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.     

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 analysis of transformation induced plasticity in multiphase steels

“To what extent do plastic strain gradients affect the strengthening resulting from the transformation of small metastable inclusions into hard inclusions within a plastically deforming matrix?” is the central question addressed here. Though general in the approach, the focus is on the behavior of TRIP-assisted multiphase steels. A two-dimensional embedded cell model of a simplified microstructure composed of a single metastable austenitic inclusion surrounded by a soft ferritic matrix is considered. The cell is inserted in a large homogenized medium. The transformation of a fraction of the austenite into a hard martensite plate is simulated, accounting for a transformation strain, and leading to complex elastic and plastic accommodation. The size of a transforming plate in real multiphase steels is typically between 0.1 and 2 μm, a range of size in which plastic strain gradient effects are expected to play a major role. The single parameter version of the Fleck–Hutchinson strain gradient plasticity theory is used to describe the plasticity in the austenite, ferrite and martensite phases. The higher order boundary conditions imposed on the plastic flow have a large impact on the predicted strengthening. Using realistic values of the intrinsic length parameter setting the scale at which the gradients effects have an influence leads to a noticeable increase of the strengthening on top of the increase due to the transformation of a volume fraction of the retained austenite. The geometrical parameters such as the volume fraction of retained austenite and of the transforming zone also bring significant strengthening. Strain gradient effects also significantly affect the stress state inside the martensite plate during and after transformation with a potential impact on the damage resistance of these 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 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 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.     

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.     

A Finite Element approach with patch projection for strain gradient plasticity formulations

Several strain gradient plasticity formulations have been suggested in the literature to account for inherent size effects on length scales of microns and submicrons. The necessity of strain gradient related terms render the simulation with strain gradient plasticity formulation computationally very expensive because quadratic shape functions or mixed approaches in displacements and strains are usually applied. Approaches using linear shape functions have also been suggested which are, however, limited to regular meshes with equidistanced Finite Element nodes. As a result the majority of the simulations in the literature deal with plane problems at small strains. For the solution of general three dimensional problems at large strains an approach has to be found which has to be computationally affordable and robust.     

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 hardening plasticity with bauschinger effect

Summary The plastic behaviour of metals is studied, taking into account the strain bardening and the Bauschinger effect.Basic assumptions go back to the slip theory of plasticity and many features of this theory are developed and thoroughly examined. The Bauschinger effect is introduced on every couple of planes and the stress-strain relations are given in incremental form. The plastic strain properties corresponding to the theoretical model are studied with special reference to the evolution of the yield surface.The paper concludes by applying the proposed theory to the tensile test with stress reversal and to the characterization of the subsequent yield surfaces in an ideal tension-torsion test. The results agree, at least qualitatively, with the experimental results.

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Sommario In questa Nota viene studiato il comportamento plastico dei metalli prendendo in conto l'influenza dell'incrudimento e l'effetto Bauschinger.Partendo dai punti di base della teoria degli slip si sviluppano ed approfondiscono molti aspetti di questa, tra l'altro introducendo l'effetto Bauschinger su ogni coppia di piani, e si forniscono in forma incrementale le relazioni tensioni-deformazioni.Successivamente vengono studiate le proprietà della deformazione plastica che consegue a tale formulazione con particolare riferimento al problema dell'evoluzione della superficie di plasticizzazione.Il lavoro si conclude con l'applicazione della teoria al caso della prova a trazione con inversione di sforzo e alla caratterizzazione delle superfici susseguenti di plasticizzazione nella prova ideale combinata di trazione e torsione ritrovando accordo, almeno dal punto di vista qualitativo, con ricerche sperimentali in merito.

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Study supported by the C.N.R.     

Post-necking behaviour modelled by a gradient dependent plasticity theory

The delay of the onset of localization and the post-necking behaviour for stretched thin sheets are determined by three-dimensional effects. Thus, a 2-D finite element analysis based on a local plasticity theory will give a physically unrealistic mesh dependent solution. This, in spite of the fact that the stress state, is essentially two-dimensional. By incorporating a length scale with relation to the thickness of the sheet, it is demonstrated how a 2-D finite element analysis based on a gradient dependent plasticity theory can give a good approximation of the post-necking behaviour. This is illustrated by numerical comparison of results from a full 3-D finite element analysis, with results from a 2-D finite element model based on a finite strain version of a gradient dependent J₂-flow theory. Some numerical problems in the modeling will be discussed briefly.     

A study of microbend test by strain gradient plasticity

Metallic materials display strong size effect when the characteristic length associated with plastic deformation is on the order of microns. This size effect cannot be explained by classical plasticity theories since their constitutive relations do not have an intrinsic material length. Strain gradient plasticity has been developed to extend continuum plasticity to the micron or submicron regime. One major issue in strain gradient plasticity is the determination of the intrinsic material length that scales with strain gradients, and several microbend test specimens have been designed for this purpose. We have studied different microbend test specimens using the theory of strain gradient plasticity. The pure bending specimen, cantilever beam, and the microbend test specimen developed by Stolken and Evans (Stolken, J.S., Evans, A.G., 1998. A microbend test method for measuring the plasticity length scale Acta Mater. 46, 5109–5115) are found suitable for the determination of intrinsic material length in strain gradient plasticity. However, the double cantilever beam (both ends clamped) is unsuitable since its deformation is dominated by axial stretching. The strain gradient effects significantly increase the bending stiffness of a microbend test specimen. The deflection of a 10-μm thick beam is only a few percent of that estimated by classical plasticity.     

Nonlocal and gradient rate plasticity

This paper deals with a formulation of nonlocal and gradient plasticity with internal variables. The constitutive model complies with local internal variables which govern kinematic hardening and isotropic softening and with a nonlocal corrective internal variable defined either as the sum between a new internal variable and its spatial weighted average or as the gradient of a measure of plastic strain. The rate constitutive problem is cast in the framework provided by the convex analysis and the potential theory for monotone multivalued operators which provide the suitable tools to perform a theoretical analysis of such nonlocal and gradient problems. The validity of the maximum dissipation theorem is assessed and constitutive variational formulations of the rate model are provided. The structural rate problem for an assigned load rate is then formulated. The related variational formulation in the complete set of state variable is contributed and the methodology to derive variational formulations, with different combinations of the state variables, is explicitly provided. In particular the generalization to the present nonlocal and gradient model of the principles of Prager–Hodge, Greenberg and Capurso–Maier is presented. Finally nonlocal variational formulations provided in the literature are derived as special cases of the proposed model.     

Wedge indentation of thin films modelled by strain gradient plasticity

A plane strain study of wedge indentation of a thin film on a substrate is performed. The film is modelled with the strain gradient plasticity theory by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379–1406] and analysed using finite element simulations. Several trends that have been experimentally observed elsewhere are captured in the predictions of the mechanical behaviour of the thin film. Such trends include increased hardness at shallow depths due to gradient effects as well as increased hardness at larger depths due to the influence of the substrate. In between, a plateau is found which is observed to scale linearly with the material length scale parameter. It is shown that the degree of hardening of the material has a strong influence on the substrate effect, where a high hardening modulus gives a larger impact on this effect. Furthermore, pile-up deformation dominated by plasticity at small values of the internal length scale parameter is turned into sink-in deformation where plasticity is suppressed for larger values of the length scale parameter. Finally, it is demonstrated that the effect of substrate compliance has a significant effect on the hardness predictions if the effective stiffness of the substrate is of the same order as the stiffness of the film.     

A physically based gradient plasticity theory

The intent of this work is to derive a physically motivated mathematical form for the gradient plasticity that can be used to interpret the size effects observed experimentally. The step of translating from the dislocation-based mechanics to a continuum formulation is explored. This paper addresses a possible, yet simple, link between the Taylor’s model of dislocation hardening and the strain gradient plasticity. Evolution equations for the densities of statistically stored dislocations and geometrically necessary dislocations are used to establish this linkage. The dislocation processes of generation, motion, immobilization, recovery, and annihilation are considered in which the geometric obstacles contribute to the storage of statistical dislocations. As a result, a physically sound relation for the material length scale parameter is obtained as a function of the course of plastic deformation, grain size, and a set of macroscopic and microscopic physical parameters. Comparisons are made of this theory with experiments on micro-torsion, micro-bending, and micro-indentation size effects.     

Modeling plastic shocks in periodic laminates with gradient plasticity theories

Steady plastic shocks generated by planar impact on metal-polymer laminate composites are analyzed in the framework of gradient plasticity theories. The laminate material has a periodic structure with a unit cell composed of two layers of different materials. First- and second-order gradient plasticity theories are used to model the structure of steady plastic shocks. In both theories, the effect of the internal structure is accounted for at the macroscopic level by two material parameters that are dependent upon the layer's thickness and the properties of constituents. These two structure parameters are shown to be uniquely determined from experimental data. Theoretical predictions are compared with experiments for different cell sizes and for various shock intensities. In particular, the following experimental features are well-reproduced by the modeling:

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the shock width is proportional to the cell size;

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the magnitude of strain rate is inversely proportional to cell size and increases with the amplitude of applied stress following a power law.

While these results are equally described by both the plasticity theories, the first gradient plasticity approach seems to be favored when comparing the structure of the shock front to the experimental data.     

Thermodynamics-based gradient plasticity theories with an application to interface models

In the framework of small deformations, the so-called residual-based gradient plasticity theory is reconsidered and improved. Using the notion of moving geometrically necessary dislocations (GNDs), suitable micromechanics interpretations are heuristically given for the higher order boundary conditions and the long distance particle interactions. Also, a comparison is made between this theory and the analogous virtual work principle (VWP)-based one, whereby their respective conceptual and methodological features are pointed out. The conditions under which the two theories lead to a same constitutive model are investigated, showing that, correspondingly, a certain indeterminacy exhibited by the VWP-based theory disappears. A phenomenological interface model is used to better point out differences and analogies of the two theories.     

Strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic–plastic material

An analytical solution is presented for an internally pressurized thick-walled spherical shell of an elastic strain-hardening plastic material. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. The solution gives explicit expressions for the stress, strain and displacement components. The inner radius of the shell enters these expressions not only in non-dimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect. The classical plasticity-based solution of the same problem is shown to be a special case of the present solution. Numerical results for the maximum effective stress in the shell wall are also provided to illustrate applications of the newly derived solution. The new solution can be used to construct improved expanding cavity models in indentation mechanics that incorporate both the strain-hardening and indentation size effects.