Fretting fatigue case

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Introdution

This case is an extension of the classical Hertzian contact, as defined in Pereira et al. (2016) and described in detail in Hojjati-Talemi et al. (2014)

Description of the case and parameters

Case definition

We are solving the equation $$(\lambda + \mu) \nabla (\nabla \cdot \b{u}) + \mu \nabla^2 \b{u} = 0$$ on the domain $D = [-L/2, L/2] \times [-H/2, 0]$. Componentwise $\b{u} = [u; v]$. The boundary conditions are:

• top: $\vec{t}(x) = (q(x), -p(x))$ or componentwise $\mu \frac{\partial u}{\partial y}(x, 0) + \mu \frac{\partial v}{\partial x}(x, 0) = q(x)$ and $(2 \mu+\lambda) \frac{\partial u}{\partial x}(x, 0) + \lambda \frac{\partial v}{\partial y}(x, 0) = -p(x)$ .
• left: $\b{u} = 0$ or componentwise $u(-L/2, y) = v(-L/2, y) = 0$.
• bottom: up-down symmetry conditions: componentwise $\frac{\partial u}{\partial y}(x, -H/2) = 0, v(x, -H/2) = 0$.
• right: this part is traction free, ie. $\vec{t}(y) = 0$ or componentwise $\mu \frac{\partial u}{\partial y}(L/2, y) + \mu \frac{\partial v}{\partial x}(L/2, y) = 0$ and $(2 \mu+\lambda) \frac{\partial v}{\partial y}(L/2, y) + \lambda \frac{\partial u}{\partial x}(L/2, y) = 0$.

Illustration of the computational domain is presented below.

Phsyical parameters

Basic parameters are:

• $E = \unit{72.1}{GPa}$, Youngs modulus
• $\nu = 0.33$, Poissons ratio
• $L = \unit{40}{mm}$, length, width of the pad
• $H = \unit{4}{mm}$, height, thickness of the pad
• $\sigma_{ax} = \unit{100}{MPa}$, axial pressure
• $F = \unit{543}{N}$, normal force
• $Q = \unit{155}{N}$, tangential force
• $R = \unit{10}{mm}$ or $\unit{50}{mm}$, raduis of curvature of the cylindrical pad
• $COF = \mu = 0.3$ or $0.85$ or $2$, coefficient of friction

Derived parameters are, for choice of $R = \unit{10}{mm}$ and $\mu = 0.3$:

• $E^\ast = \frac{E}{2(1-\nu^2)} = \unit{40.4}{GPa}$, combined Young's modulus
• $p_{max} = p_0 = \sqrt{\frac{FE^\ast}{HR \pi}} = \unit{418.10407}{MPa}$, maximal normal pressure
• $a = 2 \sqrt{\frac{FR}{HE^\ast \pi}} = \unit{0.2067}{mm}$, semi contact width, contact region is $[-a, a] \times \{0\}$
• $c = a \sqrt{1 - \frac{Q}{\mu F}} = \unit{0.04504}{mm}$, stick zone semi width
• $e = \frac{a \sigma_{ax}}{4 \mu p_{max}} = \unit{0.041197}{mm}$, eccentricity due to axial load

Numerical solution

Comparison of this solution against FEM solution published in Pereira et al. (2016) is presented below.

File:Fwo cases.png

The data for this plot (only MLSM data) is available to download here: File:fwo_cases_data.csv.

A contour plot of von Mises stress in the specimen is presented below.

Note that contour of stress are similar to the ones in Hertzian contact above, but a significant concentracion of stresses under the contact area is present.