Hertzian contact - Revision history

E6WikiAdmin at 19:57, 22 October 2022

2022-10-22T19:57:11Z

2022-10-22T19:56:47Z

‎MLSM numerical solution for halfplane contact

2018-03-19T21:13:09Z

‎FreeFem++ numerical solution

2017-10-09T23:11:11Z

‎Contact of Cylinders - the Hertz problem

2017-10-03T06:23:02Z

2017-10-03T06:20:45Z

‎Contact of cylinders under partial slip

2017-10-03T06:20:22Z

‎MLSM numerical solution for bulk load

2017-09-08T08:38:52Z

‎Numerical solution

2017-09-08T07:55:49Z

‎MLSM numerical solution for halfplane contact

2017-09-07T10:37:41Z

‎MLSM numerical solution for bulk load

← Older revision		Revision as of 19:57, 22 October 2022
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		+	{{Box-round\|title= Related papers \|
		+	[https://e6.ijs.si/ParallelAndDistributedSystems/publications/32230439.pdf J. Slak, G. Kosec; Adaptive radial basis function-generated finite differences method for contact problems, International journal for numerical methods in engineering, vol. 119, 2019 [DOI: 10.1002/nme.6067]]
		+	}}
		+
		+
	Click on [[Solid Mechanics]] to go back.		Click on [[Solid Mechanics]] to go back.

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 [[File:mss_contours.png|900px]]
-= MLSM numerical solution for halfplane contact =
+= Meshless numerical solution for halfplane contact =
 We are solving the equation $$(\lambda + \mu) \nabla (\nabla \cdot \b{u}) + \mu \nabla^2 \b{u} = 0$$

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 *Specimen length: $L = 40$ mm
 *Specimen height: $H = 10$ mm
 We assume that both the pad and specimen are from the same material, therefore the combined modulus is \[E^* = \frac{E}{2(1-\nu^2)}.\] According to Pereira et al. (2016) contact width is now defined with the specimen height in the denominator:
-\[b = 2\sqrt{\frac{PR}{H\pi E^*}}\]
+\[b = 2\sqrt{\frac{PR}{t\pi E^*}}\]
-\[p_0 = \sqrt{\frac{PE^*}{H\pi R}}\]
+\[p_0 = \sqrt{\frac{PE^*}{t\pi R}}\]
 Out of simplicity we only use traction boundaries at the top surface. On the left, right and bottom sides of the specimen Dirichlet boundaries are used. The center of the coordinate system is placed at $(x,y) = (L/2,0)$. The axises therefore go from $-L/2$ to $L/2$ in $x$-direction and from $0$ to $H$ in $y$-direction.

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 The solutions were verified and graphs were generated using this mathematica notebook: [[:File:hertzian_analytical.nb]]
 Copy-pasteable Matlab code:
 <syntaxhighlight lang="matlab" line>
 function [sxx, syy, sxy] = analytical(x, z, b, p0)
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 sxy = p0 / b * n .* ((m2 - z.^2) ./ mpn);
 end
 </syntaxhighlight>

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 = FreeFem++ numerical solution =
-For the numerical solution in FreeFem++ we choose parameters similar to those in Pereira et al. (2016): <ref name=Pereira></ref>
+For the numerical solution in FreeFem++ we choose parameters similar to those in Pereira et al. (2016): <ref> K. Pereira et al., On the convergence of stresses in fretting fatigue, Materials
 *Modulus of elasticity: $E = 72.1$ GPa

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 $s_{zz} = \frac{b p_0 z^3}{x^4}+O(x^{-5})$,
 $s_{xz} = \frac{b p_0 z^2}{x^3}+O(x^{-5})$.
 = FreeFem++ numerical solution =

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 [[File:hertzian_psd_contour_refined.png|800px]] -->
 =References=
 <references/>

← Older revision		Revision as of 08:38, 8 September 2017
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	[[File:fwo_cases.png\|800px]]		[[File:fwo_cases.png\|800px]]
		+
		+	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.		A contour plot of von Mises stress in the specimen is presented below.

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 [[File:hertzian_solution_deformed_vm.png|600px]]
 <!--
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 [[File:hertzian_midplane_nonrefined.png|800px]]
 The error decreases as the density of nodes increases, up to some limit when the system becomes too unstable.

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 * 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 $(2 \mu+\lambda) \frac{\partial v}{\partial y}(L/2, y) + \lambda \frac{\partial u}{\partial x}(L/2, y) = 0$.
 == Phsyical parameters ==
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 * $e = \frac{a \sigma_{ax}}{4 \mu p_{max}} = \unit{0.041197}{mm}$, eccentricity due to axial load
-== MLSM solution ==
+== Numerical solution ==
-[[File:fwo_cases.png|800px]][[File:fwo_bc.png|600px]]
+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.
 =References=
 <references/>