Problems of the Flamant Boussinesq and Kelvin Type in Dipolar Gradient...

Problems of the Flamant–Boussinesq and Kelvin Typein Dipolar Gradient Elasticity

H. G. Georgiadis & D. S. Anagnostou

Received: 7 January 2007 /Accepted: 29 June 2007 / Published online: 5 September 2007# Springer Science + Business Media B.V. 2007

Abstract This work studies the response of bodies governed by dipolar gradient elasticity toconcentrated loads. Two-dimensional configurations in the form of either a half-space(Flamant–Boussinesq type problem) or a full-space (Kelvin type problem) are treated andthe concentrated loads are taken as line forces. Our main concern is to determine possibledeviations from the predictions of plane-strain/plane-stress classical linear elastostatics whena more refined theory is employed to attack the problems. Of special importance is thebehavior of the new solutions near to the point of application of the loads where pathologicalsingularities and discontinuities exist in the classical solutions. The use of the theory ofgradient elasticity is intended here to model material microstructure and incorporate sizeeffects into stress analysis in a manner that the classical theory cannot afford. A simple butyet rigorous version of the generalized elasticity theories of Toupin (Arch. Ration. Mech.Anal. 11:385–414, 1962) and Mindlin (Arch. Ration. Mech. Anal. 16:51–78, 1964) isemployed that involves an isotropic linear response and only one material constant (the so-called gradient coefficient) additional to the standard Lamé constants (Georgiadis et al., J.Elast. 74:17–45, 2004). This theory, which can be viewed as a first-step extension of theclassical elasticity theory, assumes a strain-energy density function, which besides itsdependence upon the standard strain terms, depends also on strain gradients. The solutionmethod is based on integral transforms and is exact. The present results show departure fromthe ones of the classical elasticity solutions (Flamant–Boussinesq and Kelvin plane-strainsolutions). Indeed, continuous and bounded displacements are predicted at the points ofapplication of the loads. Such a behavior of the displacement fields is, of course, morenatural than the singular behavior present in the classical solutions.

Keywords Concentrated loads . Flamant–Boussinesq problem . Kelvin problem .

Green’s functions . Gradient elasticity . Microstructure . Dipolar stresses . Laplace transforms

Mathematics Subject Classifications (2000) 35Q72 . 44A10 . 74B99 . 74J15

J Elasticity (2008) 90:71–98DOI 10.1007/s10659-007-9129-x

H. G. Georgiadis (*) : D. S. AnagnostouMechanics Division, National Technical University of Athens, Zographou Campus, Athens 15773,Greecee-mail: [email protected]

1 Introduction

The Flamant–Boussinesq and Kelvin problems are considered two of the most celebratedproblems in the classical linear elastostatic theory. In the first problem, a half-space underplane-strain/plane-stress conditions is acted upon by a concentrated line force on its surface(see e.g. Timoshenko and Goodier [4], Fung [5], Barber [6]). In the plane-strain/plane-stressversion of the second problem, a 2D full-space is acted upon by a concentrated line force inits interior (see, e.g., Timoshenko and Goodier [4], Barber [6]). The problems enjoyimportant applications in various areas, e.g., in Contact Mechanics and Tribology. Also, theFlamant–Boussinesq and Kelvin solutions serve as pertinent Green’s functions in amultitude of problems analyzed by the Boundary Element Method.

A well-known feature of the classical elasticity solutions to the above problems is alogarithmic singularity for the displacement field at the point of application of the load.Besides the fact that this is a physically unsatisfactory result, there occurs a displacementfield at infinity which is also logarithmically unbounded at infinity (i.e., in these elastostatic2D source solutions, a pathological behavior appears both at the source point and at infinity).However, as Fung [5] pointed out, the logarithmically unbounded behavior at infinity is apeculiarity in the 2D elastostatic problems with concentrated loads — the corresponding 3Dor dynamical solutions do not exhibit such a peculiarity. Still, the displacement field remainsunbounded at the point of application of the load in all cases within classical elasticity. Inaddition, the classical elasticity Flamant–Boussinesq solution exhibits a discontinuity in thetangential (normal) surface displacement in the case of a normal (tangential) load.

It is obvious, therefore, that the classical solutions do not reflect at all the actualsituation. In fact, unbounded displacements occur at the point of application of the load, nomatter how small the load intensity is. Also, the use of the classical Flamant–Boussinesqsolution as a Green’s function to solve contact problems leads to the well-knownsingularities in contact stresses and strains in the 2D problems of indentation of a half-plane by a rigid wedge and indentation by a rigid flat punch (Sadowsky problem) [6]. So,the present work by using a more sophisticated theory than classical elasticity aims atproviding exact solutions to the plane-strain Flamant–Boussinesq and Kelvin problems thatare exempt from the aforementioned deficiencies. Indeed, use of a three-parameter lineargradient theory of the Toupin–Mindlin type [1, 2, 3] leads to solutions that do not exhibitthe singular behavior described above. In particular, continuous and bounded displacementsare predicted at the points of application of the loads. In addition, these displacementsdepend now upon the load intensity allowing, within the new context, a sound meaning ofthe load-carrying capacity of the medium. As discussed below, the Toupin–Mindlingradient theory models the mechanical response of solids with microstructure allowing thepresence of characteristic material lengths. On the other hand, it is well documented fromexperimental observations (see References given below) that the response of certainmaterials is significantly affected by their microstructure. Therefore, classical elasticityseems to be inadequate to describe the actual response of these materials.

In the present study, the simplest version of Mindlin’s gradient theory is employed, i.e.form II of Ref. [2]. According to this version: (i) each material particle has three degrees offreedom (the displacement components — just as in the classical theory), (ii) the micro-density does not differ from the macro-density, (iii) the Euler–Cauchy principle assumes itsclassical form (see, e.g., Fung [5]) with vanishing couple-stress vector, and (iv) the strain-energy density depends not only upon the strain tensor (as in standard elasticity) but alsoupon the gradient of strain tensor. In this way, form II of Mindlin’s theory is a first-stepextension of classical elasticity.

72 H.G. Georgiadis, D.S. Anagnostou

Our solution method is based on integral transforms and is exact. Bounded displace-ments are predicted at the point of application of the loads and, thus, the behavior of thesolutions is more natural. This may have important implications for more general contactproblems and the Boundary Element Method. The occurrence of bounded displacementfields here seems to have an explanation because, in general, materials governed bygeneralized continuum theories tend to behave in a more rigid way (having increasedstiffness) as compared to materials governed by classical continuum theories (see, e.g.,Weitsman [7], Georgiadis [8]).

A brief discussion and literature review of the theory of dipolar gradient elasticity is nowin order. More details can be found in some recent papers of the first author (see, e.g.,Georgiadis et al. [3], Georgiadis [8], Grentzelou and Georgiadis [9], Georgiadis andGrentzelou [10]). The theory of dipolar gradient elasticity was introduced by Toupin [1] andMindlin [2] in an effort to model the mechanical behavior of solids with microstructure. Forthe generalized continuum theory of Mindlin [2], especially, the basic concept lies in theconsideration of a medium containing elements or particles (called macro-media), which arein themselves deformable media. This behavior can easily be realized if such a macro-particle is viewed as a collection of smaller sub-particles (called micro-media). In this way,each particle of the continuum is endowed with an internal displacement field, which canbe expanded as a power series in internal coordinate variables. Within the above context,the lowest-order theory (dipolar theory) is the one obtained by retaining only the first(linear) term of the foregoing series. In its most general form, Mindlin’s theory withmicrostructure contains the linear Cosserat (or micropolar) theory as a special case.However, one would face formidable difficulties trying to attack even simple boundary valueproblems with such a complicated theory. Further simplifications introduced by Mindlinhimself [2] led to the so-called gradient theory, of which there exist three versions—thegeneral gradient form, form II, and form III (a particular case of which is the well-knowncouple-stress elasticity).

The Toupin–Mindlin theory had already some successful applications on stressconcentration problems concerning holes and inclusion, during the sixties and seventies(see e.g. Cook and Weitsman [11], and Eshel and Rosenfeld [12, 13]). More recently, thisapproach and related extensions for microstructured materials have been employed toanalyze various problems involving, among other areas, wave propagation (Georgiadis etal. [3], Vardoulakis and Georgiadis [14]), fracture (Georgiadis [8], Grentzelou andGeorgiadis [9], Shi et al. [15]), mechanics of defects (Lazar and Maugin [16]), finiteelasticity (Fosdick and Royer-Carfagni [17]), plasticity (Fleck et al. [18], Vardoulakis andSulem [19], Begley and Hutchinson [20], Huang et al. [21]), and numerical techniques (Shuet al. [22], Amanatidou and Aravas [23], Tsepoura et al. [24], Giannakopoulos et al. [25],Tsamasphyros et al. [26]). Based on the existing results, it is concluded that the Mindlintheory does extend the range of applicability of the ‘continuum’ concept in an effort tobridge the gap between classical continuum theories and atomic-lattice theories. Finally, itis interesting that this theory allows for the emergence of boundary layer effects that cancapture related phenomena (see, e.g., Georgiadis et al. [3], Georgiadis [8], Shi et al. [15]).

Regarding now appropriate length scales for strain gradient theories, although straingradient effects are associated with geometrically necessary dislocations in plasticity, theymay also be important for the elastic range in microstructured materials. Indeed, Chen et al.[27] developed a continuum model for cellular materials and found out that the continuumdescription of these materials obey an elasticity theory with strain gradient effects. In thelatter study, the intrinsic material length was naturally identified with the cell size. Otherexamples of the size effect in elastically deformed solids include bending of a

Problems of concentrated loads in gradient elasticity 73

polycrystalline aluminum beam (Kakunai et al. [28]) and buckling of elastic fibers incomposites (Fleck and Shu [29]). Since the strengthening effects arising from straingradients become important when these gradients are large enough, these effects will besignificant when the material is deformed in very small volumes, such as in the immediatevicinity of concentrated loads, crack tips, notches, small holes and inclusions, andmicrometer indentations. Also, in wave propagation dealing with electronic-deviceapplications, surface-wave frequencies on the order of gigahertz are often used andtherefore wavelengths on the micron order appear (see, e.g., White [30]). In such situations,dispersion phenomena at high frequencies can only be explained on the basis of straingradient theories (Georgiadis et al. [3]). In addition, the latter study provides estimates for asingle microstructural parameter—the gradient coefficient c (i.e., an additional materialparameter to the standard Lamé constants λ and μ of classical linear isotropic elasticity)employed in the simplest possible form of gradient elasticity (this form was used in theanalysis of the crack problem [8] and in the present study). We should also mention thework by Chang et al. [31], who provide estimates for the microstructural constants ingranular materials modeled as strain-gradient continua.

Finally, we give a brief literature review of related efforts trying to alleviate singularitiesat the points of application of concentrated loads by the use of generalized continuumtheories. Firstly, within the context of the Toupin–Mindlin theory and for a constitutivemodel of form II in Mindlin’s paper [2], the Flamant–Boussinesq problem was analyzed byExadaktylos [32], Zhou and Jin [33], and Lazar and Maugin [34]. However, all of the latterstudies use approximate boundary conditions and, therefore, they did not provide exactsolutions. The work by Polyzos et al. [35] on Boundary Element formulations for gradientelasticity provides also approximate results for a Kelvin type problem that show anasymptotic bounded behavior for the displacement as the point of application of the force isapproached. Secondly, both analyses by Nowinski [36] and Walsh and Tordesillas [37]were not successful in eliminating the singularity of the displacement field in the Flamant–Boussinesq problem. The former work utilized the Kroener–Eringen theory of non-localelasticity of integral type, whereas the latter one utilized the Cosserat (or micropolar)theory. In particular, an explanation of why the Cosserat theory is not able to provide abounded solution is given in Sect. 8.

2 Fundamentals of the Dipolar Gradient Elasticity

In the present study, the simplest version of Mindlin’s gradient theory is employed, i.e. formII of Mindlin [2]. According to this version: (i) each material particle has three degrees offreedom, (ii) the micro-density does not differ from the macro-density, (iii) the classicalform of the Euler–Cauchy principle prevails, and (iv) the strain-energy density dependsboth upon the strain tensor and its gradient. In this way, this form is a first-step extension ofclassical elasticity.

It is noticed that form II of Mindlin’s theory is different from both the couple-stresstheory (assuming an augmented form of the Euler–Cauchy principle with a non-vanishing couple-stress vector and a strain-energy density that depends upon the straintensor and the gradient of rotation vector) and the Cosserat (or micropolar) theory that,in addition to the aforementioned features of the couple-stress theory, takes materialparticles with six independent degrees of freedom (three displacement components andthree rotation components, the latter involving rotation of a micro-medium w.r.t. itssurrounding medium). It is mentioned, finally, that the general form of Mindlin’s


gradient theory (employed, e.g., by the first author in his recent works on solutionuniqueness in crack problems [9] and on energy theorems [10]) adopts the augmentedform of the Euler–Cauchy principle with a non-vanishing couple traction and takes astrain-energy density that depends upon the strain tensor and upon the gradient of bothstrain and rotation.

The particular case of the dipolar gradient elasticity described above (i.e., form II ofMindlin’s theory) is best introduced by the following form of the first law ofthermodynamics

rE� ¼ tpq"

�pq þ mrpq@r"

�pq; ð1Þ

where small strains and displacements are assumed, and a Cartesian rectangular coordinatesystem Ox1,x2,x3 is considered for a 3D continuum (indicial notation and the summationconvention will be used throughout). In the above equation, ∂p( )≡∂( )/∂xp, a superposeddot denotes time derivative, the Latin indices span the range (1–3), ρ is the mass density ofthe continuum, E is the internal energy per unit mass, "pq ¼ 1=2ð Þ @puq þ @qup

� � ¼ "qp isthe linear strain tensor, uq is the displacement vector, τpq is the monopolar stress tensor, andmrpq is the dipolar (or double) stress tensor (a third-rank tensor) expressed in dimensions of[force][length]−1. For reference, we also write the definitions of the rotation tensor 5 pq ¼1=2ð Þ @puq � @qup

� �and the rotation vector 5 q ¼ 1=2ð Þeqkl@kul, with eqkl being the Levi–

Civita permutation symbol.Clearly, the above form of the first law of thermodynamics can be viewed as a more

accurate description of the material response than that provided by the standard theory (caseof rE

� ¼ tpq"�pq), if one thinks of a series expansion for rE

�containing higher-order gradients

of strain. For instance, the additional terms may become significant in the vicinity of stress-concentration points where the strain undergoes very steep variations.

The dipolar stress tensor now follows from the notion of dipolar forces, which are anti-parallel forces acting between the micro-media contained in the continuum withmicrostructure (see, e.g., Fig. 1 in Georgiadis et al. [3]). As explained by Green andRivlin [38], Jaunzemis [39], and Gronwald and Hehl [40], the notion of multipolar forcesarises from a series expansion of the mechanical power M containing higher-order velocitygradients, i.e. M ¼ Fqu

�q þ Fpq @pu

�q

� �þ Frpq @r@pu

�q

� �þ . . ., where Fq are the usual

(monopolar) forces of classical continuum mechanics and (Fpq, Frpq,...) are the multipolarforces within the framework of generalized continuum mechanics.

Regarding the notation of the dipolar forces and stresses, the first index of the forceindicates the orientation of the lever arm between the forces and the second one theorientation of the pair of forces itself. The same holds true for the last two indices ofthe dipolar stresses, whereas the first index denotes the orientation of the normal to thesurface upon which the stress acts. Also, the dipolar forces Fpq have dimensions of[force][length]; their diagonal terms are double forces without moment and their off-diagonal terms are double forces with moment. In particular, the anti-symmetric partF½pq� ¼ 1=2ð Þ xpFq � xqFp

� �gives rise to couple-stresses (here, [] as a subscript will

denote the anti-symmetric part of a matrix or tensor). Here, we do not consider couple-stress effects assuming the existence of dipolar forces with vanishing anti-symmetric part.It is emphasized that this is compatible with the particular choice of the form of rE

�in (1),

i.e., a form dependent upon strain gradient but completely independent upon rotationgradient. Finally, across a section with its outward unit normal in the positive direction,


the force at the positive end of the lever arm is positive if it acts in the positive direction.‘Positive’ refers to the positive sense of the coordinate axis parallel to the lever arm orforce.

Next, in accord with (1), the following form is taken for the strain-energy density Wstored in the continuum

W � W "pq; @r"pq� �

: ð2Þ

In what follows, we assume the existence of a positive definite function W "pq; @r"pq� �

.Of course, (2) allows for non-linear constitutive behavior as well. Further, stresses can bedefined in terms of W in the standard variational manner

tpq � @W

@"pq; mrpq � @W

@ @r"pq� � ; ð3a; bÞ

where the following symmetries for the monopolar and dipolar stress tensors arenoticed: C pq=C qp and mrpq=mrqp. By definition (3b), we have that mr[pq]=0, wheremr pq½ � � 1=2ð Þ mrpq � mrqp

� �. This fact and the general definition of the couple-stress

tensor μrl � 1=2ð Þelpqmr pq½ � (see, e.g., Jaunzemis [39]) leads to the immediate conclusionthat the couple-stress tensor μpq vanishes identically when (2) is chosen (i.e. when form IIof Mindlin [2] is considered).

Then, the equations of equilibrium (global equilibrium) and the traction boundaryconditions along a smooth boundary (local equilibrium) can be obtained from variationalconsiderations (Mindlin [2], Bleustein [41]). The appropriate expression of the Principle ofVirtual Work is written as [10, 41]Z

Vtpqδ"pq þ mrpqδ @r"pq

� �� dV ¼

ZSt nð Þq δuqdS þ

ZST nð Þqr @q δurð ÞdS; ð4Þ

where the symbol δ denotes weak variations and it acts on the quantity existing on its right.In the above equation, t nð Þ

q is the true force surface traction, T nð Þpq is the true double force

surface traction, and np is the outward unit normal to the boundary along a section insidethe body or along the surface of it. Examples of the latter tractions along the surface of a 2Dhalf-space are given in Fig. 1.

)2(

11Τ

x1

x2)2(

21Τ)2(

2t

Fig. 1 Positively oriented truemonopolar and dipolar tractionson the surface of a half-space


In the present study, we generally assume the absence of body forces, in which case theequations of equilibrium and the traction boundary conditions take the following form

@p tpq � @rmrpq

� � ¼ 0 in V ; ð5Þ

np tpq � @rmrpq

� �� Dp nrmrpq

� �þ Djnj� �

nrnpmrpq ¼ Pq on bdy; ð6Þ

nrnpmrpq ¼ Rq on bdy; ð7Þ

where V is the region (open set) occupied by the body, bdy denotes any boundary along asection inside the body or along the surface of it, DpðÞ � @pðÞ � npDðÞ is the surfacegradient operator, DðÞ � nr@rðÞ is the normal gradient operator, Pq � t nð Þ

q þ Drnrð ÞnpT nð Þpq �

DpT nð Þpq is the auxiliary force traction, and Rq � npT nð Þ

pq is the auxiliary double force traction.It is also noticed that the matrix T nð Þ

qr should be symmetric. If T nð Þqr were not symmetric, an

inconsistent situation would arise, because the first member of (4) does not depend uponrotation. The restriction that T nð Þ

qr be symmetric also follows from the condition in (7),nqnpmpqr ¼ nqT nð Þ

qr , and the symmetry mpqr=mprq noticed before.Concrete boundary value problems can be found in, e.g., Georgiadis et al. [3],

Georgiadis [8], and Grentzelou and Georgiadis [9]. We should mention finally that in thecase of non-vanishing monopolar and dipolar body forces, the equations of global and localequilibrium are given in the paper by Bleustein [41].

In the present study, we will deal only with traction boundary conditions. However, for thesake of completeness, we also quote the pertinent kinematical boundary conditions. The latterconditions were derived by Georgiadis and Grentzelou [10] in the context of the Principle ofComplementary Virtual Work for dipolar theory. Let Ss be the portion of the surface S of thebody on which external tractions are prescribed and Su be the portion of the surface S onwhich kinematical conditions are prescribed. Of course, Ss [ Su ¼ S and Ss \ Su ¼ ; holdtrue. The kinematical boundary conditions are then as follows

uq: given on Su; ð8Þ

D uq� �

: given on Su: ð9Þ

Now, we proceed to the particular form of constitutive relations that will be utilizedbelow in analyzing problems of the Flamant–Boussinesq and Kelvin type. In this study, weconfine interest only to a linear constitutive behavior and consider the simplest possibleisotropic expression for W [3]. This strain-energy density function reads

W ¼ 1=2ð Þl"pp"qq þ μ"pq"pq þ c 1=2ð Þl @r"pp� �

@r"qq� �þ cμ @r"pq

� �@r"pq� �

; ð10Þ

where c is the gradient coefficient having dimensions of [length]2, and (l,μ) are thestandard Lamé constants with dimensions of [force][length]−2. In this way, only one new


material constant is introduced with respect to classical linear isotropic elasticity.Combining (3) with (10) provides the constitutive equations

tpq ¼ ldpq"jj þ 2m"pq ; ð11Þ

mrpq ¼ c@r ldpq"jj þ 2m"pq� �

; ð12Þ

where δpq is the Kronecker delta. Equations (11) and (12) written for a general 3D state willbe employed below only for a plane-strain state.

As Lazar and Maugin [16] pointed out, the particular choice of (10) is physically justifiedand possesses a noticeable symmetry. To expose this symmetry, we first consider the generalexpression (definition) of the strain-energy density W � R "pq

0 tpqd"pqþR @r"pq0 mrpqd @r"pq

� �,

which for a linear constitutive law takes the form W ¼ 1=2ð Þtpq"pq þ 1=2ð Þmrpq@r"pq. Then,noting from (11) and (12) that mrpq ¼ c@rtpq, the strain-energy density in (10) takes the formW ¼ 1=2ð Þtpq"pq þ c 1=2ð Þ @rtpq

� �@r"pq� �

, which exhibits symmetry with respect to bothstrain and standard stress. This simple form of the Toupin–Mindlin gradient elasticity istherefore a strain gradient theory as well as a stress gradient theory.

In summary, (5), (6), (7), (11) and (12) are the governing equations for the isotropiclinear gradient elasticity with no couple stresses. Combining (5) with (11) and (12) leads tothe system of field equations. It is noticed that uniqueness theorems have been proved onthe basis of positive definiteness of the strain-energy density in cases of regular and singular(crack problems) fields in the recent works of Georgiadis and Grentzelou [10], andGrentzelou and Georgiadis [9], respectively. As shown by Georgiadis et al. [3], therestriction of positive definiteness of W requires the following inequalities for the materialconstants appearing in the theory employed here: 3lþ 2mð Þ > 0 ; m > 0 ; c > 0. Inaddition, stability for the field equations in the general inertial case was proved inGeorgiadis et al. [3] and to accomplish this the condition c>0 is a necessary one.

3 Basic Equations in Plane Strain

We present here the basic equations of linear isotropic gradient elasticity for a plane-strainstate. A body occupying a domain in the x � x1; y � x2ð Þ -plane is considered with the z-axis being normal to this plane. All tractions are assumed to act ‘inside’ the plane (x, y) andare independent upon z. The following 2D displacement field is generated

ux � ux x; yð Þ 6¼ 0; uy � uy x; yð Þ 6¼ 0; uz � 0; ð13Þ

whereas the strain-energy density in (10) takes the form

W ¼ 1 =2ð Þ "xx þ "yy� �2þμ "2xx þ 2"2xy þ "2yy

� �þ c1 =2ð Þ @"xx

@x þ @"yy@x

� �2þ @"xx

@y þ @"yy@y

� �2�

þ

þcμ @"xx@x

� �2þ2 @"xy@x

� �2þ @"yy

@x

� �2þ @"xx

@y

� �2þ2 @"xy

@y

� �2þ @"yy

@y

� �2�

:

ð14Þ


In the plane-strain state, the independent components of the stress tensors that act‘inside’ the plane (x, y) and that do not vanish identically are three for C pq and six for mrpq.The components of C pq are as follows

txx ¼ l @xux þ @yuy� �þ 2m@xux; ð15aÞ

tyy ¼ l @xux þ @yuy� �þ 2m@yuy; ð15bÞ

txy ¼ m @yux þ @xuy� � ð15cÞ

whereas the components of mrpq are

mxxx ¼ c@

@xl @xux þ @yuy� �þ 2m@xux

� �; ð16aÞ

mxxy ¼ cm@

@x@yux þ @xuy� �

; ð16bÞ

mxyy ¼ c@

@xl @xux þ @yuy� �þ 2m @yuy

� �; ð16cÞ

myxx ¼ c@

@yl @xux þ @yuy� �þ 2m @xux

� �; ð16dÞ

myyx ¼ cm@

@y@yux þ @xuy� �

; ð16eÞ

myyy ¼ c@

@yl @xux þ @yuy� �þ 2m @yuy

� �; ð16fÞ

where @xðÞ � @ðÞ=@x and @yðÞ � @ðÞ[email protected] now (15a–15c) and (16a–16f) in the equations of equilibrium (5) leads to

the following system of coupled PDEs of the fourth order for the displacement components

1� cr2� �

l @2x ux þ @x@yuy

� �þ m @2y ux þ @x@yuy

� �þ 2m @2

x uxh i

¼ 0; ð17aÞ

1� cr2� �

l @2y uy þ @x@yux

� �þ m @2

x uy þ @x@yux� �þ 2m @2

y uyh i

¼ 0; ð17bÞ

where @2x ðÞ � @x@xðÞ, @2

y ðÞ � @y@yðÞ, and r2ðÞ � @2x ðÞ þ @2

y ðÞ is the 2D Laplace operator.Also, due to the assumed continuity of the displacement gradient, the change in the order ofdifferentiations ∂x∂yux=∂y∂xux and ∂x∂yuy=∂y∂xuy is permissible. Now, as expected, in the


limit c → 0 the Navier–Cauchy equations of classical linear isotropic elasticity arerecovered from (17a, 17b). Indeed, the fact that equations (17a, 17b) have an increasedorder w.r.t. their limit case (recall that the Navier–Cauchy equations are PDEs of the secondorder) and the coefficient c multiplies the higher-order term reveal the singular-perturbationcharacter of the gradient theory and the emergence of associated boundary-layer effects. Inview of the latter observation, it is expected that the influence of dipolar stresses hingescrucially on the relative size of the characteristic length-parameter c1/2.

In any specific boundary value problem, one has to solve (17a, 17b) supplied bypertinent boundary conditions. This will be done in what follows for the gradient-elasticityanalogues of the Flamant–Boussinesq and Kelvin problems.

4 Formulation and Transformed Solution for the Problem of the Flamant–BoussinesqType

We consider a body occupying the half-plane �1 < x < 1; 0 < y < 1ð Þ under plane-strain conditions. The body is acted upon by a normal line load P and a tangential line loadS at a point on its surface. This point is taken as the origin x ¼ 0; y ¼ 0ð Þ of a Cartesianrectangular coordinate system (see Fig. 2). The intensities of the concentrated loads areexpressed in dimensions of [force][length]−1. The traction boundary conditions along thesurface y=0 follow from (6) and (7) and are written as

tyy � @xmxyy � @ymyyy � @xmyxy ¼ �Pd xð Þ; ð18Þ

tyx � @xmxyx � @ymyyx � @xmyxx ¼ �Sd xð Þ; ð19Þ

myyy ¼ 0; ð20Þ

myyx ¼ 0; ð21Þwhere δ( ) is the Dirac delta distribution. It appears above with the standard ‘symbolic’sense.

x

y

P

Sy=0

Fig. 2 Half-space acted upon bynormal and tangential line forcesalong its surface


The boundary value problem will be attacked with the two-sided Laplace transform (see,e.g., van der Pol and Bremmer [42], Carrier et al. [43], and Bracewell [44]) in order tosuppress the x-dependence in the equations (15a–21). The transform pair (direct and inverseoperation) is defined as

f � p; yð Þ ¼Z 1

�1f x; yð Þ � e�pxdx ; f x; yð Þ ¼ 1

2πi

ZBrf � p; yð Þ � epxdp ; ð22a; bÞ

where Br denotes the Bromwich path of inversion within the region of analyticity of thefunction f *(p, y) in the complex p-plane. This region is an infinite strip parallel to the Im(p)-axis. Transforming (17a, 17b) with (22a) gives a system of ODEs for u�x ; u

�y

� �written in

the following compact form

K½ � u�xu�y

� ¼ 0

0

� ; ð23Þ

where the differential operator [K] is given as

K½ � ¼μd2 þ 1 þ 2μð Þp2h i

1� c d2 þ p2� �h i

ð1 þ μÞpd 1� c d2 þ p2� �h i

1 þ μð Þpd 1� c d2 þ p2� �h i

μp2 þ 1 þ 2μð Þd2h i

1� c d2 þ p2� �h i24 35; ð24Þ

with dðÞ � dðÞ=dy, d2ðÞ � d2ðÞdy2, etc.The set of homogeneous differential equations in (23) has a solution different than the

trivial one if and only if the determinant of [K] is zero. Hence,

m lþ 2mð Þ d2 þ p2� �2

1� c d2 þ p2� �h i2

¼ 0; ð25Þ

which has two double roots: d ¼ � �pð Þ1=2 and d ¼ � 1=cð Þ � p2½ �1=2. The first pair is thesame as in classical elasticity, whereas the second pair reflects the presence of gradienteffects. The general solution of (23) is obtained after some rather extensive algebra and ithas the following form that is bounded as y→+∞

u�x p; yð Þ ¼ C1 exp �βyð Þ þ � lþ 3μð ÞBþ lþ μð Þ Aþ Byð Þβp

exp �+ yð Þ for y � 0;

ð26aÞ

u�y p; yð Þ ¼ C2 exp �βyð Þ þ Aþ Byð Þ exp �+ yð Þ for y � 0; ð26bÞ

Fig. 3 Branch cuts for the func-tion β(p)


where (A, B, C1, C2) are yet unknown functions of p (in each specific problem, they may beobtained from the boundary conditions), β � β pð Þ ¼ "2 � p2ð Þ1=2 with ε being a realnumber such that ε→+0, and + � + pð Þ ¼ 1=cð Þ � p2½ �1=2� a2 � p2ð Þ1=2 with a=(1/c)1/2.In fact, introducing ε facilitates the introduction of branch cuts for the complex function β=(−p2)1/2 (see, e.g., Carrier et al. [43] for this convenient way of defining branch cuts).

It is emphasized that in order to obtain a bounded solution as y → +∞, the p-planeshould be cut in the way shown in Figs. 3 and 4. This introduction of branch cuts securesthat the functions (β, + ) are single-valued and that Re(β)≥0 and Re(+ )≥0 along theBromwich path. In view of (26a, 26b) and the form of the function β(p), the Bromwich pathin our case is restricted to coincide with the Im(p)-axis. The transformed generalexpressions for the stresses that enter the boundary conditions are quoted in Appendix A.The transformed boundary conditions themselves for our specific problem follow from theaction of (22a) on (18–21) and are written as

t�yy � pm�xyy �

dm�yyy

dy� pm�

yxy ¼ �P; ð27aÞ

t�yx � pm�xyx �

dm�yyx

dy� pm�

yxx ¼ �S; ð27bÞ

m�yyy ¼ 0; ð27cÞ

m�yyx ¼ 0: ð27dÞ

Next, combining (27a–27d) with (79–85) provides a system of algebraic equations for(A, B, C1, C2). After some algebra involving manipulations by hand and the symbolic

Fig. 4 Branch cuts for the func-tion + (p)


program MATHEMATICA (version 5.0), a solution is obtained and is given in Appendix B.The transformed displacements are found to be

u� Pð Þx p; yð Þ ¼ �P

�1 + 2þ 1þ2μð Þp2½ �p2cμK exp �+ yð Þþ

þP�μ 1þ2μð Þ+ 4�21 p2 + 2þ 1þ2μð Þp4½ �þ4 1þμð Þ2p4 + y

2cμ 1þμð ÞpK exp �βyð ÞþþP

1þμð Þβ 41 p2 +þ 1þ2μð Þ+ 4y�21 p2 + 2yþ 1þ2μð Þp4y½ �2cμ 1þμð ÞpK exp �βyð Þ;

ð28Þ

u� Pð Þy p; yð Þ ¼ P

�1 + 2þ 1þ2μð Þp2½ �p2cμ+K exp �+ yð Þþ

þP�4 1þμð Þ 21þ3μð Þp4 +þ 1þ2μð Þβ + 4 1þ2μð Þ�21 p2 + 2þ 1þ2μð Þp4½ �

2cμ 1þμð Þp2K exp �βyð ÞþþP 1þ2μð Þ+ 4�2 1 +�2 1þμð Þβ½ �p2 +þ 1þ2μð Þp4

2cμp2K y � exp �βyð Þ;ð29Þ

u� Sð Þx p; yð Þ ¼ S

�2 1þ2μð Þβ+ 3þμp p2�+ 2ð Þ½ �cμ+K exp �+ yð Þ�

�S� 1þ2μð Þ+ 2þ1 p2 β � 1þ2μð Þ+ 2þ1 p2ð Þ�c�1 1þμð Þp2y½ �

2cμ 1þμð Þp2K exp �βyð Þ;ð30Þ

u� Sð Þy p; yð Þ ¼ �S

μ + 2�p2ð Þþ21 β+½ �pcμK exp �+ yð Þþ

þS1þ2μð Þ+ 2�1 p2½ � μ+ 2þ21 p2þ3μp2þc�1 1þμð Þβy½ �

2cμ 1þμð ÞK exp �βyð Þ;ð31Þ

where K � K pð Þ ¼ 8 lþ mð Þp2bg3 þ lþ 2mð Þg4 � 2lp2g2 þ lþ 2mð Þp4½ � g2 � p2ð Þ, andthe superscript P or S in parentheses in a transformed displacement component denotesthat the particular component is generated by the concentrated load P or S, respectively.

Finally, the following evenness/oddness of the transformed displacement components isobserved in the above equations

u� Pð Þx �p; yð Þ ¼ �u� Pð Þ

x p; yð Þ; ð32Þ

u� Pð Þy �p; yð Þ ¼ u� Pð Þ

y p; yð Þ; ð33Þ

u� Sð Þx �p; yð Þ ¼ u� Sð Þ

x p; yð Þ; ð34Þ

u� Sð Þy �p; yð Þ ¼ �u� Sð Þ

y p; yð Þ: ð35Þ

Certainly, it is inferred from the theory of integral transforms (see, e.g., Bracewell[44]) that the same properties should hold in the physical domain as well, i.e.,u Pð Þx �x; yð Þ ¼ �u Pð Þ

x x; yð Þ, u Pð Þy �x; yð Þ ¼ u Pð Þ

y x; yð Þ, u Sð Þx �x; yð Þ ¼ u Sð Þ

x x; yð Þ, a n du Sð Þy �x; yð Þ ¼ �u Sð Þ

y x; yð Þ. In fact, such a behavior of the displacement components is


corroborated here since implied from either the symmetry (load P acting alone) or anti-symmetry (load S acting alone) of the problem in the physical domain. We will exploitrelations (32–35) in the course of transform inversions that follow.

5 Laplace Transform Inversion

In what follows, we focus attention only on the surface displacements (y=0) with a viewtowards to obtain an explicit analytical solution. Such a solution is intended to determinethe behavior of the displacement field near to the point of application of the concentratedloads and will allow detecting possible deviations from the predictions of classical theory.Certainly, this is our main concern here. Notice, however, that determining the field atpoints ‘inside’ the half-space (y≠0) may follow along the same general lines of the presentanalysis but it will involve much additional numerical work because of the presence of theterms exp(−βy) and exp(−γy).

We next exploit the properties in (32–35) along with the fact that the inversion path in (22b)is restricted to be the whole Im(p) -axis. To this end, we change the integration variable in(22b) setting p=−iξ, where x 2 <, so that the new inversion path be the whole real axis. Anyinversion of the type (22b) will be written now as f x; y ¼ 0ð Þ ¼ 1=2πð Þ R1

�1 f � ξ; y ¼ 0ð Þ � e�iξxdξ.Along this inversion path, we have β(ξ)=ξ and + ξð Þ ¼ 1=cð Þ þ ξ2

� �1=2. Also, by taking intoaccount Euler’s identity, eiϕ ¼ cosϕþ i sinϕ, and the elementary result thatZ 1

�1f xð Þdx ¼ 0; f : even

2R10 f xð Þdx; f : odd

�; ð36Þ

we are able to write the following expressions for the inverse functions

u Pð Þx x; y ¼ 0ð Þ ¼ � i

π

Z 1

0u� Pð Þx ξ; y ¼ 0ð Þ � sin ξxð Þdξ; ð37Þ

u Pð Þy x; y ¼ 0ð Þ ¼ 1

π

Z 1

0u� Pð Þy ξ; y ¼ 0ð Þ � cos ξxð Þdξ; ð38Þ

u Sð Þx x; y ¼ 0ð Þ ¼ 1

π

Z 1

0u� Sð Þx ξ; y ¼ 0ð Þ � cos ξxð Þdξ; ð39Þ

u Sð Þy x; y ¼ 0ð Þ ¼ � i

π

Z 1

0u� Sð Þy ξ; y ¼ 0ð Þ � sin ξxð Þdξ: ð40Þ

In view of the above results and by using partial-fraction decompositions, thetransformed surface displacements take the following form

u� Pð Þx ξ; y ¼ 0ð Þ ¼ � iP

2 1þμð Þ1ξ þ icP

2 1þμð Þξ

1þcξ2ð Þ þ

þ i 1þ2μð ÞP2cμ 1þμð Þ

ξ + 2 21þ3μð Þþ2+μξþμξ2½ �+ 2N ;

ð41Þ


u� Pð Þy ξ; y ¼ 0ð Þ ¼ 1þ2μð ÞP

2μ 1þμð Þ1ξ � c 1þ2μð ÞP

2μ 1þμð Þξ

1þcξ2ð Þ �

� 1þ2μð ÞP2cμ 1þμð Þ

ξ + 2 1þ2μð Þþ2+μξþξ2 1þ2μð Þ½ �+ 2N ;

ð42Þ

u� Sð Þx ξ; y ¼ 0ð Þ ¼ 1þ2μð ÞS

2μ 1þμð Þ1ξ � c 1þ2μð ÞS

2μ 1þμð Þξ

1þcξ2ð Þ �

�Sξ1þ2μð Þ 51þ6μð Þ+ 2þ2 1 2þ31 μþ3μ2ð Þξ+þ 1þ2μð Þ2ξ2

2cμ 1þμð Þ+ 2N ;ð43Þ

u� Sð Þy ξ; y ¼ 0ð Þ ¼ iS

2 1þμð Þ1ξ � icS

2 1þμð Þξ

1þcξ2ð Þ �

�iS1þ2μð Þξ + 2 21þ3μð Þþ2+μξþμξ2½ �

2cμ 1þμð Þ+ 2N ;ð44Þ

where the function N(ξ) is given by

N � N xð Þ ¼ lþ 2mð Þg4 þ 2 lþ 2mð Þg3xþ 2 3lþ 4mð Þg2x2 þ 2 lþ 2mð Þgx3

þ lþ 2mð Þx4: ð45ÞFurther, each displacement component can be written in terms of three integrals, in the

following form

u Pð Þx x; y ¼ 0ð Þ ¼ I Pð Þ

class: þ I Pð Þgrad�1 þ I Pð Þ

grad�2; ð46Þ

u Pð Þy x; y ¼ 0ð Þ ¼ J Pð Þ

class: þ J Pð Þgrad�1 þ J Pð Þ

grad�2; ð47Þ

u Sð Þx x; y ¼ 0ð Þ ¼ I Sð Þ

class: þ I Sð Þgrad�1 þ I Sð Þ

grad�2; ð48Þ

u Sð Þy x; y ¼ 0ð Þ ¼ J Sð Þ

class: þ J Sð Þgrad�1 þ J Sð Þ

grad�2: ð49Þwhere

I Pð Þclass: ¼ � P

2π 1 þ μð ÞZ 1

0

1

ξsin ξxð Þdξ; ð50aÞ

I Pð Þgrad�1 ¼

P

2π 1 þ μð ÞZ 1

0

cξ

1þ cξ2sin ξxð Þdξ; ð50bÞ

I Pð Þgrad�2 ¼

1 þ 2μð ÞP2πμ 1 þ μð Þ

Z 1

0

ξ γ2 21 þ 3μð Þ þ 2γμξ þ μξ2� �

γ2cNsin ξxð Þdξ; ð50cÞ


J Pð Þclass: ¼

1 þ 2μð ÞP2πμ 1 þ μð Þ

Z 1

0

1

ξcos ξxð Þ dξ; ð51aÞ

J Pð Þgrad�1 ¼ � 1 þ 2μð ÞP

2πμ 1 þ μð ÞZ 1

0

cξ

1þ cξ2cos ξxð Þ dξ; ð51bÞ

J Pð Þgrad�2 ¼ � 1 þ 2μð ÞP


0

ξ + 2 1 þ 2μð Þ þ 2+μξ þ ξ2 1 þ 2μð Þ� �+ 2cN

cos ξxð Þ dξ; ð51cÞ

I Sð Þclass: ¼

1 þ 2μð ÞS2πμ 1 þ μð Þ

Z 1

0

1

ξcos ξxð Þ dξ; ð52aÞ

I Sð Þgrad�1 ¼ � 1 þ 2μð ÞS


0

cξ

1þ cξ2cos ξxð Þ dξ; ð52bÞ

I Sð Þgrad�2 ¼

S


0

ξ+ 2cN

�+ 2 1 þ 2μð Þ 51 þ 6μð Þ � 1 þ 2μð Þ2ξ2��2+ 1 2 þ 31 μþ 3μ2

� �� cos ξ xð Þ dξ;

ð52cÞ

J Sð Þclass: ¼

S

2π 1 þ μð ÞZ 1

0

1

ξsin ξxð Þ dξ; ð53aÞ

J Sð Þgrad�1 ¼ � S

2π 1 þ μð ÞZ 1

0

cξ

1þ cξ2sin ξxð Þ dξ; ð53bÞ

J Sð Þgrad�2 ¼ � 1 þ 2μð ÞS


0

ξ + 2 21 þ 3μð Þ þ 2+μξ þ μξ2� �

+ 2cNsin ξxð Þdξ: ð53cÞ

Integrals (50a), (50b), (51a), (51b), (52a), (52b), (53a) and (53b) can be obtained inclosed form. But, integrals (50c), (51c), (52c) and (53c) have to be evaluated numerically.

Now, the integral in (50a) and (53a) is a convergent one giving the resultR10 sin ξxð Þ=ξ½ � dξ ¼ π=2ð Þ sgn xð Þ, where sgn( ) is the signum function. On the contrary,the integral in (51a) and (52a) is a divergent one. Nevertheless, by invoking results from thetheory of distributions (see, e.g., Zemanian [45], and Hoskins [46]), we are able to writeR10 cos ξxð Þ=ξ½ � dξ ¼ � ln xj j þ arbitrary constantð Þ. Notice that, as in analogous situationsof classical elasticity problems with concentrated loadings (Sosa and Bahar [47], andGeorgiadis and Lykotrafitis [48]), the arbitrary constant in the RHS of the latter equationcan be omitted since it is attributed to a rigid body motion.


At this point, we should mention that the paper by Sosa and Bahar [47] provides anextensive analysis of the integral

R10 cos ξxð Þ=ξ½ � exp �yξð Þ dξ (from which our integral

results by setting y=0) in dealing with the integral-transform solution of the classicalFlamant–Boussinesq problem. Indeed, they handle this integral by using Laplace transformsand the concept of Hadamard finite-part integrals. In the general case of y≠0, Sosa and Bahar[47] found that

R10 cos ξxð Þ=ξ½ � exp �yξð Þdξ ¼ � 1=2ð Þ ln x2 þ y2ð Þ þ arbitrary constantð Þ.

Also, the book by Zemanian [45] provides the latter result both in the text and in its Tableof Laplace transforms of distributions.

Now, the integrals in (50b) and (52b) are obtained by employing the MATHEMATICAsoftware (version 5.0). These are given in terms of the special transcendental functionMeijerG[], which is a tabulated function.

Finally, in the limit case of classical linear elasticity, the second and third terms in theRHS of each one of equations (46–49) vanish and the surface displacements are thenprovided by the following inversion integrals

limc!0

u Pð Þx x; y ¼ 0ð Þ ¼ � P

2π 1 þ μð ÞZ 1

0

sin ξxð Þξ

dξ; ð54Þ

limc!0

u Pð Þy x; y ¼ 0ð Þ ¼ 1 þ 2μð ÞP


0

cos ξxð Þξ

dξ; ð55Þ

limc!0

u Sð Þx x; y ¼ 0ð Þ ¼ 1 þ 2μð ÞS


0

cos ξxð Þξ

dξ; ð56Þ

limc!0

u Pð Þy x; y ¼ 0ð Þ ¼ S

2π 1 þ μð ÞZ 1

0

sin ξxð Þξ

dξ: ð57Þ

The above expressions provide the following classical elasticity solution

u Pð Þx class:ð Þ x; y ¼ 0ð Þ ¼ � P

4 lþ mð Þ sgn xð Þ; ð58Þ

u Pð Þy class:ð Þ x; y ¼ 0ð Þ ¼ � lþ 2mð ÞP

2pm lþ mð Þ ln xj j; ð59Þ

u Sð Þx class:ð Þ x; y ¼ 0ð Þ ¼ � lþ 2mð ÞS

2pm lþ mð Þ ln xj j; ð60Þ

u Sð Þy class:ð Þ x; y ¼ 0ð Þ ¼ S

4 lþ mð Þ sgn xð Þ; ð61Þ

i.e., a solution which is discontinuous and unbounded at the point of application of theloads (cf. Barber [6]).


6 Solution and Numerical Results for the Problem of the Flamant–Boussinesq Type

From equations (46–53c), the final results for the surface displacements may follow. Weprovide them below in normalized form

bu Pð Þx x0; y ¼ 0ð Þ ¼ �sgn x0ð Þ þ exp � x0j jð Þ � sgn x0ð Þþ

þ 2 1þ2μð Þπμ

R10 ρ

21 1þρ2ð Þþμ 3þ2ρ 2ρþ 1þρ2ð Þ1=2� ��

1þρ2ð Þ�0 ρð Þ sin ρx0ð Þ dρ;ð62Þ

bu Pð Þy x0; y ¼ 0ð Þ ¼ � 1

π ln x0j jð Þ � 12π1=2

MeijerG 0f g; fgf g; 0; 0f g; 12

� �; x0

4

� �2h i�

� 1π

R10 ρ

2μρ 1þρ2ð Þ1=2þ 1þ2μð Þ 1þ2ρ2ð Þ1þρ2ð Þ�0 ρð Þ cos ρx0ð Þdρ;

ð63Þ

bu Sð Þx x0; y ¼ 0ð Þ ¼ � 1

πln x0j jð Þ � 1

2π1=2MeijerG 0f g; fgf g; 0; 0f g; 1

2

� �� ;

x0

4

� �2" #

�

� 1

πðlþ 2μÞZ 1

0

ρ

ð1þ ρ2Þ � 0 ρð Þ� 1 þ 2μð Þ 51 þ 6μþ 2ð 31 þ 4μð Þρ2Þ��2 1 2 þ 31 μþ 3μ2

� �ρ 1þ ρ2ð Þ1=2

( )cos ρx0ð Þdρ;

ð64Þ

bu Sð Þy x0; y ¼ 0ð Þ ¼ sgn x0ð Þ � exp � x0j jð Þ � sgn x0ð Þ�

� 2 1þ2μð Þπμ

R10 ρ

21 1þρ2ð Þþμ 3þ2ρ 2ρþ 1þρ2ð Þ1=2� ��

1þρ2ð Þ�Λ ρð Þ sin ρx0ð Þ dρ;ð65Þ

where the superposed caret in the displacements denotes normalized quantities that will bedefined below, x0 ¼ c�1=2x denotes normalized distance (dimensionless), the specialtranscendental function MeijerG[] is tabulated and can be found, e.g., in the MATHEMA-TICA software (version 5.0), and the function Λ(ρ) is given by

Λ ρð Þ ¼ 1 þ 2μþ 81 ρ2 þ 12μρ2 þ 81 ρ4 þ 12μρ4 þ 2 1 þ 2μð Þ 1þ ρ2� �1=2

ρþ 2ρ3� �

:

ð66ÞAlso, the normalized displacements are dimensionless quantities defined as follows

bu Pð Þx ¼ 4 lþ mð Þ

Pu Pð Þx ; ð67Þ

bu Pð Þy ¼ 2m lþ mð Þ

lþ 2mð ÞP u Pð Þy ; ð68Þ

bu Sð Þx ¼ 2m lþ mð Þ

lþ 2mð ÞS u Sð Þy ; ð69Þ

bu Sð Þy ¼ 4 lþ mð Þ

Su Sð Þy : ð70Þ


The integrals in (62–65) are convergent and are evaluated numerically employingMATHEMATICA algorithms. Alternatively, one could evaluate these integrals bynumerical integration employing Longman’s method and either Euler’s transformation orthe epsilon algorithm (to speed up convergence).

Now, observing (62) and (65), one can immediately see that the first and second terms inthe RHS of each particular equation cancel each other out at x′=0, so elimination of thediscontinuity is obtained. The same happens in (63) and (64) because the MeijerG[]function behaves like the negative of the ln() function as |x′|→0. Therefore, elimination ofthe singularity is also obtained.

The elimination of the singularity in the displacement field (i.e., occurrence of abounded field), within the context of the Toupin–Mindlin gradient elasticity, can beexplained by the fact that this theory generally provides solutions exhibiting materialresponse of increased stiffness as compared to classical elasticity. Two other observationson the solution (62–65) are quoted. The relation P � u Sð Þ

y

�� ¼ S � u Pð Þx

�� holds true, and this isto be expected in view of a reciprocal theorem (of the Betti-Rayleigh type) proved recentlyby Georgiadis and Grentzelou [10] for gradient elasticity. Finally, in the limit as c→0, ourresults degenerate into the classical Flamant–Boussinesq solution.

Two graphs are provided in Fig. 5 showing displacement profiles, i.e., the variation ofthe surface normal and tangential displacements due to a normal load with the distancefrom the point of application of the load. The numerical results were derived for ageomaterial that exhibits microstructure (Dionysos marble). This material is characterizedby the constants λ=μ=30.5×109 Nm−2 and h=4×10−4 m (half grain-size). We noteincidentally that the half grain-size h has been interrelated with the gradient coefficient c inthe work by Georgiadis et al. [3]. The latter study interrelates the two characteristic lengthsc1/2 (this enters the strain-energy density) and h (this enters the kinetic-energy density in adynamic analysis), through a comparison of the forms of dispersion curves of Rayleighwaves obtained by the dipolar gradient approach with the ones obtained by atomic-latticeanalyses. The estimate that c is of the order of (0.1 h)2 was given in Georgiadis et al. [3].Since normalized distance is used, the value of gradient coefficient c is not needed to obtainthe graphs. However, the normalized distance x′=c−1/2x [with c=(0.1h)2 and h=4×10−4 m]is used to obtain the plot corresponding to the classical elasticity solution.

The first graph clearly depicts that the gradient solution exhibits a bounded normaldisplacement at the point of application of the load. Thus, elimination of the undesirablelogarithmic singularity of the classical Flamant–Boussinesq solution is achieved with ourapproach. In addition, the second graph shows a smoother (and, therefore, more natural)profile for the tangential displacement in the gradient case as compared with thediscontinuous profile given by the classical solution. Plotting graphs similar to those inFig. 5 shows that the discrepancy between the two solutions increases as the characteristiclength c1/2 increases.

7 Problem of the Kelvin Type

We now consider a body occupying the full plane (−∞<x<∞,−∞<y<∞) under plane-strainconditions. The body is acted upon by a concentrated line load F situated at the origin (x=0,y=0). There is no loss of generality if we assume that the direction of F coincides with they-axis of the coordinate system (see Fig. 6). The intensity of the load is expressed indimensions of [force][length]−1.


-10 -5 0 5 10

normalized distance

-0.8

-0.4

0

0.4

0.8

norm

aliz

ed v

erti

cal d

ispl

acem

ent

gradient classical

normalload

-10 -5 0 5 10

normalized distance

-1

-0.5

0

0.5

1

norm

aliz

ed t

ange

ntia

l dis

plac

emen

t

classicalgradient

normalload

Fig. 5 Normalized surface displacements due to a concentrated normal load as provided by gradient andclassical theories of elasticity


y=0

Fig. 7 Reformulation of Kelvin’sproblem

x

y

y=0

F

Fig. 6 2D full-space acted uponby a line force


Considerable simplification is gained in the analysis of the problem in question byobserving that the problem is anti-symmetric w.r.t. the plane y=0. In this way, as Fig. 7shows, the original problem can be split into two auxiliary problems – one involving thelower half-plane (y>0) and the other the upper half-plane (y<0) – and be reformulated inone of these half-planes. This greatly facilitates application of integral transforms. Ofcourse, such a solution strategy could also be followed in the classical 2D Kelvin problemas well. However, the solution to the latter problem was easily achieved by the use of theAiry stress function and this is why an integral-transform solution is absent in the standardtreatises on Elasticity. But, a stress-function formulation is unavailable for the gradienttheory considered here. Due to the aforementioned anti-symmetry of the original problem,it suffices to solve a boundary value problem in the half-plane y>0 with the followingboundary conditions along the surface (−∞<x<∞, y=0)

C yy � @xmxyy � @ymyyy � @xmyxy ¼ � 1=2ð ÞFδ xð Þ; ð71Þ

myyy ¼ 0; ð72Þ

ux ¼ 0; ð73Þ

@2ux@y2

¼ 0; ð74Þ

where, in particular, conditions (73) and (74) guarantee the property of anti-symmetry in theoriginal problem.

-10 -5 0 5 10

normalized distance

-0.8

-0.4

0

0.4

0.8

norm

aliz

ed v

erti

cal d

ispl

acem

ent

gradientclassical

Fig. 8 Classical and gradientelasticity solutions (vertical dis-placement profiles) at the level ofapplication of the load for theKelvin problem


The solution procedure for the above problem runs as in the Flamant–Boussinesq case.We omit, therefore, any detail and just present the final expression for the verticaldisplacement uy(x, y=0)

uy x0; y ¼ 0ð Þ ¼ � 1

πln x0j jð Þ � 1

2π1=2MeijerG 0f g; fgf g; 0; 0f g; 1

2

� �� ;

x0

4

� �2" #

þ 1

π lþ 3μð ÞZ 1

0L ρð Þ � cos ρx0ð Þdρ ; ð75Þ

where uy is a normalized dimensionless quantity defined as

uy ¼ 4m lþ 2mð Þlþ 3mð ÞF uy; ð76Þ

x′=c−1/2x is again the normalized distance (dimensionless) from the origin, and the functionL(ρ) is given as

L ρð Þ ¼ �ρμ lþ 3μð Þ 2lþ 3μð Þρþ lþ 2μð Þ l2 þ 2lμþ 3μ2

� �1þ ρ2ð Þ1=2

1þ ρ2ð Þ � G ρð Þ ; ð77Þ

with G ρð Þ ¼ lþ 2μð Þ 1þ ρ2� �1=2

lþ 2μþ 2lþ 5μð Þρ2� �þ ρ 2 lþ 2μð Þ2þ 2l2 þ 10lμþ 11μ2� �

ρ2h i

.

Finally, we also record for comparison with (75) and (76) the classical Kelvinsolution

uy classicalð Þ x; y ¼ 0ð Þ ¼ � lþ 3mð ÞF4pm lþ 2mð Þ ln xj jð Þ: ð78Þ

Indeed, such a comparison is depicted by the graphs of Fig. 8 and clearly shows thebounded character of the new solution.

8 Discussion and Concluding Remarks

Our main concern here was to determine possible deviations from the predictions ofclassical linear elastostatics when a more refined theory is employed to attack the Flamant–Boussinesq and Kelvin problems. The new approach makes use of the simplest possibleversion of the linear gradient elasticity theory. This version involving totally three materialconstants is a first-step extension of classical elasticity. Two-dimensional configurations inthe form of either a half-space (Flamant–Boussinesq type problem) or a full-space (Kelvintype problem) were treated and the concentrated loads were taken as line forces.

The solution method is based on integral transforms and is exact. Of special importanceis the behavior of the new solutions near to the point of application of the loads wherepathological singularities and discontinuities exist in the classical solutions. Our resultsshow indeed departure from the ones of the classical elasticity solutions. Bounded andcontinuous displacements are predicted at the points of application of the loads. Such abehavior of the displacement fields seems to be more natural than the logarithmicallysingular and discontinuous behavior present in the classical solutions.


The occurrence of a bounded field, within the context of the Toupin–Mindlin gradientelasticity, can be explained by the fact that this theory generally provides solutionsexhibiting material response of increased stiffness as compared to classical elasticity. Forexample, in the 2D crack problem (whose solution, within the context of gradient elasticity,is well-documented in the literature — see e.g. Shi et al. [15], Georgiadis [8], andGrentzelou and Georgiadis [9]), the gradient solution gives a near-tip displacement field ofthe r3/2-variation (polar coordinates are attached to the crack tip), whereas the classicalsolution gives an r1/2-variation. This means that the crack-face displacement closes moresmoothly (in a cusp-like manner) when the material is governed by the gradient theory ascompared with the classical solution. In addition, the near-tip strain field is bounded in thegradient solution exhibiting an r1/2-variation [9] contrary to the singular r−1/2-variation ofclassical elasticity. Another example of material response that is more rigid in the gradientsolution than the respective classical solution is the stress concentration around inclusions(Weitsman [7]). Now, the Cosserat (or micropolar) theory utilized by Walsh and Tordesillas[37] in solving the Flamant–Boussinesq problem does not provide such a ‘stiffening’ effect.On the contrary, the latter theory relies on a bigger number of degrees of freedom of eachmaterial particle (three displacements and three rotations) than the gradient theory, so itseems to be natural the persistence of singularities in the displacement field of the Flamant–Boussinesq problem.

It seems reasonable for one to test the physical meaning and usefulness of mechanicaltheories on the basis of their analytical results for concrete boundary value problems thatsimulate practical situations. In this regard, based on the present results for the Flamant–Boussinesq and Kelvin problems and on results for crack problems [8, 9, 15], we can statethat there is a serious indication that gradient elasticity is superior to both classical elasticityand Cosserat elasticity. This also holds hope for future application of the gradient theoryemployed here to ‘correct’ in a boundary-layer sense other classical singular solutions.

Finally, although our efforts that deal with several Contact Mechanics problems – e.g.,indentation of a half-plane by a rigid wedge and indentation by a flat punch (Sadowskyproblem) – are still underway with no definite results yet, it is expected that the strain fieldwill be bounded at the ‘discontinuity’ points like the wedge apex and the corner points.This expectation is based on the results of crack problems mentioned before.

Achnowledgement This paper is a partial result of the Project PYTHAGORAS II / EPEAEK II(Operational Programme for Educational and Vocational Training II) [Title of the individual program:‘Micro-mechanics of contacts and diffusion of humidity in granular geomaterials’]. This Project is co-fundedby the European Social Fund (75%) of the European Union and by National Resources (25%) of the GreekMinistry of Education.

Appendix A

Here, the transformed general expressions for the stresses that enter the boundaryconditions are given

t�yy p; yð Þ ¼ �gC2 lþ 2mð Þ þ C1lp½ � � exp �gyð Þ

� 2m �Bmþ Ab lþ mð Þ þ bB lþ mð Þy½ �lþ m

exp �byð Þ; ð79Þ


t�yx p; yð Þ ¼ m C2p� gC1ð Þ � exp �gyð Þ þ 2m Aþ Byð Þpþ 2m lþ 2mð ÞbBlþ mð Þp

� � exp �byð Þ; ð80Þ

m�yxy p; yð Þ ¼ m gcC2pþ C1 cp2 � 1

� �� exp �gyð Þ

� 2cmp Ab lþ mð Þ þ B �2l� 3mþ b lþ mð Þyð Þ½ �lþ m


m�yxx p; yð Þ ¼ C2l 1� cp2

� �� gcC1 lþ 2mð Þp� � � exp �gyð Þ

þ 2mc lþ mð Þ Aþ Byð Þp2 þ bB 3lþ 4mð Þ½ �lþ m


m�yyy p; yð Þ ¼ �gcC1lpþ C2 lþ 2mð Þ 1� cp2

� �� exp �gyð Þ

� 2cm lþ mð Þ Aþ Byð Þp2 � bB lþ 2mð Þ½ �lþ m


m�xyy p; yð Þ ¼ cp C1lp� gC2 lþ 2mð Þ½ � � exp �gyð Þ

� 2mcp �Bmþ Ab lþ mð Þ þ bB lþ mð Þy½ �lþ m


m�xxy p; yð Þ ¼ cm C2p� gC1ð Þp � exp �gyð Þ þ 2cm lþ mð Þ Aþ Byð Þp2 þ bB lþ 2mð Þ½ �

lþ m

exp �byð Þ: ð85Þ

Appendix B

The solution to the system (27a–27d) is given here. The functions (A,B,C1,C2) are functionsof p and dependent upon the material parameters:

A Pð Þ ¼ � �4g lþ mð Þ 2lþ 3mð Þp4 þ lþ 2mð Þb g4 lþ 2mð Þ � 2g2lp2 þ lþ 2mð Þp4½ �2cm lþ mð Þp2K P;

ð86Þ

B Pð Þ ¼ g4 lþ 2mð Þ � 2g2lp2 þ lþ 2mð Þp4 þ 4g lþ mð Þp2b2cmK

P; ð87Þ


C Pð Þ1 ¼ � p g2 lþ 2mð Þ � lp2½ �

cmKP; ð88Þ

C Pð Þ2 ¼ p2 �g2lþ lþ 2mð Þp2½ �

gcmKP; ð89Þ

A Sð Þ ¼ g2 lþ 2mð Þ � lp2½ � mg2 þ 2lþ 3mð Þp2½ �2cm lþ mð ÞpK S; ð90Þ

B Sð Þ ¼ � b g2 þ p2ð Þ �g2 lþ 2mð Þ þ lp2½ �2cmpK

S; ð91Þ

C Sð Þ1 ¼ � mp2 g2 � p2ð Þ þ 2g3 lþ 2mð Þb

gcmKS; ð92Þ

C Sð Þ2 ¼ � p g2 � p2ð Þmþ 2glb½ �

cmKS; ð93Þ

where the superscript P or S in parentheses in the above functions denotes that the particularfunction is generated by the concentrated load P or S, respectively.

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