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Annular inhomogeneities with eigenstrain and interphase modeling
Journal of the Mechanics and Physics of Solids 64 (2014) 468–482
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Annular inhomogeneities with eigenstrain and interphase modeling Xanthippi Markenscoff a,n, John Dundurs b a b
Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, United States Northwestern University, Evanston, IL 60201, United States
a r t i c l e in f o
abstract
Article history: Received 12 August 2013 Received in revised form 21 November 2013 Accepted 4 December 2013 Available online 18 December 2013
Two and three-dimensional analytical solutions for an inhomogeneity annulus/ring (of arbitrary thickness) with eigenstrain are presented. The stresses in the core may become tensile (for dilatational eigenstrain in the annulus) depending on the relative shear moduli. For shear eigenstrain, an “interface rotation” and rotation jumps at the interphase also occur, consistent with the Frank–Bilby interface model. A Taylor series expansion for small thickness of the annulus is obtained to the second-order as to model thin interphases, with the limit agreeing with the Gurtin–Murdoch surface membrane, but also accounting for curvature effects.. The Eshelby “driving forces” on a boundary with eigenstrain are calculated, and for small, but finite, interphase thicknesses they account for the interaction of the two interfaces of the layer, and the next order term may induce instabilities, for some bimaterial combinations, if it becomes large enough to render the driving force zero. It is also proven that for 2-D inhomogeneities with eigenstrain the stresses have reduced material dependence for any geometry of the inhomogeneity. The case when the outer boundary of the inhomogeneity annulus with eigenstrain is a free surface is also analyzed and agrees with classical surface tension results in the limit, but, moreover, the thick free surface terms (next order in the expansion depending on the radius) are also obtained and may induce instabilities depending on the bimaterial combinations. Applications of inhomogeneity annuluses with eigenstrain are wide and include interphases in thermal barrier coatings and coated particles in electrically/ thermally conductive adhesives. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Inhomogeneity Eigenstrain Interphase Surface tension Driving force
1. Introduction The Eshelby (1957) ellipsoidal inclusion solution and the Eshelby equivalent inclusion method for inhomogeneities have had widespread applications to composite materials over the last 50 years. Here we present solutions and properties of inhomogeneities with transformation strain, or eigenstrain (a term introduced by Dundurs, 1967), when the inhomogeneity is an annulus embedded in a matrix, which cannot be solved by the Eshelby (1957, 1961) equivalent inclusion method through an algebraic system of linear equations. The inhomogeneity annulus cannot be solved by superposition of Eshelby inclusions, since the Eshelby equivalent inclusion method would require to have a position dependent loading (of the outer field of the inner inclusion) in the equivalent eigenstrain system of equations, rather than the constant coefficients that are due to the Eshelby property for the ellipsoid. For an updated treatise on inclusions and inhomogeneities with eigenstrain
n
Corresponding author. Tel.: þ1 858 534 1337; fax: þ1 858 534 5698. E-mail addresses: [email protected], [email protected] (X. Markenscoff), [email protected] (J. Dundurs).
0022-5096/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmps.2013.12.003
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
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and applications we refer to Qu and Cherkaoui (2006). Attention is drawn to a related very recent paper on spherical multiinhomogeneous inclusions by Shodja and Khorshidi (2013), with which we strongly disagree. To quote them, at the end of Section 5, “ Remark 4: Therefore, interestingly, the uniform eigenstrain fields … give rise to a center of dilatation at r ¼ 0. Nonuniform eigenstrain fields produce stress and strain fields … with logarithmic singularities at r ¼ 0 as well.” By the very principles of analytic function theory, finite eigenstrain cannot induce stresses and strain fields with singularities, except at points of geometric discontinuities, such as corners of inclusions (Rodin, 1996), or corners at strips of eigenstrain meeting a free surface (Dundurs and Markenscoff, 2009). These singularities have been routinely excluded a priori in all the literature (e.g. Duan et al., 2005 for the coated inclusion problem, Duan et al., 2008; Hashin, 2002). The current paper contains the correct solution without singularities. The applications of annular inhomogeneities with eigenstrain range from modeling interphases at grain boundaries (e.g. Hirth et al., 2012) to thermal barrier coatings (Pollock et al., 2012), and to coated particles in thermally/electrically conductive adhesives (e.g. Morris and Liu, 2007). The literature is vast on coated inhomogeneities and interphase layers between the inclusions and the matrix, or surface layers (but no eigenstrain loading in the inhomogeneous annulus has been previously considered), and we will selectively reference Steigmann and Ogden (1997, 1999), Sharma and Ganti (2004), Duan et al. (2005, 2008), Hashin (2002), Dingerville et al. (2005), Dingreville and Qu (2008), Benveniste (2013), and Mohammadi et al. (2013), while a more comprehensive list of references on the material aspects of interphases is provided in the long article of Hirth et al. (2012). Interesting features of the analytical solutions presented here are, that, for dilatational eigenstrain in the inhomogeneous ring/annulus, the stress in the core actually depends on the difference (μ1 μ2 ) of the shear moduli of the matrix and the annulus, both in 3-D (called “annulus”) and in 2-D (“ring”), and, thus, the inhomogeneous annulus induces hydrostatic tension inside the core when the annulus is stiffer in shear than the matrix. Thus, if the interphase thickness is appreciable with respect to the core radius, the tensile stress in the core induced by the interphase may be significant and have implications for nanomaterials. This inhomogeneity solution elucidates further the Eshelby inclusion result for dilatational eigenstrain that the value of the bulk modulus of the matrix is “irrelevant” (Eshelby, 1961) (and only the shear modulus matters). The annular inhomogeneities with eigenstrain may be useful to model interphases of finite thickness. For small inhomogeneity thicknesses h, the stresses and rotation inside the interphase are also obtained by a Taylor series expansion in the thickness parameter, and, as the thickness h goes to zero, the stresses are found to depend only on the elastic constants of the annulus/ring, which is consistent with the membrane interface (or free surface) models (such as Gurtin and Murdoch, 1975). However, the presence of shear eigenstrain in a layer of vanishingly small thickness also produces a nonzero “interface rotation”, not obtainable otherwise. The interface rotation is not a feature of other surface stress models (such as the one of Gurtin and Murdoch, 1975), but has been experimentally observed, and is a feature of the Frank–Bilby interface model in the recent seminal paper of (Hirth et al., 2012). The eigenstrain in the annulus gives rise to “driving forces” (Eshelby forces) on the interfaces between the matrix and the interphase, which are computed according to f ¼ o sij 4½½εnij (Eshelby, 1977), and same in dynamics (Markenscoff and Ni, 2010), on each interface of the finite interphase layer thickness, accounting also for the interaction with the other interface of the layer. As the thickness tends to zero, the hoop stresses and the driving force depend on the layer material constants only, but, for small finite thickness h the driving force has the next order h=a positive term which is proportional to the (μ1 μ2 ) difference of the matrix/interphase shear moduli and the bimaterial Dundurs constants α, β. This term may induce instabilities of the interface if it becomes large enough to be of the order of the leading term due to hoop tension so that the driving force becomes zero. We also present solutions for a thin spherical inhomogeneous annulus with eigenstrain, where, again, the hoop stresses depend only on the material constants of the annulus. The leading term of the driving force at the interface is found to be equal in value to the driving force for a spherical inclusion of material 2 (in a matrix of material 2) with dilatational eigenstrain in the core, as derived by Gavazza (1977), Eshelby (1977), and Markenscoff and Ni (2010). This should be expected physically, since the thin annulus stress field “knows” only the material of the annulus and the eigenstrain in it (there is no other characteristic length), and the Gavazza–Eshelby’s result is valid for inclusions of any shape (Eshelby, 1957, 1961). Moreover, the presence of eigenstrain gives rise to a driving force, which controls the stability of the interface. The obtained finite interphase solution for the driving force also accounts for the interaction of the driving force on one interface of the layer with the other interface. In addition, the “excess energy” (Dingreville and Qu, 2008) inside the interphase, here associated with the eigenstrain, can be calculated according to Eshelby (1961). We also present here the solutions for a finite radius ring/annulus of eigenstrain enclosing a matrix and being free of traction on the outside. The generated stresses are hoop stresses of “surface tension”, which, for vanishingly small thickness, again tend to a constant depending only on the elastic moduli of the thin shell. The driving force on both the free surface and the interface to the leading order depends on the surface moduli and the eigenstrain, but the next order h=a (where a is the core radius) term may become large (for some bimaterial combination differences) and be of the order of the surface tension term, in which case the free surface and matrix/free surface interfaces may become unstable. In the obtained solutions for 2-D ring inhomogeneities with eigenstrain, the stresses are expressed in terms of the Dundurs bimaterial constants (α β) with a coefficient En , as common multiplier in all the fields. It is shown here that, by applying the CLM theorem (Cherkaev et al., 1992), this coefficient En has a reduced dependence on the elastic constants. In addition, we show that the reduced material constants dependence of the stresses is valid for arbitrary shapes of
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X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
Fig. 1. Annulus/ring inhomogeneity with eigenstrain embedded in a matrix.
inhomogeneities with eigenstrain. This result had not been previously obtained in the literature, as no step function discontinuities of the eigenstrain on the boundaries between the inclusion and the matrix were considered in the literature (in Jasiuk and Boccara, 2002). 2. Embedded cylindrical ring inhomogeneity with eigenstrain 2.1. Dilatational eigenstrain (εnij ¼ δij e) in the ring In an infinite matrix of elastic constants μ1 , κ1 (where κ is the Kolosov constant) ( 3 4v plain strain κ¼ 3 v plain stress we consider (Fig. 1) an embedded ring of an inhomogeneity of elastic constants μ2 , κ 2 in the region a or o b subjected to voluminal eigenstrain εnij ¼ δij e. The problem is reduced to finding the pressure p and q, between the core and the inhomogeneity at r ¼ a, and the inhomogeneity and the matrix at r ¼ b. The stress function for the regions 1 (core), 2 (ring), and 3 (matrix) are given in Appendix A. The unknown p and q are solved from the displacement boundary where the superscript ðiÞ in uðiÞ r refers to the regions i ¼ 1; 2; 3; of core, annulus and matrix respectively. Displacement boundary conditions: ð2;pÞ on r ¼ a; uð1Þ þuð2;qÞ r ¼ ea þ ur r
ð1aÞ
þ uð2;qÞ ¼ urð3Þ on r ¼ b; eb þuð2;pÞ r 2
ð1bÞ
The solution of the system of Eqs. (1a) and (1b) in which the displacements are being expressed in terms of p and q, according to Appendix A, gives 2
p ¼ 2En ðα βÞðb a2 Þ=Δ1
ð2Þ
2
q ¼ En ð1 αþ 2βÞðb a2 Þ=Δ1
ð3Þ
where En ¼
8eμ1 μ2 μ2 ðκ1 þ 1Þ þ μ1 ðκ2 þ 1Þ
and
2
Δ1 ¼ ð1 þ α 2βÞð1 α þ 2βÞb 4βðα βÞa2 :
ð4Þ
The solution (2) reveals that the srr and sθθ stresses in the core change sign with ðα βÞ, which is equivalent to (μ1 μ2 ), and become tensile for stiffer shear modulus of the ring (with dilatational eigenstrain). This implies that interface eigenstrains (as in thermal barrier coatings) may create damage in the core due to the tensile stresses that they induce there, as well as in nanomaterials if the interphase layer (modeled as a ring of eigenstrain) is of considerable thickness in comparison to the core. 2.2. Shear eigenstrain in the ring The ring contains shear eigenstrain (εnxx ¼ εnyy ¼ e, εnxy ¼ 0) with corresponding “eigen-displacements” unx ¼ ex ¼ er cos θ;
uny ¼ ey ¼ er sin θ;
ð5Þ
and in polar coordinates unr ¼ er cos 2θ;
unθ ¼ er sin 2θ:
ð6Þ
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471
In this case the stress functions for regions 1, 2, 3, respectively are U ð1Þ ¼ Ar 2 cos 2θ þBr 4 cos 2θ
ð7Þ
U ð2Þ ¼ Cr 2 cos 2θ þDr 4 cos 2θ þE U ð3Þ ¼ G
cos 2θ þF cos 2θ r2
ð8Þ
2
b cos 2θ þH cos 2θ r2
ð9Þ
where the coefficients are found from the boundary condition of continuity of tractions at the interfaces and continuity of displacements ð2Þ n At r ¼ a; uð1Þ r ¼ ur þur
At r ¼ b;
and
urð2Þ þunr ¼ urð3Þ
ð1Þ ð2Þ ð1Þ uθð1Þ ¼ uθð2Þ þunθ ; srr ¼ sð2Þ rr ; srθ ¼ srθ
ð10Þ
ð3Þ ð2Þ ð3Þ n ð2Þ ð3Þ uð2Þ θ þuθ ¼ uθ ; srr ¼ srr ; srθ ¼ srθ
and
ð11Þ
The solution to the algebraic linear system of 8 8 equations is (obtained by Mathematica) 2
4
6
2
A ¼ En ða2 b Þð3a2 b ðα βÞð1 þ βÞ2 þ a6 ðα βÞðα þ βÞ2 þ b ðα þβÞð 1 þ β2 Þ 3a4 b ðα3 α2 β þαβð2 þ βÞ β2 ð2 þ βÞÞÞ=Δ2 n 2 2
2
2
2
B ¼ 2E a b ða b Þð 1 þ α Þðα βÞ=Δ2 8 4 C ¼ En b ð 1 þ βÞ2 ð1 þ βÞ þ a8 ðα βÞðα þβÞ2 3a4 b ð1 þβÞð α þα2 þ β 3αβ þ2β2 Þ 6 2 þ 2a2 b ð1 þ βÞð αð2 þ βÞ þ βð1 þ 2βÞÞ þ 2a6 b ð 3αβð1 þβÞ þα2 ð2 þβÞ þβ2 ð1 þ 2βÞÞ =Δ2 2
2
D ¼ 2En a2 b ða2 b Þð1 þ αÞðα βÞð1 þβÞ=Δ2 n 4 4
4
E ¼ E a b ð1 þ αÞða4 ðα2 β2 Þ þ b ð 1 þβ2 ÞÞ=Δ2 2
6
F ¼ 2En a2 b ð1 þαÞða6 ðα2 β2 Þ þb ð 1 þ β2 ÞÞ=Δ2 n 4
2
4 2
6
4
G ¼ E b ða2 b Þð3a b ðα βÞ2 ð1 þ βÞ þ b ð 1 þ βÞ2 ð1 þ βÞ þ a6 ð 1 þβÞðα2 β2 Þ a2 b ð1 þ βÞð 1 þ4α2 6αβ þ 3β2 ÞÞ=Δ2 2
2
2
4
6
H ¼ 2En b ða2 b Þð3a4 b ðα βÞ2 ð1 þ βÞ 3a2 b ðα βÞ2 ð1 þβÞ þb ð 1 þ βÞ2 ð1 þβÞ þa6 ð 1 þ βÞðα2 β2 ÞÞ=Δ2
ð12Þ
where En
8eμ1 μ2 8em μ2 ðκ1 þ 1Þ þμ1 ðκ 2 þ1Þ
ð13Þ
and 4
8
2
Δ2 ¼ 4ð6a4 b ðα βÞ2 ð1 þ βÞ2 þ a8 ðα2 β2 Þ2 þ b ð 1 þβ2 Þ2 2a6 b ð1 þ βÞð 3αβð1 þ βÞ þ α2 ð2 þ βÞ þ β2 ð1 þ 2βÞÞ 2 6
2a b ð1 þ βÞð 3αβð1 þ βÞ þ α2 ð2 þ βÞ þ β2 ð1 þ 2βÞÞ
ð14Þ
The stresses are in Appendix B. The stresses in the core are not constant, as in the dialatational eigenstrain case, but depend on r and θ. It may be noted that only certain components of them change sign with the shear moduli difference of the matrix/ring, unlike the dilatational eigenstrain case where all components do. 3. Material rotation due to shear eigenstrain in the ring In the case of shear eigenstrain in the ring/annulus there is material rotation in the core, ring and outer matrix that is calculated here. Inside the core the rotation is ωz ¼
3 Bðκ1 þ 1Þr 2 sin 2θ 2μ1
Inside the ring the rotation is 1 ðκ2 þ 1Þ 3Dðκ2 þ 1Þr 2 sin 2θ þ F sin 2θ ωz ¼ 2μ2 r2
ð15Þ
ð16Þ
and, in the limit of vanishingly small thickness h-0, the rotation in the layer is ωz ¼
a2 ε sin 2θðμ1 μ2 Þ 4a2 ε2 sin 2θ ¼ 2 r ðα βÞμ1 μ2 En r2
ð17Þ
The rotation in the outer matrix is ωz ¼
1 ðκ 1 þ1Þ H sin 2θ 2μ1 r2
ð18Þ
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We next compute the jumps (denoted by brackets) of the rotations at the interfaces At r ¼ a, 2
2
6
½½ωZ ¼ b ð1 α2 Þð 2a6 ðα 2βÞðα βÞ þ3a4 b ðα βÞ2 b ð1 β2 ÞÞe sin 2θ=Δ3
ð19Þ
For small interphase thickness b ¼ a þ h, the Taylor series expansion gives ½½ωZ ¼ e sin 2θ
4eð2α2 3αβ þβ2 Þ sin 2θðh=aÞ þ O½h=a2 1 α2
ð20Þ
The jump at r ¼ b is 4
6
8
½½ωZ ¼ ð1 þ αÞð3a4 b ð 1 þ 3α 2βÞðα βÞð1 þ βÞ 2a2 b ð 1 þ 3α 2βÞðα βÞð1 þβÞ þb ð1 βÞ2 ð1 þ βÞ 2
þa8 ðα βÞ2 ðα þβÞ 2a6 b ðα βÞðαð2 þ βÞ βð1 þ 2βÞÞÞe sin 2θ=Δ3
ð21Þ
with 4
8
2
Δ3 ¼ 6a4 b ðα βÞ2 ð1 þ βÞ2 þ a8 ðα2 β2 Þ2 þ b ð1 β2 Þ2 2a6 b ðα βÞð1 þ βÞðαð2 þ βÞ βð1 þ 2βÞÞ 6
2a2 b ðα βÞð1 þ βÞðαð2 þ βÞ βð1 þ 2βÞÞ
ð22Þ
For small interphase thickness b ¼ a þ h, the jump of the rotation is ½½ωZ ¼ e sin 2θ
4eðα βÞð2 þ βÞ sin 2θðh=aÞ þ O½h=a2 1 α2
ð23Þ
Thus, for small thickness, to the leading order, the jumps of the rotations at r ¼ a and r ¼ b are constant depending on the eigenstrain only, as expected. 4. Embedded spherical annulus inhomogeneity with dilatational eigenstrain We also solve the three-dimensional problem of a spherical inhomogeneity annulus with dilatational eigenstrain. The solution follows Sokolnikof (1956) with the displacement being of the form ui ¼ φðrÞxi
and
φðrÞ ¼ A1 þ
A2 : r3
ð24Þ
The stresses and displacements are shown in Appendix C. The pressures p and q at r ¼ a and r ¼ b are determined from the equations for the displacements at r ¼ a and r ¼ bat r ¼ a, urð1Þ ¼ ea þ uð2;p;qÞ r
3
3
or
p pa3 qb b ðp qÞ þ ¼ eþ 3 3 3λ1 þ2μ1 ð3λ2 þ2μ2 Þðb a3 Þ 4μ2 ðb a3 Þ
or
eþ
ð25Þ
and at r ¼ b, ¼ urð3Þ eb þuð2;p;qÞ r
pa3 qb
3
3
ð3λ2 þ 2μ2 Þðb a3 Þ
þ
a3 ðp qÞ 3
4μ2 ðb a3 Þ
¼
q 4μ1
ð26Þ
The solutions to Eqs. (25) and (26) are 3
ð27Þ
3
ð28Þ
p ¼ 4eð3λ1 þ 2μ1 Þðμ1 μ2 Þð3λ2 þ 2μ2 Þða3 b Þ=Δ4 q ¼ 4eμ1 ð3λ2 þ 2μ2 Þð3λ1 þ 2μ1 þ 4μ2 Þða3 b Þ=Δ4 with 3
3
3
Δ4 ¼ 8ða3 b Þμ21 4ð4a3 þ 5b Þμ1 μ2 þ8ða3 b Þμ22 3
3
3
þ 6λ2 ð ð2a3 þ b Þμ1 þ2ða3 b Þμ2 Þ 3λ1 ð4a3 ð μ1 þμ2 Þ þ b ð3λ2 þ 4μ1 þ 2μ2 ÞÞ
ð29Þ
The solution for p also reveals the same behavior as for the 2-D problem, namely that the hydrostatic stress in the core changes sign with the difference of the moduli (μ1 μ2 ) and can become tensile when the annulus is stiffer in the shear modulus (for positive dilatational eigenstrain in the annulus). It may be also noted that the value of q and stresses induced by it coincides with the one for the Eshelby (1957, 1961) spherical inhomogeneity solution when the radius q goes to zero. 5. Interphases of small thickness The previously obtained stress fields are now expanded into Taylor series for small layer thickness h, with b ¼ a þ h.
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
473
5.1. Thin ring (a or o b) inhomogeneity under dilatational eigenstrain 5.1.1. Stresses in the core srr ¼ sθθ ¼ p ¼ En
2
ð 2Þðα βÞðb a2 Þ
ð30Þ
2
ð1 þ α 2βÞð1 α þ2βÞb 4βðα βÞa2
For b ¼ a þ h, srr ¼ sθθ ¼
4En ðα βÞðh=aÞ 2En ð 3 þðα 4βÞð3α 4βÞÞðα βÞðh=aÞ2 þ þ O½h=a3 1 α2 ð1 α2 Þ2
5.1.2. Stresses in the inhomogeneous thin ring (a or o b) ! ! 2 b a2 2 =Δ5 srr ¼ En 2ðα βÞa2 1 2 b ð1 αþ 2βÞ 1 þ 2 r r sθθ ¼ E
n
2
b 1þ 2 r
2
2ðα βÞa
!
2
ð1 α þ 2βÞb
a2 1 2 r
! =Δ5
2
Δ5 ¼ 4a2 ðα βÞβ þ b ð1 þ α 2βÞð1 α þ2βÞ
ð31Þ
ð32Þ
ð33Þ ð34Þ
For b ¼ a þ h, at r ¼ a 4En ðα βÞðh=aÞ 2En ð 3 þ ðα 4βÞð3α 4βÞÞðα βÞðh=aÞ2 þ þ O½h=a3 1 α2 ð1 α2 Þ2
srr ¼
sθθ ¼
ð35Þ
8eμ2 4En ð1 α þ 4βÞðα βÞðh=aÞ þ κ2 þ 1 ð1 αÞ2 ð1 þ αÞ 2
þ
2En ð 3 þðα 4βÞð3α 4βÞÞð 1 þα 4βÞðα βÞðh =a2 Þ 3
ð 1 þ αÞ ð1 þαÞ
2
þ O½h=a3
ð36Þ
If we define the “effective surface tension” in the interphase membrane as s0θθ sθθ h
ð37Þ
then (36) to the leading order gives s0θθ sθθ h ¼
8eμ02 8eðhμ2 Þ ¼ κ2 þ 1 κ2 þ 1
ð38Þ
where we defined as a surface shear modulus μ02 ¼ hμ2
ð39Þ
to be considered as the “effective” shear modulus in the membrane, as in the Gurtin and Murdoch (1975) surface elasticity model. Thus, in the leading order (zeroth) term, only the sθθ stress exists in the ring, and it depends only on the material constants of the ring, while all other stresses in the core, matrix and ring are of order ðh=aÞ. The next order terms in (35) and (36) show the correction effects to this model due to curvature. 5.2. Stresses in the thin ring inhomogeneity under shear eigenstrain The stresses of the finite thickness inhomogeneity ring are expanded for thin ring of thickness h, with b ¼ a þh. By Taylor series expansion we have at r ¼ a srr ¼ En
8e cos 2θðα 2βÞðh=aÞ þ O½h=a2 1 α2
ð40Þ
srθ ¼ En
8e sin 2θð2α βÞðh=aÞ þ O½h=a2 1 α2
ð41Þ
sθθ ¼
32e2 μ2 cos 2θ 8eðαð5 þ 3αÞ 4ð1 αÞβ 8β2 Þ cos 2θðh=aÞ En þ O½h=a2 κ 2 þ1 ð1 αÞ2 ð1 þαÞ
ð42Þ
which shows that also in the shear eigenstrain case only the stress sθθ is of order (1) in the ring, and all other stresses everywhere are of order h.
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5.3. Stresses in the thin spherical annulus under dilatational eigenstrain In the annulus material of the sphere, for b ¼ a þ h, at r ¼ a srr ¼
4eð3λ1 þ 2μ1 Þðμ1 μ2 Þð3λ2 þ2μ2 Þðh=aÞ þ O½h=a2 3ðλ1 þ 2μ1 Þðλ2 þ 2μ2 Þ
ð43Þ
sθθ ¼
2eμ2 ð3λ2 þ 2μ2 Þ 4eðμ1 μ2 Þð3λ2 þ2μ2 Þð3λ1 λ2 þ 4μ22 þ2λ2 ðμ1 þ 3μ2 ÞÞðh=aÞ þ O½h=a2 λ2 þ 2μ2 3ðλ1 þ 2μ1 Þðλ2 þ 2μ2 Þ2
ð44Þ
Thus, also in the thin spherical annulus the stresses are to the leading order depending on the annulus material properties only and the effective interface surface tension stresses can be defined as before.
6. Driving forces on interfaces 6.1. Cylindrical rings with eigenstrain The driving force (per unit area) on an interface separating a region with eigenstrain from the matrix is given by Eshelby (1977) as f ¼ 〈sij 〉½½εnij (also, Markenscoff and Ni, 2010, for dynamics) where the force is positive along the outward normal to the surface enclosing the region with eigenstrain, and where the brackets denote the jump. In polar coordinates it is hh ii ð45Þ f ¼ 〈sij 〉 εnij ¼ 12 ðsðrrþ 1Þ þ sðrr 1Þ Þeþ 12 ðsðθθþ 1Þ þ sðθθ 1Þ Þe with positive along the outward normal of the region containing the eigenstrain, and is evaluated for the interfaces considered above. Negative driving force implies that external energy (Markenscoff, 2010) is needed to move the interface outwards. The driving forces were also evaluated for spherical inhomogeneities (Markenscoff, 2012) and strips with eigenstrain (Dundurs and Markenscoff, 2009). In the case of an inhomogeneous ring of finite thickness with dilatational eigenstrain (i) At the interface r ¼ a the driving force is 2
f¼
En ð4a2 ðα βÞ þ b ð1 3αþ 4βÞÞe
ð46Þ
2
4a2 ðα βÞβ b ð1 α2 þ 4αβ 4β2 Þ
For thin interphase layer (thin ring), b ¼ a þ h, and the driving force is f¼
4e2 μ2 8eEn ð1 α þβÞðα βÞðh=aÞ þ O½h=a2 þ κ 2 þ1 ð1 αÞ2 ð1 þ αÞ
ð47Þ
which shows that the leading term depends only on the material constants of the inhomogeneous ring, and is due to the stress sθθ , while the pressure contributes to the order ðh=aÞ term. The second term, proportional to the thickness, depends on the bimaterial combinations and, if it becomes positive because of high material mismatch, and of the same order as the leading term, then the interface will become unstable when f becomes zero. This second term is containing the interaction with the other boundary of the interphase layer. For a thick ring a=b ¼ 0, the driving force depends on the bimaterial combination and has reduced bimaterial dependence as is shown in the next section f¼
En ð1 3α þ 4βÞe 1 α2 þ4αβ 4β2
where En ¼
8eμ1 μ2 μ2 ðκ 1 þ 1Þ þ μ1 ðκ2 þ 1Þ
ð48Þ
(ii) At the interface r ¼ b, the driving force is for thick layer 2
f¼
En ð b ð1 α þ 2βÞ þ2a2 ð α þβÞÞe 2 4a2 ðα βÞβ b ð1 α2 þ 4αβ 4β2 Þ
ð49Þ
For a thin ring b ¼ a þ h, f¼
4e2 μ2 4eEn ð1 α þ2βÞðα βÞðh=aÞ þ þ O½h=a2 κ 2 þ1 ð1 αÞ2 ð1 þαÞ
ð50Þ
For a=b ¼ 0 (thick ring)
f¼
En e 1 þ α 2β
where En ¼
8eμ1 μ2 μ2 ðκ 1 þ1Þ þ μ1 ðκ2 þ 1Þ
ð51Þ
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
475
6.2. For sphere (3D) with dilatational eigenstrain The driving force is þ 1Þ 1Þ f ¼ 12 ðsðrrþ 1Þ þsðrr 1Þ Þe þ 12 ðsðθθþ 1Þ þsðθθ 1Þ Þe þ 12 ðsðφφ þsðφφ Þe
ð52Þ
The driving force, f , evaluated at r ¼ a yields 3
3
3
f ¼ 6e2 ð3λ2 þ 2μ2 Þð 2μ1 ða3 b Þð3λ1 þ 2μ1 Þ þ μ2 ðð6a3 3b Þλ1 þ2μ1 ð2a3 þ b ÞÞÞ=Δ6
ð53Þ
For b ¼ a þ h (thin annulus) f¼
2e2 μ2 ð3λ2 þ 2μ2 Þ λ2 þ 2μ2 4e2 ðμ1 μ2 Þð3λ2 þ2μ2 Þð6λ2 ðμ1 þμ2 Þ þ 4μ2 ð2μ1 þ μ2 Þ þ 3λ1 ð3λ2 þ 4μ2 ÞÞðh=aÞ 3ðλ1 þ2μ1 Þðλ2 þ 2μ2 Þ2
þ O½h=a2
ð54Þ
It may be noted that in the first term of (54) ð3λ þ 2μÞ=ðλ þ 2μÞ ¼ ð1 þvÞ=ð1 vÞ. The driving force f , evaluated at r ¼ a, yields 3
3
f ¼ 6e2 ð3λ2 þ 2μ2 Þð μ1 ða3 b Þð3λ1 þ2μ1 Þ þμ2 ð3a3 λ1 þ 2μ1 ða3 þ 2b ÞÞÞ=Δ6
ð55Þ
where 3
3
3
Δ6 ¼ 8μ21 ða3 b Þ 4μ1 μ2 ð4a3 þ 5b Þ þ 8μ22 ða3 b Þ 3
3
3
3
3
þ6λ2 ð μ1 ð2a þ b Þ þ 2μ2 ða b ÞÞ 3λ1 ð4a3 ð μ1 þμ2 Þ þ b ð3λ2 þ4μ1 þ 2μ2 ÞÞ
ð56Þ
For b ¼ a þ h (thin annulus) f¼
2e2 μ2 ð3λ2 þ 2μ2 Þ 2e2 ðμ1 μ2 Þð3λ2 þ2μ2 Þ2 ð3λ1 þ 2μ1 þ 4μ2 Þðh=aÞ þO½h=a2 λ2 þ 2μ2 3ðλ1 þ2μ1 Þðλ2 þ 2μ2 Þ2
ð57Þ
The leading term of the driving force at the interface for small thickness (in (54) and (57)) is found to be equal to the value of the driving force for a spherical inclusion of material 2 (in a matrix of material) with dilatational eigenstrain, as derived by Gavazza (1977), Eshelby (1970), and Markenscoff and Ni (2010). 7. Finite thickness surface layer with eigenstrain: surface tension and driving forces 7.1. Surface layer of same material as the matrix We now consider a free surface layer of finite thickness and of the same material as the matrix, containing eigenstrain εnij ¼ δij e for a or o b, εnij ¼ 0 for r o a,and at r ¼ b; srr jr ¼ b ¼ 0
ð58Þ
The solution in terms of the stress functions for the core (with index1) and the surface layer (with index 2) are U ð1Þ ¼ 2μAer2 ;
U ð2Þ ¼ 2μefBr 2 þ Ca2 log rg;
ð59Þ
with the coefficients obtained from the boundary conditions ð2Þ ð1Þ ð2Þ srr ¼ sð2Þ and srr ¼0 uð1Þ r r ¼ a ¼ ea þ ur r ¼ a ; rr r ¼ a r¼a r¼b
ð60Þ
as 2
A¼
a2 b 2
b ðκ þ 1Þ
;
ð1Þ sð1Þ rr ¼ sθθ ¼
sð2Þ θθ ¼
B¼
a2 2
b ðκ þ 1Þ
;
C¼
2
4eμ ðb a2 Þ ; 2 ðκ þ 1Þ b 2
4eμ a2 ðb þ r 2 Þ ; 2 ðκ þ 1Þ b r2
2 κþ1
ð2Þ srr ¼
sð2Þ θθ
r¼a
ð61Þ 2
4eμ a2 ðb r 2 Þ ; 2 ðκ þ 1Þ b r2
2 4eμ ðb a2 Þ sð2Þ rr r ¼ a ¼ 2 ðκ þ1Þ b
ð62Þ
2
¼
4eμ ðb þa2 Þ 2 ðκ þ1Þ b
ð63Þ
For small thickness h approximation, (i) the stresses in the core
ð1Þ sð1Þ rr ¼ sθθ ¼
8eμðh=aÞ 12eμðh=aÞ2 þ þ O½h=a3 κ þ1 κþ1
ð64Þ
476
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
(ii) the stresses in the surface layer 8eμðh=aÞ 12eμðh=aÞ2 ð2Þ þ þO½h=a3 srr ¼ r¼a κ þ1 κ þ1 8eμ 8eμðh=aÞ 12eμðh=aÞ2 ð2Þ þ 2 þ O½h=a3 sθθ ¼ r¼a κþ1 κþ1 a ðκ þ1Þ
ð65Þ
Eq. (65b) provides the relation between the eigenstrain and the surface tension, sθθ , and inversely, it associates an eigenstrain to a surface tension (Gao, 1994). We may define, as in (37) the “effective surface tension” as s0θθ ¼ sθθ h and μ0 ¼ hμ; so that from (65) to the leading term 0 sθθ ¼ 8eμ0 =ðκ þ 1Þ. The driving force f at r ¼ a 2
f¼
4ð 2a2 þ b Þe2 μ
ð66Þ
2
b ðκ þ 1Þ
and, for a thin surface layer, b ¼ a þ h, at r ¼ a f¼
4e2 μ 16e2 μðh=aÞ þ þ O½h=a2 κþ1 κþ1
ð67Þ
The driving force f , at r ¼ b f¼
4a2 e2 μ
ð68Þ
2
b ðκ þ 1Þ
and, for a thin layer, b ¼ a þh, f¼
4e2 μ 8e2 μðh=aÞ þ þO½h=a2 κþ1 κ þ1
ð69Þ
7.2. For a surface layer of different material constants The stresses are obtained from the stress functions in the core (1) and surface inhomogeneous layer (2): U ð1Þ ¼ 2μ1 Aer 2 ;
U ð2Þ ¼ 2μ2 efBr2 þ Ca2 log rg;
and boundary conditions urð1Þ r ¼ a ¼ ea þ urð2Þ r ¼ a ;
ð2Þ sð1Þ rr r ¼ a ¼ srr r ¼ a
ð70Þ ð2Þ srr ¼0 r¼b
and
ð71Þ
2
ða2 b Þμ2
A¼
B¼
2
ð72Þ
2
μ2 ða2 b Þðκ 1 1Þ μ1 ða2 κ2 a2 þ2b Þ a2 μ1 2
ð73Þ
2
μ2 ð a2 þ b Þðκ1 1Þ þ μ1 ða2 κ2 a2 þ 2b Þ 2
C¼
2b μ1 μ2
2 ða2 b Þðκ
ð74Þ
2 2 2 1 1Þ μ1 ða κ 2 a þ 2b Þ
and ð1Þ srr ¼ sð1Þ θθ ¼
ð2Þ srr ¼
2
2
μ2 ða2 b Þðκ1 1Þ þ μ1 ð a2 κ2 þ a2 2b Þ
2 4eμ1 μ2 a2 ð b þr 2 Þ 2 2 2 2 r ðμ2 ð a þ b Þðκ1 1Þ þμ1 ða2 κ 2 a2 þ 2b ÞÞ
ð2Þ srr ¼ r¼a
ð2Þ sθθ ¼
2
4eμ1 μ2 ða2 b Þ
ð75Þ
ð76Þ
2
4eμ1 μ2 ða2 b Þ 2
2
μ2 ða2 b Þðκ1 1Þ þ μ1 ð a2 κ 2 þa2 2b Þ
2 4eμ1 μ2 a2 ðb þ r 2 Þ 2 2 2 2 r μ2 ðð a þ b Þðκ1 1Þ þμ1 ða2 κ 2 a2 þ 2b ÞÞ
ð77Þ
ð78Þ
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
sð2Þ θθ
477
2
r¼a
¼
4eμ1 μ2 ða2 þ b Þ μ2
2 ða2 b Þðκ
1 1Þ μ1
ða2 κ
ð79Þ
2 2 2 a þ 2b Þ
For small thickness approximation, (i) the stresses in the core ð1Þ sð1Þ rr ¼ sθθ ¼
8eμ2 ðh=aÞ 4eμ2 ð μ1 ðκ2 7Þ þ4μ2 ðκ1 1ÞÞðh=aÞ2 þ þ O½h=a3 κ2 þ 1 μ1 ð1 þ κ2 Þ2
ð80Þ
(ii) the stresses in the surface layer 8eμ2 ðh=aÞ 4eμ2 ðμ1 ð κ 2 þ7Þ þ 4μ2 ðκ1 1ÞÞðh=aÞ2 þ sð2Þ þ O½h=a3 rr r ¼ a ¼ κ 2 þ1 μ1 ðκ 2 þ1Þ2 8eμ2 2ð 1 þ α 4βÞðh=aÞ þEn sð2Þ ¼ θθ r¼a κ2 þ 1 ð1 αÞ2 þ
4eμ2 ðμ1 ðκ2 7Þ 4μ2 ðκ1 1ÞÞðμ1 ðκ2 3Þ 2μ2 ðκ1 1ÞÞðh=aÞ2 μ21 ðκ2 þ 1Þ3
þ O½h=a3
ð81Þ
The driving force f at r ¼ a 2
f¼
4e2 μ1 μ2 ð2a2 b Þ μ2
2 ð a2 þ b Þðκ
ð82Þ
2 2 2 1 1Þ þμ1 ð a þ 2b þa κ 2 Þ
for a thin surface layer, b ¼ a þh, f¼ ¼
4e2 μ2 8e2 μ2 ðμ1 ðκ2 þ 3Þ þμ2 ðκ 1 1ÞÞðh=aÞ þ þ O½h=a2 κ 2 þ1 ðκ2 þ 1Þ2 μ1 4e2 μ2 4eð1 αþ βÞðh=aÞ þEn þ O½h=a2 κ2 þ 1 ð1 αÞ2
ð83Þ
and the driving force f , at r ¼ b, 2
f¼
4e2 μ1 μ2 ð2a2 b Þ 2
ð84Þ
2
μ2 ð a2 þ b Þðκ1 1Þ þμ1 ða2 κ 2 a2 þ2b Þ
for a thin layer, b ¼ a þ h, f¼ ¼
4e2 μ2 8e2 μ2 ð2μ1 þ μ2 ðκ1 1ÞÞðh=aÞ þ þ O½h=a2 κ 2 þ1 μ1 ðκ2 þ 1Þ2 4e2 μ2 2eð1 αþ 2βÞðh=aÞ þEn þ O½h=a2 κ2 þ 1 ð1 αÞ2
ð85Þ
Eqs. (83) and (85) show the coupling between the driving forces of the surfaces of the layer. While the leading order terms in (83) and (85) are the same, the O½h=a terms are different due to coupling. 7.3. Inhomogeneous spherical surface layer with eigenstrain ð86Þ
p ¼ e=Δ7 where 3
Δ7 ¼
1 b a3 3 3 3λ1 þ 2μ1 4μ2 ða3 b Þ ða3 b Þð3λ2 þ2μ2 Þ
ð87Þ
ð1Þ ð1Þ sð1Þ rr ¼ sθθ ¼ sφφ ¼ p ¼ e=Δ7
ð88Þ
3 3 3 sð2Þ rr ¼ 4eμ2 a ð b þ r Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8 r
ð89Þ
3
3 3 sð2Þ rr r ¼ a ¼ 4eμ2 ða b Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8 ;
ð2Þ srr ¼ 0; r¼b
ð90Þ
478
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482 ð2Þ 3 3 3 sφφ ¼ sð2Þ θθ ¼ 2eμ2 a ðb þ 2r Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8 r
ð91Þ
ð2Þ sφφ
¼ sð2Þ θθ
ð92Þ
ð2Þ sφφ
ð2Þ ¼ sθθ
3
r¼a
r¼b
3
r¼a
r¼b
¼ 2eμ2 ðb þ 2a3 Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8 3ea3
¼
ð93Þ
3
2ð a3 þ b ÞΔ7
where 3
3
3
3
Δ8 ¼ 3b λ2 ð3λ1 þ2μ1 Þ þ2ð3ð2a3 þ b Þλ1 þ 6ð a3 þ b Þλ2 þ 2ð2a3 þ b Þμ1 Þμ2 3
þ 8ð a3 þ b Þμ22
ð94Þ
For small thickness approximation, (i) the stresses in the core ð1Þ ð1Þ srr ¼ sð1Þ θθ ¼ sφφ ¼
4eμ2 ð3λ2 þ 2μ2 Þðh=aÞ þO½h=a2 λ2 þ 2μ2
ð95Þ
(ii) the stresses in the surface 4eμ2 ð3λ2 þ 2μ2 Þðh=aÞ ð2Þ srr ¼ þ O½h=a2 r¼a λ2 þ2μ2 ð2Þ sφφ
r¼a
¼ sð2Þ θθ
2eμ2 ð3λ2 þ2μ2 Þ λ2 þ 2μ2 4eμ2 ð3λ2 þ 2μ2 Þð3λ1 λ2 þ4μ22 þ 2λ2 ðμ1 þ3μ2 ÞÞðh=aÞ r¼a
ð96Þ
¼
ð3λ1 þ 2μ1 Þðλ2 þ2μ2 Þ2
þO½h=a2
ð97Þ
Again, the leading term defines the surface tension with μ02 ¼ hμ2 (defined as a surface shear modulus) 1 þv2 s0φφ ¼ s0θθ r ¼ a hsφφ hsθθ ¼ 2μ02 e r¼a 1 v2 in agreement with Gao (1994). The next order terms give the curvature dependence of the surface tension stresses. The driving force f , at r ¼ a layer 3
f ¼ 6e2 μ2 ð2a3 b Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8
ð98Þ
for thin surface layer, b ¼ a þh, f¼
2e2 μ2 ð3λ2 þ2μ2 Þ λ2 þ2μ2
þ
4e2 μ2 ð3λ2 þ 2μ2 Þð6λ2 ðμ1 þ μ2 Þ þ 4μ2 ð2μ1 þ μ2 Þ þ3λ1 ð3λ2 þ4μ2 ÞÞðh=aÞ ð3λ1 þ 2μ1 Þðλ2 þ 2μ2 Þ2
þO½h=a2
ð99Þ
The driving force f , at r ¼ b, for thick layer is f¼
3e2 a3 3
2ð a3 þb ÞΔ7
ð100Þ
for thin layer, b ¼ a þ h, f¼
2e2 μ2 ð3λ2 þ2μ2 Þ 2e2 μ2 ð3λ2 þ 2μ2 Þ2 ð3λ1 þ 2μ1 þ4μ2 Þðh=aÞ þ þ O½h=a2 λ2 þ2μ2 ð3λ1 þ 2μ1 Þðλ2 þ 2μ2 Þ2
ð101Þ
where, again, in (99) and (101) in the leading term we can set ð3λ þ2μÞ=ðλ þ2μÞ ¼ ð1 þ vÞ=ð1 vÞ, and, thus, have explicit dependence in terms of the shear modulus and Poisson0 s ratio of the interface. From Eqs. (67) and (69), (83) and (85), and (99) and (101), we observe that the driving forces at the free surface r ¼ b and the interface r ¼ a are equal to each other in the zeroth order term for a thin layer and they depend only on the surface moduli and the eigenstrain, as expected, but the next order ðh=aÞ (where a is the core radius) term may become large (for some bimaterial combination differences) and be of the order of the leading surface tension term, in which case, the free surface and matrix/free surface interfaces may become unstable if f becomes zero.
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
479
8. Reduction of constants in stress dependence for inhomogeneities with eigenstrain The CLM theorem (Cherkaev et al., 1992) has established a general way to prove reducibilty of material constants in the dependence of the stress in composites, and it was extended further by Dundurs and Markenscoff (1993) for linear dependence of the material constants on the position, and by Markenscoff and Jasiuk (1998) to include body force loadings. For loading of eigenstrain, Jasiuk and Boccara (2002) have obtained some general conditions based on compatibility, but still leave unresolved the case where the eigenstrain is a step function discontinuity on the boundary of the inhomogeneity, as with constant eigenstrain inclusions. Here we show that the interface conditions (in the presence of eigenstrain on one side) allow reduction of constants for general geometry of interfaces. The interface conditions of continuity of tractions and displacement are equivalently expressed (Dundurs, 1990; Markenscoff and Wheeler, 1996) as continuity of the tangential strains and continuity of the change of curvature at the interface, respectively ð1Þ ð2Þ εss ¼ εss
ð102Þ
Δκ1 ¼ Δκ 2
ð103Þ
where the change of curvature (Dundurs, 1990) (also verified by Markenscoff and Wheeler, 1996) is in terms of normal and tangential coordinates, n, s, respectively, and has the form Δκ ¼ 2
∂εns ∂εss kεnn ∂s ∂n
ð104Þ
The above conditions can be, in turn, expressed in terms of stresses and the eigenstrains, since, with the use of the stress– strain relation (Dundurs, 1990) 1 1 sss ð3 kÞðsnn þsss Þ þεnss ð105Þ εss ¼ 2μ 4 so that the change of curvature in (104) is written in terms of stresses and eigenstrains 1 ∂sns ∂sss ∂εn ∂εn ð5 þkÞ ðk þ 1Þ 2κðk 1Þsnn þ2 ns ss κεnnn Δκ ¼ 8μ ∂s ∂n ∂s ∂s
ð106Þ
Thus, the interface compatibility conditions can be written in terms of the Dundurs constants as (Dundurs, 1990) ð2Þ nð2Þ nð1Þ nð2Þ nð1Þ ð1 αÞsss ð1 þ αÞsð1Þ ss þ2ðα 2βÞsnn ¼ 16mfεss εss þη2 εzz η1 εzz g ð2Þ ∂sss ∂sð1Þ ∂sns ð1 þαÞ ss þ 2ð3α 2βÞ 4κβsnn ∂n ∂n ∂s ( ) ! ! ! nð2Þ nð1Þ nð2Þ nð1Þ nð2Þ nð1Þ ∂εns ∂εns ∂εss ∂εss ∂εzz ∂εzz nð2Þ nð1Þ nð2Þ nð1Þ η1 ¼ 16m 2 κðεnn εnn Þ η2 κðη2 εzz η1 εzz Þ ∂s ∂s ∂n ∂n ∂n ∂n
ð107Þ
ð1 αÞ
ð108Þ
with m¼
μ1 μ2 μ2 ðκ 1 þ1Þ þ μ1 ðκ 2 þ 1Þ
ð109Þ
Eqs. (91) and (92) show that the stresses are functions of the Dundurs constants α and β with the m multiplier in front. We will not consider here the presence of εzz , when the constants reduction does not go through, and we will show that the CLM theorem goes through for m, by writing m¼
μ1 μ2 1 1 ¼ ¼ ðκ 1 þ1Þ=μ1 þðκ 2 þ 1Þ=μ2 ðA1 þ S1 Þ þðA2 þ S2 Þ μ2 ðκ 1 þ1Þ þ μ1 ðκ 2 þ 1Þ
ð110Þ
where A and S are, respectively, the voluminal and shear compliances A¼
κ1 ; 2μ
S¼
1 μ
ð111Þ
For the CLM “shift” transformation in the compliances (Cherkaev et al., 1992) Ai ¼ Ai þ a;
Si ¼ Si a;
i ¼ 1; 2
ð112Þ
and, where a is an arbitrary constant. The value of m in (110) remains unchanged under the transformations (112), so that m has reduced bimaterial dependence. Since already the Dundurs constants are a particular “reduced constants” combination, we have shown that the stresses have reduced dependence on elastic constants for inhomogeneities with in-plane eigenstrain loading for general geometrical shape of the inhomogeneities. Solutions for loading with the out of plane eigenstrain εzz are currently investigated by Dundurs.
480
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
9. Conclusions Solutions for annular inhomogeneities with eigenstrain embedded in a matrix of different elastic constants have been obtained. A feature of our problem/solution is the material rotation inside the inclusion (a feature of the Frank–Bilby interface model) induced by shear eigenstrain. The stresses in the core vary with the difference of the shear moduli of the annulus and matrix both in 2 and 3-D, and they may become tensile (with dilatational eigenstrain in the annulus) for a stiffer annulus in the shear modulus. This inhomogeneity solution elucidates further the Eshelby inclusion result for dilatational eigenstrain that the value of the bulk modulus of the matrix is “irrelevant” (Eshelby, 1961) (and only the shear modulus matters). For small interphase thicknesses, by Taylor series expansion, the terms up to the second order are obtained for all quantities. The driving (Eshelby) forces on the interfaces (due to the eigenstrain) are also computed, and, for finite thickness, they account for the interaction of one interface of the annulus with the other. This interaction may become significant for large material mismatch and may render the interface unstable. In the limit of zero thickness the model reduces to a Gurtin–Murdoch membrane model, but for shear surface eigenstrain it may also possess “surface rotation”, of the Frank–Bilby type. The next order terms in the expansion provide the curvature “corrections” to the Gurtin–Murdoch model. While the presented solutions are general and applicable to all scales and sizes of the annulus, applications, in particular, include the modeling of interphases in coated particles used as “re-enforcement” in thermal/electric conductive adhesives, where the surface strengthening properties become important for reliability in electronic packaging.
Acknowledgment X.M. is grateful for the support of the National Science Foundation, Grant CMS #1129888. Appendix A. Dilatational eigenstrains in the inhomogeneous cylindrical ring Core (r o a) under pressure p: U ð1Þ ¼
E¼
1 2 pr ; 2
srr ¼ sθθ ¼ p;
ur ¼
p 2μ1
1 ðk1 1Þr 2
ðA:1Þ
8eμ1 μ2 μ1 ðk2 þ 1Þ þμ2 ðk1 þ 1Þ
ðA:2Þ
Ring (a or o b) under pressure p at r ¼ a: U ð2;pÞ ¼ pa2 =ðb a2 Þfð1=2Þr 2 b log rg ( ) ( ) ( ) 2 2 2 pa2 b pa2 b p a2 1 b ð2;pÞ r ð2;pÞ ðk sð2;pÞ ¼ 1 ¼ 1 þ ¼ 1Þ þ ; s ; u 2 rr r θθ 2 2 2μ2 b2 a2 2 r2 r2 r2 b a2 b a2 2
2
Ring (a or o b) under pressure q at r ¼ b: 2 2 2 qb 1 qb a2 qb a2 r 2 þ a2 log r ; sð2;qÞ ¼ 2 1 þ 2 ; sð2;qÞ ¼ 2 1 2 U ð2;qÞ ¼ 2 rr θθ 2 r r b a2 b a2 b a2 ( ) 2 2 p a2 1 b q b 1 a2 ðk2 1Þr þ 2 ; uð2;qÞ ¼ ¼ ðk2 1Þb 2 : uð2;pÞ r r 2 2 2μ2 b a2 2 2μ2 b a2 2 r b
ðA:3Þ
ðA:4Þ
ðA:5Þ
Matrix under pressure q at r ¼ b: U ð3Þ ¼ qb log r; 2
ð3Þ srr ¼
2
qb ; r2
sð3Þ θθ ¼
2
qb ; r2
q urð3Þ r ¼ b ¼ b 2μ1
ðA:6Þ
where α, β are the Dundurs (1969) constants
α¼
μ2 ðκ 1 þ 1Þ μ1 ðκ2 þ 1Þ ; μ2 ðκ 1 þ 1Þ þ μ1 ðκ2 þ 1Þ
β¼
μ2 ðκ1 1Þ μ1 ðκ 2 1Þ μ2 ðκ1 þ 1Þ þμ1 ðκ 2 þ1Þ
ðA:7Þ
Appendix B. Shear eigenstrains in the ring B.1. Stresses for the core material 2
4
6
srr ¼ En 2ða2 b Þð3a2 b ðα βÞð1 þ βÞ2 þ a6 ðα βÞðα þ βÞ2 þ b ðα þβÞð 1 þ β2 Þ 2
3a4 b ðα βÞðα2 þ βð2 þβÞÞÞε cos 2θ=Δ7
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482 2
6
481
2
2
2
2
srθ ¼ En 2ða2 b Þð a6 ðα βÞðα þ βÞ2 b ðα þβÞð 1 þ β2 Þ 3a2 b ðα βÞð2r 2 ð 1 þ α2 Þ þb ð1 þβÞ2 Þ 4 2
2
þ3a b ðα βÞðα þβð2 þ βÞÞÞε sin 2θ=Δ7 2
6
sθθ ¼ En 2ða2 b Þð a6 ðα βÞðαþ βÞ2 b ðα þ βÞð 1 þβ2 Þ 3a2 b ðα βÞð4r 2 ð 1 þα2 Þ þ b ð1 þ βÞ2 Þ 2
þ3a4 b ðα βÞðα2 þβð2 þ βÞÞÞε cos 2θ=Δ7
ðB:1Þ
B.2. Stresses for the ring material The expressions of the stresses are two long to be presented, but can be easily computed from the stress functions. The thin ring/annulus approximations are presented in Section 5. B.3. Stresses for the outer material 2
2
2
2
6
2
srr ¼ En 2ða2 b Þb ð3a4 b ð3b 4r 2 Þðα βÞ2 ð1 þβÞ þb ð3b 4r 2 Þð 1 þ βÞ2 ð1 þβÞ 2
2 4
2
þa ð3b 4r Þðα β Þð 1 þ βÞ 3a b ð1 þ βÞð 4r ðα βÞ2 þb ð 1 þ 4α2 6αβ þ3β2 ÞÞÞε cos 2θ=ðr 4 Δ7 Þ 6
2
2
2
n
2
2
4 2
2
2
6
2
srθ ¼ E 2ða2 b Þb ð3a b ð3b 2r 2 Þðα βÞ2 ð1 þ βÞ þ b ð3b 2r 2 Þð 1 þβÞ2 ð1 þ βÞ 2
4
2
þa6 ð3b 2r 2 Þðα2 β2 Þð 1 þ βÞ 3a2 b ð1 þ βÞð 2r 2 ðα βÞ2 þb ð 1 þ 4α2 6αβ þ3β2 ÞÞÞε sin 2θ=ðr 4 Δ7 Þ 4
n
2
2
4 2
6
2
2
sθθ ¼ E 6b ða b Þð3a b ðα βÞ ð1 þ βÞ þ b ð 1 þβÞ ð1 þ βÞ 4
þa6 ðα2 β2 Þð 1 þ βÞ a2 b ð1 þ βÞð 1 þ 4α2 6αβ þ 3β2 ÞÞε cos 2θ=ðr 4 Δ7 Þ
ðB:2Þ
where 4
8
Δ7 ¼ 6a4 b ðα βÞ2 ð1 þ βÞ2 þ a8 ðα2 β2 Þ2 þ b ð 1 þ β2 Þ2 6 2
6
2a b ðα βÞð1 þ βÞðαð2 þ βÞ βð1 þ 2βÞÞ 2a2 b ðα βÞð1 þβÞðαð2 þβÞ βð1 þ 2βÞÞ
ðB:3Þ
Appendix C. Dilatational eigenstrains in the inhomogeneous spherical annulus (a) Spherical core under pressure: ui ¼ φðrÞxi for 0 o r oa φðrÞ ¼
p ; 3λ þ 2μ
srr ¼ p;
sφφ ¼ sθθ ¼ p;
ui ¼
p x 3λ þ 2μ i
ðC:1Þ
3
ðC:2Þ
3
ðC:3Þ
p ¼ 4eð3λ1 þ2μ1 Þðμ1 μ2 Þð3λ2 þ 2μ2 Þða3 b Þ=Δ4 q ¼ 4eμ1 ð3λ2 þ 2μ2 Þð3λ1 þ 2μ1 þ 4μ2 Þða3 b Þ=Δ4 (b) Spherical annulus under pressure p and q: ui ¼ φðrÞxi for a or o b φðrÞ ¼
pa3 qb
3
3
3
ð3λ2 þ 2μ2 Þðb a3 Þ 3
þ
1 a3 b ðp qÞ r 3 4μ2 ðb3 a3 Þ
ðC:4Þ
3
a3 b p q 3 3 r 3 b3 a3 b a 3 3 3 3 ¼ 4eð3λ2 þ 2μ2 Þð ða3 b Þr 3 μ1 ð3λ1 þ 2μ1 Þ þ ð3a3 ð b þ r 3 Þλ1 þ 2ð2b r 3 þa3 ð 3b þ r 3 ÞÞμ1 Þμ2 Þ=ðr 3 Δ4 Þ
srr ¼
pa3 qb
3
3
a3
ðC:5Þ
3
a3 b p q 2r 3 b3 a3 b 3 3 3 3 ¼ 2eð3λ2 þ 2μ2 Þð 2ða3 b Þr 3 μ1 ð3λ1 þ2μ1 Þ þð3a3 ðb þ 2r 3 Þλ1 þ 2ð3a3 b þ2ða3 þ 2b Þr 3 Þμ1 Þμ2 Þ=ðr 3 Δ4 Þ
sθθ ¼
pa3 qb
þ
ðC:6Þ
3
(c) Infinite matrix under pressure q at r ¼ b, a2 ¼ 1: ui ¼ φðrÞxi , φðrÞ ¼ ðqb =4μ1 Þð1=r 3 Þ 3
srr ¼
qb ; r3
3
sφφ ¼ sθθ ¼
qb ; 2r 3
3
ui ¼
qb 1 xi 4μ1 r 3
ðC:7Þ
References Benveniste, Y., 2013. Models of thin interphases with variable moduli in plane–strain elasticity. J. Math. Mech. Sol. 18, 119–134. Cherkaev, A., Lurie, K., Milton, G.W., 1992. Invariant properties of the stress in plane elasticity and equivalence classes of composites. Proc. R. Soc. Lond. A438, 519–529.
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Dingerville, R., Qu, J., Charkaoui, M., 2005. Surface free energy and its effects on the elastic behavior of nano-sized particles, wires and films. J. Mech. Phys. Sol. 53, 1827–1854. Dingreville, R., Qu, J., 2008. Interfacial excess energy, excess stress and excess strain in elastic solids:planar interfaces. J. Mech. Phys. Sol. 56, 1944–1954. Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L., 2005. Size-dependent effective elastic constants of solids ccontaining nano-inhomogeneities with interface stress. J. Mech. Phys. Sol. 53, 1574–1596. Duan, H.L., Wang, J., Karihaloo, B.L., 2008. Theory of elasticity at the nano-scale. Adv. Appl. Mech. 42, 1–68. Dundurs, J., 1967. On the interaction of a screw dislocation with inhomogeneities. In: Eringen, A.C. (Ed), Recent Advances in Engineering Science, vol. 2. Gordon & Breach, pp. 222–233. Dundurs, J., 1969. Discussion of a paper by D. B. Bogy. J. Appl. Mech. ASME 36, 650–652. Dundurs, J., 1990. Boundary conditions at interface. In: Weng, G.J., Taya, M., Abe, H. (Eds.), Micromechanics and Inhomogeneity, The Toshio Mura Anniversary volume. Springer-verlag, New York, pp. 109–114. Dundurs, J., Markenscoff, X., 1993. Invariance of stresses under a change in elastic compliances. Proc. R. Soc. A438, 531–544. Dundurs, J., Markenscoff, X., 2009. Stress fields and Eshelby forces on half-plane inhomogeneities with eigenstrain and strip inclusions meeting a free surface. Int. J. Sol. Struct. 46, 2481–2485. Eshelby, J.D., 1961. Elastic inclusions and inhomogeneities. Prog. Solid Mech. 2, 89–140. Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A241, 376–396. Eshelby, J.D., 1970. Energy relation of the energy–momentum tensor in continuum mechanics. In: Kanninen, M.F. et al. (Eds), Inelastic Behavior of Solids, pp. 77–115. Eshelby, J.D., 1977. Lectures on the elastic energy–momentum tensor. In: Markenscoff, X., Gupta, A. (Eds.), vol. 2006 Collected Works of J.D., Eshelby: The Mechanics of Defects and Inhomogeneities. Springer, The Netherlands, pp. 907–931. Gao, H., 1994. “Some general properties of stress-driven surface evolution in a heteroepitaxial thin film structure. J. Mech. Phys. Sol. 42, 741–772. Gavazza, S.D., 1977. Forces on pure inclusion and Somigliana dislocations. Scr. Metal. 11, 979–981. Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57, 291–323. Hashin, Z., 2002. The interphase/imperfect interface in elasticity with application to coated fiber composites. J. Mech. Phys. Sol. 50, 2309–2537. Hirth, J.P., Pond, R.C., Hoagland, R.G., Liu, X.-Y., Wang, J., 2012. Interface defects, reference spaces, and the Frank–Bilby equation. Prog. Mater. Sci.. Jasiuk, I., Boccara, S.D., 2002. On the reduction of constants in plane elasticity with eigenstrain. Arch. Mech 54, 425–437. Markenscoff, X., 2010. Driving forces on phase boundaries: ‘The Eshelby Principle for an Interface’. Int. J. Fract. 165, 223–227. Markenscoff, X., 2012. The expanding spherical inhomogeneity with transformation strain. Int. J. Fract. 174, 41–48. Markenscoff, X., Wheeler, L., 1996. On conditions at an interface between two materials in plane deformation. J. Elast. 45, 33–44. Markenscoff, X., Jasiuk, I., 1998. On multiple connectivity and the reduction of constants for composites with body forces. Proc. R. Soc. A454, 1357–1369. Markenscoff, X., Ni, L., 2010. The energy release rate and self-force of dynamically expanding spherical and plane inclusion boundaries with dilatational Eigenstrain. J. Mech. Phys. Sol. 58, 1–11. Mohammadi, P., Liu, L.P., Sharma, P., Kutka, R.V., 2013. Surface energy, elasticity, and the homogenization of rough surfaces. J. Mech. Phys. Sol. 61, 325–340. Morris, J.E., Liu, J., 2007. Electrically Conductive Adhesives: A Research Status Review. In: Wong, C.P., Li, Y., Lu, D., (Eds.), Springer, 2010, e-book. Pollock, T.M., Lipkin, D.M., Hemker, K.J., 2012. Multifunctioning coating interlayers for thermal-barrier systems. MRS Bull. 37 (10). Qu, J., Cherkaoui, M., 2006. Fundamentals of Micromechanics of Solids. John Wiley and Sons Inc., Hoboken, NJ. Rodin, G., 1996. Eshelby0 s inclusion problem for polygons and polyhedra. J. Mech. Phys. Sol. 44, 1977–1995. Sharma, P., Ganti, S., 2004. Size-dependent Eshelby0 s tensor for embedded nano-inclusions incorporating surface/interface. J. Appl. Mech. 71, 663–671. Shodja, H.M., Khorshidi, A., 2013. Tensor spherical harmonics theories on the exact nature of the elastic fields of a spherically anisotropic multiinhomogeneous inclusion. J. Mech. Phys. Solids 61 (4), 1124–1143. Sokolnikof, I.N., 1956. The Mathematical Theory of Elasticity. Mc Graw Hill, New York. Steigmann, D.J., Ogden, R.W., 1997. Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453, 853. Steigmann, D.J., Ogden, R.W., 1999. Elastic surface substrate interactions. Proc. R. Soc. Lond. A455, 437.
Contents lists available at ScienceDirect
Journal of the Mechanics and Physics of Solids journal homepage: www.elsevier.com/locate/jmps
Annular inhomogeneities with eigenstrain and interphase modeling Xanthippi Markenscoff a,n, John Dundurs b a b
Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, United States Northwestern University, Evanston, IL 60201, United States
a r t i c l e in f o
abstract
Article history: Received 12 August 2013 Received in revised form 21 November 2013 Accepted 4 December 2013 Available online 18 December 2013
Two and three-dimensional analytical solutions for an inhomogeneity annulus/ring (of arbitrary thickness) with eigenstrain are presented. The stresses in the core may become tensile (for dilatational eigenstrain in the annulus) depending on the relative shear moduli. For shear eigenstrain, an “interface rotation” and rotation jumps at the interphase also occur, consistent with the Frank–Bilby interface model. A Taylor series expansion for small thickness of the annulus is obtained to the second-order as to model thin interphases, with the limit agreeing with the Gurtin–Murdoch surface membrane, but also accounting for curvature effects.. The Eshelby “driving forces” on a boundary with eigenstrain are calculated, and for small, but finite, interphase thicknesses they account for the interaction of the two interfaces of the layer, and the next order term may induce instabilities, for some bimaterial combinations, if it becomes large enough to render the driving force zero. It is also proven that for 2-D inhomogeneities with eigenstrain the stresses have reduced material dependence for any geometry of the inhomogeneity. The case when the outer boundary of the inhomogeneity annulus with eigenstrain is a free surface is also analyzed and agrees with classical surface tension results in the limit, but, moreover, the thick free surface terms (next order in the expansion depending on the radius) are also obtained and may induce instabilities depending on the bimaterial combinations. Applications of inhomogeneity annuluses with eigenstrain are wide and include interphases in thermal barrier coatings and coated particles in electrically/ thermally conductive adhesives. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Inhomogeneity Eigenstrain Interphase Surface tension Driving force
1. Introduction The Eshelby (1957) ellipsoidal inclusion solution and the Eshelby equivalent inclusion method for inhomogeneities have had widespread applications to composite materials over the last 50 years. Here we present solutions and properties of inhomogeneities with transformation strain, or eigenstrain (a term introduced by Dundurs, 1967), when the inhomogeneity is an annulus embedded in a matrix, which cannot be solved by the Eshelby (1957, 1961) equivalent inclusion method through an algebraic system of linear equations. The inhomogeneity annulus cannot be solved by superposition of Eshelby inclusions, since the Eshelby equivalent inclusion method would require to have a position dependent loading (of the outer field of the inner inclusion) in the equivalent eigenstrain system of equations, rather than the constant coefficients that are due to the Eshelby property for the ellipsoid. For an updated treatise on inclusions and inhomogeneities with eigenstrain
n
Corresponding author. Tel.: þ1 858 534 1337; fax: þ1 858 534 5698. E-mail addresses: [email protected], [email protected] (X. Markenscoff), [email protected] (J. Dundurs).
0022-5096/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmps.2013.12.003
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
469
and applications we refer to Qu and Cherkaoui (2006). Attention is drawn to a related very recent paper on spherical multiinhomogeneous inclusions by Shodja and Khorshidi (2013), with which we strongly disagree. To quote them, at the end of Section 5, “ Remark 4: Therefore, interestingly, the uniform eigenstrain fields … give rise to a center of dilatation at r ¼ 0. Nonuniform eigenstrain fields produce stress and strain fields … with logarithmic singularities at r ¼ 0 as well.” By the very principles of analytic function theory, finite eigenstrain cannot induce stresses and strain fields with singularities, except at points of geometric discontinuities, such as corners of inclusions (Rodin, 1996), or corners at strips of eigenstrain meeting a free surface (Dundurs and Markenscoff, 2009). These singularities have been routinely excluded a priori in all the literature (e.g. Duan et al., 2005 for the coated inclusion problem, Duan et al., 2008; Hashin, 2002). The current paper contains the correct solution without singularities. The applications of annular inhomogeneities with eigenstrain range from modeling interphases at grain boundaries (e.g. Hirth et al., 2012) to thermal barrier coatings (Pollock et al., 2012), and to coated particles in thermally/electrically conductive adhesives (e.g. Morris and Liu, 2007). The literature is vast on coated inhomogeneities and interphase layers between the inclusions and the matrix, or surface layers (but no eigenstrain loading in the inhomogeneous annulus has been previously considered), and we will selectively reference Steigmann and Ogden (1997, 1999), Sharma and Ganti (2004), Duan et al. (2005, 2008), Hashin (2002), Dingerville et al. (2005), Dingreville and Qu (2008), Benveniste (2013), and Mohammadi et al. (2013), while a more comprehensive list of references on the material aspects of interphases is provided in the long article of Hirth et al. (2012). Interesting features of the analytical solutions presented here are, that, for dilatational eigenstrain in the inhomogeneous ring/annulus, the stress in the core actually depends on the difference (μ1 μ2 ) of the shear moduli of the matrix and the annulus, both in 3-D (called “annulus”) and in 2-D (“ring”), and, thus, the inhomogeneous annulus induces hydrostatic tension inside the core when the annulus is stiffer in shear than the matrix. Thus, if the interphase thickness is appreciable with respect to the core radius, the tensile stress in the core induced by the interphase may be significant and have implications for nanomaterials. This inhomogeneity solution elucidates further the Eshelby inclusion result for dilatational eigenstrain that the value of the bulk modulus of the matrix is “irrelevant” (Eshelby, 1961) (and only the shear modulus matters). The annular inhomogeneities with eigenstrain may be useful to model interphases of finite thickness. For small inhomogeneity thicknesses h, the stresses and rotation inside the interphase are also obtained by a Taylor series expansion in the thickness parameter, and, as the thickness h goes to zero, the stresses are found to depend only on the elastic constants of the annulus/ring, which is consistent with the membrane interface (or free surface) models (such as Gurtin and Murdoch, 1975). However, the presence of shear eigenstrain in a layer of vanishingly small thickness also produces a nonzero “interface rotation”, not obtainable otherwise. The interface rotation is not a feature of other surface stress models (such as the one of Gurtin and Murdoch, 1975), but has been experimentally observed, and is a feature of the Frank–Bilby interface model in the recent seminal paper of (Hirth et al., 2012). The eigenstrain in the annulus gives rise to “driving forces” (Eshelby forces) on the interfaces between the matrix and the interphase, which are computed according to f ¼ o sij 4½½εnij (Eshelby, 1977), and same in dynamics (Markenscoff and Ni, 2010), on each interface of the finite interphase layer thickness, accounting also for the interaction with the other interface of the layer. As the thickness tends to zero, the hoop stresses and the driving force depend on the layer material constants only, but, for small finite thickness h the driving force has the next order h=a positive term which is proportional to the (μ1 μ2 ) difference of the matrix/interphase shear moduli and the bimaterial Dundurs constants α, β. This term may induce instabilities of the interface if it becomes large enough to be of the order of the leading term due to hoop tension so that the driving force becomes zero. We also present solutions for a thin spherical inhomogeneous annulus with eigenstrain, where, again, the hoop stresses depend only on the material constants of the annulus. The leading term of the driving force at the interface is found to be equal in value to the driving force for a spherical inclusion of material 2 (in a matrix of material 2) with dilatational eigenstrain in the core, as derived by Gavazza (1977), Eshelby (1977), and Markenscoff and Ni (2010). This should be expected physically, since the thin annulus stress field “knows” only the material of the annulus and the eigenstrain in it (there is no other characteristic length), and the Gavazza–Eshelby’s result is valid for inclusions of any shape (Eshelby, 1957, 1961). Moreover, the presence of eigenstrain gives rise to a driving force, which controls the stability of the interface. The obtained finite interphase solution for the driving force also accounts for the interaction of the driving force on one interface of the layer with the other interface. In addition, the “excess energy” (Dingreville and Qu, 2008) inside the interphase, here associated with the eigenstrain, can be calculated according to Eshelby (1961). We also present here the solutions for a finite radius ring/annulus of eigenstrain enclosing a matrix and being free of traction on the outside. The generated stresses are hoop stresses of “surface tension”, which, for vanishingly small thickness, again tend to a constant depending only on the elastic moduli of the thin shell. The driving force on both the free surface and the interface to the leading order depends on the surface moduli and the eigenstrain, but the next order h=a (where a is the core radius) term may become large (for some bimaterial combination differences) and be of the order of the surface tension term, in which case the free surface and matrix/free surface interfaces may become unstable. In the obtained solutions for 2-D ring inhomogeneities with eigenstrain, the stresses are expressed in terms of the Dundurs bimaterial constants (α β) with a coefficient En , as common multiplier in all the fields. It is shown here that, by applying the CLM theorem (Cherkaev et al., 1992), this coefficient En has a reduced dependence on the elastic constants. In addition, we show that the reduced material constants dependence of the stresses is valid for arbitrary shapes of
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Fig. 1. Annulus/ring inhomogeneity with eigenstrain embedded in a matrix.
inhomogeneities with eigenstrain. This result had not been previously obtained in the literature, as no step function discontinuities of the eigenstrain on the boundaries between the inclusion and the matrix were considered in the literature (in Jasiuk and Boccara, 2002). 2. Embedded cylindrical ring inhomogeneity with eigenstrain 2.1. Dilatational eigenstrain (εnij ¼ δij e) in the ring In an infinite matrix of elastic constants μ1 , κ1 (where κ is the Kolosov constant) ( 3 4v plain strain κ¼ 3 v plain stress we consider (Fig. 1) an embedded ring of an inhomogeneity of elastic constants μ2 , κ 2 in the region a or o b subjected to voluminal eigenstrain εnij ¼ δij e. The problem is reduced to finding the pressure p and q, between the core and the inhomogeneity at r ¼ a, and the inhomogeneity and the matrix at r ¼ b. The stress function for the regions 1 (core), 2 (ring), and 3 (matrix) are given in Appendix A. The unknown p and q are solved from the displacement boundary where the superscript ðiÞ in uðiÞ r refers to the regions i ¼ 1; 2; 3; of core, annulus and matrix respectively. Displacement boundary conditions: ð2;pÞ on r ¼ a; uð1Þ þuð2;qÞ r ¼ ea þ ur r
ð1aÞ
þ uð2;qÞ ¼ urð3Þ on r ¼ b; eb þuð2;pÞ r 2
ð1bÞ
The solution of the system of Eqs. (1a) and (1b) in which the displacements are being expressed in terms of p and q, according to Appendix A, gives 2
p ¼ 2En ðα βÞðb a2 Þ=Δ1
ð2Þ
2
q ¼ En ð1 αþ 2βÞðb a2 Þ=Δ1
ð3Þ
where En ¼
8eμ1 μ2 μ2 ðκ1 þ 1Þ þ μ1 ðκ2 þ 1Þ
and
2
Δ1 ¼ ð1 þ α 2βÞð1 α þ 2βÞb 4βðα βÞa2 :
ð4Þ
The solution (2) reveals that the srr and sθθ stresses in the core change sign with ðα βÞ, which is equivalent to (μ1 μ2 ), and become tensile for stiffer shear modulus of the ring (with dilatational eigenstrain). This implies that interface eigenstrains (as in thermal barrier coatings) may create damage in the core due to the tensile stresses that they induce there, as well as in nanomaterials if the interphase layer (modeled as a ring of eigenstrain) is of considerable thickness in comparison to the core. 2.2. Shear eigenstrain in the ring The ring contains shear eigenstrain (εnxx ¼ εnyy ¼ e, εnxy ¼ 0) with corresponding “eigen-displacements” unx ¼ ex ¼ er cos θ;
uny ¼ ey ¼ er sin θ;
ð5Þ
and in polar coordinates unr ¼ er cos 2θ;
unθ ¼ er sin 2θ:
ð6Þ
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
471
In this case the stress functions for regions 1, 2, 3, respectively are U ð1Þ ¼ Ar 2 cos 2θ þBr 4 cos 2θ
ð7Þ
U ð2Þ ¼ Cr 2 cos 2θ þDr 4 cos 2θ þE U ð3Þ ¼ G
cos 2θ þF cos 2θ r2
ð8Þ
2
b cos 2θ þH cos 2θ r2
ð9Þ
where the coefficients are found from the boundary condition of continuity of tractions at the interfaces and continuity of displacements ð2Þ n At r ¼ a; uð1Þ r ¼ ur þur
At r ¼ b;
and
urð2Þ þunr ¼ urð3Þ
ð1Þ ð2Þ ð1Þ uθð1Þ ¼ uθð2Þ þunθ ; srr ¼ sð2Þ rr ; srθ ¼ srθ
ð10Þ
ð3Þ ð2Þ ð3Þ n ð2Þ ð3Þ uð2Þ θ þuθ ¼ uθ ; srr ¼ srr ; srθ ¼ srθ
and
ð11Þ
The solution to the algebraic linear system of 8 8 equations is (obtained by Mathematica) 2
4
6
2
A ¼ En ða2 b Þð3a2 b ðα βÞð1 þ βÞ2 þ a6 ðα βÞðα þ βÞ2 þ b ðα þβÞð 1 þ β2 Þ 3a4 b ðα3 α2 β þαβð2 þ βÞ β2 ð2 þ βÞÞÞ=Δ2 n 2 2
2
2
2
B ¼ 2E a b ða b Þð 1 þ α Þðα βÞ=Δ2 8 4 C ¼ En b ð 1 þ βÞ2 ð1 þ βÞ þ a8 ðα βÞðα þβÞ2 3a4 b ð1 þβÞð α þα2 þ β 3αβ þ2β2 Þ 6 2 þ 2a2 b ð1 þ βÞð αð2 þ βÞ þ βð1 þ 2βÞÞ þ 2a6 b ð 3αβð1 þβÞ þα2 ð2 þβÞ þβ2 ð1 þ 2βÞÞ =Δ2 2
2
D ¼ 2En a2 b ða2 b Þð1 þ αÞðα βÞð1 þβÞ=Δ2 n 4 4
4
E ¼ E a b ð1 þ αÞða4 ðα2 β2 Þ þ b ð 1 þβ2 ÞÞ=Δ2 2
6
F ¼ 2En a2 b ð1 þαÞða6 ðα2 β2 Þ þb ð 1 þ β2 ÞÞ=Δ2 n 4
2
4 2
6
4
G ¼ E b ða2 b Þð3a b ðα βÞ2 ð1 þ βÞ þ b ð 1 þ βÞ2 ð1 þ βÞ þ a6 ð 1 þβÞðα2 β2 Þ a2 b ð1 þ βÞð 1 þ4α2 6αβ þ 3β2 ÞÞ=Δ2 2
2
2
4
6
H ¼ 2En b ða2 b Þð3a4 b ðα βÞ2 ð1 þ βÞ 3a2 b ðα βÞ2 ð1 þβÞ þb ð 1 þ βÞ2 ð1 þβÞ þa6 ð 1 þ βÞðα2 β2 ÞÞ=Δ2
ð12Þ
where En
8eμ1 μ2 8em μ2 ðκ1 þ 1Þ þμ1 ðκ 2 þ1Þ
ð13Þ
and 4
8
2
Δ2 ¼ 4ð6a4 b ðα βÞ2 ð1 þ βÞ2 þ a8 ðα2 β2 Þ2 þ b ð 1 þβ2 Þ2 2a6 b ð1 þ βÞð 3αβð1 þ βÞ þ α2 ð2 þ βÞ þ β2 ð1 þ 2βÞÞ 2 6
2a b ð1 þ βÞð 3αβð1 þ βÞ þ α2 ð2 þ βÞ þ β2 ð1 þ 2βÞÞ
ð14Þ
The stresses are in Appendix B. The stresses in the core are not constant, as in the dialatational eigenstrain case, but depend on r and θ. It may be noted that only certain components of them change sign with the shear moduli difference of the matrix/ring, unlike the dilatational eigenstrain case where all components do. 3. Material rotation due to shear eigenstrain in the ring In the case of shear eigenstrain in the ring/annulus there is material rotation in the core, ring and outer matrix that is calculated here. Inside the core the rotation is ωz ¼
3 Bðκ1 þ 1Þr 2 sin 2θ 2μ1
Inside the ring the rotation is 1 ðκ2 þ 1Þ 3Dðκ2 þ 1Þr 2 sin 2θ þ F sin 2θ ωz ¼ 2μ2 r2
ð15Þ
ð16Þ
and, in the limit of vanishingly small thickness h-0, the rotation in the layer is ωz ¼
a2 ε sin 2θðμ1 μ2 Þ 4a2 ε2 sin 2θ ¼ 2 r ðα βÞμ1 μ2 En r2
ð17Þ
The rotation in the outer matrix is ωz ¼
1 ðκ 1 þ1Þ H sin 2θ 2μ1 r2
ð18Þ
472
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
We next compute the jumps (denoted by brackets) of the rotations at the interfaces At r ¼ a, 2
2
6
½½ωZ ¼ b ð1 α2 Þð 2a6 ðα 2βÞðα βÞ þ3a4 b ðα βÞ2 b ð1 β2 ÞÞe sin 2θ=Δ3
ð19Þ
For small interphase thickness b ¼ a þ h, the Taylor series expansion gives ½½ωZ ¼ e sin 2θ
4eð2α2 3αβ þβ2 Þ sin 2θðh=aÞ þ O½h=a2 1 α2
ð20Þ
The jump at r ¼ b is 4
6
8
½½ωZ ¼ ð1 þ αÞð3a4 b ð 1 þ 3α 2βÞðα βÞð1 þ βÞ 2a2 b ð 1 þ 3α 2βÞðα βÞð1 þβÞ þb ð1 βÞ2 ð1 þ βÞ 2
þa8 ðα βÞ2 ðα þβÞ 2a6 b ðα βÞðαð2 þ βÞ βð1 þ 2βÞÞÞe sin 2θ=Δ3
ð21Þ
with 4
8
2
Δ3 ¼ 6a4 b ðα βÞ2 ð1 þ βÞ2 þ a8 ðα2 β2 Þ2 þ b ð1 β2 Þ2 2a6 b ðα βÞð1 þ βÞðαð2 þ βÞ βð1 þ 2βÞÞ 6
2a2 b ðα βÞð1 þ βÞðαð2 þ βÞ βð1 þ 2βÞÞ
ð22Þ
For small interphase thickness b ¼ a þ h, the jump of the rotation is ½½ωZ ¼ e sin 2θ
4eðα βÞð2 þ βÞ sin 2θðh=aÞ þ O½h=a2 1 α2
ð23Þ
Thus, for small thickness, to the leading order, the jumps of the rotations at r ¼ a and r ¼ b are constant depending on the eigenstrain only, as expected. 4. Embedded spherical annulus inhomogeneity with dilatational eigenstrain We also solve the three-dimensional problem of a spherical inhomogeneity annulus with dilatational eigenstrain. The solution follows Sokolnikof (1956) with the displacement being of the form ui ¼ φðrÞxi
and
φðrÞ ¼ A1 þ
A2 : r3
ð24Þ
The stresses and displacements are shown in Appendix C. The pressures p and q at r ¼ a and r ¼ b are determined from the equations for the displacements at r ¼ a and r ¼ bat r ¼ a, urð1Þ ¼ ea þ uð2;p;qÞ r
3
3
or
p pa3 qb b ðp qÞ þ ¼ eþ 3 3 3λ1 þ2μ1 ð3λ2 þ2μ2 Þðb a3 Þ 4μ2 ðb a3 Þ
or
eþ
ð25Þ
and at r ¼ b, ¼ urð3Þ eb þuð2;p;qÞ r
pa3 qb
3
3
ð3λ2 þ 2μ2 Þðb a3 Þ
þ
a3 ðp qÞ 3
4μ2 ðb a3 Þ
¼
q 4μ1
ð26Þ
The solutions to Eqs. (25) and (26) are 3
ð27Þ
3
ð28Þ
p ¼ 4eð3λ1 þ 2μ1 Þðμ1 μ2 Þð3λ2 þ 2μ2 Þða3 b Þ=Δ4 q ¼ 4eμ1 ð3λ2 þ 2μ2 Þð3λ1 þ 2μ1 þ 4μ2 Þða3 b Þ=Δ4 with 3
3
3
Δ4 ¼ 8ða3 b Þμ21 4ð4a3 þ 5b Þμ1 μ2 þ8ða3 b Þμ22 3
3
3
þ 6λ2 ð ð2a3 þ b Þμ1 þ2ða3 b Þμ2 Þ 3λ1 ð4a3 ð μ1 þμ2 Þ þ b ð3λ2 þ 4μ1 þ 2μ2 ÞÞ
ð29Þ
The solution for p also reveals the same behavior as for the 2-D problem, namely that the hydrostatic stress in the core changes sign with the difference of the moduli (μ1 μ2 ) and can become tensile when the annulus is stiffer in the shear modulus (for positive dilatational eigenstrain in the annulus). It may be also noted that the value of q and stresses induced by it coincides with the one for the Eshelby (1957, 1961) spherical inhomogeneity solution when the radius q goes to zero. 5. Interphases of small thickness The previously obtained stress fields are now expanded into Taylor series for small layer thickness h, with b ¼ a þ h.
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
473
5.1. Thin ring (a or o b) inhomogeneity under dilatational eigenstrain 5.1.1. Stresses in the core srr ¼ sθθ ¼ p ¼ En
2
ð 2Þðα βÞðb a2 Þ
ð30Þ
2
ð1 þ α 2βÞð1 α þ2βÞb 4βðα βÞa2
For b ¼ a þ h, srr ¼ sθθ ¼
4En ðα βÞðh=aÞ 2En ð 3 þðα 4βÞð3α 4βÞÞðα βÞðh=aÞ2 þ þ O½h=a3 1 α2 ð1 α2 Þ2
5.1.2. Stresses in the inhomogeneous thin ring (a or o b) ! ! 2 b a2 2 =Δ5 srr ¼ En 2ðα βÞa2 1 2 b ð1 αþ 2βÞ 1 þ 2 r r sθθ ¼ E
n
2
b 1þ 2 r
2
2ðα βÞa
!
2
ð1 α þ 2βÞb
a2 1 2 r
! =Δ5
2
Δ5 ¼ 4a2 ðα βÞβ þ b ð1 þ α 2βÞð1 α þ2βÞ
ð31Þ
ð32Þ
ð33Þ ð34Þ
For b ¼ a þ h, at r ¼ a 4En ðα βÞðh=aÞ 2En ð 3 þ ðα 4βÞð3α 4βÞÞðα βÞðh=aÞ2 þ þ O½h=a3 1 α2 ð1 α2 Þ2
srr ¼
sθθ ¼
ð35Þ
8eμ2 4En ð1 α þ 4βÞðα βÞðh=aÞ þ κ2 þ 1 ð1 αÞ2 ð1 þ αÞ 2
þ
2En ð 3 þðα 4βÞð3α 4βÞÞð 1 þα 4βÞðα βÞðh =a2 Þ 3
ð 1 þ αÞ ð1 þαÞ
2
þ O½h=a3
ð36Þ
If we define the “effective surface tension” in the interphase membrane as s0θθ sθθ h
ð37Þ
then (36) to the leading order gives s0θθ sθθ h ¼
8eμ02 8eðhμ2 Þ ¼ κ2 þ 1 κ2 þ 1
ð38Þ
where we defined as a surface shear modulus μ02 ¼ hμ2
ð39Þ
to be considered as the “effective” shear modulus in the membrane, as in the Gurtin and Murdoch (1975) surface elasticity model. Thus, in the leading order (zeroth) term, only the sθθ stress exists in the ring, and it depends only on the material constants of the ring, while all other stresses in the core, matrix and ring are of order ðh=aÞ. The next order terms in (35) and (36) show the correction effects to this model due to curvature. 5.2. Stresses in the thin ring inhomogeneity under shear eigenstrain The stresses of the finite thickness inhomogeneity ring are expanded for thin ring of thickness h, with b ¼ a þh. By Taylor series expansion we have at r ¼ a srr ¼ En
8e cos 2θðα 2βÞðh=aÞ þ O½h=a2 1 α2
ð40Þ
srθ ¼ En
8e sin 2θð2α βÞðh=aÞ þ O½h=a2 1 α2
ð41Þ
sθθ ¼
32e2 μ2 cos 2θ 8eðαð5 þ 3αÞ 4ð1 αÞβ 8β2 Þ cos 2θðh=aÞ En þ O½h=a2 κ 2 þ1 ð1 αÞ2 ð1 þαÞ
ð42Þ
which shows that also in the shear eigenstrain case only the stress sθθ is of order (1) in the ring, and all other stresses everywhere are of order h.
474
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
5.3. Stresses in the thin spherical annulus under dilatational eigenstrain In the annulus material of the sphere, for b ¼ a þ h, at r ¼ a srr ¼
4eð3λ1 þ 2μ1 Þðμ1 μ2 Þð3λ2 þ2μ2 Þðh=aÞ þ O½h=a2 3ðλ1 þ 2μ1 Þðλ2 þ 2μ2 Þ
ð43Þ
sθθ ¼
2eμ2 ð3λ2 þ 2μ2 Þ 4eðμ1 μ2 Þð3λ2 þ2μ2 Þð3λ1 λ2 þ 4μ22 þ2λ2 ðμ1 þ 3μ2 ÞÞðh=aÞ þ O½h=a2 λ2 þ 2μ2 3ðλ1 þ 2μ1 Þðλ2 þ 2μ2 Þ2
ð44Þ
Thus, also in the thin spherical annulus the stresses are to the leading order depending on the annulus material properties only and the effective interface surface tension stresses can be defined as before.
6. Driving forces on interfaces 6.1. Cylindrical rings with eigenstrain The driving force (per unit area) on an interface separating a region with eigenstrain from the matrix is given by Eshelby (1977) as f ¼ 〈sij 〉½½εnij (also, Markenscoff and Ni, 2010, for dynamics) where the force is positive along the outward normal to the surface enclosing the region with eigenstrain, and where the brackets denote the jump. In polar coordinates it is hh ii ð45Þ f ¼ 〈sij 〉 εnij ¼ 12 ðsðrrþ 1Þ þ sðrr 1Þ Þeþ 12 ðsðθθþ 1Þ þ sðθθ 1Þ Þe with positive along the outward normal of the region containing the eigenstrain, and is evaluated for the interfaces considered above. Negative driving force implies that external energy (Markenscoff, 2010) is needed to move the interface outwards. The driving forces were also evaluated for spherical inhomogeneities (Markenscoff, 2012) and strips with eigenstrain (Dundurs and Markenscoff, 2009). In the case of an inhomogeneous ring of finite thickness with dilatational eigenstrain (i) At the interface r ¼ a the driving force is 2
f¼
En ð4a2 ðα βÞ þ b ð1 3αþ 4βÞÞe
ð46Þ
2
4a2 ðα βÞβ b ð1 α2 þ 4αβ 4β2 Þ
For thin interphase layer (thin ring), b ¼ a þ h, and the driving force is f¼
4e2 μ2 8eEn ð1 α þβÞðα βÞðh=aÞ þ O½h=a2 þ κ 2 þ1 ð1 αÞ2 ð1 þ αÞ
ð47Þ
which shows that the leading term depends only on the material constants of the inhomogeneous ring, and is due to the stress sθθ , while the pressure contributes to the order ðh=aÞ term. The second term, proportional to the thickness, depends on the bimaterial combinations and, if it becomes positive because of high material mismatch, and of the same order as the leading term, then the interface will become unstable when f becomes zero. This second term is containing the interaction with the other boundary of the interphase layer. For a thick ring a=b ¼ 0, the driving force depends on the bimaterial combination and has reduced bimaterial dependence as is shown in the next section f¼
En ð1 3α þ 4βÞe 1 α2 þ4αβ 4β2
where En ¼
8eμ1 μ2 μ2 ðκ 1 þ 1Þ þ μ1 ðκ2 þ 1Þ
ð48Þ
(ii) At the interface r ¼ b, the driving force is for thick layer 2
f¼
En ð b ð1 α þ 2βÞ þ2a2 ð α þβÞÞe 2 4a2 ðα βÞβ b ð1 α2 þ 4αβ 4β2 Þ
ð49Þ
For a thin ring b ¼ a þ h, f¼
4e2 μ2 4eEn ð1 α þ2βÞðα βÞðh=aÞ þ þ O½h=a2 κ 2 þ1 ð1 αÞ2 ð1 þαÞ
ð50Þ
For a=b ¼ 0 (thick ring)
f¼
En e 1 þ α 2β
where En ¼
8eμ1 μ2 μ2 ðκ 1 þ1Þ þ μ1 ðκ2 þ 1Þ
ð51Þ
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
475
6.2. For sphere (3D) with dilatational eigenstrain The driving force is þ 1Þ 1Þ f ¼ 12 ðsðrrþ 1Þ þsðrr 1Þ Þe þ 12 ðsðθθþ 1Þ þsðθθ 1Þ Þe þ 12 ðsðφφ þsðφφ Þe
ð52Þ
The driving force, f , evaluated at r ¼ a yields 3
3
3
f ¼ 6e2 ð3λ2 þ 2μ2 Þð 2μ1 ða3 b Þð3λ1 þ 2μ1 Þ þ μ2 ðð6a3 3b Þλ1 þ2μ1 ð2a3 þ b ÞÞÞ=Δ6
ð53Þ
For b ¼ a þ h (thin annulus) f¼
2e2 μ2 ð3λ2 þ 2μ2 Þ λ2 þ 2μ2 4e2 ðμ1 μ2 Þð3λ2 þ2μ2 Þð6λ2 ðμ1 þμ2 Þ þ 4μ2 ð2μ1 þ μ2 Þ þ 3λ1 ð3λ2 þ 4μ2 ÞÞðh=aÞ 3ðλ1 þ2μ1 Þðλ2 þ 2μ2 Þ2
þ O½h=a2
ð54Þ
It may be noted that in the first term of (54) ð3λ þ 2μÞ=ðλ þ 2μÞ ¼ ð1 þvÞ=ð1 vÞ. The driving force f , evaluated at r ¼ a, yields 3
3
f ¼ 6e2 ð3λ2 þ 2μ2 Þð μ1 ða3 b Þð3λ1 þ2μ1 Þ þμ2 ð3a3 λ1 þ 2μ1 ða3 þ 2b ÞÞÞ=Δ6
ð55Þ
where 3
3
3
Δ6 ¼ 8μ21 ða3 b Þ 4μ1 μ2 ð4a3 þ 5b Þ þ 8μ22 ða3 b Þ 3
3
3
3
3
þ6λ2 ð μ1 ð2a þ b Þ þ 2μ2 ða b ÞÞ 3λ1 ð4a3 ð μ1 þμ2 Þ þ b ð3λ2 þ4μ1 þ 2μ2 ÞÞ
ð56Þ
For b ¼ a þ h (thin annulus) f¼
2e2 μ2 ð3λ2 þ 2μ2 Þ 2e2 ðμ1 μ2 Þð3λ2 þ2μ2 Þ2 ð3λ1 þ 2μ1 þ 4μ2 Þðh=aÞ þO½h=a2 λ2 þ 2μ2 3ðλ1 þ2μ1 Þðλ2 þ 2μ2 Þ2
ð57Þ
The leading term of the driving force at the interface for small thickness (in (54) and (57)) is found to be equal to the value of the driving force for a spherical inclusion of material 2 (in a matrix of material) with dilatational eigenstrain, as derived by Gavazza (1977), Eshelby (1970), and Markenscoff and Ni (2010). 7. Finite thickness surface layer with eigenstrain: surface tension and driving forces 7.1. Surface layer of same material as the matrix We now consider a free surface layer of finite thickness and of the same material as the matrix, containing eigenstrain εnij ¼ δij e for a or o b, εnij ¼ 0 for r o a,and at r ¼ b; srr jr ¼ b ¼ 0
ð58Þ
The solution in terms of the stress functions for the core (with index1) and the surface layer (with index 2) are U ð1Þ ¼ 2μAer2 ;
U ð2Þ ¼ 2μefBr 2 þ Ca2 log rg;
ð59Þ
with the coefficients obtained from the boundary conditions ð2Þ ð1Þ ð2Þ srr ¼ sð2Þ and srr ¼0 uð1Þ r r ¼ a ¼ ea þ ur r ¼ a ; rr r ¼ a r¼a r¼b
ð60Þ
as 2
A¼
a2 b 2
b ðκ þ 1Þ
;
ð1Þ sð1Þ rr ¼ sθθ ¼
sð2Þ θθ ¼
B¼
a2 2
b ðκ þ 1Þ
;
C¼
2
4eμ ðb a2 Þ ; 2 ðκ þ 1Þ b 2
4eμ a2 ðb þ r 2 Þ ; 2 ðκ þ 1Þ b r2
2 κþ1
ð2Þ srr ¼
sð2Þ θθ
r¼a
ð61Þ 2
4eμ a2 ðb r 2 Þ ; 2 ðκ þ 1Þ b r2
2 4eμ ðb a2 Þ sð2Þ rr r ¼ a ¼ 2 ðκ þ1Þ b
ð62Þ
2
¼
4eμ ðb þa2 Þ 2 ðκ þ1Þ b
ð63Þ
For small thickness h approximation, (i) the stresses in the core
ð1Þ sð1Þ rr ¼ sθθ ¼
8eμðh=aÞ 12eμðh=aÞ2 þ þ O½h=a3 κ þ1 κþ1
ð64Þ
476
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
(ii) the stresses in the surface layer 8eμðh=aÞ 12eμðh=aÞ2 ð2Þ þ þO½h=a3 srr ¼ r¼a κ þ1 κ þ1 8eμ 8eμðh=aÞ 12eμðh=aÞ2 ð2Þ þ 2 þ O½h=a3 sθθ ¼ r¼a κþ1 κþ1 a ðκ þ1Þ
ð65Þ
Eq. (65b) provides the relation between the eigenstrain and the surface tension, sθθ , and inversely, it associates an eigenstrain to a surface tension (Gao, 1994). We may define, as in (37) the “effective surface tension” as s0θθ ¼ sθθ h and μ0 ¼ hμ; so that from (65) to the leading term 0 sθθ ¼ 8eμ0 =ðκ þ 1Þ. The driving force f at r ¼ a 2
f¼
4ð 2a2 þ b Þe2 μ
ð66Þ
2
b ðκ þ 1Þ
and, for a thin surface layer, b ¼ a þ h, at r ¼ a f¼
4e2 μ 16e2 μðh=aÞ þ þ O½h=a2 κþ1 κþ1
ð67Þ
The driving force f , at r ¼ b f¼
4a2 e2 μ
ð68Þ
2
b ðκ þ 1Þ
and, for a thin layer, b ¼ a þh, f¼
4e2 μ 8e2 μðh=aÞ þ þO½h=a2 κþ1 κ þ1
ð69Þ
7.2. For a surface layer of different material constants The stresses are obtained from the stress functions in the core (1) and surface inhomogeneous layer (2): U ð1Þ ¼ 2μ1 Aer 2 ;
U ð2Þ ¼ 2μ2 efBr2 þ Ca2 log rg;
and boundary conditions urð1Þ r ¼ a ¼ ea þ urð2Þ r ¼ a ;
ð2Þ sð1Þ rr r ¼ a ¼ srr r ¼ a
ð70Þ ð2Þ srr ¼0 r¼b
and
ð71Þ
2
ða2 b Þμ2
A¼
B¼
2
ð72Þ
2
μ2 ða2 b Þðκ 1 1Þ μ1 ða2 κ2 a2 þ2b Þ a2 μ1 2
ð73Þ
2
μ2 ð a2 þ b Þðκ1 1Þ þ μ1 ða2 κ2 a2 þ 2b Þ 2
C¼
2b μ1 μ2
2 ða2 b Þðκ
ð74Þ
2 2 2 1 1Þ μ1 ða κ 2 a þ 2b Þ
and ð1Þ srr ¼ sð1Þ θθ ¼
ð2Þ srr ¼
2
2
μ2 ða2 b Þðκ1 1Þ þ μ1 ð a2 κ2 þ a2 2b Þ
2 4eμ1 μ2 a2 ð b þr 2 Þ 2 2 2 2 r ðμ2 ð a þ b Þðκ1 1Þ þμ1 ða2 κ 2 a2 þ 2b ÞÞ
ð2Þ srr ¼ r¼a
ð2Þ sθθ ¼
2
4eμ1 μ2 ða2 b Þ
ð75Þ
ð76Þ
2
4eμ1 μ2 ða2 b Þ 2
2
μ2 ða2 b Þðκ1 1Þ þ μ1 ð a2 κ 2 þa2 2b Þ
2 4eμ1 μ2 a2 ðb þ r 2 Þ 2 2 2 2 r μ2 ðð a þ b Þðκ1 1Þ þμ1 ða2 κ 2 a2 þ 2b ÞÞ
ð77Þ
ð78Þ
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
sð2Þ θθ
477
2
r¼a
¼
4eμ1 μ2 ða2 þ b Þ μ2
2 ða2 b Þðκ
1 1Þ μ1
ða2 κ
ð79Þ
2 2 2 a þ 2b Þ
For small thickness approximation, (i) the stresses in the core ð1Þ sð1Þ rr ¼ sθθ ¼
8eμ2 ðh=aÞ 4eμ2 ð μ1 ðκ2 7Þ þ4μ2 ðκ1 1ÞÞðh=aÞ2 þ þ O½h=a3 κ2 þ 1 μ1 ð1 þ κ2 Þ2
ð80Þ
(ii) the stresses in the surface layer 8eμ2 ðh=aÞ 4eμ2 ðμ1 ð κ 2 þ7Þ þ 4μ2 ðκ1 1ÞÞðh=aÞ2 þ sð2Þ þ O½h=a3 rr r ¼ a ¼ κ 2 þ1 μ1 ðκ 2 þ1Þ2 8eμ2 2ð 1 þ α 4βÞðh=aÞ þEn sð2Þ ¼ θθ r¼a κ2 þ 1 ð1 αÞ2 þ
4eμ2 ðμ1 ðκ2 7Þ 4μ2 ðκ1 1ÞÞðμ1 ðκ2 3Þ 2μ2 ðκ1 1ÞÞðh=aÞ2 μ21 ðκ2 þ 1Þ3
þ O½h=a3
ð81Þ
The driving force f at r ¼ a 2
f¼
4e2 μ1 μ2 ð2a2 b Þ μ2
2 ð a2 þ b Þðκ
ð82Þ
2 2 2 1 1Þ þμ1 ð a þ 2b þa κ 2 Þ
for a thin surface layer, b ¼ a þh, f¼ ¼
4e2 μ2 8e2 μ2 ðμ1 ðκ2 þ 3Þ þμ2 ðκ 1 1ÞÞðh=aÞ þ þ O½h=a2 κ 2 þ1 ðκ2 þ 1Þ2 μ1 4e2 μ2 4eð1 αþ βÞðh=aÞ þEn þ O½h=a2 κ2 þ 1 ð1 αÞ2
ð83Þ
and the driving force f , at r ¼ b, 2
f¼
4e2 μ1 μ2 ð2a2 b Þ 2
ð84Þ
2
μ2 ð a2 þ b Þðκ1 1Þ þμ1 ða2 κ 2 a2 þ2b Þ
for a thin layer, b ¼ a þ h, f¼ ¼
4e2 μ2 8e2 μ2 ð2μ1 þ μ2 ðκ1 1ÞÞðh=aÞ þ þ O½h=a2 κ 2 þ1 μ1 ðκ2 þ 1Þ2 4e2 μ2 2eð1 αþ 2βÞðh=aÞ þEn þ O½h=a2 κ2 þ 1 ð1 αÞ2
ð85Þ
Eqs. (83) and (85) show the coupling between the driving forces of the surfaces of the layer. While the leading order terms in (83) and (85) are the same, the O½h=a terms are different due to coupling. 7.3. Inhomogeneous spherical surface layer with eigenstrain ð86Þ
p ¼ e=Δ7 where 3
Δ7 ¼
1 b a3 3 3 3λ1 þ 2μ1 4μ2 ða3 b Þ ða3 b Þð3λ2 þ2μ2 Þ
ð87Þ
ð1Þ ð1Þ sð1Þ rr ¼ sθθ ¼ sφφ ¼ p ¼ e=Δ7
ð88Þ
3 3 3 sð2Þ rr ¼ 4eμ2 a ð b þ r Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8 r
ð89Þ
3
3 3 sð2Þ rr r ¼ a ¼ 4eμ2 ða b Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8 ;
ð2Þ srr ¼ 0; r¼b
ð90Þ
478
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482 ð2Þ 3 3 3 sφφ ¼ sð2Þ θθ ¼ 2eμ2 a ðb þ 2r Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8 r
ð91Þ
ð2Þ sφφ
¼ sð2Þ θθ
ð92Þ
ð2Þ sφφ
ð2Þ ¼ sθθ
3
r¼a
r¼b
3
r¼a
r¼b
¼ 2eμ2 ðb þ 2a3 Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8 3ea3
¼
ð93Þ
3
2ð a3 þ b ÞΔ7
where 3
3
3
3
Δ8 ¼ 3b λ2 ð3λ1 þ2μ1 Þ þ2ð3ð2a3 þ b Þλ1 þ 6ð a3 þ b Þλ2 þ 2ð2a3 þ b Þμ1 Þμ2 3
þ 8ð a3 þ b Þμ22
ð94Þ
For small thickness approximation, (i) the stresses in the core ð1Þ ð1Þ srr ¼ sð1Þ θθ ¼ sφφ ¼
4eμ2 ð3λ2 þ 2μ2 Þðh=aÞ þO½h=a2 λ2 þ 2μ2
ð95Þ
(ii) the stresses in the surface 4eμ2 ð3λ2 þ 2μ2 Þðh=aÞ ð2Þ srr ¼ þ O½h=a2 r¼a λ2 þ2μ2 ð2Þ sφφ
r¼a
¼ sð2Þ θθ
2eμ2 ð3λ2 þ2μ2 Þ λ2 þ 2μ2 4eμ2 ð3λ2 þ 2μ2 Þð3λ1 λ2 þ4μ22 þ 2λ2 ðμ1 þ3μ2 ÞÞðh=aÞ r¼a
ð96Þ
¼
ð3λ1 þ 2μ1 Þðλ2 þ2μ2 Þ2
þO½h=a2
ð97Þ
Again, the leading term defines the surface tension with μ02 ¼ hμ2 (defined as a surface shear modulus) 1 þv2 s0φφ ¼ s0θθ r ¼ a hsφφ hsθθ ¼ 2μ02 e r¼a 1 v2 in agreement with Gao (1994). The next order terms give the curvature dependence of the surface tension stresses. The driving force f , at r ¼ a layer 3
f ¼ 6e2 μ2 ð2a3 b Þð3λ1 þ 2μ1 Þð3λ2 þ 2μ2 Þ=Δ8
ð98Þ
for thin surface layer, b ¼ a þh, f¼
2e2 μ2 ð3λ2 þ2μ2 Þ λ2 þ2μ2
þ
4e2 μ2 ð3λ2 þ 2μ2 Þð6λ2 ðμ1 þ μ2 Þ þ 4μ2 ð2μ1 þ μ2 Þ þ3λ1 ð3λ2 þ4μ2 ÞÞðh=aÞ ð3λ1 þ 2μ1 Þðλ2 þ 2μ2 Þ2
þO½h=a2
ð99Þ
The driving force f , at r ¼ b, for thick layer is f¼
3e2 a3 3
2ð a3 þb ÞΔ7
ð100Þ
for thin layer, b ¼ a þ h, f¼
2e2 μ2 ð3λ2 þ2μ2 Þ 2e2 μ2 ð3λ2 þ 2μ2 Þ2 ð3λ1 þ 2μ1 þ4μ2 Þðh=aÞ þ þ O½h=a2 λ2 þ2μ2 ð3λ1 þ 2μ1 Þðλ2 þ 2μ2 Þ2
ð101Þ
where, again, in (99) and (101) in the leading term we can set ð3λ þ2μÞ=ðλ þ2μÞ ¼ ð1 þ vÞ=ð1 vÞ, and, thus, have explicit dependence in terms of the shear modulus and Poisson0 s ratio of the interface. From Eqs. (67) and (69), (83) and (85), and (99) and (101), we observe that the driving forces at the free surface r ¼ b and the interface r ¼ a are equal to each other in the zeroth order term for a thin layer and they depend only on the surface moduli and the eigenstrain, as expected, but the next order ðh=aÞ (where a is the core radius) term may become large (for some bimaterial combination differences) and be of the order of the leading surface tension term, in which case, the free surface and matrix/free surface interfaces may become unstable if f becomes zero.
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482
479
8. Reduction of constants in stress dependence for inhomogeneities with eigenstrain The CLM theorem (Cherkaev et al., 1992) has established a general way to prove reducibilty of material constants in the dependence of the stress in composites, and it was extended further by Dundurs and Markenscoff (1993) for linear dependence of the material constants on the position, and by Markenscoff and Jasiuk (1998) to include body force loadings. For loading of eigenstrain, Jasiuk and Boccara (2002) have obtained some general conditions based on compatibility, but still leave unresolved the case where the eigenstrain is a step function discontinuity on the boundary of the inhomogeneity, as with constant eigenstrain inclusions. Here we show that the interface conditions (in the presence of eigenstrain on one side) allow reduction of constants for general geometry of interfaces. The interface conditions of continuity of tractions and displacement are equivalently expressed (Dundurs, 1990; Markenscoff and Wheeler, 1996) as continuity of the tangential strains and continuity of the change of curvature at the interface, respectively ð1Þ ð2Þ εss ¼ εss
ð102Þ
Δκ1 ¼ Δκ 2
ð103Þ
where the change of curvature (Dundurs, 1990) (also verified by Markenscoff and Wheeler, 1996) is in terms of normal and tangential coordinates, n, s, respectively, and has the form Δκ ¼ 2
∂εns ∂εss kεnn ∂s ∂n
ð104Þ
The above conditions can be, in turn, expressed in terms of stresses and the eigenstrains, since, with the use of the stress– strain relation (Dundurs, 1990) 1 1 sss ð3 kÞðsnn þsss Þ þεnss ð105Þ εss ¼ 2μ 4 so that the change of curvature in (104) is written in terms of stresses and eigenstrains 1 ∂sns ∂sss ∂εn ∂εn ð5 þkÞ ðk þ 1Þ 2κðk 1Þsnn þ2 ns ss κεnnn Δκ ¼ 8μ ∂s ∂n ∂s ∂s
ð106Þ
Thus, the interface compatibility conditions can be written in terms of the Dundurs constants as (Dundurs, 1990) ð2Þ nð2Þ nð1Þ nð2Þ nð1Þ ð1 αÞsss ð1 þ αÞsð1Þ ss þ2ðα 2βÞsnn ¼ 16mfεss εss þη2 εzz η1 εzz g ð2Þ ∂sss ∂sð1Þ ∂sns ð1 þαÞ ss þ 2ð3α 2βÞ 4κβsnn ∂n ∂n ∂s ( ) ! ! ! nð2Þ nð1Þ nð2Þ nð1Þ nð2Þ nð1Þ ∂εns ∂εns ∂εss ∂εss ∂εzz ∂εzz nð2Þ nð1Þ nð2Þ nð1Þ η1 ¼ 16m 2 κðεnn εnn Þ η2 κðη2 εzz η1 εzz Þ ∂s ∂s ∂n ∂n ∂n ∂n
ð107Þ
ð1 αÞ
ð108Þ
with m¼
μ1 μ2 μ2 ðκ 1 þ1Þ þ μ1 ðκ 2 þ 1Þ
ð109Þ
Eqs. (91) and (92) show that the stresses are functions of the Dundurs constants α and β with the m multiplier in front. We will not consider here the presence of εzz , when the constants reduction does not go through, and we will show that the CLM theorem goes through for m, by writing m¼
μ1 μ2 1 1 ¼ ¼ ðκ 1 þ1Þ=μ1 þðκ 2 þ 1Þ=μ2 ðA1 þ S1 Þ þðA2 þ S2 Þ μ2 ðκ 1 þ1Þ þ μ1 ðκ 2 þ 1Þ
ð110Þ
where A and S are, respectively, the voluminal and shear compliances A¼
κ1 ; 2μ
S¼
1 μ
ð111Þ
For the CLM “shift” transformation in the compliances (Cherkaev et al., 1992) Ai ¼ Ai þ a;
Si ¼ Si a;
i ¼ 1; 2
ð112Þ
and, where a is an arbitrary constant. The value of m in (110) remains unchanged under the transformations (112), so that m has reduced bimaterial dependence. Since already the Dundurs constants are a particular “reduced constants” combination, we have shown that the stresses have reduced dependence on elastic constants for inhomogeneities with in-plane eigenstrain loading for general geometrical shape of the inhomogeneities. Solutions for loading with the out of plane eigenstrain εzz are currently investigated by Dundurs.
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9. Conclusions Solutions for annular inhomogeneities with eigenstrain embedded in a matrix of different elastic constants have been obtained. A feature of our problem/solution is the material rotation inside the inclusion (a feature of the Frank–Bilby interface model) induced by shear eigenstrain. The stresses in the core vary with the difference of the shear moduli of the annulus and matrix both in 2 and 3-D, and they may become tensile (with dilatational eigenstrain in the annulus) for a stiffer annulus in the shear modulus. This inhomogeneity solution elucidates further the Eshelby inclusion result for dilatational eigenstrain that the value of the bulk modulus of the matrix is “irrelevant” (Eshelby, 1961) (and only the shear modulus matters). For small interphase thicknesses, by Taylor series expansion, the terms up to the second order are obtained for all quantities. The driving (Eshelby) forces on the interfaces (due to the eigenstrain) are also computed, and, for finite thickness, they account for the interaction of one interface of the annulus with the other. This interaction may become significant for large material mismatch and may render the interface unstable. In the limit of zero thickness the model reduces to a Gurtin–Murdoch membrane model, but for shear surface eigenstrain it may also possess “surface rotation”, of the Frank–Bilby type. The next order terms in the expansion provide the curvature “corrections” to the Gurtin–Murdoch model. While the presented solutions are general and applicable to all scales and sizes of the annulus, applications, in particular, include the modeling of interphases in coated particles used as “re-enforcement” in thermal/electric conductive adhesives, where the surface strengthening properties become important for reliability in electronic packaging.
Acknowledgment X.M. is grateful for the support of the National Science Foundation, Grant CMS #1129888. Appendix A. Dilatational eigenstrains in the inhomogeneous cylindrical ring Core (r o a) under pressure p: U ð1Þ ¼
E¼
1 2 pr ; 2
srr ¼ sθθ ¼ p;
ur ¼
p 2μ1
1 ðk1 1Þr 2
ðA:1Þ
8eμ1 μ2 μ1 ðk2 þ 1Þ þμ2 ðk1 þ 1Þ
ðA:2Þ
Ring (a or o b) under pressure p at r ¼ a: U ð2;pÞ ¼ pa2 =ðb a2 Þfð1=2Þr 2 b log rg ( ) ( ) ( ) 2 2 2 pa2 b pa2 b p a2 1 b ð2;pÞ r ð2;pÞ ðk sð2;pÞ ¼ 1 ¼ 1 þ ¼ 1Þ þ ; s ; u 2 rr r θθ 2 2 2μ2 b2 a2 2 r2 r2 r2 b a2 b a2 2
2
Ring (a or o b) under pressure q at r ¼ b: 2 2 2 qb 1 qb a2 qb a2 r 2 þ a2 log r ; sð2;qÞ ¼ 2 1 þ 2 ; sð2;qÞ ¼ 2 1 2 U ð2;qÞ ¼ 2 rr θθ 2 r r b a2 b a2 b a2 ( ) 2 2 p a2 1 b q b 1 a2 ðk2 1Þr þ 2 ; uð2;qÞ ¼ ¼ ðk2 1Þb 2 : uð2;pÞ r r 2 2 2μ2 b a2 2 2μ2 b a2 2 r b
ðA:3Þ
ðA:4Þ
ðA:5Þ
Matrix under pressure q at r ¼ b: U ð3Þ ¼ qb log r; 2
ð3Þ srr ¼
2
qb ; r2
sð3Þ θθ ¼
2
qb ; r2
q urð3Þ r ¼ b ¼ b 2μ1
ðA:6Þ
where α, β are the Dundurs (1969) constants
α¼
μ2 ðκ 1 þ 1Þ μ1 ðκ2 þ 1Þ ; μ2 ðκ 1 þ 1Þ þ μ1 ðκ2 þ 1Þ
β¼
μ2 ðκ1 1Þ μ1 ðκ 2 1Þ μ2 ðκ1 þ 1Þ þμ1 ðκ 2 þ1Þ
ðA:7Þ
Appendix B. Shear eigenstrains in the ring B.1. Stresses for the core material 2
4
6
srr ¼ En 2ða2 b Þð3a2 b ðα βÞð1 þ βÞ2 þ a6 ðα βÞðα þ βÞ2 þ b ðα þβÞð 1 þ β2 Þ 2
3a4 b ðα βÞðα2 þ βð2 þβÞÞÞε cos 2θ=Δ7
X. Markenscoff, J. Dundurs / J. Mech. Phys. Solids 64 (2014) 468–482 2
6
481
2
2
2
2
srθ ¼ En 2ða2 b Þð a6 ðα βÞðα þ βÞ2 b ðα þβÞð 1 þ β2 Þ 3a2 b ðα βÞð2r 2 ð 1 þ α2 Þ þb ð1 þβÞ2 Þ 4 2
2
þ3a b ðα βÞðα þβð2 þ βÞÞÞε sin 2θ=Δ7 2
6
sθθ ¼ En 2ða2 b Þð a6 ðα βÞðαþ βÞ2 b ðα þ βÞð 1 þβ2 Þ 3a2 b ðα βÞð4r 2 ð 1 þα2 Þ þ b ð1 þ βÞ2 Þ 2
þ3a4 b ðα βÞðα2 þβð2 þ βÞÞÞε cos 2θ=Δ7
ðB:1Þ
B.2. Stresses for the ring material The expressions of the stresses are two long to be presented, but can be easily computed from the stress functions. The thin ring/annulus approximations are presented in Section 5. B.3. Stresses for the outer material 2
2
2
2
6
2
srr ¼ En 2ða2 b Þb ð3a4 b ð3b 4r 2 Þðα βÞ2 ð1 þβÞ þb ð3b 4r 2 Þð 1 þ βÞ2 ð1 þβÞ 2
2 4
2
þa ð3b 4r Þðα β Þð 1 þ βÞ 3a b ð1 þ βÞð 4r ðα βÞ2 þb ð 1 þ 4α2 6αβ þ3β2 ÞÞÞε cos 2θ=ðr 4 Δ7 Þ 6
2
2
2
n
2
2
4 2
2
2
6
2
srθ ¼ E 2ða2 b Þb ð3a b ð3b 2r 2 Þðα βÞ2 ð1 þ βÞ þ b ð3b 2r 2 Þð 1 þβÞ2 ð1 þ βÞ 2
4
2
þa6 ð3b 2r 2 Þðα2 β2 Þð 1 þ βÞ 3a2 b ð1 þ βÞð 2r 2 ðα βÞ2 þb ð 1 þ 4α2 6αβ þ3β2 ÞÞÞε sin 2θ=ðr 4 Δ7 Þ 4
n
2
2
4 2
6
2
2
sθθ ¼ E 6b ða b Þð3a b ðα βÞ ð1 þ βÞ þ b ð 1 þβÞ ð1 þ βÞ 4
þa6 ðα2 β2 Þð 1 þ βÞ a2 b ð1 þ βÞð 1 þ 4α2 6αβ þ 3β2 ÞÞε cos 2θ=ðr 4 Δ7 Þ
ðB:2Þ
where 4
8
Δ7 ¼ 6a4 b ðα βÞ2 ð1 þ βÞ2 þ a8 ðα2 β2 Þ2 þ b ð 1 þ β2 Þ2 6 2
6
2a b ðα βÞð1 þ βÞðαð2 þ βÞ βð1 þ 2βÞÞ 2a2 b ðα βÞð1 þβÞðαð2 þβÞ βð1 þ 2βÞÞ
ðB:3Þ
Appendix C. Dilatational eigenstrains in the inhomogeneous spherical annulus (a) Spherical core under pressure: ui ¼ φðrÞxi for 0 o r oa φðrÞ ¼
p ; 3λ þ 2μ
srr ¼ p;
sφφ ¼ sθθ ¼ p;
ui ¼
p x 3λ þ 2μ i
ðC:1Þ
3
ðC:2Þ
3
ðC:3Þ
p ¼ 4eð3λ1 þ2μ1 Þðμ1 μ2 Þð3λ2 þ 2μ2 Þða3 b Þ=Δ4 q ¼ 4eμ1 ð3λ2 þ 2μ2 Þð3λ1 þ 2μ1 þ 4μ2 Þða3 b Þ=Δ4 (b) Spherical annulus under pressure p and q: ui ¼ φðrÞxi for a or o b φðrÞ ¼
pa3 qb
3
3
3
ð3λ2 þ 2μ2 Þðb a3 Þ 3
þ
1 a3 b ðp qÞ r 3 4μ2 ðb3 a3 Þ
ðC:4Þ
3
a3 b p q 3 3 r 3 b3 a3 b a 3 3 3 3 ¼ 4eð3λ2 þ 2μ2 Þð ða3 b Þr 3 μ1 ð3λ1 þ 2μ1 Þ þ ð3a3 ð b þ r 3 Þλ1 þ 2ð2b r 3 þa3 ð 3b þ r 3 ÞÞμ1 Þμ2 Þ=ðr 3 Δ4 Þ
srr ¼
pa3 qb
3
3
a3
ðC:5Þ
3
a3 b p q 2r 3 b3 a3 b 3 3 3 3 ¼ 2eð3λ2 þ 2μ2 Þð 2ða3 b Þr 3 μ1 ð3λ1 þ2μ1 Þ þð3a3 ðb þ 2r 3 Þλ1 þ 2ð3a3 b þ2ða3 þ 2b Þr 3 Þμ1 Þμ2 Þ=ðr 3 Δ4 Þ
sθθ ¼
pa3 qb
þ
ðC:6Þ
3
(c) Infinite matrix under pressure q at r ¼ b, a2 ¼ 1: ui ¼ φðrÞxi , φðrÞ ¼ ðqb =4μ1 Þð1=r 3 Þ 3
srr ¼
qb ; r3
3
sφφ ¼ sθθ ¼
qb ; 2r 3
3
ui ¼
qb 1 xi 4μ1 r 3
ðC:7Þ
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