Void growth and interaction in crystalline materials

An inelastic rate-dependent crystalline constitutive formulation and specialized computational schemes have been developed and used to obtain a detailed understanding of the interrelated physical mechanisms that can result in ductile material failure in rate-dependent porous crystalline materials subjected to finite inelastic deformations. The effects of void growth and interaction and specimen necking on material failure have been investigated for a single material cell, with a discrete cluster of four voids, where geometrical parameters have been varied to result in seven unique periodic and random void arrangements. The interrelated effects of void distribution and geometry, strain hardening, geometrical softening, localized plastic strains and slip-rates, and hydrostatic stresses on failure paths and ligament damage in face centered cubic (f.c.c.) crystalline materials have been studied. Results from this study are consistent with experimental observations that ductile failure can occur either due to void growth parallel to the stress axis, which results in void coalescence normal to the stress axis, or void interaction along bands, which are characterized by intense shear-strain localization and that intersect the free surface at regions of extensive specimen necking.     

Neck development in metal/elastomer bilayers under dynamic stretchings

Dynamic stretching of bilayer plates comprised of an elastomer layer bonded to a metal layer is studied with emphasis on neck development under plane strain. Neck retardation and multiple necks in the metal layer are promoted by the elastomer layer under quasi-static conditions and also by inertia in the all-metal layer under dynamic stretching. The present study explores the interaction of these two effects under the assumption that ductility is unlimited. The study reveals the roles of the elastomer modulus, the metal strength and the rate of stretching in giving rise to necking retardation.     

An Element Free Galerkin analysis of steady dynamic growth of a mode i crack in elastic–plastic materials

An Element Free Galerkin (EFG) method based formulation for steady dynamic crack growth in elastic–plastic materials is developed. A domain convecting parallel to the steadily moving crack tip is employed. The EFG methodology eliminates the stringent mesh requirements of the Finite Element Method (FEM) for such problems. Both rate-independent materials and rate-dependent materials are considered. The material is characterized by von Mises yielding condition and an associated flow rule. For rate-independent materials, both the influence of crack speeds and that of strain hardening on the mechanics of steady dynamic crack growth are investigated. For rate-dependent materials, only a non-hardening material is considered with emphasis on determining the influence of viscous properties of materials and crack speeds. The influence of strain hardening on steady dynamic crack growth shows the same trends as for steady quasi-static crack growth. The simplifications used in the literature in deriving analytical solutions for high strain-rate crack growth have been examined thoroughly using the numerical results.     

Dynamic neck development in a polymer tube under internal pressure loading

The initiation and growth of necks in polymer tubes subjected to rapidly increasing internal pressure is analyzed numerically. Plane strain conditions are assumed to prevail in the axial direction. The polymer is characterized by a finite strain elastic–viscoplastic constitutive relation and the calculations are carried out using a dynamic finite element program. Numerical results for neck development are illustrated and discussed for tubes of various thicknesses. The sensitivity to the wave number of the thickness imperfections is studied with a focus on comparing a long wave length imperfection and a short wave length imperfection. After some thinning down at the necks, the mode of deformation switches to neck propagation along the circumference of the tube. A case is shown in which the necks have propagated along the entire tube wall, so that network locking in the polymer results in high stiffness against further expansion of the tube. The rate dependence of the necking behavior gives noticeable differences in neck development for slow loading versus fast loading.     

Experiments and Material Parameter Identification Using Finite Elements. Uniaxial Tests and Validation Using Instrumented Indentation Tests

In this article, we focus our attention on the relation between instrumented indentation tests and the prediction by means of finite element calculations. To this end, a finite strain viscoplasticity model of Perzyna-type with non-linear isotropic and kinematic hardening is calibrated at experimental data of steel S690QL. A particular concept for conducting uniaxial tensile and compression tests is taken up in order to represent the basic rate-dependent material behavior. In this respect, an algorithmic framework of material parameter identification using finite elements is proposed leading to a two-stage procedure in the case of the underlying rate-dependent constitutive model. On the basis of the termination points of relaxation the rate-independent equilibrium stress state can be identified and all viscous parts of the model are obtained using rate-dependent loading paths. Finally, use is made of finite elements for predicting indentation experiments, which results in a critical view on modeling and parameter identification on the basis of experimental results occurring in instrumented indentation tests.     

Identification of Post-Necking Hardening Phenomena in Ductile Sheet Metal

A combined theoretical/experimental approach accurately quantifying post-necking hardening phenomena in ductile sheet materials that initially exhibit diffuse necking in tension is presented. The method is based on the minimization of the discrepancy between the internal and the external work in the necking zone during a quasi-static tensile test. The main focus of this paper is on the experimental validation of the method using an independent material test. For this purpose, the uniaxial tube expansion test is used to obtain uniaxial strain hardening behavior beyond the point of maximum uniform strain in a tensile test. The proposed method is used to identify the post-necking hardening behavior of a cold rolled interstitial-free steel sheet. It is demonstrated that commonly adopted phenomenological hardening laws cannot accurately describe all hardening stages. An alternative phenomenological hardening model is presented which enables to disentangle pre- and post-necking hardening behavior. Additionally, the influence of the yield surface on the identified post-necking hardening behavior is scrutinized. The results of the proposed method are compared with the hydraulic bulge test. Unlike the hydraulic bulge test, the proposed method predicts a decreased hardening rate in the post-necking regime which might be associated with probing stage IV hardening. While inconclusive, the discrepancy with the hydraulic bulge test suggests differential work hardening at large plastic strains.     

Effect of the rate dependence of nonlinear kinematic hardening rule on relaxation behavior

Relaxation experiments for metallic materials and solid polymers have exhibited nonlinear dependence of stress relaxation on prior loading rate; the relaxed stress associated with the fastest prior strain rate has the smallest magnitude at the end of the same relaxation periods. Modeling capability for the basic feature of relaxation behavior is qualitatively investigated in the context of unified state variable theory. Unified constitutive models are categorized into three general classes according to the rate dependence of kinematic hardening rule, which defines the evolution of the back (equilibrium) stress and is the major difference among constitutive models. The first class of models adopts the nonlinear kinematic hardening rule proposed by Armstrong and Frederick. In this class, the back stress appears to be rate-independent under loading and subsequent relaxation conditions. In the second class of models, a stress rate term is incorporated into the Armstrong–Frederick rule and the back stress then becomes rate-dependent during relaxation condition even though it remains rate-independent under loading condition. The final class proposed here includes a new nonlinear kinematic hardening rule that causes the back stress to be rate-dependent all the time. It is shown that the apparent rate dependence of the back stress during relaxation enables constitutive models to predict the influence of prior loading rate on relaxation behavior.     

Damage of ductile materials deforming under multiple plastic or viscoplastic mechanisms

This work presents a model to represent ductile failure (i.e. failure controlled by nucleation, growth and coalescence) of materials whose irreversible deformation is controlled by several plastic or viscoplastic deformation mechanisms. In addition work hardening may result from both isotropic and kinematic hardening. Damage is represented by a single variable representing void volume fraction. The model uses an additive decomposition of the plastic strain rate tensor. The model is developed based on the definition of damage dependant effective scalar stresses. The model is first developed within the generalized standard material framework and expressions for Helmholtz free energy, yield potential and dissipation potential are proposed. In absence of void nucleation, the evolution of the void volume fraction is governed by mass conservation and damage does not need to be represented by state variables. The model is extended to account for void nucleation. It is implemented in a finite element software to perform structural computations. The model is applied to three case studies: (i) failure by void growth and coalescence by internal necking (pipeline steel) where plastic flow is either governed by the Gurson–Tvergaard–Needleman model or the Thomason model, (ii) creep failure (Grade 91 creep resistant steel) where viscoplastic flow is controlled by dislocation creep or diffusional creep and (iii) ductile rupture after pre-compression (aluminum alloy) where kinematic hardening plays an important role.     

Effects of inertia on dynamic neck formation in tensile bars

Neck localization during high rate extension of round bars is analyzed numerically. An axisymmetric problem formulation is given and the material is described as rate independent and elastic–plastic. The time history of neck development is investigated and effects of geometry and initial thickness imperfections are visualized. Studies of both weakly and strongly developed necks are performed, revealing the occurrence of multiple necks in some loading cases. The influence on neck formation from background inertia (lateral inertia) corresponding to the inertia originating from homogeneous deformation of the bar is examined somewhat qualitatively by introducing an artificial volume load. Calculations show that in the present analysis, background inertia does not have any noticeable influence on the necking pattern unless the effect is artificially magnified by three orders of magnitude so that it becomes comparable to the yield stress.     

Comparison of bifurcation and imperfection analyses of localized necking in rate-independent polycrystalline sheets

Plastic flow localization into a narrow band is investigated for a rate-independent polycrystalline sheet. Strain localization is considered to initiate at a point of bifurcation from the homogeneous deformation state or to result from the growth of a preexisting imperfection. Bifurcation analysis based on the rate-independent crystal plasticity model is found to give limit strains at a realistic strain level in the biaxial-stretching range. Limit strains predicted by the imperfection analysis are lower than those for bifurcation analysis and tend to approach those strain levels with decreasing amount of initial imperfection. Then, the bifurcation analysis is applied to the simulation of the forming limit for strongly textured sheets. The bifurcation analysis reveals the same trend of a texture dependence of limit strains as that for the imperfection analysis employing a rate-dependent model reported in the literature.     

Effect of yield surface curvature and void nucleation on plastic flow localization

The effect of void nucleation is incorporated in a recently proposed material model that accounts for a combination of kinematic hardening and isotropic hardening of a porous ductile material. Since each of plastic dilatancy, void nucleation and yield surface curvature have a strong influence on predictions of plastic flow localization, the present material model can be used to study the interaction of these effects. Nucleation controlled by the plastic strain as well as nucleation controlled by the maximum normal stress on the particle-matrix interface are modelled. The predictions of the material model, for various combinations of parameters, are illustrated by analyses of shear band formation under plane strain or axisymmetric conditions, and by analyses of necking in biaxially stretched sheets.     

Benchmark experiments and characteristic cyclic plasticity deformation

Key issues in cyclic plasticity modeling are discussed based upon representative experimental observations on several commonly used engineering materials. Cyclic plasticity is characterized by the Bauschinger effect, cyclic hardening/softening, strain range effect, nonproporitonal hardening, and strain ratcheting. Additional hardening is identified to associate with ratcheting rate decay. Proper modeling requires a clear distinction among different types of cyclic plasticity behavior. Cyclic hardening/softening sustains dependent on the loading amplitude and loading history. Strain range effect is common for most engineering metallic materials. Often, nonproportional hardening is accompanied by cyclic hardening, as being observed on stainless steels and pure copper. A clarification of the two types of material behavior can be made through benchmark experiments and modeling technique. Ratcheting rate decay is a common observation on a number of materials and it often follows a power law relationship with the number of loading cycles under the constant amplitude stress controlled condition. Benchmark experiments can be used to explore the different cyclic plasticity properties of the materials. Discussions about proper modeling are based on the typical cyclic plasticity phenomena obtained from testing several engineering materials under various uniaxial and multiaxial cyclic loading conditions. Sufficient experimental evidence points to the unambiguous conclusion that none of the hardening phenomena (cyclic hardening/softening, strain range effect, nonproportional hardening, and strain hardening associated with ratcheting rate decay) is isotropic in nature. None of the hardening behavior can be properly modeled with a change in the yield stress.     

On neck propagation in amorphous glassy polymers under plane strain tension

Unlike metals, necking in polymers under tension does not lead to further localization of deformation, but to propagation of the neck along the specimen. Finite element analysis is used to numerically study necking and neck propagation in amorphous glassy polymers under plane strain tension during large strain plastic flow. The constitutive model used in the analyses features strain-rate, pressure, and temperature dependent yield, softening immediately after yield and subsequent orientational hardening with further plastic deformation. The latter is associated with distortion of the underlying molecular network structure of the material, and is modelled here by adopting a recently proposed network theory developed for rubber elasticity. Previous studies of necking instabilities have almost invariably employed idealized prismatic specimens; here, we explicitly account for the unavoidable grip sections of test specimens. The effects of initial imperfections, strain softening, orientation hardening, strain-rate as well as of specimen geometry and boundary conditions are discussed. The physical mechanisms for necking and neck propagation, in terms of our constitutive model, are discussed on the basis of a detailed parameter study.     

A Finite Element Analysis of a Tapered Flat Sheet Tensile Specimen

A uniaxial tension sheet metal coupon with a tapered instead of a straight gage section has been used for centering the location of diffuse neck and for measuring sheet stretchability in a non-uniform strain field. A finite element analysis of such a tensile coupon made of automotive steel sheet metals has been carried out to assess the effect of the tapered gage section geometry and material plastic strain hardening characteristics on the development of local plastic deformation pattern and local stress state, especially beyond the onset of diffuse necking but before localized necking. In particular, the finite element analysis was used in this study to evaluate the accuracy and reliability of an experimental data analysis method for estimating the post-necking effective plastic stress-strain curve based on the direct local surface axial plastic strain measurements for base metal, heat-affected zone, and weld metals of a dual-phase steel DP600. It is concluded that the estimated lower and upper bounds of the effective stress-strain curve at large strains are not satisfactory for low strain-hardening materials such as heat-affected zone and weld metals with the tapered tension coupons. A simple correction method utilizing only the additional local surface strain measurement in the transverse direction is proposed and it is shown to be effective in correcting the estimated effective stress-strain curve of dual-phase steel weld metals obtained for two tapered gage section geometries.     

Thermal softening induced plastic instability in rate-dependent materials

A linear perturbation analysis is performed for a class of rate-dependent materials, such as the Johnson-Cook model, in which the rate contribution to the stress can be separated from that of the plastic strain and temperature and in which the temperature rises adiabatically. The analysis is facilitated by perturbing both the rate of momentum equation and the momentum equation. An identical material stability/instability criterion is deduced from the characteristic spectral equations for one-dimensional deformation, one-dimensional shearing, and general three-dimensional field equations, and thus shows that the instability derived here is a material constitutive instability.The criteria indicate that the materials become unstable once the thermal softening overcomes the strain hardening, regardless of the strain rate. The strain rate enters the criteria through its effects on the accumulated temperature and the current stress. Based on the criterion, the three-dimensional instability surface is established in the space of plastic strain, plastic strain rate, and temperature. Instability surface is shown as a material property and independent of deformation histories or modes. Both necking and shear banding are simulated to validate the excellent predictive capability of the criterion.     

The dynamic growth of a single void in a viscoplastic material under transient hydrostatic loading

We have examined the problem of the dynamic growth of a single spherical void in an elastic-viscoplastic medium, with a view towards addressing a number of problems that arise during the dynamic failure of metals. Particular attention is paid to inertial, thermal and rate-dependent effects, which have not previously been thoroughly studied in a combined setting. It is shown that the critical stress for unstable growth of the void in the quasistatic case is strongly affected by the thermal softening of the material (in adiabatic calculations). Thermal softening has the effect of lowering the critical stress, and has a stronger influence at high strain hardening exponents. It is shown that the thermally diffusive case for quasistatic void growth in rate-dependent materials is strongly affected by the initial void size, because of the length scale introduced by the thermal diffusion. The effects of inertia are quantified, and it is demonstrated that inertial effects are small in the early stages of void growth and are strongly dependent on the initial size of the void and the rate of loading. Under supercritical loading for the inertial problem, voids of all sizes achieve a constant absolute void growth rate in the long term. Inertia first impedes, but finally promotes dynamic void growth under a subcritical loading. For dynamic void growth, the effect of rate-hardening is to reduce the rate of void growth in comparison to the rate-independent case, and to reduce the final relative void growth achieved.     

Characterization of the post-necking strain hardening behavior using the virtual fields method

The present study aims at characterizing the post-necking strain hardening behavior of three sheet metals having different hardening behavior. Standard tensile tests were performed on sheet metal specimens up to fracture and heterogeneous logarithmic strain fields were obtained from a digital image correlation technique. Then, an appropriate elasto-plastic constitutive model was chosen. Von Mises yield criterion under plane stress and isotropic hardening law were considered to retrieve the relationship between stress and strain. The virtual fields method (VFM) was adopted as an inverse method to determine the constitutive parameters by calculating the stress fields from the heterogeneous strain fields. The results show that the choice of a hardening law which can describe the hardening behavior accurately is important to derive the true stress–strain curve. Finally, post-necking hardening behavior was successfully characterized up to the initial stage of localized necking using the VFM with Swift and modified Voce laws.     

Predicting failure modes and ductility of dual phase steels using plastic strain localization

Ductile failure of metals is often treated as the result of void nucleation, growth and coalescence. Various criteria have been proposed to capture this failure mechanism for various materials. In this study, ductile failure of dual phase steels is predicted in the form of plastic strain localization resulting from the incompatible deformation between the harder martensite phase and the softer ferrite matrix. Microstructure-level inhomogeneity serves as the initial imperfection triggering the instability in the form of plastic strain localization during the deformation process. Failure modes and ultimate ductility of two dual phase steels are analyzed using finite element analyses based on the actual steel microstructures. The plastic work hardening properties for the constituent phases are determined by the in-situ synchrotron-based high-energy X-ray diffraction technique. Under different loading conditions, different failure modes and ultimate ductility are predicted in the form of plastic strain localization. It is found that the local failure mode and ultimate ductility of dual phase steels are closely related to the stress state in the material. Under plane stress condition with free lateral boundary, one dominant shear band develops and leads to final failure of the material. However, if the lateral boundary is constrained, splitting failure perpendicular to the loading direction is predicted with much reduced ductility. On the other hand, under plane strain loading condition, commonly observed necking phenomenon is predicted which leads to the final failure of the material. These predictions are in reasonably good agreement with experimental observations.     

Neck propagation

Propagation of a neck along the length of a tensile specimen as occurs in certain polymeric materials and in a few metals is studied. Two material models are considered: a nonlinear elastic solid, and an inelastic flow theory solid with both rate-dependent and rate-independent behaviors. For the elastic solid the states ahead and behind the neck transition can be obtained fairly simply from just the jump conditions governing continuity of mass, momentum and energy. For the inelastic solid a full three-dimensional analysis must be performed to obtain the same information, and an analysis of axisymmetric neck propagation is carried out.     

Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper

Cyclic plasticity experiments were conducted on a pure polycrystalline copper and the material was found to display significant cyclic hardening and nonproportional hardening. An effort was made to describe the cyclic plasticity behavior of the material. The model is based on the framework using a yield surface together with the Armstrong–Frederick type kinematic hardening rule. No isotropic hardening is considered and the yield stress is assumed to be a constant. The backstress is decomposed into additive parts with each part following the Armstrong–Frederick type hardening rule. A memory surface in the plastic strain space is used to account for the strain range effect. The Tanaka fourth order tensor is used to characterize nonproportional loading. A set of material parameters in the hardening rules are related to the strain memory surface size and they are used to capture the strain range effect and the dependence of cyclic hardening and nonproportional hardening on the loading magnitude. The constitutive model can describe well the transient behavior during cyclic hardening and nonproportional hardening of the polycrystalline copper. Modeling of long-term ratcheting deformation is a difficult task and further investigations are required.