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

Harm Askes

Koiter Institute Delft / Delft University of Technology, Faculty of Civil and Geosciences, P.O. Box 5048, 2600 GA Delft, The Netherlands

ABSTRACT

Discretisation of and 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 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

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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.

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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. More elaborated formulations exist, but simple models suffice for the present purpose of showing the differences between a phenomenological gradient elasticity model and a gradient elasticity model derived from a microstructure.

3.1. Formulation

Below, restriction is made to one-dimensional formulations for reasons of clarity. Extension towards multiple spatial dimensions is straightforward and can be found, e.g., in References /20,35/. In order to introduce higher-order strain gradients, an additional term can be postulated in the strain

153 Vol. 13, Νos. 3-4, 2002 Higher-Order Gradient Models

energy density function W as

ζ 2 ,21 fV = jE ε +r\ — & (9)

where the /2-term is the enhancement with respect to the classical continuum, and I is an additional (material) parameter with the dimension of length. In the format of Equation (9), the higher-order term is always non- negative, and therefore stability of the gradient-enhanced model is guaranteed. The stress σ is found via σ = dfV/de. To arrive at a constitute relation in the conventional format, Equation (9) is rewritten as

f 2n 2 2 (Λ] W UE ε +1 d£ = άε (10) de J 2 J 2 < / I X ) whereby integration by parts has been used for the higher-order term and boundary terms have been neglected. Via W = JV d ε the stress-strain relation can be retrieved as

2£V σ=Ε ε-Ι (11)

Note that in Equation (11) the higher-order term is preceded by a negative sign, which is a direct consequence of the positive sign of the higher-order term in Equation (9). As an alternative to derive a gradient elasticity model, a discrete chain of particles and springs can be homogenised. Assume for simplicity that all inter-particle springs have the same stiffness K, that all particles have the same mass M, and that the interparticle distance d is constant in the domain. Then, the equation of motion for particle η reads

Mün + Κ (-Un+ι + 2un - κ*.,) = 0 (12)

In the homogenisation procedure, the continuous displacement u(x) equals the discrete displacement un at position xn. Furthermore, u(x ± d) = u„±l can be expressed in terms of u(x) using Taylor series expansions, leading to

- = + + (13) with the mass density ρ = M/Ad and Young's modulus Ε = Kd/A, where A is the cross-sectional area. When the kinematic relation ε = du/dx is used, and the equation of motion of the continuum is expressed

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in terms of stresses as ρ κ = dalebe, the constitutive relation can be retrieved as

(14)

It can be seen that the second-order truncation of Equation (14) differs from Equation (11) in one significant aspect, namely the sign of the higher-order term. Consequently, the higher-order term in Equation (14) leads to a negative contribution to the strain energy density, and therefore this term is destabilising.

3.2. Discretisation aspects

For the numerical analyses, the gradient-enhanced constitutive equations must be discretised together with the equilibrium equations and the kinematic relations. The starting point is the weak formulation of the three-dimensional equation of motion

J <5uT (p ü - l/σ - b) dfi = 0 (15) in which 5u contains the virtual displacements that correspond to the stress field σ, b contains the body forces, and L is the :

d_ d d 0 0 0 dx dy dz d_ d_ d LT = 0 υπ (16) dy dx dz d d_ d Au 0 dz dy dx

Integrating the stress term by parts yields

J 6uTpüdü + J δετσάη = J

with t the prescribed tractions on the free part of the boundary Γ„ and ε= Lu Next, the general constitutive equation σ= D (ε- cV2e) is substituted, in which D contains the elastic moduli. Furthermore, c = I1 according to Equation (11) and c = -j^d2 in case the second-order truncation

of Equation (14) is considered. Integrating the higher-order term by parts leads to

de JSuTpü

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where a summation over ξ = xt, x2, x3 is implied. Due to the higher-order terms that have appeared in the constitutive relation, also higher-order boundary conditions are needed. Here, it is assumed that the deri- vatives of evanish on the boundary. Hence, the last boundary integral in Eq. (18) cancels. Discretising the test and trial functions via 5u = H5u and u = Hu, and requiring that Eq. (18) holds for any admissible 5u leads to

Μ ii + [K0 + Kj] μ = f"" (19) in which the mass matrix

M. = J HTp Η <1Ω (20) n and the external force vector fxt is written as

r< = JnTbdn + JnTtdr (21) Ω Γ,

The standard stiffness matrix K0 and the higher-order stiffness matrix K| are given by

T K0 = J B DB dn (22)

ξ = χ1,*2,*3 (23) with Β = LH. From Eq. (19) it can be seen that the only difference with a classical continuum model concerns the higher-order stiffness matrix. For this term, second (rather than first) derivatives of the shape functions have to be computed.

3.3. Wave propagation

Firstly, a one-dimensional problem is studied in which a shock wave propagates through a bar. The geometry and loading conditions are given in Figure 2. The force amplitude is F0 = 5 N, Young's modulus is 2 4 taken as Ε = 1000 MPa and the mass density is ρ = 1000 Ns /mm , so that the elastic bar velocity is ce = yjE / ρ = 1 mm/s. For the EFG discretization 81 equally-spaced nodes and 400 equally-spaced integration points have been used. For the time discretization the implicit Newmark scheme is used /22,36/ with time step At = 0.2 s and Newmark parameters γ = 0.5 and β = 0.25. For a straightforward comparison between the models of Equations (11) and (14) the same internal length parameter is used for both models, putting simply d=lj\2 .

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A =1 mm'

->x -> F

20 mm

Fig. 2: Wave propagation in a bar - geometry and loading conditions (top) and load-time diagram (bottom).

Firstly, the gradient elasticity model according to Equation (11) is studied. In Figure 3, the strain profiles at time / = 4 s, ί = 8 s, / = 12s and t= 16 s are plotted for different values of the internal length scale d. For non-zero values of the internal length scale, it can be seen that the shape of the wave front smoothens with increasing time. This effect is more pronounced for larger values of d, and it is absent for d = 0. Indeed, the need to describe wave dispersion properly has been one of the motivations to include higher-order gradients in the classical continuum. However, it must be noted that due to the specific format of this model, the wave front travels with a velocity that is higher than the elastic bar velocity ce, which is not entirely realistic /20,37/. Next, the model of Equation (14) is investigated. The same loading conditions and material parameters as above have been used, with an internal length scale d = 2 mm. In Figure 4 the propagation of the wave is plotted for the first two time steps. It can be seen that after the first time increment the influence of the shock wave is present in the entire bar, which is unrealistic. After the second time increment, the amplitude of the strain profile increases in an unphysical manner to unrealistically large values. This is a manifestation of the unstable character of this model /37/, and it must be concluded that this model is not suitable for dynamic analyses.

157 Vol. 13, Νos. 3-4, 2002 Higher-Order Gradient Models

> (/>

— II

i w > CO -

' (It ' 2> CM ? < ι

( -

in in § §

[-] 3 UIEJJS [-] 3 U|EJ|S

[-] 3 UIEJ1S [-] 3 uiejis

158 Harm Askes Journal of the Mechanical Behavior of Materials

χ [mm]

Fig. 4: Wave propagation in a bar for model of Equation (14) - strain profiles for first two time steps.

3.4. Modelling of size effect

The second context in which the two gradient elasticity models are compared is that of a static, two- dimensional plane strain example as depicted in Figure 5. Also the EFG discretisation is given. For reasons of only one quarter of the square plate is modelled. An imposed displacement of 0.05 mm has been applied. Young's modulus Ε = 1000 MPa and Poisson's ratio ν = 0.25. Firstly, the models of Equations (11) and (14) are compared for a hole radius R = 3 mm and an internal length scale l = d / -Jl2 = 1 mm. In Figure 6 the contours of the strain component ε^ are plotted for the two models. The two responses differ in a number of aspects. Using the model of Equation (11) leads to a smooth strain field, whereby a peak in the strain field occurs directly right of the hole. In an infinite plate modelled with a classical continuum, the strain intensity factor would have been 3; however, due to the higher-order gradients the peak strain is reduced. Conversely, with the model of Equation (14) an oscillatory strain field is obtained, in which the extreme values are significantly higher than with the model of Equation (11) and in which maxima and minima in the strain field appear throughout the domain. This oscillatory character of the response is another manifestation of the instability of this model /21,33/. As a consequence, also in static analyses the applicability of the model of Equation (14) is limited. The model of Equation (11) is explored further in the context of modelling size effects /38,39/. As an indicator for the strength of the structure, the ratio of nominal strain and maximum strain is taken. The

159 Vol. 13, Nos. 3-4, 2002 Higher-Order Gradient Models

Fig. 5: Square plate with circular hole - geometry and loading conditions (left) and EFG discretization of upper right quarter with nodes Ο and integration points x (right).

Fig. 6: Square plate with circular hole - contours of strain component ε^, for the model of Equation (11) (top) and of Equation (14) (bottom).

160 Harm Askes Journal of the Mechanical Behavior of Materials

nominal strain simply equals the imposed displacement divided by the length of (half of) the specimen. Thus,

. . . "imposed / , „ strain ratio = - (24) ^max

The maximum strain occurs directly right of the hole, see also Figure 6. In Figure 7 the strain ratio is plotted as a function of the characteristic size of the specimen R, normalised with respect to the internal length scale /. It can be verified that the strain ratio increases for decreasing dimensions of the specimen. Thus, it can be concluded that the model of Equation (11) is capable of describing size effects in structural strength in a qualitative satisfactory manner.

Fig. 7: Square plate with circular hole - size effect for nominal strength obtained with the model of Equation (11).

4. GRADIENT DAMAGE VERSUS NONLOCAL DAMAGE

Another example where the EFG method can be used to compare various higher-order models is in damage mechanics. Two series of strain gradients are derived from the underlying integral (nonlocal) model and compared by means of a one-dimensional and a two-dimensional test.

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4.1. Formulation

In an isotropic damage model the stresses are related to the strains as

σ=(ϊ-ώ)Όε (25) where ω is the damage parameter that ranges from 0 initially to 1 when all load-carrying capacity is exhausted. Damage growth is driven by a scalar state variable η which is taken here as an invariant of the strain tensor. To account for the difference in tensile and compressive loading which is needed for instance to model concrete, we adopt

Σ < >2 (26) Μ with < ε, > the positive part of the principal strain component ε,. Damage growth is specified by defining a loading function f as

/= η - κ with κ= max (η, κ0) (27) where κ is the history parameter and Kb is a material parameter that bounds the elastic regime. Loading takes place when/= 0 and / = 0. A linear softening damage evolution law is written as

o) (28)

while ω = 0 for κ < KQ and always ω < 1. In Equation (28) Η is the hardening/softening modulus. In the above model a length scale is lacking. Therefore, the width of the band in which damage localises is zero /3,4/. This is physically unrealistic, and a length-scale parameter is added by taking a spatial average η of η as /3,4/

fs(i)>7(x+f)

where ξ is the vector pointing to the infinitesimal volume dQ and g is a weight function taken here as g^) =

τ exp [~ξ ξ/2/?t) with /mt a parameter that sets the averaging volume in Equation (29). The nonlocal variable η replaces the local variable η in Equation (27). With this amendment, mathematical well-

posedness is ensured in the entire loading process. Gradient formulations are found when η (χ + ξ) in Equation (29) is expanded in a Taylor series around x. This results in/10,40,41/

ij(x) = ^(x) + i&VVx) + + ££,vyx) +... (30)

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The first approximation of Equation (30) is to truncate after the second-order term. In a finite element context this formulation has the severe disadvantage that C1 shape functions must be used. Therefore, it has been proposed to modify Equation (30) in that the second derivative of Equation (30) is taken, multiplied with -j/?t and backsubstituted into Equation (30). Thus,

m - V25?(x) = r?(x) - i&VVx) - ii«,V6r,(x) +... (31)

This procedure can be repeated so that all terms on the right-hand side of Equation (31) are replaced by terms on the left-hand side. Elaboration then yields /11/

2 m ~ R,V 57(x) + §&ν^(χ) - ±llV%x) +... = TJ(X) (32)

Thus, two series of gradient-enhancements can be distinguished, namely the original series (30) where the nonlocal variable is given explicitly as a function of the local variable, and the series (32) where the nonlocal variable is implicitly given as a function of the local variable. Accordingly, the two series are denoted as the explicit and the implicit gradient-enhancement series, respectively. The coefficients are the same in both equations, but they appear with uniform signs in the explicit series and with alternating signs in the implicit series. Since no truncations have been made, the two series are completely equivalent and equal to the integral expression (29). However, differences arise when the two series are truncated.

4.2. Discretisation aspects

Three different truncations of the gradient series are considered, namely

nd 2 explicit 2 order: f?(x) = r?(x) + §Z?nlV 7(x) (33)

implicit 2nd order: r?(x) - ^.V^x) = r?(x) (34)

implicit 4th order: η (χ) - l&V2 r?(x) + |£,ν4τ/(χ) = τ?(χ) (35)

They are discretised in a unified manner. Firstly, consider the explicit formulation (33). Its weak format reads

Jsv^-v-^n) dn =o (36) Ω

with δν a test function corresponding to η . Integration by parts yields

J δν (η - η) dfi + J iß, mT VrjdQ — / tfjv nrVr, dT = 0 (37) η η Γ 163 Vol. 13, Nos. 3-4, 2002 Higher-Order Gradient Models

Due to the higher-order of the formulation, additional boundary conditions are needed. Following the literature /10,11,41/, a zero value is imposed on the boundary integral in Equation (37). Equation (37) is discretised via η = Η η η and<5v= as

T T δι? jHlH,dn5+ 6u j (-κϊη + ill (VH,) Vr,) dÜ = 0 (38) n n

Compared with the explicit formulation the implicit gradient enhancements (Equations (34-35)) are true differential equations in the sense that the unknown η with one or more of its derivatives is given as a function of the known η . Therefore, boundary conditions for η must be provided. Analogous to the explicit model, vanishing odd derivatives of η are imposed on the boundary:

= 0 on Y,k =0,2 (39)

The weak format of Equation (35) is found as

JSv (η- LllV'n + i^V^-η)όΩ=0 (40) Ω where Equation (34) can be found by neglectihg the term. Integrating the /?t term by parts once and the

/^j term twice yields

J Sv (η - η) <ΚΪ+J1 ll (VSvf V5?dΩ + / ^V'SvV^dQ - f 5νητνηάΓ n n Ω -

According to Equation (39) the first and third boundary integral in Equation (41) cancel. However, the second boundary integral does not cancel. Discretisation of Equation (41) then yields

Γ T r T s r 2 ίϋ j HjUfdnj + Su J (VH„) VHfjdOg+ Su I ill (V H,) V HTdn 5 n n

T r T τ -Sii f\lL (n VH5) dr g - δα JHZ η άΏ. = 0 (42) r n

A consistent incremental-iterative scheme is obtained through the linearisation

= + = (43)

Together with the weak form of the equilibrium equation (not elaborated here) and Equations (38) and (42), a global system of equations is obtained as /10,11/

164 Harm Askes Journal of the Mechanical Behavior of Materials

fext _ f int Kuu Κ Δυ' (44) stt K-TJU Κ-ψ) Δ5 . f - where

Kuu = J — u)DB„ dfi (45) Ω

(46)

Kjju — (47) impl.:

η ' σε

expl.: /H^H,ydn Kw = (48)

2 7 impl.: f {H^H, + (VH^ VH, + |ifnl (V ^) V'H,} dfi

r r 2 -/14 (n VH?) V Hff dr

f«· = J Hjbdii + J H^tdr (49) f) r,

fim = J Bladü (50) Ω

(51) impl.: J Ηϊ7/dQ

The subscripts u and η in Η and Β denote that shape functions of the displacement or of the nonlocal variable are concerned. More details on the discretization and the implementation can be found in Reference

4.3. Localisation analysis

The first numerical example is a one-dimensional tensile bar of which the geometry and loading conditions are shown in Figure 8. Strain localisation is initiated due to a geometrical imperfection: in the central part of the bar the cross-section area is reduced by 10 percent. A displacement of 0.05 mm is imposed incrementally at the right end, while the left end is fixed. Young's modulus is taken as £ = 20000 MPa, the

= 4 4 damage threshold kö 1 x 10' , the hardening modulus H = -1/62 xlO MPa and the internal length scale /int

165 Vol. 13, Nos. 3-4, 2002 Higher-Order Gradient Models

2 A =10 mm A = 9mm'

-> u 45 mm 10 mm 45 mm 1

Fig. 8: Strain localization in a bar - problem statement.

mm. The EFG nodal spacing h = 0.5 mm in the central 28 mm of the bar and h = 2 mm elsewhere. Figure 9 shows the damage profiles along the bar for the three gradient enhancement formalisms at the stage where the load-canying capacity has been exhausted. The load-displacement curves are depicted in Figure 10. Also the load-displacement curve of the nonlocal (integral) model has been included. The nonlocal model, for which a secant stiffness matrix has been used, fails to converge after the snap-back point. When examining the differences in the damage profiles, it can be seen that the two implicit models give very similar results. Also the explicit model shows a good agreement with the other two curves, although the damaged zone has developed more. Consequently, more energy has been dissipated, which can be seen from the load-displacement curves. Here, the difference between the explicit model and the two implicit models is more significant. The explicit model shows a much more ductile response than the two implicit models. As a consequence, the snap-back behaviour that occurs with the implicit models is not present in the explicit

0 J · ^ ' -50 -25 0 25 50 χ [mm]

Fig. 9: Strain localization in a bar - final damage profiles for 2nd order explicit (solid), 2nd order implicit (dotted) and 4th order implicit (dashed) model.

166 Harm Askes Journal of the Mechanical Behavior of Materials

0.02 0.04 0.06 displacement [mm]

Fig. 10: Strain localization in a bar - load-displacement curves for 2nd order explicit (solid), 2nd order implicit (dotted), 4th order implicit (dashed) and nonlocal (dash-dotted) model.

formulation. Obviously, the fourth-order implicit model gives the best approximation of the nonlocal model. Only in a relatively late stage of the loading process a difference between the nonlocal model and the fourth- order implicit model occurs. Further, it is obvious that the second-order implicit model is a closer approximation of the nonlocal model than the explicit model is. As a second example, a three-point bending test is analysed of which the geometry, loading conditions and EFG node distribution are shown in Figure 11. For reasons of symmetry only the right half of the beam has been modelled. A plane strain configuration has been taken. The load is applied by prescribing the conjugate displacement. The nodal spacing is 25 mm in both directions for the part under the loading and 0.1 m for the outermost part, with an intermediate zone where the nodal spacing is 50 mm. The material 2 parameters are taken as Ε = 20000 N/mm , ν = 0.2 and /int = 20 mm. To test the gradient-enhancement framework with different softening laws, an exponential rather than linear softening curve has been used:

ω = 1 - ^ ((1 - ori) + Qi exp(-a2(/c - /«>))) (52)

with k0 = 0.0001, αϊ = 0.99 and a2 = 500. The beam has been modelled with the second-order implicit model and with the fourth-order implicit model. Analyses with the second-order explicit model showed strong fluctuations in the averaged strain field. It is known from the literature that the second-order explicit model amplifies oscillations, whereas oscillations are attenuated in the second-order implicit formulation /41/.

167 Vol. 13, Nos. 3-4, 2002 Higher-Order Gradient Models

U \l 300 mm Δ A

1000 mm 1000 mm

ι I I

lOOOOOOOO lOOOOOOOO lOOOOOOOO lOOOOOOOO ω lOOOOOOOO lOOOOOOOO 300 ε lOOOOOOOO mm ε lOOOOOOOO lOOOOOOOO Χ/1 lOOOOOOOO lOOOOOOOO

| 200 mm | 100 700 mm

Fig. 11: Three-point bending test - geometry and loading conditions (top) and EFG node distribution of right half (bottom).

Attempts to overcome this difficulty of the second-order explicit model are beyond the scope of this study, and only results of the implicit models are shown. Damage contour plots at three different load stages can be found in Figure 12. The differences between the two models are very small. This holds also for the load- displacement curves shown in Fig. 13.

5. DISCUSSION

The Element-Free Galerkin (EFG) method has been proposed as a method to discretise higher-order gradient models. Due to the high order of continuity that can easily be obtained with EFG shape functions, the implementation of higher-order gradient models is straightforward, which facilitates the comparison of different gradient models.

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0.1 0.3 0.5 0.7 0.9

Fig. 12: Three-point bending test - damage contours in central part for implicit second-order model (top) and implicit fourth-order model (bottom), u = 0.5 mm (left), u = 1 mm (centre) and u = 2 mm (right).

Two gradient elasticity models have been compared, one of which is derived from a discrete microstructure while in the other model the higher-order term is postulated in the strain energy functional. The microstructural gradient model has been found to be unstable in dynamic analyses, while strange oscillations are found in static analyses. On the other hand, the phenomenological gradient elasticity does not bear these deficiencies, and is well capable of describing size effects in structural strength.

169 Vol. 13, Nos. 3-4, 2002 Higher-Order Gradient Models

Fig. 13: Three-point bending test - load-displacement curves for implicit second-order model (dashed) and implicit fourth-order model (solid).

Two different gradient-enhancement series have been compared in the context of damage mechanics. Both series can be derived from the underlying integral (nonlocal) model. It has been found that the so-called second-order implicit model has a better correspondence with the nonlocal model than the second-order explicit model. Adding a fourth-order term to the second-order implicit model does not have a significant influence. The EFG method does not seem to be a numerical tool that can easily be used in engineering practice, since it lacks efficiency. As such, it should be used in the development phase of gradient models. Once a specific gradient model is identified that suits the specific needs of the problems under consideration, a finite element implementation of the model should be made.

ACKNOWLEDGEMENTS

Fruitful discussions with Bert Sluys, Akke Suiker, Jerzy Pamin, Ioannis Tsagrakis, Elias Aifantis, Andrei Metrikine, Rend de Borst and Ching Chang are gratefully acknowledged. Support of EC under the TMR Network project FMRX-CT96-0062 is acknowledged.

170 Harm Askes Journal of the Mechanical Behavior of Materials

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