During hot extrusion of copper alloys, extrusion tools have to withstand cyclic thermal and mechanical loads. To enhance the service life of the tools, materials with high temperature strength are designed as well as an optimised process control is performed. To characterise the tool damage evolution during service and to improve process guiding, modelling and simulation are appropriate means. The extrusion process of copper billets at three different temperatures was simulated by an FE-program to obtain the temporal boundary conditions, i.e. stress and temperature distributions at the interface billet-liner. Those boundary conditions were used to simulate the elastic-viscoplastic behaviour of the tool steel Bohler W750 in service by means of Abaqus Standard (TM) v.6.8-3 software in conjunction with Z-Mat package. A lifetime rule was added in order to compute the lifetime consumption and the cycles to failure. (C) 2010 Elsevier BM. All rights reserved.

A hierarchy of higher order continua is presented that introduces additional degrees of freedom accounting for volume changes, rotation and straining of an underlying microstructure. An increase in the number of degrees of freedom represents a refinement of the material description. In addition to available nonlinear Cosserat and micromorphic theories, general formulations of elastoviscoplastic behaviour are proposed for microdilatation and microstretch continua. A microstrain theory is introduced that is based on six additional degrees of freedom describing the pure straining of the microstructural element. In each case, balance equations and boundary conditions are derived, decompositions of the finite strain measures into elastic and plastic parts are provided. The formulation of finite deformation elastoviscoplastic constitutive equations relies on the introduction of the free energy and dissipation potentials, thus complying with requirements of continuum thermodynamics. Some guidelines for the selection of a suitable higher order model for a given material close the discussion.

This work deals with the formulation of a three-dimensional crystallographic time-incremental lifetime rule for face-centered cubic (fcc) single crystals used for gas-turbine blade applications. The damage contribution rate of each slip system to the total damage is governed by the current values of the resolved shear stress and the slip rate on the corresponding slip system. The damage rule is combined with a crystallographic viscoplastic deformation model. For the nickel-base single-crystal superalloy CMSX4 at 950°C, various strain- and stress-controlled uniaxial cyclic tests with and without hold-times can be described for different crystal orientations by one set of material parameters. For verification, simulation results for a single-crystal specimen with a notch have been compared with corresponding experimental results. The predicted lifetime is within the factor of two of the measured one.

In this work, the influence of the development of geometrically necessary dislocations (GNDs) at a crack tip in single crystals on the hardening and crack propagation behaviour is investigated. In particular, we are interested in examining the effect of such additional hardening on the development of glide and kink bands at the crack tip as well as on the process of crack opening. To this end, following Nye and many others, local deformation incompatibility in the material is adopted as a measure of the density of GNDs. Their development results in additional energy being stored in the material, leading to additional kinematic-like hardening. A thermodynamic formulation of the model in the context of the dissipation principle facilitates the derivation of the corresponding hardening relation. Results suggest that this additional hardening retards kink-band development, but has little or no influence on glide-band development. It also influences the crack tip opening displacement (CTOD). It turns out that the simulated CTOD correlates well with experimentally determined crack-propagation rates for different crack growth direction in the crystal.

Laird and Smith [(1962). Philosophical Magazine 8, 847–857] proposed a plastic sliding-off mechanism for the stage II fatigue crack growth via striation formation. In their view, the fatigue crack extension results solely from the changing character of deformation at the crack tip during loading and unloading. In particular, the crack tip blunts during the loading stage and folds into a double notch during the unloading stage, resulting in striation formation. In order to verify Laird’s plastic blunting mechanism for ductile polycrystals as well as for ductile fcc single crystals, FE calculations were performed for a rectangular plate with an initially sharp crack under plane strain conditions. The plate was subjected to a fully reversed tension-to-pressure cyclic load perpendicular to the crack plane (Mode 1). In the single crystal case the crack propagation simulations were carried out for cracks with crack plane (001) for two different crack growth orientations [110] and [100]. No initial radius for the crack tip was assumed. The actual shape of the crack tip followed from an initially sharp crack by repeated remeshing. To model the constitutive behavior typical for polycrystalline ductile metals, J2 hypo-elasto-plasticity model with Armstrong–Frederick kinematic hardening was used. To model the constitutive behavior typical for ductile fcc single crystals, a geometrically nonlinear version of Cailletaud’s model based on the multiplicative elasto-plastic decomposition of the deformation gradient was implemented into the FE program ABAQUS. For simplicity, only octahedral slip systems were considered. Using repeated remeshing for severely distorted elements at the advancing crack tip, deformation patterns in the sense of Laird’s mechanism for fatigue crack propagation with striation formation were obtained in the case of the polycrystal simulation as well as in the case of the single crystal simulation for [110] crack growth direction. The simulation for [100] crack growth direction with the same stress level as for [110] direction also yielded crack extension by progressive large deformations but without striation formation. The dependence of the fatigue striation formation on the crack growth direction as predicted by the simulation of crack propagation in single crystals is verified by the experimental results of Neumann [(1974). Acta Metallurgica 22, 1155–1165] on pure copper single crystals.

A unifying thermomechanical constitutive framework for generalized continua including additional degrees of freedom or/and the second gradient of displacement is presented. Based on the analysis of the dissipation, state laws, flow rules and evolution equations are proposed for Cosserat, strain gradient and micromorphic continua. The case of the gradient of internal variable approach is also incorporated by regarding the nonlocal internal variable as an actual additional degree of freedom. The consistency of the continuum thermodynamical framework is ensured by the introduction of a viscoplastic pseudo–potential of dissipation, thus extending the classical class of so–called standard material models to generalized continua.

Theaim of homogenizationmethodsis to replace a heterogenous materialby a homogenous equivalent one endowed with effective properties. The now well-establishedresultsof classicalhomogenizationtheory [1] aregener ally basedon theassumptionof slowly varying meanfields [2]. It means that the typical sizeof the heterogeneitiesis muchsmallerthan the wave-length Lw of the applied loading conditions. If this hypothesisis dropped or, equivalently, when the heterogenous material is subjectedto strong overall deformation gradients, homogeneous substitutionmedia can also be constructedbut they usually are generalizedcontinua [3,4,5]. In particular, it canbeadequatelydescribedby a secondgrade medium as proposedin [6,7], which requires the useof the first and secondgradients of thedisplacementfield u [8,9]. Strong overall deformationgradientscanalsobeinducedby strongly non-homogeneous temper aturefields. Let us considerfor instancea periodically perforatedthermoelasticplateunderplanestrainconditionssubjectedto aprescribedbilinear temper aturefield (figure1) andto stringentboundaryconditions at thebottom. The rapid variationof the meandeformationfrom cell to cell canbe seenon figure 1. However figure3ashows thattheperforatedplateunder suchconditions canbesuccessfully replaced by a homogeneousthermoelasticmediumendowedwith theeffective thermoelastic properties according to classical periodic homogenization. If the wave length of the temper aturevariation is notmuchbiggerthanthecell size(likefor thesinusoidal temperature field of figure 2), the solutionpredicted by the classicalhomogeneous mediumis seento depart from theactualresponse(figure 3b).

Article Strain Gradient Crystal Plasticity: Thermomechanical Formulations and Applications was published on August 1, 2002 in the journal Journal of the Mechanical Behavior of Materials (volume 13, issue 3-4).

The asymptotic stress-strain field at a stationary crack tip in elastic-ideally plastic f.c.c. and b.c.c. single crystals, as determined by Rice et al. [1], turns out to be locally constant within angular sectors. It involves shear displacement discontinuities at sector boundaries, that can be interpreted as strain localization bands. The numerical analysis of the same problem using finite strain crystal elastoplasticity in [2] reveals that the condition of constant stress state in each sector must be relieved because of possible local unloading, but also that the strain localization patterns pertain. Discrete models based on dislocation dynamics also lead to strongly localized dislocation distributions near the crack tip and to the progressive formation of the predicted sectors [3]. Experimental observations in a b.c.c. single crystal in [4] confirms the existence of such intense deformation bands radiating from the crack tip. The fact that the strong strain gradients developing in the vicinity of the crack tip may affect the local mechanical response of a crystalline solid suggests that generalized continuum theories, including Cosserat, strain gradient and non local models, could be helpful for computing more realistic local stress-strain fields. The strain gradient model used in [5] results in a substantial increase of the tractions ahead of the tip of a mode I crack within a domain of characteristic size related to the constitutive intrinsic length. This monitoring of the local strain field enables one to improve the prediction of subsequent crack growth. The use of a generalized continuum model also strongly affects the localized deformation modes as demonstrated in [6] in the case of single crystals. In particular, classical crystal plasticity theory predicts two types of deformation bands in single crystals undergoing single slip:slip bands lying in the slip plane of the locally activated slip system, or kink bands lying in a plane normal to the slip direction of the slip system [7]. The formation of a kink band is associated with the development of strong lattice rotation gradients at its boundary and may therefore be precluded if the model incorporates additional hardening due to lattice curvature [6]. The present work investigates the effect of a generalized continuum theory on the localized deformation patterns arising at the crack tip in elastoplastic f.c.c. single crystals. Important consequences are expected regarding crack branching after stable crack growth. Scripta mater. 44 (2001) 953–958

Earthquake faults are the result of severe strain localization in rocks deep down in the earth's crust. This localized deformation is controlled by the size of the microstructure and various Thermo-Hydro-Mechanical (THM) couplings, whose modeling is central for understanding earthquake nucleation and seismic energy release. We model this challenging system using the Cosserat theory and by considering large shear deformations during seismic slip. Our numerical results justify the rate and state friction law (Dieterich, 1992; Ruina, 1983a), which describes empirically the fault tribology. This shows the importance of the underlying physics of our model for fault friction. In our analyses traveling shear bands along the thickness of the fault are present, leading to oscillations in the fault's frictional response. Existing numerical analyses presented in Rattez et al. (2018a, 2018b, 2018c) do not capture this behavior, which goes beyond the established models of uniform shear (Lachenbruch, 1980) and shear on a mathematical plane (Rice, 2006a). Recent experimental results, that insulate thermal pressurization from other weakening mechanisms (Badt et al., 2020), corroborate our numerical results. Our results motivate us to extend the classical model of thermal pressurization in Mase & Smith (1987) and Rice (2006b) to incorporate different strain localization modes, temperature and pore fluid pressure boundary conditions.In particular, we start our analysis by using the normalized coupled system of partial differential equations that include the THM couplings for the case of a Cosserat continuum. We then perform a bifurcation analysis, which indicates that traveling shear bands are possible inside the fault gouge. Next, we derive our non linear mesh independent numerical results accounting for the influence of large displacements by using and Adaptive Lagrangian Eulerian (ALE) procedure. We introduce viscosity in our numerical analyses for the rate and state phenomenology to emerge. We corroborate our numerical results comparing them to similar laboratory experiments. Furthermore, we modify the Volterra integral equation of the classical model of thermal pressurization in Mase & Smith (1987) and Rice (2006b), which we solve by a semi analytical procedure, in order to capture the effect of isothermal drained boundary conditions and traveling shear bands. Our results reappraise and extend the established models of frictional weakening due to thermal pressurization during coseismic slip.

The method of virtual power and continuum thermodynamics are used to incorporate temperature and temperature gradients into the theory of second grade solids settled by [Germain, 1973] in the isothermal case. In a second part, it is shown that heterogeneous classical materials submitted to slowly-varying mean fields can be replaced by a homogeneous equivalent medium including higher order gradients of displacement and temperature. For that purpose, an asymptotic analysis of thermoelastic heterogeneous periodic materials is performed. The form of the derived effective properties are compared to the previous phenomenological framework.

This work is concerned with the formulation of hyperelastic-thermodynamic-based models for associated elastoplasticity with non-linear isotropic and kinematic hardening valid for both large elastic and large plastic deformation. On this basis, one can then introduce explicitly the assumptions of (1), small incremental plastic deformation, and (2), small elastic strain, into the general model and obtain special cases whose behaviour corresponds to that of various classical hypoelastic formulations. In particular, these are obtained on the basis of two different thermodynamic formulations for kinematic hardening with respect to the intermediate configuration. The simplest of these, in which the plastic part of the free energy does not depend explicitly on the plastic deformation, leads for example to Jaumann-or Green-Naghdi-hypoelastic-type behaviour for linear kinematic hardening in simple shear. In particular, the former case is obtained in this context when the plastic spin is assumed constant and equal to zero, and the latter case when the plastic rotation is assumed constant and equal to the identity. Allowing the plastic part of the free energy to depend explicitly on the plastic deformation yields the second thermodynamic model for kinematic hardening considered in this work. Here, again in the special case of linear hardening, Oldroyd-like behaviour for the shear stress and back stress, but not for the normal stress, is obtained in simple shear.

Recently, several proposals were made for the enlargement of the classical field equations in order to solve locally inhomogeneous deformation problems (e.g. at shear banding and damage localization or of composites). In the present paper basic issues of this topic will be treated: i) the implicit dependence within the gradient plasticity theory, ii) the more open micromorphic view-point of the gradient of internal variable approach, iii) a derivation of an elastic-plastic decomposition of the Cosserat strain measures, iv) its application to the description of lattice curvature in crystals. Finally, finite element simulations of strain localization in Cosserat single crystals are presented.

In this note, we generalize the usual algorithmic formulation of large-deformation, associated, multiplicative hyperelastoplasticity to the case when the plastic spin is not assumed to be zero. In particular, starting from the general backward-Euler exponential algorithmic form for the evolution of the plastic deformation, we show that, in the context of small incremental plastic deformations, and small elastic strains, one obtains algorithmic update relation for both the elastic strain and elastic rotation, the latter depending upon the plastic spin.

IN THIS WORK, displacement and lattice rotation are regarded as independent degrees of freedom. They are connected only on the constitutive level and by the balance equations. The description of plastic deformation is based on the slip theory. Elastic lattice curvature and torsion are associated with couple-stresses. The continuum theory of dislocations has been revisited to derive the kinematics of plastic lattice torsion-curvature. Explicit constitutive equations and hardening rules are proposed to close the theory in the case of elastoviscoplasticity. The thermodynamical formulation of the model involves internal variables which are similar to the densities of statistically stored dislocations and the densities of geometrically necessary dislocations. Accordingly, the proposed Cosserat theory can be regarded, on the one hand, as the classical crystal plasticity theory complemented by lattice curvature and torsion variables and, on the other hand, as the continuum theory of dislocations closed by the missing hardening variables and constitutive equations within the appropriate micropolar framework. A generalization of Mandel's elastoviscoplastic decomposition of strain is used especially for the torsion-curvature measure at finite deformation.