Adhesively bonded disk under compressive diametrical load

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(2) ARTICLE IN PRESS. JID: SAS. [m5G;August 28, 2018;12:47]. International Journal of Solids and Structures xxx (2018) xxx–xxx. Contents lists available at ScienceDirect. International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr. Adhesively bonded disk under compressive diametrical load E. Radi∗, E. Dragoni, A. Spaggiari. Q1. Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, via G. Amendola 2, 42122, Reggio Emilia, Italy. a r t i c l e. i n f o. Article history: Received 16 November 2017 Revised 30 March 2018 Available online xxx Keywords: Airy stress function Bipolar coordinates Plane elasticity Singular integral equation Perturbation analysis Adhesive bonding Mode mixity Collocation method. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18. a b s t r a c t A closed-form full-field solution is presented for stresses and displacement in a circular disk containing a diametrical adhesive thin layer induced by two opposite compressive loads acting along an arbitrary diametrical direction. For the sake of simplicity, the adhesive layer is treated as a tangential displacement discontinuity between the two disk halves. The problem is split into symmetric and skew-symmetric loading conditions. No contribution is expected from the layer for the symmetric problem. For the skewsymmetric loading condition, a general integral solution in bipolar coordinates has been assumed for the Airy stress function in the form of a Fourier sine transform. The imposition of the boundary conditions then allows us to reduce the problem to a Fredholm integral equation of the first kind defined on the half-line or equivalently to a singular integro-differential equation defined on a bounded interval. A preliminary asymptotic analysis of the stress and displacement fields at the edges of the adhesive thin layer shows that the stress field is finite therein, but the rotation displays a logarithmic singularity. A numerical solution of the singular integro-differential equation is then provided by assuming a power series expansion for the shear stress, whose coefficients are determined by using a collocation method. An approximate closed-form solution is also derived by exploiting a perturbation method that assumes the ratio between the shear modulus of the disk material and the shear stiffness of the adhesive thin layer as small parameter. The shear stress distribution along the thin layer turns out to be more and more uniform as the adhesive shear stiffness decreases. In order to validate the analytical results, FE investigations and also experimental results obtained by using Digital Image Correlation (DIC) techniques are presented for varying loading orientation and material parameters. © 2018 Elsevier Ltd. All rights reserved.. 1. Introduction Adhesive bond joints find an increasing number of applications in aircraft, marine and civil structures even if their strength cannot be accurately predicted by theoretical modeling, due to stress singularities that may arise at their edges. The occurring of stress singularity is a consequence of the assumed linear elastic behavior of the materials and even if not realistic it may however cause initiation and propagation of cracks and debonding. Therefore, a strong request comes from structural engineers and designers for the definition of a reliable experimental methodology that enables one to characterize the adhesive bonding strength properties up to failure, under various loading and physical conditions. Adhesive research over the years has strived to identify specimen designs of bonded joints where shear and tensile stresses occur without stress concentration at the edges, due to materials elastic mismatch. Testing the adhesive in thin film provides more realistic information than retrieving its properties from tests on bulk specimen, due to several reasons. For thin film, indeed, the. ∗. Corresponding author. E-mail address: eradi@unimore.it (E. Radi).. chemical effect of the adhesive-adherend interface plays a significant role, as suggested by Krogh et al. (2015). Moreover, for test on bulk the curing conditions may differ because of uncontrolled exothermic reactions in massive form, as shown by Adnan and Sun (2008), and the stress triaxiality promoted in-situ by the adherends cannot take place, as stated by Wang and Rose (1997). Therefore, the mechanical and chemical properties of the interphase region differ from the bulk. All bond properties are affected but the influence on Young’s modulus is especially significant, Peretz (1978). In technical literature we can trace some standard specimen that lead to non-singular stress distribution in thin film adherends, such as: thick adherend shear test, prepared according to the ASTM D5656-10, the napkin ring torsion test, as shown in the ASTM E229-97. This test specimen was analyzed by Grant (1987) then by Castagnetti et al (2011) and improved by Spaggiari et al. (2012,2013) and the solid rod butt-joint in torsion, proposed by Adams (1977). Moreover, there exist other interesting options which exploit rigid adherends bonded together such as: the Arcan test, proposed by Arcan et al. (1987), the buttbonded notched beam under four-point antisymmetric bending, studied by Wycherley et al. (1990) and Iosipescu (1967) and, more recently, the three-point bend test of a sandwich beam made by. https://doi.org/10.1016/j.ijsolstr.2018.05.021 0020-7683/© 2018 Elsevier Ltd. All rights reserved.. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40.

(3) JID: SAS 2. 41 42 43 44 45 46 47 48 49 50 51 Q2 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106. ARTICLE IN PRESS. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. bonding two thick flats on top of each other, as suggested by Moussiaux et al.(1987), He et al. (2001) and Brinson et al. (1995). These promising models were studied analytically and numerically by Dragoni and Brinson (2015) and validated experimentally using Digital Image Correlation by Spaggiari et al. (2016). The bonded Brazilian disc is a very simple test coupon, easier to be manufactured than the previous ones. It has been proposed by Martin et al. (2012) for evaluating the shear strength of epoxy joined ceramics. Moreover, unlike the napkin ring or tubular specimens, it does not need a torsional testing machine and, compared to the three or four point bending specimens, it has no need of ad hoc fixtures. The bonded Brazilian disc can be simply compressed with a universal testing machine and it is able to create the desired stress field in the adhesive under test, with a variable degree of mode mixity, and without a strong edge effect at the corners as already observed by Martin et al. (2012). Due to its simple specimen preparation and experimental performance, the Brazilian test is a widely used test for the determination of the tensile strength and fracture toughness of brittle materials like rocks and concrete (Berenbaum and Brodie, 1958; Awaji and Sato, 1978; Dong et al., 2004). In the standard test, a homogeneous thin disk is diametrically compressed and that loading causes an almost uniform tensile stress perpendicular to the loading direction, which yields a tensile failure. This problem has attracted the interest of many researchers (Kourkoulis et al., 2012, 2013; Markides et al., 2011), who approached it by using the method of complex potentials of plane elasticity (Muskhelishvili, 1954) or the most general solution of the biharmonic equation in polar coordinates (Mitchell, 1965). An exhaustive depiction of the problem can be found in the review paper on the Brazilian test and its generalization by Li and Wong (2013). A number of generalizations of the test to bonded bimaterial disks or rings has been also proposed in the literature in order to determine the toughness of the interface under pure or mixed-mode conditions (Wong and Suo, 1990; Soares and Tianxi, 1998; Banks-Sills and Schwartz; 2002; Budzik and Jensen, 2014). These investigations provide only numerical results or asymptotic analyses of the crack tip fields in a bonded bimaterial disk or ring performed by using the linear elastic theory of interface fracture mechanics. However, no full-field analytical solutions have been supplied for a disk with an adhesive thin layer subject to compressive load along an arbitrary direction, which induces both normal and shear stress along the bonding joint. Such a configuration may indeed be used for the characterization of the adhesive bonding strength proprieties. Therefore, an accurate evaluation of the stress distribution along the adhesive thin layer under mixed loading conditions is a prerequisite to assure the structural integrity of many bonded components. In the present work, we propose the use of the Brazilian test on a bonded disk for the characterization of adhesion properties of the adhesive bonding. The main advantage of this test is that any combination of shear and normal loading can be achieved by appropriate choice of the bonding inclination angle with respect to the loading direction. To this aim, we first investigate the twodimensional problem of the stress and displacement distribution in the two disk halves, by treating the adhesive thin layer as a tangential displacement discontinuity between the two halves. A similar assumption was extensively adopted in the conventional analysis of bonded joints (Klabring, 1991; Koutosov, 2007). For the sake of simplicity, only shear compliance of the adhesive layer is considered, leaving the contribution of normal compliance for future analysis. Namely, the adhesive joint is represented as an imperfect elastic interface whose shear stress is assumed to depend linearly on the tangential displacement discontinuity across the thin layer (Mishuris, 1997,1999,2001), whereas a perfect bond between the two semi-disks is considered for the normal displacement.. Under these assumptions, the shear stress along the bondline is finite, being proportional to the tangential displacement discontinuity. Then, the stress and displacement fields near the right wedges at the bonding edges subject to uniform shear loading on one side and vanishing traction on the other side are non-singular, as observed by England (1971) and later by Barber (2010). However, an unbounded rotation is expected at the corner. Unfortunately, the Mitchell solution for a biharmonic function in polar coordinates is not general enough to include the solution of the present problem due to the non-classical boundary condition along the common diameter and to the boundary layer which may arise at both edges of the adhesive bonding. The method of complex potentials also requires additional terms to properly model the boundary layer which may arise at the bonding edges. In the present investigation we instead adopt bipolar coordinates to describe the geometry of the problem. The most general form of a biharmonic stress function in bipolar coordinates was found out by Jeffery (1921). The approach was then generalized by Ling (1948a,b), who solved the problem of two overlapping holes in an infinite plate under general far-field stresses by introducing a general integral solution of the biharmonic equation in bipolar coordinates. Then, he determined the parameters involved in the solution from the given boundary conditions with the aid of Fourier transforms. The use of bipolar coordinates has been proved to be effective for investigating mechanical problems whose geometries are defined by two circles or two circular arches, e.g. by using the Jeffery solution, Radi and Strozzi (2009) solved the problem of a circular disk containing a sliding eccentric circular inclusion; Radi (2011) presented an analytic solution for stresses induced in an infinite plate with two unequal circular holes by remote uniform loadings and arbitrary hole pressures and calculated the corresponding stress concentration factors; Lanzoni et al. (2017) investigated the effective thermal properties of composites reinforced with fibers whose cross section is defined by two circular arches. The problem is split into symmetric and skew-symmetric loading conditions, thus reducing the study to a quarter of disk. Here, we focused only on the skew-symmetric loading conditions because the symmetric problem trivial if the adhesive layer is modeled by an imperfect interface displaying only elastic shear compliance and no normal displacement discontinuity is admitted between the two halves. In this case, indeed, the solution of the symmetric problem coincides with the solution already available for an integer disk (Dong et al., 2004) since no tangential displacement discontinuity occurs between the two halves under symmetric loading conditions. Therefore, the present approach is not limited to the skew-symmetric loading condition but it provides the solution for the general loading case. In particular, the shear stress distribution along the bondline in the general case coincides with the solution for the skew-symmetric loading condition provided here. The solution is obtained here by exploiting the integral representation of the Airy stress function in bipolar coordinates in the form of a Fourier sine or cosine transform. The imposition of the boundary conditions on the tractions along the circular contour, the symmetry and bonding conditions along the adhesive thin layer then allows to reduce the problem to a Fredholm integral equation of the first kind defined on the half-line. After a preliminary asymptotic investigation of the stress and displacement fields at the bonding edges, which turn out to be finite therein, an approximate solution is provided by reducing the problem to a singular integro-differential equation defined on a bounded interval, and assuming a power series expansion for the shear stress distribution, whose coefficients are found by means of a collocation method. Another approximate closed-form solution is also derived by using a perturbation method that assumes the ratio between the shear modulus of the disk material and the shear stiffness of the adhesive layer as small parameter. In order to validate the. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172.

(4) ARTICLE IN PRESS. JID: SAS. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. 3. Fig. 1. Elastic disk adhesively bonded along a diameter and subject to compressive load P acting along an arbitrary direction defined by the angle θ 0 (a). The problem is split into the sum of symmetric (b) and skew-symmetric (c) loading conditions.. In particular, the circular and straight boundaries of the semidisk are defined by the coordinates β = π /2 and β = π , respectively (Fig. 2). The polar angle θ around the circle defined by the corresponding value of α is associated with the bipolar coordinate β by the following relations:. sin θ =. y sinhα = , R cosh α − cos β. cos. θ=. x sin β = . R cosh α − cos β. σx =. P. π Rb. 1 − 4cos2 θ0. 1 + 2y − y (1 − 2 cos 2θ0 ) 2. 4. (1 + 2y2 cos 2θ0 + y4 )2. 196 197 198 199. (3). For an intact disk with thickness b, the normal and shear stresses along a diametrical direction whose normal is inclined of an angle θ 0 with respect to the compressive loads P (Fig. 1(a)) are given by Dong et al. (20 04) and Sadd (20 05) and reported here:. . 195. . 200 201 202 203. ,. 2. τxy = Fig. 2. Semicircular elastic disk elastically constrained in the tangential direction along its straight boundary under skew-symmetric loading condition.. 176. analytical results, FE investigations and also experimental results obtained by using Digital Image Correlation (DIC) techniques are presented for varying loading orientation angle and material parameters.. 177. 2. Problem description in bipolar coordinates. 178. The problem considered here consists in a disk of radius R with an adhesive thin layer along a diameter (Fig. 1(a)), under plane strain or plane stress loading conditions. The disk is loaded by two opposite loads P acting along an arbitrary diametrical direction. The problem is split into the sum of symmetric and skewsymmetric loading conditions, as depicted in Fig. 1(b) and (c), respectively, thus reducing the analysis to a semidisk bonded to the adhesive thin layer (Fig. 2). A Cartesian coordinate system (0, x, y) is introduced with the y-axis direction coinciding with the bondline. Let θ 0 denote the angle between the direction of loading and the x-axis. Following Jeffery (1921), use is made of the bipolar coordinates (α , β ) with the poles located on the y-axis at y = ±R. The relations between Cartesian and bipolar coordinates then write. 173 174 175. 179 180 181 182 183 184 185 186 187 188 189 190. R sin β x= , cosh α − cos β 191 192 193. R sinh α y= . cosh α − cos β. (1). Elimination of α from equations (1) then provides the equation of a family of circles passing for the two poles with centers on the x axis at distance R cotβ from the origin and radius R/sinβ , namely. 194. (x − R cot β )2 + y2 = (R/ sin β )2 .. (2). 2P (1 − y2 ) (1 + y2 ) sin 2θ0 , π Rb (1 + y4 + 2y2 cos 2θ0 )2. (4). where the normal stress σ x is positive if tensile. The stress distributions provided by Eq. (4) are plotted in Fig. 3 for various values of the loading angle θ 0 . We chose to plot the stresses for θ 0 = 30°, 60°, 70°, 80°, namely with decreasing interval as θ 0 increases, because the stress distribution becomes more sensitive to the variation of θ 0 as θ 0 approaches 90°. From Eq. (4.2) it can be calculated that for θ 0 ≤ 52.2° the maximum of the shear stress is attained at the disk center (y = 0) where the normal stress is compressive, whereas for θ 0 > 52.2° the peak of the shear stress increases and its location moves toward the outer border (y = ±R). Correspondingly, the (peeling) normal stress becomes tensile at the disk center and attains a compressive peak near the bonding edges. As θ 0 approaches 90° the shear stress (4.2) tends to vanish along the loading line, whereas the normal stress (4.1) approaches the uniform tensile value P/π Rb. The latter result has been exploited for indirect tension testing. Similar trends are expected also for an adhesively bonded disk, as observed by the numerical investigations performed by Martin et al. (2012) on a epoxy-jointed ceramic disk. In general, for an adhesively bonded disk under symmetric loading conditions (Fig. 1(b)) shear stress may occur along the bondline in order to generate the same tangential strain both in the disk and in the adhesive layer (Lanzoni and Radi, 2009). However, if the adhesive layer is modelled as an imperfect shear interface then under symmetric loading conditions (Fig. 1(b)) the normal stress along the adhesive bonding coincides with (4.1), whereas the shear stress vanishes therein. Conversely, under skewsymmetric loading conditions (Fig. 1(c)) the normal stress vanishes, whereas the shear stress coincides with the solution of the general problem (Fig. 1(a)). Therefore, in the following we will focus only on the problem of skew-symmetric loading of a disk with an. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234.

(5) ARTICLE IN PRESS. JID: SAS 4. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. Fig. 3. Normal stress (a) and shear stress (b) distributions along the diameter of an intact disk whose normal is inclined of the angle θ 0 with respect to the loading direction.. 236. imperfect shear interface (Fig. 2), which has never been investigated previously.. 237. 2.1. Stress fields. 235. 238 239 240. 241 242 243 244 245 246 247. The problem is formulated using an Airy stress function χ (x, y), which satisfies the biharmonic equation χ = 0. In bipolar coordinates, this condition becomes (Jeffery, 1921):. . χ. =. ∞. 0. F (β ) sin sα ds,. (6). P. π Rb. {[ f1 (s ) sin β + f2 (s ) cos β ] sinh sβ. + [ f3 (s ) sin β + f4 (s ) cos β ] cosh sβ}.. (7). The corresponding in-plane stress components are given by:. ∂2 χ ∂ ∂ −sinh α −sin β + cosh α ] , ∂α ∂β J ∂β2 2 ∂ χ ∂ ∂ σβ = [(cosh α −cos β ) 2 −sinh α −sin β + cos β ] , ∂α ∂β J ∂α ∂2 χ τβα = −(cosh α − cos β ) . (8) ∂β ∂α J σα = [(cosh α −cos β ). 250 251 252. The introduction of the stress function (6) in Eq. (8) then provides the following integral relations for the stress components in bipolar coordinates:. σα =. . ∞ 0. {[F (β ) cos β − s2 (cosh α − cos β )F (β ) − F (β ) sin β ]. sin sα − s F (β ) sinh α cos sα} ds, ∞ τβα = −(cosh α − cos β ) s F (β ) cos sα ds.. . ∞ 0. {[F (β ) cosh α − F (β ) sin β + F (β )(cosh α − cos β )]. sin sα − s F (β ) sinh α cos sα} ds,. (9). 0. where. 253. P F (β ) = {[( f1 + s f4 ) cos β − ( f2 − s f3 ) sin β ] sinh sβ + π Rb + [( f3 − s f2 ) cos β − ( f4 − s f1 ) sin β ] cosh sβ}, P F (β ) = {[2s f3 + (s2 − 1 ) f2 ] cos β π Rb − [2s f4 − (s2 − 1 ) f1 ] sin β} sinh sβ P + {[2s f1 + (s2 − 1 ) f4 ] cos β π Rb − [2s f2 − (s2 − 1 ) f3 ] sin β} cosh sβ , (10) according to (7). The traction boundary conditions on the outer semicircle at β = π /2 then require:. ταβ (α , π /2 ) = 0, σβ (α , π /2 ) = p(α ),. where. F (β ) =. 249. (5). where J = R/(cosh α − cos β ). Being the expected stress distribution skew-symmetric with respect to the x-axis, the stress function χ can be assumed as the most general integral solution of the biharmonic equation in bipolar coordinates (5) that contains only odd terms in α . An integral solution of Eq. (4) that is odd in α then takes the following form. J 248. ∂4 ∂4 ∂2 ∂2 ∂4 χ + 2 + + 2 − 2 + 1 = 0, J ∂β4 ∂ α2∂ β 2 ∂ α4 ∂β2 ∂ α2. σβ =. 256 257 258. p( α ) = −. (12). being sin θ = tanh α for β = π /2, as it follows from Eq. (3), and thus. cosh α0 = 1/ cos θ0 ,. sinh α0 = tan θ0 .. Along the adhesive thin layer, namely at β = π , we must require vanishing of normal stress due to the skew-symmetry of the problem and proportionality between shear stress and radial displacement occurring for the imperfect interface condition and due to the shear compliance of the adhesive thin layer, namely. σβ ( α , π ) = 0 , u α ( α , π ) +. h τ ( α , π ) = 0, 2G αβ. 259 260. (13). (14). Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 255. (11). where p(α ) is the normal stress along the curved boundary at β = π /2 consisting in a concentrated radial force P/2 applied at θ = θ 0 . By introducing the Dirac delta function δ it writes. P P δ ( θ − θ0 ) = − δ (α − α0 ) cosh α0 2Rb 2Rb ∞ P =− cosh α0 sin sα0 sin sα ds, π Rb 0. 254. 261 262 263 264 265.

(6) ARTICLE IN PRESS. JID: SAS. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. 266 267 268 269 270. where G and h are the shear modulus and the thickness of the adhesive layer, respectively. The case of an intact disk is thus recovered as G tends to infinity or as h becomes vanishing small. For G = μ, the stress distribution along the bondline at distance h/2 from the mean line of the adhesive layer is instead recovered.. 271. 2.2. Displacement fields. 272. According to Jeffery (1921), the displacements uα and uβ along the directions orthogonal to the curves α = const and β = const (Fig. 2) are given by. 273 274. . 275. . 276 278. v v/ ( 1/v ). Q = (1 − ν¯ ) J. 282. . . ∂2 ∂2 − −1 2 ∂α ∂β2. . χ J. dα dβ. [(s2 + 1 ) F (β ) + F (β )] sin sα dα dβ .. (17). Introduction of (10) in (17) then yields. Q = J. 280. (16). ∞ P {2(1 − ν¯ ) [( f2 cos β + f1 sin β ) cosh sβ π Rb 0 + ( f4 cos β + f3 sin β ) sinh sβ ] cos sα ds + D sin β},. 2μ uα ( α , π ) = − R. . ∞ 0. . ν¯ s cos sα + (1 − 2ν¯ ). F (π ) + (1 − ν¯ )[F (π ) + F (π )]. sinh α sin sα 1 + cosh α. cos sα s. ds+. . P. π Rb. D.. (19) 283. 3. Imposition of the boundary conditions. 284. The functions fi (s), for i = 2, 3, 4, introduced in Eq. (7) can be determined in terms of f1 (s) by imposing the boundary conditions (11), (12) and (14) at β = π /2 and β = π . In particular, from conditions (11) and (12) one obtains. 285 286 287. F ( π /2 ) = 0, 288. . ∞. =. P cosh α0 2Rb. δ (α − α0 ).. (21). By taking the derivative with respect to α of Eq. (21). 289. . ∞ 0. 291. s [s cosh α sin sα + sinh α cos sα ]F (π /2 ) ds. s (s2 + 1 ) F (π /2 ) cos sα ds =. P δ (α − α0 ) cosh α0 , (22) 2Rb cosh α. and, then, the inverse Fourier cosine transform of Eq. (22), one obtains. (cosh α + 1 ) P. 0. ∞. 293. s F (π ) cos sα ds +. D. π Rb 2 ε (1 − ν¯ ). 1 2ε. 294. . ∞ 0. F (π ). cos sα s. ,. (25). where the following parameter has been introduced. h μ . 2R (1 − ν¯ ) G. 295. (26). Let us define. 296. cossα. coshα0 ∂ ssinsα0 + tanhα0 cossα0 0 q (s ) = =− , 2 2 s (1 + s ) s (1 + s ) ∂ α0 coshα0 (27) then, by considering the general expression (7) of function F(s) and its derivatives (10), the boundary conditions (20), (23) and (24) then yield. sπ + f1 (s )], 2 sπ sπ f3 (s ) = q(s ) sech − f1 (s ) tanh , 2 2 sπ f4 (s ) = 2s[q(s ) sinh + f1 (s )]. 2. 297 298 299. f2 (s ) = −2s coth sπ [q(s ) sinh. (28). The introduction of Eqs. (28) into (10) then provides. P 2 2 sπ [(s2 − sinh ) f 1 (s ) π Rb sinh sπ 2 sπ + (s2 − cosh sπ ) sinh q(s )] 2.

(7) P sπ F (π ) = − 2s f1 (s ) + 2q(s ) sinh . π Rb 2. 300. F ( p) =. (29). Finally, the introduction of Eqs. (29) into (25) yields the following integral equation for the unknown function f1 (s):. . 2s 2 sπ [(s2 − sinh ) f 1 (s ) sinh s π 2 0 sπ + (s2 − coshsπ )q(s ) sinh ] cos sα ds 2 ∞ 1 sπ D − [ f1 (s ) + 2q(s ) sinh ] cos sα ds = , ε 0 2 2ε (1 − ν¯ ). (cosh α + 1 ). (30). for 0 ≤ α ≤ ∞. In order to simplify the previous integral equation, let us define the function. sπ D + δ (s ), 2 2 (1 − ν¯ ). ∞. . 2s (cosh α + 1 ). 0. −. t (s ). ε. sinh sπ. (s2 −sinh2. 303 304. (31). then, the introduction of (31) into (30) provides the following Fredholm integral equation of the first kind for the unknown function t( s). . 301 302. ∞. 305 306 307.

(8) sπ. sπ )t (s ) − (1 + s2 )q(s ) sinh 2 2. cos sα ds = 0,. that can be written in the more compact form:. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 292. (24). t (s ) = f1 (s ) + 2q(s ) sinh 0. 290. (20). (23). F ( π ) = 0,. (18). where the last term defines an arbitrary rigid body motion. In particular, the radial displacement along the adhesive thin layer at β = π follows from (15.1) and (18) as. (23). Moreover, the introduction of the stress and displacement components (9) and (19) into the boundary conditions (14) yields. ε=. plane strain plane stress. . = −(1 − ν¯ ). 281. (15). being ν the Poisson ratio of the disk material. The function Q is determined from the Airy stress function χ by the condition (Jeffery, 1921). 279. dα. ds =. respectively, where μ is the elastic shear modulus and. v¯ =. 277. (1 − 2ν¯ ). ∞ P δ (α − α0 ) cosh α0 cos sα π Rb cosh α 0 P = (s sin sα0 + tanh α0 cos sα0 ). π Rb. s ( s2 + 1 ) F ( π /2 ) =. . ∂χ ∂ Q − , ∂α ∂β R ∂χ ∂ Q uβ = (1 − 2ν¯ ) + , 2μ J ∂β ∂α R 2μ J. uα =. 5. 308.

(9) ARTICLE IN PRESS. JID: SAS 6. . 309 310. 2s 2 sπ s2 − sinh sinh s π 2 0 = G(α ), for 0 ≤ α ≤ ∞, ∞. 311 313 314 315 316. 317. −.

(10). 1 t (s ) cos sα ds (1 + cosh α ) ε (32). . to. Gradshtein. s sin sα0 + tanh α0 cos sα0 cos sα ds cosh(sπ /2 ) 0 4 coshα sinh2α0 cosh α0 = = (cosh2α + cosh 2α0 )2 ∂ 2 coshα coshα0 = −cosh α0 . ∂ α0 cosh2α + cosh 2α0 ∞. ∞ 0. . ∞. 0. T (α ) cos sα dα ,. (34). T (z ) dz. . ∞ 0. 2s 2 sπ s2 − sinh sinh sπ 2. 321 322.

(11). ∞ 0. = + 323 324. (35). for 0 ≤ α ≤ ∞. By using contour integration and calculus of residues, one can evaluate the inner definite integral in (35) as the sum of the residues of the integrating function in the upper halfplane of the complex variable s multiplied by π i, namely. . 2s cos sα cos sz 2 2 sπ s − sinh ds sinh sπ 2 2 cosh(α − z ) − 1 [1 + cosh(α − z )] sinh (α − z ) 2 cosh(α + z ) − 1 2. [1 + cosh(α + z )] sinh (α + z ) 2. .. (36). Moreover, the inversion of the Fourier cosine transform of the Dirac delta function δ (α − z) yields. . ∞ 0. cos sα cos sz ds =. π 2. δ ( α − z ).. (37). By using the results (36) and (37), then Eq. (35) becomes. 325. . ∞. [2 cosh(α − z ) − 1] T (z ) dz. −∞. [1 + cosh(α − z )]sinh (α − z ) 2. −. π T (α ) = G ( α ). 2 ε 1 + cosh α (38). 326. . Integration of Eq. (38) with respect to α then provides ∞ −∞. =. cosech(α − z ) π T (z ) d z − 1 + cosh(α − z ) 2ε. . 2 sinh α sinhα0 + arctan cosh 2α + cosh 2α0. sinh α cosh α0. . T (α ) d α 1 + cosh α. . tanh α0 , 327 328. (39). for −∞ ≤ α ≤ ∞. Let us now introduce the following transformations. z = log. 1−t , 1+t. α = log. Fig. 4. Normalized angular variation of the leading order stress field in polar coordinates near the semidisk corner.. 1 − cos sα cos sz ds = G(α ), (1 + cosh α ) ε 320. (33). then by introducing (34) in Eq. (32) and changing the order of integration one obtains. . 319. and. The integral Eq. (32) will be solved in Section 6 by using a perturbation approach that assumes ε as a small parameter (Kanwal, 1971). In the following, we introduce some further transformations in order to reduce the integral Eq. (32) to a more familiar form that allow us for its numerical treatment. To this aim, let us denote with T(α ) the Fourier cosine transform of t(s). t (s ) =. 318. where, by using (27) and according Ryzhik (1965), the right hand term is. G (α ) =. 312. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. 1+y , 1−y. (40). then, considering that T(y) = T(−y) due to symmetry condition, Eq. (39) becomes. . 330. . 1 (1 − t 2 ) T (t ) π 2 (1 − y2 ) p.v. dt − T (y ) d y = 3 ε −1 (t − y ) (1 − ty ) 2y. 2y (1 − y2 )coshα0 1 =π + arctan 2 1 − y2 cosh α0 4y2 + (1 − y2 ) cosh2 α0 tanh α0 , (41). for − 1 ≤ y ≤ 1, where p.v. denotes the Cauchy principal value of the integral. The singular integro-differential Eq. (41) will be solved for the unknown function T(y) in Section 5 by using an approximate procedure based on a power series expansion and the collocation methods (Erdogan et al., 1973; Monegato, 1987; Badr, 2001). The shear stress along the adhesive thin layer follows from (9.3) for β = π , by using (27), (29), (31), (33) and (34), as. ταβ (α , π ) =. 331 332 333 334 335 336 337. ∞. P 2s (1 + cosh α ) G(α ) − π Rb sinh sπ 0.

(12) 1 P 2 sπ s2 − sinh t (s ) cos sα ds = − T ( α ). 2 ε π Rb (42). 4. Asymptotic analysis near the corner of the semicircle A preliminary asymptotic analysis of the stress and displacement fields near a right-angle elastic wedge with one face free of tractions and the other one subject to the imperfect interface conditions between the shear stress and the radial displacement (Mishuris 1999, 2001) is performed in order to investigate the behavior of these fields at the bonding edges, namely at y = ±R. The results will then be useful for the proper treatment of the integral Eq. (32) that will be performed in Section 5 and, in particular, for the definition of a power series expansion for the function T(y). With respect to a polar coordinate system (0, r, θ ) centred at the corner of the wedge (Fig. 4), the considered boundary conditions write:. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 329. 338 339 340 341 342 343 344 345 346 347 348 349 350. Q3.

(13) JID: SAS. ARTICLE IN PRESS. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. 7. Fig. 5. Normalized shear stress distribution along the adhesive thin layer for different values of the relative stiffness parameter ε and four different loading orientation angle θ 0.. σ θ θ ( 0 ) = 0 , τr θ ( 0 ) = 0 σ θ θ ( π / 2 ) = 0 , τr θ ( π / 2 ) + K u r ( 0 ) = 0 . (43) 351 352 353 354 355 356 357. A similar stress analysis for a quarter of plane subject to uniform shear loading on one face can be found in Section. 11.1.1 of Barber (2010). Finite values of radial displacement and shear stress are expected at the right-angle corner. Therefore, the leading order terms of the biharmonic Airy stress function in polar coordinates can be assumed in the form. χ (r, θ ) = r2 (c0 + c1 θ + c2 cos 2θ + c3 sin 2θ ) + r3 (d0 cos θ + d2 cos 3θ ) + o(r 3 ), as r → 0. (44) 358 359. The corresponding stress and displacement components near the wedge corner are given by (Williams, 1952):. ∂ 2χ 1 ∂χ 1 ∂ 2χ ∂ 1 ∂χ σθ θ = 2 , σrr = + 2 , τr θ = − , r ∂r ∂ r r ∂θ ∂r r ∂θ2. (45). 1 ∂χ ∂χ ∂ψ ∂ψ 2μ ur = − + (1 − ν¯ ) r , 2μ uq = − + (1 − ν¯ ) r 2 , ∂r ∂θ r ∂θ ∂r (46) where the displacement function ψ must be an harmonic function obeying the following condition. ∂ ∂r. . r. ∂ψ ∂θ. . = χ ,. namely:. ψ (r, θ ) = b(cos θ )/r + 2c1 (θ 2 − ln2 r ) + 4c0 θ + 4r d0 sin θ + o(r ), as r → 0.. 361 362. (47) 363. (48). Then, the constants ci and di (i = 0, 1, 2, 3) can be determined by using conditions (43), thus providing the following asymptotic. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 360. 364 365.

(14) ARTICLE IN PRESS. JID: SAS 8. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. Fig. 6. Shear stress distribution along the adhesive thin layer obtained by the perturbation method for small values of the relative stiffness parameter ε , for two different loading orientation angle θ 0 .. Correspondingly, the asymptotic stress field in the elastic half plane takes the following form. 368 369. σθ θ = 12 (π sin2 θ + sin 2θ − 2θ ) + 34πεr cos θ sin2 θ + o(r ), π r (cos θ + 3 cos 3θ ) + o(r ), σrr = 12 (π cos2 θ − sin 2θ − 2θ ) + 16 ε τrθ = 12 (2 sin θ − π cos θ ) sin θ − π8 εr sin θ (1 + 3 cos 2θ ) + o(r ). (51) where the normalization condition τ r θ = 1 for θ = π /2 and r = 0 has been introduced, as usual in the asymptotic analysis of wedge and crack problems. The normalized angular variation of the leading order stress components (51) are plotted in Fig. 4. There, it can be noted that the stress field is finite at the corner and has no logarithmic singularity under the boundary conditions (43), even if the principle of shear stress reciprocity is not met at the rightangle corner (Barber, 2010). According to (51) the shear stress at the edges of the adhesive layer, namely for θ = π /2, behaves as. τr θ ( r ) = ( 1 + Fig. 7. Specimen considered in the FE analysis.. 366. r2 (π sin2 θ + sin 2θ − 2θ ) 4 π r3 + (cos θ − cos 3θ ) + o(r3 ), 32 ε. ur = −. 373 374 375 376 377 378 379. 380 381 382. sin θ r + [(π − 4θ )(1 − 2ν¯ ) + π cos 2θ − 2 sin 2θ ] K 8μ. π r2 (1 + 2ν¯ − 3 cos 2θ ) cos θ , +o(r2 ), 32 με cos θ r uθ = − + [8(1 − ν¯ ) ln r − π sin 2θ − 2 cos 2θ ] K 8μ −. as r → 0,. (49). +. 367. ψ (r, θ ) =. 371 372. (52). The displacement field within the half plane corresponding to the Airy stress function (49) (see Barber (2010), Tables 8.I and 9.I) reads:. series expansion of the Airy stress function. χ (r, θ ) =. πr )τ (0 ) + o(r ). 4 ε rθ. 370. 2ε π 2 cos θ + ln r + θ − θ 2 r 2 πr + sin θ + o(r ), as r → 0. 8ε. (50). π r2 (1 − 2ν¯ − 3 cos 2θ ) sin θ + o(r2 ). 32 με. (53). Therefore, the displacement field is also finite at the corner. However, the rigid rotation Wr θ turns out to be singular as ln r as r → 0 due to the term r ln r occurring in uθ , as observed by England (1971) and Barber (2010), namely. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 383 384 385 386.

(15) ARTICLE IN PRESS. JID: SAS. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. 9. Fig. 8. Finite element mesh, loading (red arrows) and constraints (green arrows). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). Wrθ =. 1 2r. . ∂u 1 ∂ ( r uθ ) − r = [3 − 4ν¯ + 4(1 − ν¯ ) ln r] + o(ln r ). ∂r ∂θ 4μ (54). 387 388 389 390 391 392 393 394 395 396. ∞ . cn y2n .. (55). n=1. 397. 2. + 2y2n ln. The kernel of the integral Eq. (32) is almost general and singular as s tends to infinity, and thus no exact closed form solution for the function t(s) can be expected; it can be calculated only numerically. As shown in Section 4, the shear stress is found to be finite at the bonding edges. Moreover, it is symmetrically distributed with respect to the x-axis, as expected for skew-symmetric loading conditions. Then, according to (42) we can assume a regular power series expansion for the even function T(y), namely:. Therefore, the singular integro-differential Eq. (41) provides the following relation ∞ . . cn Hn (y ) −. n=1. =π. . π y2n+1 ε 2n + 1. . 2y (1 − y )coshα0. 4y2 + ( 1 − tanh α0 ,. 2. 2 y2 cosh2. ). α0. + arctan. (1 −. y2. 2y ) cosh α0. (56). 399. Hn (y ) = (1 − y2 ) p.v.. 5. Numerical solution of the singular integro-differential equation. T (y ) =. 398. for −1 ≤ y ≤ 1, where. +. . 1 −1. (1 − t ) t 1−y dt = 2n 3 y (t − y )(1 − ty ) 2. 2n. 2. 1+y + y (y2 , 1, 1/2 − n ) + 1−y. 1 [ ( 1 − 2n )n y4 + ( 1 + 4n2 )y2 y. − n(1 + 2n )](y2 , 1, 1/2 + n ),. (57). where is the Lerch transcendent function Gradshtein and Ryzhik, 1965). Note that the conditions Hn (±1) = 0 and Hn (0) = 0 follow from ((57). In order to evaluate the unknown coefficients cn of the power series expansion (55) from Eq. (56) we consider a finite number N of terms in the series expansion. Then, by adopting a collocation method we evaluate Eq. (56) at N + 1 points chosen as the Gauss points within the interval [0, 1], namely at. . yk = sin. k+1 π N+1 2. . ,. for k = 0, 1, ..., N,. 400 401 402 403 404 405 406 407. (58). thus obtaining a linear system of N + 1 equations for the unknowns cn (n = 0, 1,…, N). Once the function T(y) has been calculated, the shear stress distribution along the adhesive bonding follows from (42). The results for the shear stress distribution obtained for N = 30 are plotted in Fig. 5 for various values of the relative stiffness parameter ε rang-. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. Q4. 408 409 410 411 412 413.

(16) JID: SAS 10. ARTICLE IN PRESS. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. Fig. 9. Analytical (solid line) and FEM (markers) results for the shear stress distribution along the bondline for the same elastic modulus of the disk E = 206 GPa and four different elastic moduli of the adhesive layer.. vided in (4.2) and plotted in Fig. 3(b) for an intact disk, namely for an infinite stiffness of the adhesive thin layer. In this case, the shear stress is vanishing at both joint edges and attains its maximum value at the disk center for a small angle θ 0 or at increasing distance to the centre for large loading angle θ 0 , thus recovering the results of Dong et al. (2004) plotted in Fig. 3(b). As the shear stiffness of the interface decreases, namely for increasing values of ε, the shear stress distribution along the interface becomes more and more uniform. Note that the area under the curves in each figure is constant and coincides with the overall load component along the bonding direction, namely. 0. Fig. 10. Close-up of the broken specimen after the experimental test. Yellow polyurethane upper layer was used to avoid excessive concentrated loads. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). 414 415 416. ing between 0 and 5 and for four loading orientation angles θ 0 , namely 30°, 60°, 70° and 80° From these plots it can be observed that, as ε tends to 0, the shear distribution approaches that pro-. R. ταβ (α , π ) dy =. P sin θ0 . 2b. 419 420 421 422 423 424 425 426 427. (59). As the loading inclination angle θ 0 increases the shear stress along the bonded joint attains a larger peak closer to the edges. In particular, for θ 0 = 90° and ε = 0 the shear stress becomes unbounded therein. However, for ε > 0 the shear stress displays a smooth peak and non-singular behavior at the edges of the bonding line, where it approaches the asymptotic prediction (52). Note also that the tangential displacement along the bondline turns out to be proportional to the shear stress, according to the condition (14.2). 6. Perturbation approach for small ε A closed form approximate solution of the integral Eq. (32) can be derived by using a perturbation method as suggested in Chapter 11 of the book of Kanwal (1971) if the parameter ε defined. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 417 418. 428 429 430 431 432 433 434 435 436. 437 438 439 440.

(17) ARTICLE IN PRESS. JID: SAS. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. 11. Fig. 11. Force-displacement curve for the bonded disk (θ 0 = 60°).. 441 442 443 444 445 446 447. in (26) is sufficiently small, namely if the stiffness of the elastic spring is relatively large. In this case, the lowest order term for ε = 0 corresponds to an infinite shear stiffness of the adhesive thin layer and thus it coincides with the solution of the Brazilian test for an intact elastic disk (Dong et al., 2004), whereas the higher order contributions can be calculated by assuming a power series expansion in ε of the function t(s), namely. t (s ) =. ∞ . ε n tn ( s ).. (60). n=1. 448 449 450 451. 452. By introducing the series expansions (60) in the integral Eq. (32) and equating equal powers of ε , the problem can be reduced to the following system of integral equations with the same kernel:. . ∞ 0 ∞ 0. 453. t1 (s ) cos sα ds = −G(α ), tn+1 (s ) cos sα ds = (1 + cosh. for 0 ≤ α ≤ ∞, where. Gn ( α ) = 454 455 456. . ∞. 0. (61). α )Gn ( α ), ( n ≥ 1 ). sπ. 2s 2 s2 − sinh sinh sπ 2. tn (s ) cos sα ds.. (62). (63). The functions tn (s), for n ≥ 1, can be explicitly obtained by taking the inverse Fourier cosine transforms of (61) and (62), by using the result (33) and the definition (63), namely. . (1 + cosh α ) coshα cos sα dα = cosh2α + cosh 2α0 sinh α0 sin sα0 d cos sα0 = cosh α0 − , dα0 coshα0 cosh (sπ /2 ) coshα0 sinh (sπ /2 ). t1 ( s ) =. 4. π. cosh α0. d dα0. tn+1 (s ) =. 2. π. . ∞ 0. . tba (N ) (α , π ) =. . P (1 + cosh α ) G(α ) − ε n Gn ( α ) , π Rb N. 458 459. (66). n=1. where N is the number of terms considered in the power series expansion (60). The function g1 (α ), t2 (s) and g2 (α ) have been calculated by using contour integration and calculus of residues according to their definition (63) and (65). For the sake of conciseness, their expressions and derivations have been reported in the Appendix. The results obtained for N = 2, namely up to terms of order ε 2 in the expression of the shear stress field (66), have been reported in Fig. 6 for some small values of the parameter ε . It can be observed that for ε = 0 the shear stress distribution recovers exactly the solution reported by Dong et al. (2004). In this case indeed, by using (33), Eq. (66) provides the result (4.2). As ε increases the shear stress distribution becomes more uniform. However, the three-terms perturbed solution starts losing its validity already for small values of ε , namely for ε = 0.15 in the case θ 0 = 30° and for ε = 0.10 in the case θ 0 = 60°, thus denoting that a larger number of terms is required for obtaining an accurate approximation when the parameter ε is larger. Moreover, an unrealistic shear stress concentration is predicted at the bonding edges (y = ±R) as the value of ε increases.. 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479. ∞. 7. Verification by the finite element method. 480. A Finite Element analysis of a disk with radius R = 50 mm bonded along a diameter with a h = 0.1 mm thin adhesive layer has been performed by using the LUSAS Finite Element system. Mesh geometry and loading are shown in Figs. 7 and 8. The disk halves were discretized with quadratic 2D plane-strain/stress elements with out of plane width b = 1 mm. Two elements were. 481. 0. (64) 457. Finally, the shear stress along the adhesive bonding follows from (42), by using (60) and (61), as. (1 + cosh α ) Gn (α ) cos sα dα. (n ≥ 1 ). (65). Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 482 483 484 485 486.

(18) JID: SAS 12. ARTICLE IN PRESS. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. placed through the thickness of the adhesive layer while each adherend was partitioned in different regions with a finer mesh in proximity of the adhesive layer. The two regions near the bondline were uniform without space grading, resulting in quasi-square elements with side of 0.1 mm for the adherends and rectangular elements with sides 0.1×0.05 mm for the adhesive, as shown in Fig. 8. The total number of elements (adherends + adhesive) is 184,454 (8-noded quadrilaterals), with 20 0 0 elements in adhesive layer (2 elements through the thickness). We decided to adopt a quite refined mesh since we were interested in getting a fairly precise results on the bondline, especially at y = ±R. The materials were assumed elastic with the constitutive properties of steel for the semidisks (E = 206 GPa, ν = 0.3) and various elastic properties for the adhesive layer, in order to verify the effect of the elastic mismatch (E0 = 206 GPa, ν 0 = 0.3, E1 = 2060 MPa, ν 1 = 0.3, E2 = 206 MPa, ν 2 = 0.3, E3 = 20.6 MPa, ν 3 = 0.3). Both adhesive and adherends are modelles as linear elastic and isotropic materials and no failure criterion is adopted. The purely elastic response, consistent with the assumptions of the theoretical model, makes the results applicable to brittle adhesives. Compared to Fig. 7 the model in Fig. 8 is rotated in order to place the bondline along the vertical y-axis and the external load is decomposed according to the vertical and horizontal directions. The centre node O of the adhesive layer was constrained horizontally to eliminate any rigid-body motions (see Fig. 8). The results of FEM analyses are reported in terms of shear stress along the adhesive bondline. Since the analytical framework considered a perfect bond in normal direction and the normal strains within the adhesive thin layer have been neglected, then the normal stress along the bondline are expected to coincide with those occurring in an intact disk (14). Future work will be aimed at estimating the stress field in case of an elastic bond in normal direction. The shear stress is symmetrically distributed with respect to the center of the bondline and four curves are reported in Fig. 9 for the four configurations analyzed. It is immediate to verify that the FE predictions (single markers) for all the considered configurations coincide with the analytical results (solid lines) obtained from the analytical approach developed in Section 5. Some discrepancy has been found for the finite elements placed at both edges of the bonding line, due to the infinite rigid rotation predicted by the asymptotic analysis in Section 4, which may cause some numerical instability for the FE code. 8. Experimental results from digital image correlation. Fig. 12. Digital image correlation of the bonded Brazilian disk at point 11,735 N (a), at 13,201 N (b) and the onset of failure at 14,427 N (c).. The Digital Image Correlation (DIC) was used to validate experimentally the theoretical and numerical analyses of the bonded disk. DIC is a powerful optical-numerical method which measures full-field surface displacements by means of a non-invasive contactless techniques. Several works about the application of DIC techniques in adhesive bonded joints analysis can be traced in technical literature. Comer et al. (2013) and Kumar et al. (2013a,b) exploited them to analyze composite joints and kissing bonds (2013b). Guo et al. (2011) used them to study the adhesive interface, Colavito et al. (2013) showed some refinements to get reliable results in the adhesive joints analyses. The review work of Hild and Roux (2006) shows the main experimental techniques about this methods and the work of Pan et al. (2009) helps in choosing the subset size and the region of interest. The DIC system is able to assess the full displacement field by using a contactless optical method, which exploits a high resolution camera and a coherent light source. The specimen is painted using a random net of dots, called speckles, which are used by the software to correlate the images and to determine the displacement field. The camera records the image of the unloaded specimen and then the loading. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550. Q5. Q6.

(19) JID: SAS. ARTICLE IN PRESS E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx. [m5G;August 28, 2018;12:47] 13. Fig. 13. Shear strain distribution along the bondline for the three acquisition points of Fig. 12. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591. procedure starts, synchronized with the CCD acquisition. The system records a set of deformed images as the load increases and compares them with the reference configuration. Since the initial position of each subset is known, and its final position can be estimated, it is possible to compute the displacement fields either in the plane (2D DIC, one camera needed) or in the space, by using two cameras and recovering the out of plane displacement by a triangulating procedure (3D DIC, two or more cameras needed). The experimental test were carried out by using a DIC Software by Dantech Dynamics called Istra 4D, in the 2D configuration, since no measure of the out of plane displacement is needed, as stated by Pan et al. (2009). The camera is a 5MP CCD with a C-mount lens with 50 mm focal length. The experimental set-up is made of two flat plates used to compress the speckled Brazilian bonded disk. The speckle patterns were acquired by randomly spraying black and white paints on the two flat semi-disk surfaces, according to Pan et al. (2008). An image captured at the end of the experimental procedure is reported in Fig. 10. The test rig implements the conceptual loading configuration reported in Fig. 7 for a loading inclination angle θ 0 = 60°, in order to demonstrate the feasibility of the analytical procedure developed. A polyurethane rigid sheet is inserted on the top plate in order to give a smoother compression to the disk and it is visible in Fig. 10 in yellow and to ensure the correct position of the specimen, preventing possible slips. We decided to add this layer only on the top to be able to verify if the steel-steel contact on the bottom part could cause problems during testing. This additional elasticity does not affect the maximum failure force of the joint but only the stroke measured by the universal machine, which is not used since the full field acquisition by the DIC apparatus is available. The specimen is made by two mild steel adherends (E = 206 GPa) with thickness b = 10 mm, and radius R = 25 mm. The two adherends are bonded with the high strength epoxy Loctite Hysol 3422 (Henkel, 2003), which ensures fast curing and good material strength. The adhesive elastic modulus, according to the producer datasheet, is 1298 MPa, with a nominal tensile strength of 28.6 MPa and an elongation at break of 3.3%. All properties were retrieved from technical data sheets provided by the producer and comply with the ASTM D882 Standard. Some other experimental test Carbas et al. (2013) reveals that this adhesive tends to have brittle failure when cured at room temperature. The tests were carried out on a MTS Bionix 858 universal machine equipped. with a 25 kN load cell at quasi static loading rate of 1 mm/min. The force-displacement curves were sampled at 102.5 Hz while the DIC acquisition was set at 1 Hz, in order to limit the amount of data to process. The force-displacement curve derived from the experimental set-up in the previous section (Fig. 10) is reported in Fig. 11. The curve shows a monotonic non-linear behaviour up to failure at 14,427 N. The non-linearity is due to the soft layer of polyurethane needed to avoid misplacement of the specimen and excessive contact pressure. As soon as the first crack in the adhesive bondline occurs an immediate abrupt failure of the joint takes place. The crack was easily detectable using the DIC equipment. The primary output of the digital image correlation is the displacement full field, from which the DIC software is able to compute all the other interesting quantities, such as the shear strains, through simple differentiation. Fig. 12 shows the engineering shear strain map for three salient points highlighted in Fig. 11. The strains, expressed in millistrain, are calculated using a subset of 17 × 17 pixel, following the work of Pan et al. (2008) in addition to the Dantech Dynamics manual and using a reference system consistent with the bondline, visible in Fig. 12(c). The frame reported in Fig. 12 shows a loading state where it is possible to see the increasing shear strains in the bondline, which reach the maximum at the onset of the crack, shown in Fig. 12(c), where it is clearly visible an almost uniform and regular strain distribution in the adhesive layer. The shear stress distribution along the bondline at the three points analyzed is reported in Fig. 13. The first part of the bondline (around 7 mm) is not reported since the correlation is prevented by the deformation of the PU sheet. After the frame showed in Fig. 12(c), the correlation is lost, since the cracked part of the image cannot be computed due to the excessive displacement. The average value of the shear strain along the bondline is 33.5 mstrain, (blue solid line in Fig. 13), consistent with the producer datasheet. The experimental tests demonstrates a good behaviour of the proposed joint in terms of easiness of manufacture and simple set-up. The bonded Brazilian disk is very easy to be obtained with minimum machining of a bar, and it is simply compressed between flat plates with a universal testing machine, with no need of special fixtures.. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632.

(20) ARTICLE IN PRESS. JID: SAS 14. 633. 9. Conclusions. 634. A simple test for the characterization of adhesively bonded joint has been proposed here, as a generalization of the widely used Brazilian test for homogeneous brittle specimens. Due to the simple geometry of the test, mixed-mode loading conditions along the adhesive layer can be easily achieved by properly selecting the loading inclination angle. To this aim, analytical and numerical solutions for the stress and displacement fields in a bonded elastic disk with a diametrical adhesive thin layer, subjected to compression along an arbitrary direction, have been derived herein. In particular, the shear stress distribution along the adhesive thin layer as well as the stress and displacement fields within the elastic disk have been calculated for various loading orientation angles. Then, the analytical results have been successfully compared with FE numerical predictions. Finally, the DIC technique was also employed in order to demonstrate the feasibility of the proposed test. The present investigation thus provides some fundamental understandings of the effects of adhesive compliance on the distribution of the shear stress along the adhesive bonding and near the edges and makes clear that a reduction of the stress peaks can be achieved by decreasing the bonding shear stiffness. The closed form solution presented here is significant especially if the adhesive layer is thin enough, due to the assumption of continuity of the normal displacement between the two disk halves. Indeed, in most real applications the thickness of the adhesive layer is made as small as possible. It may be considered particularly valuable, since it allows for the validation of numerical methods as well as for a preliminary design of adhesively bonded connections employed in many structural engineering applications.. 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 Q7 662. π 2 cos sα cos sα0 ds = 2 sinh sπ sinh (sπ /2 ) 0 π − 2 cosh (α − α0 ) + 2(α − α0 ) sinh (α − α0 ) = π cosh3 (α − α0 ) π − 2 cosh (α + α0 ) + 2(α + α0 ) sinh (α + α0 ) + . π cosh3 (α + α0 ) . J2 (α ,α0 ) =. Uncited references. 664. Dantech Dynamic Digitial Image correlation Q-400, Nov. 2017; Kaya and Erdogan, 1987; Timoshenko and Goodier, 1970.. 665. Acknowledgments. 666 667. Useful discussion with Prof. Gennady Mishuris of the Aberystwyth University (UK) is gratefully acknowledged.. 668. Appendix. 663. [m5G;August 28, 2018;12:47]. E. Radi et al. / International Journal of Solids and Structures xxx (2018) xxx–xxx ∞. 2s. s2 − sinh. (A4) The integrals in A3) and (A4) have been calculated by using contour integration as the sum of the residues of the integrating function in the upper halfplane of the complex variable s multiplied by π i. Then, from ((65) and (A2). t2 ( s ) =. . 2. cosh α0. π. where. ∂ 1 ∂ h1 ∂ h2 + tanh α0 , ∂ α0 coshα0 ∂ α0 ∂ α0. ∞ 0. (1 + cosh α ) Ji (α , α0 ) cos sα dα ,. namely. 678. 2π. 2s cosh. 2. 670. 671. By introducing (64) into (63) for n = 1, it follows:. ∞. ∂ 2s cos sα 2 sπ G1 (α ) = cosh α0 s2 − sinh ∂ α0 0 coshα0 sinh sπ 2 sinh α0 sin sα0 cos sα0 − ds, cosh (sπ /2 ) sinh (sπ /2 ). (A1). namely. ∂ 1 ∂ J1 ∂ L2 G1 (α ) = cosh α0 + tanh α0 , ∂ α0 coshα0 ∂ α0 ∂ α0 where. . π. (A2). 2 cos sα sin sα0 ds = 2 sinh sπ cosh (sπ /2 ) 0 2(α − α0 ) − π sinh (α − α0 ) + sinh 2(α − α0 ) = π cosh3 (α − α0 ) 2(α + α0 ) − π sinh (α + α0 ) + sinh 2(α + α0 ) − , π cosh3 (α + α0 ) (A3). J1 (α ,α0 ) =. ∞. 2s. s2 − sinh. 672. (A7) 679. . sπ h2 ( s ) = −2s sinh coshα0 2 2 sinh sπ

(21) sπ 2 sπ + (s2 − sinh ) cosh cos sα0 2 2 sπ 2 sπ + sinh α0 (s2 + sinh ) sinh sin sα0 . 2 2 Finally from (65) and (A5) one gets. 669. (i = 1, 2 ) (A6). . sinh sπ. 2π. G2 ( α ) =. 2. π . cosh α0. 1 coshα0. ∂ ∂ α0. . ∞. 680. sπ. 2s cos sα 2 s2 − sinh coshα0 sinh sπ 2. ∂ h1 ∂ h2 + tanh α0 ds, ∂ α0 ∂ α0 0. (A8). (A9). which can be calculated by using contour integration also. The result is quite a long expression which has not reported here for the sake of conciseness.. 681. References. 684. Adams, R.D., Coppendale, J., 1977. The elastic moduli of structural adhesives. In: Allen, K.W. (Ed.), Adhesion. Applied Science Publishers, London, United Kingdom, pp. 1–17. Adnan, A., Sun, C.T., 2008. Effect of adhesive thickness on joint strength: a molecular dynamics perspective. J. Adhes. 84 (5), 401–420. Arcan, L, Arcan, M, Daniel, L., 1987. SEM fractography of pure and mixed mode interlaminar fracture in graphite/epoxy composites. ASTM Tech Publ. 948, 41–67. ASTM D5656-10, 2010. Standard Test Method for Thick-Adherend Metal Lap-Shear Joints For Determination of the Stress-Strain Behavior of Adhesives in Shear by Tension Loading. ASTM International, West Conshohocken, PA doi:10.1520/ D5656-10. ASTM Standard E229-97, 1997. Standard Test Method For Shear Strength and Shear Modulus of Structural Adhesives. ASTM International, West Conshohocken, PA doi:10.1520/E0229-97. Awaji, H., Sato, S., 1978. Combined mode fracture toughness measurement by the disk test. ASME Trans. J. Eng. Mater. Technol. 100, 175–182. Badr, A.A., 2001. Integro-differential equation with Cauchy kernel. J. Comp. Appl. Math. 134, 191–199. Banks-Sills, L., Schwartz, J., 2002. Fracture testing of Brazilian disk sandwich specimens. Int. J. Fract. 118, 191–209. Barber, J.R., 2010. Elasticity, third ed Springer, New York. Berenbaum, R., Brodie, I., 1958. Measurement of the tensile strength of brittle materials. Br. J. Appl. Phys. 10, 281–287.. 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707. Please cite this article as: E. Radi et al., Adhesively bonded disk under compressive diametrical load, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.05.021. 676. (A5). sπ coshα0 2

(22) sπ 2 sπ − (s2 − sinh ) sinh sin sα0 2 2 sπ 2 sπ + sinh α0 (s2 + sinh ) cosh cos sα0 , 2 2. h1 ( s ) = −. 674 675. 677. . hi (s,α0 ) =. . 673. 682 683.