J Elast (2011) 104:45–70 DOI 10.1007/s10659-011-9308-7
Plane-Strain Problems for a Class of Gradient Elasticity Models—A Stress Function Approach Nikolaos Aravas
Received: 11 October 2010 / Published online: 26 January 2011 © Springer Science+Business Media B.V. 2011
Abstract The plane strain problem is analyzed in detail for a class of isotropic, compressible, linearly elastic materials with a strain energy density function that depends on both the strain tensor ε and its spatial gradient ∇ε. The appropriate Airy stress-functions and doublestress-functions are identified and the corresponding boundary value problem is formulated. The problem of an annulus loaded by an internal and an external pressure is solved. Keywords Gradient elasticity · Stress functions Mathematics Subject Classification (2000) 74A35
1 Introduction Theories with intrinsic- or material-length-scales find applications in the modeling of sizedependent phenomena. In elasticity, length scales enter the constitutive equations through the elastic strain energy function, which in this case depends not only on the strain tensor but also on gradients of the rotation and strain tensors; in such cases we refer to “gradient elasticity” theories. A first attempt to incorporate length scale effects in elasticity was made by Mindlin [34], Koiter [27, 28] and Toupin [42]. They solved also a number of problems and demonstrated the effects of the material length scales that enter the strain-gradient elasticity theories (Mindlin and Tiersten [37], Mindlin [34, 35], Koiter [28]). Several theoretical issues related to strain-gradient elasticity were addressed later by Germain [23–25]. More recently,
The paper is dedicated to the memory of Professor Donald Carlson. N. Aravas () Department of Mechanical Engineering, University of Thessaly, Volos 38334, Greece e-mail:
[email protected] N. Aravas The Mechatronics Institute, Center for Research and Technology—Thessaly (CERETETH), 1st Industrial Area, Volos 38500, Greece
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N. Aravas
a variety of “non-local” or “gradient-type” theories have been used in order to introduce material length scales into constitutive models (Aifantis [1], Pijaudier-Cabot and Bazant [39], Vardoulakis et al. [44–46], de Borst [9], Fleck et al. [16–18], Leblond et al. [32], Tvergaard and Needleman [43]). A common feature of all the aforementioned theories is the nonsymmetry of the true stress tensor, and the existence of couple and higher order stresses. Some of the early developments in the theory are summarized in the Nowacki’s book [38] that was published in the 80s. Lazar and Maugin [29–31] have solved a series of problems including dislocations, disclinations, and line forces in the context of gradient elasticity. A summary of the applicability of gradient elasticity to certain micro/nano problems has been given recently by Aifantis [3]. In the present work, the plane strain problem is analyzed in detail for a class of isotropic, linearly elastic materials with a strain energy density function that depends on both the strain tensor ε and its spatial gradient ∇ε. The appropriate Airy stress-functions and double-stressfunctions are identified and the corresponding boundary value problem is formulated. The problem of an annulus loaded by an internal and an external pressure is solved and the effects of strain gradients are examined. The special cases of a thin-walled tube and an infinite body with a cylindrical hole are also analyzed. Standard notation is used throughout. Boldface symbols denote tensors the orders of which are indicated by the context. The usual summation convention is used for repeated Latin and Greek indices of tensor components with respect to a fixed Cartesian coordinate system with base vectors ei (i = 1, 2, 3). Latin indices take the values (1, 2, 3), whereas Greek indices range over the integers (1, 2), unless indicated otherwise. A comma followed by a subscript, say i, denotes partial differentiation with respect to the spatial coordinate xi , i.e., f,i = ∂f/∂xi . Let a, b, c be vectors, A, B second-order tensors, and κ, μ thirdorder tensors; the following products are used in the text: (ab)ij = ai bj , (abc)ij k = ai bj ck , (aA)ij k = ai Aj k , (a · A)i = aj Aj i , (aA)ij k = ai Aj k , A : B = Aij Bij , (a · μ)ij = ak μkij , κ : μ · ∇)ij = μij k,k , (a × A)i = eipq ap Aqj , (∇ × A)ij = μ = κij k μij k , (∇ · μ )ij = μkij,k ,1 (μ eipq Aqj,p , (A × ∇)ij = −ejpq Aiq,p , and (∇ × κ )ij k = eipq κqj k,p , where eij k is the alternating symbol.
2 The Constitutive Model We consider the class of elastic materials in which the elastic strain energy density function W depends on the infinitesimal strain tensor and its spatial gradient. Mindlin and Eshel [36] have shown that for an isotropic, compressible, linearly elastic material, the general form of W is ν εii εjj W (εε ,κκ ) = G εij εij + 1 − 2ν + a1 κiik κkjj + a2 κijj κikk + a3 κiik κjj k + a4 κij k κij k + a5 κij k κkj i ,
(1)
where ε = 12 (u∇ + ∇u) is the infinitesimal strain tensor, u the displacement field, κ = ∇εε (κij k = κikj = εj k,i ) the strain gradient third-order tensor, G the elastic shear modulus, ν Poisson’s ratio, and (a1 , a2 , a3 , a4 , a5 ) material constants. 1 With respect to a Cartesian system x with unit base vectors e , the gradient operator is written in the form i i ∇ = ei ∂x∂ = ei ∂i . i
Plane-Strain Problems for a Class of Gradient Elasticity Models
47
We consider the special case in which a1 = a3 = a5 = 0,
a2 =
ν G2 , 1 − 2ν
a4 = G2 ,
where is a material length, so that ν ν 2 εii εjj + κij k κij k + κijj κikk . W (εε ,κκ ) = G εij εij + 1 − 2ν 1 − 2ν We define the stress-like and double-stress-like quantities τ and μ as follows: ∂W ν = 2G εij + εkk δij , τij = ∂εij 1 − 2ν and
∂W ν 2 κipp δj k . μij k = = 2G κij k + ∂κij k 1 − 2ν
The above equation can be inverted to yield 1 ν τkk δij εij = and τij − 2G 1+ν
1 ν μipp δj k . κij k = μij k − 2G2 1+ν
(2)
(3)
(4)
(5)
(6)
Equations (5) imply also that μij k = 2 τj k,i and μij k,i = 2 τj k,ii = 2 ∇ 2 τj k , or μ = 2 ∇ττ
and
∇ · μ = 2 ∇ 2τ .
(7)
The quantity τ − ∇ · μ that enters (20) below can be written as τ − ∇ · μ = λεkk δ + 2με − 2 ∇ 2 (λεkk δ + 2με) .
(8)
The right hand side of the above equation is formally similar to the expression used for the stress tensor by Aifantis [2] and Altan and Aifantis [4] in their version of a gradient elasticity theory. The strain energy density function given in (3) can be written also in the form 1 1 2 1 W = τ : ε + μ : κ = τ : ε + ∇ττ : ∇εε . 2 2 2 2
(9)
3 The Compatibility Conditions The compatibility conditions for the strain tensor have the well know form S ≡ ∇ × ε × ∇ = −eipm ej qn εpq,mn ei ej = 0.
(10)
The above equations are the necessary and sufficient conditions for the existence of a displacement field u(x) such that the relation ε = (1/2)(u∇ + ∇u) is satisfied. If the region of interest B is simply connected, then the ε-compatibility conditions (10) guarantee also that u(x) is single valued. An interesting discussion of the strain compatibility conditions and their relationship to a Stokes theorem for second-order tensor fields has been given by Fosdick and Royer-Carfagni [19].
48
N. Aravas
Fig. 1 Triply-connected region (N = 2)
The definition of the strain gradient κ = ∇ε implies the κ-compatibility conditions: P ≡ ∇ × κ = eipq κqj k,p ei ej ek = 0.
(11)
Conversely, satisfaction of the κ-compatibility conditions implies the existence of a tensor field ε(x) such that κ = ∇ε. Furthermore, if the region is simply connected, ε(x) is singlevalued (Courant and John [14]). If ε and κ are related to τ and μ though (6), the above compatibility conditions can be written in the form 1 ν 2 S= ∇ τkk δ − ∇ (∇τkk ) ∇ ×τ × ∇ + 2G 1+ν 1 ν 2 = ∇ τkk δij − τkk,ij ei ej = 0, (12) −eipm ej qn τpq,mn + 2G 1+ν and 1 ν × μ e δ ∇ × μ − (∇ ) i irr 2G2 1+ν eipq ν = μ − δ μ qj k,p qrr,p j k ei ej ek = 0. 2G2 1+ν
P=
(13)
We conclude this section with a discussion of the compatibility conditions in multiply connected domains. Assume that the region B is (N + 1)-tuply connected, i.e., it has N “holes”. Starting with B , we can form a simply-connected domain by inserting N interior surfaces Sn (n = 1, 2, . . . , N ) between the outside boundary and each hole (doted lines in Fig. 1). For u(x) to be single valued, in addition to the ε-compatibility conditions (10), the following 6N conditions are required (e.g., see Boley and Weiner [7] pp. 92–95 and 97–100,
Plane-Strain Problems for a Class of Gradient Elasticity Models
49
Fung [21] pp. 104–107, Fraeijs de Veubeke [20] pp. 90–97):
Su ≡ [ε + x × (∇ × ε)] · dx = εij − xk erik erpq εpj,q dxj ei Cn
Cn
εij − xk εij,k − εkj,i dxj ei
= Cn
=−
xk εik,j + εij,k − εkj,i dxj ei = 0,
Cn
(14)
n = 1, 2, . . . , N,
(15)
(∇ × ε) · dx =
Sω ≡
n = 1, 2, . . . , N,
Cn
eipq εpj,q dxj ei = 0, Cn
where Cn (n = 1, 2, . . . , N ) are simple closed curves in B that surround only a single hole or alternatively, each cut only a single surface Sn as introduced above (see Fig. 1). Equations (15) guarantee that the infinitesimal rotation field is single valued and, under these conditions, (14) guarantee that the displacement filed u(x) is single valued. Similarly, it can be shown readily that for ε(x) to be single valued, in addition to the κ-compatibility conditions (11), the following 6N conditions are required: ε (16) P ≡ (dx · κ) = κkij dxk ei ej = 0, n = 1, 2, . . . , N. Cn
Cn
If ε and κ are related to τ and μ though (6), the compatibility conditions on Cn (14)–(16) can be written in the form 1 ν ν τkk δ + x × ∇ × τ + eijp τkk,p ei ej Su = τ− · dx, (17) 2G 1+ν 1+ν Cn
1 s = 2G
ω
Cn
Pε =
1 2G2
ν eijp τkk,p ei ej x× ∇ ×τ + 1+ν dx · μ −
Cn
ν μipp ei δ , 1+ν
· dx,
(18)
(19)
where n = 1, 2, . . . , N in all three equations above.
4 The Boundary Value Problem Consider a homogeneous elastic body that occupies a region B in a fixed reference configuration and obeys the constitutive equations (4) and (5). The field equations in B are ∇ · (ττ − ∇ · μ ) + b = 0,
(20)
1 (u∇ + ∇u) , 2
(21)
ε=
κ = ∇εε ,
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N. Aravas
τ = 2G ε +
ν εkk δ , 1 − 2ν
μ = 2G2 κ +
ν κipp ei δ = 2 ∇τ , 1 − 2ν
(22)
where b is the body force per unit volume. Let ∂ B be the smooth boundary of B . The corresponding boundary conditions are (Mindlin [34], Mindlin and Eshel [36]) u = u¯ on ∂ Bu ,
(23)
n · (ττ − ∇ · μ ) − D · (n · μ ) + (D · n) n · (n · μ ) = P on ∂ BP ,
(24)
Du = v¯ on ∂ Bv ,
(25)
n · (n · μ ) = R on ∂ BR ,
(26)
∂ ¯ v¯ , P, R) are known functions, D = n · ∇ = ∂n where (u, is the normal derivative to ∂ B , D = ∇ − nD is the “surface gradient” on ∂ B , ∂ Bu ∪ ∂ BP = ∂ Bv ∪ ∂ BR = ∂ B , and ∂ Bu ∩ ∂ BP = ∂ Bv ∩ ∂ BR = ∅. The boundary value problem defined by (20)–(26) determines u, ε, κ, τ , and μ. Mindlin and Eshel [36] have shown that the conditions G > 0 and −1 < ν < 1/2 are sufficient for a unique solution (see also Georgiadis et al. [22]). If the boundary ∂ B is not smooth, additional boundary conditions are required on the edges of ∂ B [34, 36]. The quantities P and R in (24) and (26) are work conjugate to u and Du, and are referred to as the “generalized” or “auxiliary” or “mathematical” forces and double-forces respectively. It should be emphasized that τ and μ are defined as work-conjugate quantities to ε and κ , i.e.,
∂W ∂W and μ = , (27) ∂εε ∂κκ and then (20) and the boundary conditions (23)–(26) are derived from Hamilton’s principle (Mindlin [34], Mindlin and Eshel [36]). The principles of conservation of linear and angular momentum introduce the true stress σ and the true couple-stress μ¯¯ second-order tensors. The relation of σ and μ¯¯ to τ and μ is discussed in Sect. 5. The true force and moment per unit area are defined by t = n · σ and m = n · μ¯¯ and not by P and R, which appear in (24) and (26) (see Mindlin and Eshel [36], p. 124). We consider next the traction boundary value problem in which ∂ Bu = ∂ Bv = ∅ and ∂ BP = ∂ BR = ∂ B . Let (u(0) , ε (0) ,ττ (0) ) be the solution of the classical elasticity problem ( = 0) that corresponds to loads b and P, i.e., the solution of the following boundary value problem τ=
∇ · τ (0) + b = 0, 1 (0) ε (0) = u ∇ + ∇u(0) , 2 ν (0) (0) (0) ε δ , τ = 2G ε + 1 − 2ν kk
(28) (29) and
(30)
n · τ (0) = P on ∂ B.
(31)
μ(1) ) so that the solution of the We introduce the “gradient correction” (u(1) , ε (1) ,ττ (1) ,μ boundary value problem defined by (20)–(26) is written as u = u(0) + u(1) ,
ε = ε (0) + ε (1) ,
τ = τ (0) + τ (1) ,
μ = μ (0) + μ (1) ,
(32)
Plane-Strain Problems for a Class of Gradient Elasticity Models
51
where μ (0) = 2 ∇ττ (0) . Then, the gradient correction is defined by the following boundary value problem:
∇ · τ (1) − ∇ · μ (1) = ∇ · μ (0) , 1 (1) u ∇ + ∇u(1) , 2 ν (1) (1) (1) ε δ , τ = 2G ε + 1 − 2ν kk
(33)
ε (1) =
(34) μ (1) = 2 ∇ 2τ (1) ,
(35)
and
n · τ (1) − ∇ · μ(1) − D · n · μ(1) + (D · n) n n · μ(1)
= n · ∇ · μ (0) + D · n · μ (0) − (D · n) n n · μ (0) on ∂ B,
n · n · μ (1) = R − n · n · μ (0) on ∂ B.
(36) (37)
Equations (33)–(37) show that if R = 0, then the “loads” on the right hand side of (33), (36), and (37) are all proportional to μ(0) (which equals 2 ∇τ (0) ). Therefore, when R = 0, μ(1) ) is if μ(0) is proportional to a parameter, then the gradient correction (u(1) , ε (1) ,ττ (1) ,μ proportional to the same parameter. An application of this property will be discussed in Sect. 7.1, where the problem of an annulus loaded by internal and external pressure is solved.
5 The “True” Stress and the “True” Couple Stress The quantities τ and μ have dimensions of stress and double-stress respectively and are defined by (4) and (5) as work-conjugate to ε and κ (Mindlin [34]). However, the physical interpretation of τ and μ is not clear. Mindlin and Eshel [36] established the relationship between τ and μ and the true stress tensor σ and the true couple stress tensor μ¯¯ as described briefly in the following. Mindlin and Eshel [36] consider the standard true stress vector t and true couple-stress vector m, defined the usual way as force and moment per unit area. Then, the standard argument of force and moment equilibrium of an infinitesimal tetrahedron leads to the well know relations t=n·σ
and
¯¯ m = n · μ,
(38)
¯¯ are the usual stress where n is the unit vector normal to the infinitesimal area, and (σ , μ) and couple-stress second-order tensors. The principles of conservation of linear and angular momentum lead to the well known equations ∇ ·σ +b=0
and
∇ · μ¯¯ + s + c = 0,
(39)
where c is the body moment per unit volume and si = eij k σj k . Let be the double-force per unit volume. Mindlin and Eshel [36] identify the symmetric part of with the body double-force without moment per unit volume, and the antisymmetric part of with the body moment per unit volume with its components defined as
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N. Aravas
[ij ] = 12 eij k ck or ci = eij k [j k] . Mindlin and Eshel [36] have shown that (see also Germain [23–25], Aravas and Giannakopoulos [6]) 1 2 σ = τ − μ · ∇ − ∇ · μ − or 3 3
2 1 σij = τij − μij k,k − μkij,k − ij , 3 3
(40)
and 2 μ¯¯ ij = μpqi epqj . 3
(41)
6 Plane Strain Problems We consider the plane displacement field u = uα (x1 , x2 ) eα .
(42)
The corresponding strain ε and strain-gradient κ fields are of the form ε = εαβ eα eβ
and κ = ∇εε = καβγ eα eβ eγ ,
(43)
with εαβ =
1
uα,β + uβ,α 2
καβγ = εβγ ,α =
and
1
uβ,γ α + uγ ,βα . 2
(44)
The plane strain conditions ε33 =
1 ν τkk δ33 = 0 τ33 − 2μ 1+ν
κα33 =
and
1 ν μ − δ μ α33 αpp 33 = 0, 22 μ 1+ν (45)
imply that τ33 = νταα
and
μα33 = νμαββ ,
(46)
and τ and μ are of the form τ = ταβ eα eβ + νταα e3 e3
and μ = μαβγ eα eβ eγ + νμαββ eα e3 e3 .
(47)
The constitutive equations can be written now in the form
ταβ
ν εγ γ δαβ , = 2G εαβ + 1 − 2ν
μαβγ = 2 G καβγ 2
ν καδδ δβγ , + 1 − 2ν
(48)
with ταα =
2G εαα 1 − 2ν
and
μαββ =
22 G καββ . 1 − 2ν
(49)
The above equations can be inverted to yield εαβ =
1
ταβ − ντγ γ δαβ , 2G
καβγ =
1
μαβγ − νμαδδ δβγ . 2 2 G
(50)
Plane-Strain Problems for a Class of Gradient Elasticity Models
53
6.1 The Compatibility Equations The compatibility equations (12) and (13) now take the form
S = 2ε21,12 − ε11,22 − ε22,11 e3 e3 =
1
2τ21,12 − τ11,22 − τ22,11 − ν∇ 2 ταα e3 e3 = 0, 2G
(51)
and
P = κ2βγ ,1 − κ1βγ ,2 e3 eβ eγ =
1 μ2βγ ,1 − μ1βγ ,2 − ν μ2αα,1 − μ1αα,2 δβγ e3 eβ eγ = 0. 2 2 G
(52)
The above compatibility equations can be written in the form 2τ21,12 − τ11,22 − τ22,11 − ν∇ 2 ταα = 0,
(53)
(1 − ν)(μ211,1 − μ111,2 ) − ν(μ222,1 − μ122,2 ) = 0,
(54)
(1 − ν)(μ222,1 − μ122,2 ) − ν(μ211,1 − μ111,2 ) = 0,
(55)
μ212,1 − μ112,2 = 0.
(56)
and
For ν = 1/2, the κ-compatibility equations (54)–(56) are equivalent to μ211,1 − μ111,2 = 0,
(57)
μ222,1 − μ122,2 = 0,
(58)
μ212,1 − μ112,2 = 0.
(59)
If the region of interest is (N + 1)-tuply connected, the following additional 6N conditions should be satisfied on the closed curves Cn (n = 1, 2, . . . , N ) introduced in Sect. 3 above:
εaβ − xζ eαζ eγ δ εγβ,δ dxβ Sαu = Cn
=
1 2G
ταβ − ντγ γ δαβ − xζ eαζ eγ δ τγβ,δ − ντηη,δ δγβ = 0,
n = 1, 2, . . . , N,
Cn
S3ω =
eβγ εβα,γ dx α = Cn
e Pαβ =
κγ αβ dxγ = Cn
1 2G
1 2G Cn
(60)
eβγ ταβ,γ − ντδδ,γ δαβ dxα ,
n = 1, 2, . . . , N,
(61)
Cn
ταβ,γ − ντδδ,γ δαβ dxγ ,
n = 1, 2, . . . , N.
(62)
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N. Aravas
6.2 The Plane-Strain Traction Boundary Value Problem in Terms of τ and μ For simplicity we assume for the rest of the paper that the body forces vanish (b = 0). Then the plane strain boundary value problem becomes
(63) ταβ − μγ αβ,γ ,β = 0, 2τ21,12 − τ11,22 − τ22,11 − ν∇ 2 ταα = 0,
(64)
μγ αβ = ταβ,γ ,
(65)
2
and
nβ τβα − μγβα,γ − Dβ nγ μγβα + (Dδ nδ ) nγ nβ μγβα = Pα nγ nβ μγβα = Rα
on ∂ B,
on ∂ B,
(66) (67)
where B is now a two-dimensional domain and ∂ B its smooth boundary. Equations (63)–(65) form a linear second-order system of equations, which together with the boundary conditions (66) and (67), define (τ11 , τ22 , τ12 ) and (μ111 , μ122 , μ112 , μ211 , μ222 , μ212 ). We can eliminate the double-stress μ from the above equations and restate the boundary value problem in the following form:
(68) ταβ − 2 ∇ 2 ταβ ,β = 0, 2τ21,12 − τ11,22 − τ22,11 − ν∇ 2 ταα = 0
(69)
and
nβ ταβ − 2 ∇ 2 ταβ − 2 Dβ nγ ταβ,γ + 2 (Dδ nδ ) nγ nβ ταβ,γ = Pα 2 nγ nβ ταβ,γ = Rα
on ∂ B.
on ∂ B,
(70) (71)
Equations (68) and (69) form a linear third-order system of equations, which together with the boundary conditions (70) and (71), define the three unknowns (τ11 , τ22 , τ12 ). If the region of interest is (N + 1)-tuply connected, the following additional conditions should be satisfied on the closed curves Cn (n = 1, 2, . . . , N ) introduced in Sect. 3 above:
1 ταβ − ντγ γ δαβ − xζ eαζ eγ δ τγβ,δ − ντηη,δ δγβ = 0, (72) Sαu = 2G Cn
S3ω =
1 2G
eβγ ταβ,γ − ντδδ,γ δαβ dxα ,
(73)
Cn e Pαβ =
1 2G
ταβ,γ − ντδδ,γ δαβ dxγ .
(74)
Cn
6.3 Stress and Double-Stress Functions Schaefer [40], Mindlin [33] and Carlson [10] derived the appropriate stress functions for plane problems with couple stresses. For the particular case of a homogeneous,
Plane-Strain Problems for a Class of Gradient Elasticity Models
55
isotropic, linearly elastic constrained-Cosserat material, they used the equilibrium and the κcompatibility conditions to show that the stress and couple-stress components can be written in terms of two stress functions (see also Boresi and Chong [8], pp. 390–398). We follow a similar approach here, starting with the κ-compatibility conditions (57)– (59): μ211,1 = μ111,2 ,
μ222,1 = μ122,2 ,
μ212,1 = μ112,2 .
(75)
According to the theory of total differentials (Courant and John [14]), the above equations imply the existence of functions F11 , F22 , and F12 (single valued if B is simply connected) such that μ111 =
∂F11 , ∂x1
μ211 =
∂F11 , ∂x2
(76)
μ122 =
∂F22 , ∂x1
μ222 =
∂F22 , ∂x2
(77)
μ112 =
∂F12 , ∂x1
μ212 =
∂F12 . ∂x2
(78)
The above equations can be written compactly in the form μαβγ =
∂Fβγ ∂xα
or μ = ∇F,
(79)
with F21 = F12 . Comparison of (79) with the constitutive equation (7), i.e., μ = 2 ∇ττ
(80)
leads to the conclusion that F can be identified with 2 τ , i.e., the second order tensor τ can be viewed as a double-stress function in the present model. We consider next equations (68):
ταβ − 2 ∇ 2 ταβ
,β
= 0,
(81)
which imply the existence of the well-known Airy [5] stress function f (single valued if B is simply connected) such that τ11 − 2 ∇ 2 τ11 = f,22 ,
τ22 − 2 ∇ 2 τ22 = f,11 ,
τ12 − 2 ∇ 2 τ12 = −f,12 .
(82)
The above equations can be written in compact form as ταβ − 2 ∇ 2 ταβ = ∇ 2 f δαβ − f,αβ = eαγ eβδ f,γ δ ,
(83)
where eαβ is the alternating symbol in two dimensions that takes the values e11 = e22 = 0 and e12 = −e21 = 1. It should be noted that (82) are a consequence of (81) and are independent of any constitutive equations τ and μ may be required to satisfy. However, (79) are valid only for the particular gradient elasticity model described in Sect. 2. Günther [26], Schaefer [40], and Carlson [11–13] presented stress functions for couple and dipolar stresses appropriate for all materials.
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N. Aravas
The plane-strain boundary value problem can be written now in terms of τij and the stress function f as follows τ11 − 2 ∇ 2 τ11 = f,22 ,
(84)
τ22 − ∇ τ22 = f,11 ,
(85)
2
2
τ12 − 2 ∇ 2 τ12 = −f,12 ,
(86)
2τ21,12 − τ11,22 − τ22,11 − ν∇ ταα = 0, 2
(87)
and
nβ ∇ 2 f δαβ − f,αβ − 2 Dβ nγ ταβ,γ + 2 (Dδ nδ ) nγ nβ ταβ,γ = Pα 2 nγ nβ ταβ,γ = Rα
on ∂ B.
on ∂ B,
(88) (89)
Equations (84)–(87) form a linear second-order system of equations, which together with the boundary conditions (88) and (89), define the four unknowns (f, τ11 , τ22 , τ12 ). The stress function f is defined to within a linear function of the form kα xα + c that causes no stress. If the domain is multiply connected, conditions (72)–(74) are imposed as well. If polar coordinates are used, equations (84)–(87) have the form 1 ∂ 2 τrr 1 ∂ 2f 1 ∂f 1 ∂τrr ∂τrθ ∂ 2 τrr + − 2 + 2 2 , (90) = + − 4 − τ (τ ) rr θθ 2 2 2 ∂r r ∂r r ∂θ ∂θ r ∂r r ∂θ 2 2 2 1 ∂τθθ ∂τrθ 1 ∂ τθθ ∂ f ∂ τθθ τθθ − 2 + +4 (91) + 2 + 2 (τrr − τθθ ) = 2 , 2 2 ∂r r ∂r r ∂θ ∂θ ∂r 2 1 ∂ 2 τrθ 1 ∂τrθ ∂ ∂ 1 ∂f 2 ∂ τrθ τrθ − + 2 + + 2 (τrr − τθθ ) − 4τrθ = − , (92) ∂r 2 r ∂r r ∂θ 2 ∂θ ∂r r ∂θ
τrr − 2
−
1 ∂ 1 ∂ 2 τrr 2 ∂ 2 τrθ 2 ∂τrθ ∂ 2 τθθ + − 2 + + ν∇ 2 (τrr + τθθ ) = 0. (93) (τrr − 2τθθ ) + 2 2 ∂r r ∂r∂θ r ∂r r ∂θ r ∂θ 2
The boundary conditions (88) and (89) can be stated in polar coordinated, if we take into account that 1 ∂2 ∂2 1 ∂ ∂ 1 ∂ + eθ , ∇2 = 2 + + 2 2, ∂r r ∂θ ∂r r ∂r r ∂θ ∂ 1 ∂ ∂ nr ∂ D = er nθ nθ − nr − nθ + eθ nr , ∂r r ∂θ r ∂θ ∂r ∇ = er
(94) (95)
and ∇ττ =
∂τθθ ∂τrθ ∂τrr er er er + er eθ eθ + er (er eθ + eθ er ) ∂r ∂r ∂r 1 ∂τrr 1 ∂τθθ − 2τrθ eθ er er + + 2τrθ eθ eθ eθ + r ∂θ r ∂θ 1 ∂τrθ + τrr − τθθ eθ (er eθ + eθ er ) . + r ∂θ
(96)
Plane-Strain Problems for a Class of Gradient Elasticity Models
57
In particular, on a circular arc of radius r, n = ±er (nr = ±1, nθ = 0) and D = eθ
1 ∂ , r ∂θ
1 D · n = Dα n α = ± . r
(97)
The polar components of (72)–(74), required in multiply connected domains, are given in Appendix A. 6.4 Axisymmetric Problems We consider plane-strain axisymmetric problems, in which the solution is independent of θ and uθ = 0,
τrθ = 0.
(98)
Equations (90)–(93) reduce now to 2 2 dτrr 2 1 df d 2 τrr + 1 + 2 , − τrr − 2 2 τθθ = 2 2 dr r dr r r r dr 2 2 dτθθ 2 d 2f d 2 τθθ −2 + 1 + 2 − τθθ − 2 2 τrr = 2 , 2 2 dr r dr r r dr
−2
ν
d 2 τrr 2 − ν dτθθ d 2 τθθ 1 + ν dτrr − = 0. − − ν) + (1 dr 2 dr 2 r dr r dr
(99) (100) (101)
On a circular boundary of radius r with outward unit normal n = ±er , the boundary conditions (88) and (89) become dτrr 1 df 2 dτθθ (102) + = Pr and 2 = Rr on ∂ B. ± r dr dr dr Equations (99)–(102) form a second-order system of linear ordinary differential equations which defines the Airy stress function f and the “double stress functions” τrr and τθθ . If the domain is multiply connected, conditions (72)–(74) should be imposed as well; the form of these conditions on circular boundaries for axisymmetric problems is given in Appendix A. In the following we present the general solution of the aforementioned system of equations. The form of (99) and (100) suggests that the problem can be simplified if we add and subtract them. In fact, if we define S = τrr + τθθ ,
D = τrr − τθθ ,
and
F=
so that τrr =
S +D , 2
τθθ =
S −D , 2
1 df , r dr
(103)
rF dr,
(104)
and
f=
addition and subtraction of (99) and (100) yield −2
1 d 2 d 2 S 2 dS +S = r F − dr 2 r dr r dr
(105)
58
N. Aravas
and 2
2 d 2 D 2 dD dF + 1 + 4 , − D = −r 2 2 dr r dr r dr
whereas (101) becomes (1 − 2ν)
d dr
r
dS dr
=
1 d r 2 dr
r3
dD . dr
(106)
(107)
Elimination of D and F from (105)–(107) leads to the following fourth-order ordinary differential equation for S 4 22 d 3 S 2 d 2 S 1 2 dS 2d S + − 1+ 2 − 1− 2 = 0, (108) dr 4 r dr 3 r dr 2 r r dr and rest of the solution is defined by
1 dD dS 2 d = 3 (1 − 2ν) r r dr + c5 , dr r dr dr 3 d S d 2 S 1 dS 1 dS dD 2 + D= 2 (1 − ν) r 3 + 2 − −r , 2 dr dr r dr dr dr S +D S −D , τθθ = , τrr = 2 2 2 dτrr 2 dτ 2 + τrr + 2 2 (τrr − τθθ ), F = −2 rr2 − 2 dr r dr r
f = rF dr.
(109) (110) (111) (112) (113)
where c5 is an arbitrary constant. The general solution of (108) is S(r) = c1 + c2 K0
r
+ c3 I0
r
r + c4 ln ,
(114)
where c1 , c2 , c3 , and c4 are arbitrary constants, and (In , Kn ) are modified Bessel functions of the first and second kind. The rest of the solution is determined by (109)–(113): r 1 c6 c2 r c1 − 2 + K0 + (1 − 2ν) K1 2 r 2 r c r c3 r 1 4 + I0 + (1 − 2ν) I1 + ln − , 2 2 2 r 1 c6 c2 r c1 + 2 + K0 − (1 − 2ν) K1 τθθ (r) = 2 r 2 r c r c3 r 1 4 + I0 − (1 − 2ν) I1 + ln + , 2 2 2 c6 r2 r c1 , f (r) = r 2 − ln r − c4 2 ln r + 2 1 − ln 4 2 4 τrr (r) =
where c6 = c5 + 4 (1 − 2ν) c2 2 .
(115)
(116) (117)
Plane-Strain Problems for a Class of Gradient Elasticity Models
59
The terms that involve c4 in the above solution are not consistent with the assumption of an axisymmetric solution;2 therefore we set c4 = 0. The corresponding displacement is ur (r) =
1 2G
r r c6 1 − 2ν c1 r + − (1 − 2ν) c2 K1 − c3 I1 . 2 2r
(119)
Finally, the boundary conditions (102) on a circular boundary with outward unit normal n = ±er become r r 1 c6 c1 − 2 − c2 νK1 − (1 − 2ν) K2 Pr (r) = ± 2 r r r c 2 r 6 + c3 νI1 + (1 − 2ν) I2 − , (120) r 2 r4 and r r Rr (r) = −c2 (1 − ν) K1 + (1 − 2ν) K2 r r 2 − (1 − 2ν) I2 + c6 3 . + c3 (1 − ν) I1 r
(121)
Assuming that the double-force per unit volume vanishes, i.e. = 0, and using (40) and (41), we conclude that the only non-zero in-plane components the double-stress μ , the true stress σ , and true couple stress μ¯¯ are as follows (see also Appendix A in Aravas and Giannakopoulos [6]) μrrr = 2
dτrr , dr
μrθθ = 2
dτθθ , dr
μθrθ = μθθr = 2
σrr = τrr −
1 dμrrr 2 4 − μrrr − μrθθ − μθrθ , dr r 3 3
σθθ = τθθ −
1 dμrθθ 2 dμθrθ μrθθ + 2μθrθ − − , 3 dr 3 dr r
τrr − τθθ , r
(122) (123) (124)
and 2 μ¯¯ θ3 = −μ¯¯ 3θ = (μrθθ − μθrθ ) . 3
(125)
2 The strains corresponding to the c -terms are 4
c εrr = − 4 [1 − 2 (1 − 2ν) ln r] 4G
c and εθ θ = 4 [1 + 2 (1 − 2ν) ln r] . 4G
(118)
In axisymmetric problems with uθ = 0, the radial and hoop strains are εrr = dur /dr and εθ θ = ur /r. Therefore, if we evaluate ur = rεθ θ and substitute it into εrr = dur /dr, we conclude that c4 = 0. The c4 -terms are known to correspond to uθ displacements that are not single-valued and cannot be used in multiply connected domains (such as an annulus) expect in those special problems involving “dislocations” (see following Sect. 7.1 and Soutas-Little [41], p. 159).
60
N. Aravas
Fig. 2 Annulus loaded by internal and external pressure
7 Examples 7.1 An Annulus Subjected to Internal and External Pressure We consider the problem of an annulus loaded by an internal pressure pi and an external pressure po under conditions of plane strain (Fig. 2). The domain is doubly connected; therefore, the conditions (72)–(74) should be imposed on the inner and outer circular boundaries of the annulus. These conditions together with the strain compatibility equation (101) guarantee that the corresponding displacement field is single valued in the doubly connected domain. As discussed in Appendix A, conditions (72) and (74) are satisfied identically, whereas (73) requires that the following condition be satisfied dτθθ τrr − τθθ dτrr + (1 − ν) − = 0. (126) −ν dr dr r r=a Using (115) and (116) for τrr and τθθ in the above equation, we find c4
1−ν = 0, a
(127)
i.e., it is required that c4 = 0 for the displacement field to be single valued. This finding is consistent with the remark made immediately after (116) concerning the vanishing of c4 . In the case of the classical “local” linear isotropic elasticity ( = 0), the solution is of the form 1 f (0) = Ar 2 + B ln r, 2
σrr(0) = τrr(0) = A +
and u(0) r
B , r2
(0) (0) σθθ = τθθ =A−
1 B = , (1 − 2ν) Ar − 2G r
B , r2
(128)
(129)
where A=
pi a 2 − po b 2 , b2 − a 2
B = (po − pi )
a 2 b2 , b2 − a 2
and (pi , po ) are the inner and outer applied pressure loads as shown in Fig. 2.
(130)
Plane-Strain Problems for a Class of Gradient Elasticity Models
61
In order to make connection with the aforementioned “local” solution, we set c1 = c7 + 2A,
c6 = c8 − 2B,
(131)
so that the gradient elasticity solution is written in the form c7 2 c8 r − ln r, 4 2 r 1 c8 c2 r τrr (r) = τrr(0) (r) + c7 − 2 + K0 + (1 − 2ν) K1 2 r 2 r c3 r + I0 + (1 − 2ν) I1 , 2 r 1 c8 c2 r (0) τθθ (r) = τθθ c7 + 2 + K0 − (1 − 2ν) K1 (r) + 2 r 2 r c3 r + I0 − (1 − 2ν) I1 , 2 f (r) = f (0) (r) +
(132)
(133)
(134)
and 1 1 − 2ν c8 c7 − 2 2G 2 2r r r r r + K1 + c3 I0 − I1 , + (1 − 2ν) c2 K0 r r 1 1 − 2ν c8 (0) εθθ (r) = εθθ c7 + 2 (r) + 2G 2 2r r r − c3 I1 , − (1 − 2ν) c2 K1 r 1 1 − 2ν c8 ur (r) = rεθθ = u(0) c7 r + (r) + r 2G 2 2r r r − c3 I1 . − (1 − 2ν) c2 K1 (0) (r) + εrr (r) = εrr
(135)
(136)
(137)
The only non-zero in-plane components of the true stress σ and the true couple-stress μ¯¯ are r r 1 c8 2ν c7 − 2 − c2 K1 − c3 I1 , 2 r 3 r r 1 c8 2ν r (0) σθθ (r) = σθθ c7 + 2 + + K1 (r) + c2 K0 2 r 3 r r r + c3 I0 − I1 , r σrr (r) = σrr(0) (r) +
(138)
(139)
and r r 2ν μ¯¯ 3θ (r) = −μ¯¯ θ3 (r) = c2 K1 − c3 I1 . 3
(140)
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N. Aravas
Fig. 3 Spatial variation of normalized displacement (|ur |/a)/(pi /G) for ν = 0.3, /a = 0.2, b = 2a, and po = 2pi
The terms that involve c2 , c3 , c7 , and c8 define now the “gradient correction”, where the constants c2 , c3 , c7 , and c8 are determined from the following boundary conditions: on r = a
(n = −er ) :
on r = b
(n = er ) :
Pr (a) = pi ,
Rr (a) = 0,
(141)
Pr (b) = −po ,
Rr (b) = 0.
(142)
The boundary conditions (141) and (142) define a system of four algebraic linear equations that is solved for c2 , c3 , c7 and c8 . The resulting expressions for c2 , c3 , c7 and c8 are lengthy and are listed in Appendix B. We note that the solution is of the form b r , ν, , f (r) = f (0) (r) + (po − pi ) Q , (143) a a a b r (0) , ν, , τrr (r) = τrr (r) + (po − pi )Trr , (144) a a a b r (0) , ν, , (r) + (po − pi )Tθθ τθθ (r) = τθθ , (145) a a a p o − pi r b (0) U , ν, , , (146) ur (r) = ur (r) + G/a a a a where (Q, Trr , Tθ,θ , U ) are dimensionless functions defined in Appendix B. It is interesting to note that, whereas the classical solution depends on the individual values of pi and po though A and B, the “gradient correction” is proportional to the difference po − pi (see (143)–(146)). An explanation of this is given at the end of Sect. 4, where is noted that, if R = 0, the “gradient correction” is proportional to the magnitude of μ (0) . In the present problem R = 0, and (128) and (130) show that μ (0) = 2 ∇ττ (0) ∼ B ∼ p0 − pi .
(147)
Therefore, the “gradient correction” is proportional to p0 − pi , i.e.,
f (1) , u(1) ,ττ (1) ∼ p0 − pi .
(148)
Figure 3 shows the radial variation of the normalized radial displacement ur . In Fig. 3 and in all following figures of this section ν = 0.3, /a = 0.3, b/a = 2, and po /pi = 2.
Plane-Strain Problems for a Class of Gradient Elasticity Models
63
Fig. 4 Spatial variation of normalized radial and hoop stresses for ν = 0.3, /a = 0.2, b = 2a, and po = 2pi . All stress components shown in the figure are compressive
Fig. 5 Spatial variation of normalized strains εrr /(pi /G) and |εθ θ |/(pi /G) for ν = 0.3, /a = 0.2, b = 2a, and po = 2pi
gr
As shown in Fig. 3 |ur (r)| < |uclr (r)| ∀r, where the superscripts “gr” and “cl” denote the gradient and classical solution respectively; i.e., the gradient elastic material appears to be gr stiffer. Also, |ur (r)| varies monotonically with r, whereas the classical solution is such that cl |ur (r)| has a minimum at r 1.2a. Figure 4 shows the spatial variation of the normalized radial and hoop stress components of τ , the true stress σ , and the classical solution σ (0) . All stress components shown in Fig. 4 are compressive. The gradient solution predicts a higher compressive radial stress and a lower compressive hoop stress. It is interesting to note that τrr is substantially different from the true stress σrr , whereas τθθ σθθ . Figure 5 shows the spatial variation of the normalized strain components εrr /(pi /G) and |εθθ |/(pi /G) for both the classical and the gradient solution. The radial strain εrr in the gradient solution remains compressive everywhere in the annulus, whereas both tensile and compressive radial strains appear in the classical solution, reflecting the fact that |uclr (r)| has a minimum in the range a ≤ r ≤ b (see Fig. 3). We consider next the case of a thin-walled annulus. Let t be the thickness and R the mean radius of the annulus, i.e., t = b − a,
R=
a+b , 2
t t so that a = R − , b = R + , 2 2
(149)
64
N. Aravas
where t ≡ 1. R
(150)
If we set a = R − t/2, b = R + t/2, evaluate (133), (134), (139), and (140) at r = R, and expand the solution in = t/R, we find that τrr (R) = − (po − pi ) τθθ (R) = − (po − pi )
3−2ν 2 ( ) 1−ν R 1 + 2 3−2ν ( )2 1−ν R
R p o + pi − + O (p) , t 2
1+
3−2ν 2 ( ) 1−ν R 3−2ν 1 + 2 1 − ν ( R )2
R p o + pi − + O (p) , t 2
(151)
(152)
and σrr (R) = − (po − pi )
σθθ (R) = − (po − pi )
3−2ν 2 ( ) 3(1−ν) R 3−2ν 2 1 + 2 1−ν ( R )
R p o + pi − + O (p) , t 2
1+
(3−2ν)(4−5ν) 2 (R) 3(1−ν)2 3−2ν 2 1 + 2 1−ν ( R )
R p o + pi − + O (p) , t 2
(153)
(154)
where p is a typical pressure of order po or pi . The corresponding thin-wall solution of the classical theory ( = 0) is σrr(0) (R) = τrr(0) = −
p o + pi + O (p) , 2
(0) (0) σθθ (R) = τθθ = − (po − pi )
(155)
R p o + pi − + O (p) . t 2
(156)
It is interesting to note that the gradient theory compared to the classical local theory predicts higher values for the radial stress and lower values for the hoop stress. In particular, whereas the classical theory predicts a radial stress of order p, the gradient theory predicts a radial stress of order p ( R )2 . 7.2 Infinite Body with Cylindrical Hole We consider the problem of a infinite body with a cylindrical hole of radius a (Fig. 6). The hole is loaded by an internal pressure pi and a pressure po is applied at infinity (Fig. 6). The solution to this problem can be found from the solution developed in Sect. 7.1 by considering the limit b → ∞. In the limit as b → ∞, the constants that enter the solution take the values A = −po ,
B = (po − pi )a 2 ,
and c2 = −
p o − pi , c
c8 = 2
c3 = c7 = 0,
a po − pi aK1 , c
(157)
(158)
Plane-Strain Problems for a Class of Gradient Elasticity Models
65
Fig. 6 Infinite body with circular hole
with c=
a 1 − ν 1 − 2ν K0 + 2 2
a a +4 K1 a
(159)
and the solution becomes a2 τrr (r) = −po + (po − pi ) 2 r r r po − pi a a + (1 − ν) K0 + (1 − 2ν) K1 , (160) − K1 c r2 r a2 τθθ (r) = −po − (po − pi ) 2 r r r po − pi a a − νK + − 2ν) K , + K (1 1 0 1 c r2 r a p o − pi 1 aK1 ln r. f (r) = − po r 2 + (po − pi )a 2 ln r − 2 c
(161) (162)
The corresponding displacement field is ur (r) = − (1 − 2ν)
r po − p i a 2 p o − p i a a po r− + K1 + (1 − 2ν) K1 . 2G 2G r 2Gc r (163)
This solution was communicated to the author by Prof. Exadaktylos [15] in 2001.
66
N. Aravas
The corresponding non-zero in-plane true stresses and true couple-stresses are a r 2ν r − K , (164) K 1 1 r2 3 r r a 2 p0 − pi a a 2ν r − + K1 , (165) = −p0 − (p0 − pi ) 2 + K1 K0 r c r2 3 r
σrr = −p0 + (p0 − pi ) σθθ
a 2 p0 − pi − r2 c
and r 2ν p0 − pi K1 . μ¯¯ 3θ = −μ¯¯ θ3 = − 3 c
(166)
Acknowledgements Fruitful discussions with Prof. A.E. Giannakopoulos of the University of Thessaly are gratefully acknowledged. The author would like also to thank Prof. D. Panagiotounakos of the National Technical University of Athens for helpful discussions and material.
Appendix A: Plane Strain Compatibility Equations in Polar Coordinates for Multiply-Connected Regions In multiply connected domains the additional compatibility conditions are Su =
[εε + x × (∇ × ε )] · dx = 0,
(167)
(∇ × ε ) · dx = 0,
(168)
(dx · κ ) = 0.
(169)
Cn
Sω = Cn
Pε = Cn
The polar coordinates of the quantities that appear in gradient elasticity theories can be found in Appendix A of Aravas and Giannakopoulos [6]. Here, we present the form of the above compatibility equations in polar coordinates. We consider the plane-strain problem of Sect. 6 and introduce polar coordinates (r, θ ). The position vector x, its differential dx, and the gradient operator on the plane are x = rer ,
dx = dr er + r dθ eθ ,
∇=
∂ 1 ∂ er + eθ , r r ∂θ
(170)
where (er , eθ ) are the unit vectors of the polar coordinate system. The strain tensor ε and the strain gradient tensor κ can be written as ε = εrr er er + εθθ eθ eθ + εrθ (er eθ + eθ er ) ,
(171)
κ = κrrr er er er + κrθθ er eθ eθ + κrrθ er (er eθ + eθ er ) + κθrr eθ er er + κθθθ eθ eθ eθ + κθrθ eθ (er eθ + eθ er ) .
(172)
Plane-Strain Problems for a Class of Gradient Elasticity Models
67
We can evaluate the quantities that appear in (167)–(169) as follows: [εε + x × (∇ × ε )] · dx ∂εrr ∂εrθ − + εrθ eθ dr = εrr er − r ∂r ∂θ ∂εrθ ∂εθθ − + 2εθθ − εrr eθ dθ, + r εrθ er − r ∂r ∂θ
(173)
(∇ × ε ) · dx 1 ∂εrr 2εrθ ∂εrθ ∂εrθ ∂εθθ − + − + εθθ − εrr dθ e3 , (174) = dr + r ∂r r ∂θ r ∂r ∂θ dx · κ = [κrrr er er + κrθθ eθ eθ + κrrθ (er eθ + eθ er )] dr + [κθrr er er + κθθθ eθ eθ + κθrθ (er eθ + eθ er )] r dθ.
(175)
In the above equations the unit vectors (er , eθ ) are functions of θ : er (θ ) = cos θ e1 + sin θ e2 ,
eθ (θ ) = − sin θ e1 + cos θ e2 ,
(176)
where (e1 , e2 ) are the base vectors of a fixed Cartesian coordinate system in the plane where the polar system is defined. On a circular contour of radius r, dr = 0 and the conditions (167)–(169) can be written as follows:
2π S =r u
∂εrθ ∂εθθ − + 2εθθ − εrr eθ dθ = 0, εrθ er − r ∂r ∂θ
(177)
0
2π S =r ω
1 ∂εrθ εrr − εθθ ∂εθθ − − ∂r r ∂θ r
dθ e3 = 0,
(178)
0
2π Pε = r
[κθrr er er + κθθθ eθ eθ + κθrθ (er eθ + eθ er )] dθ = 0,
(179)
0
where er (θ ) and eθ (θ ) are defined by (176). All the above equations can be written in terms of the components of τ if we use the constitutive equations εαβ =
1
ταβ − ντγ γ δαβ , 2G
καβγ =
1 (∇ττ )αβγ − ν (∇τδδ )α δβγ , 2 2G
(180)
where the polar components of ∇ττ are defined by (96) and ∇τδδ =
∂(τrr + τθθ ) 1 ∂(τrr + τθθ ) er + eθ . ∂r r ∂θ
(181)
In axisymmetric problems we have that εrθ = 0,
κrrθ = κrθr = κθrr = κθθθ = 0,
∂ = 0. ∂θ
(182)
68
N. Aravas
In that case, if we take into account (176) and that the axisymmetric solution is independent of θ , we conclude that (177) and (179) are satisfied identically, i.e., Su = Pε = 0, whereas (178) takes the form dτrr dτθθ τrr − τθθ 2π Sω = r −ν + (1 − ν) − e3 = 0. G dr dr r
(183)
(184)
Appendix B: Constants in the Solution of the Annulus Problem The constants c2 , c3 , c7 , and c8 that enter (132)–(140) of the solution of the annulus problem take the values 1 − ν a3 a b b c2 a2 a − I − I = I I − , (185) 1 2 2(po −pi ) a 3 1 2 2 1 − 2ν b b b b2 1 − ν a3 a b b c3 a2 a − K1 − K2 = K K + , (186) 2(po −pi ) a 3 1 2 2 1 − 2ν b b b b2 a a a 2 2 a a b b c7 + I K = 1 + − I K 2 1 2 1 2 2 4(po −pi ) a 2 b b b b b 2 2
b(b −a )
a a a b a2 − I1 K K 1 2 2 2 b b 2 a b b 1−ν a , + 1 − 2 K1 + K2 I1 1 − 2ν b b a3 a a 2 a b b + 3 I2 K1 =2 3 − I2 K1 b b b
−
c8 4(po −pi ) a 2 2 b2 −a 2
1−ν 1 − 2ν
1−
1−ν + 1− 1 − 2ν 1−ν − 1− 1 − 2ν
a3 a a b a4 − 4 K2 I1 K1 b4 b a b b a4 , K1 + K2 I1 4 b b
where a a2 2 b a4 a K1 + − ν) − 1 − ν + 2 I2 (1 5 2 2 2 b b b b 2 2 2 a a a K1 − 2 2 + (1 − ν) 2 1 − 2 b b b b a a a2 b + (1 − 2ν) 2 1 − 2 K2 I2 b b 2 2 2 a a 2 1−ν a 2 2 + 2 1−ν +2 2 K1 + 1− 2 1 − 2ν b b b b
=4
(187)
(188)
Plane-Strain Problems for a Class of Gradient Elasticity Models
69
a a a2 2 b I1 1 − ν − 1 − ν + 2 K2 b2 b2 b2 2 2 a a b a K2 + (1 − 2ν) 3 1 − 2 I2 b b 2 2 2 a 2 b 1−ν a 2 2 + 2 1−ν +2 2 K1 − 1− 2 1 − 2ν b b b b 2 a a2 a2 b . K2 + 2 2 + (1 − ν) 2 1 − 2 I1 b b b b +
(189)
In the limiting case of an incompressible material (ν → 1/2), the constants take the following values 4 3
c2 → (po − pi )
8 b [ ab3 I1 ( b ) − I1 ( a )] 2
2
2
( ab2 − 1)[1 + 4 b2 ( ab2 + 1)][I1 ( a )K1 ( b ) − I1 ( b )K1 ( a )]
,
(190)
,
(191)
3
c3 → (po − pi )
4 b [ ab3 K1 ( b ) − K1 ( a )] 2
2
2
( ab2 − 1)[1 + 4 b2 ( ab2 + 1)][I1 ( a )K1 ( b ) − I1 ( b )K1 ( a )] 2 2
c7 → (po − pi )
8 a 4b 2
2
2
2
c8 → (po − pi )
2
( ab2 − 1)[ ab2 + 4 a2 ( ab2 + 1)] 2
(192)
.
(193)
2
8 2 ab2 ( ab2 + 1) 2
,
2
2
( ab2 − 1)[ ab2 + 4 a2 ( ab2 + 1)]
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