On Compressible Materials Capable of Sustaining Axisymmetric Shear Deformations. Part 4: Helical Shear of Anisotropic Hyperelastic Materials

Conditions on the form of the strain energy function in order that homogeneous, compressible and isotropic hyperelastic materials may sustain controllable static, axisymmetric anti-plane shear, azimuthal shear, and helical shear deformations of a hollow, circular cylinder have been explored in several recent papers. Here we study conditions on the strain energy function for homogeneous and compressible, anisotropic hyperelastic materials necessary and sufficient to sustain controllable, axisymmetric helical shear deformations of the tube. Similar results for separate axisymmetric anti-plane shear deformations and rotational shear deformations are then obtained from the principal theorem for helical shear deformations. The three theorems are illustrated for general compressible transversely isotropic materials for which the isotropy axis coincides with the cylinder axis. Previously known necessary and sufficient conditions on the strain energy for compressible and isotropic hyperelastic materials in order that the three classes of axisymmetric shear deformations may be possible follow by specialization of the anisotropic case. It is shown that the required monotonicity condition for the isotropic case is much simpler and less restrictive. Restrictions necessary and sufficient for anti-plane and rotational shear deformations to be possible in compressible hyperelastic materials having a helical axis of transverse isotropy that winds at a constant angle around the tube axis are derived. Results for the previous case and for a circular axis of transverse isotropy are included as degenerate helices. All of the conditions derived here have essentially algebraic structure and are easy to apply. The general rules are applied in several examples for specific strain energy functions of compressible and homogeneous transversely isotropic materials having straight, circular, and helical axes of material symmetry.     

Finite strain––beam theory

An appropriate strain energy density for an isotropic hyperelastic Hookean material is proposed for finite strain from which a constitutive relationship is derived and applied to problems involving beam theory approximations. The physical Lagrangian stress normal to the surfaces of a element in the deformed state is a function of the normal component of stretch while the shear is a function of the shear component of stretch. This paper attempts to make a contribution to the controversy about who is correct, Engesser or Haringx with regard to the buckling formula for a linear elastic straight prismatic column with Timoshenko beam-type shear deformations. The derived buckling formula for a straight prismatic column including shear and axial deformations agrees with Haringx’s formula. Elastica-type equations are also derived for a three-dimensional Timoshenko beam with warping excluded. When the formulation is applied to the problem of pure torsion of a cylinder no second-order axial shortening associated with the Wagner effect is predicted which differs from conventional beam theory. When warping is included, axial shortening is predicted but the formula differs from conventional beam theory.     

Effect of viscoplastic relations on the instability strain,shear band initiation strain,the strain corresponding to the minimum shear band spacing,and the band width in a thermoviscoplastic material

We study thermomechanical deformations of a steel block deformed in simple shear and model the thermoviscoplastic response of the material by four different relations. We use the perturbation method to analyze the stability of a homogeneous solution of the governing equations. The smallest value of the average strain for which the perturbed homogeneous solution becomes unstable is called the critical strain or the instability strain. For each one of the four viscoplastic relations, we investigate the dependence upon the nominal strain-rate of the critical strain, the shear band initiation strain, the shear band spacing and the band width. It is found that the qualitative responses predicted by the Wright–Batra, Johnson–Cook and the power law relations are similar but these differ from that predicted by the Bodner–Partom relation. The computed band width is found to depend upon the specimen height.     

An experimental and numerical investigation of different shear test configurations for sheet metal characterization

Simple shear tests are widely used for material characterization especially for sheet metals to achieve large deformations without plastic instability. This work describes three different shear tests for sheet metals in order to enhance the knowledge of the material behavior under shear conditions. The test setups are different in terms of the specimen geometry and the fixtures. A shear test setup as proposed by Miyauchi, according to the ASTM standard sample, as well as an in-plane torsion test are compared in this study. A detailed analysis of the experimental strain distribution measured by digital image correlation is discussed for each test. Finite element simulations are carried out to evaluate the effect of specimen geometries on the stress distributions in the shear zones. The experimental macroscopic flow stress vs. strain behavior shows no significant influence of the specimen geometry when similar strain measurements and evaluation schemes are used. Minor differences in terms of the stress distribution in the shear zone can be detected in the numerical results. This work attempts to give a unique overview and a detailed study of the most commonly used shear tests for sheet metal characterization. It also provides information on the applicability of each test for the observation of the material behavior under shear stress with a view to material modeling for finite element simulations.     

In-plane buckling of circular arches and rings with shear deformations

A finite strain formulation is developed for elastic circular arches and rings in which the effects of shear deformations are included. Timoshenko beam hypothesis is adopted for incorporating shear. Finite strains are defined in terms of the normal and shear component of the longitudinal stretch. The constitutive relations for stress and finite strain are based on a hyperelastic constitutive model. Virtual work and equilibrium equations are derived. Closed-form in-plane buckling solutions are developed for circular rings and high arches under hydrostatic pressure. The effects of axial deformation prior to buckling as well as shear deformations are included in the buckling analysis. The formulation developed is compared with solutions in the literature and to the predictions of the finite element package ANSYS. The importance of including the effects of shear deformations for deep arches is investigated.     

Investigation of shear damage considering the evolution of anisotropy

The damage that occurs in shear deformations in view of anisotropy evolution is investigated. It is widely believed in the mechanics research community that damage (or porosity) does not evolve (increase) in shear deformations since the hydrostatic stress in shear is zero. This paper proves that the above statement can be false in large deformations of simple shear. The simulation using the proposed anisotropic ductile fracture model (macro-scale) in this study indicates that hydrostatic stress becomes nonzero and (thus) porosity evolves (increases or decreases) in the simple shear deformation of anisotropic (orthotropic) materials. The simple shear simulation using a crystal plasticity based damage model (meso-scale) shows the same physics as manifested in the above macro-scale model that porosity evolves due to the grain-to-grain interaction, i.e., due to the evolution of anisotropy. Through a series of simple shear simulations, this study investigates the effect of the evolution of anisotropy, i.e., the rotation of the orthotropic axes onto the damage (porosity) evolution. The effect of the evolutions of void orientation and void shape onto the damage (porosity) evolution is investigated as well. It is found out that the interaction among porosity, the matrix anisotropy and void orientation/shape plays a crucial role in the ductile damage of porous materials.     

Instabilities of wrinkled membranes with pressure loadings

Wrinkling can affect the functionality of thin membranes subjected to various loadings or boundary conditions. The concept of relaxed strain energy was studied for isotropic, hyperelastic, axisymmetric membranes pressurized by gas or fluid. Non-intuitive instabilities were observed when axisymmetric wrinkled membranes were perturbed with angle dependent displacement fields. A linearized theory showed that static equilibrium states of pressurized membranes, modelled by a relaxed strain energy formulation, are unstable, when the wrinkled surface is subjected to pressure loadings. The theory is extended to the non-axisymmetric membranes and it is shown that these instabilities are local phenomena. Simulations for the pressurized cylindrical membranes with non-uniform thickness and hemispherical membranes support the claims in both theoretical and numerical contexts including finite element simulations.     

Mechanics of rubber shear springs

FEA calculations have been carried out for a model rubber shear spring, consisting of a block of a highly elastic material, bonded between two rigid parallel plates and sheared by displacing one of the plates parallel to the other in its own plane. The block was prevented from deforming in the perpendicular direction, and thus was deformed in plane strain. Stress distributions along the bond-line and the center-line are reported and compared with those expected from the theory of large elastic deformations. Unexpected tensile stresses were found to develop in the interior of the sheared block. They are attributed to the absence on the end surfaces of the stresses needed to maintain a simple shear, causing a pronounced change in the reference pressure—a consequence that is usually overlooked. Because the internal stresses are governed by the boundary conditions, they were strongly affected by the shape of the end surfaces. In addition, they were reduced markedly by assigning values to Poisson's ratio slightly lower than 0.5, thus allowing some volume expansion of the rubber. Strain energy release rates were also evaluated for growth of a crack along the bond-line, starting at the edges, and compared with those reported previously by Lindley and Teo [Energy for crack growth at the bonds of rubber springs, Plast. Rubber Mat. Appl. 4 (1979) 29-37], Muhr et al. [A fracture mechanics study of natural rubber-to-metal bond failure, J. Adhes. Sci. Technol. 10 (1996) 593-616], Gregory and Muhr [Stiffness and fracture analysis of bonded rubber blocks in simple shear, in: D. Boast, V.A. Coveny (Eds.), Finite Element Analysis of Elastomers, Professional Engineering Publications, Bury St. Edmunds, UK, 1999, pp. 265-274] and Gough and Muhr [Initiation of failure of rubber close to bondlines, in: Proceedings of the International Rubber Conference, Maastricht, Netherlands, June 2005, IOM Communications Ltd., London, 2005, pp. 165-174]. They confirm that a long crack at the compression edge will grow faster than one at the tension edge, but the results for short cracks were inconclusive.     

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

Consider a cylinder (not necessarily of circular cross-section) that is composed of a hyperelastic material and which is stretched parallel to its axis of symmetry. Suppose that the elastic material that constitutes the cylinder is homogeneous, transversely isotropic, and incompressible and that the deformed length of the cylinder is prescribed, the ends of the cylinder are free of shear, and the sides are left completely free. In this paper it is shown that mild additional constitutive hypotheses on the stored-energy function imply that the unique absolute minimizer of the elastic energy for this problem is a homogeneous, isoaxial deformation. This extends recent results that show the same result is valid in 2-dimensions. Prior work on this problem had been restricted to a local analysis: in particular, it was previously known that homogeneous deformations are strict (weak) relative minimizers of the elastic energy as long as the underlying linearized equations are strongly elliptic and provided that the load/displacement curve in this class of deformations does not possess a maximum.     

On Deformations with the Same Principal Stretches

In the context of the finite strain theory, plane isochoric homogeneous deformations are considered. Inspired by two examples (plane elliptical and plane hyperbolic deformations), it is seen that for any such isochoric deformation the corresponding principal stretches are equal to those of simple shear provided there is a certain relation between the amount of shear of the simple shear and the parameters of the general plane deformation. Then, the link is established between any two homogeneous deformations which have identical principal stretches. It involves two rotations, one in the undeformed state and the other in the deformed state. These rotations are determined explicitly for an arbitrary isochoric homogeneous deformation and the simple shear with the same principal stretches.     

Modeling adiabatic shear failure from energy considerations

The numerical simulation of dynamic structural failure by localized shear is quite complex in terms of constitutive models and choice of adequate failure criteria, along with a pronounced mesh-sensitivity. As a result, the existing numerical procedures are usually quite sophisticated, so that their application for design purposes is still limited. This study is based on the implementation of a simple energy-based criterion, which was developed on experimental considerations (Rittel et al., 2006), and uses a minimal number of adjustable parameters. According to this criterion, a material point starts to fail when the total strain energy density reaches a critical value. Thereafter, the strength of the element decreases gradually to zero to mimic the actual structural behavior. The criterion was embedded into commercial finite element software and tested by simulating numerically four typical high-rate experiments. The first is the dynamic torsion test of a tubular specimen. The second concerns the failure mode transition in mode II fracture of an edge crack in plain strain. The last two involve dynamic shear localization under high rate compression of a cylindrical and a shear compression specimen. A very good adequation was found both qualitatively and quantitatively. Qualitatively, in terms of failure path selection, and quantitatively, in terms of local strains, temperatures and critical impact velocity. The proposed approach is enticing from an engineering perspective aimed at predicting the onset and propagation of dynamic shear localization in actual structures.     

试样形状对轴承钢绝热剪切带微观组织的影响

绝热剪切带(ASB)的微观组织受试样几何形状的影响。对圆柱、帽形和剪切压缩型三种不同形状的试样进行分离式霍普金森压杆高速冲击试验,研究试样形状对轴承钢绝热剪切带的形成和微观组织的影响。结果表明,在应变率为1 800～3 100 s^-1的范围内,材料对应变率的敏感性很低。圆柱试样呈现明显的应变硬化,而帽形试样和剪切压缩型试样(SCS)在不同应变率下分别出现应变硬化和无应变硬化的特征,但流变应力并未因应变硬化而提高。试样形状对ASB的微观形貌和组织有很大影响。圆柱试样上产生了窄且细长的ASB,只发生了应变诱发的晶粒细化,属于形变ASB;帽形试样和SCS则形成大片状的ASB,由等轴晶组成,且发生了体心立方体(BCC)马氏体转变为面心立方体(FCC)奥氏体的相变,属于相变ASB。尤其是SCS中ASB的等轴晶,有非常清晰的晶界,是典型的动态再结晶晶粒。温升计算结果显示,圆柱试样ASB的温升远低于奥氏体相变温度,而帽形试样和SCS的温升高于马氏体的熔点,导致局部熔融。     

Effects of texture on shear band formation in plane strain tension/compression and bending

In this study, effects of typical texture components observed in rolled aluminum alloy sheets on shear band formation in plane strain tension/compression and bending are systematically studied. The material response is described by a generalized Taylor-type polycrystal model, in which each grain is characterized in terms of an elastic–viscoplastic continuum slip constitutive relation. First, a simple model analysis in which the shear band is assumed to occur in a weaker thin slice of material is performed. From this simple model analysis, two important quantities regarding shear band formation are obtained: i.e. the critical strain at the onset of shear banding and the corresponding orientation of shear band. Second, the shear band development in plane strain tension/compression is analyzed by the finite element method. Predictability of the finite element analysis is compared to that of the simple model analysis. Third, shear band developments in plane strain pure bending of a sheet specimen with the typical textures are studied. Regions near the surfaces in a bent sheet specimen are approximately subjected to plane strain tension or compression. From this viewpoint, the bendability of a sheet specimen may be evaluated, using the knowledge regarding shear band formation in plane strain tension/compression. To confirm this and to encompass overall deformation of a bent sheet specimen, including shear bands, finite element analyses of plane strain pure bending are carried out, and the predicted shear band formation in bent specimens is compared to that in the tension/compression problem. Finally, the present results are compared to previous related studies, and the efficiency of the present method for materials design in future is discussed.     

Finite telescopic shear of a compressible hyperelastic tube

Finite telescopic shear of a compressible hyperelastic tube is considered. It is shown that solutions with isochoric deformation fields exist for a class of strain energy functions. A numerical method is proposed for the analysis of the problem when a solution with an isochoric deformation field does not exist. Numerical results obtained by using a programmable desk calculator are presented graphically for two strain energy functions.     

921A 钢纯剪切帽状试件绝热剪切变形的数值模拟

结合相关实验,通过一系列基于921A 钢纯剪切帽状试件的SHPB数值模拟,研究试件的绝热剪 切行为,分析试件内绝热剪切带(ASB)的产生、发展以及相应的试件温度场分布。研究发现:ASB是通过剪 切区两端高温高应变的不稳定区域的扩展而形成;ASB的扩展速率与加载速率相关;在本文加载速率范围 内,ASB带宽无明显变化,均为约70m,基本与所设计的试件剪切区宽度一致;且对应所有加载速率,ASB 均为形变带。     

The shear fracture of dual-phase steel

Unexpected fractures at high-curvature die radii in sheet forming operations limit the adoption of advanced high strength steels (AHSS) that otherwise offer remarkable combinations of high strength and tensile ductility. Identified as “shear fractures” or “shear failures,” these often show little sign of through-thickness localization and are not predicted by standard industrial simulations and forming limit diagrams. To understand the origins of shear failure and improve its prediction, a new displacement-controlled draw-bending test was developed, carried out, and simulated using a coupled thermo-mechanical finite element model. The model incorporates 3D solid elements and a novel constitutive law taking into account the effects of strain, strain rate, and temperature on flow stress. The simulation results were compared with companion draw-bend tests for three grades of dual-phase (DP) steel over a range of process conditions. Shear failures were accurately predicted without resorting to damage mechanics, but less satisfactorily for DP 980 steel. Deformation-induced heating has a dominant effect on the occurrence of shear failure in these alloys because of the large energy dissipated and the sensitivity of strain hardening to temperature increases of the order of 75 °C. Isothermal simulations greatly overestimated the formability and the critical bending ratio for shear failures, thus accounting for the dominant effect leading to the inability of current industrial methods to predict forming performance accurately. Use of shell elements (similar to industrial practice) contributes to the prediction error, and fracture (as opposed to strain localization) contributes for higher-strength alloys, particularly for transverse direction tests. The results illustrate the pitfall of using low-rate, isothermal, small-curvature forming limit measurements and simulations to predict the failure of high-rate, quasi-adiabatic, large-curvature industrial forming operations of AHSS.     

Flow localization in the plane strain tensile test

The deformations in a plane strain tensile test are analyzed numerically, both for a solid characterized by a phenomenological corner theory of plasticity and for a nonlinear elastic solid. As opposed to the simplest flow theory of plasticity with a smooth yield surface, both these material models exhibit shear band instabilities at a realistic level of strain. Initial imperfections are specified in the form of thickness inhomogeneities. A long-wavelength imperfection grows into the well-known necking mode and subsequently, at a sufficiently high local strain level, bands of intense shear deformations develop in the necking region. The location of these shear bands is strongly influenced by the location of small strain concentrations near the surface, induced by various short-wave patterns of initial thickness imperfections. In accord with the non-uniform straining in the neck it is found that the intensity of the localized deformations varies along the bands, and some of the shear bands end inside the material.     

921A钢纯剪切帽状试件在SHPB实验中的动态变形

应用ANSYS/LS-DYNA软件,开展了一系列基于921A钢纯剪切帽状试件的SHPB数值模拟.结合SHPB系统应力波理论,研究不同加载速率v0(或应力脉冲I(t))下,特别是高应变率(约106 s-1)下的压杆轴向应变波形以及相应的试件动态变形特性,并对高速撞击下压杆中应变波形的适用性作了相关讨论.     

Effect of material parameters on shear band spacing in work-hardening gradient dependent thermoviscoplastic materials

We study thermomechanical deformations of a viscoplastic body deformed in simple shear. The effect of material elasticity is neglected but that of work hardening, strain-rate hardening, thermal softening, and strain-rate gradients is considered. The consideration of strain-rate gradients introduces a material characteristic length into the problem. A homogeneous solution of the governing equations is perturbed at different values t₀ of time t, and the growth rate at time t₀ of perturbations of different wavelengths is computed. Following Wright and Ockendon's postulate that the wavelength of the dominant instability mode with the maximum growth rate at time t₀ determines the minimum spacing between shear bands, the shear band spacing is computed. It is found that for the shear band spacing to be positive, either the thermal conductivity or the material characteristic length must be positive. Approximate analytical expressions for locally adiabatic deformations of dipolar (strain-rate gradient-dependent) materials indicate that the shear band spacing is proportional to the square-root of the material charateristic length, and the fourth root of the strain-rate hardening exponent. The shear band spacing increases with an increase in the strain hardening exponent and the thermal conductivity of the material.