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Discretisation Aspects of and Numerical Simulations with Higher-Order Gradient Models*

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Discretisation Aspects of and Numerical Simulations with Higher-Order Gradient Models* Discretisation Aspects of and Numerical Simulations with Higher-Order Gradient Models* Harm Askes Koiter Institute Delft / Delft University of Technology, Faculty of Civil Engineering and Geosciences, P.O. Box 5048, 2600 GA Delft, The Netherlands ABSTRACT Discretisation of and numerical analysis with higher-order gradient models can be cumbersome, since high continuity requirements may be put on the shape functions of a discretisation method. Thus, the comparison of different gradient models in boundary value problems is not straightforward. In this paper, the Element-Free Galerkin (EFG) method is proposed as a tool to discretise higher-order gradient models due to the high order of continuity that is easily attained with EFG shape functions. The methodology is illustrated by means of examples with gradient elasticity models and gradient damage models. Keywords: Element-Free Galerkin method, meshless methods, higher-order continuum, gradient elasticity, gradient damage. 1. INTRODUCTION The impact of microstructural processes on macrostructural behavior is nowadays widely recognised. Classical continua, in which the constitutive equation relates stresses to strains without any higher-order terms, do not contain parameters that are related to the microstructure. Thus, microstructural effects cannot be modelled straightforwardly with a classical continuum, unless each microstructural component is modelled separately. As an alternative, it has been proposed to enhance the classical continuum by higher-order terms. Various types of strategies can be distinguished. For instance, one can make a classification between the format of the "Presented at the Mini Symposium on Gradient Theory and Material Instabilities, 8-9 January 2001, Aristotle University of Thessaloniki - Polytechnic School, Thessaloniki, Greece. 149 Vol. 13, Nos. 3-4, 2002 Higher-Order Gradient Models higher-order term, being of the integral (or nonlocal) type /1-4/ or of the gradient type /5-8/. It can be argued that from a mechanical standpoint the two types of models are similar, in the sense that the spatial integral of a nonlocal model can be rewritten as an infinite series of spatial gradients /9-11/. However, from an implementational point of view gradient-dependent models appear to be preferable over nonlocal models. Namely, in nonlocal models the constitutive update has to be performed in two steps, consistent tangent matrices are difficult to implement in existing numerical codes /12,13/, and the combination between nonlocal models and adaptivity is cumbersome since the nonlocal connectivity between nodes and integration points has to be determined again every time the discretisation is changed. A second classification concerns the methodology with which the higher-order terms are added to the classical continuum. For example, the higher-order terms can simply be postulated, either in the strain energy functional /8,14/ or in the constitutive equations /7,10,15,16/. Consequently, relatively simple models can be developed that give a qualitative description of the physical processes under consideration. However, the link with the underlying microstructure remains phenomenological. As an alternative, so-called microstructural approaches can be taken in which homogenisation procedures of a (discrete) microstructure naturally lead to higher-order terms /17-19/. This yields a direct link with the underlying microstructure. Unfortunately, this approach can lead to unstable models for certain truncations of gradient series /20,21/. Due to the various strategies and motivations with which higher-order terms have been introduced, a straightforward comparison between different higher-order models is often difficult. A tool that may be used to compare different models is the analysis of dispersive waves /11,33,36,40/, but this implies that certain assumptions must be made, such as a homogeneous state and infinitesimal perturbations. For a more general comparison, boundary value problems and/or initial value problems should also be considered. For the general (multidimensional, nonlinear) case, this implies that numerical solution strategies must be taken. Especially for models with higher-order spatial gradients, restrictions may be put on the discretisation. Namely, the appearance of higher-order gradients increases the order of continuity that is required for the shape functions. For instance, the straightforward incorporation of second-order strain gradients in the constitutive equation leads to the necessity of C'-continuity, rather than C°-continuity, of the shape functions /7,10/ (a similar situation occurs in the discretisation of Bernoulli-Euler beam theory, where fourth-order derivatives appear in the field equations 1221). When finite elements are used, the need for C1 continuity of the shape functions dictates that special constructions such as mixed formulations 1211, Hermitian formulations /16,24/ or penalty functions /16/ should be employed. As an alternative, other discretisation methods may be used in which the shape functions intrinsically have a higher order of continuity. For instance, in the Element-Free Galerkin (EFG) method the shape functions can straightforwardly be formulated with an arbitrary order of continuity /25-27Λ Consequently, locking phenomena can easily be attenuated using EFG shape functions /25,28,29/ and the description of fields with steep gradients that are nonetheless smooth is excellent /30/. The higher-order of continuity also makes the EFG method a suitable tool to compare different higher-order gradient models in a simple manner /11,21,31-33/. This aspect of the EFG method will be explored in this contribution. After 150 Harm Askes Journal of the Mechanical Behavior of Materials reviewing briefly the formulation of EFG shape functions in Section 2, several numerical analyses with higher-order models will be carried out, and it will be shown that a comparison between different models is straightforward when the EFG method is used. In Section 3, two types of gradient elasticity models are formulated and compared in a wave propagation problem and a static problem where the occurrence of a size effect is studied. One of the models is of the phenomenological type, while the other is obtained through the microstructural approach, see above. In Section 4, three types of gradient damage models are compared to the underlying nonlocal (integral) damage model. A summary and perspectives for future research are given in Section 5. Boldface lowercase symbols denote vectors, and boldface uppercase symbols represent matrices. An underlined symbol denotes that this symbol has been discretised. 2. EFG SHAPE FUNCTIONS The Element-Free Galerkin (EFG) method is a meshless method, in which the shape functions are formulated in terms of a set of nodes without elements. Instead of employing elements to set a connectivity between nodes, each node is assigned a so-called domain of influence. Inside this domain of influence, a weight function corresponding to the associated node is nonzero, while it is zero outside the domain of influence. In this study a circular domain of influence with radius ίή„π is taken as 725,34/ f s2 ] f d2 exp -exp "infl 2 2 k Mi nfl) ( k Minfl) , d2 w(s) = "infl if s s d .•inf l 1-exp (1) Knfl)2 if s>d infl with s = |x - x,| and α = '/« a numerical parameter to set the relative weights inside the domain of influence. The weight function w(s) decays with increasing distance to the associated node, and it is non-negative in the entire domain of influence. The EFG shape functions are constructed in the following manner. The approximant function uh· is assumed to be the inner product of a monomial base vector ρ and a vector with coefficients a: uh = pr (x)a(x) (2) Vol. 13, Nos. 3-4, 2002 Higher-Order Gradient Models where for instance a complete quadratic monomial base vector in two dimensions is given by pr(x) = [l,*,;>>,x\xy,/] (3) The (yet unknown) coefficients a are solved for by minimising the moving least squares sum J given by j= i^wip^yi«-«,)2 (4) i=l where η is the number of nodes, w, is the weight function of node i, and u, is the discrete parameter of node /'. Minimisation of Equation (4) with respect to a yields (pr W(x)p) a(x) = (ργ W(x)) u (5) where u contains u, and Γ Ρ =[p(X1).P(x2)'-'P(X„)] (6) W(x) = diag [w|(x), w2(x), ...,w„ (x)] (7) From Eq. (5) the coefficients a(x) can be obtained, which are then substituted into Eq. (2). Then, EFG shape functions Η can be derived as /25,27/ h r r r T u = p (x)a(x) = p (x) (p W(x)p) '(p^Wix))« = H (x) u (8) The two main ingredients of the EFG shape functions are the monomial base vector ρ (or, more precisely, the terms that are included in p) and the weight functions w. The governing parameter in the weight function is the size of the domain of influence dmi\ as compared to the nodal spacing h. Smooth one-dimensional EFG r 2 3 shape functions and derivatives can be obtained, for instance, by taking p (x) = [1, x, x , x ] and dmS]/h = 5, as has been argued in Reference /11/. Figure 1 shows the corresponding EFG shape function and its first two derivatives. It can be verified that all derivatives are continuous. More details on the EFG method and its implementation can be found in References /25-27Λ In References /11,31/ specific details concerning the discretisation of higher-order gradient models are treated. In the sequel, a complete cubic monomial base vector ρ has been used for the discretisation of the displace- ments. Lagrange multipliers have been used to enforce the essential boundary conditions, and they have been discretised using a complete quadratic monomial base vector p. 152 Harm Askes Journal of the Mechanical Behavior of Materials Fig. 1: EFG shape function and its first and second derivative. 3. GRADIENT ELASTICITY In the gradient elasticity models treated here, the stresses are related to the strains and to the second derivatives of the strains.
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