Gradient-dependent Plasticity Research Articles

Parametric Study on an Integral-Type Nonlocal Elastoplasticity Model Regularized with Tikhonov-Phillips Method

An integral-type nonlocal elastoplasticity model is proposed and formulated in a one-dimensional scenario to address the strain-softening problem. This nonlocal model includes not only the nonlocal plasticity but also the nonlocal elasticity. The resulting elastic constitutive equation and the plastic consistency condition are ill-posed Fredholm integral equations of the first kind whose solutions are instable without appropriate regularization. In this paper, the Tikhonov-Phillips regularization method is used to reformulate the integral-differential consistency condition to obtain an approximate solution of the local internal variable, which is stable and smooth. A detailed parametric study is carried out to examine the influences of the regularization parameter, the internal length scale, the plastic modulus ratio, and the tolerance of the nonlocal yield condition on the regularized solutions. A numerical example shows that the solutions from the proposed model and the regularization method are mesh-independent. A comparison of the results from the proposed model with those from the nonlocal plasticity model and from the gradient-dependent plasticity model shows good agreement.

Strain-gradient elasticity and gradient-dependent plasticity with hierarchical refinement of NURBS

Higher-order strain-gradient models are relevant for engineering materials which exhibit size-dependent behaviour as observed from experiments. Typically, this class of models incorporate a length scale - related to micro-mechanical material properties - to capture size effects, remove stress singularities, or regularise an ill-posed boundary value problem resulting from localisation of deformation. The higher-order continuity requirement on shape functions can be met using NURBS discretisation, as is considered herein. However, NURBS have a tensor-product nature which makes selective refinement cumbersome. To maintain accuracy and efficiency in analysis, a finer mesh may be required, to capture a localisation band, certain geometrical features, or in regions with high gradients. This work presents strain-gradient elasticity and strain-gradient plasticity, both of second-order, with hierarchically refined NURBS. Refinement is performed based on a multi-level mesh with element-wise hierarchical basis functions interacting through an inter-level subdivision operator. This ensures a standard finite-element data structure. Suitable marking strategies have been used to select elements for refinement. The capability of the numerical schemes is demonstrated with two-dimensional examples.

A linear complementarity formulation of meshfree method for elastoplastic analysis of gradient-dependent plasticity

This work presents a linear complementarity formulation for elastoplastic analysis of the gradient-dependent plasticity including large deformation problems. The formulation is based on the meshfree smoothed radial point interpolation method, where the parametric variational principle (PVP) is used in the form of linear complementarity and the gradient-dependent plasticity is represented by the linearization of yield criterion. The yield stress is linearly evolved through equivalent plastic strain as well as its Laplacian (namely second gradient). The global discretized system equations are transformed into a standard linear complementarity problem (LCP), which can be solved readily using the Lemke method. The proposed approach is capable of simulating material hardening/softening and strain localization. An extensive numerical study is performed to validate the proposed method and to investigate the effects of various parameters. The numerical results demonstrate that the proposed approach is accurate and stable for the elastoplastic analysis of 2D solids with gradient-dependent plasticity on strain localization.

Smoothed Point Interpolation Method for Elastoplastic Analysis

This work formulates the node-based smoothed radial point interpolation method (NS-RPIM), a typical model of smoothed point interpolation method, for the elastoplastic analysis of two-dimensional solids with gradient-dependent plasticity. The NS-RPIM uses radial point interpolation shape functions for field approximation and node-based gradient smoothing for strain field construction. The formulation is based on the parametric variational principle (PVP) in the form of complementarity with the gradient-dependent plasticity being represented by means of the linearization of the yield criterion and the flow rule. Numerical study results have demonstrated the accuracy and stability of the proposed approach for elastoplastic analysis.

Microstructural modeling of dual phase steel using a higher-order gradient plasticity–damage model

This paper focuses on the application of a higher-order gradient-dependent plasticity–damage model for microstructural modeling of dual-phase (DP) steels. Damage evolution is governed by the evolution of a nonlocal plasticity measure which is a function of the local equivalent plastic strain rate and its corresponding first-order gradient. Different two-dimensional representative volume elements of the DP microstructure are virtually generated by varying the martensite phase volume fraction, distribution, and size, and volume fraction and size of dispersed hard inclusions. It is shown that the employed modeling framework is capable of addressing three main issues that are not considered by the current studies on microstructural modeling of advanced high strength steels (AHSS); finite element mesh-dependency, size effects, and additional hardening due to plastic strain gradients. It is concluded that based on the employed mechanical properties of each phase in the DP steel, strength and ductility is governed by damage evolution and not necessarily by plastic strain localization alone. Therefore, it is shown that including nonlocal damage evolution is critical for accurate prediction of the strength and ductility of DP steels. It is also shown that dispersing 5% volume fraction of hard inclusions in the DP steel optimizes both strength and ductility such that a new generation of AHSS might be attained.

On the numerical implementation of the higher-order strain gradient-dependent plasticity theory and its non-classical boundary conditions

The higher-order gradient plasticity theory is successful in explaining the size effects encountered at the micron and submicron length scale. Due to the incorporation of spatial gradients of one or more internal variables in these theories and the associated non-classical boundary conditions, special types of elements in the finite element method maybe necessary. This makes the numerical implementation of this higher-order theory not straightforward. In this paper, a robust but straightforward numerical implementation of higher-order gradient-dependent plasticity theories is presented. The novelty of this paper is in (1) the application of the meshless methods, particularly the moving weighted least square method, combined with the finite element method for the numerical computation of plastic strain gradients, and (2) the numerical implementation of different types of higher-order microscopic boundary conditions at internal/external surfaces, interfaces, and moving elastic–plastic boundaries. The proposed numerical implementation algorithms can be easily adapted in the implementation of any form of higher-order gradient-dependent constitutive models. Examples of applying the current numerical approach is demonstrated for capturing mesh-objective shear band formation and size effect and boundary layer formation in thin films on elastic substrates and metal matrix composites with embedded elastic inclusions through the consideration of stiff, intermediate, and soft interfaces.

A model for the shear displacement distribution of a flow line in the adiabatic shear band based on gradient-dependent plasticity

A model for the shear displacement distribution of a flow line in the adiabatic shear band based on gradient-dependent plasticity

Adiabatic shear sensitivity of ductile metal based on gradient-dependent JOHNSON-COOK model

Based on the expression proposed by WANG for the local plastic shear deformation distribution in the adiabatic shear band (ASB) using gradient-dependent plasticity, the effects of 10 parameters on the adiabatic shear sensitivity were studied. The experimental data for a flow line in the ASB obtained by LIAO and DUFFY were fitted by use of the curve-fitting least squares method and the proposed expression. The critical plastic shear strains corresponding to the onset of the ASB for Ti-6Al-4V were assessed at different assigned ASB widths. It is found that the proposed expression describes well the non-linear deformation characteristics of the flow line in the ASB. Some parameters in the JOHNSON-COOK model are back-calculated using different critical plastic shear strains. The adiabatic shear sensitivity decreases as initial static yield stress, work to heat conversion factor and strain-rate parameter decrease, which is opposite to the effects of density, heat capacity, ambient temperature and strain-hardening exponent. The present model can predict the ASB width evolution process. The predicted ASB width decreases with straining until a stable value is reached. The famous model proposed by DODD and BAI only can predict a final stable value.

A mixed finite element and mesh-free method using linear complementarity theory for gradient plasticity

A mixed finite element (FE) and mesh-free (MF) method for gradient-dependent plasticity using linear complementarity theory is presented. The assumed displacement field is interpolated in terms of its discrete values defined at the nodal points of the FE mesh with the FE shape functions, whereas the assumed plastic multiplier field required to express its Laplacian is interpolated in terms of its discrete values defined at the integration points of the FE mesh with the MF interpolation functions. A standard form of linear complementarity problem is constructed by combining the weak form of momentum conservation equation and pointwise enforcements of both non-local constitutive equation and non-local yield criterion. The discrete values of the plastic multiplier are taken as the only primary unknowns to be determined. The numerical results demonstrate the validity of the proposed method in the simulation of the strain localization phenomenon due to strain softening.

Modelling size effects in micro/nano-systems by including interfacial effects in a gradient plasticity framework

The effect of the material microstructural interfaces increases as the surface-to-volume ratio increases. It is shown in this work that interfacial effects have a profound impact on the scale-dependent plasticity encountered in micro/nano-systems. This is achieved by developing a physically-based higher-order gradient-dependent plasticity theory that enforces microscopic boundary conditions at interfaces and free surfaces. These non-standard boundary conditions relate the micro traction stress at the interface to the interfacial energy. Application of the proposed framework to size effects in biaxial tension of a thin-film on an elastic substrate is presented. Three film-interface conditions are modelled: soft, intermediate, and hard interfaces.

Phenomenon of transformed adiabatic shear band surrounded by deformed adiabatic shear band of ductile metal

The coexistent phenomenon of deformed and transformed adiabatic shear bands(ASBs) of ductile metal was analyzed using the JOHNSON-COOK model and gradient-dependent plasticity(GDP). The effects of melting point, density, heat capacity and work to heat conversion factor were investigated. Higher work to heat conversion factor, lower density, lower heat capacity and higher melting point lead to wider transformed ASB and higher local plastic shear deformation between deformed and transformed ASBs. Higher work to heat conversion factor, lower density, lower heat capacity and lower melting point cause higher local plastic shear deformation in the deformed ASB. Three reasons for the scatter in experimental data on the ASB width were pointed out and the advantages of the work were discussed. If the transformed ASB width is used to back-calculate the internal length parameter in the GDP, undoubtedly, the parameter will be extremely underestimated.

Effects of Temperature and Strain Rate on the Evolution of Thickness of Transformed Adiabatic Shear Band

The coexistent phenomenon of deformed and transformed adiabatic shear bands (ASBs) is analyzed using Johnson-Cook model and gradient-dependent plasticity for heterogeneous ductile metal material. The size of deformed ASB is described by the internal length reflecting the heterogeneity of material. Microstructural effect leads to a nonuniform distribution of temperature rise in deformed ASB. When the peak temperature in deformed ASB exceeds the transformation temperature, a transformed ASB appears at the center of deformed ASB. With a decrease of flow shear stress, the width of transformed ASB increases until its upper bound, i.e., the size of deformed ASB, is reached. The effects of initial temperature and strain rate on the occurrence of transformation, evolution of the thickness of transformed ASB, distributions of local temperature and plastic shear deformation in ASB are investigated. Lower initial temperature results in higher peak shear stress, later occurrence of shear strain localization, lower shear stress when transformation occurs, later occurrence of transformation, thinner transformed ASB, lower peak temperature in ASB, and lower value of local plastic shear deformation in the boundaries of transformed ASB. At higher strain rates, the transformed ASB is wider; the peak temperature in ASB is higher; the value of local plastic shear deformation in the boundaries of transformed ASB is higher; the flow shear stress that corresponds to transformation is higher; earlier occurrence of transformation and higher peak shear stress will be expected.

Grain Size Effect in Sheet Metal Microforming Simulation Adopting Strain Gradient Concept

A strain gradient dependent crystal plasticity approach is adopted to model the size effect in the microforming process of sheet metal. To take into account the grain size effect in the simulation, the total slip resistance in each active system was assumed to be due to a mixed population of forest obstacles arising from both statistically stored and geometrically necessary dislocations. The non-local crystal plasticity has been established by directly incorporating the above slip resistance into the conventional rate-dependent crystal plasticity and implemented into the Abaqus/Standard FE platform by developing the user subroutine UMAT. The formulation has been recapitulated and followed by presentation of the numerical examples employing both the local and non-local formulation. The comparison of the counterpart simulation results reveals the grain size effect in the microforming process and demonstrates the availability of the code developed.

Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals

The effect of the material microstructural interfaces increases as the surface-to-volume ratio increases. It is shown in this work that interfacial effects have a profound impact on the scale-dependent yield strength and strain hardening of micro/nano-systems even under uniform stressing. This is achieved by adopting a higher-order gradient-dependent plasticity theory [Abu Al-Rub, R.K., Voyiadjis, G.Z., Bammann, D.J., 2007. A thermodynamic based higher-order gradient theory for size dependent plasticity. Int. J. Solids Struct. 44, 2888–2923] that enforces microscopic boundary conditions at interfaces and free surfaces. Those nonstandard boundary conditions relate a microtraction stress to the interfacial energy at the interface. In addition to the nonlocal yield condition for the material’s bulk, a microscopic yield condition for the interface is presented, which determines the stress at which the interface begins to deform plastically and harden. Hence, two material length scales are incorporated: one for the bulk and the other for the interface. Different expressions for the interfacial energy are investigated. The effect of the interfacial yield strength and interfacial hardening are studied by analytically solving a one-dimensional Hall–Petch-type size effect problem. It is found that when assuming compliant interfaces the interface properties control both the material’s global yield strength and rates of strain hardening such that the interfacial strength controls the global yield strength whereas the interfacial hardening controls both the global yield strength and strain hardening rates. On the other hand, when assuming a stiff interface, the bulk length scale controls both the global yield strength and strain hardening rates. Moreover, it is found that in order to correctly predict the increase in the yield strength with decreasing size, the interfacial length scale should scale the magnitude of both the interfacial yield strength and interfacial hardening.

Distributions of Local Damage Variable and Local Plastic Tensile Strain and Precursors to Failure of Quasi-Brittle Pure Bending Beam

For many quasi-brittle materials (such as rock, ceramic and concrete) in pure bending state, the material on the tensile side will fail firstly since the compressive strength can be ten times the tensile strength. After tensile strain localization zone is initiated in the midspan of the beam, its propagation direction will be perpendicular to the neutral axis. In the paper, using nonlocal theory or gradient-dependent plasticity, the distributions of local plastic tensile strain and local damage variable in tensile strain localization zone of a pure bending beam are analyzed theoretically. The evolutions of the maximum local plastic tensile strain, the maximum local damage variable and the bending moment with tensile stress acting on the tensile side are presented through examples. The distributions of local plastic tensile strain and local damage variable in tensile strain localization zone are highly nonuniform due to microstructural effect. When the maximum bending moment is reached, the maximum local damage variable is proportional to the ratio of elastic modulus to elastoplastic modulus, while the maximum local plastic tensile strain is inversely proportional to elastic modulus and elastoplastic modulus. For quasi-brittle materials, the elastoplastic modulus that is a constitutive parameter equal to the absolute value of the slope of tensile stress-tensile strain curve in strain-softening stage is much higher. The present theoretical results mean that the precursors to failure are less apparent for extremely brittle materials.

Peak and Average Temperatures in Adiabatic Shear Band for Thermo-Viscoplastic Metal Materials

Gradient-dependent plasticity considering the microstructural effect is introduced into Johnson-Cook model to calculate the nonuniform temperature distribution in adiabatic shear band (ASB) and the evolutions of average and peak temperatures in ASB. Effects of initial static yield stress, strain-hardening coefficient, strain-hardening exponent, strain-rate parameter and thermal-softening parameter are numerically investigated. The calculated peak temperature in ASB considering both the plastic work and the microstructural effect is always greater than the average temperature calculated only using the plastic work. For much lower flow shear stress, the peak temperature approaches two times the average temperature. The occurrence of phase transformation in ASB is easier in metal material with higher initial static yield stress, strain-hardening coefficient, strain-rate parameter and thermal-softening parameter. At much lower flow shear stress or much higher average plastic shear strain, the phase transformation occurs more easily in material with a lower strain-hardening exponent. Traditional elastoplastic theory without the microstructural effect underestimates the peak temperature in ASB so that the experimentally observed phase transformations cannot be explained.

Quantitative calculation of local shear deformation in adiabatic shear band for Ti-6Al-4V

JOHNSON-COOK(J-C) model was used to calculate flow shear stress—shear strain curve for Ti-6Al-4V in dynamic torsion test. The predicted curve was compared with experimental result. Gradient-dependent plasticity(GDP) was introduced into J-C model and GDP was involved in the measured flow shear stress—shear strain curve, respectively, to calculate the distribution of local total shear deformation(LTSD) in adiabatic shear band(ASB). The predicted LTSDs at different flow shear stresses were compared with experimental measurements. J-C model can well predict the flow shear stress—shear strain curve in strain-hardening stage and in strain-softening stage where flow shear stress slowly decreases. Beyond the occurrence of ASB, with a decrease of flow shear stress, the increase of local plastic shear deformation in ASB is faster than the decrease of elastic shear deformation, leading to more and more apparent shear localization. According to the measured flow shear stress — shear strain curve and GDP, the calculated LTSDs in ASB are lower than experimental results. At earlier stage of ASB, though J-C model overestimates the flow shear stress at the same shear strain, the model can reasonably assess the LTSDs in ASB. According to the measured flow shear stress — shear strain curve and GDP, the calculated local plastic shear strains in ASB agree with experimental results except for the vicinity of shear fracture surface. In the strain-softening stage where flow shear stress sharply decreases, J-C model cannot be used. When flow shear stress decreases to a certain value, shear fracture takes place so that GDP cannot be used.

Nonlocal Gradient-Dependent Thermodynamics for Modeling Scale-Dependent Plasticity

This article is concerned with formulating the thermodynamics of nonlocal gradient-dependent plasticity based on the nonlocality energy residual introduced by Eringen and Edelen (1972). A thermodynamic based theory for small strain gradient plasticity is developed by introducing gradients for variables associated with kinematic and isotropic hardening. This theory is a three-nonlocal-parameter theory that takes into consideration large variations in the plastic strain, large variations in the accumulated plastic strain, and accumulation of plastic strain gradients. It is shown that the presence of higher-order gradients in the plastic strain enforces the presence of a corresponding history variable brought by the accumulation of the plastic strain gradients. Gradients in the plastic strain introduce anisotropy in the form of kinematic hardening and are attributed to the net Burgers vector, whereas gradients in the accumulation of the plastic strain introduce isotropic hardening attributed to the additional storage of geometrically necessary dislocations. The equilibrium, or so-called microforce balance, between the internal Cauchy stress and the microstresses that are conjugates to the higher-order gradients turns out to be the yield criterion, which can be simply retrieved from the principle of virtual power. The classical macroscopic boundary conditions are supplemented by nonclassical microscopic boundary conditions associated with plastic flow. The developed nonlocal theory preserves the classical assumption of the local plasticity theory such that the plastic flow direction is governed by the deviatoric Cauchy stress. However, it is also argued here that plastic flow direction is the same as if it is governed by the nonlocal microstress. This is not in line with Gurtin (2003), who argued that the plastic flow direction is governed by a microstress and not the deviatoric Cauchy stress. Some generalities and the utility of this theory are discussed, and comparisons with other gradient theories are given. Applications of the proposed theory for size effects in thin films are presented.

Effects of constitutive parameters on adiabatic shear localization for ductile metal based on JOHNSON-COOK and gradient plasticity models

By using the widely used JOHNSON-COOK model and the gradient-dependent plasticity to consider microstructural effect beyond the occurrence of shear strain localization, the distributions of local plastic shear strain and deformation in adiabatic shear band(ASB) were analyzed. The peak local plastic shear strain is proportional to the average plastic shear strain, while it is inversely proportional to the critical plastic shear strain corresponding to the peak flow shear stress. The relative plastic shear deformation between the top and base of ASB depends on the thickness of ASB and the average plastic shear strain. A parametric study was carried out to study the influence of constitutive parameters on shear strain localization. Higher values of static shear strength and work to heat conversion factor lead to lower critical plastic shear strain so that the shear localization is more apparent at the same average plastic shear strain. Higher values of strain-hardening exponent, strain rate sensitive coefficient, melting point, thermal capacity and mass density result in higher critical plastic shear strain, leading to less apparent shear localization at the same average plastic shear strain. The strain rate sensitive coefficient has a minor influence on the critical plastic shear strain, the distributions of local plastic shear strain and deformation in ASB. The effect of strain-hardening modulus on the critical plastic shear strain is not monotonous. When the maximum critical plastic shear strain is reached, the least apparent shear localization occurs.

Volume change of heterogeneous quasi-brittle materials in uniaxial compression

The volumetric strain was categorized into elastic and plastic parts. The former composed of axial and lateral strains is uniform and determined by Hooke's law; however, the latter consisting of axial and lateral strains is a function of thickness of shear band determined by gradient-dependent plasticity by considering the heterogeneity of quasi-brittle materials. The non-uniform lateral strain due to the fact that shear band was formed in the middle of specimen was averaged within specimen to precisely assess the volumetric strain. Then, the analytical expression for volumetric strain was verified by comparison with two earlier experimental results for concrete and rock. Finally, a detailed parametric study was carried out to investigate effects of constitutive parameters (shear band thickness, elastic and softening moduli) and geometrical size of specimen (height and width of specimen) on the volume dilatancy.