- Email: [email protected]
Evaluation of stress concentration in multi-wedge systems with functionally graded wedges
International Journal of Engineering Science 61 (2012) 87–93
Contents lists available at SciVerse ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Evaluation of stress concentration in multi-wedge systems with functionally graded wedges Alexander Linkov ⇑, Liliana Rybarska-Rusinek Rzeszow University of Technology, Powstancow Warszawy 12, 35-959 Rzeszow, Poland
a r t i c l e
i n f o
Article history: Received 29 February 2012 Accepted 2 April 2012 Available online 24 July 2012 Keywords: Functionally graded material Multi-wedge systems Stress concentration
a b s t r a c t A common apex of structural elements with different physical properties is a point of stress concentration leading to dangerous effects like fracture, corrosion and fatigue. As the concentration may be reduced by proper functional grading of the elements, there is need in evaluating the influence of various grading on the singularity of a field at the apex. In the paper, we present and compare two efficient methods (exact and approximate) to reach this goal. The exact method employs eigenfunctions of the ordinary differential equation, to which the partial differential equation of a considered problem is reduced after the Mellin’s transform. The approximate method consists in piece-wise constant approximation of the physical modules of a graded wedge. Numerical examples for antiplane strain illustrate high efficiency of the methods and beneficial influence of functional grading when the latter provides continuity of shear modulus. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Singular points of a medium, such as crack tips, corners of notches and common apexes of grains, are sources of high field concentration. The concentration manifests itself in unfavorable effects like sparkling, energy leak off, fracture, corrosion, fatigue, and in the specific geometry of nano-structures at intersections of dislocation arrays. In view of the significance of singular points, the (multi-) wedge systems have been the subject of numerous studies starting from the papers by Tranter (1948) and Williams (1952). Comprehensive reviews may be found in papers of Blinova and Linkov (1995), Dempsey and Sinclair (1979), Linkov and Koshelev (2006a), Linkov and Rybarska-Rusinek (2008), Paggi and Carpinteri (2008) and Sinclair (1999). Most of the work concerns with homogeneous wedges and ideal, or smooth, or Coulomb friction contacts. Meanwhile, there is need in accounting for more complicated contact conditions and inhomogeneity of wedges. Studying of the first complicating factor, initiated by Mishuris and Kuhn (2001), has resulted in the general method (Linkov & Rybarska-Rusinek, 2008, 2010), applicable to a wide range of interfacial conditions. The importance of the second factor (inhomogeneity of wedges) was clearly recognized for cracks by Erdogan (1983). Its influence on the singularity at a crack tip has been studied in the papers by Eischen (1987), Erdogan and Wu (1997) and Konda and Erdogan (1994). Investigation of an arbitrary functionally graded material (FGM) near a (multi-) wedge point has been started in the papers of Carpinteri and Paggi (2005), Carpinteri and Paggi (2006). The authors wrote explicitly the equation for a FG wedge with the Young modulus depending on the polar angle. They outlined that from the mathematical point of view the only difference, as compared with a homogeneous wedge, is that the eigenfunctions differ from those for a homogeneous wedge. This suggests straightforward application of the mathematical techniques developed for homogeneous wedges to FGM. The authors also noted that for the modulus exponentially depending on the angle, the equation has constant coefficients what allows one to easily find needed ⇑ Corresponding author. E-mail address: [email protected] (A. Linkov). 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2012.06.012
88
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
eigenfunctions. For this particular dependence, they studied the cases of a crack inside an angular FGM and trimaterial with a single FG transitional wedge. The numerical results obtained clearly show that ‘‘the presence of a FG intermediate material is favorable since it significantly reduces the order of the stress singularity as compared with the same trimaterial junction involving homogeneous different materials’’ (Paggi and Carpintery, 2008, p. 020801-19). This stimulates further researches in this area and development of efficient general methods for accounting for FGM. The present paper aims to reach this goal. We consider the general case when physical properties, for instance shear modulus and/or Poisson’s ratio of a FGM, change arbitrary with the angle. Then finding of the eigenfunctions becomes a part of the problem. They may be found using power series and working out a subroutine to determine the coefficients of the expansions numerically. An alternative way to account for FGM may be based on the favorable feature of the method, developed for homogenous wedges (Linkov & Koshelev, 2006a, 2006b), to provide accurate (at least three correct digits) results for systems of very thin wedges (with angles of angular minutes). This suggests replacing a FG wedge by a system of homogeneous wedges what means piece-wise constant approximation of the modules of the FGM. Below we employ the both approaches and show that the second one provides very accurate evaluation of the singularity. For brevity, we consider antiplane strain. Obviously, the piece-wise constant approximation is also applicable to plane strain and plain stress, for which one can use the mentioned efficient method (Linkov & Koshelev, 2006a).
2. Problem formulation Consider antiplane strain of a system of m elastic wedges. The system may be open (Fig. 1a) or closed (Fig. 1b). The angle of a wedge is denoted H, its shear modulus is l. One of the wedges, marked by the subscript 0, is functionally graded (FG). Its angle is H0 and the shear modulus l0(#) changes as a continuous function of the polar angle #. In each of the wedges, the component uz(r, #) of the displacement vector, considered in polar coordinates (r,#), satisfies the equation:
@ 2 uz @uz @ 2 uz 1 dl @uz þ þ þ ¼ 0: @r 2 r@r r 2 @#2 r 2 ld# @#
ð1Þ
z z The stresses r#z and rrz are expressed via the displacement as r#z ¼ l 1r @u ; rrz ¼ l @u . For a homogeneous wedge we have @# @r l = const, and (1) becomes the Laplace equation. Assume for simplicity that both displacements and tractions are continuous across a contact:
uþz ðrÞ ¼ uz ðrÞ;
rþ#z ðrÞ ¼ r#z ðrÞ:
ð2Þ
Herein, the superscript plus (minus) refers to the limiting value from the wedge, for which the external normal is directed clock-wise (counter clock-wise). For a closed system, we have m pairs of the conditions (2). For an open system, there are m 1 pairs, and two additional equations are given by boundary conditions at the external boundaries OA and OB of the system. In a sufficiently general case, these boundary conditions may be written as
au uz ðrÞ þ ar r r#z ðrÞ ¼ bu fu ðrÞ þ br rfr ðrÞ;
ð3Þ
where au, ar, bu, br are prescribed constants; fu(r), fr(r) are prescribed functions. Obviously, Eq. (3) includes the case of prescribed displacements, when ar = 0, br = 0, and the case of prescribed tractions, when au = 0, bu = 0.
Fig. 1. A system of m wedges (a) open, (b) closed with a functionally graded wedge.
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
89
3. Exact solution As usual, we employ the Mellin’s transform for each of the wedges:
Z
uz ðs; #Þ ¼
1
uz ðr; #Þr s1 dr:
ð4Þ
0
It reduces the Eq. (1) to the linear ordinary differential equation 2
d uz ðs; #Þ 1 dlð#Þ duz ðs; #Þ þ s2 uz ðs; #Þ ¼ 0: þ d# l d# d#2
ð5Þ
Its general solution is:
uz ðs; #Þ ¼ C 1 ðsÞf1 ðs; #Þ þ C 2 ðsÞf2 ðs; #Þ;
ð6Þ
where f1(s, #),f2(s, #) are two linearly independent eigenfunctions. In particular cases, the eigenfunctions have a simple analytical form. Specifically, (i) for a homogeneous (l = const) wedge, the eigenfunctions are f1(s, #) = sin (s#), f2(s, #) = cos (s#), (ii) for a FG material with the shear modulus changing exponentially as l(#) = bexp{2a#} with a > 0, b > 0, the eigenfunctions are:
( " f1 ðs; #Þ ¼ exp a 1 þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ) s 2 # ; 1 a
( " f 2 ðs; #Þ ¼ exp a 1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ) s 2 # if s – a 1 a
and
f1 ðs; #Þ ¼ expfa#g;
f 2 ðs; #Þ ¼ # expfa#g if s ¼ a:
In other cases of a FG material, the eigenfunctions f1(s,#) and f2(s,#) may be found in the form of power series (for instance, ð#Þ when the function dlld# is analytical), or numerically. When having the eigenfunctions, the solution (6) may be re-written in terms of the Mellin transformed boundary values of displacements and tractions rather than in terms of the constants C1(s) and C2(s):
ut ðsÞ ub ðsÞ
pt ðsÞ ¼ RðsÞ : pb ðsÞ
ð7Þ
Herein, ut ðsÞ ¼ uz s; H20 is the displacement of the top boundary of the wedge; ub ðsÞ ¼ uz s; H20 is the displacement at its H0 H0 bottom boundary; pt ðsÞ ¼ r#z s; 2 ; pb ðsÞ ¼ r#z s; 2 are the traction at the top and bottom boundary, respectively, and the matrix R(s) is defined as
RðsÞ ¼
Rtt
Rtb
Rbt
Rbb
¼
ð8Þ
! f1 s; H20 f2 s; H20 f1 s; H20 f2 s; H20
H0 0 H0 f1 s; 2 2 H20 f10 s; H20
l
l
!1 l H20 f20 s; H20 ; H 0 l 20 f2 s; H20
ð9Þ
where the prime denotes the derivative with respect to polar angle. The form (9) allows us to use the efficient method suggested for multi-wedge systems by Blinova and Linkov (1995) and successfully employed in the papers by Linkov and Koshelev (2006a) and Linkov and Rybarska-Rusinek (2008, 2010). The only (non-principal) difference consists in expressions for the shear modulus and eigenfunctions: for a FG material, l changes with the polar angle, while the eigenfunctions f1, f2 and their derivatives f10 ; f10 are not given by sine and cosine. 4. Approximate solution The piece-wise linear approximation of l(#) may be used when we do not have analytical expressions for the eigenfunctions f1(s, #) and f2(s, #) or when avoiding numerical evaluation of these functions. Actually, it corresponds to representation of an inhomogeneous wedge by the sum of homogeneous wedges with ideal contacts. We represent a FG wedge (Fig. 2a) as the sum of n wedges with sufficiently small angles (Fig. 2b). Each of the thin wedges is assumed homogeneous with a constant shear modulus equal to
li ¼ l
H0 2
þ ð2i 1Þ
H0 2n
;
i ¼ 1; . . . ; n:
ð10Þ
90
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
Fig. 2. Approximation of FGM by n homogeneous wedges: (a) FGM, (b) piece-wise homogeneous approximation.
By using procedures developed for an arbitrary system of homogeneous wedges (Linkov & Koshelev, 2006a; Linkov & Rybarska-Rusinek, 2008), we find parameters, which characterize the concentration of physical fields near the common apex of the total system of wedges. Below the accuracy of this simple straightforward approach is checked by (i) comparing the results with those, obtained without the approximation in particular cases when the eigenfunctions are available, and (ii) inspecting of evaluated parameters, when the number n of thin wedges grows, while their angles decrease the convergence Hi ¼ Hn0 ; i ¼ 1; . . . ; n . 5. Numerical examples
Example 1. Consider a system of three wedges with a crack between the first and last wedges (Fig. 3a). The intermediate wedge with the angle H0 ¼ p6 is inhomogeneous. Its shear modulus changes as l(#) = bexp{2a#}, a = 2, b = 1. The eigenfunctions of the differential Eq. (4) are given above. Then the matrix R0(s) has the entries:
! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 Rtt ¼ 2 expfaH0 g 1 1 coth ~s ; a bs ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 a Rbb ¼ 2 expfaH0 g 1 þ 1 coth ~s ; a bs rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 a 1 Rtb ¼ Rbt ¼ 2 ; a bs sinh~s a
ð11Þ ð12Þ ð13Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where ~s ¼ a 1 as H0 : p The external wedges are located symmetrically. They have the same angle H1 ¼ H2 ¼ 11 12 and the same shear module l1 = l2 = 1. Thus for the external wedges, the matrices Ri(s), i = 1, 2, are:
Ri ðsÞ ¼
Ritt
Ritb
Ribt
Ribb
! ¼
l1s cotðsHi Þ
1
l1s i
1 sinðsHi Þ
1
li s sinðsHi Þ
i
1
li s
cotðsHi Þ
! :
Having the matrices, we employ algorithms and procedures developed by Linkov and Rybarska-Rusinek (2008) to find the roots s⁄ of the characteristic determinant in the interval (1, 0). In the example considered, there are two such roots. Their
Fig. 3. Schemes to Examples 2 and 3: (a) plane with a crack and functionally graded wedge, (b) single functionally graded wedge.
91
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
bench-mark values, obtained by using the analytical expressions of the eigenfunctions, are: s1 ¼ 0:4918324966; s2 ¼ 0:9823963572. Below, to compare the results with those given Carpinteri and Paggi (2005) and Pageau, Joseph, and Biggers (1994), we shall use the values k1 ¼ 1 þ s1 and k2 ¼ 1 þ s2 . They characterize the strength of the stress singularity at the common apex of wedges: near an apex, the stresses behave as r#z 1/rk, rrz 1/rk. Compare the accurate results k1 = 0.5081675034, k2 = 0.0176036428 with those obtained by using the approximate approach based on the piece-wise approximation of the shear modulus. We wedge as the sum represent the inhomogeneous H0 0 of n thin homogeneous wedges with the shear modules li ¼ expf2a H20 þ ð2i 1Þ H 2n g, and angles n ; i ¼ 1; . . . n. We change the number of the thin wedges n = 1(3, 10, 30, 60) to inspect convergence. The results are summarized in Table 1. It contains the values of k1 and k2 found for various numbers of thin homogeneous wedges approximating the inhomogeneous wedge. As could be expected, the accuracy grows with increasing number (decreasing angles) of thin wedges. When the angles equal to 3o(n = 10), the accuracy of both k1, k2 is three significant digits. For h0 = 0.5o (n = 60) it is five digits. Example 2. Consider an inhomogeneous wedge with the angle H0 ¼ 53p and the shear modulus changing as l (#) = exp{2a#}, where 56p 6 # 6 53p ; a ¼ 1:5 (Fig. 3b). The exact value of the characteristic determinant D(s) = R(s) is s12 , and it is the same for an arbitrary angle H0 and for an arbitrary real a. Hence, there are no roots in the interval (1, 0). Compare this exact result with approximate solution obtained by using the piece-wise linear approximation. We again represent the inhomogeneous wedge aso the sum of n thin homogeneous wedges: n = 2(3, 4, 5, 6, 7, 8, 9). Their n 0 shear modules are li ¼ exp 2a H20 þ ð2i 1Þ H ; i ¼ 1; . . . n. Table 2 contains the approximate values of k found for 2n various n. In accordance with the analytical result, the value of k decreases; for n = 9, in complete agreement with the analytical solution, there are no roots in the interval (1,0). Example 3. Consider a plane with a crack: H0 = 2p. The plane is inhomogeneous with the shear modulus changing linearly as l (#) = a#, 0 6 # 6 2p, a = 2. We use the approximate method to find k. Data of Table 3 show convergence when the number of thin homogeneous wedges grows. When n P 24 the results are reproduced with three significant digits at least. Example 4. As the last example, we consider the tri-material junction composed of two homogeneous dissimilar (l1 – l2) quarter-planes (H1 = H2 = p/2) joined by the FG half-plane (Fig. 4a). This geometry has been considered by Pageau et al. (1994) for the case of homogeneous half-plane and by Carpinteri and Paggi (2006) for the case when elasticity modulus changes exponentially from the value l1 at the boundary with the wedge 1 to the value l2 at the boundary with the wedge 2. Carpinteri and Paggi (2006) considered plane-stress conditions and assumed the Poisson’s ratio the same in each of the wedges. Therefore, the ratio of the Young modules equals to the ratio of the shear modules. In the pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi case of the FG half-plane, the exponential dependence of the shear modulus is: l3(#) = bexp{2a#}, where b ¼ l2 = l1 =l2 ; a ¼ 21p ln ll1 . In further dis2 cussion we use the same ratio ll1 ¼ 10 as that in the papers by Carpinteri and Paggi (2006) and Pageau et al. (1994). Then the 2 strength of stress singularity k, calculated by the exact method, is k = 0.4408366840. With the purpose to compare stress concentration in the case of the FG half-plane with that in the case, when the halfplane is homogeneous, we use the average value of the shear modulus. For the exponentially changing modulus, it is:
l3a ¼
1
Z 3p=2
p
p=2
l3 ð#Þd# ¼
l2 ðl1 =l2 1Þ ; lnðl1 =l2 Þ
what for ll1 ¼ 10 gives ll3a ¼ 3:9086. Table 4 presents the strength of stress singularity k for the case of the homogeneous half2 2 plane 3 when the ratio ll3 takes different values: 0.1, 1, 2, 3.9086, 5.5, 7, 10, 20, 60, 100. 2 We see that for ll3 ¼ 3:9086 the strength of the stress singularity is kh = 0.4753712570. It is greater than the value 2 k = 0.4408366840, corresponding to the FG half-plane. This means that the functionally graded wedge, providing continuous change of the shear modulus in the junction, may serve for decreasing the stress concentration. Now we want to learn if this conclusion is true for other dependencies of the shear modulus on the angle. Since in general there are no analytical expressions for eigenfunctions, we firstly study when the piece-wise constant approximation of the shear modulus is acceptable for the FG half-plane. The results are presented in Table 5. It contains the values of k, calculated for various number of thin wedges, which approximate the half-plane with exponentially changing shear modulus. It can be seen, that even for twenty thin wedges (n = 20, H0 = 9o), the approximate method provides three correct significant digits. Consider the case of linearly changing modulus (Fig. 5):
l3 ð#Þ ¼ l2
1 ð3 l1 =l2 Þ þ ðl1 =l2 1Þ#=pÞ : 2
Table 1 Values of k = 1 + s⁄ for Example 1.
k1 k2
n=1
n=3
n = 10
n = 30
n = 60
.5000000000
.5071533824 .0153494258
.5080750093 .0173973088
.5081572141 .0175806826
.5081649308 .0175979019
92
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
Table 2 Values of k = 1 + s⁄ for Example 2. n
2
3
4
5
k n k
.4000000000 6 .1026334451
.2457976326 7 .0868760453
.1691462578 8 .0762927013
.1274067962 9 -
Table 3 Values of k = 1 + s⁄ for Example 3. n
2
3
4
8
9
k n k
.5000000000 12 .3930082168
.4440310977 24 .3908571164
.4195693767 36 .3904707276
.3967859769 100 .3902044647
.3953338662 180 .3901772099
Fig. 4. Example of tri-material junction: (a) a plane with a crack and functionally graded wedge, (b) exponential dependence of shear modulus on polar angle in FG half-plane.
Table 4 Values of k = 1 + s⁄ for Example 4 (for different value of l⁄ = l3/l1).
l⁄
0.1
1
2
3.9086
5.5
k
.8558432731 7 .4074202086
.6341541107 10 .3658458893
.5533003790 20 .2875187785
.4753712570 60 .1821147736
.4355832945 100 .1441567269
l⁄ k
Table 5 Approximate values of k = 1 + s⁄ for Example 4 (shear modulus changing exponentially). n
2
3
5
10
20
k n k
.4535126165 30 .4408889225
.4462471562 40 .4408660634
.4427413718 60 .4408497400
.4413082164 100 .4408413839
.4409542749 150 .4408387728
The strength of stress singularity k, evaluated for various number of thin wedges, approximating the half-plane with linearly changing shear modulus, is presented in Table 6. Obviously the results converge with growing number of thin wedges. They confirm that for n = 20 (H0 = 9o), there are three correct digits. To this accuracy, k1 = 0.371. Note that it is even less than that in the case of exponentially changing modulus (k1 = 0.441). Compare the value k1 = 0.371 with that, corresponding to the homogenenous half-plane. In the considered case of linearly changing modulus, its average value is l3a ¼ l2 1þl21 =l2 . Hence for ll1 ¼ 10, we have ll3a ¼ 5:5. From Table 4 we see that 2 2 for the homogeneous half-plane with ll3 ¼ 5:5, the strength of stress singularity is kh = 0.4355832945. Again we see that the 2 strength of stress singularity for the FG half-plane (k1 = 0.371) is weaker than that for the homogeneous half-plane (kh = 0.436). This confirms the beneficial influence of angular grading.
93
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
Fig. 5. Example of tri-material junction: (a) a plane with a crack and functionally graded wedge, (b) linear dependence of shear modulus on polar angle in FG half-plane. Table 6 Approximate values of k = 1 + s⁄ for Example 4 (shear modulus changing linearly). n
2
3
5
10
20
k1 k2 n k1 k2
.3910803297 30 .3712564981 .0228484180
.3793791684 .0093529534 40 .3712372386 .0229508745
.3736627571 .0166910496 60 .3712238395 .0230248402
.3716852924 .0211465305 100 .3712170996 .0230629731
.3713144828 .0225621950 150 .3712150105 .0230749270
Numerous other examples of FG wedges with arbitrary angular dependencies of modules may be considered by using the suggested method based on the piece-wise constant approximation of the modules. In all the studied cases, employing a FG wedge, which provides smooth transition of modules in a junction, appeared beneficial for decreasing the stress concentration. Acknowledgement The authors gratefully acknowledge the support of the Polish Ministry of Science and Higher Education (Scientific Project N N519 440739 for 2010–2012). References Blinova, V., & Linkov, A. (1995). A method of finding asymptotic forms at the common apex of elastic wedges. Journal of Applied Mathematics and Mechanics, 59, 187–195. Carpinteri, A., & Paggi, M. (2005). On the asyptotic stress field in angularly nonhomegenous materials. International Journal of Fracture, 135, 267–283. Carpinteri, A., & Paggi, M. (2006). Influence of the intermediate material on the singular stress field in tri-material junctions. Materials Science, 42, 95–101. Dempsey, J. P., & Sinclair, G. B. (1979). On the stress singularities in the plate elasticity of the composite wedge. Journal of Elasticity, 94, 373–391. Eischen, J. W. (1987). Fracture of Nonhomogeneous Materials. International Journal of Fracture, 34, 3–22. Erdogan, F. (1983). Stress intensity factors. ASME Journal of Applied Mechanics, 25, 992–1002. Erdogan, F., & Wu, B. H. (1997). The surface crack problem for a plate with functionally graded properties. ASME Journal of Applied Mechanics, 64, 449–456. Konda, N., & Erdogan, F. (1994). The mixed-mode crack problem in a nonhomogeneous elastic medium. Engineering Fracture Mechanics, 47, 533–545. Linkov, A., & Koshelev, V. (2006a). Multi-wedge points and multi-wedge elements in computational mechanics: evaluation of exponent and angular distribution. International Journal of Solids and Structures, 43, 5909–5930. Linkov, A., & Koshelev, V. (2006b). Multi-wedge singular points in materials: theory, numerical techniques and applications. In: Proceedings of fourth International Conference ’’Mathematical Modeling and Computer Simulation of Materials Technologies, MMT-2006’’. The College of Judea and Samaria, Vol. 1, pp. I-48–I-56. Linkov, A., & Rybarska-Rusinek, L. (2008). Numerical methods and models for anti-plane strain of a system with a thin elastic wedge. Archive of Applied Mechanics, 78, 821–831. Linkov, A., & Rybarska-Rusinek, L. (2010). Plane elasticity problem for a multi-wedge system with a thin wedge. International Journal of Solids and Structures, 47, 3297–3304. Mishuris, G., & Kuhn, G. (2001). Comparative study of an interface crack for different wedge-interface models. Archive of Applied Mechanics, 71, 764–780. Pageau, S., Joseph, P., & Biggers, S. J. (1994). The order of stress singularities for bonded and disbonded three-material junctions. ASME Journal of Applied Mechanics, 31, 2979–2997. Paggi, M., & Carpinteri, A. (2008). On the stress singularities at multimaterial interfaces and related analogies with fluid dynamics and diffusion. Applied Mechanics Reviews, 61, 020801, pages 22. Sinclair, G. B. (1999). Logarithmic stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of Applied Mechanics, 66, 556–560. Tranter, C. J. (1948). The use of Mellin transform in finding the stress distribution in an infinite wedge. Quarterly Journal of Mechanics and Applied Mathematics, 1(2), 125–130. Williams, M. L. (1952). Stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of Applied Mechanics, 19, 526–528.
Contents lists available at SciVerse ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Evaluation of stress concentration in multi-wedge systems with functionally graded wedges Alexander Linkov ⇑, Liliana Rybarska-Rusinek Rzeszow University of Technology, Powstancow Warszawy 12, 35-959 Rzeszow, Poland
a r t i c l e
i n f o
Article history: Received 29 February 2012 Accepted 2 April 2012 Available online 24 July 2012 Keywords: Functionally graded material Multi-wedge systems Stress concentration
a b s t r a c t A common apex of structural elements with different physical properties is a point of stress concentration leading to dangerous effects like fracture, corrosion and fatigue. As the concentration may be reduced by proper functional grading of the elements, there is need in evaluating the influence of various grading on the singularity of a field at the apex. In the paper, we present and compare two efficient methods (exact and approximate) to reach this goal. The exact method employs eigenfunctions of the ordinary differential equation, to which the partial differential equation of a considered problem is reduced after the Mellin’s transform. The approximate method consists in piece-wise constant approximation of the physical modules of a graded wedge. Numerical examples for antiplane strain illustrate high efficiency of the methods and beneficial influence of functional grading when the latter provides continuity of shear modulus. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Singular points of a medium, such as crack tips, corners of notches and common apexes of grains, are sources of high field concentration. The concentration manifests itself in unfavorable effects like sparkling, energy leak off, fracture, corrosion, fatigue, and in the specific geometry of nano-structures at intersections of dislocation arrays. In view of the significance of singular points, the (multi-) wedge systems have been the subject of numerous studies starting from the papers by Tranter (1948) and Williams (1952). Comprehensive reviews may be found in papers of Blinova and Linkov (1995), Dempsey and Sinclair (1979), Linkov and Koshelev (2006a), Linkov and Rybarska-Rusinek (2008), Paggi and Carpinteri (2008) and Sinclair (1999). Most of the work concerns with homogeneous wedges and ideal, or smooth, or Coulomb friction contacts. Meanwhile, there is need in accounting for more complicated contact conditions and inhomogeneity of wedges. Studying of the first complicating factor, initiated by Mishuris and Kuhn (2001), has resulted in the general method (Linkov & Rybarska-Rusinek, 2008, 2010), applicable to a wide range of interfacial conditions. The importance of the second factor (inhomogeneity of wedges) was clearly recognized for cracks by Erdogan (1983). Its influence on the singularity at a crack tip has been studied in the papers by Eischen (1987), Erdogan and Wu (1997) and Konda and Erdogan (1994). Investigation of an arbitrary functionally graded material (FGM) near a (multi-) wedge point has been started in the papers of Carpinteri and Paggi (2005), Carpinteri and Paggi (2006). The authors wrote explicitly the equation for a FG wedge with the Young modulus depending on the polar angle. They outlined that from the mathematical point of view the only difference, as compared with a homogeneous wedge, is that the eigenfunctions differ from those for a homogeneous wedge. This suggests straightforward application of the mathematical techniques developed for homogeneous wedges to FGM. The authors also noted that for the modulus exponentially depending on the angle, the equation has constant coefficients what allows one to easily find needed ⇑ Corresponding author. E-mail address: [email protected] (A. Linkov). 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2012.06.012
88
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
eigenfunctions. For this particular dependence, they studied the cases of a crack inside an angular FGM and trimaterial with a single FG transitional wedge. The numerical results obtained clearly show that ‘‘the presence of a FG intermediate material is favorable since it significantly reduces the order of the stress singularity as compared with the same trimaterial junction involving homogeneous different materials’’ (Paggi and Carpintery, 2008, p. 020801-19). This stimulates further researches in this area and development of efficient general methods for accounting for FGM. The present paper aims to reach this goal. We consider the general case when physical properties, for instance shear modulus and/or Poisson’s ratio of a FGM, change arbitrary with the angle. Then finding of the eigenfunctions becomes a part of the problem. They may be found using power series and working out a subroutine to determine the coefficients of the expansions numerically. An alternative way to account for FGM may be based on the favorable feature of the method, developed for homogenous wedges (Linkov & Koshelev, 2006a, 2006b), to provide accurate (at least three correct digits) results for systems of very thin wedges (with angles of angular minutes). This suggests replacing a FG wedge by a system of homogeneous wedges what means piece-wise constant approximation of the modules of the FGM. Below we employ the both approaches and show that the second one provides very accurate evaluation of the singularity. For brevity, we consider antiplane strain. Obviously, the piece-wise constant approximation is also applicable to plane strain and plain stress, for which one can use the mentioned efficient method (Linkov & Koshelev, 2006a).
2. Problem formulation Consider antiplane strain of a system of m elastic wedges. The system may be open (Fig. 1a) or closed (Fig. 1b). The angle of a wedge is denoted H, its shear modulus is l. One of the wedges, marked by the subscript 0, is functionally graded (FG). Its angle is H0 and the shear modulus l0(#) changes as a continuous function of the polar angle #. In each of the wedges, the component uz(r, #) of the displacement vector, considered in polar coordinates (r,#), satisfies the equation:
@ 2 uz @uz @ 2 uz 1 dl @uz þ þ þ ¼ 0: @r 2 r@r r 2 @#2 r 2 ld# @#
ð1Þ
z z The stresses r#z and rrz are expressed via the displacement as r#z ¼ l 1r @u ; rrz ¼ l @u . For a homogeneous wedge we have @# @r l = const, and (1) becomes the Laplace equation. Assume for simplicity that both displacements and tractions are continuous across a contact:
uþz ðrÞ ¼ uz ðrÞ;
rþ#z ðrÞ ¼ r#z ðrÞ:
ð2Þ
Herein, the superscript plus (minus) refers to the limiting value from the wedge, for which the external normal is directed clock-wise (counter clock-wise). For a closed system, we have m pairs of the conditions (2). For an open system, there are m 1 pairs, and two additional equations are given by boundary conditions at the external boundaries OA and OB of the system. In a sufficiently general case, these boundary conditions may be written as
au uz ðrÞ þ ar r r#z ðrÞ ¼ bu fu ðrÞ þ br rfr ðrÞ;
ð3Þ
where au, ar, bu, br are prescribed constants; fu(r), fr(r) are prescribed functions. Obviously, Eq. (3) includes the case of prescribed displacements, when ar = 0, br = 0, and the case of prescribed tractions, when au = 0, bu = 0.
Fig. 1. A system of m wedges (a) open, (b) closed with a functionally graded wedge.
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
89
3. Exact solution As usual, we employ the Mellin’s transform for each of the wedges:
Z
uz ðs; #Þ ¼
1
uz ðr; #Þr s1 dr:
ð4Þ
0
It reduces the Eq. (1) to the linear ordinary differential equation 2
d uz ðs; #Þ 1 dlð#Þ duz ðs; #Þ þ s2 uz ðs; #Þ ¼ 0: þ d# l d# d#2
ð5Þ
Its general solution is:
uz ðs; #Þ ¼ C 1 ðsÞf1 ðs; #Þ þ C 2 ðsÞf2 ðs; #Þ;
ð6Þ
where f1(s, #),f2(s, #) are two linearly independent eigenfunctions. In particular cases, the eigenfunctions have a simple analytical form. Specifically, (i) for a homogeneous (l = const) wedge, the eigenfunctions are f1(s, #) = sin (s#), f2(s, #) = cos (s#), (ii) for a FG material with the shear modulus changing exponentially as l(#) = bexp{2a#} with a > 0, b > 0, the eigenfunctions are:
( " f1 ðs; #Þ ¼ exp a 1 þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ) s 2 # ; 1 a
( " f 2 ðs; #Þ ¼ exp a 1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ) s 2 # if s – a 1 a
and
f1 ðs; #Þ ¼ expfa#g;
f 2 ðs; #Þ ¼ # expfa#g if s ¼ a:
In other cases of a FG material, the eigenfunctions f1(s,#) and f2(s,#) may be found in the form of power series (for instance, ð#Þ when the function dlld# is analytical), or numerically. When having the eigenfunctions, the solution (6) may be re-written in terms of the Mellin transformed boundary values of displacements and tractions rather than in terms of the constants C1(s) and C2(s):
ut ðsÞ ub ðsÞ
pt ðsÞ ¼ RðsÞ : pb ðsÞ
ð7Þ
Herein, ut ðsÞ ¼ uz s; H20 is the displacement of the top boundary of the wedge; ub ðsÞ ¼ uz s; H20 is the displacement at its H0 H0 bottom boundary; pt ðsÞ ¼ r#z s; 2 ; pb ðsÞ ¼ r#z s; 2 are the traction at the top and bottom boundary, respectively, and the matrix R(s) is defined as
RðsÞ ¼
Rtt
Rtb
Rbt
Rbb
¼
ð8Þ
! f1 s; H20 f2 s; H20 f1 s; H20 f2 s; H20
H0 0 H0 f1 s; 2 2 H20 f10 s; H20
l
l
!1 l H20 f20 s; H20 ; H 0 l 20 f2 s; H20
ð9Þ
where the prime denotes the derivative with respect to polar angle. The form (9) allows us to use the efficient method suggested for multi-wedge systems by Blinova and Linkov (1995) and successfully employed in the papers by Linkov and Koshelev (2006a) and Linkov and Rybarska-Rusinek (2008, 2010). The only (non-principal) difference consists in expressions for the shear modulus and eigenfunctions: for a FG material, l changes with the polar angle, while the eigenfunctions f1, f2 and their derivatives f10 ; f10 are not given by sine and cosine. 4. Approximate solution The piece-wise linear approximation of l(#) may be used when we do not have analytical expressions for the eigenfunctions f1(s, #) and f2(s, #) or when avoiding numerical evaluation of these functions. Actually, it corresponds to representation of an inhomogeneous wedge by the sum of homogeneous wedges with ideal contacts. We represent a FG wedge (Fig. 2a) as the sum of n wedges with sufficiently small angles (Fig. 2b). Each of the thin wedges is assumed homogeneous with a constant shear modulus equal to
li ¼ l
H0 2
þ ð2i 1Þ
H0 2n
;
i ¼ 1; . . . ; n:
ð10Þ
90
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
Fig. 2. Approximation of FGM by n homogeneous wedges: (a) FGM, (b) piece-wise homogeneous approximation.
By using procedures developed for an arbitrary system of homogeneous wedges (Linkov & Koshelev, 2006a; Linkov & Rybarska-Rusinek, 2008), we find parameters, which characterize the concentration of physical fields near the common apex of the total system of wedges. Below the accuracy of this simple straightforward approach is checked by (i) comparing the results with those, obtained without the approximation in particular cases when the eigenfunctions are available, and (ii) inspecting of evaluated parameters, when the number n of thin wedges grows, while their angles decrease the convergence Hi ¼ Hn0 ; i ¼ 1; . . . ; n . 5. Numerical examples
Example 1. Consider a system of three wedges with a crack between the first and last wedges (Fig. 3a). The intermediate wedge with the angle H0 ¼ p6 is inhomogeneous. Its shear modulus changes as l(#) = bexp{2a#}, a = 2, b = 1. The eigenfunctions of the differential Eq. (4) are given above. Then the matrix R0(s) has the entries:
! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 Rtt ¼ 2 expfaH0 g 1 1 coth ~s ; a bs ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 a Rbb ¼ 2 expfaH0 g 1 þ 1 coth ~s ; a bs rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 a 1 Rtb ¼ Rbt ¼ 2 ; a bs sinh~s a
ð11Þ ð12Þ ð13Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where ~s ¼ a 1 as H0 : p The external wedges are located symmetrically. They have the same angle H1 ¼ H2 ¼ 11 12 and the same shear module l1 = l2 = 1. Thus for the external wedges, the matrices Ri(s), i = 1, 2, are:
Ri ðsÞ ¼
Ritt
Ritb
Ribt
Ribb
! ¼
l1s cotðsHi Þ
1
l1s i
1 sinðsHi Þ
1
li s sinðsHi Þ
i
1
li s
cotðsHi Þ
! :
Having the matrices, we employ algorithms and procedures developed by Linkov and Rybarska-Rusinek (2008) to find the roots s⁄ of the characteristic determinant in the interval (1, 0). In the example considered, there are two such roots. Their
Fig. 3. Schemes to Examples 2 and 3: (a) plane with a crack and functionally graded wedge, (b) single functionally graded wedge.
91
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
bench-mark values, obtained by using the analytical expressions of the eigenfunctions, are: s1 ¼ 0:4918324966; s2 ¼ 0:9823963572. Below, to compare the results with those given Carpinteri and Paggi (2005) and Pageau, Joseph, and Biggers (1994), we shall use the values k1 ¼ 1 þ s1 and k2 ¼ 1 þ s2 . They characterize the strength of the stress singularity at the common apex of wedges: near an apex, the stresses behave as r#z 1/rk, rrz 1/rk. Compare the accurate results k1 = 0.5081675034, k2 = 0.0176036428 with those obtained by using the approximate approach based on the piece-wise approximation of the shear modulus. We wedge as the sum represent the inhomogeneous H0 0 of n thin homogeneous wedges with the shear modules li ¼ expf2a H20 þ ð2i 1Þ H 2n g, and angles n ; i ¼ 1; . . . n. We change the number of the thin wedges n = 1(3, 10, 30, 60) to inspect convergence. The results are summarized in Table 1. It contains the values of k1 and k2 found for various numbers of thin homogeneous wedges approximating the inhomogeneous wedge. As could be expected, the accuracy grows with increasing number (decreasing angles) of thin wedges. When the angles equal to 3o(n = 10), the accuracy of both k1, k2 is three significant digits. For h0 = 0.5o (n = 60) it is five digits. Example 2. Consider an inhomogeneous wedge with the angle H0 ¼ 53p and the shear modulus changing as l (#) = exp{2a#}, where 56p 6 # 6 53p ; a ¼ 1:5 (Fig. 3b). The exact value of the characteristic determinant D(s) = R(s) is s12 , and it is the same for an arbitrary angle H0 and for an arbitrary real a. Hence, there are no roots in the interval (1, 0). Compare this exact result with approximate solution obtained by using the piece-wise linear approximation. We again represent the inhomogeneous wedge aso the sum of n thin homogeneous wedges: n = 2(3, 4, 5, 6, 7, 8, 9). Their n 0 shear modules are li ¼ exp 2a H20 þ ð2i 1Þ H ; i ¼ 1; . . . n. Table 2 contains the approximate values of k found for 2n various n. In accordance with the analytical result, the value of k decreases; for n = 9, in complete agreement with the analytical solution, there are no roots in the interval (1,0). Example 3. Consider a plane with a crack: H0 = 2p. The plane is inhomogeneous with the shear modulus changing linearly as l (#) = a#, 0 6 # 6 2p, a = 2. We use the approximate method to find k. Data of Table 3 show convergence when the number of thin homogeneous wedges grows. When n P 24 the results are reproduced with three significant digits at least. Example 4. As the last example, we consider the tri-material junction composed of two homogeneous dissimilar (l1 – l2) quarter-planes (H1 = H2 = p/2) joined by the FG half-plane (Fig. 4a). This geometry has been considered by Pageau et al. (1994) for the case of homogeneous half-plane and by Carpinteri and Paggi (2006) for the case when elasticity modulus changes exponentially from the value l1 at the boundary with the wedge 1 to the value l2 at the boundary with the wedge 2. Carpinteri and Paggi (2006) considered plane-stress conditions and assumed the Poisson’s ratio the same in each of the wedges. Therefore, the ratio of the Young modules equals to the ratio of the shear modules. In the pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi case of the FG half-plane, the exponential dependence of the shear modulus is: l3(#) = bexp{2a#}, where b ¼ l2 = l1 =l2 ; a ¼ 21p ln ll1 . In further dis2 cussion we use the same ratio ll1 ¼ 10 as that in the papers by Carpinteri and Paggi (2006) and Pageau et al. (1994). Then the 2 strength of stress singularity k, calculated by the exact method, is k = 0.4408366840. With the purpose to compare stress concentration in the case of the FG half-plane with that in the case, when the halfplane is homogeneous, we use the average value of the shear modulus. For the exponentially changing modulus, it is:
l3a ¼
1
Z 3p=2
p
p=2
l3 ð#Þd# ¼
l2 ðl1 =l2 1Þ ; lnðl1 =l2 Þ
what for ll1 ¼ 10 gives ll3a ¼ 3:9086. Table 4 presents the strength of stress singularity k for the case of the homogeneous half2 2 plane 3 when the ratio ll3 takes different values: 0.1, 1, 2, 3.9086, 5.5, 7, 10, 20, 60, 100. 2 We see that for ll3 ¼ 3:9086 the strength of the stress singularity is kh = 0.4753712570. It is greater than the value 2 k = 0.4408366840, corresponding to the FG half-plane. This means that the functionally graded wedge, providing continuous change of the shear modulus in the junction, may serve for decreasing the stress concentration. Now we want to learn if this conclusion is true for other dependencies of the shear modulus on the angle. Since in general there are no analytical expressions for eigenfunctions, we firstly study when the piece-wise constant approximation of the shear modulus is acceptable for the FG half-plane. The results are presented in Table 5. It contains the values of k, calculated for various number of thin wedges, which approximate the half-plane with exponentially changing shear modulus. It can be seen, that even for twenty thin wedges (n = 20, H0 = 9o), the approximate method provides three correct significant digits. Consider the case of linearly changing modulus (Fig. 5):
l3 ð#Þ ¼ l2
1 ð3 l1 =l2 Þ þ ðl1 =l2 1Þ#=pÞ : 2
Table 1 Values of k = 1 + s⁄ for Example 1.
k1 k2
n=1
n=3
n = 10
n = 30
n = 60
.5000000000
.5071533824 .0153494258
.5080750093 .0173973088
.5081572141 .0175806826
.5081649308 .0175979019
92
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
Table 2 Values of k = 1 + s⁄ for Example 2. n
2
3
4
5
k n k
.4000000000 6 .1026334451
.2457976326 7 .0868760453
.1691462578 8 .0762927013
.1274067962 9 -
Table 3 Values of k = 1 + s⁄ for Example 3. n
2
3
4
8
9
k n k
.5000000000 12 .3930082168
.4440310977 24 .3908571164
.4195693767 36 .3904707276
.3967859769 100 .3902044647
.3953338662 180 .3901772099
Fig. 4. Example of tri-material junction: (a) a plane with a crack and functionally graded wedge, (b) exponential dependence of shear modulus on polar angle in FG half-plane.
Table 4 Values of k = 1 + s⁄ for Example 4 (for different value of l⁄ = l3/l1).
l⁄
0.1
1
2
3.9086
5.5
k
.8558432731 7 .4074202086
.6341541107 10 .3658458893
.5533003790 20 .2875187785
.4753712570 60 .1821147736
.4355832945 100 .1441567269
l⁄ k
Table 5 Approximate values of k = 1 + s⁄ for Example 4 (shear modulus changing exponentially). n
2
3
5
10
20
k n k
.4535126165 30 .4408889225
.4462471562 40 .4408660634
.4427413718 60 .4408497400
.4413082164 100 .4408413839
.4409542749 150 .4408387728
The strength of stress singularity k, evaluated for various number of thin wedges, approximating the half-plane with linearly changing shear modulus, is presented in Table 6. Obviously the results converge with growing number of thin wedges. They confirm that for n = 20 (H0 = 9o), there are three correct digits. To this accuracy, k1 = 0.371. Note that it is even less than that in the case of exponentially changing modulus (k1 = 0.441). Compare the value k1 = 0.371 with that, corresponding to the homogenenous half-plane. In the considered case of linearly changing modulus, its average value is l3a ¼ l2 1þl21 =l2 . Hence for ll1 ¼ 10, we have ll3a ¼ 5:5. From Table 4 we see that 2 2 for the homogeneous half-plane with ll3 ¼ 5:5, the strength of stress singularity is kh = 0.4355832945. Again we see that the 2 strength of stress singularity for the FG half-plane (k1 = 0.371) is weaker than that for the homogeneous half-plane (kh = 0.436). This confirms the beneficial influence of angular grading.
93
A. Linkov, L. Rybarska-Rusinek / International Journal of Engineering Science 61 (2012) 87–93
Fig. 5. Example of tri-material junction: (a) a plane with a crack and functionally graded wedge, (b) linear dependence of shear modulus on polar angle in FG half-plane. Table 6 Approximate values of k = 1 + s⁄ for Example 4 (shear modulus changing linearly). n
2
3
5
10
20
k1 k2 n k1 k2
.3910803297 30 .3712564981 .0228484180
.3793791684 .0093529534 40 .3712372386 .0229508745
.3736627571 .0166910496 60 .3712238395 .0230248402
.3716852924 .0211465305 100 .3712170996 .0230629731
.3713144828 .0225621950 150 .3712150105 .0230749270
Numerous other examples of FG wedges with arbitrary angular dependencies of modules may be considered by using the suggested method based on the piece-wise constant approximation of the modules. In all the studied cases, employing a FG wedge, which provides smooth transition of modules in a junction, appeared beneficial for decreasing the stress concentration. Acknowledgement The authors gratefully acknowledge the support of the Polish Ministry of Science and Higher Education (Scientific Project N N519 440739 for 2010–2012). References Blinova, V., & Linkov, A. (1995). A method of finding asymptotic forms at the common apex of elastic wedges. Journal of Applied Mathematics and Mechanics, 59, 187–195. Carpinteri, A., & Paggi, M. (2005). On the asyptotic stress field in angularly nonhomegenous materials. International Journal of Fracture, 135, 267–283. Carpinteri, A., & Paggi, M. (2006). Influence of the intermediate material on the singular stress field in tri-material junctions. Materials Science, 42, 95–101. Dempsey, J. P., & Sinclair, G. B. (1979). On the stress singularities in the plate elasticity of the composite wedge. Journal of Elasticity, 94, 373–391. Eischen, J. W. (1987). Fracture of Nonhomogeneous Materials. International Journal of Fracture, 34, 3–22. Erdogan, F. (1983). Stress intensity factors. ASME Journal of Applied Mechanics, 25, 992–1002. Erdogan, F., & Wu, B. H. (1997). The surface crack problem for a plate with functionally graded properties. ASME Journal of Applied Mechanics, 64, 449–456. Konda, N., & Erdogan, F. (1994). The mixed-mode crack problem in a nonhomogeneous elastic medium. Engineering Fracture Mechanics, 47, 533–545. Linkov, A., & Koshelev, V. (2006a). Multi-wedge points and multi-wedge elements in computational mechanics: evaluation of exponent and angular distribution. International Journal of Solids and Structures, 43, 5909–5930. Linkov, A., & Koshelev, V. (2006b). Multi-wedge singular points in materials: theory, numerical techniques and applications. In: Proceedings of fourth International Conference ’’Mathematical Modeling and Computer Simulation of Materials Technologies, MMT-2006’’. The College of Judea and Samaria, Vol. 1, pp. I-48–I-56. Linkov, A., & Rybarska-Rusinek, L. (2008). Numerical methods and models for anti-plane strain of a system with a thin elastic wedge. Archive of Applied Mechanics, 78, 821–831. Linkov, A., & Rybarska-Rusinek, L. (2010). Plane elasticity problem for a multi-wedge system with a thin wedge. International Journal of Solids and Structures, 47, 3297–3304. Mishuris, G., & Kuhn, G. (2001). Comparative study of an interface crack for different wedge-interface models. Archive of Applied Mechanics, 71, 764–780. Pageau, S., Joseph, P., & Biggers, S. J. (1994). The order of stress singularities for bonded and disbonded three-material junctions. ASME Journal of Applied Mechanics, 31, 2979–2997. Paggi, M., & Carpinteri, A. (2008). On the stress singularities at multimaterial interfaces and related analogies with fluid dynamics and diffusion. Applied Mechanics Reviews, 61, 020801, pages 22. Sinclair, G. B. (1999). Logarithmic stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of Applied Mechanics, 66, 556–560. Tranter, C. J. (1948). The use of Mellin transform in finding the stress distribution in an infinite wedge. Quarterly Journal of Mechanics and Applied Mathematics, 1(2), 125–130. Williams, M. L. (1952). Stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of Applied Mechanics, 19, 526–528.