Peel Stress Bonded Joints

Peel Stress in Bonded Joints (structures/composites/peel-stress-bonded-joints)

Use when the task is the peel stress at the overlap end of a single-lap bonded joint between two identical adherends of a composite or metallic structure: the classical Goland-Reissner bending moment factor, the edge moment that the load path eccentricity puts into the adherends, and the maximum peel stress that the adhesive bondline sees at the overlap end from the Winkler-foundation beam model. This leaf implements the Goland-Reissner analysis in pure Python, stdlib only, and reports the peel margin that gates the peel-critical joint check in the CMH-17 composite context. The sibling structures/composites/adhesive-bonded-joints owns the in-plane bondline shear transfer of the same single-lap geometry and self-declares peel and adherend bending out of scope; this leaf fills that gap for peel-critical designs. It also pairs with structures/composites/composite-bolted-joints, the mechanical fastener alternative, and with laminate-stiffness for the adherend side of the joint.

Domain quick reference

• Geometry: balanced single-lap joint, two identical adherends of thickness t and modulus E with Poisson ratio nu, bonded over an overlap of total length 2 c (c the overlap half length) by an adhesive layer of modulus E_a and thickness t_a. All quantities are per unit width: the load per unit width is P_pw = P / b.
• Goland-Reissner bending moment factor: u^2 = (3 (1 - nu^2) / 2) * P_pw / (E t^3), then k = cosh(u c) / (cosh(u c) + 2 sqrt(2) sinh(u c)). k lies in (0, 1]: it tends to 1 as the load or the overlap half length tends to zero (the no-bending limit) and falls toward the classical floor of about 0.261 for very long overlaps, so the edge moment never grows without bound with the overlap.
• Edge moment at the overlap end: M0 = k * P_pw * t / 2, the moment that the eccentric load path applies to each adherend at the edge of the bondline.
• Adherend flexural rigidity: D = E t^3 / (12 (1 - nu^2)).
• Winkler-foundation beam parameter: lambda^4 = 3 (1 - nu^2) E_a / (E t^3 t_a) with lambda in 1/m.
• Peel stress at the overlap end: the edge moment M0 deflects the adherend beam on the adhesive foundation, w0 = M0 / (2 lambda^2 D), and the adhesive strains with that deflection, sigma_peel = (E_a / t_a) * w0 in Pa.
• Peel decay coefficient: beta = sqrt(6 E_a / (t_a E t)) in 1/m, the classical exponential peel-decay parameter that measures how fast the peel stress dies away from the overlap end.
• Peel margin: margin = peel_strength_allowable / sigma_peel. The joint passes the peel-critical check when the margin is at least one, that is when the peak peel stress does not exceed the allowable.
• Sensitivity: sigma_peel scales with sqrt(E_a) and with 1 / sqrt(t_a), so doubling the adhesive modulus raises the peel stress by about sqrt(2) and doubling the adhesive thickness lowers it by the same factor; a stiffer or thinner adhesive layer is not automatically a safer bondline.
• SI units throughout: N, m, Pa. CMH-17 frames the bonded joint data context; the relations above are standard engineering methodology, summary-only.
• The model boundary: this is the Goland-Reissner peel analysis of the overlap end only; the in-plane bondline shear transfer along the overlap belongs to the adhesive-bonded-joints leaf, and full finite-element peel modeling is out of scope here.

Workflow

1. Fix the joint geometry and materials: load per unit width P_pw (load_per_unit_width), adherend thickness t (adherend_thickness), adherend modulus E (adherend_modulus), Poisson ratio nu (poisson_ratio), overlap half length c (overlap_half_length), adhesive modulus E_a (adhesive_modulus), adhesive thickness t_a (adhesive_thickness), and the peel strength allowable (peel_strength_allowable).
2. Compute the Goland-Reissner bending moment factor with bending_moment_factor; k falls from 1 toward the 0.261 floor as the load or the overlap grows, so it carries the whole bending state of the joint into the peel check.
3. Compute the peel decay coefficient with peel_decay_coefficient; beta sets the exponential decay of the peel stress away from the overlap end.
4. Resolve the peel stress at the overlap end with peel_stress_at_overlap_end, which forms the edge moment M0 = k P_pw t / 2, solves the adhesive Winkler-foundation beam response at the overlap end, and returns the dict {peel_stress, edge_moment, lambda}.
5. Rate the joint with peel_margin against the peel strength allowable; the margin is allowable over peel stress, and a margin below one fails the peel-critical joint check.
6. Confirm the deterministic checks by running the contract test scripts/test_peel_stress_bonded_joints.py.

Worked example

Aluminum adherends E = 70 GPa, nu = 0.33, t = 1.6 mm; epoxy adhesive E_a = 1.5 GPa, t_a = 0.25 mm; overlap 25 mm (half length c = 12.5 mm); load P = 4000 N over a 25 mm width, so P_pw = 1.6e5 N/m.

• Average adherend stress sigma_avg = P_pw / t = 100 MPa.
• bending_moment_factor(1.6e5, 1.6e-3, 70e9, 0.33, 0.0125) = 0.518201.
• Edge moment M0 = k P_pw t / 2 = 66.3298 N m/m.
• lambda = 486.335 1/m, and the peel stress at the overlap end is 3.13768e7 Pa = 31.3768 MPa, about 0.31 of the average adherend stress for this joint.
• peel_decay_coefficient(1.5e9, 0.25e-3, 70e9, 1.6e-3) = 566.947 1/m.
• peel_margin(31.3768e6, 35e6) = 1.11547: the 35 MPa allowable clears the peak peel stress. peel_margin(31.3768e6, 25e6) = 0.796767: the 25 MPa allowable fails the peel-critical check even though the same joint passes an in-plane shear check.

Lower-load case P_pw = 8.0e4 N/m (sigma_avg = 50 MPa): k = 0.598868, M0 = 38.3275 N m/m, and the peel stress falls to 18.1306 MPa, so the peel stress is monotone increasing with the load.

Pitfalls

• Reading the peel stress as uniform through the bondline: the peel stress is strongly concentrated at the overlap end where the edge moment acts; the margin must be checked against the peak value at the end, not against any average across the overlap.
• Stopping at the in-plane shear analysis: a joint can clear the adhesive shear allowable and still fail in peel, because the adherend bending from the eccentric load path adds a through- thickness stress the shear model never sees; peel-critical designs need this Goland-Reissner check.
• Expecting a stiffer or stronger adhesive to always help: the peel stress grows with sqrt(E_a), so doubling the adhesive modulus raises the peak peel stress by about sqrt(2) and can consume the margin the stronger adhesive was meant to buy.
• Ignoring adhesive thickness control: the peel stress scales with 1 / sqrt(t_a), so a thin bondline region drives up the peel stress at the overlap end; the worked example's 0.5 mm bondline halves the 31.38 MPa peak down to 22.19 MPa.
• Assuming the bending moment factor saturates at one: for long overlaps k falls toward the classical floor near 0.261, and the edge moment M0 = k P_pw t / 2 follows it down; using k = 1 overstates the moment for a long joint.
• Treating a margin of one as a comfortable pass: margin = allowable / peel stress, so one means the peak peel stress exactly equals the allowable with no reserve; only a margin above one passes with margin to spare.
• Feeding non-physical joint inputs: negative load per unit width, zero or negative thicknesses and moduli, a zero overlap half length, Poisson ratios outside (-1, 0.5), and non-positive allowables all raise ValueError instead of producing a stress.

Verification

• Confirm bending_moment_factor(1.6e5, 1.6e-3, 70e9, 0.33, 0.0125) returns 0.518201 within 1e-5, that the factor is 1.0 at zero load and 0.999924 at a near-zero load, and that it reaches 0.261204 at a 1 m overlap half length, the classical long-overlap floor.
• Confirm the factor is monotone decreasing in the overlap half length (0.928308 at 1 mm down through 0.287155 at 50 mm) and in the load per unit width (0.805922 at 10 N/mm, 0.518201 at 160 N/mm, 0.417725 at 400 N/mm).
• Confirm peel_stress_at_overlap_end returns 31.3768 MPa peel stress, 66.3298 N m/m edge moment and 486.335 1/m lambda within the spec tolerances, that the peel stress grows monotonically with the load (18.1306, 31.3768 and 63.23 MPa at 80, 160 and 400 N/mm), that doubling E_a raises it to 44.37 MPa and doubling t_a lowers it to 22.19 MPa, and that the dict keys are exactly peel_stress, edge_moment and lambda.
• Confirm peel_decay_coefficient(1.5e9, 0.25e-3, 70e9, 1.6e-3) = 566.947 1/m within 0.01 and peel_margin(31.3768e6, 35e6) = 1.11547 within 1e-3.
• Confirm negative loads, zero or negative thicknesses and moduli, zero overlap half length, Poisson ratio 0.6, and zero allowables all raise ValueError.
• Spec note recorded as an assumption: the prep checkpoint 0.287155 for the overlap traverse is realized at a 50 mm overlap half length, and the checkpoint 0.805922 at 10 N/mm, not at the 80 N/mm level of the worked example; the module follows the formula exactly and the tests assert the values at the half lengths and loads where the formula actually produces them.
• Run the contract test offline: python3 scripts/test_peel_stress_bonded_joints.py (34 tests, deterministic).

Related leaves

• structures/composites/adhesive-bonded-joints: the in-plane bondline shear transfer of the same single-lap geometry, which self-declares peel and adherend bending out of scope; this leaf supplies the peel check those joints need.
• 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.

Behavior contract (gate 3)

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

python3 scripts/test_peel_stress_bonded_joints.py

The test covers the worked example anchors (moment factor 0.518201, edge moment 66.3298 N m/m, lambda 486.335 1/m, peel stress 31.3768 MPa, decay coefficient 566.947 1/m, margins 1.11547 and 0.796767), the no-bending limit k to 1 at zero load and the classical 0.261 floor at a long overlap, the monotone traverses of k in the overlap half length and in the load, the monotone peel growth with the load, the sqrt(2) sensitivity to doubled adhesive modulus and thickness, the exact dict keys, the peel margin round trip, the determinism of repeated calls, and ValueError rejection of negative loads, zero thicknesses and moduli, zero overlap half length, Poisson ratio 0.6 and zero allowables.

Compliance

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