Harm Askes

One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure: Part 1: Generic formulation

This paper is the first in a series of two that focus on gradient elasticity models derived from a discrete microstructure. In this first paper, a new continualization method is proposed in which each higher-order stiffness term is accompanied by a higher-order inertia term. As such, the resulting models are dynamically consistent. A new parameter is introduced that accounts for the nonlocal interaction between variables of the discrete model and of the continuous model. When this parameter is set to proper values, physically realistic behavior is obtained in statics as well as in dynamics. In this sense, the proposed methodology is superior to earlier approaches to derive gradient elasticity models, in which anomalies in the dynamic behavior have been found. A generic formulation of field equations and boundary conditions is given based on Hamiltons principle. In the second paper, analytical and numerical results of static and dynamic response of the second-order model and the fourth-order model will be treated.

Dispersion analysis and element‐free Galerkin solutions of second‐ and fourth‐order gradient‐enhanced damage models

Gradient-dependent damage formulations incorporate higher-order derivatives of state variables in the constitutive equations. Different formulations have been derived for this gradient enhancement, comparison of which is difficult in a finite element context due to higher-order continuity requirements for certain formulations. On the other hand, the higher-order continuity requirements are met naturally by element-free Galerkin (EFG) shape functions. Thus, the EFG method provides a suitable tool for the assessment of gradient enhanced continuum models. Dispersion analyses have been carried out to compare different gradient enhanced models with the non-local damage model. The formulation of the additional boundary conditions is addressed. Numerical examples show the objectivity with respect to the discretization and the differences between various gradient formulations with second- and fourth-order derivatives. It is shown that with the same underlying internal length scale, very different results can be obtained. Copyright

One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure: Part 2: Static and dynamic response

This paper is the second in a series of two that focus on gradient elasticity models retrieved from a discrete mictrostructure. In the first paper, the governing equations for a second-order model and a fourth-order model have been derived. In this paper, the proposed models are studied by means of static and dynamic examples, both from an analytical point of view and a numerical point of view. Details on the spatial discretization are provided. Finally, an experiment is proposed by which the newly introduced parameter can be determined that is responsible for the nonlocal relation between the continuous and the discrete field variables.

Two gradient plasticity theories discretized with the element-free Galerkin method

Abstract In this paper, the element-free Galerkin method is exploited to analyze gradient plasticity theories. The use of this discretization method has the advantage that higher-order continuity, which can be required for the plastic multiplier field, is provided for. In passing, we show that the regularization properties of the higher-order gradients are necessary, since, similar to finite element methods, a severe discretization sensitivity is encountered otherwise. The mesh-free discretization method is first applied to an established, stress-space gradient plasticity theory. Next, a strain-space gradient plasticity theory, which employs an implicit averaging of an invariant strain measure, is proposed and elaborated. This theory is insensitive to the order of the interpolation polynomials and, as a result, the convergence of the Newton–Raphson procedure is better. The performance of both gradient plasticity models is examined for the mesh-free discretization scheme. A comparative study is carried out for a one-dimensional bar in tension, while the influence of the discretization in biaxial compression is studied next, including that of the numerical parameters of the discretization scheme and that of the internal length scale contained in the gradient plasticity theories.

Numerical modeling of size effects with gradient elasticity - Formulation, meshless discretization and examples

A theory of gradient elasticity is used and numerically implemented by a meshless method to model size effects. Two different formulations of this model are considered, whereby the higher-order gradients are incorporated explicitly and implicitly, respectively. It turns out that the explicit gradient dependence leads to a straightforward spatial discretization, while use of the implicit gradient dependence can result in an awkward form of the stiffness matrix. For the numerical analyses the Element-Free Galerkin method has been used, due to its ability to incorporate higher-order gradients in a straightforward manner. Two boundary value problems have been considered, which show the capability of the gradient elasticity theory to capture size effects. In a follow-up paper, the formulation developed herein will be used to analyze additional configurations with attention to comparison with available experimental data on size effects and verification of available scaling laws for structural components.

Four simplified gradient elasticity models for the simulation of dispersive wave propagation

Gradient elasticity theories can be used to simulate dispersive wave propagation as it occurs in heterogeneous materials. Compared to the second-order partial differential equations of classical elasticity, in its most general format gradient elasticity also contains fourth-order spatial, temporal as well as mixed spatial-temporal derivatives. The inclusion of the various higher-order terms has been motivated through arguments of causality and asymptotic accuracy, but for numerical implementations it is also important that standard discretization tools can be used for the interpolation in space and the integration in time. In this paper, we will formulate four different simplifications of the general gradient elasticity theory. We will study the dispersive properties of the models, their causality according to Einstein and their behavior in simple initial/boundary value problems.

Remeshing strategies for adaptive ALE analysis of strain localisation

Computational analyses of strain localisation must be carried out in a proper, accurate, and efficient manner. While a continuum material model with an intrinsic length scale parameter can guarantee mesh-objective results, the ratio between accuracy and efficiency can be improved through the application of mesh adaptivity. In this paper, the Arbitrary Lagrangian Eulerian (ALE) technique is applied to strain localisation phenomena. Nodes are detached from the material, so that they can be used optimally in the spatial discretisation. A new remesh indicator is proposed that concentrates nodes in inelastic zones as well as in zones where strain localisation is likely to occur. Thus, the formation of new cracks is anticipated, which is crucial for accurate remeshing. Two algorithms have been tested and compared to equidistribute this remesh indicator. The algorithms give similar results, but the parabolic formulation can be solved explicitly. Multi-dimensional examples are presented. The proposed strategies are able to capture complicated crack patterns with multiple curved cracks. Two and three-dimensional examples illustrate this.

Higher-order strain/higher-order stress gradient models derived from a discrete microstructure, with application to fracture

Higher-order gradient models are derived from a discrete particle structure. A general set of constitutive equations is found in which zeroth and higher-order stress terms are related to zeroth and higher-order strain terms. As special cases, a model with only higher-order strain terms is considered as well as a model with both higher-order stress and higher-order strain terms. The model without higher-order stress is found to be unstable. The model with higher-order stress, on the other hand, is stable. The model with a higher-order stress term and a higher-order strain term can successfully be used to model softening phenomena.

Conditions for locking-free elasto-plastic analyses in the Element-Free Galerkin method

The use of finite elements with linear displacement fields in volume-preserving elasto-plastic problems can lead to volumetric locking. Due to the inability of the shape functions to describe certain displacement modes, a response of the structure can be obtained which is much too stiff. The Element-Free Galerkin (EFG) method can provide shape functions that have an intrinsic richer nature and do not exhibit volumetric locking. It is shown that the finite element (FE) method can be conceived as a specialisation of the EFG method, and that for a specific choice of the EFG parameters the FE shape functions are retrieved. Then, volumetric locking behaviour is found again. However, for other choices of the EFG parameters shape functions are obtained that are able to describe isochoric elasto-plasticity without volumetric locking.

An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice

This paper presents a derivation of a second-order isotropic continuum from a 2D lattice. The derived continuum is isotropic and dynamically consistent in the sense that it is unconditionally stable and prohibits the infinite speed of energy propagation. The Lagrangian density of the continuum is obtained from the Lagrange function of the underlying lattice. This density is used to obtain the expressions for standard and higher-order stresses in direct correspondence with the equations of the continuum motion. The derived continuum is characterized by two additional parameters relative to the classical elastic continuum. These are the characteristic lengthscale and a dimensionless continualization parameter, which characterizes indirectly the timescale of the derived continuum. The margins for the latter parameter are found from the stability analysis. It is envisaged that the continualization parameter could be measured employing a high-frequency pulse propagating along the surface of the continuum. Excitation and propagation of such pulse is studied theoretically in this paper.