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The mechanics of bondline thickness in balanced sandwich structures
International Journal of Adhesion and Adhesives 78 (2017) 4–12
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International Journal of Adhesion and Adhesives journal homepage: www.elsevier.com/locate/ijadhadh
The mechanics of bondline thickness in balanced sandwich structures
MARK
E.H. Wong Nanyang Technological University, Energy Research Institute, 1 Cleantech Loop, Singapore 637141, Singapore
A R T I C L E I N F O
A B S T R A C T
Keywords: Adhesive bonded Single lap joint Electronic assemblies Closed-form solutions Singular stress
Assuming negligible in-plane stress, non-varying shear stress, and linearly varying transverse stress along the thickness of the adhesive, the closed-form solutions for the interfacial shear and the interfacial peeling stresses in balanced bonded sandwich structures was derived. The nil-shear-stress condition at the free edge of the adhesive was enforced through the use of a decay function. The solutions for the shear and the peel stresses along the interfaces right up to the free edge were validated with the finite element analysis. The solutions were used to investigate (i) the observed increase susceptibility of single lap joints with increase bondline thickness and (ii) the nil report of such susceptibility for electronic assemblies experiencing differential thermal strain. The first investigation was compromised by the inability of the solutions to model the singular stress/strain field at the free-edge of the interfaces. The second investigation revealed a far higher rate of reduction in the magnitudes of the interfacial shear and peeling stresses in structures (electronic assemblies) that experience differential thermal strain than in structures (single lap joints) that experience bending strain.
1. Introduction A large class of adhesively bonded structures have adherends that are structurally stiff relative to the adhesive. Ignoring the in-plane stress in the adhesive reduces substantially the complexity of the solutions and allows the formulation of closed-form solutions. The first of such analyses was presented by Volkersen (1938) [1] who modelled the adhesive as having only shear stiffness and the adherends as capable of only in-plane stretching. A much more sophisticated analysis was presented by Goland & Reissner (1944) [2] who modelled the adhesive as having stiffness in transverse stretching besides shearing and the adherends as capable of flexing besides in-plane stretching. However, their solution was restricted to balanced structures of single lap-joints. Delale et al. (1981) [3] and Bigwood & Crocombe (1989) [4] generalised the analysis of Goland & Reissner to unbalanced bonded structures experiencing a general state of edge loadings. Delale et al. [3] also incorporated shear deflection into the adherends by modeling the adherends as Timoshenko beams. In the meanwhile, the electronic assembly community was focusing on stresses in electronic assemblies due to mismatch thermal expansion of adherends. Chen & Nelson, (1979) [5] extended the solution of Goland & Reissner to unbalanced electronic assemblies experiencing differential thermal strain. The above authors have all assumed that the adherends experienced negligible shear and transverse-normal strains, which is not necessary true especially if the thickness of the adherends are significantly larger than that of the adhesive. Suhir (1986, 1989) [6,7] included approximately
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijadhadh.2017.06.003 Accepted 18 June 2017 Available online 24 June 2017 0143-7496/ © 2017 Elsevier Ltd. All rights reserved.
the shear and the transverse-normal compliances of adherends in the solution. Pao & Eisele [8] extended the solution of Suhir [6] to multilayer structures. The above authors have conveniently assumed the shear and the transverse stresses to be non-varying over the thickness of the adhesive, which however violates the differential equation of equilibrium:
∂τxz ∂σx + =0 ∂x ∂z ∂τxz ∂σz + =0 ∂z ∂x
(1)
In light of the inability of the above solutions to satisfy the equilibrium equation, these solutions are frequently referred to as the “strength of material” solutions. These strength of material approaches have also failed to satisfy the condition that there shall be nil shear stress at the free edges of the adhesive. A “theory of elasticity” solution was presented by Allman (1977) [9]. The adhesive was assumed to have nil in-plane stress, a nonvarying shear stress over its thickness, and a linearly varying transverse stress along its thickness. The assumed stresses satisfy the equilibrium Eq., Eq. (1). Allman described the stress/strain field in the adhesive and the adherends using two independent stress functions ensuring that the equilibrium condition, Eq. (1), and the free edge condition, τfree-edge=0, were satisfied. The differential equations for the stress functions were established by invoking the principal of complementary energy. Similar approach was followed by Chen & Cheng (1983) [10] and Adams &
International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
Nomenclature
σpi(x)
Subscript #i Subscript #1 & #2 are adherends and #3 is adhesive. Di, Ei, Gi, hi Flexural rigidity, elastic modu4lus, shear modulus, thickness of member #i. D Effective flexural compliance of the sandwich structure. M±il, N±il, Q±il Moment, sectional stretching force, sectional shear force applied to adherend #i at x=±l l Half of the length of the sandwich structure α, β Characteristic parameters of the sandwich structure. αi Coefficient of thermal expansion of adherend #i. εT, εNl, εMl Differential strain between adherends #2 and #1 at x=l due to temperature, due to edge stretching; collective fibre strain due to edge bending at x=l. κ±il Edge curvature of adherend #i at x=l. κsi, κs Shear compliance of adherend #i between its interface with the adhesive and its centroid plane, of the sandwich structure between the centroid planes of adherends #1 and #2. λxi, λx x-compliance of adherend #i, of the sandwich structure. λxθ Additional x-compliance of a sandwich structure due to it undergoes flexural deformation. λzi, λz z-compliance of adherend #i between its interface with the adhesive and its centroid plane, of the sandwich structure between the centroid planes of adherends #1 and #2. σm(x), σa(x) The mean, the amplitude of peeling stress at the two interfaces.
τ(x) ΔT
Peeling stress along the interface between adhesive and adherend #i Shear stress within the adhesive (assuming uniform) and along the interfaces Temperature change.
Basic formulas
4α 4
=
D ; 4λ z λx κs Ei hi3 (plane stress); 12 1 1 + D D1 2 κs1 + κs2 + κ s3 h1 , 4G1 h2 , 4G 2 h3 G3
β2
=
Di
=
D κs κs1
= = ≈
κs2
≈
κs3 λx λ x1
= = λ x1 + λ x 2 + λ xθ 1 ≈ Eh ,
λ x2
≈
λ xθ
=
λz λ z1
= λ z1 + λ z 2 + λ z3 11h ≈ 32E1 ,
λ z2
≈
λ z3
=
1 1
1 , E2 h2 1 h1 (h1 + h3) ⎡ D1 4⎣
+
h2 (h2 + h3) ⎤ D2 ⎦
1
11h2 , 32E2 h3 E3
bondline thickness in the SLJ and (ii) the nil-report of such a phenomenon for electronic assemblies that experience differential thermal strain.
Mallick (1992) [11]; the latter modelled the adhesive to be having a linearly varying in-plane stress, a parabolically varying shear stress and a cubicically varying transverse stress along its thickness. Four independent stress functions were needed, and these could only be solved using numerical analysis. Yin (1991) [12] generalised the stress function approach to multi-layer structures. Interested readers may refer to the article of Wong & Liu [13] for a more elaborated review of the classical works in lap-joints and in microelectronic assemblies. The main critique of the closed-form solutions is their inability to correlate with the experimental observations that the susceptibility of single lap shear joint (SLJ) to failure could increase with increasing bondline thickness [14–23]. Gleich et al. [24] have shown, with the aid of finite element analysis (FEA), that the interfacial stresses immediately adjacent to the free edge of the bonding interface decreased before they increased with increasing bondline thickness. Gleich et al. [25] and Yang et al. [26] have shown, with the aid of FEA, that the generalized stress intensity factors, H1 and H2, that describe the singular stress field at the free edge of the bonding interface, σij (r ) = H1 r −λ1 + H2 r −λ2 , decreased and then increased with increasing bondline thickness. In contrast to the feverish interest in SLJ, there has been calm in electronic community – there has been no report of increase vulnerability with increasing bondline thickness. It is the objective of this article to investigate, using improved analytical solutions, (i) the observed increase vulnerability with increasing
2. Interfacial stresses in balanced structures Despite its inferior accuracy over the theory of elasticity solution, the strength of material solution continued to be favoured by practising engineers because of its simplicity. However, its inability to model the nil-shear-stress condition at the free edge could lead to solutions that are too erroneous to be useful. An improved strength of material solution that is incorporated with a decay function to enforce the nilshear-stress condition at the free edge [27,28] is used for this study. Fig. 1 shows a bonded structure made up of adherend #1, adherend #2, and adhesive #3. The structure experiences a mismatched thermal expansion between the adherends; and experiences stretching, shearing, and bending at their edges. Note the notations and the positive directions of the in-plane stretching forces N ± il, the section shear forces Q ± il, the moments M ± il, and the curvature κil, at x= ± l, where l is the half-length of the adhesive. The height, Young’s modulus, shear modulus, flexural stiffness, and thermal coefficient of expansion of member #i are denoted as hi, Ei, Gi, Di, and αi, respectively. The shear, the in-plane, the transverse, and the flexural compliances of the structure are denoted as κs, λx, λz, and D , respectively. The corresponding compliances of member #i are denoted as κsi, λxi, λzi, and Di , Fig. 1. Schematic of a bonded structure experiencing general condition of edge loadings and thermal strain.
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International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
respectively. The formula for computing these compliances are collected under the heading “basic formula” at the front of this article. The derivations of these formula are presented in Appendix B. The adhesive is assumed to experience negligible in-plane stress, σx. The differential equation of equilibrium, Eq. (1), then suggests an unvarying shear stress, τxz, and a linearly varying transverse stress, σz, along the thickness of the adhesive. Denoting σm and σa as the mean and the amplitude of variation, respectively, of the transverse stress along the thickness of the adhesive, the peeling stress at the interface between the adhesive and adherend #i is given by
σpi = σm ∓ σa, i = 1.2.
A3 =
n ≈ e nφβh3 − 1 (3)
φ ≈ 0.407(βh30.88)−0.26
d3τ dτ − β2 = 0, dx 2 dx
Referring to Eq. (A6) and with the parameter, μτ, equates to nil for a balanced bonded structure, the differential equation for the mean of the peeling stress is given by
= λ x / κs . The solution to Eq. (4) is given by sinh βx cosh βx + Ae + A3 , cosh βl sinh βl
d4σm + 4α 4σm = 0 dx 4 (5)
1 (ε 2βκs Nl
− ε−Nl + εMl − ε−Ml )
A3 =
1 2l
(N
2Ae β
2l
σm (x ) = Ce1 ξe1x + Ce2 ξe2x + Co1 ξo1x + Co2 ξo2x ,
,
)
(6)
where
εT = (α2 − α1) ΔT ε ± Nl = N ± 2l λ x 2 − N ± 1l λ x1 1 ε ± Ml = ((h1 + h3) κ ± 1l + (h2 + h3) κ ± 2l ), 2
(7)
Do note that while the fibre strain on adherend #i at x = ± l is indeed described by the simple bending relation: ε ± il = hi M ± il /(2Di ) = hi κ ± il/2 ; the condition of compatibility dictates that the effective bending strain, ε ± Ml, ought to include the thickness of the adhesive [27]. For structures of reasonably large length, say βl > 3, Eq. (5) may be expressed more concisely as
τ (x ) = As e β (x − l) + A3 , x > 0
(8)
where
As =
εT + εNl + εMl βκs
(15)
wherein the first two terms are associated with symmetric deformation and the last two terms with anti-symmetric deformation. The coefficients and the functions are given by
+ εNl + ε−Nl + εMl + ε−Ml )
− N−2l −
(14)
where 4α 4 = D / λ z . The solution to Eq. (14) is given by
wherein the first term is associated with symmetric deformation and the second term with anti-symmetric deformation about the mid length of the structure at x=0. The coefficients Ao, Ae, and A3 are given by
Ae =
(13)
2.2. The mean of peeling stress, σm(x)
(4)
1 (2εT 2βκs
(12)
for 0.018≤βh3≤1.73.
Referring to Eq. (A5) and with the parameter, μσ, equates to nil for a balanced bonded structure, the differential equation for the shear stress in the adhesive and along the interface is given by
Ao =
(11)
The “constant” φ is indeed dependent on β and h3, which has been established through collocating with the finite element analysis as
2.1. Interfacial shear stress, τ(x)
τ (x ) = Ao
(10)
wherein n is a positive real number with a magnitude significantly larger than unity such that the decay function diminishes rapidly from the free edge towards x = 0. Assuming the stationary point of τ(x) occurs at a distance φh3 from the free edge, where φ is assumed to be a constant, the magnitude of n may be evaluated by differentiating τ (x) with respect to x followed by equating the stationary point with l-φh3. This yields
The derivations of the differential equations for the shear stress, τ, and the mean of the transverse stress, σm, in the adhesive are elaborated in Appendix A.
where
for anti−symmetric deformation
τ (x ) = (As e β (x − l) + A3 )(1 − e nβ (x − L) ) , x > 0
(2)
h3 dτ 2 dx
β2
for symmetric deformation εNl + εMl λx l
Nevertheless, Eq. (8) suffers from a critical setback in that it does not satisfy the free-edge condition: τ(l) = 0. This is enforced arbitrary by multiplying the equation with a decay function; that is [27],
Unless otherwise stated in this manuscript, the upper and the lower signs in ∓ is associated with adherend #1 and adherend #2, respectively. These stresses, together with the interfacial shear stress, τ, on an elemental representation of a bonded structure are shown in Fig. 2. Substituting σa/(2h3) as ∂σz/∂z into Eq. (1) gives
σa = −
⎧0 N ⎨ 22l l − ⎩
Fig. 2. Elemental representation of a bonded structure.
(9)
and 6
International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
magnitude of σa,max is the highest of the three components of stress.
21l − Q −21l) + α (ξ − ξ )(M 21l + M −21l) ξe1l (Q o1l o2l
Ce1 =
2α3λ z (sinh(2αl) + sin(2αl)) 21l − Q −21l) + α (ξ + ξ )(M 21l + M −21l) ξe2l (Q o1l o2l
Ce2 =
2α3λ z (sinh(2αl) + sin(2αl)) 21l + Q −21l) + α (ξ − ξ )(M 21l − M −21l) ξo1l (Q e1l e2l
Co1 =
2α3λ z (sinh(2αl) − sin(2αl))
Co2 =
3. The effects of bondline thickness
,
The maximum shear stress occurs at x = l-φh3 while the maximum peeling stress occurs at x = l. Substituting these into the respective equations for the shear and the peeling stresses give
21l + Q −21l) + α (ξ + ξ )(M 21l − M −21l) ξo2l (Q e1l e2l 2α3λ z (sinh(2αl) − sin(2αl))
ξe1x = cos αx cosh αx , ξe2x = sin αx sinh αx
where
τmax = (As e−φh3 + A3 ) (1 − e−nφh3) 1 21l / α + M 21l ) σm, max = (Q
± 21l = M ± 2l / D2 − M ± 1l / D1 M ± 21l = Q ± 2l / D2 − Q ± 1l / D1 Q
σa, max = 2 3 (As + A3 ) σp, max = σm, max + σa, max
ξo1x = cos αx sinh αx , ξo2x = sin αx cosh αx
(16)
D λz
nβh
(17)
, for x > 0 (20)
For structures of reasonably large length, say αl > 3, Eq. (15) may be expressed more concisely as
σm =
1 D λz
3.1. Single lap joints
21l / α + M 21l )cos α (x − l) + M 21l sin α (x e α (x − l) [(Q
− l)], x > 0
For single lap joints, the coefficients As and A3 and the edge loads are given by
(18)
2.3. The amplitude of peeling stress, σa(x) Substituting Eq. (11), the reduced function embedded with a decay function, into Eq. (3) yields
σa =
h3 [As βe β (x − l) ((n + 1) e nβ (x − l) − 1) + A3 nβe nβ (x − l) ] for x > 0. 2
(19)
It can be shown that the reduced Eq., Eq. (8), will yield an expression of σa whose maximum magnitude, which occurs at x = l, is merely 1/[(n+A3/As)] that of Eq. (19), and whose direction is opposite to that of Eq. (19); thus highlighting the importance of enforcing the nil-shear-stress condition at the free edge of the adhesive. 2.4. Validation The analytical equations are validated against the solutions from FEA using a balanced bonded structure (Fig. 3(a)). Adherends #1 and #2 of the model are assigned with identical thicknesses and elastic moduli. The adherends experience thermal strain and unequal stretching, shearing, and bending at the edges. The magnitudes of individual components of the applied loads are chosen such that each will lead to noticeable contribution to the interfacial stresses; thus, any errors in the closed-form solutions will be readily revealed. The domain of the bonded structure was modelled with 100,000 eight-node quadrilateral elements. The domain around the free edge was discretised at 75 divisions per mm. The adhesive was assigned with anisotropic properties with negligible Ex so as to be consistent with the assumption in the analytical solutions. The mean, σm(x), and the amplitude of variation, σa(x), for the FEA are evaluated from the interfacial peel stresses, σp1 and σp2 using Eq. (2). The compliances (assuming plane stress) of the structure: κs, λx, and λz, the characteristic constants: α and β, and the free-edge parameters: φ and n, are tabulated in Table 1. Fig. 3(b) shows the interfacial shear stress, τ(x), for three solutions: the comprehensive solution, Eq. (5), the reduced solution, Eq. (11), that has been enhanced with the decay function, 1 − e nβ (x − L) , and the FEA solution. Eq. (5) agrees well with the FEA except near the free edges. On the other hand, Eq. (11) agrees well with the FEA right up to the free edge. The free-edge parameters, φ = 0.50, computed using Eq. (13) agrees well with that extracted from FEA at φFEA = 0.54. Fig. 3(c) shows the mean of the interfacial peel stress, σm(x), for the comprehensive solution Eq. (15), the reduced solution, Eq. (18), and the FEA solution. The three have agreed well. Fig. 3(d) shows the amplitude of the interfacial peel stress, σa(x), for the reduced solution, Eq. (19), and the FEA solution. The two have agreed well. It is further noted that the
Fig. 3. Interfacial stresses in validation exercise (a) model, (b) τ(x), (c) σm(x) and (d) σa(x).
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International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
τmax=Ase-φβh3 decreases monotonically with increasing hR.
Table 1 Characteristics of the bonded structure used in the validation exercise. Compliances
Characteristic constants (mm-1)
Free edge parameters
κs λx λz
α β
φ n
7.04 × 10-4 4.58 × 10-4 2.79 × 10-4
0.97 0.81
3.1.2. Interfacial peeling stress Differentiating σm,max with respect to h3 yields
0.50 11.9
dσm, max ≈− dh3
λx (N-1.mm), λz,κs (N-1.mm3), α,β (mm-1).
εNl + εMl N A , A3 = 2l − s βκs βl 2l (h2 + h3) M2l εNl = N2l λ x 2 , εMl = 2D2
M2l
σa, max ≈
,
≈
(27)
(28)
3.2. Electronic assemblies experiencing differential thermal strain
(22)
For an electronic assembly that experiences differential thermal strain between its adherends, the coefficients As and A3 and the edge loads are given by
εT , A3 = 0 βκs 21l , M 21l = 0 Q εT = (α2 − α1) ΔT As =
4M2l [2GR − 3(4 − GR)(1 + hR) − 4(1 − hR)] dAs , = (GR + 4hR)(4D2 + E2 h22 (h2 + h3)) 3.2 dh3 ⎤ 2 D2 E2 G3 ⎡ D2 E2 G3 ⎣ ⎦ (23)
Substituting Eq. (29) into Eq. (20) and assuming
τmax
wherein GR = G3/G2, hR = h3/h2. The conditions for positive dτmax/dh3 are given by
16.5GR 8 for GR < hR < 3 −8.3GR 16.5GR 8 for GR > hR > 3 −8.3GR
(26)
The parameter β is in general a weak function of h3; thus, the product β-0.12h3-0.17 decreases monotonically with increasing h3. It was shown in the last section that dAs/dh3 is either negative or a weak positive (for GR > 8/3); thus, σa,max tends to decrease with increasing h3. This is illustrated in Fig. 5(d), which shows the variations of σm,max and σa,max with hR for GR = 3 and GR = 4.
3.1.1.1. Assuming τmax occurs at x = l. If we ignore the exponential term, e−φβh3 , which is equivalent to assuming that τmax = As at x = l. The gradient dτmax/dh3 is simply assu min g τmax occurs at x = l
3.2
))
σa, max ≈ 1.55β −0.12h3−0.17 As
3.1.1. Interfacial shear stress
dτmax dh3
h3 E3
Substituting it into Eq. (22) yields
(21)
for x > 0
nβh3 As 2
+
nβh3 ≈ 3.1β −0.12h3−0.17
(h2 + h3) M2l −φβh3 e 2D2 λ x κs
D λ z D2
3h2 4E2
as
In order to present an analysis that is not clustered by mathematics, 21/ α , As » A3, εMl »εNl, and e−nβφh3 ≈ 1. Eq. 21 ≫ Q we shall assume that M (20) is reduced to
σm, max ≈
( (
2D2 E3 D
It is noted that dσm,max/dh3 is always negative. Substituting Eq. (13) into (12), the factor nβh3 may be approximated
As =
τmax ≈ As e−φβh3 =
M2l D
≈
σm, max = σa, max ≈
(29)
e−nβφh3
≈ 1 yields
(α2 − α1) ΔT −φβh3 e λ x κs for x > 0
As e−φβh3 = 0, nβh3 As 2
(30)
Unlike Eq. (22), the nominator of As in Eq. (30) contains no h3. Since λx and κs increase monotonically with increasing h3, the magnitude of As, and hence τmax and σa,max, decrease monotonically and strongly with increasing h3.
(24)
This is graphed in Fig. 4, which suggests that in the special circumstances (possibly for SLJ whose adherends are made of laminated fibre reinforced polymeric composites) such that the magnitude of GR is greater than 8/3, the gradient dτmax/dh3 transits from being negative to being positive as hR transits from smaller to larger than (16-5GR)/(3GR8). This is illustrated in Fig. 5(b) and (c) for GR = 3 and GR = 4, respectively. It is noted that (a) for the case of GR = 3, τmax = As decreases before increases with increasing hR; the point of transition occurs at hR = 1; and (b) for the case of GR = 4, the point of transition occurs at hR = -1; hence, τmax = As increases monotonically with increasing hR for all practical values of hR.
4. Discussions 4.1. SLJ – failure to correlate with experiment The assumption that “the in-plane stress in the adhesive is negligibly small” has greatly simplified the analysis and led to very concise closed-
3.1.1.2. τmax occurs at x = l-φh3. Referring to Section 2.1, the interfacial shear stress acquires its maximum magnitude at x = l-φh3. The expression for τmax is given by Eq. (22). The gradient dτmax/dh3 is given by
dτmax dA ≈ ⎛ s − As φβ ⎞ e−φβh3 dh3 ⎝ dh3 ⎠ ⎜
⎟
(25)
In general, the second term in the above equation is significantly larger than the first term. Thus, the gradient dτmax/dh3 is always negative. This is illustrated in Fig. 5(b) and (c) in which it is showed that
Fig. 4. Domains in which the gradient dτmax/dh3 is positive assuming τmax at x = l.
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International Journal of Adhesion and Adhesives 78 (2017) 4–12
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investigating the phenomenon of increased susceptibility of the structure to failure with increasing bondline thickness, there has been little interest among the electronic assembly community. It is either the electronic assembly community has somehow omitted to notice this phenomenon or there is no such phenomenon for electronic assemblies. The answer appears to lie with the coefficient As. It has been presented in Section 3.2 that the coefficient As decreases monotonically and strongly with increasing h3 for an electronic assembly that experience pure differential thermal strain. It is noted from Eq. (30) that τmax and σa,max are linearly proportional to As. This must imply that τmax and σa,max decrease monotonically and strongly with increasing h3. It is postulated that the rate of reduction in τmax and σa,max are so high that the singular stress intensities of shearing and peeling too decrease monotonically with increasing h3. This explains the silence among the electronic assembly community. 4.3. The relative magnitude of peeling and shearing stresses From Eqs. (22) or (30), the ratio σa,max/τmax is given by
σa, max nβh3 φβh3 ≈ e ≈ nχe χ , τmax 2
(31) χ
where χ = βh3/2 assuming φ ≈ 0.5. The product χe increases monotonically from nil at χ = 0. Assuming n = 5, which is a conservative assumption, the condition for σa,max/τmax = nχeχ > 1 is given by χ = βh3/2 > 0.17. This condition is satisfied by almost all practical bonded structures. In other words, the magnitude of σa,max is larger than τmax for most practical bonded structures. This again highlights the importance of enforcing the nil-shear-stress condition at the free edge of the adhesive. Fig. 5. Variations of interfacial shear and peeling stresses with hR.
5. Conclusion The closed-form solutions for the interfacial shear and the interfacial peeling stresses in balanced bonded sandwich structures have been derived. The solutions were enhanced with a decay function that enforces the nil-shear-stress condition at the free edge of the adhesive. The closed-form solutions for the shear and the peel stresses along the interface, especially around the free edge, have been successfully validated with the finite element analysis. However, the solutions have failed to correlate with the observed increased susceptibility of the single lap joints with increase bondline thickness. This has been attributed to the inability of the solutions to model the singular stress/ strain field at the free-edge of the interfaces. The interfacial shear and peel stresses in electronic assemblies that experience differential thermal strain have been found to decrease with increasing bondline thickness, at a rate that is significantly higher than that in single lap joint. This explained the nil report of increased susceptibility of electronic assemblies with increased bondline thickness.
form solutions as presented in Section 2. The concise solutions have facilitated insights into the mechanics of bonded structures as presented in Section 3. However, the concise solutions did not correlate well with the observed increased susceptibility of SLJ to failure with increasing bondline thickness. On the other hand, good correlation has been found between the intensity of the singular stress at the free-edge of the interfaces of a SLJ (obtained using finite element analysis) and the observed increased susceptibility of SLJ to failure with increasing bondline thickness [24–26]. Unfortunately, closed-form solutions incorporating singular stress field are possible only for structures made of simple geometry and experiencing simple loads [29–33]. 4.2. Electronic assemblies – absence of turning point In contrast to the heightened enthusiasm among the SLJ users in Appendix A. The fundamental equations
Refers to Fig. 2, assuming the traction Ni passes through the centroid plane of adherend #i, the equilibrium of the differential elements gives
dNi = ∓ τdx dQi = (∓ σm + σa) dx i = 1.2, dQ3 = −2σa dx dMi =
(
τhi 2
)
− Qi dx
(A1)
where the upper and the lower signs in “∓” refers to adherend #1 and #2, respectively. The x-directional traction-stretching relation of the centroid plane of adherend #i is given by:
dui − αi ΔT = Ni λ xi , dx
(A2)
The moment-rotation relation of the centroid plane of adherend #i is assumed to obey simple beam: 9
International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
d3wi d 2θi 1 dMi 1 ⎛ hi τ = = = − Qi ⎞ dx 3 dx 2 Di dx Di ⎝ 2 ⎠
(A3)
The differential displacement of the centroid planes of adherend #1 and #2 in the u-direction and in the z-direction are given by:
u2 − u1 = κs τ −
h1 θ1 + h2 θ2 2 ,
w2 − w1 = λ z σm
(A4)
where θi is the rotation of the neutral axis of member #i. Differentiating Eq. (A2) twice with respect to x followed by taking the difference between adherend #2 and #1, and equating with the third differential of the first equation of (A4) gives the differential equation:
μ d3τ dτ − β2 = − σ σm dx 2 dx κs
(A5)
Differentiating Eq. (A3) with respect to x followed by taking the difference between adherends #2 and #1, and equating with the fourth differential of the second equation of (A4) gives the differential equation
μ dτ d4σm + 4α 4σm = τ λ z dx dx 4
(A6)
The parameters and the coefficients for Eqs. (A5) and (A6) are given by
β2 =
λx D , 4α 4 = κs λz 2
λ x = λ x1 + λ x 2 + ∑i = 1 3
hi (hi + h3) 4Di
3
2
κs = ∑i = 1 κsi,λ z = ∑i = 1 λ zi ,D = ∑i = 1 μσ =
h 1 ⎛ h2 − 1⎞ D1 ⎠ 2 ⎝ D2
μτ =
h + h3 ⎞ 1 ⎛ h2 + h3 − 1 D1 ⎠ 2 ⎝ D2
⎜
1 Di
⎟
⎜
⎟
(A7)
The compliances of member #i are derived in Appendix B. Appendix B. The compliances of Adherents Fig. B1 shows a differential element of adherend #1 and adherend #2; only differential tractions and moments are shown. The equilibrium of the segment z=0 to z=z in the x-direction gives
τxz dx + dFN + dFM = 0
(B1)
Assuming the differential traction dN acts uniformly over the section then,
dFN = dN
z = ZdN h
(B2)
Assuming εx to be independent of σz, the differential fibre stresses dσb due to the differential moment dM is given by
Fig. B1. Equilibrium of a segment in the differential element of adherends #1 and #2.
10
International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
dM ⎛ h − z⎞ I ⎝2 ⎠
dσb =
(B3)
The resultant differential force, dFM, over the segment z=0 to z=z is then given by
∫0
dFM =
z
6dM (Z − Z 2) h
dσb dz =
(B4)
Substituting dN = −τdx and dM = (hτ /2 − Q) dx from Eq. (A1) into Eqs. (B2) and (B4), respectively, and then substitute them into Eq. (B1) gives
τxz (Z ) = τ (3Z 2 − 2Z ) +
6Q (Z − Z 2) h
(B5)
The relative magnitudes of the two terms in Eq. (B5) are dependent on the nature of the loading. However, for the sole purpose of arriving at a shear compliance that is independent of x, and in keeping with the spirit of the strength of material approach, we shall assume
τxz (Z ) ≈ 2τ (3Z 2 − 2Z )
(B6)
Assuming the shear deformation in the z-direction, dw/dx, is fully constraint by the neighbouring fibres in the adherend, the shear strain γxy is given by γxy = du/dz. The shear-induced displacement of the shear surface at z = h relative to its centroid plane at z = h/2 is given by:
Δus
Z = 1.0.5
=
h G
∫1.2 2τ (3Z 2 − 2Z ) dZ = 4hτG 1
(B7)
Defining the shear compliant κsi of adherend #i as the differential x-displacement between its shear surface and its centroid axis per unit shear stress τ, then
κsi =
Δus
Z = 1.0.5
=
τ
hi 4Gi
(B8)
Substituting Eq. (B5) into Eq. (1) followed by definite integration between x = l to x and z = 0 to z, respectively, yield
[σx ]lx = [σz ]0Z =
[N ]lx
(6Z − 2) +
h 2hσa (Z 3 h3
6 (2Z h2
x
− 1) ∫l Qdx
− Z 2) + (−σm + σa)(2Z 3 − 3Z 2)
, (B9)
– Nl/h = σz. Defining the x-compliance λxi of an adherend as the strain of its centroid axis per unit sectional traction N, and the z-compliance λzi as the differential z-displacement between its bonding interface and its centroid axis per unit stress σm, then
where [σx ]lx =σx
λx =
εx
and [σz ]0Z
Z = 1.2
N Δw Z = 1 − 1.2 h 1 λz = = ∫ εz dZ σm σm 1.2
(B10)
For the case of plane stress and assuming εx is independent of σz and εz independent of σx,
λ xi =
εxi Z = 1.2 1 = Ei N Ei hi
λ zi =
hi 13 hi h σ 11 hi 13 ⎞ 1 − i a⎛ + ∫ σz dZ = Ei σm 1.2 32 Ei Ei σm ⎝ 96 h3 32 ⎠ ⎜
⎟
(B11)
The compliance λzi is a function of σa/σm, which is in turn a function of x. For reason of simplicity, the second term in the equation is ignored. In most cases, σa and σm acts in the same direction, the ignorance of the second term therefore tends to overestimate the compliance λzi. This error becomes more evident in sandwich structure with large ratios of hi/h3.
1992;38:199–217. [12] Yin WL. Thermal stresses and free-edge effects in laminated beams: a variational approach using stress functions. J Electron Packag 1991;113:68–75. [13] Wong EH, Liu J. Interface and interconnection stresses in electronic assemblies – A critical review of analytical solutions. Microelect Reliab 2017. http://dx.doi.org/ 10.1016/j.microrel.2017.03.010. [14] Kinloch AJ. Adhesion and Adhesives. London: Chapman and Hall; 1987. [15] Adams RD, Grant LDR. Adhesive Layer Thickness as a Variable in the Strength of Bonded Joints. In: Proceedings of the 16th Annual Meeting of the Adhesion Society, pp. 91–93; 1993. [16] Sawa T, Uchida H. A two dimensional stress analysis and strength evaluation of band adhesive butt joints subjected to tensile loads. J Adhes Sci Technol 1997;11:811–33. [17] Objois A, Gilbert Y, Fargette B. Theoretical and experimental analysis of the scrf joint bonded structure: influence of the adhesive thickness on the micro-mechanical behaviour. J Adhes 1999;70:13–32. [18] Tomblin J, Harter P, Seneviratne W, Yang C. Characterization of bondline thickness effects in adhesive joints. J Compos Tech Res 2002;24(2):332–44. [19] van Tooren MJL, Gleich DM, Beukers A. Experimental verification of a stress singularity model to predict the effect of bondline thickness on joint strength. J Adhes Sci Technol 2004;18(4):395–412. [20] Kim KS, Yoo JS, Yi YM, Kim CG. Failure mode and strength of uni-directional composite single lap bonded joints with different bonding methods. Compos Struct 2006;72(4):477–85.
References [1] Volkersen Q. Die NietKraftverteilung in Zugbeanspruchten Nietverbindungen mit Konstentan Laschenquerschnitten. Luftfahrtforchung 1938;15:41–7. [2] Goland M, Reissner E. Stresses in cemented joints. ASME J Appl Mech 1944;11:A17–27. [3] Delale F, Erdogan F, Aydinoglu MN. Stresses in adhesively bonded joints: a closedform solution. J Compos Mater 1981;15:249–71. [4] Bigwood DA, Crocombe AD. Elastic analysis and engineering design formulae for bonded joints. Int J Adhes Adhes 1989;9:229. [5] Chen WT, Nelson CW. Thermal Stress in Bonded Joint. IBM J res Dev 1979;23(2):179–88. [6] Suhir E. Stresses in bi-metal thermostats. ASME J Appl Mech 1986;53:657–60. [7] Suhir E. Interfacial stresses in bimetal thermostats. ASME J Appl Mech 1989;56:595–600. [8] Pao YH, Eisele E. Interfacial shear and peel stresses in multi-layered thin stacks subjected to uniform thermal loading. ASME J Electron Packag 113. 1991; 1991. p. 172. [9] Allman D. A theory for elastic stresses adhesive bonded lap joints. Quart J Mech Appl Math 1977;30(4):10–5. [10] Chen D, Cheng S. An analysis of adhesive-bonded single-lap joints. J Appl Mech 1983;50:109–15. [11] Adams RD, Mallick V. A method for the stress analysis of lap joints. J Adhes
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International Journal of Adhesion and Adhesives 78 (2017) 4–12
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[26] Tang JH, Sridhar I, Srikanth N. Static and fatigue failure analysis of adhesively bonded thick composite single lap joints. Compos Sci Technol 2013;86:18–25. [27] Wong EH. Interfacial stresses in sandwich structures subjected to temperature and mechanical loads. Comp Struct 2015;134:226–36. [28] Wong EH. Design analysis of sandwiched structures experiencing differential thermal expansion and differential free-edge stretching. Int J Adhes Adhes 2016;2016(65):19–27. [29] Bogy DB. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. ASME J Appl Mech 1968;35:460–6. [30] Bogy DB. On the problem of edge-bonded elastic quarter-planes loaded at the boundary. Int J Solids Struct 1970;6:1287–313. [31] Dundurs J. Discussion on Bogy (1968). ASME J Appl Mech 1969;36:650–2. [32] Hein VL, Erdogan F. Stress singularities in a two-material wedge. Int J Fract Mech 1971;7:317–30. [33] Kuo A. Thermal stress at the edge of a bi-metallic thermostat. ASME J Appl Mech 1989;56:585–9.
[21] da Silva LFM, Rodrigues TNSS, Figueiredo MAV, de Moura MFSF, Chousal JAG. Effect of adhesive type and thickness on the lap shear strength. J Adhes 2006;82(11):1091–115. [22] Kahraman R, Sunar M, Yilbas B. Influence of adhesive thickness and filler content on the mechanical performance of aluminum single-lap joints bonded with aluminum powder filled epoxy adhesive. J Mater Process Technol 2008;205(1–3):183–9. [23] Castagnetti D, Spaggiari A, Dragoni E. Effect of bondline thickness on the static strength of structural adhesives under nearly-homogeneous shear stresses. J Adhes 2011;87(7–8):780–803. [24] Gleich DM, van Tooren MJL, Beukers A. Analysis and evaluation of bondline thickness effects on failure load in adhesively bonded structures. J Adhes Sci Technol 2001;15(9):1091–101. [25] Gleich DM, van Tooren MJL, Beukers A. A stress singularity approach to failure initiation in a bonded joint with varying bondline thickness. J Adhes Sci Technol 2001;15(10):1247–59.
12
Contents lists available at ScienceDirect
International Journal of Adhesion and Adhesives journal homepage: www.elsevier.com/locate/ijadhadh
The mechanics of bondline thickness in balanced sandwich structures
MARK
E.H. Wong Nanyang Technological University, Energy Research Institute, 1 Cleantech Loop, Singapore 637141, Singapore
A R T I C L E I N F O
A B S T R A C T
Keywords: Adhesive bonded Single lap joint Electronic assemblies Closed-form solutions Singular stress
Assuming negligible in-plane stress, non-varying shear stress, and linearly varying transverse stress along the thickness of the adhesive, the closed-form solutions for the interfacial shear and the interfacial peeling stresses in balanced bonded sandwich structures was derived. The nil-shear-stress condition at the free edge of the adhesive was enforced through the use of a decay function. The solutions for the shear and the peel stresses along the interfaces right up to the free edge were validated with the finite element analysis. The solutions were used to investigate (i) the observed increase susceptibility of single lap joints with increase bondline thickness and (ii) the nil report of such susceptibility for electronic assemblies experiencing differential thermal strain. The first investigation was compromised by the inability of the solutions to model the singular stress/strain field at the free-edge of the interfaces. The second investigation revealed a far higher rate of reduction in the magnitudes of the interfacial shear and peeling stresses in structures (electronic assemblies) that experience differential thermal strain than in structures (single lap joints) that experience bending strain.
1. Introduction A large class of adhesively bonded structures have adherends that are structurally stiff relative to the adhesive. Ignoring the in-plane stress in the adhesive reduces substantially the complexity of the solutions and allows the formulation of closed-form solutions. The first of such analyses was presented by Volkersen (1938) [1] who modelled the adhesive as having only shear stiffness and the adherends as capable of only in-plane stretching. A much more sophisticated analysis was presented by Goland & Reissner (1944) [2] who modelled the adhesive as having stiffness in transverse stretching besides shearing and the adherends as capable of flexing besides in-plane stretching. However, their solution was restricted to balanced structures of single lap-joints. Delale et al. (1981) [3] and Bigwood & Crocombe (1989) [4] generalised the analysis of Goland & Reissner to unbalanced bonded structures experiencing a general state of edge loadings. Delale et al. [3] also incorporated shear deflection into the adherends by modeling the adherends as Timoshenko beams. In the meanwhile, the electronic assembly community was focusing on stresses in electronic assemblies due to mismatch thermal expansion of adherends. Chen & Nelson, (1979) [5] extended the solution of Goland & Reissner to unbalanced electronic assemblies experiencing differential thermal strain. The above authors have all assumed that the adherends experienced negligible shear and transverse-normal strains, which is not necessary true especially if the thickness of the adherends are significantly larger than that of the adhesive. Suhir (1986, 1989) [6,7] included approximately
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijadhadh.2017.06.003 Accepted 18 June 2017 Available online 24 June 2017 0143-7496/ © 2017 Elsevier Ltd. All rights reserved.
the shear and the transverse-normal compliances of adherends in the solution. Pao & Eisele [8] extended the solution of Suhir [6] to multilayer structures. The above authors have conveniently assumed the shear and the transverse stresses to be non-varying over the thickness of the adhesive, which however violates the differential equation of equilibrium:
∂τxz ∂σx + =0 ∂x ∂z ∂τxz ∂σz + =0 ∂z ∂x
(1)
In light of the inability of the above solutions to satisfy the equilibrium equation, these solutions are frequently referred to as the “strength of material” solutions. These strength of material approaches have also failed to satisfy the condition that there shall be nil shear stress at the free edges of the adhesive. A “theory of elasticity” solution was presented by Allman (1977) [9]. The adhesive was assumed to have nil in-plane stress, a nonvarying shear stress over its thickness, and a linearly varying transverse stress along its thickness. The assumed stresses satisfy the equilibrium Eq., Eq. (1). Allman described the stress/strain field in the adhesive and the adherends using two independent stress functions ensuring that the equilibrium condition, Eq. (1), and the free edge condition, τfree-edge=0, were satisfied. The differential equations for the stress functions were established by invoking the principal of complementary energy. Similar approach was followed by Chen & Cheng (1983) [10] and Adams &
International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
Nomenclature
σpi(x)
Subscript #i Subscript #1 & #2 are adherends and #3 is adhesive. Di, Ei, Gi, hi Flexural rigidity, elastic modu4lus, shear modulus, thickness of member #i. D Effective flexural compliance of the sandwich structure. M±il, N±il, Q±il Moment, sectional stretching force, sectional shear force applied to adherend #i at x=±l l Half of the length of the sandwich structure α, β Characteristic parameters of the sandwich structure. αi Coefficient of thermal expansion of adherend #i. εT, εNl, εMl Differential strain between adherends #2 and #1 at x=l due to temperature, due to edge stretching; collective fibre strain due to edge bending at x=l. κ±il Edge curvature of adherend #i at x=l. κsi, κs Shear compliance of adherend #i between its interface with the adhesive and its centroid plane, of the sandwich structure between the centroid planes of adherends #1 and #2. λxi, λx x-compliance of adherend #i, of the sandwich structure. λxθ Additional x-compliance of a sandwich structure due to it undergoes flexural deformation. λzi, λz z-compliance of adherend #i between its interface with the adhesive and its centroid plane, of the sandwich structure between the centroid planes of adherends #1 and #2. σm(x), σa(x) The mean, the amplitude of peeling stress at the two interfaces.
τ(x) ΔT
Peeling stress along the interface between adhesive and adherend #i Shear stress within the adhesive (assuming uniform) and along the interfaces Temperature change.
Basic formulas
4α 4
=
D ; 4λ z λx κs Ei hi3 (plane stress); 12 1 1 + D D1 2 κs1 + κs2 + κ s3 h1 , 4G1 h2 , 4G 2 h3 G3
β2
=
Di
=
D κs κs1
= = ≈
κs2
≈
κs3 λx λ x1
= = λ x1 + λ x 2 + λ xθ 1 ≈ Eh ,
λ x2
≈
λ xθ
=
λz λ z1
= λ z1 + λ z 2 + λ z3 11h ≈ 32E1 ,
λ z2
≈
λ z3
=
1 1
1 , E2 h2 1 h1 (h1 + h3) ⎡ D1 4⎣
+
h2 (h2 + h3) ⎤ D2 ⎦
1
11h2 , 32E2 h3 E3
bondline thickness in the SLJ and (ii) the nil-report of such a phenomenon for electronic assemblies that experience differential thermal strain.
Mallick (1992) [11]; the latter modelled the adhesive to be having a linearly varying in-plane stress, a parabolically varying shear stress and a cubicically varying transverse stress along its thickness. Four independent stress functions were needed, and these could only be solved using numerical analysis. Yin (1991) [12] generalised the stress function approach to multi-layer structures. Interested readers may refer to the article of Wong & Liu [13] for a more elaborated review of the classical works in lap-joints and in microelectronic assemblies. The main critique of the closed-form solutions is their inability to correlate with the experimental observations that the susceptibility of single lap shear joint (SLJ) to failure could increase with increasing bondline thickness [14–23]. Gleich et al. [24] have shown, with the aid of finite element analysis (FEA), that the interfacial stresses immediately adjacent to the free edge of the bonding interface decreased before they increased with increasing bondline thickness. Gleich et al. [25] and Yang et al. [26] have shown, with the aid of FEA, that the generalized stress intensity factors, H1 and H2, that describe the singular stress field at the free edge of the bonding interface, σij (r ) = H1 r −λ1 + H2 r −λ2 , decreased and then increased with increasing bondline thickness. In contrast to the feverish interest in SLJ, there has been calm in electronic community – there has been no report of increase vulnerability with increasing bondline thickness. It is the objective of this article to investigate, using improved analytical solutions, (i) the observed increase vulnerability with increasing
2. Interfacial stresses in balanced structures Despite its inferior accuracy over the theory of elasticity solution, the strength of material solution continued to be favoured by practising engineers because of its simplicity. However, its inability to model the nil-shear-stress condition at the free edge could lead to solutions that are too erroneous to be useful. An improved strength of material solution that is incorporated with a decay function to enforce the nilshear-stress condition at the free edge [27,28] is used for this study. Fig. 1 shows a bonded structure made up of adherend #1, adherend #2, and adhesive #3. The structure experiences a mismatched thermal expansion between the adherends; and experiences stretching, shearing, and bending at their edges. Note the notations and the positive directions of the in-plane stretching forces N ± il, the section shear forces Q ± il, the moments M ± il, and the curvature κil, at x= ± l, where l is the half-length of the adhesive. The height, Young’s modulus, shear modulus, flexural stiffness, and thermal coefficient of expansion of member #i are denoted as hi, Ei, Gi, Di, and αi, respectively. The shear, the in-plane, the transverse, and the flexural compliances of the structure are denoted as κs, λx, λz, and D , respectively. The corresponding compliances of member #i are denoted as κsi, λxi, λzi, and Di , Fig. 1. Schematic of a bonded structure experiencing general condition of edge loadings and thermal strain.
5
International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
respectively. The formula for computing these compliances are collected under the heading “basic formula” at the front of this article. The derivations of these formula are presented in Appendix B. The adhesive is assumed to experience negligible in-plane stress, σx. The differential equation of equilibrium, Eq. (1), then suggests an unvarying shear stress, τxz, and a linearly varying transverse stress, σz, along the thickness of the adhesive. Denoting σm and σa as the mean and the amplitude of variation, respectively, of the transverse stress along the thickness of the adhesive, the peeling stress at the interface between the adhesive and adherend #i is given by
σpi = σm ∓ σa, i = 1.2.
A3 =
n ≈ e nφβh3 − 1 (3)
φ ≈ 0.407(βh30.88)−0.26
d3τ dτ − β2 = 0, dx 2 dx
Referring to Eq. (A6) and with the parameter, μτ, equates to nil for a balanced bonded structure, the differential equation for the mean of the peeling stress is given by
= λ x / κs . The solution to Eq. (4) is given by sinh βx cosh βx + Ae + A3 , cosh βl sinh βl
d4σm + 4α 4σm = 0 dx 4 (5)
1 (ε 2βκs Nl
− ε−Nl + εMl − ε−Ml )
A3 =
1 2l
(N
2Ae β
2l
σm (x ) = Ce1 ξe1x + Ce2 ξe2x + Co1 ξo1x + Co2 ξo2x ,
,
)
(6)
where
εT = (α2 − α1) ΔT ε ± Nl = N ± 2l λ x 2 − N ± 1l λ x1 1 ε ± Ml = ((h1 + h3) κ ± 1l + (h2 + h3) κ ± 2l ), 2
(7)
Do note that while the fibre strain on adherend #i at x = ± l is indeed described by the simple bending relation: ε ± il = hi M ± il /(2Di ) = hi κ ± il/2 ; the condition of compatibility dictates that the effective bending strain, ε ± Ml, ought to include the thickness of the adhesive [27]. For structures of reasonably large length, say βl > 3, Eq. (5) may be expressed more concisely as
τ (x ) = As e β (x − l) + A3 , x > 0
(8)
where
As =
εT + εNl + εMl βκs
(15)
wherein the first two terms are associated with symmetric deformation and the last two terms with anti-symmetric deformation. The coefficients and the functions are given by
+ εNl + ε−Nl + εMl + ε−Ml )
− N−2l −
(14)
where 4α 4 = D / λ z . The solution to Eq. (14) is given by
wherein the first term is associated with symmetric deformation and the second term with anti-symmetric deformation about the mid length of the structure at x=0. The coefficients Ao, Ae, and A3 are given by
Ae =
(13)
2.2. The mean of peeling stress, σm(x)
(4)
1 (2εT 2βκs
(12)
for 0.018≤βh3≤1.73.
Referring to Eq. (A5) and with the parameter, μσ, equates to nil for a balanced bonded structure, the differential equation for the shear stress in the adhesive and along the interface is given by
Ao =
(11)
The “constant” φ is indeed dependent on β and h3, which has been established through collocating with the finite element analysis as
2.1. Interfacial shear stress, τ(x)
τ (x ) = Ao
(10)
wherein n is a positive real number with a magnitude significantly larger than unity such that the decay function diminishes rapidly from the free edge towards x = 0. Assuming the stationary point of τ(x) occurs at a distance φh3 from the free edge, where φ is assumed to be a constant, the magnitude of n may be evaluated by differentiating τ (x) with respect to x followed by equating the stationary point with l-φh3. This yields
The derivations of the differential equations for the shear stress, τ, and the mean of the transverse stress, σm, in the adhesive are elaborated in Appendix A.
where
for anti−symmetric deformation
τ (x ) = (As e β (x − l) + A3 )(1 − e nβ (x − L) ) , x > 0
(2)
h3 dτ 2 dx
β2
for symmetric deformation εNl + εMl λx l
Nevertheless, Eq. (8) suffers from a critical setback in that it does not satisfy the free-edge condition: τ(l) = 0. This is enforced arbitrary by multiplying the equation with a decay function; that is [27],
Unless otherwise stated in this manuscript, the upper and the lower signs in ∓ is associated with adherend #1 and adherend #2, respectively. These stresses, together with the interfacial shear stress, τ, on an elemental representation of a bonded structure are shown in Fig. 2. Substituting σa/(2h3) as ∂σz/∂z into Eq. (1) gives
σa = −
⎧0 N ⎨ 22l l − ⎩
Fig. 2. Elemental representation of a bonded structure.
(9)
and 6
International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
magnitude of σa,max is the highest of the three components of stress.
21l − Q −21l) + α (ξ − ξ )(M 21l + M −21l) ξe1l (Q o1l o2l
Ce1 =
2α3λ z (sinh(2αl) + sin(2αl)) 21l − Q −21l) + α (ξ + ξ )(M 21l + M −21l) ξe2l (Q o1l o2l
Ce2 =
2α3λ z (sinh(2αl) + sin(2αl)) 21l + Q −21l) + α (ξ − ξ )(M 21l − M −21l) ξo1l (Q e1l e2l
Co1 =
2α3λ z (sinh(2αl) − sin(2αl))
Co2 =
3. The effects of bondline thickness
,
The maximum shear stress occurs at x = l-φh3 while the maximum peeling stress occurs at x = l. Substituting these into the respective equations for the shear and the peeling stresses give
21l + Q −21l) + α (ξ + ξ )(M 21l − M −21l) ξo2l (Q e1l e2l 2α3λ z (sinh(2αl) − sin(2αl))
ξe1x = cos αx cosh αx , ξe2x = sin αx sinh αx
where
τmax = (As e−φh3 + A3 ) (1 − e−nφh3) 1 21l / α + M 21l ) σm, max = (Q
± 21l = M ± 2l / D2 − M ± 1l / D1 M ± 21l = Q ± 2l / D2 − Q ± 1l / D1 Q
σa, max = 2 3 (As + A3 ) σp, max = σm, max + σa, max
ξo1x = cos αx sinh αx , ξo2x = sin αx cosh αx
(16)
D λz
nβh
(17)
, for x > 0 (20)
For structures of reasonably large length, say αl > 3, Eq. (15) may be expressed more concisely as
σm =
1 D λz
3.1. Single lap joints
21l / α + M 21l )cos α (x − l) + M 21l sin α (x e α (x − l) [(Q
− l)], x > 0
For single lap joints, the coefficients As and A3 and the edge loads are given by
(18)
2.3. The amplitude of peeling stress, σa(x) Substituting Eq. (11), the reduced function embedded with a decay function, into Eq. (3) yields
σa =
h3 [As βe β (x − l) ((n + 1) e nβ (x − l) − 1) + A3 nβe nβ (x − l) ] for x > 0. 2
(19)
It can be shown that the reduced Eq., Eq. (8), will yield an expression of σa whose maximum magnitude, which occurs at x = l, is merely 1/[(n+A3/As)] that of Eq. (19), and whose direction is opposite to that of Eq. (19); thus highlighting the importance of enforcing the nil-shear-stress condition at the free edge of the adhesive. 2.4. Validation The analytical equations are validated against the solutions from FEA using a balanced bonded structure (Fig. 3(a)). Adherends #1 and #2 of the model are assigned with identical thicknesses and elastic moduli. The adherends experience thermal strain and unequal stretching, shearing, and bending at the edges. The magnitudes of individual components of the applied loads are chosen such that each will lead to noticeable contribution to the interfacial stresses; thus, any errors in the closed-form solutions will be readily revealed. The domain of the bonded structure was modelled with 100,000 eight-node quadrilateral elements. The domain around the free edge was discretised at 75 divisions per mm. The adhesive was assigned with anisotropic properties with negligible Ex so as to be consistent with the assumption in the analytical solutions. The mean, σm(x), and the amplitude of variation, σa(x), for the FEA are evaluated from the interfacial peel stresses, σp1 and σp2 using Eq. (2). The compliances (assuming plane stress) of the structure: κs, λx, and λz, the characteristic constants: α and β, and the free-edge parameters: φ and n, are tabulated in Table 1. Fig. 3(b) shows the interfacial shear stress, τ(x), for three solutions: the comprehensive solution, Eq. (5), the reduced solution, Eq. (11), that has been enhanced with the decay function, 1 − e nβ (x − L) , and the FEA solution. Eq. (5) agrees well with the FEA except near the free edges. On the other hand, Eq. (11) agrees well with the FEA right up to the free edge. The free-edge parameters, φ = 0.50, computed using Eq. (13) agrees well with that extracted from FEA at φFEA = 0.54. Fig. 3(c) shows the mean of the interfacial peel stress, σm(x), for the comprehensive solution Eq. (15), the reduced solution, Eq. (18), and the FEA solution. The three have agreed well. Fig. 3(d) shows the amplitude of the interfacial peel stress, σa(x), for the reduced solution, Eq. (19), and the FEA solution. The two have agreed well. It is further noted that the
Fig. 3. Interfacial stresses in validation exercise (a) model, (b) τ(x), (c) σm(x) and (d) σa(x).
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τmax=Ase-φβh3 decreases monotonically with increasing hR.
Table 1 Characteristics of the bonded structure used in the validation exercise. Compliances
Characteristic constants (mm-1)
Free edge parameters
κs λx λz
α β
φ n
7.04 × 10-4 4.58 × 10-4 2.79 × 10-4
0.97 0.81
3.1.2. Interfacial peeling stress Differentiating σm,max with respect to h3 yields
0.50 11.9
dσm, max ≈− dh3
λx (N-1.mm), λz,κs (N-1.mm3), α,β (mm-1).
εNl + εMl N A , A3 = 2l − s βκs βl 2l (h2 + h3) M2l εNl = N2l λ x 2 , εMl = 2D2
M2l
σa, max ≈
,
≈
(27)
(28)
3.2. Electronic assemblies experiencing differential thermal strain
(22)
For an electronic assembly that experiences differential thermal strain between its adherends, the coefficients As and A3 and the edge loads are given by
εT , A3 = 0 βκs 21l , M 21l = 0 Q εT = (α2 − α1) ΔT As =
4M2l [2GR − 3(4 − GR)(1 + hR) − 4(1 − hR)] dAs , = (GR + 4hR)(4D2 + E2 h22 (h2 + h3)) 3.2 dh3 ⎤ 2 D2 E2 G3 ⎡ D2 E2 G3 ⎣ ⎦ (23)
Substituting Eq. (29) into Eq. (20) and assuming
τmax
wherein GR = G3/G2, hR = h3/h2. The conditions for positive dτmax/dh3 are given by
16.5GR 8 for GR < hR < 3 −8.3GR 16.5GR 8 for GR > hR > 3 −8.3GR
(26)
The parameter β is in general a weak function of h3; thus, the product β-0.12h3-0.17 decreases monotonically with increasing h3. It was shown in the last section that dAs/dh3 is either negative or a weak positive (for GR > 8/3); thus, σa,max tends to decrease with increasing h3. This is illustrated in Fig. 5(d), which shows the variations of σm,max and σa,max with hR for GR = 3 and GR = 4.
3.1.1.1. Assuming τmax occurs at x = l. If we ignore the exponential term, e−φβh3 , which is equivalent to assuming that τmax = As at x = l. The gradient dτmax/dh3 is simply assu min g τmax occurs at x = l
3.2
))
σa, max ≈ 1.55β −0.12h3−0.17 As
3.1.1. Interfacial shear stress
dτmax dh3
h3 E3
Substituting it into Eq. (22) yields
(21)
for x > 0
nβh3 As 2
+
nβh3 ≈ 3.1β −0.12h3−0.17
(h2 + h3) M2l −φβh3 e 2D2 λ x κs
D λ z D2
3h2 4E2
as
In order to present an analysis that is not clustered by mathematics, 21/ α , As » A3, εMl »εNl, and e−nβφh3 ≈ 1. Eq. 21 ≫ Q we shall assume that M (20) is reduced to
σm, max ≈
( (
2D2 E3 D
It is noted that dσm,max/dh3 is always negative. Substituting Eq. (13) into (12), the factor nβh3 may be approximated
As =
τmax ≈ As e−φβh3 =
M2l D
≈
σm, max = σa, max ≈
(29)
e−nβφh3
≈ 1 yields
(α2 − α1) ΔT −φβh3 e λ x κs for x > 0
As e−φβh3 = 0, nβh3 As 2
(30)
Unlike Eq. (22), the nominator of As in Eq. (30) contains no h3. Since λx and κs increase monotonically with increasing h3, the magnitude of As, and hence τmax and σa,max, decrease monotonically and strongly with increasing h3.
(24)
This is graphed in Fig. 4, which suggests that in the special circumstances (possibly for SLJ whose adherends are made of laminated fibre reinforced polymeric composites) such that the magnitude of GR is greater than 8/3, the gradient dτmax/dh3 transits from being negative to being positive as hR transits from smaller to larger than (16-5GR)/(3GR8). This is illustrated in Fig. 5(b) and (c) for GR = 3 and GR = 4, respectively. It is noted that (a) for the case of GR = 3, τmax = As decreases before increases with increasing hR; the point of transition occurs at hR = 1; and (b) for the case of GR = 4, the point of transition occurs at hR = -1; hence, τmax = As increases monotonically with increasing hR for all practical values of hR.
4. Discussions 4.1. SLJ – failure to correlate with experiment The assumption that “the in-plane stress in the adhesive is negligibly small” has greatly simplified the analysis and led to very concise closed-
3.1.1.2. τmax occurs at x = l-φh3. Referring to Section 2.1, the interfacial shear stress acquires its maximum magnitude at x = l-φh3. The expression for τmax is given by Eq. (22). The gradient dτmax/dh3 is given by
dτmax dA ≈ ⎛ s − As φβ ⎞ e−φβh3 dh3 ⎝ dh3 ⎠ ⎜
⎟
(25)
In general, the second term in the above equation is significantly larger than the first term. Thus, the gradient dτmax/dh3 is always negative. This is illustrated in Fig. 5(b) and (c) in which it is showed that
Fig. 4. Domains in which the gradient dτmax/dh3 is positive assuming τmax at x = l.
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International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
investigating the phenomenon of increased susceptibility of the structure to failure with increasing bondline thickness, there has been little interest among the electronic assembly community. It is either the electronic assembly community has somehow omitted to notice this phenomenon or there is no such phenomenon for electronic assemblies. The answer appears to lie with the coefficient As. It has been presented in Section 3.2 that the coefficient As decreases monotonically and strongly with increasing h3 for an electronic assembly that experience pure differential thermal strain. It is noted from Eq. (30) that τmax and σa,max are linearly proportional to As. This must imply that τmax and σa,max decrease monotonically and strongly with increasing h3. It is postulated that the rate of reduction in τmax and σa,max are so high that the singular stress intensities of shearing and peeling too decrease monotonically with increasing h3. This explains the silence among the electronic assembly community. 4.3. The relative magnitude of peeling and shearing stresses From Eqs. (22) or (30), the ratio σa,max/τmax is given by
σa, max nβh3 φβh3 ≈ e ≈ nχe χ , τmax 2
(31) χ
where χ = βh3/2 assuming φ ≈ 0.5. The product χe increases monotonically from nil at χ = 0. Assuming n = 5, which is a conservative assumption, the condition for σa,max/τmax = nχeχ > 1 is given by χ = βh3/2 > 0.17. This condition is satisfied by almost all practical bonded structures. In other words, the magnitude of σa,max is larger than τmax for most practical bonded structures. This again highlights the importance of enforcing the nil-shear-stress condition at the free edge of the adhesive. Fig. 5. Variations of interfacial shear and peeling stresses with hR.
5. Conclusion The closed-form solutions for the interfacial shear and the interfacial peeling stresses in balanced bonded sandwich structures have been derived. The solutions were enhanced with a decay function that enforces the nil-shear-stress condition at the free edge of the adhesive. The closed-form solutions for the shear and the peel stresses along the interface, especially around the free edge, have been successfully validated with the finite element analysis. However, the solutions have failed to correlate with the observed increased susceptibility of the single lap joints with increase bondline thickness. This has been attributed to the inability of the solutions to model the singular stress/ strain field at the free-edge of the interfaces. The interfacial shear and peel stresses in electronic assemblies that experience differential thermal strain have been found to decrease with increasing bondline thickness, at a rate that is significantly higher than that in single lap joint. This explained the nil report of increased susceptibility of electronic assemblies with increased bondline thickness.
form solutions as presented in Section 2. The concise solutions have facilitated insights into the mechanics of bonded structures as presented in Section 3. However, the concise solutions did not correlate well with the observed increased susceptibility of SLJ to failure with increasing bondline thickness. On the other hand, good correlation has been found between the intensity of the singular stress at the free-edge of the interfaces of a SLJ (obtained using finite element analysis) and the observed increased susceptibility of SLJ to failure with increasing bondline thickness [24–26]. Unfortunately, closed-form solutions incorporating singular stress field are possible only for structures made of simple geometry and experiencing simple loads [29–33]. 4.2. Electronic assemblies – absence of turning point In contrast to the heightened enthusiasm among the SLJ users in Appendix A. The fundamental equations
Refers to Fig. 2, assuming the traction Ni passes through the centroid plane of adherend #i, the equilibrium of the differential elements gives
dNi = ∓ τdx dQi = (∓ σm + σa) dx i = 1.2, dQ3 = −2σa dx dMi =
(
τhi 2
)
− Qi dx
(A1)
where the upper and the lower signs in “∓” refers to adherend #1 and #2, respectively. The x-directional traction-stretching relation of the centroid plane of adherend #i is given by:
dui − αi ΔT = Ni λ xi , dx
(A2)
The moment-rotation relation of the centroid plane of adherend #i is assumed to obey simple beam: 9
International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
d3wi d 2θi 1 dMi 1 ⎛ hi τ = = = − Qi ⎞ dx 3 dx 2 Di dx Di ⎝ 2 ⎠
(A3)
The differential displacement of the centroid planes of adherend #1 and #2 in the u-direction and in the z-direction are given by:
u2 − u1 = κs τ −
h1 θ1 + h2 θ2 2 ,
w2 − w1 = λ z σm
(A4)
where θi is the rotation of the neutral axis of member #i. Differentiating Eq. (A2) twice with respect to x followed by taking the difference between adherend #2 and #1, and equating with the third differential of the first equation of (A4) gives the differential equation:
μ d3τ dτ − β2 = − σ σm dx 2 dx κs
(A5)
Differentiating Eq. (A3) with respect to x followed by taking the difference between adherends #2 and #1, and equating with the fourth differential of the second equation of (A4) gives the differential equation
μ dτ d4σm + 4α 4σm = τ λ z dx dx 4
(A6)
The parameters and the coefficients for Eqs. (A5) and (A6) are given by
β2 =
λx D , 4α 4 = κs λz 2
λ x = λ x1 + λ x 2 + ∑i = 1 3
hi (hi + h3) 4Di
3
2
κs = ∑i = 1 κsi,λ z = ∑i = 1 λ zi ,D = ∑i = 1 μσ =
h 1 ⎛ h2 − 1⎞ D1 ⎠ 2 ⎝ D2
μτ =
h + h3 ⎞ 1 ⎛ h2 + h3 − 1 D1 ⎠ 2 ⎝ D2
⎜
1 Di
⎟
⎜
⎟
(A7)
The compliances of member #i are derived in Appendix B. Appendix B. The compliances of Adherents Fig. B1 shows a differential element of adherend #1 and adherend #2; only differential tractions and moments are shown. The equilibrium of the segment z=0 to z=z in the x-direction gives
τxz dx + dFN + dFM = 0
(B1)
Assuming the differential traction dN acts uniformly over the section then,
dFN = dN
z = ZdN h
(B2)
Assuming εx to be independent of σz, the differential fibre stresses dσb due to the differential moment dM is given by
Fig. B1. Equilibrium of a segment in the differential element of adherends #1 and #2.
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International Journal of Adhesion and Adhesives 78 (2017) 4–12
E.H. Wong
dM ⎛ h − z⎞ I ⎝2 ⎠
dσb =
(B3)
The resultant differential force, dFM, over the segment z=0 to z=z is then given by
∫0
dFM =
z
6dM (Z − Z 2) h
dσb dz =
(B4)
Substituting dN = −τdx and dM = (hτ /2 − Q) dx from Eq. (A1) into Eqs. (B2) and (B4), respectively, and then substitute them into Eq. (B1) gives
τxz (Z ) = τ (3Z 2 − 2Z ) +
6Q (Z − Z 2) h
(B5)
The relative magnitudes of the two terms in Eq. (B5) are dependent on the nature of the loading. However, for the sole purpose of arriving at a shear compliance that is independent of x, and in keeping with the spirit of the strength of material approach, we shall assume
τxz (Z ) ≈ 2τ (3Z 2 − 2Z )
(B6)
Assuming the shear deformation in the z-direction, dw/dx, is fully constraint by the neighbouring fibres in the adherend, the shear strain γxy is given by γxy = du/dz. The shear-induced displacement of the shear surface at z = h relative to its centroid plane at z = h/2 is given by:
Δus
Z = 1.0.5
=
h G
∫1.2 2τ (3Z 2 − 2Z ) dZ = 4hτG 1
(B7)
Defining the shear compliant κsi of adherend #i as the differential x-displacement between its shear surface and its centroid axis per unit shear stress τ, then
κsi =
Δus
Z = 1.0.5
=
τ
hi 4Gi
(B8)
Substituting Eq. (B5) into Eq. (1) followed by definite integration between x = l to x and z = 0 to z, respectively, yield
[σx ]lx = [σz ]0Z =
[N ]lx
(6Z − 2) +
h 2hσa (Z 3 h3
6 (2Z h2
x
− 1) ∫l Qdx
− Z 2) + (−σm + σa)(2Z 3 − 3Z 2)
, (B9)
– Nl/h = σz. Defining the x-compliance λxi of an adherend as the strain of its centroid axis per unit sectional traction N, and the z-compliance λzi as the differential z-displacement between its bonding interface and its centroid axis per unit stress σm, then
where [σx ]lx =σx
λx =
εx
and [σz ]0Z
Z = 1.2
N Δw Z = 1 − 1.2 h 1 λz = = ∫ εz dZ σm σm 1.2
(B10)
For the case of plane stress and assuming εx is independent of σz and εz independent of σx,
λ xi =
εxi Z = 1.2 1 = Ei N Ei hi
λ zi =
hi 13 hi h σ 11 hi 13 ⎞ 1 − i a⎛ + ∫ σz dZ = Ei σm 1.2 32 Ei Ei σm ⎝ 96 h3 32 ⎠ ⎜
⎟
(B11)
The compliance λzi is a function of σa/σm, which is in turn a function of x. For reason of simplicity, the second term in the equation is ignored. In most cases, σa and σm acts in the same direction, the ignorance of the second term therefore tends to overestimate the compliance λzi. This error becomes more evident in sandwich structure with large ratios of hi/h3.
1992;38:199–217. [12] Yin WL. Thermal stresses and free-edge effects in laminated beams: a variational approach using stress functions. J Electron Packag 1991;113:68–75. [13] Wong EH, Liu J. Interface and interconnection stresses in electronic assemblies – A critical review of analytical solutions. Microelect Reliab 2017. http://dx.doi.org/ 10.1016/j.microrel.2017.03.010. [14] Kinloch AJ. Adhesion and Adhesives. London: Chapman and Hall; 1987. [15] Adams RD, Grant LDR. Adhesive Layer Thickness as a Variable in the Strength of Bonded Joints. In: Proceedings of the 16th Annual Meeting of the Adhesion Society, pp. 91–93; 1993. [16] Sawa T, Uchida H. A two dimensional stress analysis and strength evaluation of band adhesive butt joints subjected to tensile loads. J Adhes Sci Technol 1997;11:811–33. [17] Objois A, Gilbert Y, Fargette B. Theoretical and experimental analysis of the scrf joint bonded structure: influence of the adhesive thickness on the micro-mechanical behaviour. J Adhes 1999;70:13–32. [18] Tomblin J, Harter P, Seneviratne W, Yang C. Characterization of bondline thickness effects in adhesive joints. J Compos Tech Res 2002;24(2):332–44. [19] van Tooren MJL, Gleich DM, Beukers A. Experimental verification of a stress singularity model to predict the effect of bondline thickness on joint strength. J Adhes Sci Technol 2004;18(4):395–412. [20] Kim KS, Yoo JS, Yi YM, Kim CG. Failure mode and strength of uni-directional composite single lap bonded joints with different bonding methods. Compos Struct 2006;72(4):477–85.
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[26] Tang JH, Sridhar I, Srikanth N. Static and fatigue failure analysis of adhesively bonded thick composite single lap joints. Compos Sci Technol 2013;86:18–25. [27] Wong EH. Interfacial stresses in sandwich structures subjected to temperature and mechanical loads. Comp Struct 2015;134:226–36. [28] Wong EH. Design analysis of sandwiched structures experiencing differential thermal expansion and differential free-edge stretching. Int J Adhes Adhes 2016;2016(65):19–27. [29] Bogy DB. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. ASME J Appl Mech 1968;35:460–6. [30] Bogy DB. On the problem of edge-bonded elastic quarter-planes loaded at the boundary. Int J Solids Struct 1970;6:1287–313. [31] Dundurs J. Discussion on Bogy (1968). ASME J Appl Mech 1969;36:650–2. [32] Hein VL, Erdogan F. Stress singularities in a two-material wedge. Int J Fract Mech 1971;7:317–30. [33] Kuo A. Thermal stress at the edge of a bi-metallic thermostat. ASME J Appl Mech 1989;56:585–9.
[21] da Silva LFM, Rodrigues TNSS, Figueiredo MAV, de Moura MFSF, Chousal JAG. Effect of adhesive type and thickness on the lap shear strength. J Adhes 2006;82(11):1091–115. [22] Kahraman R, Sunar M, Yilbas B. Influence of adhesive thickness and filler content on the mechanical performance of aluminum single-lap joints bonded with aluminum powder filled epoxy adhesive. J Mater Process Technol 2008;205(1–3):183–9. [23] Castagnetti D, Spaggiari A, Dragoni E. Effect of bondline thickness on the static strength of structural adhesives under nearly-homogeneous shear stresses. J Adhes 2011;87(7–8):780–803. [24] Gleich DM, van Tooren MJL, Beukers A. Analysis and evaluation of bondline thickness effects on failure load in adhesively bonded structures. J Adhes Sci Technol 2001;15(9):1091–101. [25] Gleich DM, van Tooren MJL, Beukers A. A stress singularity approach to failure initiation in a bonded joint with varying bondline thickness. J Adhes Sci Technol 2001;15(10):1247–59.
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