A thermodynamic approach to analyze shear localization in semi-solid materials

A theoretical framework of the shear localization analysis was developed for semi-solid materials taking into account a non-equilibrium relationship between viscous deformation, pressure and interfacial surface energy. Considering a shear layer model, the necessary condition of perturbation growth and subsequent shear localization was derived. The results revealed that the localization phenomenon in the semi-solid deformation strongly depends on the difference between irreversible viscous work done on pores and grains and the reversible viscous deformational work stored as the interfacial surface energy. This thermodynamic quantity indicates the possibility of a perturbation growth or decade in terms of the process parameters such as dilatancy, permeability and also the fraction of the solid skeleton.     

A thermodynamical gradient theory for deformation and strain localization of porous media

In this work, a thermodynamically consistent gradient formulation for partially saturated cohesive-frictional porous media is proposed. The constitutive model includes a classical or local hardening law and a softening formulation with state parameters of non-local character based on gradient theory. Internal characteristic length in softening regime accounts for the strong shear band width sensitivity of partially saturated porous media regarding both governing stress state and hydraulic conditions. In this way the variation of the transition point (TP) of brittle-ductile failure mode can be realistically described depending on current confinement condition and saturation level. After describing the thermodynamically consistent gradient theory the paper focuses on its extension to the case of partially saturated porous media and, moreover, on the formulation of the gradient-based characteristic length in terms of stress and hydraulic conditions. Then the localization indicator for discontinuous bifurcation is formulated for both drained and undrained conditions.     

A finite deformation theory of strain gradient plasticity

Plastic deformation exhibits strong size dependence at the micron scale, as observed in micro-torsion, bending, and indentation experiments. Classical plasticity theories, which possess no internal material lengths, cannot explain this size dependence. Based on dislocation mechanics, strain gradient plasticity theories have been developed for micron-scale applications. These theories, however, have been limited to infinitesimal deformation, even though the micro-scale experiments involve rather large strains and rotations. In this paper, we propose a finite deformation theory of strain gradient plasticity. The kinematics relations (including strain gradients), equilibrium equations, and constitutive laws are expressed in the reference configuration. The finite deformation strain gradient theory is used to model micro-indentation with results agreeing very well with the experimental data. We show that the finite deformation effect is not very significant for modeling micro-indentation experiments.     

A finite deformation second gradient theory of plasticity

Extending the previous work by Chambon et al. [2] to the finite deformation regime, a local second gradient theory of plasticity for isotropic materials with microstructure is developed based on the multiplicative decomposition of the deformation gradient, the additive decomposition of the second deformation gradient and the principle of maximum dissipation.     

A finite deformation theory of higher-order gradient crystal plasticity

For higher-order gradient crystal plasticity, a finite deformation formulation is presented. The theory does not deviate much from the conventional crystal plasticity theory. Only a back stress effect and additional differential equations for evolution of the geometrically necessary dislocation (GND) densities supplement the conventional theory within a non-work-conjugate framework in which there is no need to introduce higher-order microscopic stresses that would be work-conjugate to slip rate gradients. We discuss its connection to a work-conjugate type of finite deformation gradient crystal plasticity that is based on an assumption of the existence of higher-order stresses. Furthermore, a boundary-value problem for simple shear of a constrained thin strip is studied numerically, and some characteristic features of finite deformation are demonstrated through a comparison to a solution for the small deformation theory. As in a previous formulation for small deformation, the present formulation applies to the context of multiple and three-dimensional slip deformations.     

A direct approach to fiber and membrane reinforced bodies. Part I. Stress concentrated on curves for modelling fiber reinforced materials

An approach is outlined to the equilibrium in fiber-reinforced materials in which the fibers are modeled as curves or lines with concentrated material properties. The system of forces representing the interaction of the fibers with the bulk matter is analyzed, and equilibrium of forces is derived from global laws. The displacements of the bulk matter are assumed to have continuous extension to the fibers. This forces the set of admissible deformations superquadratically integrable. This in turn forces the energy of the bulk of superquadratic growth. The material of the bulk matrix therefore cannot be linearly elastic. The energy of fibers can have a slower growth and can be quadratic. A formal set of assumptions is given under which an equilibrium state of minimum energy exists in the given external conditions. A weak form of equilibrium equations is derived for this equilibrium state. An explicitly calculable axisymmetric example is presented with an isotropic and quadratic energy of the matrix (linear elasticity) and linearly stretchable fiber. Since the superquadratic growth assumption is not satisfied, some peculiar features of the solution arise, such as the infinite limit of the radial displacement near the fiber. Nevertheless, from the obtained solution, we can compute the normal force in the fiber and the shear stress at the interface.     

A micromechanics approach to homogenizing elasto-viscoplastic heterogeneous materials

The variational asymptotic method for unit cell homogenization (VAMUCH) has emerged as a general-purpose micromechanics code capable of predicting the effective properties of heterogeneous materials and recovering the local fields. The objective of this paper is to propose a micromechanics approach enabling VAMUCH to homogenize elasto-viscoplastic heterogeneous materials. An affine formulation of the constitutive relations for an elasto-viscoplastic constituent, which exhibits viscoplastic anisotropy and combined isotropic–kinematic hardening, is derived. The weak form of the problem is derived using an asymptotic method, discretized using finite elements, and implemented into VAMUCH. The new features of VAMUCH are validated with examples such as homogenizing binary, fiber-reinforced, and particle-reinforced composites. VAMUCH is found to be capable of handling various microstructure, complex material models, complex loading conditions, and complex loading paths. More sophisticated material models can be implemented into it.     

A theory of inelastic behavior of materials Part I. Ideal inelastic materials

Archive for Rational Mechanics and Analysis -     

Modeling multi-axial deformation of planar anisotropic elasto-plastic materials,part I: Theory

In sheet metal forming processes local material points can experience multi-axial and multi-path loadings. Under such loading conditions, conventional phenomenological material formulations are not capable to predict the deformation behavior within satisfying accuracy. While micro-mechanical models have significantly improved the understanding of the deformation processes under such conditions, these models require large sets of material data to describe the micromechanical evolution and quite enormous computation expenses for industrial applications. To reduce the drawbacks of phenomenological material models under the multi-path loadings a new anisotropic elasto-plastic material formulation is suggested. The model enables the anisotropic yield surface to grow (isotropic hardening), translate (kinematic hardening) and rotate (rotation of the anisotropy axes) with respect to the deformation, while the shape of the yield surface remains essentially unchanged.Essentially, the model is formulated on the basis of an Armstrong–Frederick type kinematic hardening, the plastic spin theory for the reorientation of the symmetry axes of the anisotropic yield function, and additional terms coupling these expressions. The capability of the model is illustrated with multi-path loading simulations in ‘tension-shear’ and ‘reverse-shear’ to assess its performance with ‘cross’ hardening and ‘Bauschinger’ effects.     

A crack in functionally gradient materials under thermal shock

Summary This paper deals with the problem of two bonded semi-infinite functionally gradient material plates with a crack at the interface under thermal shock loading conditions. All material properties are supposed to be exponentially dependent on the distance from the crack line. By using both the Laplace transform and the Fourier transform, the problem is reduced to a singular integral equation which is solved numerically. The stress intensity factor versus time for various material constants is calculated. The results show that by selecting the material constants appropriately, the stress intensity factor can be lowered substantially.

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Ein Riß im Funktionalgradientenmaterial unter einem thermischen Schock

Übersicht Die Arbeit behandelt das Problem zweier beschichteter Platten aus einem Funktionalgradientenmaterial mit einem Riß entlang der Verbindungsfläche unter einer thermischen Schockbeanspruchung. Die Materialeigenschaften hängen exponentiell vom Abstand von der Bruchlinie ab. Durch kombinierte Anwendung der Laplace- und der Fourier-Transformation wird das Problem auf eine singuläre Integralgleichung reduziert, die numerisch gelöst wird. Daraufhin wird der Spannungsintensitätsfaktor als Funktion der Zeit für mehrere Sätze von Materialkonstanten berechnet. Es zeigt sich, daß der Spannungsintensitätsfaktor durch eine geeignete Wahl der Materialkonstanten beträchtlich reduziert werden kann.

     

Modelling of localization and propagation of phase transformation in superelastic SMA by a gradient nonlocal approach

In this work, a nonlocal phenomenological behavior model is proposed in order to describe the localization and propagation of stress-induced martensite transformation in shape memory alloy (SMA) wires and thin films. It is a nonlocal extension of an existing local model that was derived from a micromechanical-inspired Gibbs free energy expression. The proposed model uses, besides the local field of the internal variable, namely the martensite volume fraction, a nonlocal counterpart. This latter acts as an additional degree of freedom, which is determined by solving an additional partial differential equation (PDE), derived so as to be equivalent to the integral definition of a nonlocal quantity. This PDE involves an internal length parameter, dictating the global scale at which the nonlocal interactions of the underlying micromechanisms are manifested during phase transformation. Moreover, to account for the unstable softening behavior, the transformation yield force parameter is considered as a gradually decreasing function of the martensite fraction. Possible material and geometric imperfections that are responsible for localization initiation are also considered in this analysis. The obtained constitutive equations are implemented in the Abaqus® finite element code in one and two dimensions. This requires the development of specific finite elements having the nonlocal volume fraction variable as an additional degree of freedom. This implementation is achieved through the UEL user’s subroutine. The effect of martensitic localization on the superelastic global behavior of SMA wire and holed thin plate, subjected to tension loading, is analyzed. Numerical results show that the developed tool correctly captures the commonly observed unstable superelastic behavior characterized by nucleation and propagation of martensitic phase. In particular, they show the influence of the internal length parameter, appearing in the nonlocal model, on the size of the localization area and the stress nucleation peak.     

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

This paper develops a novel nonlinear numerical method to perform shakedown analysis of structures subjected to variable loads by means of nonlinear programming techniques and the displacement-based finite element method. The analysis is based on a general yield function which can take the form of most soil yield criteria (e.g. the Mohr–Coulomb or Drucker–Prager criterion). Using an associated flow rule, a general yield criterion can be directly introduced into the kinematic theorem of shakedown analysis without linearization. The plastic dissipation power can then be expressed in terms of the kinematically admissible velocity and a nonlinear formulation is obtained. By means of nonlinear mathematical programming techniques and the finite element method, a numerical model for kinematic shakedown analysis is developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load can be calculated. An effective, direct iterative algorithm is then proposed to solve the resulting nonlinear programming problem. The calculation is based on the kinematically admissible velocity with one-step calculation of the elastic stress field. Only a small number of equality constraints are introduced and the computational effort is very modest. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples.     

A lower bound approach to the yield loci of porous materials

A lower bound approach is proposed for the first time to solve the macroscopic yield loci of porous materials. The results are then compared with Gurson's upper bound yield loci and those of the experiments. It is shown that the present analysis is much more in accordance with the experimental results than the Gurson's.     

Suppressed plastic deformation at blunt crack-tips due to strain gradient effects

Large deformation gradients occur near a crack-tip and strain gradient dependent crack-tip deformation and stress fields are expected. Nevertheless, for material length scales much smaller than the scale of the deformation gradients, a conventional elastic–plastic solution is obtained. On the other hand, for significant large material length scales, a conventional elastic solution is obtained. This transition in behaviour is investigated based on a finite strain version of the Fleck–Hutchinson strain gradient plasticity model from 2001. The predictions show that for a wide range of material parameters, the transition from the conventional elastic–plastic to the elastic solution occurs for length scales ranging from 0.001 times the size of the plastic zone to a length scale of the same order of magnitude as the plastic zone.     

A second gradient theoretical framework for hierarchical multiscale modeling of materials

A theoretical framework for the hierarchical multiscale modeling of inelastic response of heterogeneous materials is presented. Within this multiscale framework, the second gradient is used as a nonlocal kinematic link between the response of a material point at the coarse scale and the response of a neighborhood of material points at the fine scale. Kinematic consistency between these scales results in specific requirements for constraints on the fluctuation field. The wryness tensor serves as a second-order measure of strain. The nature of the second-order strain induces anti-symmetry in the first-order stress at the coarse scale. The multiscale internal state variable (ISV) constitutive theory is couched in the coarse scale intermediate configuration, from which an important new concept in scale transitions emerges, namely scale invariance of dissipation. Finally, a strategy for developing meaningful kinematic ISVs and the proper free energy functions and evolution kinetics is presented.     

A viscoplastic strain gradient analysis of materials with voids or inclusions

A finite strain viscoplastic nonlocal plasticity model is formulated and implemented numerically within a finite element framework. The model is a viscoplastic generalisation of the finite strain generalisation by Niordson and Redanz (2004) [Journal of the Mechanics and Physics of Solids 52, 2431–2454] of the strain gradient plasticity theory proposed by Fleck and Hutchinson (2001) [Journal of the Mechanics and Physics of Solids 49, 2245–2271]. The formulation is based on a viscoplastic potential that enables the formulation of the model so that it reduces to the strain gradient plasticity theory in the absence of viscous effects. The numerical implementation uses increments of the effective plastic strain rate as degrees of freedom in addition to increments of displacement. To illustrate predictions of the model, results are presented for materials containing either voids or rigid inclusions. It is shown how the model predicts increased overall yield strength, as compared to conventional predictions, when voids or inclusions are in the micron range. Furthermore, it is illustrated how the higher order boundary conditions at the interface between inclusions and matrix material are important to the overall yield strength as well as the material hardening.