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.     

Microstructural effects on shear instability and shear band spacing

A constitutive relation that accounts for the thermally activated dislocation motion and microstructure interaction is used to study the stability of a homogeneous solution of equations governing the simple shearing deformations of a thermoviscoplastic body. An instability criterion and an upper bound for the growth rate of the infinitesimal deformations superimposed on the homogeneous solution are derived. By adopting Wright and Ockendon's postulate, i.e., the wavelength of the dominant instability mode with the maximum growth rate determines the minimum spacing between shear bands, the shear band spacing is computed. The effect of the initial dislocation density, the nominal strain-rate, and parameters describing the initial thermal activation and the initial microstructure interaction on the shear band spacing are delineated.     

Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity

Gradient theories, as a regularized continuum mechanics approach, have found wide applications for modeling strain localization failure process. This paper presents a second gradient stress–strain damage elasticity theory based upon the method of virtual power. The theory considers the strain gradient and its conjugated double stresses. Instead of introducing an intrinsic material length scale into the constitutive law in an ad hoc fashion, a microstructural granular mechanics approach is applied to derive the higher-order constitutive coefficients such that the internal length scale parameter reflects the natural granularity of the underlying material microstructure. The derivations of the required damage constitutive relationships, the strong form governing equations as well as its weak form for the second gradient model are described. The recently popularized Element-Free Galerkin (EFG) method is then employed to discretize the weak form equilibrium equation for accommodating the resultant higher-order continuity requirements and further handling the mesh sensitivity problem. Numerical examples for shear band simulations show that the proposed second gradient continuum model can produce stable, accurate as well as mesh-size independent solutions without a priori assumption of the shear band path.     

Anti-plane shear crack in a functionally gradient piezoelectric material

The main objective of this paper is to study the singular nature of the crack-tip stress and electric displacement field in a functionally gradient piezoelectric medium having material coefficients with a discontinuous derivative. The problem is considered for the simplest possible loading and geometry, namely, the anti-plane shear stress and electric displacement in-plane of two bonded half spaces in which the crack is parallel to the interface. It is shown that the square-root singularity of the crack-tip stress field and electric displacement field is unaffected by the discontinuity in the derivative of the material coefficients. The problem is solved for the case of a finite crack and extensive results are given for the stress intensity factors, electric displacement intensity factors, and the energy release rate. Project supported by the National Natural Science Foundation of China (No. 10072041), the National Excellent Young Scholar Fund, of China (No. 10125209) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P. R. C..     

Numerical simulation of shear band formation in a frictional-dilational material

Summary The explicit, finite difference code FLAC is used to model shear band development in isotropic, elastic-plastic, Coulomb, non-associated, non-hardening materials. The code reproduces shear band inclinations which depend on both the angle of friction and the angle of dilation. Localization occurs at the yield point. Shear band width is sensitive to the size of the finite difference mesh but is controlled also by the magnitudes of the friction and dilation angles. The distribution of shear bands throughout the specimen is also influenced by the values of friction and, to a lesser extent, of dilation. There is no suggestion that numerical instability or numerical truncation errors are responsible for shear band nucleation.

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Numerische Simulation von Scherfugenbildungen in druckabhängigen, dilatanten Materialien

Übersicht In numerisch simulierten Experimenten an einem gleichförmig belasteten rechteckigen Block werden die Möglichkeiten des Finite-Differenzen-Programms FLAC zur Darstellung von Scherfugenbildungen überprüft. Wir betrachten elastisch-ideal plastische Materialien vom Coulomb-Typ mit nicht-assoziierter Fließregel. Es ergeben sich Scherfugenorientierungen, die sowohl vom Wert des Reibungswinkels als auch vom Wert des Dilatanzwinkels abhängen. Die Scherfugendicken hängen sowohl von der Maschenweite des Differenzennetzes als auch vom Reibungs- und Dilatanzwinkel ab. Der Wert des Reibungswinkels und in geringem Maße auch der Wert des Dilatanzwinkels beeinflussen die Art der Scherfugenverteilung innerhalb des Blocks. Die Ergebnisse der Berechnungen werden, soweit möglich, mit analytischen Lösungen verglichen.

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Presented at the workshop on Limit Analysis and Bifurcation Theory, held at the University of Karlsruhe (FRG), February 22–25, 1988     

Direct calculations of material parameters for gradient plasticity

Length scale parameters introduced in gradient theories of plasticity are calculated in closed form with a continuum dislocation based theory. The similarity of the governing equations in both models for the evolution of plastic deformation of a constrained thin film makes it possible to identify parameters of the gradient plasticity theory with the dislocation based model. A one-to-one identification is not possible given that gradient plasticity does not account for individual dislocations. However, by comparing the mean plastic deformation across the film thickness we find that the length scale parameter, l, introduced in the gradient plasticity theory depends on the geometry as well as material constants.     

Multi-slip gradient formulation for modeling microstructure effects on shear bands in granular materials

This paper presents a higher order gradient multi-slip formulation to model the effect of inhomogeneous deformation in granular materials. The effects of heterogeneity and porosity anisotropy within the multi-slip formulation are taken into consideration through the modification of the mobilized friction. The mobilized friction is assumed to be a direct function of either the gradient of the porosity distribution or the fabric tensor. The formulation with two active slip planes was implemented into a finite element code and used to simulate biaxial shear tests on dry sand. The analysis quantifies most of the shear band characteristics observed by past experimentation. It is shown that the localization and shear band characteristics in granular materials are very much dependent on the initial fabric and slip system arrangement.     

Effect of geometrical and material parameters in nanoindentation of layered materials with an interphase

Indentation models for thin layer-substrate geometry with an interphase have been developed. The interphase can be modeled either as a nonhomogeneous layer or as a homogeneous layer. Between the two models of the interphase, contact depth and critical interfacial stresses are compared to find the effect of indentation area, film and substrate Young’s moduli, and the interphase and film thicknesses. Although contact depth is found not to be sensitive to the type of interphase model used, critical interfacial stresses are significantly different (up to 15%) for film to substrate elastic Young’s moduli ratios of more than 25. A formal sensitivity analysis based on design of experiments shows that on critical interfacial stresses, interphase to film thickness ratio and film to substrate Young’s moduli ratio has the most impact, while type of elastic moduli variation in the interphase and indentor width to film thickness ratio has the least impact.     

Constitutive model for plastic deformation of nanocrystalline materials with shear band

A phase mixture model was used to study the plastic deformation behaviors in hardening stage of nanocrystalline materials. The material was considered as a composite of grain interior phase and grain boundary (GB) phase. The constitutive equations of the two phases were determined in term of their main deformation mechanisms. In softening stage, a shear band deformation mechanism was presented and the corresponding constitutive relation was established. Numerical simulations have shown that the predications fit well with experimental data. The investigation using the finite-element method (FEM) provided a direct insight into quantifying shear localization effect in nanocrystalline materials.     

On the speed of an unconstrained shear band in a perfectly plastic material

It is shown that the speed of an adiabatic shear band may be estimated from its physical and constitutive properties within a constant, which must be obtained from the complete boundary value problem.     

Damage accumulation for growth of anti-plane crack in work-hardening material

Elastic–plastic solutions of an anti-plane crack in an infinite body are used in conjunction with a continuum damage model to describe the conditions necessary for the onset of crack instability, fatigue crack propagation due to cyclic loading, and rates of crack growth due to time dependent events. A power law relates the stress to the strain of the material. The damage, which invokes nucleation, growth and coalescence of microvoids due to elevated strain, is confined to the plastic zone surrounding the crack tip. For applied loading below the yield stress, the small-scale and large-scale yielding solutions are used to determine the influence of strain hardening on crack instability and failure. Crack growth due to cyclic loading and time-dependent deformations are studied using the small-scale yielding solution of the deformation theory of plasticity.     

Prediction of the initial thickness of shear band localization based on a reduced strain gradient theory

Plastic flow localization in ductile materials subjected to pure shear loading and uniaxial tension is investigated respectively in this paper using a reduced strain gradient theory, which consists of the couple-stress (CS) strain gradient theory proposed by Fleck and Hutchinson (1993) and the strain gradient hardening (softening) law (C–W) proposed by Chen and Wang (2000). Unlike the classical plasticity framework, the initial thickness of the shear band and the strain rate distribution in both cases are predicted analytically using a bifurcation analysis. It shows that the strain rate is obviously non-uniform inside the shear band and reaches a maximum at the center of the shear band. The initial thickness of the shear band depends on not only the material intrinsic length l_cs but also the material constants, such as the yield strength, ultimate tension strength, the linear hardening and softening shear moduli. Specially, in the uniaxial tension case, the most possible tilt angle of shear band localization is consistent qualitatively with the existing experimental observations. The results in this paper should be useful for engineers to predict the details of material failures due to plastic flow localization.     

Effect of particle type on the shear behaviour of granular materials

In discrete element method(DEM)simulations,multi-sphere(MS)clumped and convex particles are two main particle models that are used to study the mechanical behav...     

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。     

Crack shielding and material deterioration in damaged materials: an antiplane shear fracture problem

Summary A simple damage evolution model is proposed for a quasibrittle material in the framework of continuum damage mechanics. The model is used to obtain a closed form solution for a mode-III stationary crack under small scale damage conditions. It is found that the crack tip stress intensity factor is reduced, i.e., the crack is shielded by the damage. However, this shielding effect is completely offset by the material deterioration caused by the damage. It also holds for steady state crack growth. When the most effective shielding is reached for the stationary crack, the zone dominated by the stress intensity factor shrinks to the crack tip. The results for the antiplane shear problem should shed some light on the in- plane fracture problem. Received 4 August 1997; accepted for publication 7 October 1997     

Effect of transverse shear and material orthotropy in a cracked spherical cap

The elastostatic problem for a relatively thin-walled spherical cap containing a through crack is considered. The problem is formulated for a specially orthotropic material within the confines of a linearized, shallow shell theory. The theory used is equivalent to Reissner's theory of flat plates and hence permits the consideration of all five physical conditions on the shell boundaries separately. The solution of the problem is reduced to that of a pair of singular integral equations and the asymptotic stress state around the crack tips is investigated. The numerical solution of the problem is given for an isotropic shell and for two specially orthotropic shells. The results indicate that the material orthotropy as well as the shell curvature and thickness may have a considerable effect on the stress intensity factors at the crack tips.     

Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity

In the context of single-crystal strain gradient plasticity, we focus on the simple shear of a constrained strip in order to study the effects of the material parameters possibly involved in the modelling. The model consists of a deformation theory suggested and left undeveloped by Bardella [(2007). Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved. Int. J. Plasticity 23, 296–322] in which, for each glide, three dissipative length scales are considered; they enter the model through the definition of an effective slip which brings into the isotropic hardening function the relevant plastic strain gradients, averaged by means of a p-norm. By means of the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]), the model further involves an energetic material length scale. The application suggests that two dissipative length scales may be enough to qualitatively describe the size effect of metals at the microscale, and they are chosen in such a way that the higher-order state variables of the model be the dislocation densities. Moreover, we show that, depending on the crystallography, the size effect governed by the defect energy may be different from what expected (based on the findings of [Bardella, L., 2006. A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 54, 128–160] and [Gurtin et al. 2007. Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878]), leading mostly to some strengthening. In order to investigate the model capability, we also exploit a Γ-convergence technique to find closed-form solutions in the “isotropic limit”. Finally, we analytically show that in the “perfect plasticity” case, should the dissipative length scales be set to zero, the presence of the sole energetic length scale may lead, as in standard plasticity, to non-uniqueness of solutions.