Fracture behaviour in tension of viscoplastic porous metallic materials saturated with liquid

The present work extends the investigation which has been initiated in Parts I and II of this study (Martin, C.L., Favier, D., Suéry, M., 1997a. Viscoplastic behaviour of porous metallic materials saturated with liquid, part I: constitutive equations. Int. J. Plasticity 13, 215–235; Martin, C.L., Favier, D. Suéry, M., 1997b. Viscoplastic behaviour of porous metallic materials saturated with liquid, Part II: experimental identification on a Sn–Pb model alloy. Int. J. Plasticity 13, 237–259) to the tensile behaviour of viscoplastic porous metallic materials saturated with liquid. Simple tensile experiments together with ring extension tests are carried out to study the fracture behaviour of this class of material. Ring tests consist in applying an internal pressure on a specimen with a ring shape. A Sn–Pb model alloy with a dendritic microstructure is used to characterise the behaviour of the material up to fracture. The liquid presence is accounted for to derive the intrinsic behaviour of the solid skeleton. The collected data are then incorporated in the model framework presented in Part I. A simple modification of the model allows the treatment of the strong asymmetry between tension and compression which is exhibited by these materials.     

Large deformation plasticity and the Poynting effect

The aims of this paper are fourfold: (1) To develop a set of constitutive equations that are applicable to isotropic inelastic materials with large elastic and plastic strains using the multiconfigurational framework (Rajagopal, K.R., Srinivasa, A.R. Int. J. Plasticity 14 (1998) 945; Rajagopal, K.R., Srinivasa, A.R. Int. J. Plasticity 14 (1998), 948), in such a way as to generalize the central ideas (such as isotropy, constant elastic modulii, quadratic yield surfaces and non-hardening behavior) of the Prandtl–Reuss theory to finite deformations, (2) to examine the consequences of using a physically plausible criterion of maximum rate of mechanical dissipation, (3) to examine the relationship of the resulting models to the classical Prandtl–Reuss theory as well as other possible formulations (specifically those that rely on the use of a maximum plastic work postulate), and (4) to consider the effect of finite elastic strains on the response of the material subject to some simple homogenous deformations. By considering the response under simple shear, it is shown that the elastic-plastic counterpart of the well known Poynting effect in finite elasticity has a profound influence on the post-yield behavior of such materials. In particular, it is shown that this gives rise to a strain softening effect even though the overall response is that of a non-hardening material.     

Evaluation of identification methods for YLD2004-18p

Four calibration methods have been evaluated for the linear transformation-based anisotropic yield function YLD2004-18p (Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., 2005. Linear transformation-based anisotropic yield functions. Int. J. Plasticity 21, 1009–1039) and the aluminium alloy AA5083-H116. The different parameter identifications are based on least squares fits to combinations of uniaxial tensile tests in seven directions with respect to the rolling direction, compression (upsetting) tests in the normal direction and stress states found using the full-constraint (FC) Taylor model for 690 evenly distributed strain paths. An elastic–plastic constitutive model based on YLD2004-18p has been implemented in a non-linear finite element code and used in finite element simulations of plane-strain tension tests, shear tests and upsetting tests. The experimental results as well as the Taylor model predictions can be satisfactorily reproduced by the considered yield function. However, the lacking ability of the Taylor model to quantitatively reproduce the experiments calls for more advanced crystal plasticity models.     

Implicit finite element formulations for multi-phase transformation in high carbon steel

An anomalous plastic deformation observed during the phase transformation of steels was implemented into the finite element modeling. The constitutive equations for the transformation plasticity originally proposed by Greenwood and Johnson [Greenwood, G.W., Johnson, R.H., 1965. The deformation of metals under small stresses during phase transformation. Proc. Roy. Soc. A 283, 403] and further extended by Leblond et al. [Leblond, J.B., Mottet, G., Devaux, J.C., 1986a. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, I. Derivation of general relations. J. Mech. Phys. Solids 34, 395–409; Leblond, J.B., Mottet, G., Devaux, J.C., 1986b. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, II. Study of classical plasticity for ideal-plastic phases. J. Mech. Phys. Solids 34, 411–432; Leblond, J.B., Devaux, J., Devaux, J.C., 1989a. Mathematical modeling of transformation plasticity in steels, I: case of ideal-plastic phases. Int. J. Plasticity 5, 511–572; Leblond, J.B., 1989b. Mathematical modeling of transformation plasticity in steels, II: coupling with strain hardening phenomena. Int. J. Plasticity 5, 573–591] were modified to consider the thermo-mechanical response of generalized multi-phase steel during phase transformations from austenite at high temperature. An implicit numerical solution procedure to calculate the plastic deformation of each constituent phase was newly proposed and implemented into the general purpose implicit finite element program via user material subroutine. The new algorithms include efficient calculation of consistent tangent modulus for the transformation plasticity and application of general anisotropic yield functions without limitation to the isotropic yield function. Besides the thermo-elastic–plastic constitutive equations, non-isothermal transformation kinetics was characterized by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation and additivity relationship for the diffusional transformation, while the model proposed by Koistinen and Marburger was used for the diffusionless transformation. Numerical verifications for the continuous cooling experiments under various loading conditions were conducted to demonstrate the applicability of the developed numerical algorithms to the high carbon steel SK5.     

Micromechanical modeling of stress-induced phase transformations. Part 1. Thermodynamics and kinetics of coupled interface propagation and reorientation

The universal (i.e. independent of the constitutive equations) thermodynamic driving force for coherent interface reorientation during first-order phase transformations in solids is derived for small and finite strains. The derivation is performed for a representative volume with plane interfaces, homogeneous stresses and strains in phases and macroscopically homogeneous boundary conditions. Dissipation function for coupled interface (or multiple parallel interfaces) reorientation and propagation is derived for combined athermal and drag interface friction. The relation between the rates of single and multiple interface reorientation and propagation and the corresponding driving forces are derived using extremum principles of irreversible thermodynamics. They are used to derive complete system of equations for evolution of martensitic microstructure (consisting of austenite and a fine mixture of two martensitic variants) in a representative volume under complex thermomechanical loading. Viscous dissipation at the interface level introduces size dependence in the kinetic equation for the rate of volume fraction. General relationships for a representative volume with moving interfaces under piece-wise homogeneous boundary conditions are derived. It was found that the driving force for interface reorientation appears when macroscopically homogeneous stress or strain are prescribed, which corresponds to experiments. Boundary conditions are satisfied in an averaged way. In Part 2 of the paper [Levitas, V.I., Ozsoy, I.B., 2008. Micromechanical modeling of stress-induced phase transformations. Part 2. Computational algorithms and examples. Int. J. Plasticity (2008)], the developed theory is applied to the numerical modeling of the evolution of martensitic microstructure under three-dimensional thermomechanical loading during cubic-tetragonal and tetragonal-orthorhombic phase transformations.     

Macro versus micro-scale constitutive models in simulating proportional and nonproportional cyclic and ratcheting responses of stainless steel 304

A recent study by Hassan et al. [Hassan, T., Taleb, L., Krishna, S., 2008. Influences of nonproportional loading paths on ratcheting responses and simulations by two recent cyclic plasticity models. Int. J. Plasticity, 24, 1863–1889.] demonstrated that some of the nonproportional ratcheting responses under stress-controlled loading histories cannot be simulated reasonably by two recent cyclic plasticity models. Two major drawbacks of the models identified were: (i) the stainless steel 304 demonstrated cyclic hardening under strain-controlled loading whereas cyclic softening under stress-controlled loading, which depends on the strain-range and which the existing models cannot describe; (ii) the change in biaxial ratcheting responses due to the change in the degree of nonproportionality were not simulated well by the models. Motivated by these findings, two modified cyclic plasticity models are evaluated in predicting a broad set of cyclic and ratcheting response of stainless steel 304. The experimental responses used in evaluating the modified models included both proportional (uniaxial) and nonproportional (biaxial) loading responses from Hassan and Kyriakides [Hassan, T., Kyriakides, S., 1994a. Ratcheting of cyclically hardening and softening materials. Part I: uniaxial behavior. Int. J. Plasticity, 10, 149–184; Hassan, T., Kyriakides, S., 1994b. Ratcheting of cyclically hardening and softening materials. Part II: multiaxial behavior. Int. J. Plasticity, 10, 185–212.] and Hassan et al. [Hassan, T., Taleb, L., Krishna, S., 2008. Influences of nonproportional loading paths on ratcheting responses and simulations by two recent cyclic plasticity models. Int. J. Plasticity, 24, 1863–1889.] The first model studied is a macro-scale, phenomenological, constitutive model originally proposed by Chaboche et al. [Chaboche, J.L., Dang-Van, K., Cordier, G., 1979. Modelization of the strain memory effect on the cyclic hardening of 316 stainless steel. In: Proceedings of the Fifth International Conference on SMiRT, Div. L, Berlin, Germany, L11/3.]. This model was systematically modified for incorporating strain-range dependent cyclic hardening–softening, and proportional and nonproportional loading memory parameters. The second model evaluated is a polycrystalline model originally proposed by Cailletaud [Cailletaud, G., 1992. A micromechanical approach to inelastic behavior of metals. Int. J. Plasticity, 8, 55–73.] based on crystalline slip mechanisms. These two models are scrutinized against simulating hysteresis loop shape, cyclic hardening–softening, cross-effect, cyclic relaxation, subsequent cyclic softening and finally a broad set of ratcheting responses under uniaxial and biaxial loading histories. The modeling features which improved simulations for these responses are elaborated in the paper. In addition, a novel technique for simulating both the monotonic and cyclic responses with one set of model parameters is developed and validated.     

Specific hardening function definition and characterization of a multimechanism generalized potential-based viscoelastoplasticity model

Given the previous complete-potential structure framework [see Int. J. Plasticity 10(3) (1994) 263], together with the notion of strain- and stress- partitioning in terms of separate contributions of several submechanisms (viscoelastic and viscoplastic) to the thermodynamic functions (stored energy and dissipation), see [Int. J. of Plasticity 17(10) (2001) 1305], a detailed viscoelastoplastic multimechanism characterization of a specific hardening functional form of the model is presented and discussed. TIMETAL 21S is the material of choice as a comprehensive test matrix, including creep, relaxation, constant strain-rate tension tests, etc. are available at various temperatures. Discussion of these correlations tests, together with comparisons to several other experimental results, are given to assess the performance and predictive capabilities of the present model particularly with regard to the notion of hardening saturation as well as the interaction of multiplicity of dissipative (reversible/irreversible) mechanisms.     

Experimental and theoretical analysis of the limits to ductility of type 304 stainless steel sheet

The forming behaviour of type 304 stainless steel sheet has been investigated. The strain-hardening behaviour has been characterised in uniaxial tension tests, and the forming limits at necking and at fracture have been determined using the Marciniak punch test. The results, complemented by measurements of the fraction of martensite formed by plastic straining, have been compared with the predictions derived from the constitutive laws proposed by Iwamoto and Tsuta [Iwamoto, T., Tsuta, T., 2000. Computational simulation of the dependence of the austenitic grain size on the deformation behaviour of trip steels. Int. J. Plasticity 16, 791–804; Iwamoto, T., Tsuta T., 2002. Computational simulation on deformation behaviour of CT specimens of trip steel under mode I loading for evaluation of fracture toughness. Int. J. Plasticity 18, 1583–1606] for steels exhibiting transformation induced plasticity, and from a flow localisation analysis developed along the lines of the model of Marciniak and Kuczinski [Marciniak, Z., Kuczynski, K., 1967. Limit strains in the processes of stretch-forming sheet metal. Int. J. Mech. Sci. 9, 609–620]. A good account of the whole results can be obtained by considering the limits to ductility imposed by ductile fracture.     

Path-dependent failure of inflated aluminum tubes

Our recent investigation on the formability of Al alloy tubes under combined internal pressure and axial load is expanded by examining the effect of the loading path traced. A set of Al-6260-T4 tubes were loaded along orthogonal stress paths to failure and the results are compared to those of the corresponding radial paths. It is confirmed that failure strains are path-dependent, but also is demonstrated that failure stresses become path-dependent if the prestrain is significant. The experiments are simulated using the previously developed finite element models and the calibration of the Yld2000-2D [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets-part I: theory. Int. J. Plasticity 19, 1297--1319] anisotropic yield function shown in [Korkolis, Y.P., Kyriakides, S., 2008b. Inflation and burst of anisotropic aluminum tubes. Part II: an advanced yield function including deformation-induced anisotropy. Int. J. Plasticity 24, 1625–1637] to yield accurate predictions of rupture for nine radial paths. The models are shown to reproduce the path dependence of the failure stresses and strains quite well. A group of additional radial and corner paths are subsequently examined numerically to enrich the existing data on path-dependence of failure. It is again shown that the amount of plastic prestraining in either of the two directions influences the difference of the failure stresses and strains between the radial and the corner stress paths.     

Orthotropic yield criteria for description of the anisotropy in tension and compression of sheet metals

In this paper, yield functions describing the anisotropic behavior of textured metals are proposed. These yield functions are extensions to orthotropy of the isotropic yield function proposed by Cazacu et al. (Cazacu, O., Plunkett, B., Barlat, F., 2006. Orthotropic yield criterion for hexagonal close packed metals. Int. J. Plasticity 22, 1171–1194). Anisotropy is introduced using linear transformations of the stress deviator. It is shown that the proposed anisotropic yield functions represent with great accuracy both the tensile and compressive anisotropy in yield stresses and r-values of materials with hcp crystal structure and of metal sheets with cubic crystal structure. Furthermore, it is demonstrated that the proposed formulations can describe very accurately the anisotropic behavior of metal sheets whose tensile and compressive stresses are equal.     

A finite strain, finite band method for modeling ductile fracture

We present a finite deformation generalization of the finite thickness embedded discontinuity formulation presented in our previous paper [A.E. Huespe, A. Needleman, J. Oliver, P.J. Sánchez, A finite thickness band method for ductile fracture analysis, Int. J. Plasticity 25 (2009) 2349-2365]. In this framework the transition from a weak discontinuity to a strong discontinuity can occur using a single constitutive relation which is of importance in a range of applications, in particular ductile fracture, where localization typically precedes the creation of new free surface. An embedded weak discontinuity is introduced when the loss of ellipticity condition is met. The resulting localized deformation band is given a specified thickness which introduces a length scale thus providing a regularization of the post-localization response. The methodology is illustrated through several example problems emphasizing finite deformation effects including the development of a cup-cone failure in round bar tension.     

Forming of AA5182-O and AA5754-O at elevated temperatures using coupled thermo-mechanical finite element models

A temperature-dependent anisotropic material model was developed for two aluminum alloys AA5182-O and AA5754-O and their anisotropy parameters were established. A coupled thermo-mechanical finite element analysis of the forming process was then performed for the temperature range 25–260 °C (77–500 °F) at different strain rates. In the developed model, the anisotropy coefficients for Barlat’s YLD2000-2d anisotropic yield function [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets – Part 1: Theory. Int. J. Plasticity 19, 1297–1319] in the plane-stress condition and the parameters for the isotropic strain hardening were established as a function of temperature. The temperature-dependent anisotropic yield function was then implemented into the commercial FEM code LS-DYNA as a user material subroutine (UMAT) using the cutting-plane algorithm for the integration of a general class of elastoplastic constitutive models [Abedrabbo, N., Pourboghrat, F., Carsley, J., 2006b. Forming of aluminum alloys at elevated temperatures – Part 2: Numerical modeling and experimental verification. Int. J. Plasticity 22 (2), 342–737]. The temperature-dependent material model was used to simulate the coupled thermo-mechanical finite element analysis of the stamping of an aluminum sheet using a hemispherical punch under the pure stretch boundary condition (no material draw-in was allowed). Simulation results were compared with experimental data at several elevated temperatures to evaluate the accuracy of the UMAT’s ability to predict both forming behavior and failure locations. Two failure criteria were used in the analysis; the M–K strain based forming limit diagrams (ε-FLD), and the stress based forming limit diagrams (σ-FLD). Both models were developed using Barlat’s YLD2000-2d anisotropic model for the two materials at several elevated temperatures. Also, as a design tool, the Genetic Algorithm optimization program HEEDS was linked with the developed thermo-mechanical models and used to numerically predict the “optimum” set of temperatures that would generate the maximum formability for the two materials in the pure stretch experiments. It was found that a higher temperature is not needed to form the part, but rather the punch should be maintained at the lowest temperature possible for maximum formability.     

Ideal sheet forming with frictional constraints

The ideal forming theory, previously developed as a direct design method to guide iterative design practices based on analytic methods, does not properly account for frictional effects prevalent in real forming. A method for introducing the effects of the Coulomb friction in the ideal sheet forming theory was developed in this work by modifying the extremum work criterion. For numerical implementation, a rigid-plastic finite element method was used based on triangular membrane elements. Computational results were compared with experiments given in NUMISHEET'93 using a planar anisotropic strain rate potential proposed by Barlat et al. (Barlat, F., Chung, K., Richmond, O., 1993. Strain rate potential for metals and its application to minimum plastic work path calculations. Int. J. Plasticity 9, 51.) and the Coulomb friction model.     

Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets: Application to sheet springback

The constitutive model for the unusual asymmetric hardening behavior of magnesium alloy sheet presented in a companion paper (Lee, M.G., Wagoner, R.H., Lee, J.K., Chung, K., Kim, H.Y., 2008. Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheet, Int. J. Plasticity 24(4), 545–582) was applied to the springback prediction in sheet metal forming. The implicit finite element program ABAQUS was utilized to implement the developed constitutive equations via user material subroutine. For the verification purpose, the springback of AZ31B magnesium alloy sheet was measured using the unconstrained cylindrical bending test of Numisheet (Numisheet ’2002 Benchmark Problem, 2002. In: Yang, D.Y., Oh, S.I., Huh, H., Kim, Y.H. (Eds.), Proceedings of 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Jeju, Korea) and 2D draw bend test. With the specially designed draw bend test the direct restraining force and long drawn distance were attainable, thus the measurement of the springback could be made with improved accuracy comparable with conventional U channel draw bend test. Besides the developed constitutive models, other models based on isotropic constitutive equations and the Chaboche type kinematic hardening model were also considered. Comparisons were made between simulated results by the finite element analysis and corresponding experiments and the newly proposed model showed enhanced prediction capability, which was also supported by the simple bending analysis adopting asymmetric stress–strain response.     

Analytic plastic potential for porous aggregates with matrix exhibiting tension-compression asymmetry

This paper is devoted to modeling the effects of the tension-compression asymmetry of the matrix on yielding of the void-matrix aggregate. The matrix plastic behavior is described by the Cazacu et al. [2006. Orthotropic yield criterion for hexagonal closed packed metals. Int. J. Plasticity 22, 1171-1194] isotropic yield criterion, which captures strength differential effects. Using an upper-bound approach, a new analytic isotropic plastic potential for a random distribution of spherical voids is obtained. The derived analytic potential is sensitive to the third invariant of the stress deviator and displays tension-compression asymmetry. In the case when the matrix material has the same yield in tension and compression, it reduces to Gurson's [1977. Continuum theory of ductile rupture by void nucleation and growth: Part I: Yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. Trans. ASME Ser. H 99, 2-15.] criterion. Furthermore, the proposed criterion predicts the exact solution of a hollow sphere loaded in hydrostatic tension or compression. The accuracy of the proposed analytical criterion is assessed through comparisons with finite-element cell calculations.     

The mechanics of active clays circulated by salts, acids and bases

A model that accounts for electro-chemo-mechanical couplings in clays, due to the presence of dissolved salts and acids and bases, is developed and applied to simulate experimental data. Chemically sensitive clays are viewed as two-phase multi-species saturated porous media circulated by an electrolyte. To the authors’ best knowledge, no other comprehensive project to embody the effects of pH in the elastic-plastic behavior of geomaterials has been attempted so far.The developments are embedded in the framework of the thermodynamics of multi-phase multi-species porous media. This approach serves to structure the model, and to motivate constitutive equations. The present extension capitalizes upon the earlier developments by Gajo et al. [2002. Electro-chemo-mechanical couplings in saturated porous media: elastic-plastic behaviour of heteroionic expansive clays. Int. J. Solids Struct. 39, 4327-4362] and Gajo and Loret [2004. Transient analysis of ionic replacement in elastic-plastic expansive clays. Int. J. Solids Struct. 41(26), 7493-7531], which were devoted to modeling chemo-mechanical couplings at constant pH.Four transfer mechanisms between the solid and fluid phases are delineated in the model: (1) hydration, (2) ion exchange, (3) acidification, (4) alkalinization. Thus all fundamental exchanges at particle level are fully taken into account. Only mineral dissolution is neglected, since experimental observations indicate a negligible role of mineral dissolution for active clays at room temperature. In particular, the newly considered mechanisms of acidification and alkalinization directly affect the electrical charge of clay particles and thus have a key role in the electro-chemo-mechanical couplings. These four mechanisms are seen as controlling both elastic and elasto-plastic behaviors. Depending on concentrations and ionic affinities to the clay mineral, the transfer mechanisms either compete or cooperate to modify the compressibility and strength of the clay. At given stress, they induce swelling (volume expansion) or shrinking (volume contraction).The framework is sufficiently rich to allow for the simulations of recently performed laboratory experiments on clay samples submitted to intertwined mechanical and chemical loading programmes, involving large modifications in ionic strengths and pH, and leading to significant changes in volume, stiffness and strength.     

Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function

In this work, the recently proposed anisotropic yield function, Yld2004-18p [Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., 2005. Linear transformation based anisotropic yield function, Int. J. Plasticity 21, 1009], is implemented in a finite element (FE) code for application to the cup drawing simulation of a circular blank sheet. A short review of the Yld2004-18p relevant features is provided and the stress integration scheme for its implementation in FE codes is described. The simulation of the drawing process is conducted for an aluminum alloy sheet sample (AA2090-T3). The predicted and experimental cup height profiles (earing profiles) with six ears are shown to be in excellent agreement. Additional simulations on a ficticious material are performed in order to show that the yield function Yld2004-18p can lead to the prediction of cups with eight ears. In order to achieve these results, a sufficient number of input data are required to calculate the yield function coefficients. Finally, a simplified analytical approach that relates the earing profile to the r-value directionality is also presented in this paper. It is shown that this approach can be very useful as a first approximation of the earing profile of drawn cups.