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.     

Kinetic wrinkling of an elastic film on a viscoelastic substrate

A compressed elastic film on a compliant substrate can form wrinkles. On an elastic substrate, equilibrium and energetics set the critical condition and select the wrinkle wavelength and amplitude. On a viscous substrate, wrinkle grows over time and the kinetics selects the fastest growing wavelength. More generally, on a viscoelastic substrate, both energetics and kinetics play important roles in determining the critical condition, the growth rate, and the wavelength. This paper studies the wrinkling process of an elastic film on a viscoelastic layer, which in turn lies on a rigid substrate. The film is elastic and modeled by the nonlinear von Karman plate theory. The substrate is linear viscoelastic with a relaxation modulus typical of a cross-linked polymer. Beyond a critical stress, the film wrinkles by the out-of-plane displacement but remains bonded to the substrate. This study considers plane strain wrinkling and neglects the in-plane displacement. A classification of the wrinkling behavior is made based on the critical conditions at the elastic limits, the glassy and rubbery states of the viscoelastic substrate. Linear perturbation analyses are conducted to reveal the kinetics of wrinkling in films subjected to intermediate and large compressive stresses. It is shown that, depending on the stress level, the growth of wrinkles at the initial stage can be exponential, accelerating, linear, or decelerating. In all cases, the wrinkle amplitude saturates at an equilibrium state after a long time. Subsequently, both amplitude and wavelength of the wrinkle evolve, but the process is kinetically constrained and slow compared to the initial growth.     

Well-posedness of a model of strain gradient plasticity for plastically irrotational materials

The initial boundary value problem corresponding to a model of strain gradient plasticity due to [Gurtin, M., Anand, L., 2005. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. J. Mech. Phys. Solids 53, 1624–1649] is formulated as a variational inequality, and analysed. The formulation is a primal one, in that the unknown variables are the displacement, plastic strain, and the hardening parameter. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution. It is shown that this is the case when hardening takes place. A similar property does not hold for the case of softening. The model is therefore extended by adding to it terms involving the divergence of plastic strain. For this extended model the desired property of coercivity holds, albeit only on the boundary of the set of admissible functions.     

Nonlinear constitutive models from nanoindentation tests using artificial neural networks

This paper presents a new approach using Artificial Neural Networks (ANNs) models to simulate the response during nanohardness tests of a variety of materials with nonlinear behavior. The ANNs continuous input and output variables usually include material parameters, indentation deflection, and resisting force. Different ANN models, including dimensionless input/output variables, are generated and trained with discrete finite-element (FE) simulations with different geometries and nonlinear material parameters. Only the monotonic loading part of the load–displacement indentation response is used to generate the trained ANN models. This is a departure from classical indentation simulations or tests where typically the unloading portion is used to determine the stiffness and hardness. The experimental part of this study includes nanoindentation tests performed on a silicon (Si) substrate with and without a nanocrystalline copper (Cu) film. The new ANN models are used to back-calculate (inverse problem) the in situ nonlinear material parameters for different copper material systems. The results are compared with available data in the literature. The proposed FE–ANN modeling approach is very effective and can be used in calibrating and predicting the in situ inelastic material properties using the monotonic part of the indentation response and for depths above 50 nm where the overall resisting force represents a continuum response.     

Three dimensional combined fracture–plastic material model for concrete

This paper describes a combined fracture–plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. The plasticity model for concrete in compression is based on the Menétrey–Willam failure surface, the plastic volumetric strain as a hardening/softening parameter and a non-associated flow rule based on a nonlinear plastic potential function. Both models use a return-mapping algorithm for the integration of constitutive equations. Special attention is given to the development of an algorithm for the combination of the two models. The suggested combination algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The algorithm can handle cases when failure surfaces of both models are active, but also when physical changes such as crack closure occur. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The model is integrated in a general finite element package ATENA and its performance is evaluated by comparisons with various experimental results from the literature.     

Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites

The aim of this work is to construct yield surfaces to describe initial yielding and characterize hardening behavior of a highly anisotropic material. A methodology for constructing yield surfaces for isotropic materials using axial–torsion loading is extended to highly anisotropic materials. The technique uses a sensitive definition of yielding based on permanent strain rather than offset strain, and enables multiple yield points and multiple yield surfaces to be conducted on a single specimen. A target value of 20 × 10⁻⁶ is used for Al₂O₃ fiber reinforced aluminum laminates having a fiber volume fraction of 0.55. Sixteen radial probes are used to define the yield locus in the axial–shear stress plane. Initial yield surfaces for [0₄], [90₄], and [0/90]₂ fibrous aluminum laminates are well described by ellipses in the axial–shear stress plane having aspect ratios of 10, 2.5, and 3.3, respectively. For reference, the aspect ratio of the Mises ellipse for an isotropic material is 1.73. Initial yield surfaces do not have a tension–compression asymmetry. Four overload profiles (plus, ex, hourglass, and zee) are applied to characterize hardening of a [0/90]₂ laminate by constructing 30 subsequent yield surfaces. Parameters to describe the center and axes of an ellipse are regressed to the yield points. The results clearly indicate that kinematic hardening dominates so that material state evolution can be described by tracking the center of the yield locus. For a nonproportional overload of (σ, τ) = (500, 70) MPa, the center of the yield locus translated to (σ, τ) = (430, 37) MPa and the ellipse major axis was only 110 MPa.     

Gradient-enhanced softening material models

Classical constitutive models exhibit strong mesh dependency during softening and the numerical responses tend towards perfectly brittle behavior upon mesh refinements. Such sensitivity can be avoided by adopting the gradient-enhanced formulation. The implicit approach incorporates the gradient contributions indirectly via an additional Helmholtz equation and requires only C⁰ continuity. The explicit approach computes the gradient terms directly from the local field variables. Assuming a weak satisfaction of the yield function, C¹ continuity or C⁰ continuity with additional degrees of freedoms in the penalty approach is required. This makes the explicit method less attractive computationally. However, the explicit approach is able to fully regularize some material models where the standard implicit method fails to perform. Drawing analogy to the over-nonlocal integral formulation, the over-implicit-gradient framework is proposed. In addition, an alternative framework for the explicit gradient method requiring only C⁰ continuity is proposed. The regularizing effects of the abovementioned two gradient frameworks show promising applications to strain-softening materials.     

Debonding failure and size effects in micro-reinforced composites

Failure in micro-reinforced composites is investigated numerically using the strain-gradient plasticity theory of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52 (6) 1379–1406] in a plane strain visco-plastic formulation. Bi-axially loaded unit cells are used and failure is modeled using a cohesive zone at the reinforcement interface. During debonding a sudden stress drop in the overall average stress–strain response is observed. Adaptive higher-order boundary conditions are imposed at the reinforcement interface for realistically modeling the restrictions on moving dislocations as debonding occurs. It is found that the influence of the imposed higher-order boundary conditions at the interface is minor. If strain-gradient effects are accounted for a void with a smooth shape develops at the reinforcement interface while a smaller void having a sharp tip nucleates if strain-gradient effects are excluded. Using orthogonalization of the plastic strain gradient with three corresponding material length scales it is found that, the first length scale dominates the evaluated overall average stress–strain response, the second one only has a small effect and the third one has an intermediate effect. Finally, studies of reinforcement having elliptical cross-sections show rather significant gradients of stress which is not seen for the corresponding circular cross-sections. Also, an increased drop in the overall load carrying capacity is observed for cross-sections elongated perpendicular to the principal tensile direction compared to the corresponding circular cross-sections.     

A return-map algorithm for general associative isotropic elasto-plastic materials in large deformation regimes

The present paper addresses a flexible solution algorithm for associative isotropic elasto-plastic materials, i.e. for materials whose elastic and plastic behaviors are described through an isotropic free-energy function, an isotropic yield function and an associative flow rule. The discussion is relative to a large deformation regime, while no hardening mechanisms are included. The algorithm is based on a combination of the operator split method and a return map scheme. Both the algorithm linearization and the requirements for the yield criterion convexity are discussed in detail. Finally, to show the algorithm flexibility and performance, the discussion is specialized to three yield criteria and some test problems are studied.     

Electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators: experiment and finite element simulation

The effects of applied voltage on the electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators were examined in a combined experimental and numerical investigation. Experiments were performed to measure the strain near internal and surface electrodes at various electrical loading conditions. The finite element method was also used to solve the coupled electro-elastic boundary value problem. The strain, stress and electric displacement concentrations were calculated and a non-linear behavior induced by localized polarization switching was discussed. A comparison of strain concentration was made between experiment and simulation.     

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

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

Effects of characteristic material lengths on mode III crack propagation in couple stress elastic–plastic materials

The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions are investigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear and isotropic strain hardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinct material characteristic lengths. It can also capture the strong size effects arising at small scales, which results from the underlying microstructures. According to the asymptotic crack tip fields for a stationary crack provided by the indeterminate theory of couple stress elasticity, the effects of microstructure mainly consist in a switch in the sign of tractions and displacement and in a substantial increase in the singularity of tractions ahead of the crack-tip, with respect to the classical solution of LEFM and EPFM. The increase in the stress singularity also occurs for small values of the strain hardening coefficient and is essentially due to the skew-symmetric stress field, since the symmetric stress field turns out to be non-singular. Moreover, the obtained results show that the ratio η introduced by Koiter has a limited effect on the strength of the stress singularity. However, it displays a strong influence on the angular distribution of the asymptotic crack tip fields.     

Shakedown theory for elastic plastic kinematic hardening bodies

Shakedown static and kinematic theorems for elastic–plastic (generally nonlinear) kinematic hardening solids are derived in classical (path-independence) spirit with new constructions. The generally plastic-deformation-history-dependent hardening curve is assumed to be limited by the initial yield stress and ultimate yield strength, and to obey a positive hysteresis postulate for closed plastic cycles, but else can be arbitrary and unspecified. The theorems reveal that the shakedown of structures is not affected by the particular form of the hardening curve, but just by the initial and ultimate yield stresses. While the ultimate yield strength is clearly defined macroscopically and attached to the incremental collapse mode with unbounded plastic deformations, the initial yield stress, which is responsible for the bounded cyclic plasticity collapse mode, should not be taken as the convenient one at a fixed amount of plastic deformation (0.2%), but is suggested to be taken as low as the fatigue limit to preserve the classical load-history-independence spirit of the shakedown theorems. Otherwise, for our pragmatic application purpose, it may be given empirical values between the low fatigue limit and high ultimate yield stresses according to particular loading processes considered, which may range anywhere between the high-cycle and low-cycle ones. The theorems appear as simple as those of Melan and Koiter for perfect plasticity but applied to the much larger class of more realistic kinematic hardening materials.     

Micromechanical analysis of kinematic hardening in natural clay

This paper presents a micromechanical analysis of the macroscopic behaviour of natural clay. A microstructural stress–strain model for clayey material has been developed which considers clay as a collection of clusters. The deformation of a representative volume of the material is generated by mobilizing and compressing all the clusters along their contact planes. Numerical simulations of multistage drained triaxial stress paths on Otaniemi clay have been performed and compared the numerical results to the experimental ones in order to validate the modelling approach. Then, the numerical results obtained at the microscopic level were analysed in order to explain the induced anisotropy observed in the clay behaviour at the macroscopic level. The evolution of the state variables at each contact plane during loading can explain the changes in shape and position in the stress space of the yield surface at the macroscopic level, as well as the rotation of the axes of anisotropy of the material.     

Simulation and interpretation of diffuse mode bifurcation of elastoplastic solids

The diffuse mode bifurcation of elastoplastic solids at finite strain is investigated. The multiplicative decomposition of deformation gradient and the hyperelasto-plastic constitutive relationship are adapted to the numerical bifurcation analysis of the elastoplastic solids. First, bifurcation analyses of rectangular plane strain specimens subjected to uniaxial compression are conducted. The onset of the diffuse mode bifurcations from a homogeneous state is detected; moreover, the post-bifurcation states for these modes are traced to arrive at localization to narrow band zones, which look like shear bands. The occurrence of diffuse mode bifurcation, followed by localization, is advanced as a possible mechanism to create complex deformation and localization patterns, such as shear bands. These computational diffuse modes and localization zones are shown to be in good agreement with the associated experimental ones observed for sand specimens to ensure the validity of this mechanism. Next, the degradation of horizontal sway stiffness of a rectangular specimen due to plane strain uniaxial compression is pointed out as a cause of the bifurcation of the first antisymmetric diffuse mode, which triggers the tilting of the specimen. Last, circular and punching failures of a footing on a foundation are simulated.     

Modelling of the interface between a thin film and a substrate within a strain gradient plasticity framework

Interfaces play an important role for the plastic deformation at the micron scale. In this paper, two types of interface models for isotropic materials are developed and applied in a thin film analysis. The first type, which can also be motivated from dislocation theory, assumes that the plastic work at the interface is stored as a surface energy that is linear in plastic strain. In the second model, the plastic work is completely dissipated and there is no build-up of a surface energy. Both formulations introduce one length scale parameter for the bulk material and one for the interface, which together control the film behaviour. It is demonstrated that the two interface models give equivalent results for a monotonous, increasing load. The combined influence of bulk and interface is numerically studied and it is shown that size effects are obtained, which are controlled by the length scale parameters of bulk and interface.     

Yielding and intrinsic plasticity of Ti–Zr–Ni–Cu–Be bulk metallic glass

Bulk metallic glass with composition Ti₄₀Zr₂₅Ni₈Cu₉Be₁₈ exhibits considerably high compressive yield stress, significant plasticity (with a concomitant vein-like fracture morphology) and relatively low density. Yielding and intrinsic plasticity of this alloy are discussed in terms of its thermal and elastic properties. An influence of normal stresses acting on the shear plane is evidenced by: (i) the fracture angle (<45°) and (ii) finite-element simulations of nanoindentation curves, which require the use of a specific yield criterion, sensitive to local normal stresses acting on the shear plane, to properly match the experimental data. The ratio between hardness and compressive yield strength (constraint factor) is analyzed in terms of several models and is best adjusted using a modified expanding cavity model incorporating a pressure-sensitivity index defined by the Drucker–Prager yield criterion. Furthermore, comparative results from compression tests and nanoindentation reveal that deformation also causes strain softening, a phenomenon which is accompanied with the occurrence of serrated plastic flow and results in a so-called indentation size effect (ISE). A new approach to model the ISE of this metallic glass using the free volume concept is presented.     

Prediction of ductile fracture in tension by bifurcation,localization, and imperfection analyses

In this investigation, it is shown that the onset of ductile fracture in tension can be interpreted as the result of a supercritical bifurcation of homogeneous deformation and that this fact can be applied to predict ductile fracture initiation of materials with general imperfections or flaws. We focus on one dimensional quasi-static simple tension for rate-independent isotropic plastic materials. For deformation beyond the bifurcation point, multiple equilibrium paths appear. The homogeneous deformation, as one of the equilibrium paths, loses stability while the inhomogeneous paths are stable, thus indicating the occurrence of strain localization. This investigation also provides a physical example for the application of the Lambert W function in material localization analyses. Material instability is treated as the instability of a static system with dynamic perturbation. We also address the presence of microvoids in a power law plastic material as an unfolding of the supercritical pitchfork bifurcation. The imperfect system, idealized as spherical voids within the plastic matrix, is analyzed using the familiar Gurson model which is based on the presumption of a randomly voided material and characterized by the volume fraction of voids. If, in addition, the sizes of the microvoids are known, this then provides a length scale for the imperfection zone. In this manner, relevance to the sample size effects of strain-to-failure for ductile fracture initiation is addressed by considering separate zones with variations in void volume fractions. Fracture initiation predictions are presented and compare very well to existing experimental results.