# Archard Wear Equation

### Revision for “Archard Wear Equation” created on April 19, 2022 @ 17:23:36

Title Archard Wear Equation The importance of wear losses leads to considerable effort in establishing theories and predictive models of wear. Meng and Ludema  [1] have identified 182 equations for different types of wear. Among them were empirical relations, contact mechanics-based approaches, such as Archards model, and equations based on material failure mechanisms, which were found to get more popular recently according to authors. In this review, empirical equations won’t be considered, as they are applicable for very specific range of parameters. No unified fundamental theory of wear was established so far, and as a consequence, there is no unique wear model, applicable in all cases. One of the most famous and frequently used wear equations was developed by Holm and Archard in 1953[2]. The model considers adhesive wear and assumes the sliding spherical asperities to deform fully plastically in contact. The area of contact then is circular with the contact area equal to $\pi a^2$ , where $a$ is the radius. The mean contact pressure in this case equal to hardness of the softer material, and thus, $H = P/{\pi a^2}$ . After the asperity slides a distance of $2a$, it is released from the contact and there is a probability $K$, that debris will form. It is assumed, that if debris is formed, it is formed as a hemisphere with the radius $a$, having a volume of $2\pi a^3/3$. Then the wear volume per sliding distance $2a$  is $W = K\pi a^2/3$, and hence, as $\pi a^2=P/H$, $W = K*P/{3H}$. Introducing $k=K/3$, the total wear volume for a sliding distance $s$, equals to $V_T=W*s=k*P/H*s$. The coefficient $k$ is known as a wear coefficient and is frequently used to compare the material wear resistance[2,3]. Most of the times, it is estimated experimentally. Although the Archard’s equation was developed for the adhesive wear, it is widely used for modeling of abrasive, fretting and other types of wear[4]. It should be noted that Archard equation is often applied on a local level. For that, the Archard equation is first formally divided by the area $A$:

$\begin{eqnarray} \label{complete_sys1} V_T/A = k*P/A/H*s => h = k *P_c/H*s \end{eqnarray}$

where $h, P_c$ are the local wear depth and contact pressures. Further, this equation is differentiated in time and the equation takes the following form:

$\begin{eqnarray} \label{complete_sys1} \frac{\partial h}{\partial t} = k *P_c/H*\frac{\partial s}{\partial t} => \frac{\partial h}{\partial t} = k *P_c/H*v \end{eqnarray}$

where $v$ is the sliding speed. This equation can be used to calculate wear locally if the contact pressure is known and is also applied to track the evolution of the surface roughness in rough contacts. This approach was implemented in Tribology Simulator (a stand-alone free to download software).

A chart linking the specific wear coefficients and friction is given below: [caption id="attachment_29683" align="aligncenter" width="578"] Friction coefficients and specific or volumetric wear rate map of tribological materials, [5][/caption]

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April 19, 2022 @ 17:23:36 Alex First
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