Strain gradient plasticity analysis of elasto-plastic contact between rough surfaces

From a microscopic point of view, the real contact area between two rough surfaces is the sum of the areas of contact between facing asperities. Since the real contact area is a fraction of the nominal contact area, the real contact pressure is much higher than the nominal contact pressure, which results in plastic deformation of asperities. As plasticity is size dependent at size scales below tens of micrometers, with the general trend of smaller being harder, macroscopic plasticity is not suitable to describe plastic deformation of small asperities and thus fails to capture the real contact area and pressure accurately. Here we adopt conventional mechanism-based strain gradient plasticity (CMSGP) to analyze the contact between a rigid platen and an elasto-plastic solid with a rough surface. Flattening of a single sinusoidal asperity is analyzed first to highlight the difference between CMSGP and J₂ isotropic plasticity. For the rough surface contact, besides CMSGP, pure elastic and J₂ isotropic plasticity analysis is also carried out for comparison. In all cases, the contact area A rises linearly with the applied load, but with a different slope which implies that the mean contact pressures are different. CMSGP produces qualitative changes in the distributions of local contact pressures compared with pure elastic and J₂ isotropic plasticity analysis, furthermore, bounded by the two.     

Grain size effects in multiphase steels assisted by transformation-induced plasticity

The influence of the austenitic grain size on the overall stress–strain behavior in a multiphase carbon steel is analyzed through three-dimensional finite element simulations. A recently developed multiscale martensitic transformation model is combined with a plasticity model to simulate the transformation-induced plasticity effects of a grain of retained austenite embedded in a ferrite-based matrix. Grain size effects are included via a surface energy term in the Helmholtz energy. Tensile simulations for representative orientations of the grain of retained austenite show that the initial stability of the austenite increases as the grain size decreases. Consequently, the effective strength is initially higher for smaller grains. The influence of the grain size on the evolution of the transformation process strongly depends on the grain orientation. For “hard” orientations, the transformation rate is higher for larger grains. In addition, the phase transformation is partially suppressed as the grain size decreases. In contrast, for “soft” orientations, the transformation rate is lower for larger grains. The phase transformation is more homogeneous for smaller grains and, consequently, the effective transformation strain is larger. Nevertheless, in multiphase carbon steels with a relatively low percentage of retained austenite, the influence of the austenitic grain size on the overall constitutive response is smaller than the influence of the austenitic grain orientation.     

Crystallographically based model for transformation-induced plasticity in multiphase carbon steels

The microstructure of multiphase steels assisted by transformation-induced plasticity consists of grains of retained austenite embedded in a ferrite-based matrix. Upon mechanical loading, retained austenite may transform into martensite, as a result of which plastic deformations are induced in the surrounding phases, i.e., the ferrite-based matrix and the untransformed austenite. In the present work, a crystallographically based model is developed to describe the elastoplastic transformation process in the austenitic region. The model is formulated within a large-deformation framework where the transformation kinematics is connected to the crystallographic theory of martensitic transformations. The effective elastic stiffness accounts for anisotropy arising from crystallographic orientations as well as for dilation effects due to the transformation. The transformation model is coupled to a single-crystal plasticity model for a face-centered cubic lattice to quantify the plastic deformations in the untransformed austenite. The driving forces for transformation and plasticity are derived from thermodynamical principles and include lower-length-scale contributions from surface and defect energies associated to, respectively, habit planes and dislocations. In order to demonstrate the essential features of the model, simulations are carried out for austenitic single crystals subjected to basic loading modes. To describe the elastoplastic response of the ferritic matrix in a multiphase steel, a crystal plasticity model for a body-centered cubic lattice is adopted. This model includes the effect of nonglide stresses in order to reproduce the asymmetry of slips in the twinning and antitwinning directions that characterizes the behavior of this type of lattices. The models for austenite and ferrite are combined to simulate the microstructural behavior of a multiphase steel. The results of the simulations show the relevance of including plastic deformations in the austenite in order to predict a more realistic evolution of the transformation process. This work is part of the research program of the Netherlands Institute for Metals Research (NIMR) and the Stichting voor Fundamenteel Onderzoek der Materie (FOM, financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)). The research was carried out under project number 02EMM20 of the FOM/NIMR program “Evolution of the Microstructure of Materials” (P-33).     

Strain gradient plasticity effects in whisker-reinforced metals

A metal reinforced by fibers in the micron range is studied using the strain gradient plasticity theory of Fleck and Hutchinson (J. Mech. Phys. Solids 49 (2001) 2245). Cell-model analyses are used to study the influence of the material length parameters numerically, for both a single parameter version and the multiparameter theory, and significant differences between the predictions of the two models are reported. It is shown that modeling fiber elasticity is important when using the present theories. A significant stiffening effect when compared to conventional models is predicted, which is a result of a significant decrease in the level of plastic strain. Moreover, it is shown that the relative stiffening effect increases with fiber volume fraction. The higher-order nature of the theories allows for different higher-order boundary conditions at the fiber-matrix interface, and these boundary conditions are found to be of importance. Furthermore, the influence of the material length parameters on the stresses along the interface between the fiber and the matrix material is discussed, as well as the stresses within the elastic fiber which are of importance for fiber breakage.     

Strain gradient plasticity,strengthening effects and plastic limit analysis

Within the framework of isotropic strain gradient plasticity, a rate-independent constitutive model exhibiting size dependent hardening is formulated and discussed with particular concern to its strengthening behavior. The latter is modelled as a (fictitious) isotropic hardening featured by a potential which is a positively degree-one homogeneous function of the effective plastic strain and its gradient. This potential leads to a strengthening law in which the strengthening stress, i.e. the increase of the plastically undeformed material initial yield stress, is related to the effective plastic strain through a second order PDE and related higher order boundary conditions. The plasticity flow laws, with the role there played by the strengthening stress, are addressed and shown to admit a maximum dissipation principle. For an idealized elastic perfectly plastic material with strengthening effects, the plastic collapse load problem of a micro/nano scale structure is addressed and its basic features under the light of classical plastic limit analysis are pointed out. It is found that the conceptual framework of classical limit analysis, including the notion of rigid-plastic behavior, remains valid. The lower bound and upper bound theorems of classical limit analysis are extended to strengthening materials. A static-type maximum principle and a kinematic-type minimum principle, consequences of the lower and upper bound theorems, respectively, are each independently shown to solve the collapse load problem. These principles coincide with their respective classical counterparts in the case of simple material. Comparisons with existing theories are provided. An application of this nonclassical plastic limit analysis to a simple shear model is also presented, in which the plastic collapse load is shown to increase with the decreasing sample size (Hall–Petch size effects).     

Strain gradient effects on cyclic plasticity

Size effects on the cyclic shear response are studied numerically using a recent higher order strain gradient visco-plasticity theory accounting for both dissipative and energetic gradient hardening. Numerical investigations of the response under cyclic pure shear and shear of a finite slab between rigid platens have been carried out, using the finite element method. It is shown for elastic-perfectly plastic solids how dissipative gradient effects lead to increased yield strength, whereas energetic gradient contributions lead to increased hardening as well as a Bauschinger effect. For linearly hardening materials it is quantified how dissipative and energetic gradient effects promote hardening above that of conventional predictions. Usually, increased hardening is attributed to energetic gradient effects, but here it is found that also dissipative gradient effects lead to additional hardening in the presence of conventional material hardening. Furthermore, it is shown that dissipative gradient effects can lead to both an increase and a decrease in the dissipation per load cycle depending on the magnitude of the dissipative length parameter, whereas energetic gradient effects lead to decreasing dissipation for increasing energetic length parameter. For dissipative gradient effects it is found that dissipation has a maximum value for some none zero value of the material length parameter, which depends on the magnitude of the deformation cycles.     

Analysis of size effects associated to the transformation strain in TRIP steels with strain gradient plasticity

The size dependent strengthening resulting from the transformation strain in Transformation Induced Plasticity (TRIP) steels is investigated using a two-dimensional embedded cell model of a simplified microstructure composed of small cylindrical metastable austenitic inclusions within a ferritic matrix. Earlier studies have shown that within the framework of classical plasticity or of the single length parameter Fleck–Hutchinson strain gradient plasticity theory, the transformation strain has no significant impact on the overall strengthening. The strengthening is essentially coming from the composite effect with a marked inclusion size effect resulting from the appearance during deformation of new boundaries constraining the plastic flow. The three parameters version of the Fleck–Hutchinson strain gradient plasticity theory is used here in order to better capture the effect of the plastic strain gradients resulting from the transformation strain. The three parameters theory incorporates separately the rotational and extensional gradients in the formulation, which leads to a significant influence of the shear component of the transformation strain, not captured by the single-parameter theory. When the size of the austenitic inclusions decreases, the overall strengthening increases due to a combined size dependent effect of the transformation strain and of the evolving composite structure. A parametric study is proposed and discussed in the light of experimental evidences giving indications on the optimization of the microstructure of TRIP-assisted multi-phase steels.     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

Strain gradient crystal plasticity analysis of a single crystal containing a cylindrical void

The effects of void size and hardening in a hexagonal close-packed single crystal containing a cylindrical void loaded by a far-field equibiaxial tensile stress under plane strain conditions are studied. The crystal has three in-plane slip systems oriented at the angle 60° with respect to one another. Finite element simulations are performed using a strain gradient crystal plasticity formulation with an intrinsic length scale parameter in a non-local strain gradient constitutive framework. For a vanishing length scale parameter the non-local formulation reduces to a local crystal plasticity formulation. The stress and deformation fields obtained with a local non-hardening constitutive formulation are compared to those obtained from a local hardening formulation and to those from a non-local formulation. Compared to the case of the non-hardening local constitutive formulation, it is shown that a local theory with hardening has only minor effects on the deformation field around the void, whereas a significant difference is obtained with the non-local constitutive relation. Finally, it is shown that the applied stress state required to activate plastic deformation at the void is up to three times higher for smaller void sizes than for larger void sizes in the non-local material.     

Strain gradient plasticity by internal-variable approach with normality structure

This paper presents a constitutive formulation for materials with strain gradient effects by internal-variable approach with normality structure. Specific micro-structural rearrangements are assumed to account for the inelasticity deformations for this class of materials, and enter the constitutive formulations in form of internal variables. It is further assumed that the kinetic evolution of any specific micro-structural rearrangement may be fully determined by the thermodynamic forces associated with that micro-structural rearrangement, by normality relations via a flow potential. Macroscopic gradient-enhanced inelastic behaviours may then be predicted in terms of the microscopic internal variables and their conjugate forces, and thus a micro–macro bridging formulation is available for strain-gradient-characterised materials. The obtained formulations are first applied to crystallographic materials, and a crystal gradient plasticity model is developed to account for the influence of microscopic slip rearrangements on the macroscopic gradient-dependent mechanical behaviour for this class of materials. Micro-cracked geomaterials are also treated with these formulations and a gradient-enhanced damage constitutive model is developed to address the impacts of the evolutions of micro-cracks on the macroscopic inelastic deformations with strain gradient effects for these materials. The available formulations are further compared with other thermodynamic approaches of constitutive developing.     

Experimental analysis of the correlation between martensitic transformation plasticity and the austenitic grain size in steels

This work investigates the effect of the Austenite Grain Size (AGS) on the thermo-mechanical behavior of 35NiCrMo16 steel during the martensitic transformation under complex loading. Particularly in view of some thermal cycles, we analyzed the evolution of the austenite grain size under different austenitizing conditions. Using the same temperature-holding time parameters, we performed a series of experiments to assess the Transformation-Induced Plasticity (TRIP) under uniaxial (tension or torsion) and biaxial (tension + torsion) loading. The results suggest that the uniaxial torsion loading case, Transformation-Induced Plasticity does not depend on the prior Austenite Grain Size (AGS) whereas, in the uniaxial tension loading case, it is a “slightly” increasing function of the AGS.     

Experimental analysis of transformation plasticity

Transformation plasticity is an irreversible strain observed when metallurgical transformation occurs under small external stress lower than the yield stress of the weaker phase. This paper is devoted to an experimental analysis of that phenomenon in ferrous alloys. The particular case of the steel which composes vessels in French nuclear reactors is considered. This steel is called 16 MND 5 and A 533 respectively in AFNOR and ASTM norms. The main results proposed in literature concerning transformation plasticity on this material were analyzed and perfectible aspects are pointed out. The results of several performed tests are analyzed. The transformation plasticity coefficient, the kinetics as well as the dependence on the norm and the direction of the applied stress are particularly studied. The experimental results are compared to the predictions of the main existing models.     

Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures

Two recently proposed Helmholtz free energy potentials including the full dislocation density tensor as an argument within the framework of strain gradient plasticity are used to predict the cyclic elastoplastic response of periodic laminate microstructures. First, a rank-one defect energy is considered, allowing for a size-effect on the overall yield strength of micro-heterogeneous materials. As a second candidate, a logarithmic defect energy is investigated, which is motivated by the work of Groma et al. (2003). The properties of the back-stress arising from both energies are investigated in the case of a laminate microstructure for which analytical as well as numerical solutions are derived. In this context, a new regularization technique for the numerical treatment of the rank-one potential is presented based on an incremental potential involving Lagrange multipliers. The results illustrate the effect of the two energies on the macroscopic size-dependent stress–strain response in monotonic and cyclic shear loading, as well as the arising pile-up distributions. Under cyclic loading, stress–strain hysteresis loops with inflections are predicted by both models. The logarithmic potential is shown to provide a continuum formulation of Asaro's type III kinematic hardening model. Experimental evidence in the literature of such loops with inflections in two-phased FFC alloys is provided, showing that the proposed strain gradient models reflect the occurrence of reversible plasticity phenomena under reverse loading.     

Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases

Transformation plasticity in steels (i.e., the anomalous plastic flow observed during the progress of a phase transformation) is usually attributed to two distinct physical mechanisms, which have been proposed by Greenwood and Johnson and Magee. This paper proposes a theoretical approach to the problem, in the case where the Magee mechanism is negligible and the phases are ideal-plastic. An explicit expression for the transformation plastic strain rate is obtained for a steel undergoing a transformation under a small applied stress; this expression is consistent with experiments conducted on various materials. A finite element simulation provides a confirmation of the theoretical formula and allows for a detailed examination of the validity of some physical hypotheses made in the treatment. It also allows for a study of transformation plasticity under high applied stresses. Based on these results, a general (i.e., applicable for all kinds of stresses applied) model is proposed in the case of ideal-plastic phases.     

New investigations on transformation induced plasticity and its interaction with classical plasticity

In this paper, transformation induced plasticity (TRIP) in anisothermal single as well as double transformations (austenite → bainite and austenite → bainite + Martensite) in 16MND5 steel is experimentally analyzed. Several investigations have been performed related mainly on: (a) the evaluation of the physical mechanism responsible of the TRIP in bainitic transformation; (b) the kinetics of TRIP and its specificity in a double transformation; (c) the consequence when the load is applied during only a part of phase transformation; (d) the interaction between TRIP and classical plasticity and so on. The results seem indicate that Greenwood and Johnson mechanism is dominant compared to Magee mechanism. The interaction between classical plasticity and TRIP is clearly demonstrated and it seems that the strain hardening state of the parent phase plays an important role in the TRIP progress. Due to such interaction, TRIP appears even in the absence of external applied load; the behavior depends strongly on the transformation under consideration (bainitic or martensitic). From a modeling point of view, it is shown that Leblond’s model that is the only one “industrial” model which enables qualitatively to account for such interactions, fails to predict the observed phenomena especially in martensitic transformation.     

Mathematical modelling of transformation plasticity in steels II: Coupling with strain hardening phenomena

The models for the plastic behaviour of steels during phase transformations proposed in Part I and in a previous paper ( et al. [1986b]) for the case of ideal-plastic phases are extended to include strain-hardening effects (isotropic or kinematic hardening). An expression for the transformation plastic strain rate is obtained by modifying the treatment of Part I in a suitable manner. The classical plastic strain rate is also studied in a similar way. Complementary evolution equations for the hardening parameters are finally given, taking into account the possible “recovery” of strain hardening during transformations (i.e., the fact that the newly formed phase can “forget,” partially or totally, the previous hardening).     

A phenomenological description of shape memory alloy transformation induced plasticity

Transformation induced plasticity is defined as the plastic flow arising from solid state phase transformation processes involving volume and/or shape changes without overlapping the yield surface. This phenomenon occurs in shape memory alloys (SMAs) having significant influence over their macroscopic thermomechanical behavior. This contribution presents a macroscopic three-dimensional constitutive model to describe the thermomechanical behavior of SMAs including classical and transformation induced plasticity. Comparisons between numerical and experimental results attest the model capability to capture plastic phenomena. Both uniaxial and multiaxial simulations are carried out.     

Couplings between plasticity and martensitic phase transformation: overall behavior of polycrystalline TRIP steels

Several couplings between plasticity and martensitic phase transformation are at the origin of remarkable properties of ductility and toughness in the case of TRIP steels. A micromechanical model is developed to predict the conditions of nucleation and growth of a martensitic microdomain inside an inhomogeneous plastic strain field. More explicit relations are developed in the case of a simple shear test where a heterogeneous plastic strain field leads to a significant decrease of the critical stress for martensitic transformation. The obtained results are combined with a kinetics and kinematics studies to derive the constitutive equation of an austenitic single crystal from which the overall behavior of a polycrystalline steel is deduced using the self-consistent scale transition method. Comparison with experimental data shows a good agreement.     

Discontinuous bifurcation analysis in fracture energy-based gradient plasticity for concrete

Conditions for discontinuous bifurcation in limit states of selective non-local thermodynamically consistent gradient theory for quasi-brittle materials like concrete are evaluated by means of both geometrical and analytical procedures. This constitutive formulation includes two internal lengths, one related to the strain gradient field that considers the degradation of the continuum in the vicinity of the considered material point. The other characteristic length takes into account the material degradation in the form of energy release in the cracks during failure process evolution.The variation from ductile to brittle failure in quasi-brittle materials is accomplished by means of the pressure dependent formulation of both characteristic lengths as described by Vrech and Etse (2009).In this paper the formulation of the localization ellipse for constitutive theories based on gradient plasticity and fracture energy plasticity is proposed as well as the explicit solutions for brittle failure conditions in the form of discontinuous bifurcation. The geometrical, analytical and numerical analysis of discontinuous bifurcation condition in this paper are comparatively evaluated in different stress states and loading conditions.The included results illustrate the capabilities of the thermodynamically consistent selective non-local gradient constitutive theory to reproduce the transition from ductile to brittle and localized failure modes in the low confinement regime of concrete and quasi-brittle materials.