Advances in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations

This work is a review of experimental methods for observing and modeling the anisotropic plastic behavior of metal sheets and tubes under a variety of loading paths, such as biaxial compression tests; biaxial tension tests on metal sheets and tubes using closed-loop electrohydraulic testing machines; the abrupt strain path change method for detecting a yield vertex and subsequent yield loci without unloading; in-plane stress reversal tests on metal sheets; and multistage tension tests. Observed material responses are compared with the predictions of phenomenological plasticity models. Special attention is paid to the plastic deformation behavior of materials commonly used in industry, and to verifying the validity of conventional anisotropic yield criteria for those materials and associated flow rules at large plastic strains. The effects of using appropriate anisotropic yield criteria on the accuracy of simulations of forming defects, such as large springback and fracture, are also presented to highlight the importance of accurate material testing and modeling.     

Experimental observations of evolving yield loci in biaxially strained AA5754-O

Experimental measurement of the plastic biaxial mechanical response for an aluminum alloy (AA5754-O) sheet metal is presented. Traditional methods of multiaxial sheet metal testing require the use of finite element analysis (FEA) or other assumptions of material response to determine the multiaxial true stress versus true strain behavior of the as-received sheet material. The method used here strives to produce less ambiguous measurements of data for a larger strain range than previously possible, through a combination of the Marciniak flat bottom ram test and an X-ray diffraction technique for stress measurement. The study is performed in conjunction with a study of the microstructural changes that occur during deformation, and these microstructural results are briefly mentioned in this work. Issues of calibration and applicability are discussed, and results are presented for uniaxial (U), plane strain (PS), and balanced biaxial (BB) extension. The results show repeatable behavior (within quantified uncertainties) for U to 20%, PS to almost 15%, and BB to above 20% in-plane strains. The results are first compared with three common yield locus models (von Mises’, Hill’48, and Hosford’79), and show some unexpected results in the shape change of the yield locus at high strain levels (>5% strain). These changes include the rotation of the locus toward the von Mises surface and elongation in the balanced biaxial direction. Comparison with a more complex yield locus model (Yld2000-2d with eight adjustable parameters) showed that the locus elongation in the biaxial direction could be fit well (for a specific level of work), but at the detriment of fit to the plane strain data. Artificially large plastic strain ratios would be needed to match both the biaxial and plane strain behavior even with this more complex model.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

A unified treatment of strain gradient plasticity

A theoretical framework is presented that has potential to cover a large range of strain gradient plasticity effects in isotropic materials. Both incremental plasticity and viscoplasticity models are presented. Many of the alternative models that have been presented in the literature are included as special cases. Based on the expression for plastic dissipation, it is in accordance with Gurtin (J. Mech. Phys. Solids 48 (2000) 989; Int. J. Plast. 19 (2003) 47) argued that the plastic flow direction is governed by a microstress q_ij and not the deviatoric Cauchy stress σ_ij′ that has been assumed by many others. The structure of the governing equations is of second order in the displacements and the plastic strains which makes it comparatively easy to implement in a finite element programme. In addition, a framework for the formulation of consistent boundary conditions is presented. It is shown that there is a close connection between surface energy of an interface and boundary conditions in terms of plastic strains and moment stresses. This should make it possible to study boundary layer effects at the interface between grains or phases. Consistent boundary conditions for an expanding elastic-plastic boundary are as well formulated. As examples, biaxial tension of a thin film on a thick substrate, torsion of a thin wire and a spherical void under remote hydrostatic tension are investigated.     

Gradient plasticity with the tangential-subloading surface model and the prediction of shear-band thickness of granular materials

An extended gradient elastoplastic constitutive equation is formulated, which is capable of describing the plastic strain rate due to the rate of stress inside the yield surface and the inelastic strain rate due to the stress rate component tangential to the subloading surface by incorporating the tangential-subloading surface model. Based on the extended constitutive equation, the post-localization analysis of granular materials is performed to predict the shear-band thickness. It is revealed that the shear-band thickness is almost determined by the gradient coefficient characterizing the inhomogeneity of deformation, although the stress–strain curve is strongly dependent on material properties.     

Constitutive framework optimized for myocardium and other high-strain, laminar materials with one fiber family

Central to this analysis is the identification of six rotation invariant scalars α_1-6 that succinctly define the strain in materials that have one family of parallel fibers arranged in laminae. These scalars were chosen so as to minimize covariance amongst the response terms in the hyperelastic limit, and they are termed strain attributes because it is necessary to distinguish them from strain invariants. The Cauchy stress t is expressed as the sum of six response terms, almost all of which are mutually orthogonal for finite strain (i.e. 14 of the 15 inner products vanish). For small deformations, the response terms are entirely orthogonal (i.e. all 15 inner products vanish). A response term is the product of a response function with its associated kinematic tensor. Each response function is a scalar partial derivative of the strain energy W with respect to a strain attribute. Applications for this theory presently include myocardium (heart muscle) which is often modeled as having muscle fibers arranged in sheets. Utility for experimental identification of strain energy functions is demonstrated by showing that common tests on incompressible materials can directly determine terms in W. Since the described set of strain attributes reduces the covariance amongst response terms, this approach may enhance the speed and precision of inverse finite element methods.     

Multiscale modelling of the plastic anisotropy and deformation texture of polycrystalline materials

A hierarchical multilevel method is presented for the plastic deformation of polycrystalline materials with texture-induced anisotropy. It is intended as a constitutive material model for finite element codes for the simulation of metal forming processes or for the prediction of forming limits. It consists of macroscopic models of which the parameters are to be identified using the results of two-level (meso/macro) or three-level (micro/meso/macro) models. A few such two-level models are presented, ranging from the full-constraints Taylor model to the crystal-plasticity finite element models, including the grain interaction models GIA, LAMEL and ALAMEL. Validation efforts based on experimental cold rolling textures obtained for steel and aluminium alloys are shortly discussed. An assessment is also given of the assumptions of the LAMEL and ALAMEL models concerning stress and strain rate heterogeneity at grain boundaries, based on the results of a crystal plasticity finite element study. Finally a recent three-level model which also looks at the microscopic level (dislocation substructure) is discussed.     

On the fracture toughness of ferroelastic materials

The toughness enhancement due to domain switching near a steadily growing crack in a ferroelastic material is analyzed. The constitutive response of the material is taken to be characteristic of a polycrystalline sample assembled from randomly oriented tetragonal single crystal grains. The constitutive law accounts for the strain saturation, asymmetry in tension versus compression, Bauschinger effects, reverse switching, and strain reorientation that can occur in these materials due to the non-proportional loading that arises near a propagating crack. Crack growth is assumed to proceed at a critical level of the crack tip energy release rate. Detailed finite element calculations are carried out to determine the stress and strain fields near the growing tip, and the ratio of the far field applied energy release rate to the crack tip energy release rate. The results of the finite element calculations are then compared to analytical models that assume the linear isotropic K-field solution holds for either the near tip stress or strain field. Ultimately, the model is able to account for the experimentally observed toughness enhancement in ferroelastic ceramics.     

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.     

Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids

The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile porous solids is investigated. The Gologanu–Leblond–Devaux’s (GLD) analysis of an rigid-ideal plastic (von Mises) spheroidal volume containing a confocal spheroidal cavity loaded axisymmetrically is extended to the case when the matrix is anisotropic (obeying Hill’s [Hill, R., 1948. A theory of yielding and plastic flow of anisotropic solids. Proc. Roy. Soc. London A 193, 281–297] anisotropic yield criterion) and the representative volume element is subjected to arbitrary deformation. To derive the overall anisotropic yield criterion, a limit analysis approach is used. Conditions of homogeneous boundary strain rate are imposed on every ellipsoidal confocal with the cavity. A two-field trial velocity satisfying these boundary conditions are considered. It is shown that for cylindrical and spherical void geometries, the proposed criterion reduces to existing anisotropic Gurson-like yield criteria. Furthermore, it is shown that for the case when the matrix is considered isotropic, the new results provide a rigorous generalization to the GLD model. Finally, the accuracy of the proposed approximate yield criterion for plastic anisotropic media containing non-spherical voids is assessed through comparison with numerical results.     

Three dimensional combined fracture–plastic material model for concrete

This paper describes a combined fracture–plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. The plasticity model for concrete in compression is based on the Menétrey–Willam failure surface, the plastic volumetric strain as a hardening/softening parameter and a non-associated flow rule based on a nonlinear plastic potential function. Both models use a return-mapping algorithm for the integration of constitutive equations. Special attention is given to the development of an algorithm for the combination of the two models. The suggested combination algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The algorithm can handle cases when failure surfaces of both models are active, but also when physical changes such as crack closure occur. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The model is integrated in a general finite element package ATENA and its performance is evaluated by comparisons with various experimental results from the literature.     

Potential flow model of cavitation-induced interfacial fracture in a confined ductile layer

Fracture of a thin ductile layer sandwiched between stiff substrates often results from growth and coalescence of microscopic cavities ahead of an extending crack. Cavitation induced by plastic flow in a confined, ductile layer is analyzed here to evaluate the interfacial fracture toughness of such sandwich structures. For rigid-plastic materials, a new method is proposed in which the potential flow field of a fluid is used to approximate the plastic deformation. The principle of virtual work rate is applied to determine the equivalent traction-separation law. The method is demonstrated and validated for spherically symmetric cavity growth, for which an exact solution exists. We then study in detail the growth of an initially spherical cavity in a cylindrical bar of finite length subject to uniform traction at its ends. The results show that the stress-separation curves depend strongly on initial cavity size and the strain-hardening exponent, and weakly on the nominal strain. The method has clear advantages over numerical methods, such as finite-element analysis, for parametric study of cavity growth with large plastic deformation.     

Yielding and intrinsic plasticity of Ti–Zr–Ni–Cu–Be bulk metallic glass

Bulk metallic glass with composition Ti₄₀Zr₂₅Ni₈Cu₉Be₁₈ exhibits considerably high compressive yield stress, significant plasticity (with a concomitant vein-like fracture morphology) and relatively low density. Yielding and intrinsic plasticity of this alloy are discussed in terms of its thermal and elastic properties. An influence of normal stresses acting on the shear plane is evidenced by: (i) the fracture angle (<45°) and (ii) finite-element simulations of nanoindentation curves, which require the use of a specific yield criterion, sensitive to local normal stresses acting on the shear plane, to properly match the experimental data. The ratio between hardness and compressive yield strength (constraint factor) is analyzed in terms of several models and is best adjusted using a modified expanding cavity model incorporating a pressure-sensitivity index defined by the Drucker–Prager yield criterion. Furthermore, comparative results from compression tests and nanoindentation reveal that deformation also causes strain softening, a phenomenon which is accompanied with the occurrence of serrated plastic flow and results in a so-called indentation size effect (ISE). A new approach to model the ISE of this metallic glass using the free volume concept is presented.     

Multiaxial constitutive model accounting for the strength-differential in Inconel 718

The nickel-base alloy Inconel 718 exhibits a strength-differential, that is, a different plastic flow behavior in uniaxial tension and uniaxial compression. A phenomenological viscoplastic model founded on thermodynamics has been extended for material behavior that deviates from classical metal plasticity by including all three stress invariants in the threshold function. The model can predict plastic flow in isotropic materials with or without a flow stress asymmetry as well as with or without pressure dependence. Viscoplastic material parameters have been fit to pure shear, uniaxial tension, and uniaxial compression experimental results at 650°°C. Threshold function material parameters have been fit to the strength-differential. Four classes of threshold functions have been considered and nonproportional loading of hollow tubes, such as shear strain followed by axial strain, has been used to select the most applicable class of threshold function for the multiaxial model as applied to Inconel 718 at 650 °C. These nonproportional load paths containing corners provide a rigorous test of a plasticity model, whether it is time-dependent or not. A J₂J₃ class model, where J₂ and J₃ are the second and third effective deviatoric stress invariants, was found to agree the best with the experimental results.