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Two cantilevers close a gap

A ContactFEA hex-mesh solution of a double-cantilever contact test. Both arms are fixed at a rigid wall; tip loads on the upper (slave) arm close a 5 mm face-to-face clearance (not the ~17 mm tip deflection at λ = 1). The load–displacement path is the Newton–Raphson history from MATLAB.

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Tip force F280 kN
Tip U₂—
Clearance (mesh)5 mm
Upper thickness8.75 mm

Appearance only; the FEA solution is unchanged.

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U₂ min (λ=1)—
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Load–displacement (ContactFEA path)

Physical setup

  • Rigid wall and clamp plates fix both cantilevers at X = −50.
  • Upper arm (slave) and lower arm (master) share the wall.
  • Initial clearance between the facing faces is 5 mm (upper bottom at Y = 5, lower top at Y = 0; upper thickness still 8.75 mm). Do not confuse clearance with tip deflection (~17.3 mm at λ = 1).
  • Loading is seven equal tip forces on the free-end top edge (40 kN each → ΣFY = −280 kN at λ = 1), shown as an equivalent uniform tip traction block (or optional nodal arrows).

What to notice

  • Both arms bend after the gap closes — contact pressure loads the lower arm. Before contact (~λ ≈ 0.08, tip |U₂| ≈ 5.2 mm) only the upper arm deflects.
  • Tip U₂ ≈ −17.3 mm at the load nodes (λ = 1) is deflection, not the clearance.
  • The F–δ curve softens until contact (~5.2 mm tip U₂), then stiffens as the patch grows.
  • Mesh at intermediate λ matches the F–δ tip U₂: before contact only the upper arm moves; after contact both arms share the closing motion. The chart is the true N–R path.
  • Compare with the smooth Hertz elliptic pressure.

Theory — ContactFEA beam example

Formulation used by the open-source MATLAB ContactFEA solver (Wang, Bai, Lu & Zuo, Adv. Eng. Softw. 199, 2025): surface-to-surface frictional contact with large deformation, penalty regularization, and Newton–Raphson. The mesh and load path on this page are that beam example (clearance enlarged to 5 mm for visualization).

Balance

Bodies without contact

\[\nabla\cdot\boldsymbol{\sigma}+\mathbf{b}=\mathbf{0}\ \text{in }\Omega^{(1)}\cup\Omega^{(2)}\]

Cauchy balance on slave and master continua, with prescribed traction on \(\Gamma_t\) and fixed displacement on \(\Gamma_u\) (here the wall at \(X=-50\)).

Contact

Signorini–Coulomb

\[g\ge 0,\quad t_N\le 0,\quad \mathbf{t}^{(1)}=-\mathbf{t}^{(2)}\]

Non-penetration, compression-only normal traction, and action–reaction on \(\gamma^{(1)}\cup\gamma^{(2)}\). Tangential response obeys Coulomb: stick if \(|\mathbf{t}_T|<\mu|t_N|\), slip if equality holds.

Weak form

Contact virtual work

\[\delta W_c=\int_{\gamma^{(1)}}(\delta\mathbf{u}^{(1)}-\delta\mathbf{u}^{(2)})^{\mathsf{T}}\mathbf{t}^{(1)}\,\mathrm{d}a^{(1)}\]

Added to the internal and external virtual work of both bodies. When the surfaces are separated, \(\delta W_c=0\); in contact, \(g=0\) and \(t_N<0\).

Kinematics

Ray-tracing gap

\[g=\bigl(\mathbf{x}^{(2)}-\mathbf{x}^{(1)}\bigr)^{\mathsf{T}}\mathbf{n}^{(1)}\]

Master contact point found by projecting the slave point along the slave outward normal (Zimmerman surface-to-surface). Relative slip velocity defines the friction direction \(\mathbf{s}^{(1)}\).

Constitutive

Penalty (slip / stick)

\[\mathbf{t}^{(1)}_{\mathrm{slip}}=\varepsilon g\,(\mathbf{n}^{(1)}+\mu\mathbf{s}^{(1)}),\quad \mathbf{t}^{(1)}_{\mathrm{stick}}=\varepsilon\,\mathbf{g}_s\]

Slip uses the scalar gap; stick uses the vectorial gap from the previous master point. The paper uses \(\varepsilon=10^6\) (order of \(E\)); too small → penetration, too large → more NR iterations.

State

Slip criterion

\[\Phi=|\mathbf{t}_T^{(1)}|-\mu|t_n|\]

\(\Phi<0\) stick, \(\Phi=0\) slip. First contact is often started in friction slip for stability; later increments trial stick and correct with \(\Phi\).

Example

Geometry and mesh

\[\text{upper }50\times 15\times 8.75,\ \text{lower }50\times 30\times 10\ \mathrm{mm}\]

Upper (slave) 20×6×4 hexes; lower (master) 13×8×3 hexes. Left ends fixed. Paper tip load: 40 kN nodal forces on the upper free end (here seven equal forces, \(\Sigma F_Y=-280\,\mathrm{kN}\) at \(\lambda=1\)).

Example

Material and friction

\[E=2.1\times 10^5\,\mathrm{MPa},\ \nu=0.3\]

Steel-like isotropic elasticity. With \(\mu=0.6\) versus frictionless, the paper reports smaller tip deflection, lower peak Mises stress and contact traction, and less relative slip — friction enlarges the stick patch. See also Newton iteration and Contact and Friction.