1. Introduction

Usually, strength reveals the mechanical properties of a kind of material or a component to resistant to fracture and over deformation. Based on the variation characteristics of stress endured by the material or components, the strength can be classified into static strength and variable stress intensity [1, 2]. In the real application, different materials are used for manufacturing mechanical components that work under contact load. According to the contact form and stress level induced by the load, the contact form can be classified into line contact as shown in

Figure 1. Diagram for different contact forms of mechanical parts.

Figure 1(a) and (b) and point contact as shown in Figure 1(c) [1–3].The contact line or point will be transformed to load-bearing area [3, 4]. For actual application, the contact between gears, rolls in mill, rail/wheel, roller bearing and contact of bolt and holes are all under line contact form; while the ball and ring in bearing are under the point contact form [1–4].

As the strength is the capability of material enduring the load or stress, failure will occur if the load or stress is beyond the level that it can bear [4, 5]. The failure modes of material working under contact load show different forms with the different stress vibration characteristics such as plastic deformation, surface pitting, surface wear and crushing of contact surface. Typically, the plastic deformation and crushing of contact surface are both resulted from the stress which is exceeding the yield limit of the material. The surface pitting and wear usually are considered to be formed by the accumulation effect induced by varying stress. For ensuring the strength of material working under stress with different variation characteristics, different manner should be used to assess the safety of the material.

#### 2. Static contact strength

#### 2.1 Static contact stress

According to the classical elasticity theory, the deformation of material can be divided into three mechanical states: elastic state, plastic state and fracture. Some materials used for manufacturing the components works under static contact state, are not allowed have plastic deformation due to the requirement of working reliability and transmission accuracy. For the materials used in the manufacturing of the components working under contact state, most of them cannot work properly with obvious plastic deformation.

According to the different contact forms shown in Figure 1, the contact stress between two cylinders loaded by force F as shown in Figure 2(a) can be expressed as below [1, 3]:

Contact Strength of Material DOI: http://dx.doi.org/10.5772/intechopen.90228

$$
\sigma\_{\text{Hmax}} = \frac{4}{\pi} \frac{F}{2ab} = \sqrt{\frac{F}{\pi b} \frac{\left(\frac{1}{\rho\_1} \pm \frac{1}{\rho\_2}\right)}{\frac{1-\mu\_1^2}{E\_1} + \frac{1-\mu\_2^2}{E\_2}}}\tag{1}
$$

The half-width a of contact area along the circumference is [3]:

$$a = \sqrt{\frac{4F}{\pi b} \frac{\frac{1-\mu\_1^2}{E\_1} + \frac{1-\mu\_2^2}{E\_2}}{\left(\frac{1}{\rho\_1} \pm \frac{1}{\rho\_2}\right)}}\tag{2}$$

where ρ<sup>1</sup> and ρ<sup>2</sup> are the curvature radius of contact area of two parts; the plus sign '+' means outer surface contact as shown in Figure 1(a); the minus sign '�' means inner surface contact as shown in Figure 1(b); E<sup>1</sup> and E<sup>2</sup> are the Young's modulus of materials of two components in contact, respectively; μ<sup>1</sup> and μ<sup>2</sup> are the Poisson's ratio of materials of two components in contact, respectively.

The contact stress between two balls loaded by force F shown in Figure 2(b) can be expressed as equation below [3]:

$$\sigma\_{\text{Hmax}} = \frac{3}{2} \frac{F}{\pi c^2} = \frac{1}{\pi} \sqrt{6F \left( \frac{\left(\frac{1}{\rho\_1} \pm \frac{1}{\rho\_2}\right)}{\frac{1 - \mu\_1^2}{E\_1} + \frac{1 - \mu\_2^2}{E\_2}} \right)^2} \tag{3}$$

The radius c of contact area is [3]:

$$\mathcal{L} = \left[ \frac{\mathbf{3F} \frac{1 - \mu\_1^2}{E\_1} + \frac{1 - \mu\_2^2}{E\_2}}{\left( \frac{1}{\rho\_1} \pm \frac{1}{\rho\_2} \right)} \right]^{1/3} \tag{4}$$

where c is the radius of contact area; the meaning of other symbols are same to that mentioned before.

Figure 2. Diagram for stress distribution of different contact form.

#### 2.2 Static extrusion stress

For materials used in bolt connection, pin connection and key connection shown like in Figure 3 mostly transfer the load by extrusion of match surface. Under contact form like the one shown in Figure 3, the main failure mode of the material located in the contact area is crushing. Based on the assumption of the stress distribution in the contact area, the stress between two match surfaces under extrusion load can be expressed as below:

$$
\sigma\_{\overline{P}} = \frac{F}{A} \tag{5}
$$

where F is the load acting on the two components and the acting direction of it is perpendicular to the bearing surface. A is the area size of match surface between two components when the bearing surface is flat or the projection area size of match surface between two components when the bearing surface is not flat.

#### 2.3 Condition of static contact strength

For the linear state of stress, the condition of strength can be recorded as the following form:

$$\sigma \le [\sigma] = \frac{\sigma\_{\text{lim}}}{[\text{S}]} \tag{6}$$

where σ is the maximum stress active in a part or structure; [σ] is the admissible stress level which can be determined by σlim=[S]; σlim is the ultimate stress of material; and the [S] > 0 is termed the strength safety factor.

The materials for manufacturing the components' transmission load usually fails in the form of plastic deforms under static load. So the condition of materials under contact conditions, as shown in Figure 1(a) and (b) or Figure 2, is:

Contact Strength of Material DOI: http://dx.doi.org/10.5772/intechopen.90228

$$
\sigma\_{H\max} \le [\sigma\_H]\_{\max} = \frac{\sigma\_{\lim}}{[\mathcal{S}]} \tag{7}
$$

The condition of strength for the components enduring load by extrusion of match surface is:

$$
\sigma\_p \le \begin{bmatrix} \sigma\_p \end{bmatrix} = \frac{\sigma\_{\text{lim}}}{\text{[S]}} \tag{8}
$$

For different types of materials, the ultimate stress σlim is different. If a material is not allowing the transition to the plastic state, then σlim ¼ σy; where σ<sup>y</sup> is the yield strength of material. If the material is of brittle type, then the σlim ¼ σb. Here, σ<sup>b</sup> is the ultimate strength of the material, which can be obtained by standard tension test.

The safety factor [S] can be determined based on the safety requirement of components in the actual application conditions.
