Adhesive Bonded Joint: Method

Adhesive Bonded Joints: Shear Lag, the Elastic-Plastic Adhesive, Single-Lap Bending, Peel, Scarf and Stepped-Lap Joints

Why the load transfer is not uniform

If the adherends of a lap joint were rigid, every point of the adhesive would be sheared by the same amount and the stress would be the average, load over bond area. Real adherends stretch. At one end of the overlap adherend 1 carries the whole load and adherend 2 none; at the other end the roles are reversed. The two adherends therefore strain differently along the overlap, and the difference in their displacements, which is what shears the adhesive, is largest at the two ends and smallest in the middle. Volkersen (1938) wrote this down for riveted and bonded lap joints; the result is the shear-lag distribution, with two stressed end zones and a lightly loaded trough between them. Lengthening the overlap lengthens only the trough.

The shear-lag equation

Let x run along the overlap from the end where adherend 2 stops to the end where adherend 1 stops, let T₁ and T₂ be the adherend forces per unit width (T₁ + T₂ = N, the load on one bond line) and γ = (u₂ − u₁)/η the adhesive shear strain in a bond line of thickness η. Equilibrium of each adherend gives dT₂/dx = τ = −dT₁/dx, and the adherend strains give

dγ/dx = (1/η) [T₂/(E₂t₂) − T₁/(E₁t₁) + (α₂ − α₁) ΔT],   d²γ/dx² = (τ/η) [1/(E₁t₁) + 1/(E₂t₂)].

The thermal mismatch does not appear in the differential equation, only in the end conditions γ′(0) = (−N/(E₁t₁) + (α₂ − α₁)ΔT)/η and γ′(L) = (N/(E₂t₂) + (α₂ − α₁)ΔT)/η. ΔT is the operating temperature minus the stress-free temperature of the bond, which Hart-Smith takes close to the cure temperature. In a double-lap joint each bond line works with one outer adherend and half of the inner one, so E₂t₂/2 replaces E₂t₂ and N = P/2. With an elastic adhesive, τ = Gγ, the equation is γ″ = λ²γ, λ² = (G/η)[1/(E₁t₁) + 1/(E₂t₂)], and the solution is a pair of hyperbolic functions (Volkersen). For a balanced double lap the elastic peak is (P/2)(λ/2) coth(λL/2): once λL exceeds about 5 it no longer falls as the overlap grows.

The elastic-plastic adhesive

Structural adhesives are not elastic to failure. A thick-adherend shear test of a ductile film adhesive shows a short elastic rise followed by a long, nearly flat yield to a failure strain of the order of 1 to 2; a brittle high-temperature adhesive fails at a strain of about 0.1. Hart-Smith (NASA CR-112235) replaced the measured curve by an elastic-perfectly plastic one with the same failure stress, the same failure strain γ_e + γ_p and the same area under the curve, and showed with a second, two-slope idealization that only that area matters for the strength of a long joint. In a plastic zone τ = τ_p is constant, so γ is a parabola; in the elastic zone it is hyperbolic. The zone lengths follow from continuity of γ and γ′ where |γ| = γ_e and from equilibrium. This page solves the general case (unequal adherends, thermal mismatch, plastic zones of either sign at none, one or both ends) in closed form zone by zone, with a one-dimensional root for the zone lengths, and takes the joint to have failed when the larger end strain reaches γ_e + γ_p.

Three results of the report follow directly. First, a short overlap goes fully plastic before the failure strain is reached, and its strength is τ_p times the bond area. Second, beyond a transition overlap the strength approaches a plateau that no further overlap raises. Multiplying the differential equation by γ′ and integrating from the trough to the critical end gives (γ′_end)² = (2/η)[1/(E₁t₁) + 1/(E₂t₂)] τ_p(γ_e/2 + γ_p), and inserting the end conditions gives the plateau load with each end critical in turn (CR-112235 eqs 54, 55, 108, 109):

P = E₂t₂ [√(2ητ_p(γ_e/2 + γ_p)(1/(E₁t₁) + 2/(E₂t₂))) − (α₂ − α₁)ΔT],   P = 2E₁t₁ [√(2ητ_p(γ_e/2 + γ_p)(1/(E₁t₁) + 2/(E₂t₂))) + (α₂ − α₁)ΔT]

for the double lap, the lesser governing; for a balanced joint without thermal mismatch both reduce to P = 4√(E t)√(τ_pη(γ_e/2 + γ_p)) with E t that of one outer adherend (eq. 22). Third, the only adhesive property in the plateau is the strain energy per unit bond area τ_pη(γ_e/2 + γ_p): a stiff brittle adhesive and a soft ductile one of the same energy give the same joint strength. Stiffness imbalance lowers the plateau because the end from which the softer adherend extends reaches the failure strain while the other end is still lightly strained; thermal mismatch adds to the adhesive strain at one end and relieves it at the other, so tension and compression strengths differ, and a negative plateau value means that cooling from cure alone breaks a long joint.

The report's design rule for the overlap is ℓ = P/(2τ_p) + 2/λ (eq. 112): plastic zones long enough to carry the load plus an elastic allowance. Hart-Smith's later rule, quoted in MIL-HDBK-17-3F section 6.2.2.7, asks for an elastic zone of at least 6/λ so that the trough stress is about a tenth of τ_p; the lightly stressed middle resists creep under sustained and slowly cycled load and anchors the joint. The page prints both, with the plastic zone lengths of the actual state at ultimate load.

The unsupported single lap

The load path of a single lap is eccentric by t + η. If the joint is free to rotate, the adherends bend until the load line passes nearer the bond, and a bending moment M₀ = k P (t + η)/2 remains in each adherend at the end of the overlap. Goland and Reissner (1944) found k by treating the overlap as one plate of double thickness, k = 1/(1 + 2√2 tanh(ξc/(2√2))) with ξ² = P/D, D = E t³/(12(1 − ν²)) and c = L/2; this tends to 0.26 for long overlaps or high loads. Hart-Smith (CR-112236) analysed the two adherends separately and obtained k = 1/(1 + ξc + ξ²c²/6), which tends to zero, so that a long overlap does relieve the bending. Both start at k = 1; the page prints both and uses the one selected.

The moment matters in three ways. It adds a bending stress to the adherend, whose largest fibre stress at the end of the overlap is (P/t)(1 + 3k(1 + η/t)); a metal adherend usually forms a plastic hinge there before the adhesive fails. It strains the bond-line surface of each adherend, which changes the shear-lag equation to γ″ = kτ − c₀ with k = (2/(E t η))(1 + 3(1 − ν²)(1 + η/t)/k_b), end slope γ′ = (P/(E t η))(1 + 3k(1 − ν²)(1 + η/t)/k_b) and a constant c₀ fixed by equilibrium (CR-112236 eqs 12, 23 to 25 and 37); the elastic peak of that equation is the Goland-Reissner shear formula, CR-112236 eq. (40), and the same zone-by-zone solution gives the elastic-plastic state. And it peels the adherends apart: with the transverse shear V₀ = ξM₀, the two adherends joined by the transverse stiffness of the adhesive obey σ′′′′ + 4χ⁴σ = 0, χ⁴ = E_c′/(2Dη), whose end value for a long overlap is χ²M₀ + χV₀ (the form printed in MIL-HDBK-17-3F); Hart-Smith's eq. (113) keeps the first term. The page solves the equation for the entered overlap and uses the larger value. k_b is the ratio of the laminate bending stiffness to that of a homogeneous plate of the same in-plane modulus.

Peel in the double lap

A double lap has no net eccentricity, but at the end of each outer adherend the adhesive shear stress acts on the bond face, half a thickness from the adherend's middle surface. The outer adherend bends away from the inner one as a beam on the elastic foundation of the adhesive, χ⁴ = E_c′/(4Dη), and the peak peel stress is σ_c = τ (3E_c′(1 − ν²)t_o/(E_oη))^¼ (CR-112235 eq. 83), with τ the adhesive shear stress at that end (τ_p once the end is plastic). It rises with the fourth root of the outer adherend thickness and is independent of the overlap, which is why thick adherends need tapered ends, stepped laps or scarfs. E_c′ is the tensile modulus of the constrained film, softened by the transverse flexibility of the adherends (eq. 68). The peel analysis is elastic: the thin constrained film does not yield in tension, and a laminated adherend fails in interlaminar tension, at its own lower allowable, as soon as the peak is reached. Under a compressive load the transverse stress at these ends is compressive and no peel margin is formed.

Scarf joints

Resolving the adherend stress σ on a bond plane inclined at θ gives τ = σ sin θ cos θ and σ_n = σ sin²θ. Between identical adherends the shear is uniform, so the joint develops τ_p over its whole length. Between adherends of different stiffness or expansion it is not: with the stiffnesses tapering linearly the force T in adherend 2 obeys (η/G)T″ = T/(E₂t₂x/L) − (P − T)/(E₁t₁(1 − x/L)) + (α₂ − α₁)ΔT (CR-112237 section 2). The report solves this by a power series; the page uses finite differences with Richardson extrapolation and reproduces the report's printed tables. For a long scarf τ_av/τ_max tends to the stiffness ratio of the adherends (eq. 21), with the peak at the tip of the stiffer adherend.

With the elastic-perfectly plastic adhesive (CR-112237 section 3) a plastic zone of length a grows from the critical end, the end from which the softer adherend extends unless the thermal mismatch reverses it, and a second zone may form at the other end. In the plastic zone at x = 0 the adherend forces are known, T₂ = τ_px and T₁ = P − τ_px, so the strain integrates in closed form:

η[γ(0) − γ(a)] = τ_pL a [1/(E₁t₁) − 1/(E₂t₂)] − (α₂ − α₁)ΔT a + (τ_pL − P)(L/(E₁t₁)) ln(1 − a/L).

At failure the left side is ηγ_p, which gives the load as a function of the zone length, P(a) = τ_pL − E₁t₁[τ_pL a (1/(E₁t₁) − 1/(E₂t₂)) − (α₂ − α₁)ΔT a − ηγ_p] / (−L ln(1 − a/L)). The true a is fixed by the elastic zone, which has no closed form; whatever it is, the least value of P(a) over a cannot exceed the true strength. That minimum is Hart-Smith's algebraic lower bound (CR-112237 eqs 54 to 56; CR-2218 eqs 14 and 15), which the page prints with each end taken critical in turn. The elastic-plastic strength itself comes from the complete problem. Eliminating T gives the conservative form [c(x)(ηγ′ − (α₂ − α₁)ΔT) + P r(x)]′ = τ(γ), with c = S₁S₂/(S₁ + S₂) and r = S₂/(S₁ + S₂) built from the tapered stiffnesses S₁ = E₁t₁(1 − x/L) and S₂ = E₂t₂x/L; because c vanishes at both feather edges the end conditions are met identically. The page solves it on 400 and 800 cells by Newton's method with Richardson extrapolation of the end strains and finds the load at which the larger strain reaches γ_e + γ_p. The scarf margin is written on this elastic-plastic strength; the elastic strength and the lower bound are printed beside it. For a long scarf all three tend to τ_pL times the stiffness ratio, and, unlike a lap joint, the strength keeps rising with the length. The tip of an adherend carries τ per unit length on a section that grows as the slope, so the scarf angle must exceed τ/F_u (eq. 57), which the page checks with the adhesive stress at each feather edge.

Stepped-lap joints

A stepped-lap joint is a row of short uniform laps (CR-112237 sections 5 to 7). Adherend 1 carries the whole load at x = 0 and thins step by step; adherend 2 starts there and thickens. Inside step j the shear-lag equation of the double lap holds with that step's thicknesses, γ″ = (n_bτ/η)[1/(E₁t_1j) + 1/(E₂t_2j)], n_b = 2 for the joint that is symmetric about its mid-plane (the report's case, thicknesses being totals) and 1 for a one-sided joint held against bending. At a step change the adherend loads and displacements are continuous, so γ is continuous while its slope, ηγ′ = T₂/(E₂t₂) − T₁/(E₁t₁) + (α₂ − α₁)ΔT, jumps with the thicknesses; this jump is what puts a stress spike at every step end. The report marches from one end with a guessed end strain and corrects the guess, and describes how a change in the sixteenth figure of the guess changes the fourth step by a factor of ten million. The page takes the strains at the step ends as the unknowns instead. A step with both end strains given has a unique solution with at most a plastic zone at each end and an elastic zone between them, in closed form apart from one root for the elastic length, and it returns the two end slopes; the conditions T₂(0) = 0, T₂ = P at the far end and continuity of T₂ at each step change are then solved by Newton's method. The two routes give the same state.

The page prints, for every step, the plastic and elastic zone lengths, the strains and stresses at its ends (with the purely elastic stresses beside them), the adherend loads, and the net-section stress of each adherend, taken on the larger end load over the thickness at that step. The potential bond strength is the load at which the largest strain, always at a step end, reaches γ_e + γ_p, or τ_p times the bond area if every step goes plastic first; the load at which the first detail (adhesive or an adherend section at F_tut) becomes critical is printed with the detail named. The adhesive margin is written on the potential bond strength at ultimate load, the adherend margins on the net-section stresses at ultimate load (F_tu) and at limit load times FS_yld (F_ty), each from its own solution, because the internal loads are not proportional to the applied load. Hart-Smith's design observations apply: the three steps nearest the critical end set the strength; a step that already holds a full elastic trough gains nothing from extra length, so more and thinner steps are the remedy; and the end step of the stiffer adherend must be short enough that τ_p times its bond area does not exceed F t of the tab. Peel stresses are not computed for the stepped joint (the report neglects them for thin end steps).

Load levels and margins

The entered load is ultimate or limit. Adhesive shear strength, peel, interlaminar tension and F_tu are checked at ultimate load, F_ty at limit load times FS_yld; the thermal mismatch is part of the adhesive strength and is not factored. In the unsupported single lap the bending factor k is evaluated at each level's own load, because it falls as the load rises. The Volkersen elastic peak is printed as a comparison with the load at which the adhesive first reaches τ_p; Hart-Smith's recommendation is to keep the adhesive elastic at the frequently repeated load and to make the bond's shear strength at least 50 % greater than the adherend strength, so that the adherends, not the bond, are the weak link.

Checks and limits

The zone-by-zone solution agrees with an independent finite-difference solution of the same boundary-value problem to better than one part in a million on 26 seeded joints, and reproduces the non-dimensional strength tables printed in CR-112235 Appendix A.2 to the fourth decimal in the plateau, fully plastic and two-plastic-zone cells. In cells where only one end is plastic the report's program prints values up to 6 % below the exact solution of its own equations (the independent solver confirms that the end strain at the printed load is short of the failure strain); the page gives the exact value. The scarf solver reproduces the elastic table of CR-112237 Appendix A.1 to four figures.

The stepped-lap solver reproduces the report's printed cases for its titanium to graphite-epoxy joint of six steps (CR-112237 pp. 42, 101, 114 and 115): the elastic strengths, the elastic-plastic strengths limited by the adherends and the potential bond strengths at ΔT = 0, −280 and −400 °F in tension and compression, within 0.025 % (one printed potential bond strength, 45,000 lb/in at −400 °F in compression, is τ_p times the bond area and is not what the strain criterion gives, 31,476 lb/in), and the printed zone lengths, strains and adherend loads of every step of the two elastic-plastic listings to the printed digits; the report's purely elastic listing agrees in the first step and drifts in the lightly loaded steps, the loss of precision its text describes. The solver agrees with an independent solution by numerical integration and shooting on 20 seeded joints. The scarf lower bound reproduces the report's Appendix A.2 table to the fifth decimal. The elastic-plastic scarf strength agrees to five figures with an independent solution in the report's own formulation (closed-form plastic zones, the elastic zone by Chebyshev collocation) on 20 seeded joints, but not everywhere with the report's Appendix A.3 table: in the fifteen cells checked the printed values lie between 2.4 % above and 3.9 % below the solution of the report's equations, and in long-overlap cells the table prints the lower-bound value. The page prints the solution of the equations; the lower bound beside it is the conservative figure.

Not covered: peel stresses and the load-path eccentricity of stepped-lap joints, tapered or stepped ends of the outer adherends (CR-112238), unequal adherends in the unsupported single lap, in-plane shear loading, transverse shear deformation of the adherends, adherend yielding inside the overlap, fatigue, creep and environmental effects, and bond flaws. The one-dimensional model takes the shear stress uniform across the bond thickness, so it does not vanish at the free ends of the adhesive as the true stress does within a few bond thicknesses; with a yielding adhesive this is slightly conservative.