forked from 1b-agent/fieldcraft-upstream
solver: mortar contact in 2D models
Plane strain, plane stress and axisymmetric models take contact pairs and general (self) contact on the linear edges of tri3 and quad4 elements: edges oriented out of their elements, the edge mortar of fem with the thickness or 2 pi r in the weights, the element's area over the edge's length as the thickness under it, and the in-plane rows of the tangent blocks. Verified by the contact patch test in plane strain and axisymmetric (to 1e-9 small sliding, 1e-4 finite), Hertz's line contact on quad4 edges (Johnson 1985, 4.3; equal to the hex8 model's to 1e-5), and a hybrid rubber C-ring closing on itself that moves node for node as the one-layer solid ring. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo
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jyvikoandClaude Opus 5.5 authored and Irakli Kourtis committed
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{
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"title": "Contact patch test in plane strain: two blocks with non-matching edges in uniaxial compression (implicit mortar contact)",
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"kind": "theory",
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"source": "The contact patch test: a contact discretisation must transmit a uniform pressure across a non-matching interface exactly; Taylor & Papadopoulos, On a patch test for contact problems in two dimensions, Nonlinear Computational Mechanics, Springer 1991, 690–702; passed by 2D mortar methods: Yang, Laursen & Meng, IJNME 62:1183–1225, 2005. Uniaxial stress between frictionless supports is uniform in both blocks",
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"quantities": [
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{
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"id": "small_lower_corner",
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"quantity": "Small: σ_yy of element 4 over that of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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4
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "small sliding is linear and its edge mortar integrals exact (degree-3 Gauss on the overlaps): the uniform stress is the discrete solution up to the convergence tolerance"
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},
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{
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"id": "small_lower_far",
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"quantity": "Small: σ_yy of element 6 over that of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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6
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "as for the corner element"
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},
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{
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"id": "small_lower_base",
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"quantity": "Small: σ_yy of element 1 over that of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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1
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "as for the corner element"
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},
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{
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"id": "small_upper_top",
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"quantity": "Small: σ_yy of element 10 over that of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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10
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "as for the corner element"
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},
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{
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"id": "small_lateral",
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"quantity": "Small: σ_xx of element 6 over its σ_yy (uniaxial stress: 0)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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6
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]
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},
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"pointer": "/elements/0/stress/0/0",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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6
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 0.0,
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"tolerance": {
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"absolute": 1e-06
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},
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"reason": "a frictionless interface (and symmetry plane) leaves the blocks free to expand sideways; the non-matching interface must not hold them"
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},
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{
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"id": "small_pressure",
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"quantity": "Small: the contact pressure at the secondary node (0.5, 1) over −σ_yy of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"at": [
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[
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0.5,
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1.0,
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0.0
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]
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]
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},
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"pointer": "/nodes/0/contact/0/pressure",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"scale": -1.0
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface"
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},
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{
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"id": "small_force",
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"quantity": "Small: the integrated contact pressure over the reaction at the pushed top",
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"measure": {
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"from": "small",
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"pointer": "/contacts/0/normal_force",
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"over": {
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"command": "results.reactions",
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"params": {
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"step": "Small",
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"sets": [
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"top"
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]
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},
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"pointer": "/sets/0/force/1",
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"scale": -1.0
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "the pressure carries the whole load across the interface (both are totals over the thickness)"
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},
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{
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"id": "finite_lower_corner",
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"quantity": "Finite: σ_yy of element 4 over that of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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4
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 0.0002
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},
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"reason": "finite sliding is exact up to the geometric terms of the gaps the augmentation leaves (10⁻⁵ of the element thickness) and the default convergence tolerances"
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},
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{
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"id": "finite_lower_far",
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"quantity": "Finite: σ_yy of element 6 over that of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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6
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 0.0002
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},
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"reason": "as for the corner element"
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},
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{
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"id": "finite_lower_base",
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"quantity": "Finite: σ_yy of element 1 over that of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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1
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 0.0002
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},
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"reason": "as for the corner element"
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},
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{
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"id": "finite_upper_top",
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"quantity": "Finite: σ_yy of element 10 over that of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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10
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 0.0002
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},
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"reason": "as for the corner element"
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},
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{
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"id": "finite_lateral",
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"quantity": "Finite: σ_xx of element 6 over its σ_yy (uniaxial stress: 0)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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6
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]
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},
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"pointer": "/elements/0/stress/0/0",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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6
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]
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},
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"pointer": "/elements/0/stress/0/1"
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}
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},
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"value": 0.0,
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"tolerance": {
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"absolute": 0.0002
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},
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"reason": "a frictionless interface (and symmetry plane) leaves the blocks free to expand sideways; the non-matching interface must not hold them"
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},
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{
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"id": "finite_pressure",
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"quantity": "Finite: the contact pressure at the secondary node (0.5, 1) over −σ_yy of element 7",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"at": [
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[
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0.5,
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1.0,
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0.0
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]
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]
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},
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"pointer": "/nodes/0/contact/0/pressure",
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"over": {
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"command": "solve.probe",
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"params": {
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"step": "Finite",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1",
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"scale": -1.0
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 0.0002
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},
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"reason": "the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface"
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},
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{
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"id": "finite_force",
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"quantity": "Finite: the integrated contact pressure over the reaction at the pushed top",
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"measure": {
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"from": "finite",
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"pointer": "/contacts/0/normal_force",
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"over": {
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"command": "results.reactions",
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"params": {
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"step": "Finite",
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"sets": [
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"top"
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]
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},
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"pointer": "/sets/0/force/1",
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"scale": -1.0
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}
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},
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"value": 1.0,
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"tolerance": {
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"relative": 0.0002
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},
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"reason": "the pressure carries the whole load across the interface (both are totals over the thickness)"
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},
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{
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"id": "small_stress",
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"quantity": "Small: σ_yy of element 7 (plane strain uniaxial stress E ε/(1 − ν²) = −10.989 for the 1% squeeze)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Small",
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"elements": [
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7
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]
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},
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"pointer": "/elements/0/stress/0/1"
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},
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"value": -10.989010989010989,
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"tolerance": {
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"relative": 0.0001
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},
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"reason": "the squeeze less the penetration the augmented Lagrangian leaves (up to 10⁻⁵ of the element thickness)"
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}
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]
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}
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