed that the strain-hardening depends markedly on the shape of the plastic strain path, and that the strain-hardening (measured by equivalent stress amplitudes) in the saturated state is significant in the order of circular, square, stellate, cruciform and proportional paths. It was also observed that these saturated values were independent of the less significant plastic strain cycles experienced in the past. Finally, the characteristic features of strain-hardening mechanisms under non-proportional loadings were discussed in some detail on the basis of the present results.     

An expanding cavity model incorporating strain-hardening and indentation size effects

An expanding cavity model (ECM) for determining indentation hardness of elastic strain-hardening plastic materials is developed. The derivation is based on a strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic power-law hardening material. Closed-form formulas are provided for both conical and spherical indentations. The indentation radius enters these formulas with its own dimensional identity, unlike that in classical plasticity based ECMs where indentation geometrical parameters appear only in non-dimensional forms. As a result, the newly developed ECM can capture the indentation size effect. The formulas explicitly show that indentation hardness depends on Young’s modulus, yield stress, strain-hardening exponent and strain gradient coefficient of the indented material as well as on the geometry of the indenter. The new model reduces to existing classical plasticity based ECMs (including Johnson’s ECM for elastic–perfectly plastic materials) when the strain gradient effect is not considered. The numerical results obtained using the newly developed model reveal that the hardness is indeed indentation size dependent when the indentation radius is very small: the smaller the indentation, the larger the hardness. Also, the indentation hardness is seen to increase with the Young’s modulus and strain-hardening level of the indented material for both conical and spherical indentations. The strain-hardening effect on the hardness is observed to be significant for materials having strong strain-hardening characteristics. In addition, it is found that the indentation hardness increases with decreasing cone angle of the conical indenter or decreasing radius of the spherical indenter. These trends agree with existing experimental observations and model predictions.     

Two new expanding cavity models for indentation deformations of elastic strain-hardening materials

Two expanding cavity models (ECMs) are developed for describing indentation deformations of elastic power-law hardening and elastic linear-hardening materials. The derivations are based on two elastic–plastic solutions for internally pressurized thick-walled spherical shells of strain-hardening materials. Closed-form formulas are provided for both conical and spherical indentations, which explicitly show that for a given indenter geometry indentation hardness depends on Young’s modulus, yield stress and strain-hardening index of the indented material. The two new models reduce to Johnson’s ECM for elastic-perfectly plastic materials when the strain-hardening effect is not considered. The sample numerical results obtained using the two newly developed models reveal that the indentation hardness increases with the Young’s modulus and strain-hardening level of the indented material. For conical indentations the values of the indentation hardness are found to depend on the sharpness of the indenter: the sharper the indenter, the larger the hardness. For spherical indentations it is shown that the hardness is significantly affected by the strain-hardening level when the indented material is stiff (i.e., with a large ratio of Young’s modulus to yield stress) and/or the indentation depth is large. When the indentation depth is small such that little or no plastic deformation is induced by the spherical indenter, the hardness appears to be independent of the strain-hardening level. These predicted trends for spherical indentations are in fairly good agreement with the recent finite element results of Park and Pharr.     

汽车薄钢板应力应变曲线及屈服轨迹的研究

采用十字形双向拉伸的实验方法对两种汽车用薄钢板BH220和SPEN进行了不同 加载路径下的双向拉伸试验，得到了不同应力状态下的应力应变关系曲线，同时，根据单位 体积塑性功相等的原则，确定了两种钢板等效塑性应变从0.2\%$\sim$2\%的实验屈服轨迹. 结果分析表明：不同加载路径下板料的应力应变关系不同，随着加载比例由单拉到等双拉状 态，板料的硬化指数逐步增大；实验屈服轨迹呈外凸性，且以等双拉为界的上下部分屈服轨 迹不对称，随着变形程度的增加，屈服轨迹向外扩大，但单拉时强化程度最小，而等双拉 时最大. 对BH220和SPEN钢板的实验屈服轨迹与几种常用理论屈服轨迹的比较发现，Hosford各向 异性屈服准则的理论轨迹与实验结果最为接近，Hill48准则与实验结果相差最大，此外一 向被视为只适用于各向同性材料的Mises准则与实验结果也较为接近，其他几个屈服准则的 理论屈服轨迹与实验点相差较大.     

The influence of strain-hardening and grain size on early fatigue damage based on a micromechanics theory

Early fatigue damage is assumed to correspond to the build-up of local plastic shear strain. The influence of strain-hardening and grain size on the early stage of fatigue damage of a polycrystal subjected to fluctuating stress is considered. The calculations are based on a micromechanics theory proposed recently by the authors. It is shown that the increase of the strain-hardening rate and/or the decrease of the grain size decreases the rate of early fatigue damage. In order to produce 100 per cent local plastic shear strain for a given number of loading cycles, the range of the alternating stress decreases with increasing amounts of mean stress. This result is shown to be rather insensitive to the rate of strain-hardening and is found to lie between the values predicted by Gerber's parabolic law and by the modified Goodman linear law for fatigue failure.     

Local mechanical properties of the 6061-T6 aluminium weld using micro-traction and instrumented indentation

The local mechanical properties of a weld zone, in a 6061-T6 aluminium alloy subjected to the modified indirect electric arc technique have been studied. The mechanical properties of the base metal, the weld metal and the heat affected zone were determined by means of usual and instrumented indentation testing, as well as micro-traction testing. To analyse the heat input effect resulting from the welding process, the evolution of the weld zone size was evaluated by means of classical indentation under a constant applied load. The results were presented using a Vickers hardness map representation. This allows monitoring exact hardness variation while leading to the identification of the different zones of the welded joint. Instrumented indentation testing was carried out to determine the local mechanical properties, such as the yield stress, the bulk modulus and the strain-hardening exponent. Obtained results are compared to those derived from tensile tests conducted on micro-specimen cuts taken from the weld zone. It was observed that yield stress values are directly comparable for indentation and micro-traction experiments. As for the elastic properties, no comparison was possible since the bulk modulus is measured by indentation, whereas it is the Young’s modulus by tensile test. The micro-traction testing seems to be more sensitive to represent the work hardening of a material since the corresponding exponent is found to be constant by instrumented indentation.     

花岗岩单轴压缩侧向变形及脆性破坏机制实验研究

为研究花岗岩侧向变形及脆性破坏机制,对花岗岩试件进行单轴压缩实验。利用动态应变采集系统、数字散斑相关方法(DSCM)和显微观测手段,记录并分析花岗岩试件在单轴压缩过程中的宏观侧向应变、局部侧向应变以及破裂面形貌,并与水泥砂浆试件的破坏过程对比,讨论了花岗岩脆性破坏机制。实验与分析结果表明:(1)花岗岩试件在加载初期发生侧向收缩变形,产生并发展于压密阶段,消失于线弹性阶段初期,这主要由于试件内部裂纹闭合造成的;此后,宏观侧向应变持续增长,当侧向应变与轴向应变之比接近0.5时试件破坏;(2)在峰值载荷前很长一段时间内,局部侧向应变在一定范围内波动,临近试件破坏时局部侧向应变最大值和最小值均出现较大幅度的波动,二者差值迅速增大,试件不均匀程度增大,最终导致试件破坏;(3)在峰值载荷前有无塑性屈服阶段是峰值载荷后脆性破坏程度的重要影响因素,而宏观裂纹的贯通程度是峰值载荷后应力降大小的决定因素。     

Strain-rate history and temperature effects on the torsional-shear behavior of a mild steel

Previous investigations on the effects of strain-rate and temperature histories on the mechanical behavior of steel are briefly reviewed. A study is presented on the influence of strain rate and strain-rate history on the shear behavior of a mild steel, over a wide range of temperature Experiments were performed on thin-walled tubular specimens of short gage length, using a torsional split-Hopkinson-bar apparatus adapted to permit quasi-static as well as dynamic straining at different temperatures. The constant-rate behavior was first measured at nominal strain rates of 10^?3 and 10³ s^?1 for ?150, ?100, ?50, 20, 200 and 400°C. Tests were then carried out, at the same temperatures, in which the strain rate was suddenly increased during deformation from the lower to the higher rate at various large values of plastic strain. The increase in rate occurred in a time of the order of 20 μs so that relatively little change of strain took place during the jump. The low strain-rate results show a well-defined elastic limit but no yield drop, a small yield plateau is found at room temperature. The subsequent strain hardening shows a maximum at 200°C, when serrated flow occurs and the ductility is reduced. The high strain-rate results show a considerable drop of stress at yield. The post-yield flow stress decreases steadily with increasing temperature, throughout the temperature range investigated. At room temperature and below, the strain-hardening rate becomes negative at large strains. The adiabatic temperature rise in the dynamic tests was computed on the assumption that the plastic work is entirely converted to heat. This enabled the isothermal dynamic stress-strain curves to be calculated, and showed that considerable thermal softening took place. The initial response to a strain-rate jump is approximately elastic, and has a magnitude which increases with decrease of testing temperature; it is little affected by the amount of prestrain. At 200 and 400° C, a yield drop occurs after the initial stress increment. The post-jump flow stress is always greater than that for the same strain in a constant-rate dynamic test, the strain-hardening rate becoming negative at large strains or low testing temperature. This observed effect of strain-rate history cannot be explained by the thermal softening accompanying dynamic deformation. These and other results concerning total ductility under various strain-rate and temperature conditions show that strain-rate history strongly affects the mechanical behavior of the mild steel tested and, hence, should be taken into account in the formulation of constitutive equations for that material.     

Anisotropic Mechanical Properties of SAC Solder Joints in Microelectronic Packaging and Prediction of Uniaxial Creep Using Nanoindentation Creep

In this paper, the mechanical properties and creep behavior of lead-free solder joints has been characterized by nano-mechanical testing of single grain SAC305 solder joints extracted from plastic ball grid array (PBGA) assemblies. The anisotropic mechanical properties characterized include the elastic modulus, hardness, and yield stress. An approach is suggested to predict tensile creep strain rates for low stress levels using nanoindentation creep data measured at very high compressive stress levels. The uniaxial creep rate measured on similarly prepared bulk (large) specimens was found to be of the same order-of-magnitude as the creep rate observed in single-grain BGA joints, with chararacteristically (slightly) higher creep strains measured during nanoindentation. This suggests that the same creep mechanism operates in both size domains. Electron backscattered diffraction (EBSD) and nanoindentation testing further showed that the modulus, hardness, and creep properties of solder joints are highly dependent on the crystal orientation.     

Use of the hole-drilling method for measuring residual stresses in highly stressed shot-peened surfaces

The same shot-peening treatment was applied to five steels with different mechanical properties. The induced residual stress profiles were analyzed using X-ray diffraction and incremental hole drilling (IHD). The results of both techniques showed that IHD can still be successfully used for measuring shot-peening residual stresses, even if these exceed the yield strength of the bulk material. Expected errors due to the plasticity effect are reduced by the strain hardening of the surface. For an assessment of the reliability of IHD data, strain-hardening variation was quantified by microhardness measurements to estimate the yield strength of the plastified layer. All the main calculation methods for IHD were applied. The results were compared and discussed with respect to the characteristics of each method.     

Modelling and experimental characterisation of the Lüders strain in complex loaded ferritic steel compact tension specimens

The use of 3D digital image correlation (DIC) has been used to capture the Lüders strains in a low carbon ferritic steel. Results were used to calibrate and compare with finite element (FE) results based on a constitutive plasticity model, capable of yield drop behaviour and therefore Lüders strains, by Zhang et al. (2001). Tensile tests were carried out at several strain rates to characterise the material behaviour. The results of these tests were used to fit parameters in the constitutive plasticity model. The FE model was then tested on a complex loading situation of in-plane compression of a compact tension (CT) specimen. The FE model predicts the shape and formation of the Lüders bands well. This FE model, using Zhang’s constitutive plasticity model, was used to predict the residual stress profile to compare with standard elastic–plastic isotropic hardening models with no yield point. The yield point reduced both the predicted peak tensile stress, at the notch root, and the amount of plastic strain. In regions where the plastic strain was of a similar size to the Lüders strain the stress profiles were perturbed from flat profiles predicted by the standard elastic–plastic hardening models.     

Quasi-static normal indentation of an elasto-plastic substrate by a periodic array of elastic strip punches

The plane-strain, elasto-plastic, contact problem described in the title is treated for a substrate that is perfectly plastic post yield, and so simulates compression molding of some metals at elevated temperatures. The analysis uses finite elements and is verified with test problems and convergence checks. The key finding is that, in what might reasonably be viewed as a fully-plastic state, the molding pressure normalized by the yield stress is equal to a constant plus a term that increases linearly with the depth of indentation. This is in contrast to Tabor’s classical result for hardness testing that has the normalized pressure solely equal to a constant when response is fully plastic. The additional linear stiffening term found with the finite element analysis of the present configuration is confirmed experimentally. An explanation of the source of this stiffening term even with a perfectly-plastic substrate is offered. Contact stresses are also tracked as indentation proceeds. These stresses initially have high stress concentrations near the edges of the strip punches. However, these peak stresses abate rapidly with plastic flow and approach a nearly uniform distribution within the fully-plastic state. Implications for compression molding are discussed.     

Cyclic stress-strain behaviour of alloy AA6060 T4, part II: Biaxial experiments and modelling

Laboratory tests have been conducted to investigate the inelastic behaviour of aluminium alloy AA6060 T4 subjected to non-proportional cyclic loading. The results of four tests with variable strain path shapes and strain amplitudes are reported in this paper. The tests were carried out by applying combined axial force and torque to thin-walled tubular specimens, using effective strain amplitudes in the range 0.4–0.8%. Major emphasis has been put on the two important material properties: plastic anisotropy and influence of strain range and strain path shapes on cyclic hardening. A constitutive model for cyclic plasticity is used to predict the stress response of the alloy for the non-proportional strain paths applied in the experiments. The model adopts a quadratic yield function and multi-component non-linear isotropic and kinematic hardening rules to describe plastic anisotropy, the shape of the hysteresis loops and the evolution of cyclic hardening. Good agreement is obtained between the physical and correlated stress response of the alloy.     

Hybrid experimental–numerical analysis of basic ductile fracture experiments for sheet metals

A basic ductile fracture testing program is carried out on specimens extracted from TRIP780 steel sheets including tensile specimens with a central hole and circular notches. In addition, equi-biaxial punch tests are performed. The surface strain fields are measured using two- and three-dimensional digital image correlation. Due to the localization of plastic deformation during the testing of the tensile specimens, finite element simulations are performed of each test to obtain the stress and strain histories at the material point where fracture initiates. Error estimates are made based on the differences between the predicted and measured local strains. The results from the testing of tensile specimens with a central hole as well as from punch tests show that equivalent strains of more than 0.8 can be achieved at approximately constant stress triaxialities to fracture of about 0.3 and 0.66, respectively. The error analysis demonstrates that both the equivalent plastic strain and the stress triaxiality are very sensitive to uncertainties in the experimental measurements and the numerical model assumptions. The results from computations with very fine solid element meshes agree well with the experiments when the strain hardening is identified from experiments up to very large strains.     

Analysis of elastic-plastic deformation in TiAl polycrystals

A computational model is described for analyzing stress variations within polycrystals of γ-TiAl, including the effect of anisotropic yielding and small-scale plastic flow. Interlamellar (soft mode) slip behavior is controlled by a separate collection of slip systems whose properties are derived from measurements on polysynthetically twinned (PST) specimens. When used to represent several hundred randomly oriented material grains, the model provides distributions and statistical data about the local stress, strain, and plastic deformation resulting from a prescribed macroscopic loading.     

Incremental collapse of thick-walled circular cylinders under steady axial tension and torsion loads and cyclic transient heating

The accumulation of plastic strain over a number of cycles of heating and cooling of the outer surface of a loaded thick-walled circular cylinder is computed. Two types of loadings are considered, viz. pure axial torsion and a combination of torsion and tension. The cylinder material is taken to be isotropic and linear strain-hardening; its yield stress reduces with temperature and with the number of thermal cycles completed. This cyclic softening is shown to produce an incremental collapse or ratcheting behaviour in contrast to the alternating plasticity which occurs when the yield stress on first loading is maintained.     

A study on robust indentation techniques to evaluate elastic–plastic properties of metals

Spherical indentation is studied based on numerical analysis and experiment, to develop robust testing techniques to evaluate isotropic elastic–plastic material properties of metals. The representative stress and plastic strain concept is critically investigated via finite element analysis, and some conditions for the representative values are suggested. The representative values should also be a function of material properties, not only indenter angle for sharp indenter and indentation depth for spherical indenter. The pros and cons of shallow and deep spherical indentation techniques are also discussed. For an indentation depth of 20% of an indenter diameter, the relationships between normalized indentation parameters and load–depth data are characterized, and then numerical algorithm to estimate material elastic–plastic curve is presented. From the indentation load–depth curve, the new approach provides stress–strain curve and the values of elastic modulus, yield strength, and strain-hardening exponent with an average error of less than 5%. The method is confirmed to be valid for various elastic properties of indenter. Experimental validation of the approach then is performed by using developed micro-indentation system. For the material severely disobeying power law hardening, a modified method to reduce errors of predicted material properties is contrived. It is found that our method is robust enough to get ideal power law properties, and applicable to input of more complex physics.     

Shear banding in strain hardening polycrystals during rolling

Initiation and development of shear band (SB) in f.c.c. strain hardening polycrystals during rolling are modelled in terms of crystallographic texture. The constitutive law of the material is expressed in terms of the texture-dependent normalized yield surface and the critical shear stress which evolves with strain. The normalized yield surface is predicted by the Taylor model as a function of rolling texture. It is shown that a rounded vertex (RV) develops at the loading point as the rolling texture becomes more and more marked. A detailed characterization of the RV is carried out. It is found that the normalized curvature radius of the RV decreases from unity towards zero at very large strain. This allows for a small stress perturbation to induce a shear strain perturbation with a large orientation deviation of deformation. By linearized stability analysis, the condition for initiation of SB from the shear strain perturbation is obtained. Development of SB is analysed by solving the established governing equations of shear banding. It is shown that the conditions for SB initiation and saturation of shear localisation depend strongly on the texture. Based on this model problem, a long discussion is carried out which allows a better understanding of the basic physical origin and saturation of SB in strain-hardening polycrystals, as well as the effects of yield surface curvature and yield surface rotation whose general form is derived.     

纯钼烧结体压缩变形本构模型建立

基于可压缩连续材料的屈服准则和塑性本构关系,推导了粉末烧结材料的压缩变形屈服条件,采用真空烧结技术制备了不同密度的钼,通过在MTS810实验机上进行单向压缩试验,研究了在不同的初始密度,变形温度和应变速率的条件下,纯钼烧结休压缩变形流变应力的变化规律.建立了其流变应力的经验回归模型,进而确定了纯钼烧结材料压缩变形的屈服条件.在此基础上,建立了钼粉烧结材料压缩变形的弹塑性本构模型.这些研究对于提高钼粉烧结材料成形工艺模拟的精度和制定合理的塑件成形工艺具有重要意义.