INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2000; 00:1–6 Prepared using nmeauth.cls [Version: 2002/09/18 v2.02] An Advanced Boundary Element Method for Solving 2D and 3D Static Problems in Mindlin’s Strain Gradient Theory of Elasticity G. F. Karlis 1 , A. Charalambopoulos 2 and D. Polyzos 3, ∗ 1 Department of Mechanical and Aeronautical Engineering, University of Patras GR-26500 Patras, Greece 2 Department of Materials Science and Engineering University of Ioannina Dourouti, Ioannina 3 Department of Mechanical and Aeronautical Engineering, University of Patras GR-26500 Patras, Greece [email protected]SUMMARY An advanced Boundary Element Method (BEM) for solving two (2D) and three dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin’s Form II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems, is illustrated with the solution of a simple 2D problem. Copyright c 2000 John Wiley & Sons, Ltd. key words: Boundary Element Method, Enhanced Elastic Theories, Mindlin’s Form II Gradient Elasticity, Microstructure and Microstructural effects. 1. INTRODUCTION Due to the lack of internal parameters classical theory of linear elasticity fails to describe size and microstructural effects or to describe fields characterized by very high gradients of strains. However, this is possible with the use of other enhanced elastic theories where * Correspondence to: Prof. Demosthenes Polyzos, Department of Mechanical and Aeronautical Engineering, University of Patras, GR-26500 Patras, Greece, email:[email protected]Received 3 July 1999 Copyright c 2000 John Wiley & Sons, Ltd. Revised 18 September 2002
Transcript
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2000; 00:1–6 Prepared using nmeauth.cls [Version: 2002/09/18 v2.02]
An Advanced Boundary Element Method for Solving 2D and 3DStatic Problems in Mindlin’s Strain Gradient Theory of Elasticity
G. F. Karlis1, A. Charalambopoulos2 and D. Polyzos3,∗
1Department of Mechanical and Aeronautical Engineering, University of PatrasGR-26500 Patras, Greece
2Department of Materials Science and EngineeringUniversity of Ioannina
Dourouti, Ioannina3Department of Mechanical and Aeronautical Engineering, University of Patras
key words: Boundary Element Method, Enhanced Elastic Theories, Mindlin’s Form II Gradient
Elasticity, Microstructure and Microstructural effects.
1. INTRODUCTION
Due to the lack of internal parameters classical theory of linear elasticity fails to describesize and microstructural effects or to describe fields characterized by very high gradientsof strains. However, this is possible with the use of other enhanced elastic theories where
∗Correspondence to: Prof. Demosthenes Polyzos,Department of Mechanical and Aeronautical Engineering,University of Patras,GR-26500 Patras, Greece,email:[email protected]
2 G. F. KARLIS, A. CHARALAMBOPOULOS AND D. POLYZOS
internal length scale parameters correlating the microstructure with the macrostructure areinvolved in the constitutive equations of the considered elastic continuum. Such theoriesand the most general are those known in the literature as Cosserat elasticity [1], Cosserattheory with constrained rotations or couple stresses theory [2], [3], strain gradient theory [4],multipolar theory of continuum mechanics [5], higher order strain gradient elastic theory [6],[7], micromorphic, microstretch and micropolar elastic theories [8] and non-local elasticity [9].Most of the aforementioned theories have been developed in the decade of 60’s and historicalreviews as well as comments on the subject can be found in [2], [8], [9], [10], [11], [12], [13].
The present work reports a boundary element formulation of Form II strain gradient elastictheory, which is a special case of Mindlin’s general strain gradient elasticity. Mindlin [6], [7] inthe middle of 60’s proposed an enhanced elastic theory to describe linear elastic behavior withmicrostructural effects. To this end, he considered the potential energy density as a quadraticform not only of strains but also of the gradient of strains. However, in order to balancethe dimensions of the aforementioned quantities and to correlate the micro-strains with themacro-strains, Mindlin utilized eighteen new constants rendering thus his general theory verycomplicated from physical and mathematical point of view. In the sequel, considering longwave-lengths and the same deformation for macro and micro structure, Mindlin proposed threenew simplified versions of his theory, known as Form I, II and III, utilizing in the constitutiveequations seven material and internal length scale constants instead of eighteen employed inhis initial model. In Form-I, the strain energy density function is assumed to be a quadraticform of the classical strains and the second gradient of displacement; in Form-II the secondgradient displacement is replaced by the gradient of strains and in Form-III the strain energyfunction is written in terms of the strain, the gradient of rotation, and the fully symmetricpart of the gradient of strain. Although the three forms are equivalent and conclude to thesame equation of motion, the Form-II leads to a total stress tensor, which is symmetric asin the case of classical elasticity avoiding thus problems associated with non-symmetric stresstensors introduced by Cosserat and couple stress theories.
As in the case of classical elasticity, the solution of gradient elastic problems with complicatedgeometry and boundary conditions requires the use of numerical methods such as the finiteelement method (FEM) and the boundary element method (BEM). The FEM is the mostwidely used numerical method for solving applied mechanics problems. Shu et al. [14] were thefirst to use FEM for solving elastostatic problems in the framework of the gradient elasticitytheories of Mindlin. Since then, many papers dealing with FEM solutions of gradient elasticproblems have appeared in the literature [15], [16], [17], [18], [19], [20], [21], [22], [23], [24],[25], [26], [27], [28], [29]. The main problem with a conventional FEM formulation is therequirement of using elements with C(1) continuity, since the presence of higher order gradientsin the expression of potential energy leads to an equilibrium equation represented by a forthorder partial differential operator. Although a displacement formulation is conceptually simplerand the most convenient for implementation, in existing finite element codes, only the works[21], [22], [23], [24], [25], [26], [27], [28] implement C(1) elements with the later being themost comprehensive and complete, since it derives both two and three dimensional C(1) finiteelements. The other works avoid the problem mainly via mixed formulations as well as withthe use of Lagrange multipliers and penalty methods. Finally, it should be mentioned thatfrom all the above cited papers only [27], [28], [29] deal with three dimensional problems.
The BEM is a well-known and powerful numerical tool, successfully used in recent yearsto solve various types of engineering problems [30], [31]. A remarkable advantage it offers as
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compared to other numerical methods, such as the finite difference and finite element methods,is the reduction of the dimensionality of the problem by one. Thus, three dimensional problemsare accurately solved by discretizing only two-dimensional surfaces surrounding the domainof interest. In the case where the problem is characterized by an axisymmetric geometry, theBEM reduces further the dimensionality of the problem, requiring just a discretization along ameridional line of the body. These advantages in conjunction with the absent of C(1) continuityrequirements, render the BEM ideal for analyzing gradient elastic problems. Tsepoura et al.[32] were the first to use BEM for solving elastostatic problems in the framework of thegradient elastic theories of Mindlin. This work was followed by the publications [33], [34], [35],[36], [37], [38], [39], which are the only papers dealing with two and three dimensional BEMsolutions of static and dynamic gradient elastic and fracture mechanics problems. All thesepapers implement two simple gradient elastic models, with the first being the simplest possiblespecial case of Mindlin’s Form II strain gradient elastic theory and the second an enrichmentof the simple gradient elastic model with surface energy terms which affect only the boundaryconditions of the problem [13].
In the present paper the BEM in its direct form is employed for the solution of two-dimensional (2D) and three-dimensional (3D) elastostatic problems in the framework ofthe Form-II strain-gradient theory of Mindlin. Although the proposed boundary elementmethodology concerns boundary value problems with smooth and non-smooth boundaries, thenumerical examples presented here are confined to smooth boundaries. The BEM solution ofgradient elastic problems with non-smooth boundaries is the subject of a forthcoming paper.The paper consists of the following five sections: Section 2 presents in brief the Mindlin’stheory implemented in the present paper. In Section 3 the 2D and 3D fundamental solutionsof the problem are explicitly derived. Section 4 demonstrates the integral representation of thegradient elastic boundary value problem in the context of Mindlin’s simplified Form-II straingradient elastic theory. Section 5 presents the proposed BEM formulation, while in section6 three numerical examples (2D and 3D) are presented to demonstrate the accuracy of themethod and illustrate the importance of considering the correct boundary conditions imposedby Mindlin’s theory. Finally, Section 7 consists of the conclusions pertaining to this work.
2. The Form-II strain gradient elastic theory of Mindlin
Mindlin in the Form II version of his strain gradient elastic theory [6] considered that thepotential energy density W is a quadratic form of the strains εij and the gradient of strains,κijk i.e.,
with ∂i denoting space differentiation, ui being displacements and λ, µ and a1–a5 beingconstants explicitly defined in [6]. It should be noted here that the constants λ, µ are not
4 G. F. KARLIS, A. CHARALAMBOPOULOS AND D. POLYZOS
the same with the corresponding Lame constants λ, µ of classical elasticity. Both λ, µ haveunits of N/m2, while α1–α5 have units of force. Thus, this particular case of Mindlin’s theoryhas in total seven elastic constants instead of the eighteen constants of his general theory.
Strains εij and gradient of strains κijk are dual in energy with the Cauchy-like and doublestresses, respectively, defined as
where the symbol ⊗ indicates dyadic product according to a ⊗ b = aibjxi ⊗ xj , ∇ is the
gradient operator, I the unit tensor and (a⊗ b⊗ c)213
= a⊗c⊗b. The total stress tensor σij
is then defined as
σij = τij − ∂iµijk or σ = τ −∇ · µ (9)
with −∇ · µ representing the relative stresses. As it is mentioned in [41], τ represents theCauchy-like (not Cauchy) stress, while the total stress vector σ is the Cauchy stresses for thepresent enhanced elastic theory. Taking the variation of (1) and equilibrating with the workdone by external and body forces fk, one obtains the following equilibrium equation
∂j(τjk − ∂iµijk) + fk = 0 (10)
accompanied by the classical essential and natural boundary conditions where the displacementvector u and/or the traction vector p have to be defined on the global boundary S of theanalyzed domain, the non-classical essential and natural boundary conditions where the normaldisplacement vector q = ∂u/∂n and/or the double traction vector R are prescribed on S, thenon-classical boundary condition satisfied only when non-smooth boundaries are dealt with,where the jump traction vector E has to be defined at corners and edges.
p = n · τ − (n ⊗ n) : ∂nµ − n · (∇S · µ) − n · (∇S · µ213)
(∇S · n)(n ⊗ n) : µ − (∇S n) : µ (14)
R = n · µ · n ≡ (n⊗ n) : µ (15)
E = ‖(m ⊗ n) : µ‖ (16)
where n is the unit vector normal to the global boundary S, D = nl∂l and Dj = (δjl − njnl) ∂l
is the surface gradient operator written in vector form as ∇S =(
I − n⊗ n)
· ∇. The
non-classical boundary condition (13) or (16) exists only when non-smooth boundaries areconsidered. Double brackets ‖•‖ indicate that the enclosed quantity is the difference betweenits values taken on the two sides of a corner while m is a vector being tangential to thecorner line. Also it should be mentioned that Giannakopoulos et al. [22], utilize for the doubletraction vector p the expression pi = njτji −njµkji,k − [Dj − (Dpnp)nj ]nkµkji, which howeveris equivalent to (11).
Finally, taking into account the form of τ and µ, the equilibrium equation (10) in terms ofdisplacements is written as
(λ + 2µ)(1 − ℓ21∇
2)∇∇ · u + µ(1 − ℓ22∇
2)∇×∇× u + f = 0 (17)
where
ℓ21 = 2(a1 + a2 + a3 + a4 + a5)/(λ + 2µ) (18)
ℓ22 = (a3 + 2a4 + a5)/2µ (19)
ℓ21 and ℓ2
2 have units of m2 and as it is discussed in [42] and [43] both can be considered asinternal length scale parameters, which correlate the microstructure with the macrostructurein irrotational and solenoidal deformations, respectively.
3. 2D and 3D fundamental solutions
Adopting the methodology presented in [34], the 2D and 3D fundamental solutions of (17) areexplicitly derived in the present section. For an infinitely extended gradient elastic space, thefundamental solutions are represented by a second order tensor u∗ (x,y) satisfying the partialdifferential equation
(
λ + 2µ)
(
1 − ℓ21∆
)
∇∇ · u∗ (x,y) − µ(
1 − ℓ22∆
)
∇×∇× u∗ (x,y) = −δ (x,y) I (20)
where δ is the Dirac δ-function, x is the point where the displacement field u∗ (x,y) is obtaineddue to a unit force at point y and r = |x − y|.
6 G. F. KARLIS, A. CHARALAMBOPOULOS AND D. POLYZOS
The field u∗ can be decomposed into irrotational and solenoidal parts [34] acoording to
u∗ = ∇∇φ + ∇∇× A + ∇×∇× G (21)
where φ (r) is a scalar function, A (r) a vector function and G (r) a dyadic function. Due tothe radial nature of the fundamental solution, it is apparent that the vector A (r) should beequal to zero. On the other hand the Dirac δ-function can be written as
−δ (r) = ∇2g (r) = ∇∇g (r) −∇×∇×[
g (r) I]
(22)
with g (r) being the fundamental solution of the Laplace operator, having the form{
g (r) = 12π ln 1
r for 2D1
4πr for 3D(23)
Inserting (21) and (22) into (20) one obtains
∇∇{(
λ + 2µ)
[
∇2φ (r) − ℓ21∇
4φ (r)]
}
+ ∇×∇×{
µ[
∇2G (r) − ℓ21∇
4G (r)]}
= ∇∇g (r) −∇×∇×[
g (r) I] (24)
The irrotational and solenoidal nature of φ (r) and G (r) impose that (24) is satisfied if(
λ + 2µ)
[
∇2φ (r) − ℓ21∇
4φ (r)]
= g (r) (25)
µ[
∇2G (r) − ℓ21∇
4G]
= −g (r) I (26)
It is not difficult to find one that the solutions of the above two partial differential equationshave the following form
φ (r) =1
4π(
λ + 2µ)
(
r
2+
ℓ21
r−
ℓ21e
−r/l1
r
)
(27)
G (r) = −1
4πµ
(
r
2+
ℓ22
r−
ℓ22e
−r/l2
r
)
I (28)
for three dimensions and
φ (r) = −1
2π(
λ + 2µ)
[
r2
2(ln r − 1) + ℓ2
1 ln r + ℓ21K0 (r/ℓ1)
]
(29)
G (r) =1
2πµ
[
r2
2(ln r − 1) + ℓ2
2 ln r + ℓ22K0 (r/ℓ2)
]
I (30)
for two dimensions.Inserting (27)–(30) into (21), the fundamental solution u∗ (x,y) obtains the final form
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where ν = λ
2(λ+µ)is the Poisson ratio,r = x−y
|x−y| and
X (r) = −1
r+ 2 (1 − 2ν)
[(
3ℓ21
r3+
3ℓ1
r2+
1
r
)
e−r/ℓ1 −3ℓ2
1
r3
]
− 4 (1 − ν)
[(
3ℓ22
r3+
3ℓ2
r2+
1
r
)
e−r/ℓ2 −3ℓ2
2
r3
]
(33)
Ψ (r) = (3 − 4ν)1
r+ 2 (1 − 2ν)
[(
ℓ21
r3+
ℓ1
r2
)
e−r/ℓ1 −ℓ21
r3
]
− 4 (1 − ν)
[(
ℓ22
r3+
ℓ2
r2+
1
r
)
e−r/ℓ2 −ℓ22
r3
]
(34)
for three dimensions and
X (r) = −1 + 2 (1 − 2ν)
[
K2
(
r
ℓ1
)
−2ℓ2
1
r2
]
− 4 (1 − ν)
[
K2
(
r
ℓ2
)
−2ℓ2
2
r2
]
(35)
Ψ (r) = − (3 − 4ν) ln r + 2 (1 − 2ν)
[
ℓ1
rK1
(
r
ℓ1
)
−ℓ21
r2
]
− 4 (1 − ν)
[
K0
(
r
ℓ2
)
+ℓ2
rK1
(
r
ℓ2
)
−ℓ22
r2
]
(36)
for two dimensions, with Kn (r/ℓi) being the modified Bessel functions of the second kind andnth order.
Utilizing the expansions of e−r/ℓi and Kn (r/ℓi), it is easy to prove that both functions X ,Ψ given by relations (33)–(36) are regular as r → 0 according to the asymptotic relations
X (r) = O(
r2 ln r)
, Ψ (r) = O (1) for the 2-D case
X (r) = O (r) , Ψ (r) = O (1) for the 3-D case(37)
4. Integral representation of the problem
Consider a finite elastic body of volume V with microstructural effects according to the gradientelastic theory illustrated in section (2), surrounded by a surface S.
Symbolizing by u, P, R, E and u∗, P∗, R∗, E∗ two deformation states of the same body,it has been proved [34], [22] that the following reciprocal identity is valid
8 G. F. KARLIS, A. CHARALAMBOPOULOS AND D. POLYZOS
for a non-smooth boundary S, where the vectors f represent body forces and P, R, E aretraction, double traction and jump traction vectors, respectively, defined in (11)–(16). Ca
represents the edge lines formed by the intersection of two surface portions when the boundaryS is non-smooth. For a two dimensional non-smooth boundary, where parts of the globalboundary form Ca corners, it is easy to prove one that the reciprocal identity (39) obtains theform
∫
V
[f∗ · u− f · u∗] dV +
∫
S
[P∗ · u − P · u∗] dS =
∫
S
[
R ·∂u∗
∂n− R∗ ·
∂u
∂n
]
dS
+∑
Ca
[E · u∗ − E∗ · u]
(40)
whereEk = ||nitjµijk|| or E =
∣
∣
∣
∣
(
t⊗ µ)
: µ
∣
∣
∣
∣ (41)
with t being the tantential vector to the curves forming the corner.Assume that the displacement field u∗, appearing in the reciprocal identity (39), is the result
of a body force having the formf∗ (y) = δ (x− y) e (42)
with δ being the Dirac δ-function and e the direction of a unit force acting at point y. Recallingthe definition of the fundamental solution derived in section 3, it is easy to see that thedisplacement field u∗ due to f∗ can be represented by means of the fundamental displacementtensor u∗ (x,y) given by the Eqs (32) and (33)–(36), according to the relation
u∗ (y) = u∗ (x,y) · e (43)
Inserting the above expression of u∗ in (39) and assuming zero body forces f = 0, one obtains∫
A DEMONSTRATION OF THE INT. J. NUMER. METH. ENGNG CLASS FILE 9
with AT indicating the transpose of A . Considering that relation (45) is valid for any directione and taking into account the symmetry of the fundamental displacement u∗, one obtains theboundary integral equation
c (x) · u (x) +
∫
S
P∗T (x,y) · u (y) − u∗ (x,y) · P (y) dSy
=
∫
S
∂u∗T (x,y)
∂ny· R (y) − R∗T (x,y) ·
∂u (y)
∂nydSy
+∑
Ca
∮
Ca
u∗ (x,y) · E (y) − E∗T (x,y) · u (y) dCy
(46)
where c (x) is the well known jump-tensor of classical boundary integral representations [34].
Utilizing the symbols U∗, P∗, Q∗, R∗ and E∗ instead of u∗, P∗T , ∂u∗T
∂n , R∗T and , E∗T ,
respectively, as well as q instead of ∂u
∂n eq (46) recieves the form
c (x) · u (x) +
∫
S
P∗ (x,y) · u (y) − U∗ (x,y) · P (y) dSy
=
∫
S
Q∗ (x,y) ·R (y) − R∗ (x,y) · q (y) dSy
+∑
Ca
∮
Ca
U∗ (x,y) ·E (y) − E∗ (x,y) · u (y) dCy
(47)
Recalling (40) and (41), the above integral equation in two dimensions obtains the form
c (x) · u (x) +
∫
S
P∗ (x,y) · u (y) − U∗ (x,y) · P (y) dSy
=
∫
S
Q∗ (x,y) ·R (y) − R∗ (x,y) · q (y) dSy
+∑
Ca
[
U∗ (x,y) · E (y) − E∗ (x,y) · u (y)]
(48)
In case the boundary S is smooth and the point x belongs to S, then the integral equations(47) and (48) reduce to
1
2u (x) +
∫
S
P∗ (x,y) · u (y) − U∗ (x,y) ·P (y) dSy
=
∫
S
Q∗ (x,y) · R (y) − R∗ (x,y) · q (y) dSy
(49)
Observing eq (47), one easily realizes that it contains four unknown vector fields, u (x), P (x),R (x) and q (x) while the boundary conditions are two (classical and non-classical). Thus, the
10 G. F. KARLIS, A. CHARALAMBOPOULOS AND D. POLYZOS
evaluation of the unknown fields requires the existence of one more integral equation. Thisintegral equation is obtained by applying the operator ∂/∂nx on (49) and has the form
c (x) · q (x) +
∫
S
∂P∗ (x,y)
∂nx· u (y) −
∂U∗ (x,y)
∂nx·P (y) dSy
=
∫
S
∂Q∗ (x,y)
∂nx·R (y) −
∂R∗ (x,y)
∂nx· q (y) dSy
+∑
Ca
∮
Ca
∂U∗ (x,y)
∂ny·E (y) −
∂E∗
∂nx· u (y) dCy
(50)
while for two dimensional and non-smooth boundary is written as
c (x) · q (x) +
∫
S
∂P∗ (x,y)
∂nx· u (y) −
∂U∗ (x,y)
∂nx·P (y) dSy
=
∫
S
∂Q∗ (x,y)
∂nx·R (y) −
∂R∗ (x,y)
∂nx· q (y) dSy
+∑
Ca
∂U∗ (x,y)
∂ny·E (y) −
∂E∗
∂nx· u (y)
(51)
Finally, for smooth boundaries the above equations obtain the form
1
2q (x) +
∫
S
∂P∗ (x,y)
∂nx· u (y) −
∂U∗ (x,y)
∂nx·P (y) dSy
=
∫
S
∂Q∗ (x,y)
∂nx·R (y) −
∂R∗ (x,y)
∂nx· q (y) dSy
(52)
The pairs of integral equations (47) and (50), (48) and (51), (49) and (52) accompaniedby the classical and non-classical boundary conditions form the integral representation ofany Mindlin’s Form II strain gradient elastic boundary value problem with 3D non-smoothboundary, 2D non-smooth boundary and smooth boundary, respectively.
5. BEM Formulation
In the present section the boundary element formulation of the gradient elastic problem isdemonstrated. As it has been already mentioned in the introduction, the formulation is confinedto gradient elastic bodies with smooth boundaries.
The goal of the boundary element method is to solve numerically the boundary integralrepresentation of the problem presented in the previous section. To this end, the smoothboundary S is discretized into e quadratic continuous isoparametric elements each of which hasA(e) nodes, with A(e) = 3, 8, 6 when line, quadrilateral and triangular elements are considered.For a nodal point k, the discretized integral equations (49) and (52) for the three dimensional
A DEMONSTRATION OF THE INT. J. NUMER. METH. ENGNG CLASS FILE 11
case have the form
1
2u
(
xk)
+
E∑
e=1
A(e)∑
a=1
1∫
−1
1∫
−1
P∗(
xk,ye (ξ1, ξ2))
Na (ξ1, ξ2)J (ξ1, ξ2) dξ1dξ2 · uea
+
E∑
e=1
A(e)∑
a=1
1∫
−1
1∫
−1
R∗(
xk,ye (ξ1, ξ2))
Na (ξ1, ξ2)J (ξ1, ξ2) dξ1dξ2 · qea
=
E∑
e=1
A(e)∑
a=1
1∫
−1
1∫
−1
U∗(
xk,ye (ξ1, ξ2))
Na (ξ1, ξ2)J (ξ1, ξ2) dξ1dξ2 ·Pea
+
E∑
e=1
A(e)∑
a=1
1∫
−1
1∫
−1
Q∗(
xk,ye (ξ1, ξ2))
Na (ξ1, ξ2)J (ξ1, ξ2) dξ1dξ2 · Rea
(53)
1
2q
(
xk)
+
E∑
e=1
A(e)∑
a=1
1∫
−1
1∫
−1
∂
∂nxP∗
(
xk,ye (ξ1, ξ2))
Na (ξ1, ξ2)J (ξ1, ξ2) dξ1dξ2 · uea
+
E∑
e=1
A(e)∑
a=1
1∫
−1
1∫
−1
∂
∂nxR∗
(
xk,ye (ξ1, ξ2))
Na (ξ1, ξ2)J (ξ1, ξ2) dξ1dξ2 · qea
=
E∑
e=1
A(e)∑
a=1
1∫
−1
1∫
−1
∂
∂nxU∗
(
xk,ye (ξ1, ξ2))
Na (ξ1, ξ2)J (ξ1, ξ2) dξ1dξ2 ·Pea
+E
∑
e=1
A(e)∑
a=1
1∫
−1
1∫
−1
∂
∂nxQ∗
(
xk,ye (ξ1, ξ2))
Na (ξ1, ξ2)J (ξ1, ξ2) dξ1dξ2 · Rea
(54)
with Na representing shape functions, the first summation is over the elements, the secondsummation over the element nodes and J is the Jacobian of the transformation from the globalcoordinate system to the local coordinate system of the element. Finally, ue
a, qea, P e
a and Rea
are the values of the unknown fields at the nodes of element e.
12 G. F. KARLIS, A. CHARALAMBOPOULOS AND D. POLYZOS
For the two dimensional case, the corresponding discretized equations are
1
2u
(
xk)
+
E∑
e=1
A(e)∑
a=1
1∫
−1
P∗(
xk,ye (ξ))
Na (ξ)J (ξ) dξ · uea
+E
∑
e=1
A(e)∑
a=1
1∫
−1
R∗(
xk,ye (ξ))
Na (ξ)J (ξ) dξ · qea
=
E∑
e=1
A(e)∑
a=1
1∫
−1
U∗(
xk,ye (ξ))
Na (ξ)J (ξ) dξ · Pea
+E
∑
e=1
A(e)∑
a=1
1∫
−1
Q∗(
xk,ye (ξ))
Na (ξ)J (ξ) dξ ·Rea
(55)
1
2q
(
xk)
+
E∑
e=1
A(e)∑
a=1
1∫
−1
∂
∂nxP∗
(
xk,ye (ξ))
Na (ξ)J (ξ) dξ · uea
+
E∑
e=1
A(e)∑
a=1
1∫
−1
∂
∂nxR∗
(
xk,ye (ξ))
Na (ξ)J (ξ) dξ · qea
=
E∑
e=1
A(e)∑
a=1
1∫
−1
∂
∂nxU∗
(
xk,ye (ξ))
Na (ξ)J (ξ) dξ · Pea
+
E∑
e=1
A(e)∑
a=1
1∫
−1
∂
∂nxQ∗
(
xk,ye (ξ))
Na (ξ) J (ξ) dξ ·Rea
(56)
with the same notation used for eqs (53) and (54).Next, a global numbering scheme is adopted by assigning a number β to each point (e, a).
Then the above equations become
1
2uk +
L∑
β=1
Hkβ · uβ +
L∑
β=1
Kkβ · qβ =
L∑
β=1
Gkβ · Pβ +
L∑
β=1
Lkβ ·Rβ (57)
1
2qk +
L∑
β=1
Skβ · uβ +
L∑
β=1
Tkβ · qβ =
L∑
β=1
Vkβ ·Rβ +
L∑
β=1
Wkβ ·Rβ (58)
where L is the total number of nodes. Note that eqs (57) and (58) are valid for both the 2Dand the 3D case. However, the integrals are different in each case.
14 G. F. KARLIS, A. CHARALAMBOPOULOS AND D. POLYZOS
Tkβ =
1∫
−1
∂
∂nxR∗
(
xk,ye (ξ))
Na (ξ) J (ξ) dξ
∣
∣
∣
∣
∣
∣
(e,a)→β
(72)
Vkβ =
1∫
−1
∂
∂nxU∗
(
xk,ye (ξ))
Na (ξ)J (ξ) dξ
∣
∣
∣
∣
∣
∣
(e,a)→β
(73)
Wkβ =
1∫
−1
∂
∂nxQ∗
(
xk,ye (ξ))
Na (ξ) J (ξ) dξ
∣
∣
∣
∣
∣
∣
(e,a)→β
(74)
When β 6= k the above integrals are non-singular and can be easily computed through Gaussquadrature. In the case of β = k the integrals become singular and special treatment isrequired. This is accomplished by applying the methodology proposed by Guiggiani [40] forsingular integrations.
Thus, eqs (57) and (58) form the following linear system of algebraic equations[
12 I + H K
S 12 I + T
]
·
[
u
q
]
=
[
G L
V W
]
·
[
P
R
]
(75)
which after the application of the classical and non-classical boundary conditions andrearranging, a final linear system is obtained of the form A ·X = B, where the vectors X andB contain all the unknown and known nodal components of the boundary fields respectively.Finally, the above linear system is solved via a typical LU-decomposition algorithm and thevector X, comprising of all the unknown nodal values of u, q, R and P is evaluated.
6. Numerical Examples
In this section some 2D and 3D benchmarks that demonstrate the accuracy of the proposedBEM are presented.
The first numerical example concerns a 3D/2D problem dealing with the tension of acylindrical/rectangular gradient elastic bar shown in Figure 1. The analytical solution of theproblem is known for the one dimensional case [32] and for material properties where onlythe Young modulus E and internal length α4 are non-zero. Thus in the present BEM solutionthe Poisson ratio has been taken equal to zero, α1 = α2 = α3 = α5 = 0 and the height ofthe cylinder/rectangle much smaller than its diameter/width (d = 4.2m, h = 1.2m) so thatto compare analytical and numerical results across the axis of symmetry. In order to avoid aBEM formulation for non-smooth boundaries, the edges of the cylinder and the corners of therectangle have been rounded with radius re = 0.05m.
The bar is subjected to a tension of T = 2.1GPa on its top and bottom sides while itsmaterial properties are E = 2.1GPa and α4 = 10.5MNt. The non classical boundary conditionapplied to the top and bottom faces of the bar is (qx, qy, qz) = (0, 0, 0). The side surface of thebar is left traction free by imposing (Px, Py, Pz) = (0, 0, 0) and (Rx, Rx, Rx) = (0, 0, 0).
The 3D problem has been solved with octant symmetry while in the 2D case of therectangular bar only the one quarter of the domain has been discretized. For the 2D case,approximately 50 elements have been used, whereas for the 3D case up to 150 elements have
A DEMONSTRATION OF THE INT. J. NUMER. METH. ENGNG CLASS FILE 15
Figure 1. Geometry of the gradient elastic bar in three and two dimensions
0.0 0.1 0.2 0.3 0.4 0.5 0.60.0
0.1
0.2
0.3
0.4
0.5
Axi
al d
ispl
acem
ents
on
inte
rnal
poi
nts
z coordinate
3D Form II Analytical solution l
1 = 0.1
l1 = 0.05
l1 = 0.001
(a)
0.0 0.1 0.2 0.3 0.4 0.5 0.60.0
0.1
0.2
0.3
0.4
0.5
Axi
al d
ispl
acem
ents
on
inte
rnal
poi
nts
z coordinate
2D Form II Analytical Solution l1 = 0.1 l1 = 0.05 l1 = 0.001
(b)
Figure 2. Axial displacements of the internal points for (a) the 3D case and (b) the 2D case
been utilized. A set of internal points has been placed along the central vertical axis of the bar.In Figures 2 the axial displacements of the internal points are presented. For all the displayedresults, the relative error with respect to the analytical solution [32] is below 0.2%.
The second numerical example deals with a hollow cylinder subjected to internal and externalpressure as it is shown in Figure 3. The problem is solved under plane strain conditions. Thematerial properties of the cylinder are described in Table I. The internal and external radii andpressures are ri = 1.05m, ro = 2.1m and Ta = 100KPa, Tb = 200KPa, respectively. Doubletractions are considered to be equal to zero everywhere. Figure 4 shows the radial displacementson the internal points of the hollow cylinder with respect to the distance r from the centerof the cylinder. The numerical results are compared with the corresponding analytical ones[29]. Furthermore, the normal derivative of the displacements, q has been calculated on theboundary and found to be equal to the corresponding analytical result with an error of 0.14%.The relative error for the internal displacement with respect to the mesh used, is presented inTable II.
In order to demonstrate the importance of considering the correct integral equations and
A DEMONSTRATION OF THE INT. J. NUMER. METH. ENGNG CLASS FILE 17
Table II. Relative error with respect to the number of elements for the plain strain hollow cylinder
Number of elements % Error
12 0.173%17 0.120%26 0.069%50 0.028%
satisfying the correct boundary conditions in a gradient elastic problem, the problem ofFigure 3 is solved for Ta = 100KPa, Tb = 0 and first considering an artificial interfacein the circumferential direction (Figure 5(a)) and next an interface separating the cylinderinto two equal parts (Figure 5(b)). In both cases, continuity on displacements u, tractionsP, normal derivatives of displacements q and double tractions R has been considered atinterfaces. The corresponding deformation profiles of the cylinder in both cases are depicted inFigires 5(c) and 5(d), respectively. It is apparent that the obtained solutions are not the same.The displacement profile of Figure 5(c) is the same with the corresponding one taken when nointerfaces are considered in the solution of the problem. On the other hand the displacementprofile of the cylinder in Figure 5(d) violates the radial symmetry of the problem and thesolution appears significant errors near to the interface. The obvious explanation for these twodifferent solutions of the same problem is that in the first case the two considered subregionshave smooth boundaries, while in the second case they have not. Thus, for the second case,first the BEM formulation for non-smooth boundaries should be taken into account and seconda continuity condition of the jump traction vectors E at the circumferential edges is required.
The third numerical example deals with a gradient elastic sphere of radius a = 0.5m underan external uniform displacement ur = 0.01m. The material characteristics of the sphere arethe same used in the previous example (Table I).
In order to model the problem octant symmetry has been used and the same classical andnon-classical boundary conditions have been applied to all elements. Namely, (ur, uθ, uφ) =(0.01, 0, 0) and (qr, qθ, qφ) = (0, 0, 0), with the subscripts r, θ and φ indicating sphericalcoordinates. A set of internal points have been placed inside the sphere, along its radius.Figure 6 shows the radial displacement on the internal points with respect to their distancefrom its center as compared to the analytical solution provided by [35]. In Table III the relativeerror of the internal radial displacement is presented with respect to the analytical solution,for various mesh sizes.
7. Conclusions
A boundary element method for solving two and three-dimensional static, strain gradientelastic problems has been developed. Microstructural effects have been taken into account bymeans of the Mindlin’s Form II strain gradient elastic theory. The equation of equilibrium,all the possible boundary conditions (classical and non-classical), the fundamental solutionand the reciprocity identity of the gradient elastic problem are explicitly presented. Both,
18 G. F. KARLIS, A. CHARALAMBOPOULOS AND D. POLYZOS
(a) (b)
(c) (d)
Figure 5. A hollow cylinder divided into two regions (a) by a circular interface and (b) by a straightinterface, as well as the boundary displacements for the (c) circular interface case and (d) for the
straight interface.
Table III. The relative error of the internal radial displacements, with respect to the analytical solution,for the gradient elastic sphere
A DEMONSTRATION OF THE INT. J. NUMER. METH. ENGNG CLASS FILE 19
0.0 0.1 0.2 0.3 0.4 0.50.000
0.002
0.004
0.006
0.008
0.010
Rad
ial d
ispl
acem
ent u
r
Radial distance
3D Form II Analytical Solution BEM
Figure 6. The radial displacements of the internal points of a sphere subjected to radial displacement
fundamental solution and reciprocity identity have been used to establish the boundary integralrepresentation of the problem consisting of one equation for the displacement and another onefor its normal derivative. This integral representation concerns problems dealing with bothsmooth and non-smooth boundaries.
The BEM formulation is confined to boundary value problems with smooth boundaries, sincethe treatment of non-smooth boundaries is the subject of a forthcoming paper. The numericalimplementation of the problem is accomplished by discretizing the external boundary intoquadratic line or quadrilateral elements and employing advanced integration algorithms forthe highly accurate evaluation of the singular integrals. Three representative two and threedimensional numerical examples have been presented to illustrate the method and demonstrateits high accuracy. Finally, with the solution of a simple two-dimensional gradient elasticproblem, has been shown that attention should be paid on the handling of the boundaryconditions of the problem, especially when non-smooth boundaries are considered.
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