2.1. Pressure elements and surface normal deflection in an elastic half-space

Consider a body that, because of its main features, can be approached to an elastic half-space, as the one shown in Figure 1a. A Cartesian coordinate system is defined over the surface of this body, which X and Y axes define a plane that is coincident with its surface, and the Z axis points inward him. A normal pressure distribution ð Þp is applied over the surface of the body, acting over an area that is denoted by S.

Now consider a generic point C within the area S, whose position is defined by the vector r0 ð Þ x; y; z , being z ¼ 0. Consider another point H in the surface of the body, whose position is defined by the vector rð Þ x; y; z , being z ¼ 0. The normal elastic deflection produced at a point H due to a normal pressure distribution applied over the area S can be determined by the superposition of the Boussinesq relation [3]:

$$\omega(r) = \frac{1-\nu}{2\pi G} \int\_{S} \frac{p(r')}{|r-r'|} dS \tag{1}$$

Figure 1. Pressure distributions applied over an elastic half-space.

Obtaining a generic closed-form solution for Eq. (1) is not possible, since it depends on the shape of the area S and on the considered pressure distribution. However, several closed-form solutions can be found in the literature for certain pressure distributions applied over areas with a specific shape (such as triangles, rectangles, hexagons, etc.).

Let's focus on the closed-form solution for Eq. (1) that Love [4] obtained for uniform pressure distributions acting over areas with rectangular shape, as the one shown in Figure 1b. From now on, this combination of shape and pressure distribution will be called pressure element, and will be denoted by Δ<sup>j</sup> . The area of the pressure element shown in Figure 1b is Aj ¼ 2a � 2b, and the uniform pressure distribution that acts over this area is p r<sup>0</sup> ð Þ¼ pj . Under these conditions, the closed-form solution for Eq. (1) is

$$
\omega(\mathbf{r}) = f\_j(\mathbf{r}) \cdot p\_j \tag{2}
$$

Finally, the displacement at any point of the surface of the body can be determined by superposition of the displacements produced at this point by uniform pressures acting over

Adaptive Mesh Refinement Strategy for the Semi-Analytical Solution of Frictionless Elastic Contact Problems

j¼1 pj ∙f j ð Þr

This methodology can be applied with different types of pressure elements, having different shapes and pressure distributions acting over them. However, using rectangular pressure

The solutions exposed in the previous section can be used to obtain the pressure distribution that is produced when two bodies are pressed together in the absence of friction. For such a purpose, it is necessary that the two bodies can be approached to elastic half-spaces in the

Consider two bodies 1 and 2 in its undeformed contact position, contacting at the initial point of contact OL (Figure 3a). At this point, a common tangent plane Π is defined, which is assumed to be so close to the surface of the bodies in the vicinity of the contact area that the deformation of the surfaces of both bodies can be referred to it in the linear small strain theory

A Cartesian coordinate system is defined with origin at point OL, being the local axis ZL normal to the plane Π and pointing inward the body 2. Consider a generic point Q in the plane

bodies, measured along ZL axis, is denoted by the function Bð Þr , which in the first instance is

The two bodies are pressed together in the absence of friction by the effect of the force FT (Figure 3b), causing a normal approach between them that is denoted by δ. Since penetration is physically inadmissible, a contact pressure distribution pð Þr is generated in the true contact area S that deforms the contacting bodies. In this way, elastic normal deflections are produced

h i (4)

http://dx.doi.org/10.5772/intechopen.72422

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� �, being zL <sup>¼</sup> 0. The gap between the two

<sup>ω</sup>ð Þ¼ <sup>r</sup> <sup>X</sup><sup>n</sup>

each pressure element of the mesh as

of elasticity.

elements has some advantages, which are discussed in [5].

Π, whose position is defined by the vector r xL; yL; zL

assumed to be smooth and continuous.

2.2. Semi-analytical method to solve frictionless elastic contact problems

vicinity of the area in which the contact between them is produced.

Figure 3. Contact between two bodies: (a) undeformed position and (b) deformed position.

where f <sup>j</sup> ð Þr is the influence coefficient of pressure element Δ<sup>j</sup> over the point H, which can be analytically determined as

$$\begin{split} f\_{j}(\mathbf{r}) &= \frac{1-\nu}{2\pi\mathbf{G}} \left\{ \mathbb{C} \cdot \ln \left[ \frac{A + \sqrt{A^{2} + \mathbf{C}^{2}}}{B + \sqrt{B^{2} + D^{2}}} \right] + A \cdot \ln \left[ \frac{\mathbb{C} + \sqrt{A^{2} + \mathbf{C}^{2}}}{D + \sqrt{A^{2} + D^{2}}} \right] + D \cdot \ln \left[ \frac{B + \sqrt{B^{2} + D^{2}}}{A + \sqrt{A^{2} + D^{2}}} \right] \\ &+ B \cdot \ln \left[ \frac{D + \sqrt{B^{2} + D^{2}}}{\mathbb{C} + \sqrt{B^{2} + \mathbf{C}^{2}}} \right] \right\} \end{split} \tag{3}$$

where coefficients A, B, C, and D are calculated as

$$\begin{aligned} A &= d\_y + b & C &= d\_x + a \\ B &= d\_y - b & D &= d\_x - a \end{aligned}$$

These pressure elements can be useful to determine the normal displacement produced at the surface of a body due to a non-uniform pressure distribution applied over a complex area. To illustrate this methodology, consider a complex area S, as the one shown in Figure 2a, over which an arbitrary pressure distribution is acting. To determine the displacement field produced by this pressure distribution, the area S is discretized into a mesh of n rectangular pressure elements Δj, as shown in Figure 2b. Then, the arbitrary pressure distribution is approached by assigning a uniform pressure value pj to each pressure element, as shown in Figure 1c.

Figure 2. Normal deflection produced by a complex pressure distribution.

Finally, the displacement at any point of the surface of the body can be determined by superposition of the displacements produced at this point by uniform pressures acting over each pressure element of the mesh as

$$\omega(\mathbf{r}) = \sum\_{j=1}^{n} \left[ p\_j f\_j(\mathbf{r}) \right] \tag{4}$$

This methodology can be applied with different types of pressure elements, having different shapes and pressure distributions acting over them. However, using rectangular pressure elements has some advantages, which are discussed in [5].
