Adhesive Bonded Joints

Adhesive Bonded Joints (structures/composites/adhesive-bonded-joints)

Use when the task is analyzing a single-lap adhesive bonded joint between two adherends of a composite or metallic structure: the Volkersen-style shear-lag model for the bondline, the average and peak adhesive shear stresses, and the joint margin against the adhesive allowable. This leaf implements the shear-lag model in pure Python, stdlib only, and reports the peak stress concentration that gates bonded joint design in the CMH-17 composite context. It pairs with structures/composites/composite-bolted-joints, the mechanical fastener alternative in the same pack, and with failure-criteria and laminate-stiffness for the adherend side of the joint.

Domain quick reference

• Geometry: single-lap joint with overlap length L, bond width b and two identical adherends of thickness t and modulus E, bonded by an adhesive layer of thickness t_a and shear modulus G_a.
• Shear-lag parameter: beta = sqrt((G_a / t_a) * (2.0 / (E * t))) in 1/m. For identical adherends the 2.0/(Et) term is the sum 1/(E1t1) + 1/(E2*t2) with E1 = E2 = E and t1 = t2 = t.
• Average adhesive shear stress: tau_avg = P / (b * L).
• Peak shear stress at the overlap ends (Volkersen style): tau_max = tau_avg * (betaL/2) / tanh(betaL/2). The factor (betaL/2)/tanh(betaL/2) tends to 1 as betaL tends to 0 (uniform shear) and grows with betaL, so long overlaps do not lower the peak below the load divided by the bond area without the concentration.
• Concentration factor: peak to average ratio (betaL/2) / tanh(betaL/2).
• Joint margin: margin_ratio = allowable / tau_max; the MS-style margin of safety is margin_ms = margin_ratio - 1.0. The joint passes when tau_max does not exceed the adhesive allowable shear stress.
• SI units throughout: N, m, Pa.
• The model is a Volkersen-style bondline shear analysis only; peel and adherend bending are out of scope. Peel-critical designs need a Goland-Reissner or FE analysis. CMH-17 frames the bonded joint data context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the joint geometry and materials: load P (load_n), bond width b (width_m), overlap L (overlap_m), adherend E and t (adherend_E_pa, adherend_t_m), adhesive G_a and t_a (adhesive_G_pa, adhesive_t_m), and the adhesive allowable (allowable_shear_pa).
2. Compute the shear-lag parameter with shear_lag_beta; it measures how fast the shear stress peaks near the overlap ends.
3. Get the average bondline stress with avg_shear_stress = P/(b*L).
4. Apply the correction with peak_shear_stress (or scale the average by concentration_factor) to get the bondline peak stress.
5. Rate the joint with joint_margin against the adhesive allowable; the margin ratio is allowable/peak and the MS margin is that ratio minus one.
6. Run analyze to get the full dict {beta, tau_avg, tau_max, concentration, margin_ratio, margin_ms, pass} in one call.
7. Confirm the deterministic checks with the contract test scripts/test_adhesive_bonded_joints.py.

Worked example

Aluminum adherends E = 70 GPa, t = 2 mm; adhesive G_a = 0.5 GPa, t_a = 0.2 mm; bond width 25 mm; load 10 kN.

Case 1: overlap L = 25 mm, allowable 25 MPa.

• beta = sqrt((0.5e9/0.2e-3) * (2/(70e92e-3))) = 188.98 1/m, betaL = 4.72.
• tau_avg = 10000 / (0.025 * 0.025) = 16 MPa.
• tau_max = 16 MPa * 2.3623 / tanh(2.3623) = 38.47 MPa, so the concentration factor is 2.40.
• margin_ratio = 25/38.47 = 0.650, margin_ms = -0.350: FAIL.

Case 2: overlap L = 10 mm, allowable 25 MPa.

• tau_avg = 10000 / (0.025 * 0.010) = 40 MPa.
• tau_max = 51.25 MPa (beta*L = 1.89, concentration 1.28): FAIL.
• The shorter overlap raises both the average and the peak stress; the peak exceeds the 25 mm case.

Case 3: allowable 45 MPa, L = 25 mm.

• margin_ratio = 45/38.47 = 1.170, margin_ms = +0.170: PASS.

Pitfalls

• Reading the peak stress as uniform bondline shear: the average P / (b L) understates the bondline ends by the concentration factor (betaL/2)/tanh(betaL/2), which is 2.40 in the worked example; the joint margin must be checked against tau_max, not tau_avg.
• Believing a longer overlap always fixes the joint: the peak stress only falls toward the average slowly with beta*L, and the shorter 10 mm overlap in the worked example raises both the average and the peak; the concentration saturates for long overlaps.
• Using non-identical adherend data: beta assumes two identical adherends with the 2.0/(Et) sum form; dissimilar adherend pairs need the general 1/(E1t1) + 1/(E2*t2) expression.
• Forgetting the model boundary: this is Volkersen-style bondline shear only; peel and adherend bending are out of scope, and peel-critical designs need a Goland-Reissner or FE analysis.
• Treating a zero margin as acceptable: margin_ms = margin_ratio - 1, so the joint only passes when tau_max stays below the allowable (25 MPa allowable against 38.47 MPa peak gives -0.350 and FAIL).
• Feeding non-physical joint inputs: negative load, zero or negative width and overlap, and non-positive moduli, thicknesses, and allowables raise ValueError instead of producing a stress.

Verification

• Confirm shear_lag_beta(70e9, 2e-3, 0.5e9, 0.2e-3) returns 188.98 1/m within 0.5.
• Confirm avg_shear_stress(10000, 0.025, 0.025) returns 16 MPa and peak_shear_stress gives 38.47 MPa within 0.2 MPa of the 38.51 MPa spec anchor, with concentration 2.40 within 0.01 of 2.407.
• Confirm the 10 mm overlap peaks higher than the 25 mm overlap and that analyze reports pass False for the 25 MPa allowables and pass True for the 45 MPa allowable at L = 25 mm.
• Confirm negative load, zero or negative width and overlap, and non-positive modulus, thickness, adhesive shear modulus, adhesive thickness and allowable all raise ValueError.
• Run the contract test offline: python3 scripts/test_adhesive_bonded_joints.py (34 tests, deterministic).

Related leaves

• structures/composites/composite-bolted-joints: the mechanical fastener alternative for joining composite laminates.
• structures/composites/failure-criteria: ply-level failure checks of the adherends either side of the bondline.
• structures/composites/laminate-stiffness: adherend modulus and thickness inputs for laminate adherends.
• structures/composites/cmh17-allowables: composite material and joint data context behind the CMH-17 reference.
• structures/composites/sandwich-panels: skin to core bonds where the same adhesive allowables apply.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_adhesive_bonded_joints.py

The test covers the worked example anchors (beta 188.98, average 16 and 40 MPa, peak 38.47 and 51.25 MPa, concentration 2.40), the shorter overlap penalty, the margin ratio and MS margin for the 25 MPa FAIL and 45 MPa PASS cases, the uniform-shear limit for a small beta*L, the peak over average round trip, load scaling, zero-load behaviour, the finite concentration for long overlaps, and ValueError rejection of negative load, zero overlap and width, zero modulus, thickness and adhesive thickness, and non-positive allowables.

Compliance

• Standards referenced, not reproduced: CMH-17 is a SAE-published composite materials handbook (sae.org/publications/cmh-17); the single-lap shear-lag relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.