Design Analysis of Adhesively Bonded Structures

The existing analytical solutions for the peeling and shearing stresses in polymeric adhesively bonded structures are either too inaccurate or too complex for adoption by practicing engineers. This manuscript presents a closed-form solution that is reasonably accurate yet simple and concise enough to be adopted by practicing engineers for design analysis and exploration. Analysis of these concise solutions have yielded insightful design guidelines: (i) the magnitude of peeling stress is generally higher than that of shearing stress; (ii) the peeling stress in a balanced structure may be reduced most effectively by reducing the elastic modulus of the adherends or by increasing the adhesive-to-adherend thickness ratio and less effectively by reducing the elastic modulus of the adhesive; and (iii) the peeling stress in an unbalanced structure may be reduced by increasing the in-plane compliance of the structure, which may be implemented most effectively by reducing the thicknesses of the adherends and less effectively by reducing the elastic modulus of the adherends.


Introduction
The polymeric adhesive in many bonded structures is much more structurally compliant than the adherends, leading to a significantly lower magnitude of in-plane stress in the adhesive. Completely ignoring this in-plane stress in the adhesive substantially reduces the complexity of the analysis and allows the formulation of closed-form solutions. The first of such analyses was presented by Volkersen (1938) [1], who modelled the adhesive layer in a single lap-shear structure as having only shear stiffness and the adherends as capable of only in-plane stretching. A more sophisticated analysis was presented by Goland and Reissner (1944) [2], who modelled the adhesive as having stiffness in shearing and transverse stretching and the adherends as capable of in-plane stretching and bending.
Goland and Reissner [2] assumed the adherends to have negligible shear and transverse-normal compliances, which may not be valid for bonded structures with relatively large ratios of adherends-to-adhesive thicknesses. Assuming a linear distribution of shear stress along the thickness of the adherends-An overly simplistic assumption- Tsai et al. (1998) [3] incorporated shear compliance of the adherends into the formulation of Goland and Reissner [2]. But before Tsai et al. [3], Suhir (1986Suhir ( , 1989 has evaluated the shear compliance [4] and the transverse compliance [5] of the adherends using Ribiere Solution for a long-and-narrow strip [6]. Besides assuming nil in-plane stress in the adhesive, the above authors and many others [7][8][9][10][11][12] have also conveniently assumed that the shear and the transverse stresses are unvarying over the thickness of the adhesive. Ojalo and Eidinoff (1978) [13] were believed to be the first to challenge this assumption; assuming linear variation of the in-plane and the transverse deformations of the adhesive along its thickness, they concluded that the shear stress varies linearly while the transverse stress is Figure 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 stretching, shearing, and bending at their edges. Note the notations and positive directions of the in-plane stretching forces N ±il , the shear forces Q ±il , moments M ±il , and curvatures κ ±il , 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 h i , E i , G i , D i , and α i , respectively. The shear, in-plane (x), transverse (z), and flexural compliances of the bonded structure are denoted as κ s , λ x , λ z , and D, respectively. The corresponding compliances of member #i are denoted as κ si , λ xi , λ zi , and D i , respectively. The formulas for computing these compliances are collected under the heading "basic formula" at the front of this article. The derivations of these formulas may be found in Refs. [15,23]. The adhesive is assumed to experience negligible in-plane stress, σx. The differential equation of equilibrium, Equation (1), then suggests an nonvarying shear stress, τxz, and a linearly varying transverse stress, σz, along the thickness of the adhesive. Denoting σm and σa respectively as the mean and amplitude of variation of the transverse stress along the thickness of the adhesive, the peeling stress at the two interfaces of the adhesive with adherend #i is given by  The adhesive is assumed to experience negligible in-plane stress, σ x . The differential equation of equilibrium, Equation (1), then suggests an nonvarying shear stress, τ xz , and a linearly varying transverse stress, σ z , along the thickness of the adhesive. Denoting σ m and σ a respectively as the mean and amplitude of variation of the transverse stress along the thickness of the adhesive, the peeling stress at the two interfaces of the adhesive with adherend #i is given by

Analytical Equations
Unless otherwise stated in this manuscript, the upper and the lower signs in ∓ are 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 Figure 2. Substituting ∂σ z /∂z as σ a /(2h 3 ) into Equation (1) gives The adhesive is assumed to experience negligible in-plane stress, σx. The differential equation of equilibrium, Equation (1), then suggests an nonvarying shear stress, τxz, and a linearly varying transverse stress, σz, along the thickness of the adhesive. Denoting σm and σa respectively as the mean and amplitude of variation of the transverse stress along the thickness of the adhesive, the peeling stress at the two interfaces of the adhesive with adherend #i is given by Unless otherwise stated in this manuscript, the upper and the lower signs in ∓ are 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 Figure 2. Substituting ∂σz/∂z as σa/(2h3) into Equation (1) gives The derivations of the differential equations for τ and σm are elaborated upon in Appendix A.

Balanced Structures
Referring to Equation (A5) and with the parameter, µσ, equates to nil for a balanced bonded structure, the differential equation for the interfacial shear stress is given by The derivations of the differential equations for τ and σ m are elaborated upon in Appendix A.

Balanced Structures
Referring to Equation (A5) and with the parameter, µ σ , equates to nil for a balanced bonded structure, the differential equation for the interfacial shear stress is given by where β 2 = λ x /κ s . For structures of reasonably large length, say βl > 3, the solution is given approximately by The boundary conditions κ s dτ(l)/dx = ε T + ε Nl + ε Ml and Referring to Equation (A6) and with the parameter, µ τ , equates to nil for a balanced bonded structure, the differential equation for the mean of the peeling stress is given by where 4α 4 = D/λ z . For structures of reasonably large length, say αl > 3, the solution is given approximately by The boundary conditions Substituting Equation (5) into Equation (3) yields the amplitude of the peeling stress:

Non-Free Edge Solutions
Combining Equations (A5) and (A6) gives rise to a seventh-order differential equation for τ(x) and a sixth-order differential equation for σ m (x) [5,[7][8][9]11]. The resulting solutions are far too complex to be attractive to practicing engineers. Instead, solving Equations (A5) and (A6) iteratively would lead to approximate solutions of τ(x) and σ m (x) that are far simpler, thus enabling insights and encouraging application by practicing engineers. Designating the subscripts c and p as complimentary and particular solutions, respectively, the approximate solution involves solving for the complementary solutions for the interfacial shear stress, τ c , and the mean of the peeling stress, σ mc -these are given by Equations (5) and (9)-followed by substituting σ mc into Equation (A5) to obtain τ =τ c + τ p , and τ c into Equation (A6) to obtain σ m = σ mc + σ mp .
The interfacial shear and peel stresses have been evaluated as where It is worth noting that in the case of an unbalanced bonded structure experiencing only mismatched thermal expansion and/or differential stretching between the adherends, the magnitude of σ m is negligibly small compared to that of τ and σ a [23]. Ignoring σ m , the differential equation for τ is given by Equation (4); and τ(x) and σ a (x) are given by Equations (5) and (11), respectively.

Free Edge Solutions
The expression of shear stress in Equation (12) does not satisfy the free edge condition, τ(l) = 0, which is essential for accurate modelling of σ a (l). The free edge condition may be enforced artificially through the introduction of a decay function, 1 − e nβ(x−l) [15,23]. The derivation of the factor n is explained in Appendix B. The expressions of τ(x), σ m (x), and σ a (x) with the free edge condition enforced are given by where

Numerical Validations
Equations (5), (9), and (11) for the balanced structures [15] and for unbalanced bonded structure experiencing only mismatched thermal expansion and/or differential stretching between the adherends [23] have been validated in previous publications. Hence, only the equations for unbalanced structures experiencing the general state of edge loading shall be validated. The material properties, dimensions of the bonded structure, thermomechanical loads, and finite element model used in this study are shown in Figure 3a. The domain of the bonded structure was modelled using more than 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 of negligible E x , rendering it consistent with the assumption in the analytical solutions that the adhesive experiences negligible σ x . The compliances (assuming plane stress), characteristic parameters (α, β), free-edge parameters (φ, n) which are computed using Equations (A9) and (A10), coupling parameters (µ σ , µ τ ), and the coefficients  Table 1. It is noted that (i) βl, αl ≥ 4, thus justifying the use of the reduced form of solutions as presented in Section 2; (ii) the magnitudes of the coupling parameters are significantly large, reflecting the significant difference in the stiffness between the two adherends; and (iii) the relatively strong coupling has led to the relatively large magnitudes of the particular stress coefficients (A p1 , A p2 , B p ) compared to that of the complementary stress coefficients (A s , B 1n , B 2n ). Figure 3b shows the shear stress, τ(x), for (i) the analytical solution that enforces the nil-shear stress free-edge condition, Equation (14), (ii) the analytical solution that does not enforce the above free-edge condition, Equation (12), and (iii) the finite element analysis (FEA) solution. The analytical solution, Equation (14), agrees reasonably well with the FEA solution for the entire length of the bonding although the magnitude of the former is consistently at approximately 85% that of the FEA. The free-edge parameter at φ = 0.45 is in reasonable agreement with that extracted from FEA at φ FEA = 0.40 and 0.48 for −l ≤ x ≤ 0 and 0 ≤ x ≤ −l, respectively. Figure 3c shows the mean of the transverse stress, σ m (x), for (i) the analytical solution that enforces the nil-shear stress free-edge condition, Equation (15), (ii) the analytical solution that does not enforce the above free-edge condition, Equation (12), and (iii) the FEA solution. The free-edge solution, Equation (15), agrees well with the FEA solution, especially if the FEA solutions at x = ±l, which are susceptible to numerical error, are ignored. Figure 3d shows the amplitude of the transverse stress, σ a (x), for (i) the analytical solution that enforces the nil-shear stress free-edge condition, Equation (16), (ii) the analytical solution that does not enforce the above free-edge condition, Equation (12), and (iii) the FEA solution. Both Equation (16) and the FEA show the magnitude of σ a increases rapidly near the free edge. In contrast, Equation (12) shows a very mild increase in magnitude near the free edge, in the opposite direction.

Balanced Structures
The shear stress in the adhesive for a balanced structure is given by Equation (5). For simplicity, we shall assume A c » A 3 , the maximum magnitude of shear stress is given by For the same magnitude of the applied strains, the magnitude of shear stress may be reduced by increasing the x-compliance and the shear compliances of the bonded structure. We shall analyse two extreme cases; case I: the ratio h adhesive /h adherend is significant such that the shear compliance of the structure is dominated by that of the adhesive; that is, κ s ≈ h 3 /G 3 , and case II: h adhesive /h adherend is insignificantly small, such that the shear compliance of the structure is dominated by that of the adherends; that is, κ s ≈ h 1 /4G 1 . The product λ x κ s for the two extreme cases (for plane stress) are given by: Case II (19) Thus, for Case I, the magnitude of shear stress in the adhesive may be reduced by reducing the elastic moduli of the adherends and the adhesive while increasing the thickness ratio of adhesive-to-adherend. For Case II, the magnitude of shear stress in the adhesive may be reduced by reducing the elastic modulus of the adherends.
The mean of the transverse stress in the adhesive, σ m , for a balanced structure is given by Equation (9). The maximum magnitude of σ m occurs at x = l and is given by Noting that 2α 2 λ z = λ z D,Q 21l = Q 2l /D 2 − Q 1l /D 1 ,M 21l = M 2l /D 2 − M 1l /D 1 and D 1 = D 2 , for the same magnitude of the normalised edge forces, (Q 2l − Q 1l )/α, and the edge moments, M 2l -M 1l , the magnitude of σ m,max may be reduced by increasing the product λ z DD 1 2 = 2λ z D 1 ; that is, by increasing the z-compliance and the flexural stiffness of the structure. We shall analyse the same Polymers 2017, 9, 664 8 of 13 two extreme cases described for shear stress : λ z ≈ h 3 /E 3 for Case I and λ z ≈ 13h 1 /16E 1 for Case II. The product λ z D 1 for the two extreme cases (for plane stress) are given by: 3 12 Case I 13h 1 4 192 Case II (21) Thus, for Case I, the magnitude of σ m,max may be reduced most effectively by increasing the thickness of the adherends and less effectively by increasing the thickness of the adhesive or the ratio of E adherend -to-E adhesive . For Case II, the magnitude of σ m,max may be reduced very effectively by increasing the thickness of the adherends.
The amplitude of the transverse stress in the adhesive, σ a , for a balanced structure is given by Equation (11), which however ought to be modified to include the free-edge condition: The maximum magnitude of σ m occurs at x = l. For simplicity, we shall assume A c » A 3 ; the maximum magnitude of σ a , after substituting the exponential factor n with Equation B(2) is given approximately by Assuming φ to be a constant, then for the same magnitudes of the applied strains, the magnitude of σ a,max may be reduced by increasing the product λ x 3 κ s h 3 2 , which for the two extreme cases: κ s ≈ h 3 /G 3 for Case I and κ s ≈ h 1 /4G 1 for Case II -are given by (for plane stress) Thus, for Case I, the magnitude of σ a,max may be reduced most effectively by reducing the elastic modulus of the adherends or by increasing the thickness ratio of adhesive-to-adherend and less effectively by reducing the elastic modulus of the adhesive. For Case II, the magnitude of σ m,max may be reduced most effectively by reducing the elastic modulus of the adherends and less effectively by increasing the thickness ratio of adhesive-to-adherend.
It is clear from the above that the design guidelines do not concurrently minimise τ max , σ a,max , and σ m,max . In other words, the design guidelines for minimizing the magnitudes of τ max , σ a,max , and σ m,max are contradictory. We shall examine the relative magnitudes of these components of stress. Equation (25) presents the ratio of σ a,max /τ max . It is noted that the magnitude of σ a,max is almost always larger than that of τ max . This, together with the fact that the peeling strength of a bonded joint is generally weaker than its shear strength, suggests that it is more important to minimise the magnitudes of σ a,max and σ m,max than the magnitude of τ max .
for Case I ≈ 3 to 4 for E 1 h 1 /G 3 h 3 between 60 to 250 (25) Equation (26) presents the ratio of σ a,max /σ m,max . Noting that σ m,max ≈ 0 for adherends experiencing only thermal strain and/or stretching strain, σ a,max is much larger than σ m,max for Polymers 2017, 9, 664 9 of 13 these cases. Noting that σ a,max ≈ 0 for adherends experiencing only edge shear forces, Q il , σ a,max is much smaller than σ m,max for this case. Thus, σ a,max may be larger or smaller than σ m,max for a general loading condition. For the case of single lap-joints (SLJ) that are dominated by stretching and bending through a single adherend, i.e., ε T = 0,Q 21l ≈ 0, κ 1l = 0, while N 2l and M 2l are positive, σ a,max is almost always larger than σ m,max . It is therefore advisable to give more weight to the design guidelines that minimize the magnitude of σ a,max .

Unbalanced Structures
In light of the coupling of τ(x) and σ m (x) for the unbalanced structures, it is improbable to find a universal guideline for the optimum design of unbalanced structures that can be expressed in the simple forms of Equations (19), (21) and (24).
The shear stress in the adhesive of an unbalanced structure is given by Equation (14). For simplicity, we shall assume A s » A 4 while also ignoring the decay function, 1 − e nβ(x−l) , the maximum magnitude of shear stress is then given by The magnitude of A c may be reduced by increasing the x-compliance and the shear compliance of the structure. Equation (27) suggests that the magnitude of the particular solution may be reduced by increasing the product 4α 4 + β 4 λ z κ s , which implies increasing the x-compliance and the flexural compliance of the structure. While increasing the x-compliance of the structure would reduce the magnitudes of both the complimentary and particular solutions, this does not imply that it will always lead to reduced magnitude of the τ max of the unbalanced structure. This is because ε T , ε Nl , ε Ml ,Q 21l , M 21l and µ σ could be either positive or negative and hence the particular solution may not act in the same direction as the complementary solution. Nevertheless, increasing the x-compliance of the structure is more likely than not to reduce the τ max of an unbalanced structure. In the same breath, it should be noted that the magnitude of τ max in an unbalanced structure may not always be larger than that in a balanced structure assuming both structures have identical characteristic parameters.
The mean of the transverse stress in the adhesive, σ m , for an unbalanced structure is given by Equation (15). For the reason of simplicity, we shall ignore those terms associated with the decay function. The maximum magnitude of σ m occurs at x = l and is given by Noting thatQ 21l = Q 2l /D 2 − Q 1l /D 1 andM 21l = M 2l /D 2 − M 1l /D 1 , for the same magnitudes of Q il and M il , the magnitude of B 1 may be reduced by increasing the z-compliance while reducing the flexural compliance of the structures. Equation (28) suggests that the magnitude of the particular solution may be reduced by increasing the magnitudes of the characteristic parameter, α, which implies increasing the flexural compliance of the structure while reducing the z-compliance of the structure. This is in reverse to the trend for B p .
The amplitude of the transverse stress in the adhesive, σ a , for an unbalanced structure is given by Equation (16). Assuming A s » A 4 , the maximum magnitude of σ a is given by: which again suggests that the magnitude of σ a,max is likely to be reduced by increasing the x-compliance of the structure. It has been argued in the previous section that the magnitude of σ a,max is almost always larger than that of τ max for balanced structures. Comparing Equations (23) and (29), it is reasonably safe to suggest that this will also be the case for unbalanced structures. It has been established in the previous section that the ratio of σ a,max to σ m,max for balanced SLJ is larger than unity. If the ratio for the particular solutions is also larger than unity then one can safely assume that the ratio of σ a,max to σ m,max for unbalanced SLJ is also larger than unity. The ratio for the particular solutions is given by which is not necessary larger than unity. Thus, σ a,max is only conditionally larger than σ m,max for unbalanced SLJ. Given the loadings and the design space, the optimum designs of unbalanced structure that gives rise to a minimum τ max , a minimum σ m,max , and a minimum σ a,max , respectively, can be readily established using Equations (27)-(29).

Conclusions
Strength-of-material solutions for the shear stress, τ, the mean, σ m , and amplitude, σ a , of the peeling stress in both balanced and unbalanced structures have been derived and used for design analysis. Design guidelines for balanced structures have been established. The magnitude of σ a,max for balanced structures is almost always larger than that of τ max and the magnitude of σ a,max for balanced single-lap-joints is also almost always larger than σ m,max . The magnitude of σ a,max for balanced structures may be reduced most effectively by reducing the elastic modulus of the adherends or by increasing the thickness ratio of adhesive-to-adherend and less effectively by reducing the elastic modulus of the adhesive. The simple expressions of τ max , σ m,max and σ a,max established in this manuscript for the unbalanced structures will help practicing engineers find the optimum design within the given design space. κ si , κ s Shear compliance of member #i, of the bonded structure between the centroid planes of adherends #1 and #2.
λ xi , λ x x-compliance of adherend #i, of the bonded structure.
λ xθ Additional x-compliance of the bonded structure attributed to its flexural deformation.
λ zi , λ z z-compliance of member #i, of the bonded structure between the centroid planes of adherends #1 and #2.
θ i Rotation of the centroid axis of adherend #i (due to bending).
σ m (x), σ a (x) Mean, amplitude of variation, of transverse stress along the thickness of the adhesive (or between the bonded interfaces).
σ p (x) Peeling stress along the bonded interfaces.

τ(x)
Shear stress within the adhesive (and along the interfaces).
∆T Temperature change.