Continua with microstructure: second-gradient theory

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Continua with microstructure: second-gradienttheoryStefanos-Aldo Papanicolopulos a & Antonis Zervos ba Department of Mechanics, National Technical University of Athens, Zografou, 157 73,Greece E-mail:b School of Civil Engineering and the Environment University of Southampton, SO17 1BJ,U.K. E-mail:

To cite this article: Stefanos-Aldo Papanicolopulos & Antonis Zervos (2010) Continua with microstructure:second-gradient theory, European Journal of Environmental and Civil Engineering, 14:8-9, 1031-1050, DOI:10.1080/19648189.2010.9693278

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EJECE – 14/2010. Mathematical modeling in geomechanics, pages 1031 to 1050

Continua with microstructure: second-gradient theory

Theory, examples and computational issues

Stefanos-Aldo Papanicolopulos* — Antonis Zervos**

* Department of Mechanics National Technical University of Athens Zografou 157 73, Greece

[email protected]

** School of Civil Engineering and the Environment University of Southampton SO17 1BJ, U.K.

[email protected]

ABSTRACT. Second-gradient theories represent a frequently used subset of theories of continua with microstructure. This paper presents an extended overview of second-gradient theories, starting from a simple one-dimensional example, proceeding with a thorough description of gradient elasticity and additionally briefly describing some other theories of this kind. A series of characteristic examples is presented to demonstrate the main aspects and applications of second-gradient theories. Finally, the complications in the finite-element implementation of second-gradient theories are presented, along with a review of the finite elements that have been developed for this purpose.

RÉSUMÉ. Les théories du second gradient représentent un sous-ensemble fréquemment utilisé des théories des milieux continus avec microstructure. Cet article présente un aperçu étendu des théories du second gradient, à partir d’un exemple simple unidimensionnel, avec une description approfondie de l’élasticité du second gradient et une brève description de quelques autres théories de ce genre. Une série d’exemples caractéristiques est présentée pour montrer les aspects principaux et les applications des théories du second gradient. Enfin, la complexité de la mise en œuvre des éléments finis basés sur des théories du second gradient est présentée, accompagnée d’une présentation des éléments finis qui ont été développés à cette fin.

KEYWORDS: second-gradient theories, gradient elasticity, microstructure, finite elements.

MOTS-CLÉS : théories du second gradient, élasticité du second gradient, microstructure, éléments finis.

DOI:10.3166/EJECE.14.1031-1050 © 2010 Lavoisier, Paris

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1032 EJECE – 14/2010. Mathematical modeling in geomechanics

1. Introduction

The theory of continua with microstructure, or generalised continua, introduces

the effect of a material microstructure whose mechanical behaviour is distinct from

the macroscopic mechanical behaviour of the material. A special case of this theory

is the second-gradient theory, also known as strain-gradient or second-grade theory,

in which the mechanical behaviour at each material point is made to depend not only

on the first gradient of the displacement (or, actually, on the strain) but on the second

gradient of the displacement as well.

As noted by Biot (1967), the basic concept of continua with microstructure and

more precisely of second- (or higher-) gradient continua can be found as back as 1851

in a paper by Cauchy (1851). It was however a century later, with the work of Toupin

(1962), Mindlin (1964) and Mindlin et al. (1968), that a theory of second-gradient

elasticity was rigorously formulated. Since then, a large number of papers has been

published on the topic of second gradient theories. Among the more recent work

on the topic, we can mention indicatively some examples of the use of strain-gradient

models in problems of elastic deformation (Shu et al., 1999; Askes et al., 2002; Zervos

et al., 2009; Papanicolopulos et al., 2009), fracture behaviour (Amanatidou et al.,

2002; Askes et al., 2006; Papanicolopulos et al., 2010a) and plasticity (Matsushima et

al., 2002; Zervos et al., 2001a; Zervos et al., 2001b).

In the present work we start by providing, in Section 2, a simple example showing

how a second-gradient theory can be derived by using averaging methods. Gradient

elasticity is described in Section 3 as a typical example of a second-gradient theory,

while other such theories are briefly considered in Section 4. To present the main

characteristics and applications of second-gradient theories, Section 5 provides some

examples of problems involving gradient models. Section 6 presents the challenges

in the finite-element implementation of second-gradient theories and lists some of the

finite elements usually employed, while Section 7 presents the main conclusions.

2. A simple one-dimensional example

This section presents a simple derivation of a second-gradient theory (Exadaktylos

et al., 2001), introducing the concepts of non-locality and material length.

In continuum mechanics, the spatial distribution of a given quantity is obtained

by assigning to each material point the average value of the quantity measured in a

representative elementary volume (REV) centred around that point. The REV is called

“representative” as it is chosen with appropriate dimensions so that variations at very

small scales are ignored while inhomogeneities at larger scales are not suppressed.

In the simple case of a one-dimensional continuum, the REV is a length L around

each point x. The average value 〈y〉 of a field y = f(x) in this case is given by

〈y〉 =1

L

∫ L/2

−L/2

f(x+ ξ) dξ [1]

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Second-gradient theory 1033

where the value of f(x+ ξ) can be expressed as a Taylor series around ξ = 0,

f(x+ ξ) = f(x) + f ′(x)ξ + 12f

′′(x)ξ2 +O(ξ3) [2]

For a slowly varying field, the local curvature is small and f can be approximated

with just two terms of the series [2], so Equation [1] yields

〈y〉 ≈ y [3]

that is, the local value y and the average value 〈y〉 coincide, so the continuum is termed

locally homogeneous or local. If the local curvature of the field is not negligible, then

additional terms of the Taylor series must be taken into account. Including the ξ2 term

we obtain

〈y〉 ≈1

L

∫ L/2

−L/2

f(x) + f ′(x)ξ +1

2f ′′(x)ξ2 dξ = y +

L2

24

d2y

dx2[4]

Since fourth-order derivatives are neglected, differentiating twice Equation [4] yields

d2y/dx2 ≈ d2〈y〉/dx2 resulting in

y ≈ 〈y〉 −L2

24

d2〈y〉

dx2[5]

so the continuum is locally inhomogeneous or non-local and is described by a second-

gradient theory.

It is important to note that L is a length that depends on the material, not on the

geometry of the problem. The presence of at least one material parameter with di-

mensions of length is typical of second-gradient theories, allowing them to introduce

a microstructure that does not scale with the problem geometry but is intrinsic to the

material under consideration. This is a fundamental difference with respect to classical

theories which have no material parameters with dimension of length.

3. Gradient elasticity

This section presents the theory of second-gradient elasticity, or simply gradient

elasticity (Toupin, 1962; Mindlin, 1964; Mindlin et al., 1968), which is the simplest

second-gradient continuum theory that is based on a rigorous framework. Like its

classical counterpart, it is simple enough to allow for analytical treatment of some

simple, yet interesting, cases while at the same time displaying the main characteristics

of the family of continuum theories to which it belongs.

Only the static case for small deformations is considered here. Indicial tensor no-

tation is used, with indices ranging from 1 to 3, repeated indices indicating summation

and a comma in the indices indicating spatial derivatives. All tensors are assumed to be

Cartesian, so no distinction is made between covariant and contravariant components.

While Mindlin (1964) presents three different forms of gradient elasticity, depend-

ing on how the components of the second gradient of the displacement are grouped

together, only Form II is presented here as all forms are completely equivalent.

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3.1. Variational formulation

Since only the static case is considered, equilibrium requires that the variation of

the total potential energy W equals the variation of the work of the external actions P

δW = δP [6]

where the total potential energy W is obtained as the volume integral of a potential

energy density function W

W =

∫

V

W dV [7]

In second-gradient elasticity, for a given a displacement field ui, the function Wtakes the form

W = W (ǫij , κijk) [8]

where ǫij is the strain and κijk is the strain gradient defined respectively as

ǫij ≡ u(i,j) =12 (ui,j + uj,i) κijk ≡ ǫjk,i [9]

The variation of W , given by

δW =∂W

∂ǫijδǫij +

∂W

∂κijkδκijk = τijδǫij + µijkδκijk [10]

introduces the stress τij and double stress µijk, defined as

τij ≡∂W

∂ǫijµijk ≡

∂W

∂κijk[11]

which are the static quantities work-conjugate to the kinematic quantities strain ǫijand strain gradient κijk respectively 1.

Using Equations [7] and [10] the variation of the total potential energy is

δW =

∫

V

(τijδǫij + µijkδκijk) dV [12]

which, using the definitions of the kinematic quantities in Equations [9], the symmetry

of the stresses and double stresses and the divergence theorem, can be written as

δW =

∫

S

nj(τjk − µijk,i)δuk dS −

∫

V

(τjk − µijk,i),jδuk dV

+

∫

S

niµijkδuk,j dS

[13]

1. The differentiations in [11] are performed within the space of symmetric tensors, therefore

the resulting static quantities share the same symmetries as the respective kinematic quantities

(Truesdell et al., 2004; Itskov, 2007).

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3.1.1. The displacement gradient on the boundary

The form of δW given in Equation [13] cannot suggest directly a form of the

variation δP , because the displacement gradient δuk,j on the boundary S is not inde-

pendent of the displacement δuk. Using the normal derivative operatorD and surface-

gradient operator Dj defined respectively as

D ≡ nj∂

∂xjDj ≡

∂

∂xj− njD [14]

Equation [13] can be written, after some calculations, as

δW = −

∫

V


+

∫

S

(

nj(τjk − µijk,i)−Dj(niµijk) + (Dlnl)njniµijk

)

δuk dS

+

∫

S

ninjµijkDδuk dS +

∫

S

nqeqpm(emljnlniµijkδuk),p dS [15]

where Dδuk is independent of the displacement δuk on the boundary S. Due to

Stokes’ theorem, the last integral in Equation [15] is zero if S is smooth. If S has an

edge C, then∫

S

nqeqpm(emljnlniµijkδuk),p dS =

∮

C

smemlj [[nlniµijk]]δuk dC [16]

where sm is the tangent unit vector on the edge C and the double brackets [[ ]] indicate

the jump in the enclosed quantity across the edge C.

The variation of the total potential energy can therefore be written as

δW = −

∫

V


+

∫

S

(

nj(τjk − µijk,i)−Dj(niµijk) + (Dlnl)njniµijk

)

δuk dS

+

∫

S

ninjµijkDδuk dS +

∮

C

smemlj [[nlniµijk]]δuk dC [17]

3.1.2. Equilibrium equations

Equation [17] suggests the following form for the variation of the work of the

external actions

δP =

∫

V

Fkδuk dV +

∫

S

Pkδuk dS +

∫

S

RkDδuk dS +

∮

C

Ekδuk dC [18]

where Fk and Pk are body forces and surface tractions like those found in classical

theories, Rk are self-equilibrating surface double-tractions and Ek are edge tractions.

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Using Equations [6], [12] and [18] leads to the virtual work equation for gradient

elasticity, which is actually the integral equilibrium equation,

∫

V

(τijδǫij + µijkδκijk) dV =

∫

V

Fkδuk dV +

∫

S

Pkδuk dS +

∫

S

RkDδuk dS +

∮

C

Ekδuk dC [19]

while using Equations [6], [17] and [18] yields the differential equilibrium equation

(τjk − µijk,i),j + Fk = 0 [20]

together with the traction boundary conditions

Pk = nj(τjk − µijk,i)−Dj(niµijk) + (Dlnl)njniµijk [21]

Rk = ninjµijk [22]

Ek = smemlj [[nlniµijk]] [23]

Note that the surface tractions Pk are given by a different formula than the one in

classical theories (Pk = njτjk), although in both cases they are the quantities that are

work conjugate to the surface displacements.

3.2. Linear and isotropic gradient elasticity

In Subsection 3.1 the potential energy densityW was defined as a generic function

of the strains and strain gradients. To obtain further results it is necessary to consider

a specific form of W that corresponds to a specific stress-strain relation.

3.2.1. Linear gradient elasticity

The simplest non-trivial stress-strain relation is a linear one where stresses and

double stresses are linear combinations of strains and strain gradients. This can be

obtained by assuming a quadratic form for the potential energy density function

W = 12cijpqǫijǫpq + fijkpqκijkǫpq +

12aijkpqrκijkκpqr [24]

resulting in the following formulae for the stresses and double stresses

τij = cijpqǫpq + fpqrijκpqr [25]

µijk = fijkpqǫpq + aijkpqrκpqr [26]

From Equation [24] we can obtain the following major symmetries (see also

Georgiadis et al., 2006)

cijkl = cklij [27]

aijklmn = almnijk [28]

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while the symmetries of the kinematic and static quantities yield the following minor

symmetries

cijkl = cjikl = cijlk [29]

fijklm = fikjlm = fijkml [30]

aijklmn = aikjlmn = aijklnm [31]

Taking into account these symmetries, there are 21 independent material parame-

ters in cijkl, 108 in fijklm and 171 in aijklmn, for a total of 300 independent material

parameters in the general case of anisotropic linear gradient elasticity, in contrast to

the 21 independent material parameters present in the case of anisotropic linear clas-

sical elasticity.

3.2.2. Isotropic linear gradient elasticity

A significant reduction in the number of the material parameters required in linear

gradient elasticity can be achieved in the case of an isotropic material, that is a material

that exhibits the same mechanical behaviour in all spatial directions. In this case, the

general linear gradient elasticity assumes a much simpler form.

To obtain isotropic behaviour, the cijkl, fijklm and aijklmn tensors of linear gra-

dient elasticity must all be isotropic. Since there are no isotropic tensors of odd rank,

fijklm is considered here to be zero 2. There are three linearly independent isotropic

rank-four tensors and fifteen linearly independent rank-six tensors, however taking

into account the major and minor symmetries given in Subsection 3.2.1 further re-

duces the number of independent material parameters so that the potential energy

density function for isotropic linear gradient elasticity is

W = 12λǫiiǫjj + µǫijǫij + a1κiikκkjj + a2κijjκikk + a3κiikκjjk

+ a4κijkκijk + a5κijkκkji

[32]

where λ and µ correspond to the Lamé parameters while a1 . . . a5 are five additional

parameters. Substituting Equation [32] into the definitions [11] yields the stress-strain

relations

τij = λδijǫpp + 2µǫij [33]

µijk = 12a1(δijκkpp + 2δjkκppi + δikκjpp) + 2a2δjkκipp

+ a3(δijκppk + δikκppj) + 2a4κijk + a5(κkji + κjki) [34]

2. As was correctly mentioned by Mindlin (1964), but often not explicitly stated in the literature,

this is only true for centrosymmetric isotropic materials. For non-centrosymmetric isotropic

gradient elasticity, fijklm depends on a single additional material parameter (Papanicolopulos,

2008).

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3.2.3. The equilibrium equation for displacements

Using the definitions of the kinematic quantities in Equation [9] and the stress-

strain relations in Equations [33] and [34], the differential equilibrium equation given

in Equation [20] can be written in terms of the displacements as

(λ+ µ)up,pk + µuk,pp − (2a1 + 2a2 +32a3 + a4 +

32a5)up,pqqk

− ( 12a3 + a4 +12a5)uk,ppqq + Fk = 0 [35]

The above form can be written as

(λ+ 2µ)up,pk − µ(up,pk − uk,pp)− 2(a1 + a2 + a3 + a4 + a5)up,pqqk

+ (12a3 + a4 +12a5)(up,pqqk − uk,ppqq) + Fk = 0 [36]

The form [36], although seemingly more complex, has the advantage that, for

W positive definite, all coefficients of the displacement gradients are strictly pos-

itive (Mindlin et al., 1968). In Equation [35], on the other hand, the coefficient

(2a1 + 2a2 +32a3 + a4 +

32a5) can be negative. It is therefore possible to set

l21 =2(a1 + a2 + a3 + a4 + a5)

λ+ 2µl22 =

a3 + 2a4 + a52µ

[37]

to obtain the equilibrium equation for displacements as

(λ + 2µ)(1̃ − l21∇2)up,pk − µ(1̃ − l22∇

2)(up,pk − uk,pp) + Fk = 0 [38]

where ∇2 is the tensor Laplacian, ∇2(·) = (·),qq and 1̃ is the identity operator.

Note that although there are five independent parameters a1 . . . a5, they appear in

the displacement equilibrium equation only as two lengths, so that their individual

values only appear through the application of boundary conditions.

3.2.4. A simplified isotropic model

The general isotropic model presented in Subsection 3.2.2 includes, besides the

two Lamé parameters λ and µ, five additional material parameters. A simpler isotropic

model is obtained by setting

a1 = 0, a2 = 12λl

2, a3 = 0, a4 = µl2, a5 = 0 [39]

In this case l1 = l2 = l and Equation [38] can be written as

(1̃− l2∇2)(

λup,pk + µ(up,pk + uk,pp))

+ Fk = 0 [40]

Although this material model is significantly simpler in its formulation, the physi-

cal interpretation of the parameter choice in [39] is not clearly defined.

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4. Other second-gradient theories

Besides gradient elasticity, two relatively frequently used theories are gradient

plasticity and gradient damage. While there are many different specific models, their

main common characteristic is the ability to regularise mathematically ill-posed prob-

lems of localisation of deformation, e.g. due to strain-softening material behaviour.

Both plasticity and damage constitutive equations introduce an additional depen-

dence on a set of internal variables. In the case of gradient models, a significant

distinction must be made between strain-gradient models, i.e. models where the con-

stitutive equation introduces dependence on the second gradient of the displacement,

and models with gradients of the internal variables.

The first strain-gradient plasticity model (Dillon et al., 1970) was actually a third-

gradient model (i.e. it included the second gradient of strain) based on third-gradient

elasticity (Mindlin, 1965). Second-gradient plasticity models include those used by

Chambon et al. (1996; 1998), Fleck et al. (1997), Zervos et al. (2001a) and Mat-

sushima et al. (2002). Plasticity models with gradients of internal variables were used

e.g. by Mühlhaus et al. (1991), Vardoulakis et al. (1991; 1994), de Borst et al. (1992),

Vardoulakis et al. (1992), Pamin (1994) and Ramaswamy et al. (1998).

A review of gradient plasticity models of both types was recently given by Jirásek

et al. (2009a; 2009b). A comparison of various gradient plasticity and gradient dam-

age models is given by Pamin (2005).

5. Example problems in second-gradient theory

This section presents different example problems chosen so as to demonstrate the

main aspects of second gradient theories and their main applications. All examples

employ the simplified gradient elasticity model given in Subsection 3.2.4.

5.1. Thick hollow cylinder under external normal traction

Consider the plane-strain problem of a thick hollow cylinder of infinite height,

with internal radius ra and external radius rb, subject to an external normal traction

pb, as shown in the drawing of Figure 1. This problem was solved by Exadaktylos et

al. (2001) for rb → ∞ (and internally applied normal traction), and by Zervos et al.

(2009) for finite rb. The solution for the finite-height cylinder is significantly more

complicated (Papanicolopulos, 2008).

Figure 1 shows the variation of the radial strain ǫrr (normalised by pb/µ) within

the cylinder, for different ratios ra/ℓ as well as for classical elasticity (ℓ = 0), with

rb/ra = 3 and Poisson’s ratio ν = 1/4. In the classical elasticity solution there is

a strong strain gradient near the internal boundary (the strain gradient is proportional

to r−3), so there the difference from the gradient elastic solution is greater. This

difference increases as the internal length ℓ becomes larger, i.e. comparable to ra.

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-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

1.0 1.5 2.0 2.5 3.0

norm

alised

radialstrain

ε rr/(pb/

µ)

normalised dimensionless radial position r/ra

ℓ= 0

ra/ℓ= 32

ra/ℓ= 16

ra/ℓ= 8

ra/ℓ= 4

ra

rb

pb

Figure 1. Radial strain variation in a thick cylinder under pressure

5.2. Infinite layer under shear

In the first example, the influence of the strain gradient was larger near the internal

boundary due to the geometry of the problem, thus forming a kind of boundary layer.

Boundary layers are often encountered in second-gradient theories, however they are

usually due to higher-order boundary conditions, as in the following example.

Consider the problem of simple shearing in plane strain of a layer of height2H , ex-

tending infinitely in the lateral direction, as shown in Figure 2. The additional higher-

order boundary condition ∂ux/∂y = 0, usually identified with a rough boundary, is

applied on the top and bottom boundary. The analytical solution is given by Zervos et

al. (2009).

The normalised shear strain γxy/(τ/µ) in the upper half of the shear layer is plot-

ted in Figure 2 for different values of the ratio H/ℓ. In classical elasticity, where

it is not possible to impose the rough-boundary condition, the solution would be

γxy/(τ/µ) = 1. The gradient elasticity solution, on the other hand, forces the shear

strain to be 0 at the boundary and therefore creates a boundary layer, whose size scales

with ℓ. For H comparable to ℓ, the two opposite boundary layers merge.

5.3. Infinite layer with bolts

In the previous example, kinematic higher-order boundary conditions were ap-

plied, whose physical meaning was linked to the presence of a rough boundary. It is

also possible to enforce double-tractions, which again lead to the formation of bound-

ary layers, yet the way they may physically be applied must be explained. One such

explanation is given by Vardoulakis (2003) for the case of the “bolted” layer.

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0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

norm

alised

positiony/H

normalised shear strain γxy/(τ/µ)

H/ℓ= 64H/ℓ= 32H/ℓ= 16H/ℓ= 8H/ℓ= 4

τ

τ

x

y

H

H

Figure 2. Shear strain variation in a sheared infinite layer

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

norm

alised

positiony/H

normalised strain εyy/(R/ℓ/(λ +2µ))

H/ℓ= 64H/ℓ= 32H/ℓ= 16H/ℓ= 8H/ℓ= 4

R

R

x

y

H

H

Figure 3. Strain variation in a bolted infinite layer

Consider a layer of height 2H extending infinitely in the lateral direction, as shown

in Figure 3. The layer is loaded, under plane strain conditions, by a double traction

Ry = R on the upper and lower boundary. Such a traction may be imposed e.g.

in a rock mass by inserting rock bolts, hence the term bolted layer. In this case, a

compressive double traction is applied so R will be negative.

The solution for the strain ǫyy normalised by R/ℓ/(λ+ 2µ) is plotted in Figure 3.

As in the previous examples, a boundary layer is formed whose size scales with ℓ.

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0.0

0.2

0.4

0.6

0.8

1.0

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2

norm

alised

positiony/H

normalised horizontal displacement ux/uy0

H/ℓ= 64H/ℓ= 32H/ℓ= 16H/ℓ= 8H/ℓ= 4classical

uy0

uy0

x

y

H

H

H H

Figure 4. Horizontal displacement of the side of a rectangular block under plane-

strain uniaxial loading in the vertical direction

5.4. Plane-strain uniaxial loading

A characteristic of gradient elasticity which is often overlooked is the presence

of edge tractions, in addition to classical and double tractions. The effect of edge

tractions however can be significant, as seen in the following example.

Consider a square block of side 2H , as seen in Figure 4 loaded uniaxially in plane

strain conditions by imposing a displacement uy0 on the top side and −uy0 on the

bottom side. Additionally, the boundary condition duy/dy = 0 is applied on both the

top and the bottom side.

As no analytical solution is available, the problem was solved numerically using

finite elements, with H = 1, uy0 = −0.001 and ν = 0.35. Figure 4 shows the results

for the horizontal displacement ux of the upper half of the right edge, normalised by

the applied vertical displacement uy0. Figure 4 also includes the solution for classical

elasticity, which of course does not allow applying the boundary condition for the

displacement gradient. It is seen that in the gradient elastic case, tension produces a

“pincushion” deformation of the block while compression would produce a “barrel”

deformation.

Note that a solution of the form uy = f(y), ux = kx satisfies the equilibrium

differential equation of the problem and all the boundary conditions except for the

one for the edge tractions. Ignoring therefore the edge tractions would lead to an

incorrect analytical solution. It is thus seen from this example that the effect of the

presence of edges in gradient elastic bodies is present even in the absence of externally

applied edge tractions.

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0.0

0.2

0.4

0.6

0.8

1.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

norm

alised

verticaldisplacementuy/u

0

normalised distance from crack tip x/a

classicala/ℓ= 50a/ℓ= 20a/ℓ= 10a/ℓ= 5

P0

P0

x

y

2a

Figure 5. Crack opening displacement for a mode I crack (ν = 0.2) (Papanicolopulos

et al., 2010a)

5.5. Mode I crack

Gradient elasticity has been used to study problems of fracture mechanics, espe-

cially due to its ability to suppress the stress singularity at the crack tip encountered

in classical elasticity. Analytical and numerical results have been published by dif-

ferent authors (e.g. Exadaktylos, 1998; Zhang et al., 1998; Askes et al., 2006; Karlis

et al., 2007; Gourgiotis et al., 2009; Aravas et al., 2009). A detailed finite element

study of all three crack modes in gradient elasticity is given by Papanicolopulos et al.

(2010a), including the example shown here.

Figure 5 shows a crack of length 2a under mode I loading by a traction P0 normal

to the crack, applied at an infinite distance from it. The same figure shows the plots

of the crack opening displacement uy, normalised by the maximum crack opening u0

predicted by classical elasticity, for ν = 0.2 and different values of a/ℓ. We see that as

the crack becomes “smaller”, that is comparable to the internal length, it also becomes

“stiffer”. An important qualitative difference between the classical and the gradient

solution is that the displacement gradient ∂uy/∂x on the crack tip in the former case

is infinite while in the latter case it is zero, resulting in a cusped shape of the crack.

The results plotted in Figure 5 were obtained using a finite element discretisation

of a quarter of a finite square domain of side 32a, while the analytical solution for

classical elasticity refers to an infinite domain, thus the normalised numerical solution

for the maximum crack opening in classical elasticity is slightly larger than1. Figure 6

shows the strains, stresses and strain gradients along the line of the crack, fora/ℓ = 10and ν = 0.2. Contrary to the classical elasticity case, all stresses and strains are finite

and continuous (but not smooth) at the crack tip. The strain gradients, on the other

hand, are not continuous at the crack tip and tend to infinity either towards or ahead of

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the crack tip. The finite element solution is obviously unable to account for the infinite

and discontinuous strain gradients, therefore the value at the crack tip and on adjacent

nodes is not shown in the strain gradient plot.

-1

0

1

2

3

-1 -0.5 0 0.5 1

norm

alised

strain

andstress


εxxεyyεxyτxxτyy

-40

-20

0

20

40

-0.1 -0.05 0 0.05 0.1

norm

alised

strain

gradient


εxx,xεyy,xεxy,x

Figure 6. Strains, stresses and strain gradients along the line of a mode I crack

(Papanicolopulos et al., 2010a)

6. Finite-element implementation of second-gradient theories

The implementation of second-gradient theories in finite-element codes is not

trivial because, as will be shown, it is not possible to use the usual finite elements

which have been developed for classical theories. The development of appropriate

finite elements is therefore a topic of ongoing research (Shu et al., 1999; Zervos et

al., 2001a; Matsushima et al., 2002; Amanatidou et al., 2002; Askes et al., 2006; Zer-

vos, 2008; Zervos et al., 2009; Papanicolopulos et al., 2009). A detailed presentation

of the issues related to the finite-element implementation of second-gradient theories

is provided by Papanicolopulos et al. (2010b), together with the presentation of vari-

ous finite elements that have been proposed in the relevant literature and a theoretical

comparison of their relative merits.

This section refers exclusively to the numerical implementation of strain-gradient

theories. Gradient theories that only use gradients of the internal variables, mentioned

in Section 4, present a different set of problems and will not be considered here.

6.1. Displacement-only discretisation

The usual choice in finite elements for solid mechanics is to discretise only the

displacement field (Zienkiewicz et al., 1989; Bathe, 1996). The displacement field

u(x) is interpolated as

u(x) = N(x)uN [41]

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where N(x) is a matrix of shape functions and uN is the vector of all the degrees of

freedom (which are not necessarily displacements themselves).

As usual in the finite element method, matrix notation is used instead of the ten-

sor notation used in analytical calculations. In the case of second-gradient theories,

however, the strain vector actually includes both strains and strain gradients, while the

stress vector contains both stresses and double stresses. The “generic strain” vector ǫ̌

and “generic stress” vector τ̌ are therefore introduced so that their inner product gives

the work of stresses and double stresses on strains and strain gradients, i.e.

ǫ̌T τ̌ = τijǫij + µijkκijk [42]

The vector ǫ̌ contains linear combinations of the first and second order derivatives

of the displacements u, so using [41] the discretisation of the generic strain is

ǫ̌(x) = B(x)uN [43]

where the matrixB(x) contains linear combinations of the first and second derivatives

of the components of N(x).

6.2. The discretised form of the virtual work equation

Substituting Equations [41], [42] and [43] into the virtual work equation given in

Equation [19] yields

δ(uN )T∫

V

BT τ̌dV = δ(uN )T(∫

V

NTFdV

+

∫

S

(

NTP +D(NT )R)

dS +

∮

C

NTEdC

)

[44]

where the dependence of B and N on x is no longer shown, in order to simplify

the resulting expressions. This scalar equation must hold for every variation δuN

therefore it is equivalent to the vector equation

∫

V

BT τ̌dV =

∫

V

NTFdV +

∫

S

(

NTP +D(NT )R)

dS +

∮

C

NTEdC [45]

The constitutive relation linking the generic stresses to the generic strains is, in

the general case, non-linear, so Equation [45] can be solved using for example the

Newton-Raphson method. In the case of linear (but not necessarily isotropic) gradient

elasticity the stress-strain relation is

τ̌ = Dǫ̌ = DBuN [46]

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where D is a matrix of material parameters incorporating the components of tensors

cijkl, fijklm and aijklmn appearing in Equation [24]. Equation [45] then becomes

(∫

V

BTDBdV

)

uN =

∫

V

NTFdV

+

∫

S

(

NTP +D(NT )R)

dS +

∮

C

NTEdC [47]

which is a linear system that can be written in the familiar form

KuN = f [48]

where K is the stiffness matrix and f is the vector of external actions. Equation [48]

can be solved directly for the degrees of freedomuN which, using Equation [41], give

the approximate solution for the displacement field.

6.3. The C1 requirement

The displacement-only discretisation presented in the previous subsection is simi-

lar to the one used for classical theories, with the following differences:

1) The existence of surface double tractions and edge tractions leads to the pres-

ence of additional terms in the vector of external actions f .

2) Generic strain and stress vectors are used to express the constitutive relations in

matrix form, which include strain gradients and double stresses respectively.

3) Second derivatives of the shape functions are included in the B matrix, in ad-

dition to the first derivatives present in the classical theories.

It is this last difference that poses the greatest difficulty in the numerical imple-

mentation. The presence in B of second derivatives of the shape functions requires

that the first derivatives must be continuous on the whole domain in order to calculate

properly the stiffness matrix K. In other words the shape functions must ensure C1

continuity of the interpolation, resulting in C1 finite elements. This requirement is

generally not a problem within each element, as the interpolation is generally polyno-

mial, but is an actual problem at the interfaces between different elements.

In one dimension, the interface between elements is a single point, which is a node

of the element. In this case the obvious choice for C1 elements are the cubic Hermite

elements, as used for example by Chambon et al. (1996; 1998) for gradient plasticity.

In two dimensions, the inter-element interface includes not only nodes but also

shared edges, thus it is more difficult to develop C1 elements. Such elements have

been developed for plate-bending problems, where a similar continuity requirement

results from the choice of a displacement-only interpolation. A C1 triangle presented

by Argyris et al. (1968) and in a simplified form by Dasgupta et al. (1990) has been

used successfully with gradient models (Zervos, 2001; Zervos et al., 2001a; Akarapu

et al., 2006; Zervos et al., 2009; Fischer et al., 2010; Papanicolopulos et al., 2010a). A

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C1 isoparametric Hermite element (Petera et al., 1994) has also been used for gradient

elasticity problems (Zervos et al., 2009; Papanicolopulos et al., 2010a).

In three dimensions, finally, neighbouring elements can share nodes, edges and

faces, so enforcing C1 continuity becomes even more difficult. Due also to the

initial focus on plate-bending, three-dimensional C1 elements have therefore re-

ceived very little attention. One such element, an isoparametric Hermite hexahe-

dron based on the element presented by Petera et al. (1994) has been recently de-

veloped (Papanicolopulos, 2008; Papanicolopulos et al., 2009).

6.4. Alternative numerical formulations

As shown in Subsection 6.3, discretising only the displacement field leads to the

requirement for C1 continuity. A variety of techniques have however been developed

to avoid the C1 requirement—both in problems involving strain-gradient models and

in other fields such as plate bending where C1 requirements appear. The search for

alternative formulations has been motivated by the relatively small number of C1 el-

ements available, as well as because the use of C1 elements is considered by some

authors as complex and numerically expensive (see however the analysis of Papani-

colopulos et al. (2010b) for a different view on the issue of computational cost).

The C1 requirement is often avoided using mixed formulations, where multi-

ple fields are discretised, with constraints placed to impose the compatibility of the

different fields. This removes the C1 requirement, thus allowing greater freedom

in the choice of the shape functions that interpolate each field and of the resulting

elements, but introduces the overhead of multiple discretisations. Mixed formula-

tions can impose the compatibility of different fields using Lagrange multipliers (Shu

et al., 1999; Matsushima et al., 2002; Amanatidou et al., 2002) or penalty meth-

ods (Zervos, 2008). A different approach, through an implicit reformulation of gradi-

ent elasticity, is used by Askes et al. (2006).

Other numerical methods used to avoid the issue ofC1 continuity include meshless

finite elements (Askes et al., 2002) and boundary elements (Karlis et al., 2007).

7. Conclusions

This paper presents an overview of second-gradient theories. It is seen that even a

very simple one-dimensional averaging procedure can lead to a second-gradient the-

ory that contains the basic characteristics of such theories. A rigorous mathematical

framework is needed, however, to develop a general theory, especially as concerns the

determination of the applicable boundary conditions and material parameters.

A series of typical example problems is provided, whose solution has been ob-

tained either analytically or numerically, indicating the main characteristic of second-

gradient theories and their applications. It is seen, for example, that such theories can

represent phenomena linked to the microstructure, such as boundary layer formation.

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The issue of finite-element implementation of second-gradient theories is also con-

sidered, showing how a standard discretisation leads to the need for C1 elements. We

present which elements of this kind have been successfully used while also mentioning

alternative techniques that try to avoid the C1 requirement.

Acknowledgements

The research leading to these results has received funding from the European Re-

search Council under the European Community’s Seventh Framework Programme

(FP7/2007–2013) / ERC grant agreement no 228051.

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Continua with microstructure: second-gradient theory