higher-order strain/higher-order stress gradient models derived from a discrete microstructure, with...

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

with application to fracture

C.S. Chang a, H. Askes b,*, L.J. Sluys b

a Department of Civil Engineering, University of Massachusetts, Amherst, MA 01003, USAb Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, NL-2600 GA Delft, Netherlands

Received 22 March 2002; accepted 16 April 2002

Abstract

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.

� 2002 Published by Elsevier Science Ltd.

Keywords: Microstructure; Higher-order continuum; Gradient models; Localization; Damage

1. Introduction

For the modeling of granular media discrete models and continuum models can be used. In the formermethodology, each particle is modeled separately, while interparticle contacts are accounted for by for-mulating force–displacement relations for the contacts. Discrete models have the advantage that allmicroscopic effects can be accounted for in a straightforward manner. However, large specimens invari-ably lead to large systems of equations, which makes this method unsuitable for use in engineering prac-tice.

As an alternative, continuum models can be used. In this class of models, the microscopic phenomenaare accounted for in a smeared manner, that is, an average is taken of the microscopic properties. Thesimplest continuum model is the so-called classical continuum, in which the stress is related to the strain viaHooke’s law, and no higher-order terms are incorporated. It is well-known that the classical continuumwith uniform material parameters cannot be used to model phenomena that are driven by processes on

Engineering Fracture Mechanics 69 (2002) 1907–1924

www.elsevier.com/locate/engfracmech

*Corresponding author. Tel.: +31-15-2782731; fax: +31-15-2786383.

E-mail address: [email protected] (H. Askes).

0013-7944/02/$ - see front matter � 2002 Published by Elsevier Science Ltd.

PII: S0013-7944 (02 )00068-1

mail to: [email protected]

lower scales of observation. For instance, use of a classical model leads to singularities in the strain fieldnear a crack tip. Also, strain-softening behavior cannot be modeled with classical continua. Furthermore,wave propagation through granular media is characterized by dispersion effects (i.e. the propagating waveschange shape since different wavelengths travel with different phase velocities). These dispersion effectscannot be captured by classical continua.

To overcome the drawbacks of the classical continuum, it has been proposed to enhance the classicalcontinuum with higher-order terms of the state variables. Distinction can be made between higher-orderspatial terms and higher-order temporal (viscous) effects. The higher-order spatial terms are normally di-vided into two classes, namely the integral models (in which spatial averages of one or more state variablesare taken [1,2]) and gradient models (in which higher-order derivatives with respect to the spatial coor-dinates are introduced [3–6]). Integral models carry some inconveniences from an implementational pointof view (for instance, consistent tangent matrices are difficult to implement in existing finite element codes[7,8], and the combination with mesh-adaptivity is cumbersome). This study will focus on nonclassicalcontinua in which higher-order spatial derivates are considered.

Two classes of higher-order spatial gradient models have been used in the literature. In the first class, theequilibrium equation for the system involves only the Cauchy stress and no higher-order stresses are in-corporated. The Cauchy stress is related to the strain and to higher-order derivatives of the strain. Thesetype of models can be obtained by postulating the according strain gradient terms in the potential energyfunctional [9,10] or directly into the constitutive equations [11–15]. Since in this case the equilibriumequation takes the same format as in the classical models, only minor modifications are needed to trans-form the numerical solution algorithm for a classical continuum into that of an enhanced continuum.However, an unfavorable aspect of this class of models consists in an ambiguity in the definition of thepotential energy. Therefore, the potential energy due to the Cauchy stress is not guaranteed to be positive.

On the other hand, in the second class of higher-order gradient models the higher-order stress termsconjugated to the higher-order strain terms are also considered. The equilibrium equation involves not onlythe Cauchy stress, but also the higher-order stresses [16,17]. In this class of models, positiveness of thepotential energy is ensured. Since multiple stress terms are involved, the equilibrium equation(s) adopt amore complicated format than in the case of a classical continuum.

However, in either case a direct link with the underlying microstructure is lost. Furthermore, the co-efficients that accompany the higher-order terms are difficult to identify or to express in terms of micro-structural properties. To overcome this drawback, microstructural approaches can be used in order totransform a discrete particle system into a continuum [5,6,18,19]. When this approach is followed, thegoverning equations of the discrete system are homogenized, e.g. by Taylor series expansions, so that acontinuum is obtained. Any higher-order gradients that are introduced in this manner will have a direct linkwith the microstructural properties of the discrete particle system, which simplifies the identification anddetermination of the higher-order coefficients.

So far, the microstructural approach has only been applied to the first class of higher-order gradientmodels, i.e. with higher-order strains but without higher-order stresses. In this contribution, we will alsofocus on the second class of gradient models from a microstructural point of view. Starting from a discreteparticle system, a series of constitutive equations is derived which include up to the third-order straingradients as well as the corresponding stress gradients. This general case is then particularized to formulatethe simplest models of either class mentioned above, namely with and without higher-order stresses.Specifically, a model is formulated where the Cauchy stress is related to the strain and its second derivative,as well as a model where the Cauchy stress is related to the strain and the first-order stress is related to thefirst-order strain. For both types of models, governing equations are elaborated. The analysis of a linearcomparison solid in an infinitely long bar is used to underline some fundamental differences between thetwo models. The static response is investigated in the linear elastic case as well as for a strain localizationtest in a damage context.

1908 C.S. Chang et al. / Engineering Fracture Mechanics 69 (2002) 1907–1924

2. Microstructural model formulation

A granular medium is considered as a collection of particles. Under deformation, the particles mayundergo a translation. It is assumed that through the interparticle contact no moments can be transmitted.Furthermore, particle rotation is solely caused from the displacement field, i.e. no particle spin is taken intoaccount. Thus, when the representative volume is sufficiently larger than the particles, the material can beregarded as a class II nonpolar type of continuum [6].

If we focus our viewing scale to the sub-particle level and consider the elastic behavior of two parti-cles a and b that are in contact, a general expression for the constitutive relations between interparticleforce f c

q and the relative displacement of the two particles (i.e. the interparticle movement) dci can be written

as

f cq ¼ Kc

qidci ð1Þ

in which Kcqi is the stiffness of the contact between the two particles. In Refs. [6,19] a decomposition of this

stiffness term into normal (compressive) and shear terms can be found. The stored strain energy in arepresentative volume V is the summation of the energy stored over all interparticle contacts. Thus, for thestrain energy density W it can be written that

W ¼ 1

2V

XNcV

c¼1

f cq dc

q ð2Þ

where NcV is the number of contacts in the considered representative volume V.

A continuum formulation is obtained by replacing the displacement unq of particle n by the constructedcontinuum displacement field uqðxÞ at the centroid of particle n, that is

uqðxnÞ ¼ unq ð3Þ

For granular materials containing particles of finite size, it is not realistic to consider the representativevolume of the material as an ‘‘infinitesimal element’’ as in classical mechanics. Therefore, instead of thelinear continuum field, the discrete field is approximated by a continuum field with higher-order derivativesusing Taylor series expansions for the displacement at particle n. To this end, a reference point x0 is definedas the center of the representative volume. When fourth-order derivatives and higher are neglected, thisyields

uqðxnÞ ¼ uqðx0Þ þ uq;jðx0Þxj þ1

2uq;jkðx0Þxjxk þ

1

6uq;jklðx0Þxjxkxl ð4Þ

where a subscript following a comma denotes a partial derivative with respect to the corresponding spatialcoordinate. Within the representative volume, uqðx0Þ, uq;jðx0Þ, uq;jkðx0Þ and uq;jklðx0Þ are constant. Thesecond-order tensor uq;j has nine components. However, since the material is nonpolar, the nine compo-nents can be reduced to six independent components. The third-order tensor uq;jk is symmetric in the lasttwo indices and has 18 independent components. The fourth-order tensor uq;jkl is symmetric in the last threeindices and has 30 independent components.

The continuous displacements uqðxaÞ and uqðxbÞ replace their discrete counterparts in the expression fordcq. Accordingly,

dcq ¼ uqðxbÞ � uqðxaÞ ¼ uq;jLc

j þ uq;jkJ cjk þ uq;jklSc

jkl ð5Þ

where the ðx0Þ notation is dropped, and in which the geometric quantities

Lcj ¼ xbj � xaj ð6aÞ

C.S. Chang et al. / Engineering Fracture Mechanics 69 (2002) 1907–1924 1909

Jcjk ¼

1

2xbj x

bk

�� xaj x

ak

�ð6bÞ

Scjkl ¼

1

6xbj x

bkx

bl

�� xaj x

akx

al

�ð6cÞ

describe the arrangement of the two particles. The branch vector Lc represents the vector from the centroidof particle a to that of particle b. The higher-order fabric tensors Jc and Sc include descriptions of the lengthand the location of the branch.

Substituting Eq. (5) into Eq. (2) yields

W ¼ 1

2V

XNcV

c¼1

f cq L

cjuq;j

�þ 1

2f cq J

cjkuq;jk þ

1

6f cq S

cjkluq;jkl

�ð7Þ

Let the strain measures e be written as

e0ij ¼ ui;j eIijk ¼ ui;jk eIIijkl ¼ ui;jkl ð8Þ

Then the stress measures r can be defined as

r0ij ¼

oWoe0ij

rIijk ¼

oWoeIijk

rIIijkl ¼

oWoeIIijkl

ð9Þ

More specifically, by combining Eqs. (1), (5), (7)–(9), the following set of constitutive equations is found:

r0iq ¼ Aiqkle

0kl þ BiqklmeIklm þ FiqklmneIIklmn ð10aÞ

rIijq ¼ Blqkije

0kl þ DijqklmeIklm þ Gijqklmne

IIklmn ð10bÞ

rIIijpq ¼ Flqkijpe0kl þ Glmqkijpe

Iklm þ Hijpqklmne

IIklmn ð10cÞ

where the constitutive tensors are expressed in terms of the fabric measures as

Aiqkl ¼1

V

XNcV

c¼1

Lci K

cqkL

cl ð11aÞ

Biqklm ¼ 1

V

XNcV

c¼1

Lci K

cqkJ

clm ð11bÞ

Dijqklm ¼ 1

V

XNcV

c¼1

JcijK

cqkJ

clm ð11cÞ

Fiqklmn ¼1

V

XNcV

c¼1

Lci K

cqkS

clmn ð11dÞ

Gijqklmn ¼1

V

XNcV

c¼1

JcijK

cqkS

clmn ð11eÞ


Hijpqklmn ¼1

V

XNcV

c¼1

ScijpK

cqkS

clmn ð11fÞ

For a representative volume of particles, the medium can be treated as statistically homogeneous. Thus,central symmetry can be assumed. For Eqs. (10a)–(10c) this implies that Biqklm ¼ Gijqklmn ¼ 0. As such, theconstitutive equations contain one fourth-order tensor Aiqkl, two sixth-order tensors Dijqklm and Fiqklmn, andone eighth-order tensor Hijpqklmn. According to Weyl’s theory [20], the majority of the components of anisotropic tensor are dependent. In general, the 81 components of Aiqkl reduce to three independent con-stants. Similarly, the 729 components of Dijqklm can be written in terms of 15 independent constants. Bymeans of the microstructural approach, these constants can be determined by a simple summation processbased on the packing structure of the granular material.

3. Governing equations

Below, a further specialization of Eqs. (10a)–(10c) is made. A model without higher-order stress terms aswell as a model with a higher-order stress term are considered. For a clear comparison between these twoclasses of models, restriction is made to the one-dimensional (1D) case. After elaborating the constitutivecoefficients, the linear and nonlinear governing equations for the two types of models are derived. Also, thestability of the models and the formulation of the higher-order boundary conditions is treated.

3.1. Higher-order constitutive constants

For the remaining constitutive coefficients in Eqs. (10a)–(10c), the assumption of monosized particles ismade. Then, xc ¼ 1

2ðxb þ xaÞ represents the location of the contact point, where the origin of the coordinate

system (for xa, xb and xc) is located at the center of the representative volume. As such,

LcSc ¼ 1

6ðLcÞ2 ðxbÞ2

�þ xbxa þ ðxaÞ2

�¼ ðLcÞ2 1

2ðxcÞ2

�þ 1

24ðLcÞ2

�ð12aÞ

JcJ c ¼ 1

4ðLcÞ2ðxb þ xaÞ2 ¼ ðLcÞ2ðxcÞ2 ð12bÞ

ScSc ¼ 1

36ðLcÞ2 ðxbÞ2

�þ xbxa þ ðxaÞ2

�2

¼ ðLcÞ2 1

4ðxcÞ4

�þ 1

24ðxcÞ2ðLcÞ2 þ 1

576ðLcÞ4

�ð12cÞ

Furthermore, the second-order and fourth-order moments of inertia of the representative volume are in-troduced as

I2 ¼1

NcV

XNcV

c¼1

ðxcÞ2 I4 ¼1

NcV

XNcV

c¼1

ðxcÞ4 ð13Þ

By means of Eqs. (12a)–(13), the higher-order constitutive constants D, F and H in Eqs. (10a)–(10c) can beelaborated as functions of I2, I4, Lc and A, i.e.

D ¼ I2A ð14aÞ

F ¼ 1

2I2

�þ 1

24ðLcÞ2

�A ð14bÞ


H ¼ 1

4I4

�þ 1

24ðLcÞ2I2 þ

1

576ðLcÞ4

�A ð14cÞ

For a large enough representative volume, the moments of inertia can be elaborated as

I2 ¼ 13d2 I4 ¼ 1

5d4 ð15Þ

in which d is the characteristic size of the representative volume (in 1D configurations, 2d is the length of therepresentative volume). If d is substantially larger than Lc (that is, if the representative volume contains arelatively large number of particles), the ðLcÞ2 and ðLcÞ4 terms in the higher-order constitutive constantsvanish, and Eqs. (10a)–(10c) can be written in matrix-vector notation as

r0

rI

rII

24

35 ¼ E

1 0 16d2

0 13d2 0

16d2 0 1

20d4

24

35 e0

eI

eII

24

35 ð16Þ

where the Young’s modulus E ¼ A ¼ 1=VP

LcKLc.In the above formulation, it has been assumed that the size of the representative volume d is known. If

this is not the case, or if the assumption of monosized particles is dropped, more complicated expressionsfor the higher-order constitutive coefficients are found. However, in either case the higher-order coefficientsfollow directly and univocally from the microstructural properties.

3.2. Higher-order theory without higher-order stress

When higher-order stress terms are left out of consideration, only the first row in system (16) is takeninto account. For the potential energy density W it can then be written that

W ¼Z

e0r0 de0 ¼

Ze0E e0�

þ 1

6d2eII

�de0 ¼

Ze0E e0�

þ 1

6d2 o

2e0

ox2

�de0 ð17Þ

Using integration by parts for the higher-order term and neglecting boundary terms, Eq. (17) can beelaborated as

W ¼ 1

2E ðe0Þ2(

� 1

6d2 oe0

ox

� �2)

ð18Þ

The negative sign that precedes the higher-order term denotes that this term is destabilizing.The equilibrium equation is straightforwardly found via

or0

ox¼ E

o2uox2

�þ 1

6d2 o

4uox4

�¼ 0 ð19Þ

where it is assumed that body forces are absent. This type of model has been used in studying static de-formation and wave propagation for granular media [6,19,21].

The stress–strain relation that underlies Eq. (19) is

r0 ¼ E e0�

þ 16d2eII

�ð20Þ

which corresponds to the first row of Eq. (16). It is noted that this stress–strain resembles the one proposedby Altan and Aifantis [22], which is given in a 1D format as

r0 ¼ E e0�

� ceII�

ð21Þ


where c is a gradient parameter that can be related to the particle size [6]. The two models have a similarform, except from the sign preceding the higher-order strain gradient term. This is a crucial difference in aframework without the inclusion of higher-order stresses. Using stress–strain relation (21) gives a stablesolution if c > 0. On the other hand, instabilities can occur when c < 0 in Eq. (21), or (equivalently) whenstress–strain relation (20) is used [22]. The positive sign in Eq. (20) or taking c < 0 in Eq. (21) is morenatural since it is derived from the microstructure. However, the decisive issue is the form of the governingequation, not the form of the stress–strain relation. As is shown in Section 3.3, the inclusion of higher-orderstresses leads to governing equations that are unconditionally stable.

Due to the increased order of the governing equations, additional (higher-order) boundary conditionsmust be formulated. The format of the higher-order boundary conditions follows from the Lagrangeequations [23,24]. Accordingly,

essential b:c: : u specified; u;x specified ð22aÞ

natural b:c: :oWou;x

� o

oxoWou;xx

¼ r0 specified;oWou;xx

specified ð22bÞ

Note that the boundary conditions of the classical continuum are incorporated, i.e. values are prescribedfor u or for r0. The higher-order (nonstandard) boundary conditions are associated with prescribing valuesfor u;x and u;xx. On an interface, all four variables u, u;x, u;xx and r0 are continuous.

A nonlinear extension of this model is easily obtained via a damage formalism. In this context, theconstitutive coefficients are premultiplied by a factor ð1� xÞ:

r0 ¼ ð1� xÞE e0�

þ 1

6d2 o

2e0

ox2

�ð23Þ

where x is the scalar (isotropic) damage variable which ranges from 0 initially to 1 when all material co-herence is lost. Assuming that damage growth is driven by the zeroth-order strain, i.e. x ¼ xðe0Þ, thenonlinear equilibrium equation can be cast as

or0

ox¼ ð1� xÞE o2u

ox2

�þ 1

6d2 o

4uox4

�� E

ouox

�þ 1

6d2 o

3uox3

�oxoe0

o2uox2

¼ 0 ð24Þ

Remark 1. Instead of taking x ¼ xðe0Þ, it can be assumed that the damage growth depends on the higher-order strains as well. This would lead to a regularized constitutive equation in case strain-softening takesplace [4,11,14]. However, it is also interesting to study the regularizing capacities of the models withoutadding higher-order strain gradients to the damage evolution law. In that case, regularization would be theresult of higher-order gradients that have appeared in the linear constitutive relation(s).

3.3. Higher-order theory with higher-order stress

Alternatively, higher-order stress terms can be included. To this end, system (16) is reconsidered, andrestriction is made to zeroth-order and first-order stresses and strains.

For this case, the potential energy density can be elaborated as

W ¼Z

e0r0 de0 þ

ZeI

rI deI ð25Þ


The constitutive relations from system (16) can be substituted into Eq. (25). Thus,

W ¼ 1

2E ðe0Þ2�

þ 1

3d2ðeIÞ2

�ð26Þ

The unconditional positiveness of the potential energy density guarantees the stability of this model, incontrast to the model without higher-order stress term.

To derive the equilibrium equations, stationarity of the potential energy is investigated. To this end, thehigher-order term is rewritten in terms of e0 using eI ¼ oe0=ox. As such,Z

eI

1

3d2EeI deI ¼ �

Ze0

1

3d2E

o2e0

ox2de0 ð27Þ

where boundary terms have been left out of consideration. By means of Eq. (27), stationarity of the po-tential energy can be cast as

Eo2uox2

�� 1

3d2 o

4uox4

�¼ 0 ð28Þ

The striking difference with Eq. (19) is the sign of the higher-order term. As will be argued below, this isrelated to the stability of the models. The format of Eq. (28) is identical to the equilibrium equation thatcorresponds to Eq. (21), including the sign of the higher-order term. As is illustrated in Appendix B for thegeneral nonlinear case, Eq. (28) can equally be written as

or0

ox� o2rI

ox2¼ 0 ð29Þ

Thus, Eq. (29) is the equivalent of the standard equilibrium equation or0=ox ¼ 0 for models that contain ahigher-order stress term.

Again, the boundary conditions can be derived by means of the Lagrange equations:

essential b:c: : u specified; u;x specified ð30aÞ

natural b:c: :oWou;x

� o

oxoWou;xx

¼ r0 � orI

oxspecified;

oWou;xx

¼ rI specified ð30bÞ

The essential boundary conditions take the same form as in Eqs. (22a). The natural boundary conditionsnow also include the higher-order stress term.

Derivation of the nonlinear equilibrium equation is somewhat more involved than for the case without ahigher-order stress term. In a damage context we assume that all constitutive coefficients are premultipliedwith the same factor ð1� xÞ, i.e.

r0 ¼ ð1� xÞEe0 ð31aÞ

rI ¼ ð1� xÞ13d2EeI ð31bÞ

Again, we assume that x ¼ xðe0Þ. The potential energy density according to Eq. (25) can be rewrittenaccording to the product rule of differentiation for a term consisting of three factors: ðfghÞ0 ¼f 0ghþ fg0hþ fgh0. In particular,Z

eIð1� xÞ 1

3d2EeI deI ¼ �

Ze0� oxoe0

oe0

ox1

3d2EeI de0 �

Ze0ð1� xÞ 1

3d2E

oeI

oxde0 ð32Þ


whereby boundary integrals have again been left out of consideration. Thus, all integrals in the nonlinearanalogue of Eq. (25) can be elaborated as integrals in de0. Substitution of eI ¼ oe0=ox results in an ex-pression for the potential energy density as

W ¼Z

e0ð1

(� xÞE e0

�� 1

3d2 o

2e0

ox2

�þ ox

oe01

3d2E

oe0

ox

� �2)de0 ð33Þ

Minimizing the potential energy then results in the nonlinear equilibrium equation (expressed in terms ofthe displacement u, the damage x and the material parameters E and d) as

ð1� xÞE o2uox2

�� 1

3d2 o

4uox4

�� E

ouox

�� d2 o

3uox3

�oxoe0

o2uox2

¼ 0 ð34Þ

Note that in the second term of Eq. (34) no factor 13is preceding the higher-order term. However, the only

fundamental difference between Eqs. (24) and (34) resides in the sign of the higher-order terms. Further-more, it must be noted that also Eq. (34) can be cast in the format of Eq. (29).

4. Infinite bar with homogeneous state

In order to study the analytical properties of the two derived models in more detail, an infinite bar isconsidered. A homogeneous state is assumed, so that the material can be regarded as a linear comparisonsolid. A small perturbation on the displacement field is then substituted such that loading takes place.

Firstly, the model without higher-order stress term is studied. Starting point is the nonlinear equilibriumequation (24). All factors that have been kept constant using the chain rule of differentiation are re-con-sidered. If they contain second-order derivatives or higher of the displacement, they are neglected followingthe assumption of a homogeneous state. Otherwise, they are maintained with a subscript 0 as a denotationof the initial, homogeneous state. This yields

ð1� x0ÞEo2uox2

�þ 1

6d2 o

4uox4

�� Ee00

oxoe0

o2uox2

¼ 0 ð35Þ

A small perturbation du ¼ uu expðikxÞ is substituted, with uu the amplitude and k the wave number of theperturbation. This leads to

ð1� x0Þ 1

�� 1

6d2k2

�� e00

oxoe0

¼ 0 ð36Þ

which can be elaborated as

1

6ðdkÞ2 ¼ 1� e00

1� x0

oxoe0

ð37Þ

For the elastic stage, where x0 ¼ ox=oe0 ¼ 0, Eq. (37) yields dk ¼ffiffiffi6

p. The values of dk that are found

through Eq. (37) govern the static response of the model in the sense that the displacement field adopts thiswave number k. The corresponding wavelength k ¼ 2p=k is found as

k ¼ 2pd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

6

1� x0

1� x0 � e00ox=oe0

sð38Þ

Secondly, the model with the higher-order stress term is considered. In this case, the reduced equilibriumequation as follows from the homogeneous state reads


ð1� x0ÞEo2uox2

�� 1

3d2 o

4uox4

�� Ee00

oxoe0

o2uox2

¼ 0 ð39Þ

Substitution of the perturbation du ¼ uu expðikxÞ yields

ð1� x0Þ 1

�þ 1

3d2k2

�� e00

oxoe0

¼ 0 ð40Þ

or

1

3ðdkÞ2 ¼ e00

1� x0

oxoe0

� 1 ð41Þ

In contrast to the case without a higher-order stress term, no real solution for dk is found in the elasticregime. The wavelength that governs the static response of the model can be written as

k ¼ 2pd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

3

1� x0

e00ox=oe0 þ x0 � 1

sð42Þ

For a further specification of the wavelength in the nonlinear regime the damage loading function mustbe specified. A damage evolution law that corresponds to a linear softening relation is adopted [11,25]. Incase of damage growth,

x ¼ juðj � j0Þjðju � j0Þ

06x6 1 ð43Þ

whereby j ¼ e0 upon loading, j0 is the strain at which damage is initiated and ju is the strain level at whichall load carrying capacity is exhausted. With this damage evolution law, the expression for the wavelengthcan be elaborated as

k ¼ 2pd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffie00 � ju

6e00

sð44Þ

for the model without higher-order stress contributions, and

k ¼ 2pd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiju � e003e00

sð45Þ

for the model with the higher-order stress term.As long as the material coherence is still intact, it holds that e00 < ju. Therefore, no real values for the

wavelength are found through Eq. (44). This means that no meaningful static response can be obtained inthe damaging stage for the model without higher-order stress term.

Remark 2. Existence of the solution for k is lost when softening is initiated. With the current (linearsoftening) damage evolution law, softening occurs for all strain levels e00 > j0. In another contribution [26]it has been shown that solutions for k can still be found for the hardening stage.

On the other hand, real solutions for k are found for all strain levels j0 6 e00 6 ju in case the model withthe higher-order stress is used, see Eq. (45). The evolution of the wavelength (normalized with respect to thelength scale d) as a function of the strain level is plotted in Fig. 1, with j0 ¼ 10�4 and ju ¼ 10�2. Throughthe wavelength as obtained in Eq. (45) the width of the zone in which damage localizes is set [15,25]. Thus,the model with higher-order stress can be used to simulate softening processes.


Remark 3. No higher-order gradients are added to the damage-driving state variable, which is in contrast toearlier enhanced damage formulations [1,2,11,27]. In the current model with higher-order stress, both theelastic field and the damage field are regularized by the higher-order gradients that have emerged from thehomogenization of the microstructure.

5. Numerical examples

Boundary value problems have been analyzed with the proposed models. For the spatial discretizationthe element-free Galerkin (EFG) method is used [28,29]. EFG shape functions are easily formulated with anarbitrary order of continuity, which makes this method suitable for use with higher-gradient models [30,31].The formulation of EFG shape functions is treated in Appendix A. Lagrange multipliers are used to enforceessential boundary conditions. For the interpolation of the displacements EFG shape functions with apolynomial base vector pT ¼ ½1; x; x2; x3 is used, while pT ¼ ½1; x; x2 for the shape functions of the Lagrangemultipliers. The linearization and the discretization of the governing equations of the model with higher-order stress is given in Appendix B. Numerical integration is used with five integration points per internodaldistance.

5.1. Linear analysis

Firstly, the linear response of the two models is studied. To this end, the problem statement of Fig. 2 isconsidered. Due to a stepwise variation in the Young’s modulus E higher-gradient activity is triggered.The cross-sectional area A ¼ 1 mm2, and the length scale parameter d ¼ 5 mm. All boundary conditionsare assumed to be essential (cf. Eqs. (22a) and (30a)) and a displacement of 0.1 mm is imposed on the

Fig. 2. Static bar example––geometry and loading conditions for linear analysis.

Fig. 1. Normalized wavelength k=d versus the strain level e00 for the model with higher-order stress.


right-hand side of the bar. For the EFG discretization 81 equally spaced nodes are used. In Figs. 3 and 4 thezeroth-order strain e0 is plotted for the model without and with higher-order stress, respectively. A wavystrain pattern is observed for the model without higher-order stress, while a regular strain profile is ob-tained for the model with higher-order stress.

The wave pattern of the model without higher-order stress can be understood by considering Eq. (37). Inthe elastic regime, Eq. (37) is fulfilled for dk ¼

ffiffiffi6

p, which corresponds to a wavelength k ¼ 2p=k ¼ 2pd=

ffiffiffi6

p.

Taking d ¼ 5 mm leads to k 13 mm, which is indeed the wavelength that is observed in Fig. 3. Theoccurrence of a wave pattern in the linear elastic response is a manifestation of the instability of this model(cf. Eq. (18)). The instability can lead to nonuniqueness when the wavelength k exactly fits an integernumber of times in the bar length [26,32]. Furthermore, in numerical dynamic analyses this model failsaltogether [33]. Thus, the applicability of the model without higher-order stress is limited.

On the other hand, the model with higher-order stress yields regular results which are physically realistic.Note that for this model stability is guaranteed (cf. Eq. (26)). Around the location where the Young’smodulus changes, the zeroth-order strain changes smoothly, and constant values are adopted at either endof the bar. This indicates that the first-order strain (which is the derivative of the zeroth-order strain) has apeak value around the imperfection while it goes to zero at the ends of the bar.

Fig. 3. Static bar example––linear elastic response of model without higher-order stress.

Fig. 4. Static bar example––linear elastic response of model with higher-order stress.


5.2. Nonlinear analysis

Also the behavior of the models in a nonlinear damage context is investigated. To this end, the linearsoftening damage evolution law of Eq. (43) is taken. For the model without higher-order stress, the intrinsicinstability precludes that meaningful solutions can be obtained in the softening stage [26,32]. On the otherhand, the model with higher-order stress can be used in softening problems, as is shown below. A barproblem is studied as given in Fig. 5. An imperfection is placed in the center of the bar, so that higher-gradients are activated. Again, all boundary conditions are assumed to be essential. Young’s modulus istaken as E ¼ 1000 MPa, and the damage evolution parameters are given as j0 ¼ 10�4 and ju ¼ 10�2. Onthe right end of the bar a displacement of 0.025 mm is imposed.

Firstly, the capacity of the model with higher-order stress is investigated to predict a unique responseupon discretization refinement. In Fig. 6 the damage profiles along the bar are plotted for two discreti-zations consisting of 41 nodes and 81 nodes, while in both cases the length parameter d ¼ 2 mm. In bothcases the nodes are uniformly distributed over the domain. It can be seen that the two solutions are virtuallyidentical. The strong dependence on the discretization that exists in classical continua has been removed bythe inclusion of the higher-order gradients. Thus, the proposed model can successfully be used to modelsoftening phenomena.

Next, the influence of the length parameter d is studied for the 81 nodes discretization. In Fig. 7 thedamage profiles along the bar are plotted for a range of values for d. As can be seen, an increase in d leadsto an increase in the width of the damaged zone. More specifically, the wavelength as given in Eq. (45)directly sets the width of the damaging zone, and the wavelength is linearly dependent on the lengthparameter d. The influence of the length parameter on the width of the damaging zone corresponds to thatin other higher-order models [1,2,11,15].

Fig. 5. Static bar example––geometry and loading conditions for nonlinear analysis.

Fig. 6. Static bar example––damage profiles for model with higher-order stress and d ¼ 2 mm, 41 nodes (–––) and 81 nodes (–––).


6. Discussion

Higher-order gradient models have been derived from a discrete microstructure consisting of discreteparticles. By means of homogenization the discrete particle structure has been transformed into a set ofconstitutive relations, in which standard (Cauchy) stress and higher-order stresses are expressed in terms ofstandard strains and higher-order strains. As an additional material parameter, a length scale parameterhas entered the models through the homogenization process.

Next, two types of gradient models are considered as specific cases of the above results. In the firstmodel, the standard stress is related to the standard strain and the second derivative of the strain, whilehigher-order stresses are absent. In the second model, the standard stress is related to the standard strain,while the first-order stress is related to the first-order strain. Field equations and boundary conditions forboth models are formulated. Stability considerations for the linear elastic regime reveal that the modelwithout higher-order stress can become unstable, while the model with higher-order stresses is uncondi-tionally stable.

By means of boundary value problems it has been shown that the instabilities of the model withouthigher-order stress may lead to wavy strain profiles in the elastic case. On the other hand, the model withhigher-order stress predicts regular strain profiles. The extension to softening phenomena has also beenmade for the model with higher-order stress, whereby a damage context has been chosen. It has been shownthat a unique response is obtained upon discretization refinement, so that it can be concluded that thehigher-order gradients in this model serve as a ‘localization limiter’. Furthermore, the influence of thelength scale parameter of the model is similar to those in other higher-order models used to model softeningproblems. Thus, the model with higher-order stress can be considered as a properly regularized model in thesense that in the elastic stage the influence of discontinuities is smoothened, while in the softening stagedamage localization is limited to a zone of finite width.

Appendix A. Element-free Galerkin shape functions

In the EFG method only nodes and no elements are used to formulate the shape functions. Each node isassigned a so-called domain of influence in which its shape function is nonzero. This domain of influence is

Fig. 7. Static bar example––damage profiles for model with higher-order stress and 81 nodes, d ¼ 1 mm (� � �), d ¼ 2 mm (–––) and

d ¼ 4 mm (–––).


set through a so-called weight function wðsÞ. In this study a circular domain of influence with radius dinfl istaken as

wðsÞ ¼ exp � s2

ðadinflÞ2

!� exp � d2

infl

ðadinflÞ2

!" #1� exp � d2

infl

ðadinflÞ2

!" #,if s6 dinfl

0 if s > dinfl

8><>: ðA:1Þ

with s ¼ jx� xij and a a numerical parameter to set the relative weights inside the domain of influence.Here, a ¼ 1=4. A polynomial base vector p is defined for one spatial dimension as, e.g.,

pT ¼ ½1; x; x2; x3; . . . ðA:2Þ

The approximant function uh is expressed in terms of EFG shape functions / and nodal parameters u as

uh ¼ /Tu ðA:3Þ

where

/T ¼ pTA�1B ðA:4Þand

A ¼Xni¼1

wipðxiÞpTðxiÞ ðA:5Þ

B ¼ ½w1pðx1Þ;w2pðx2Þ; . . . ;wnpðxnÞ ðA:6Þ

in which n is the number of nodes and wi ¼ wðx� xiÞ.The relevant parameters that determine the quality of the EFG discretization are the radius of the

domain of influence dinfl, the nodal spacing denoted here with h, and the number of terms that is containedin the polynomial base vector p. These three parameters cannot be chosen independently. Following theliterature in which optimal sets of parameters have been found [30], it is taken that

dinfl=h ¼ 5 for pT ¼ ½1; x; x2; x3 ðA:7Þ

and

dinfl=h ¼ 4 for pT ¼ ½1; x; x2 ðA:8Þ

Appendix B. Linearization and discretization of the nonlinear equilibrium equation

For numerical implementation of the nonlinear equilibrium equation (34) the weak format is taken viapremultiplying by a test function du and integrating over the 1D domain ½0; L. Thus,Z L

0

du ð1�

� xÞE u00�

� 1

3d2u0000

�� E

oxoe0

ðu0 � d2u000Þu00�dx ¼ 0 ðB:1Þ

Integration by parts is used according toZ L

0

duð1� xÞEu00 dx ¼ duð1�

� xÞEu0�L0�Z L

0

du0ð1� xÞEu0 dxþZ L

0

duoxoe0

u00Eu0 dx ðB:2Þ


and Z L

0

duð1� xÞ 13d2Eu0000 dx ¼ duð1

�� xÞ 1

3d2Eu000

L0

�Z L

0

du0ð1� xÞ 13d2Eu000 dxþ

Z L

0

duoxoe0

u001

3d2Eu000 dx

ðB:3ÞThus, Eq. (B.1) can be rewritten asZ L

0

du0ð1� xÞE u0�

� 1

3d2u000

�dx�

Z L

0

duoxoe0

u002

3d2Eu000 dx ¼ duð1

�� xÞE u0

�� 1

3d2u000

� L0

ðB:4Þ

Further integration by parts is carried out viaZ L

0

du0ð1� xÞ 13d2Eu000 dx ¼ du0ð1

�� xÞ 1

3d2Eu00

L0

�Z L

0

du00ð1� xÞ 13d2Eu00 dxþ

Z L

0

du0oxoe0

u001

3d2Eu00 dx

ðB:5Þ

and Z L

0

duoxoe0

2

3d2Eu00u000 dx ¼ du

oxoe0

2

3d2Eu00u00

� L0

�Z L

0

du0oxoe0

2

3d2Eu00u00 dx�

Z L

0

duoxoe0

2

3d2Eu000u00 dx

ðB:6ÞSince in Eq. (B.6) the left-hand side equals the last term of the right-hand side, it can be written thatZ L

0

duoxoe0

2

3d2Eu00u000 dx ¼ 1

2du

oxoe0

2

3d2Eu00u00

� L0

� 1

2

Z L

0

du0oxoe0

2

3d2Eu00u00 dx ðB:7Þ

By means of Eqs. (B.5) and (B.7), Eq. (B.4) can be elaborated asZ L

0

du0ð1� xÞEu0 dxþZ L

0

du00ð1� xÞ 13d2Eu00 dx

¼ du0ð1�

� xÞ 13d2Eu00

L0

þ du ð1��

� xÞE u0�

� 1

3d2u000

�þ ox

oe01

3d2Eu00u00

� L0

ðB:8Þ

which is equivalent to

Z L

0

du0r0 dxþZ L

0

du00rI dx ¼ du0rI� �L

0þ du r0

�� orI

ox

� L0

ðB:9Þ

in which the right-hand side contains boundary integrals corresponding to the boundary conditions of Eqs.(30a) and (30b). Eq. (B.9) can also be written asZ L

0

duor0

ox

�� o2rI

ox2

�dx ¼ 0 ðB:10Þ

which confirms Eq. (29).Linearization is performed using

Dr0 ¼ ð1� xÞEDe0 � Ee0oxoe0

De0 ¼ ð1�

� xÞE � Ee0oxoe0

�Du0 ðB:11Þ


and

DrI ¼ ð1� xÞ 13d2EDeI ¼ ð1� xÞ 1

3d2EDu00 ðB:12Þ

Discretizing the test function and the trial function as du ¼ duT/T and u ¼ /u, and requiring that Eq. (B.9)is valid for any du results in

KDu ¼ f ext � f int ðB:13Þ

with

K ¼Z L

0

o/T

oxð1�

� xÞE � Ee0oxoe0

�o/ox

dxþZ L

0

o2/T

ox2ð1� xÞ 1

3d2E

o2/ox2

dx ðB:14Þ

f ext ¼ /Ts0� �L

0þ o/T

oxsI

� L0

ðB:15Þ

f int ¼Z L

0

o/T

oxr0 dxþ

Z L

0

o2/T

ox2rI dx ðB:16Þ

where s0 and sI represent the zeroth-order and first-order prescribed tractions according to Eqs. (30a) and(30b).

References

[1] Pijaudier-Cabot G, Ba�zzant ZP. Nonlocal damage theory. ASCE J Eng Mech 1987;113:1512–33.

[2] Ba�zzant ZP, Pijaudier-Cabot G. Nonlocal continuum damage, localization instability and convergence. ASME J Appl Mech

1988;55:287–93.

[3] Aifantis EC. The physics of plastic deformation. Int J Plast 1987;3:211–47.

[4] Schreyer HL, Chen Z. One-dimensional softening with localization. ASME J Appl Mech 1986;53:791–7.

[5] M€uuhlhaus H-B, Oka F. Dispersion and wave propagation in discrete and continuous models for granular materials. Int J Solids

Struct 1996;33:2841–58.

[6] Chang CS, Gao J. Second-gradient constitutive theory for granular material with random packing structure. Int J Solids Struct

1995;32:2279–93.

[7] Pijaudier-Cabot G, Huerta A. Finite element analysis of bifurcation in nonlocal strain softening solids. Comput Meth Appl Mech

Eng 1991;90:905–19.

[8] de Vree JHP, Brekelmans WAM, van Gils MAJ. Comparison of nonlocal approaches in continuum damage mechanics. Comput

Struct 1995;55:581–8.

[9] Triantafyllidis N, Aifantis EC. A gradient approach to localization of deformation. I. Hyperelastic materials. J Elast 1986;16:225–

37.

[10] Triantafyllidis N, Bardenhagen S. On higher gradient continuum theories in 1-D nonlinear elasticity. Derivation from and

comparison to corresponding discrete models. J Elast 1993;33:259–93.

[11] Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP. Gradient enhanced damage for quasi-brittle materials. Int J Numer

Meth Eng 1996;39:3391–403.

[12] Pamin J. Gradient-dependent plasticity in numerical simulation of localization phenomena. Dissertation, Delft University of

Technology, Delft, 1994.

[13] de Borst R, M€uuhlhaus H-B. Gradient-dependent plasticity: formulation and algorithmic aspects. Int J Numer Meth Eng

1992;35:521–39.

[14] Lasry D, Belytschko T. Localization limiters in transient problems. Int J Solids Struct 1988;24:581–97.

[15] Sluys LJ. Wave propagation, localisation and dispersion in softening solids. Dissertation, Delft University of Technology, Delft,

1992.

[16] Toupin RA. Elastic materials with couple-stresses. Arch Ration Mech Anal 1962;11:385–414.

[17] Mindlin RD. Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1965;1:417–38.


[18] Liao CL, Chang TP, Young DH, Chang CS. Stress–strain relationship for granular materials based on the hypothesis of best fit.

Int J Solids Struct 1997;34:4087–100.

[19] Suiker ASJ, de Borst R, Chang CS. Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient

micro-polar constitutive theory. Acta Mechanica 2001;149:161–80.

[20] Suiker ASJ, Chang CS. Application of higher-order tensor theory for formulating enhanced continuum models. Acta Mechanica

2000;142:223–34.

[21] Chang CS, Gao J. Wave propagation in a granular rod using high-gradient theory. ASCE J Eng Mech 1997;123:52–9.

[22] Altan BS, Aifantis EC. On some aspects in the special theory of gradient elasticity. J Mech Behav Mater 1997;8:231–82.

[23] Goldstein H. Classical mechanics. Reading: Addison-Wesly; 1964.

[24] Washizu K. Variational methods in elasticity and plasticity. Oxford: Pergamon; 1975.

[25] Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP, Spee I. Some observations on localisation in nonlocal and gradient

damage models. Eur J Mech, A/Solids 1996;15:937–53.

[26] Askes H, Suiker ASJ, Sluys LJ. Dispersion analysis and element-free Galerkin simulations of higher-order strain gradient models.

Mater Phys Mech 2001;3:12–20. Available from: http://www.ipme.ru.

[27] Comi C, Driemeier L. On gradient regularization for numerical analysis in the presence of damage. In: de Borst R, van der Giessen

E, editors. Material instabilities in solids. New York: Wiley; 1998. p. 425–40 [Chapter 26].

[28] Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. Int J Numer Meth Eng 1994;37:229–56.

[29] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments. Comput

Meth Appl Mech Eng 1996;139:3–47.

[30] Askes H, Pamin J, de Borst R. Dispersion analysis and element-free Galerkin solutions of second and fourth-order gradient-

enhanced damage models. Int J Numer Meth Eng 2000;49:811–32.

[31] Pamin J, Askes H, de Borst R. An element-free Galerkin method for gradient plasticity. In: Wunderlich W, editor. European

Conference on Computational Mechanics––Solids, Structures and Coupled Problems in Engineering. 1999.

[32] Askes H, Suiker ASJ, Sluys LJ. Dispersion and numerical analysis of higher-order gradient models in homogenized and

regularized media. Technical Report 03.21.1.31.07, T.U. Delft, 1999.

[33] Askes H, Suiker ASJ, Sluys LJ. Dispersion analysis and numerical simulations of wave propagation in homogenised granular

media. In: Wall WA, Bletzinger K-U, Schweizerhof K, editors. Trends in computational structural mechanics. Barcelona:

CIMNE; 2001. p. 59–68.


http://www.ipme.ru

higher-order strain/higher-order stress gradient models derived from a discrete microstructure, with...

Documents