A shear-compression specimen for large strain testing

A new specimen geometry, the shear-compression specimen (SCS), has been developed for large strain testing of materials. The specimen consists of a cylinder in which two diametrically opposed slots are machined at 45° with respect to the longitudinal axis, thus forming the test gage section. The specimen was analyzed numerically for two representative material models, and various gage geometries. This study shows that the stress (strain) state in the gage, is three-dimensional rather than simple shear as would be commonly assumed. Yet, the dominant deformation mode in the gage section is shear, and the stresses and strains are rather uniform. Simple relations were developed and assessed to relate the equivalent true stress and equivalent true plastic strain to the applied loads and displacements. The specimen was further validated through experiments carried out on OFHC copper, by comparing results obtained with the SCS to those obtained with compression cylinders. The SCS allows to investigate a large range of strain rates, from the quasi-static regime, through intermediate strain rates (1–100 s⁻¹), up to very high strain rates (2×10⁴s⁻¹ in the present case).     

Numerical validation of the shear compression specimen. Part I: Quasi-static large strain testing

The shear compression specimen (SCS), which is used for large strain testing, is thoroughly investigated numerically using three-dimensional elastoplastic finite element simulations. In this first part of the study we address quasi-static loading. A bi-linear material model is assumed. We investigate the effect of geometrical parameters, such as gage height and root radius, on the stress and strain distribution and concentration. The analyses show that the stresses and strains are reasonably uniform on a typical gage mid-section, and their average values reflect accurately the prescribed material model. We derive accurate correlations between the averaged von Mises stress and strain and the applied experimental load and displacement. These relations depend on the specimen geometry and the material properties. Numerical results are compared to experimental data, and an excellent agreement is observed. This study confirms the potential of the SCS for large strain testing of material.     

不同加载状态下TA2钛合金绝热剪切破坏响应特性

一般认为绝热剪切现象在宏观上表现为材料动态本构失稳,即热软化大于应变硬化.本文采用帽型受迫剪切试样研究TA2钛合金的动态力学特性和本构失稳过程.首先对剪切区加载应力状态进行理论和数值分析,通过合理设计帽型试样,剪切区变形可近似按剪切状态处理;结合二维数字图像相关法(two-dimensional digital image correlation,DIC-2D)直接测试试样剪切区应变演化,给出帽型受迫剪切实验的等效应力-应变响应曲线.进一步,利用Hopkinson压杆对TA2钛合金开展动态压缩及帽型剪切对比试验研究,比较压缩、剪切试验得到的等效应力-应变曲线,采用"冻结"试样方法分析试样中绝热剪切局域化演化过程,探讨不同加载状态下TA2钛合金的绝热剪切破坏现象及其动态力学响应特性.实验结果表明,在塑性变形初始阶段,动态压缩及剪切加载下的等效应力-应变曲线符合较好,但随塑性损伤发展及绝热剪切带形成,两者出现分离,表明损伤及绝热剪切演化过程与应力状态相关.剪切试样实验得到的本构"软化"特性能够反映绝热剪切带起始、破坏演化过程的力学响应特性,而在动态压缩实验中,即使试样中已出现双锥形的绝热剪切带及局部裂纹分布,其表观等效应力-应变曲线并不出现软化特征,动态压缩实验无法得到关于绝热剪切起始、发展以及破坏的本构软化响应特性.     

Analysis of Slot Orientation in Shear-Compression Specimen (SCS)

This paper presents an analytical treatment as well as experimental measurement of the plastic deformation field in shear-compression specimen (SCS) by using digital image processing (DIC) technique. The results provide a set of useful expressions that relates externally applied displacement and load quantities to the equivalent stress and equivalent plastic strain within the gage section. Based on the analysis, we propose modifying the slot angle of SCS geometry from its original value of 45º to 35.26º in order to enhance the uniformity of stress and strain fields in gage section. It is shown by analysis that this enhancement is essentially because the compatibility and boundary conditions that yield a homogeneous deformation field is naturally satisfied for the particular slot orientation of ???=?35.26°. This conclusion is also supported by experimental evidence that comparatively shows the edge effects for varying slot angles.     

Numerical validation of the shear compression specimen. Part II: Dynamic large strain testing

Part I of this work addressed quasi-static loading of the shear compression specimen (SCS), which has been especially developed to investigate the shear dominant response of materials at various strain rates. The stress and strain states were characterized numerically. Approximations were presented to reduce the measured load,P, and displacement,d, into equivalent stress and strain . This paper addresses dynamic loading of the SCS. Several simulations were made for representative materials, whose stress-strain behavior is assumed to be rate-independent. The results show that stress wave loading induces strong oscillations in theP-d curve. However, the curve remains smooth in the gage section. The oscillations are about the quasistatic load values, so that with suitable filtering of the dynamicP-d curves, the quasi-static ones are readily recovered. Consequently, the approach that was developed for quasi-static loading of the SCS is now extended to dynamic loading situations. The average strain rate is rather constant and scales linearly with the prescribed velocity. As the plastic modulus becomes smaller, the strain rate reaches higher values. Friction at the end pieces of the specimen is also investigated, and shown to have a small overall influence on the determined mechanical characteristics. This paper thereby confirms the potential of the SCS for large strain testing of materials, using a unified approach, over a large range of strain rates in a seamless fashion.     

Sensitivity of the shear gage to normal strains

This paper documents an experimental study that was conducted to demonstrate the sensitivity of the shear gage to the presence of normal strains. The shear gage is a specially designed strain gage rosette that measures the average shear strain in the test section of notched specimens such as the losipescu, Arcan and compact shear specimens. These specimens can have complicated stress states with high shear and normal strain gradients. To evaluate the sensitivity of the shear gage to normal strains, shear gages were tested on an Arcan specimen. The Arcan specimen is a notched specimen that can be loaded in pure shear (90 deg), pure tension (0 deg) and at intermediate 15- deg increments. The shear modulus for an aluminum specimen was determined at each of these loading angles. It was found that the gages display nearly zero sensitivity to normal strains ( _x, _y). Moiré interferometry was used to document the shear and normal strain distributions in the test section and to provide an independent method for determining the average shear strain. These results reinforce the robust nature of testing with the shear gage.     

The shear gage: For reliable shear modulus measurements of composite materials

A special strain gage called the shear gage was developed for composite materials testing with notched shear specimens. The shear-gage records the average shear strain across the entire test section between the notches of the losipescu and compact shear specimens rather than just sampling the shear strain over a small region in the center of the test section. Hence, the shear stress/strain response is obtained by dividing the average shear stress (load divided by the cross-sectional area between the notches) by the average shear strain. By placing gages on both faces of the specimen, accurate and repeatable shear-modulus measurements can be made without prior knowledge of the shear strain or stress distributions. This scheme essentially integrates the shear strain through the entire test section. Knowledge of other material properties is not required to accurately determine shear modulus values. The shear gage was tested on a variety of composite and isotropic materials resulting in more reliable shear modulus determination and less scatter than previously possible.     

The interferometric strain gage

The interferometric strain gage consists of two very shallow grooves ruled on a highly polished surface. The grooves are cut with a diamond and are 4×10^?5 in. deep and 5×10^?3 in. apart. Coherent, monochromatic light from a He?Ne gas laser incident upon these grooves will produce fringe patterns. A fringe pattern with the fringes parallel to the grooves is formed on each side of the impinging beam. The position of these patterns in space is related to the distance between the two grooves. As this distance changes, the fringes shift. Measurement of these fringe shifts enables one to determine the local strain of the specimen. In this paper, the theory of the measurement is developed first. The strain, ∈, is given by ∈=ΔFλ/d _o sin α _o where ΔF is the average fringe shift of the two patterns, λ is the wavelength of light,d _o is the initial distance between grooves, and α _o is the angle between the incident light beam and the fringe patterns. A procedure for making static measurements with the interferometric strain gage is presented. The sensitivity for these measurements is 0.5 percent strain per fringe shift, and the maximum strain is 4 percent. The method is evaluated by comparing its results with other accepted means of measuring large plastic strain. These other techniques are: post-yield foil gages, a 2-in. clip gage, and an Instron testing machine. The average percent difference among these techniques is less than 0.4 percent based on a full-scale measurement of 4-percent strain. The interferometric strain gage has the following features: a gage integral with the specimen surface, a very short gage length, relatively easy application, and the ability to measure large strains.     

Characterization of Anisotropic Polymeric Foam Under Static and Dynamic Loading

An orthotropic polymeric foam with transverse isotropy (Divinycell H250) used in composite sandwich structures was characterized at various strain rates. Uniaxial experiments were conducted along principal material axes as well as along off-axis directions under tension, compression, and shear to determine engineering constants, such as Young??s and shear moduli. Uniaxial strain experiments were conducted to determine mathematical stiffness constants, i. e., C _ij . An optimum specimen aspect ratio for these tests was selected by means of finite element analysis. Quasi-static and intermediate strain rate tests were conducted in a servo-hydraulic testing machine. High strain rate tests were conducted using a split Hopkinson Pressure Bar system built for the purpose using polymeric (polycarbonate) bars. The polycarbonate material has an impedance that is closer to that of foam than metals and results in lower noise to signal ratios and longer loading pulses. It was determined by analysis and verified experimentally that the loading pulses applied, propagated along the polycarbonate rods at nearly constant phase velocity with very low attenuation and dispersion. Material properties of the foam were obtained at three strain rates, quasi-static (10^?4 s^?1), intermediate (1 s^?1), and high (10³ s^?1) strain rates. A simple model proposed for the Young??s modulus of the foam was in very good agreement with the present and published experimental results.     

Strain gradient plasticity,strengthening effects and plastic limit analysis

Within the framework of isotropic strain gradient plasticity, a rate-independent constitutive model exhibiting size dependent hardening is formulated and discussed with particular concern to its strengthening behavior. The latter is modelled as a (fictitious) isotropic hardening featured by a potential which is a positively degree-one homogeneous function of the effective plastic strain and its gradient. This potential leads to a strengthening law in which the strengthening stress, i.e. the increase of the plastically undeformed material initial yield stress, is related to the effective plastic strain through a second order PDE and related higher order boundary conditions. The plasticity flow laws, with the role there played by the strengthening stress, are addressed and shown to admit a maximum dissipation principle. For an idealized elastic perfectly plastic material with strengthening effects, the plastic collapse load problem of a micro/nano scale structure is addressed and its basic features under the light of classical plastic limit analysis are pointed out. It is found that the conceptual framework of classical limit analysis, including the notion of rigid-plastic behavior, remains valid. The lower bound and upper bound theorems of classical limit analysis are extended to strengthening materials. A static-type maximum principle and a kinematic-type minimum principle, consequences of the lower and upper bound theorems, respectively, are each independently shown to solve the collapse load problem. These principles coincide with their respective classical counterparts in the case of simple material. Comparisons with existing theories are provided. An application of this nonclassical plastic limit analysis to a simple shear model is also presented, in which the plastic collapse load is shown to increase with the decreasing sample size (Hall–Petch size effects).     

准静态压缩应力-应变曲线测量方法的探索

本文介绍了用应变片直接测量材料的准静态应力-应变曲线的试验研究。在MTS810材料试验机上分别对93W、G50、砂浆等几种材料进行了准静态压缩试验。由于仪器的系统误差不能由MTSCod规准确的得到材料的真实的弹性变形。为此在试件的中部贴应变片得到材料的弹性变形,塑性变形仍旧由MTSCod规记录,从而得到试样的真实的应变,准确获得准静态压缩应力-应变曲线。试验结果表明:在试件中部贴应变片的方法能够准确得到该材料的杨氏模量;在试件两端垫块的刀口上安置Cod规可得到材料的应力应变曲线。两者的合理组合即可得到准确而完整的准静态压缩应力-应变曲线。试验中还发现准静态实验中试件的断面加工不平,偏心压缩等都会影响E的准确测量。     

A New Shear-Compression-Specimen for Determining Quasistatic and Dynamic Polymer Properties

A new design of the shear compression specimen (SCS) for investigating the viscoelastic shear response of polymers is presented. The specimen consists of a polymer gage section with two metal ends that remain essentially rigid during deformation. Two closed-form analytic models are developed to predict the average stress and strain in the gage section from the deformation-load histories. This new SCS design and its analytic models are thoroughly evaluated via laboratory measurements and numerical simulations. These simulations show that the deformations in the gage section are more uniform than in the original design, and the distribution of the average shear stress and strain are highly homogenous. The simulation results yield good agreement with those of closed-form analytic results and the experiments demonstrate that the new SCS geometry and its analytic models are as reliable as other commonly employed specimens. It can also generate higher strain rates under usual loading conditions because of its smaller specimen gage length. The need for care in specimen preparation is also discussed in detail as illuminated by the experimental and simulation results.

A computational study on the instrumented sharp indentations with dual indenters

Finite element analysis was performed to investigate the indentation response of elasto-plastic solids for conical indenters of half included angles of 60° and 70.3°. The interdependence indentation parameters resulting from a single indentation load–depth curve is considered. Regarding dimensional analysis, several dimensionless relationships are constructed as functions of the reduced elastic modulus-loading curvature ratio E¹/C and the strain hardening exponent n. Further, the duality between corresponding parameters with dual indenters is explored. Finally, a new method based on dual indenters is proposed to extract the strain hardening exponent and the reduced elastic modulus of an indented material. The accuracy of this method is verified and discussed with experimental data from the literature and representative materials.     

On the plastic bifurcation and post-bifurcation of axially compressed beams

The paper presents two new results in the domain of the elastoplastic buckling and post-buckling of beams under axial compression. (i) First, the tangent modulus critical load, the buckling mode and the initial slope of the bifurcated branch are given for a Timoshenko beam (with the transverse shear effects). The result is derived from the 3D J₂ flow plastic bifurcation theory with the von Mises yield criterion and a linear isotropic hardening. (ii) Second, use is made of a specific method in order to provide the asymptotic expansion of the post-critical branch for a Euler-Bernoulli beam, exhibiting one new non-linear fractional term. All the analytical results are validated by finite element computations.     

Load-Inversion Device for the High Strain Rate Tensile Testing of Sheet Materials with Hopkinson Pressure Bars

A high strain rate tensile testing technique for sheet materials is presented which makes use of a split Hopkinson pressure bar system in conjunction with a load inversion device. With compressive loads applied to its boundaries, the load inversion device introduces tension into a sheet specimen. Two output bars are used to minimize the effect of bending waves on the output force measurement. A Digital Image Correlation (DIC) algorithm is used to determine the strain history in the specimen gage section based on high speed video imaging. Detailed finite element analysis of the experimental set-up is performed to validate the design of the load inversion device. It is shown that under the assumption of perfect alignment and slip-free attachment of the specimen, the measured stress–strain curve is free from spurious oscillations at a strain rate of 1,000 s^?1. Validation experiments are carried out using tensile specimens extracted from 1.4 thick TRIP780 steel sheets. The experimental results for uniaxial tension at strain rates ranging from 200 s^?1 to 1,000 s^?1 confirm the oscillation-free numerical results in an approximate manner. Dynamic tension experiments are also performed on notched specimens to illustrate the validity of the proposed experimental technique for characterizing the effect of strain rate on the onset of ductile fracture in sheet materials.     

Analyses of steady quasistatic crack growth in plane strain tension in elastic-plastic materials with non-isotropic hardening

Steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method. The elastic-perfectly plastic solution shows that elastic unloading wedges subtended by the crack tip in the plastic wake region do exist and that the stress state around the crack tip is similar to the modified Prandtl fan solution. To demonstrate the effects of a vertex on the yield surface, the small strain version of a phenomenological J₂, corner theory of plasticity (Christoffersen, J. and Hutchinson, J. W. J. Mech. Phys. Solids,27, 465 C 1979) with a power law stress strain relation was used to govern the strain hardening of the material. The results are compared with the conventional J₂ incremental plasticity solution. To take account of Bauschinger like effects caused by the stress history near the crack tip, a simple kinematic hardening rule with a bilinear stress strain relation was also studied. The results are again compared with the smooth yield surface isotropic hardening solution for the same stress strain curve. There appears to be more potential for steady state crack growth in the conventional J₂ incremental plasticity material than in the other two plasticity laws considered here if a crack opening displacement fracture criterion is used. However, a fracture criterion dependent on both stress and strain could lead to a contrary prediction.     

Experimental strain analysis of the losipescu shear test specimen

An experimental strain analysis of the losipescu shear test specimen was performed, utilizing a 20-ply AS4/3501-6 carbon/epoxy unidirectional composite. Using three-element strain-gage rosettes, it was shown that the presence of loading-point-induced transverse normal strains in the gage section do not affect the measured shear strain. Thus, the shear modulus determined using the standard notch specimen is not affected. Likewise, modulus determination is not influenced by cracking at the notch tips, since this occurs at strains beyond the range over which modulus is determined. To further evaluate the effect of notch-tip cracking, material was removed adjacent to the standard V-notches where these cracks initiate. The measured shear strength was unaffected by removing this material, although the shear modulus was reduced slightly (by as much as eight percent for the more grossly exaggerated geometries). E.Q. Lewis, former graduate student, is now Engineer, Lockheed Corporation.     

Energy balance and identification of hardening moduli in plastic deformation processes

The hardening moduli H_r and H_d of plastic deformation associated with the free energy and dissipation function in a representative material element are defined analytically and specified experimentally for three materials. Besides the stress–strain curve and work expended during the deformation process, variation of the hardening moduli with plastic deformation is also determined for austenitic steel, austenitic-ferritic steel and Fe–Si alloy.     

In situ determination of adhesive shear moduli using strain gages

A new, convenient and cost-effective method of determining in situ adhesive shear moduli using strain gages is proposed and evaluated. Thick-adherend lap shear specimens with stacked gage rosettes at the center of the bond line are loaded in tension for adhesive shear strain measurement. Experimental and numerical results indicate that the test specimen has a nonuniform adhesive shear stress (or strain) distribution in the test section and that this distribution (except at the center point of the bond line) is greatly affected by load eccentricity. In addition to the nonuniformity in the shear stress distribution, the issue of material nonhomogeneity in the gage-covered region affects the strain gage measurement. By taking into account these two issues and assuming linear-elastic behavior, two approaches for converting the gage-measured shear strain into the adhesive shear strain are developed and verified by experiment. It is shown that the strain gage measurement associated with either of two conversion techniques can determine the in situ adhesive shear moduli, which are comparable with moiré experiment and KRG-1 extensometer measurements.     

Constitutive equations for metal powders: application to powder forming processes

A new constitutive model for cold compaction of metal powders is developed. The plastic flow of metal powders at the macroscopic level is assumed to be representable as a combination of a distortion mechanism, and a consolidation mechanism. For the distortion mechanism the model employs a pressure-sensitive, Mohr–Coulomb type yield condition, and a new physically based non-associated flow rule. For the consolidation mechanism the model employs a smooth yield function which has a quarter-elliptical shape in the mean-normal pressure and the equivalent shear stress plane, together with an associated flow rule. The constitutive model has been implemented in a finite element program. The material parameters in the constitutive model have been calibrated for MH-100 iron powder by fitting the model to reproduce data from true triaxial compression experiments, torsion ring-shear experiments, and simple compression experiments. The predictive capability of the constitutive model and computational procedure is checked by simulating two simple powder forming processes: (i) a uniaxial strain compression of a cylindrical sample, and (ii) forming of a conical shaped-charge liner. In both cases the predicted load–displacement curves and density variations in the compacted specimens are shown to compare well with corresponding experimental measurements.