A boundary element formulation for a class of non-local damage models

This paper proposes and discusses a boundary element formulation for a particular class of non-local damage models. The formulation as well as the boundary element computational code developed during this research have proven to be very simple and efficient, providing reliable information on the strains and stresses in damage-softening models. The numerical approach uses a finite-grid to estimate the damage values. Some illustrative numerical examples, which show the simplicity and versatility of the proposed approach, are included and discussed in detail.     

A general class of predation models with multiplicative Allee effect

A class of models of predator–prey interaction with Allee effect on the prey population is presented. Both the Allee effect and the functional response are modelled in the most simple way by means of general terms whose conveniently chosen mathematical properties agree with, and generalise, a number of concrete Leslie–Gower-type models. We show that this class of models is well posed in the sense that any realistic solution is bounded and remains non-negative. By means of topological equivalences and desingularization techniques, we find specific conditions such that there may be extinction of both species. In particular, the local basin boundaries of the origin are found explicitly, which enables one to determine the extinction or survival of species for any given initial condition near this equilibrium point. Furthermore, we give conditions such that an equilibrium point corresponding to a positive steady state may undergo saddle-node, Hopf and Bogdanov–Takens bifurcations. As a consequence, we are able to describe the dynamics governed by the bifurcated limit cycles and homoclinic orbits by means of carefully sketched bifurcation diagrams and suitable illustrations of the relevant invariant manifolds involved in the overall organisation of the phase plane. Finally, these findings are applied to concrete model vector fields; in each case, the particular relevant functions that define the conditions for the associated bifurcations are calculated explicitly.     

On a damage law for creep rupture of clays with accumulated inelastic deviatoric strain as a damage measure

To avoid the dependency on origin of time, an improved damage law for creep rupture of clays is proposed considering the accumulated inelastic deviatoric strain as a measure of damage, instead of incorporating time directly. This law is incorporated into an existing anisotropic elastoplastic-viscoplastic bounding surface model for clays. The performance of the damage law was demonstrated via the simulations of creep rupture tests on undisturbed clays, and generally a good agreement between model simulations and test data was obtained. Discussions on the creep rupture parameters were followed and further improvement was suggested. At present when high quality test data for creep rupture is very limited, the proposed damage law could serve as a practical way to model creep rupture of clays.     

From the second gradient operator and second class of integral theorems to Gaussian or spherical mapping invariants

By combining of the second gradient operator, the second class of integral theorems, the Gaussian-curvature-based integral theorems and the Gaussian （or spherical） mapping, a series of invariants or geometric conservation quantities under Gaussian （or spherical） mapping are revealed. From these mapping invariants important transformations between original curved surface and the spherical surface are derived. The potential applications of these invariants and transformations to geometry are discussed     

The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models

We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.     

Extension of covariant derivative (III): From classical gradient to shape gradient

This paper further extends the generalized covariant derivative from the first covariant derivative to the second one on curved surfaces. Through the linear transformation between the first generalized covariant derivative and the second one, the second covariant differential transformation group is set up. Under this transformation group, the second class of differential invariants and integral invariants on curved surfaces is made clear. Besides, the symmetric structure of the tensor analysis on curved surfaces are revealed.     

On coupled gradient-dependent plasticity and damage theories with a view to localization analysis

Combinations of gradient plasticity with scalar damage and of gradient damage with isotropic plasticity are proposed and implemented within a consistently linearized format. Both constitutive models incorporate a Laplacian of a strain measure and an internal length parameter associated with it, which makes them suitable for localization analysis.The theories are used for finite element simulations of localization in a one-dimensional model problem. The physical relevance of coupling hardening/softening plasticity with damage governed by different damage evolution functions is discussed. The sensitivity of the results with respect to the discretization and to some model parameters is analyzed. The model which combines gradient-damage with hardening plasticity is used to predict fracture mechanisms in a Compact Tension test.     

Validation and internal length scale determination for a gradient damage model: application to short glass-fibre-reinforced polypropylene

A strain-based transient-gradient damage model is used to analyse and describe the experimentally observed failure process in a Compact-Tension test carried out on short glass-fibre-reinforced polypropylene. Several aspects regarding the nonlocal character of the damage process in the material are emphasized and the intrinsic length scale is determined using available strain fields from an experimental analysis. A good agreement between theory and experiments has been found on a global and on a local level.     

Chaos for a class of linear kinetic models

In recent years it was observed that chaotic behaviour can occur in some infinite–dimensional linear systems. An example of this type, related to a kinetic model (death process), has been previously reported. In this paper we generalize these earlier results to the case of variable coefficients, showing that the property of being chaotic can be in a certain sense stable. On the other hand the ‘opposite’ birth process cannot be chaotic.     

Limit analysis for a general class of yield conditions

The paper describes a generalisation of the programming method described by Ponter and Carter (1997) for the evaluation of optimal upper bounds on the limit load of a body composed of a rigid/perfectly plastic material. The method is based upon similar principles to the `Elastic Compensation' method which has been used for design calculations for some years but re-interpreted as a non-linear programming method. A sufficient condition for convergence is derived which relates properties of the yield surface to those of the linear solutions solved at each iteration. The method is demonstrated through an application to a Drucker–Prager yield condition in terms of the Von Mises effective stress and the hydrostatic pressure. Implementation is shown to be possible using the user routines in a commercial finite element code, ABAQUS. The examples chosen indicate that stable convergent solutions may be obtained. There are, however, limits to the application of the method if isotropic linear solutions are used for an isotropic yield surface. In an accompanying paper (Ponter and Engelhardt, 2000) the method is extended to shakedown and related problems.     

A unified energy approach to a class of micromechanics models for composite materials

Several micromechanics models for the determination of composite moduli are investigated in this paper, including the dilute solution, self-consistent method, generalized self-consistent method, and Mori-Tanaka's method. These micromechanical models have been developed by following quite different approaches and physical interpretations. It is shown that all the micromechanics models share a common ground, the generalized Budiansky's energy-equivalence framework. The difference among the various models is shown to be the way in which the average strain of the inclusion phase is evaluated. As a bonus of this theoretical development, the asymmetry suffered in Mori-Tanaka's method can be circumvented and the applicability of the generalized self-consistent method can be extended to materials containing microcracks, multiphase inclusions, non-spherical inclusions, or non-cylindrical inclusions. The relevance to the differential method, double-inclusion model, and Hashin-Shtrikman bounds is also discussed. The application of these micromechanics models to particulate-reinforced composites and microcracked solids is reviewed and some new results are presented.     

Shakedown analysis for a class of strengthening materials within the framework of gradient plasticity

The classical shakedown theory is extended to a class of perfectly plastic materials with strengthening effects (Hall–Petch effects). To this aim, a strain gradient plasticity model previously advanced by Polizzotto (2010) is used, whereby a featuring strengthening law provides the strengthening stress, i.e. the increase of the yield strength produced by plastic deformation, as a degree-zero homogeneous second-order differential form in the accumulated plastic strain with associated higher order boundary conditions. The extended static (Melan) and kinematic (Koiter) shakedown theorems are proved together with the related lower bound and upper bound theorems. The shakedown limit load problem is addressed and discussed in the present context, and its solution uniqueness shown out. A simple micro-scale structural system is considered as an illustrative example. The shakedown limit load is shown to increase with decreasing the structural size, which is a manifestation of the classical Hall–Petch effects in a context of cyclic loading.     

Sensitivity of a general class of shape functionals to topological changes

The topological derivative represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations. The topological derivative has been successfully applied in the treatment of problems such as topology optimization, inverse analysis and image processing. In this paper, the calculation of the topological derivative for a general class of shape functionals is presented. In particular, we evaluate the topological derivative of a modified energy shape functional associated to the steady-state heat conduction problem, considering the nucleation of a small circular inclusion as the topological perturbation. Several methods were proposed to calculate the topological derivative. In this paper, the so-called topological-shape sensitivity method is extended to deal with a modified adjoint method, leading to an alternative approach to calculate the topological derivative based on shape sensitivity analysis together with a modified Lagrangian method. Since we are dealing with a general class of shape functionals, which are not necessarily associated to the energy, we will show that this new approach simplifies the most delicate step of the topological derivative calculation, namely, the asymptotic analysis of the adjoint state.     

Stability of a class of neural network models with delay

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On a general concept for the analysis of crack growth and material damage

For finite strain dynamics a variational model of crack evolution is formulated within the generalized oriented continuum methodology. In this approach position- and direction-dependent deformation and strain measures are used to describe the (macro)motion of the body with defects, which may evolve relative to the moving body. The inelastic behaviour of continua with evolving defects is represented by phenomenological equations including the transversal crack evolution. A strain-induced crack propagation criterion is defined by the difference between the strain energy release rate and the rate of the surface energy of the crack. A possible nucleation of microcracks in terms of the average drag coefficient of the crack configuration is proposed. Based on the crack growth criterion presented in this paper, the kinking of cracks is investigated using variational concepts. A constitutive damage model of Kachanov's type accounting for the crack density is derived in terms of the free energy functional and a dissipation potential.     

A note on strain localization for a class of non-associative plasticity rules

Summary Explicit solutions for the formation of discontinuity bands are obtained, for a class of non-associative flow rules. Specialization to particular yield functions for pressure sensitive, dilatant or compactive materials is given.

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Bemerkungen zur Lokalisierung der Verformungen für eine Klasse von nicht assoziierten plastischen Fließgesetzen

Übersicht Es werden explizite Lösungen für die Bildung von Unstetigkeitsflächen bei einer Klasse von nicht assoziierten plastischen Fließgesetzen hergeleitet. Diese werden insbesondere für einige spezielle Fließgesetze diskutiert, die zur Beschreibung von druckempfindlichen, dilatierenden oder kontrahierenden Materialien geeignet sind.

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Presented at the workshop on Numerical Methods for Localization and Bifurcation of Granular Bodies, held at the Technical University of Gdansk (Poland), September 25–30, 1989     

Finite element methods for a class of continuum models for immiscible flows with moving contact lines

In this paper, we present a finite element method for two‐phase incompressible flows with moving contact lines. We use a sharp interface Navier–Stokes model for the bulk phase fluid dynamics. Surface tension forces, including Marangoni forces and viscous interfacial effects, are modeled. For describing the moving contact lines, we consider a class of continuum models that contains several special cases known from the literature. For the whole model, describing bulk fluid dynamics, surface tension forces, and contact line forces, we derive a variational formulation and a corresponding energy estimate. For handling the evolving interface numerically, the level‐set technique is applied. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space. We apply a Nitsche technique to weakly impose the Navier slip conditions on the solid wall. A unified approach for discretization of the (different types of) surface tension forces and contact line forces is introduced. Results of numerical experiments are presented, which illustrate the performance of the solver. Copyright © 2016 John Wiley & Sons, Ltd.     

A thermodynamic method for the construction of a cohesive law from a nonlocal damage model

Several published papers deal with the possibility of replacing a damage finite element model by a combination of cohesive zones and finite elements. The focus of the paper is to show under which conditions this change of model can be done in an energy-wise manner.The objective is to build a cohesive model based on a known damage model, without making any assumption on the shape of the cohesive law. The method is characterized, on the one hand, by the use of a well-defined thermodynamic framework for the cohesive model and, on the other hand, by the idea that the main quantity which must be maintained through the change of model is the energy dissipated by the structure. An analysis of the stability criteria enables us to determine the domains of validity of the different models. Thus, we show that it is consistent to derive the cohesive law from a given nonlocal damage model because the occurrence of a discontinuity can be viewed as an alternative way to limit localization. The method is illustrated on one-dimensional examples and a numerical resolution method for the problem with a cohesive zone is presented.