2. Laws of wear

In its initial form, Archard’s law [bib1] expresses, for a configuration of adhesive wear, in sliding, a relationship between the volume used and quantities characteristic of contact:

\(V=\frac{k\text{.}\parallel {F}_{n}\parallel \text{.}L}{H}\)

where	\(V\)	:	worn volume,
	\(k\)	:	dimensionless wear coefficient,
	\(\parallel {F}_{n}\parallel\)	:	modulus of the normal contact force, assumed to be constant,
	\(L\)	:	slipped length,
	\(H\)	:	hardness.

The coefficient \(k\) is different for each of the bodies present. It depends on the geometric and thermodynamic conditions during contact.

It has been shown that Archard’s law can be extended to other mechanisms, in a dominant way. By redefining certain parameters, the preceding equation can be written as:

\(V=K\text{.}W\)

where	\(K\)	:	equals \(\frac{k}{H}\),
	\(W\)	:	equals \(\parallel {F}_{n}\parallel \text{.}L\).

\(W\) has the dimension of a job. By convention, it is called « usury work. »

In the case where the normal contact force varies over time (for example, in an impact-slip situation, \(\parallel {F}_{n}\parallel\) has very strong variations of short duration during shocks), the definition of \(W\) becomes:

\(W=\underset{{t}_{0}}{\overset{{t}_{1}}{\int }}\parallel {F}_{n}\parallel \cdot \parallel {V}_{t}\parallel \cdot \mathrm{dt}\)

where	\(W\)	:	wear and tear work,
	\(\parallel {F}_{n}\parallel\)	:	modulus of normal force during contact,
	\(\parallel {V}_{t}\parallel\)	:	modulus of the sliding speed during contact,
	\({t}_{0}\)	:	calculation start time,
	\({t}_{1}\)	:	moment of completion of the calculation.

Therefore, by analogy with the usual laws of mechanics, it is possible to define « wear power » by asking:

\(P=\parallel {F}_{n}\parallel \cdot \parallel {V}_{t}\parallel\)

where \(P\): wear power.

If a steady state is reached, the wear power is assumed to be constant over time. In order to ensure this stationarity, the interval \([{t}_{\mathrm{0,}}{t}_{1}]\) can be divided into several blocks in the POST_USURE [U4.67.03] operator. For each of these blocks, it is necessary to check that the wear power changes little (From a point of view, the use of the wear laws below assumes that the wear power is constant).

2.1. Law of usury “ARCHARD”

The law is linear [bib1]: \(V=K\cdot P\cdot t\)

where	\(V\)	: wear volume,
	\(K\)	: wear coefficient,
	\(P\)	: wear power,
	\(t\)	: time interval.

The coefficient \(K\) is provided by the user or taken from a database (see [§3]). It is different for the two bodies present and depends on the geometric and thermodynamic conditions in the contact. The time interval \(t\) used to calculate the wear does not correspond to the actual simulation time but to the time interval over which the user wishes to evaluate the wear.

2.2. Law of usury “KWU_EPRI”

The model’s approach consists in determining a wear coefficient \(K\), within the meaning of Archard’s law, taking into account the particular conditions of the contact studied [bib2].

The normal forces \({F}_{i}(N)\) are divided into 5 classes, as well as the \({V}_{j}(m/s)\) sliding speeds.

25 classes are obtained, the identification of which is indicated as follows:

For a given calculation, the percentages obtained for each of the 25 classes are determined.

The treatment is carried out by applying weighting factors appropriate for each class, which take into account its particular contribution to the overall wear process.

In the case of pure impacts (classes 1.1 to 1.5), the contribution of these classes is modelled using a weighting factor \({m}_{{h}_{\mathrm{ij}}}\) defined by:

\({m}_{{h}_{\mathrm{ij}}}={k}_{1}\cdot k\cdot {(\frac{{F}_{i}}{c})}^{3}\)

where	\({m}_{{h}_{\mathrm{ij}}}\)	:	dimensionless impact-work hardening intensity factor
	\({k}_{1}\)	:	dimensional correction coefficient
	\(k\)	:	experimental adimensional constant
	\(c\)	:	experimental adimensional constant
	\({F}_{i}\)	:	mean normal force value for class ij

In the case of slippage (class 1.1 and classes 2.1 to 5.5), the contribution of these classes is modelled using a weighting factor \({m}_{{w}_{\mathrm{ij}}}\) defined by:

\({m}_{{w}_{\mathrm{ij}}}={k}_{2}\cdot {F}_{i}\cdot {({V}_{j})}^{2}\)

where	\({m}_{{w}_{\mathrm{ij}}}\)	:	dimensionless slip wear intensity factor
	\({k}_{2}\)	:	dimensional correction coefficient
	\({F}_{i}\)	:	mean normal force value for class ij
	\({V}_{j}\)	:	mean value of the sliding speed for class ij

It is then necessary to calculate the weighted percentages for each class of the two categories impact-work hardening and sliding wear:

\({P}_{{h}_{\mathrm{ij}}}={m}_{{h}_{\mathrm{ij}}}\cdot {p}_{\mathrm{ij}}\)

\({P}_{{w}_{\mathrm{ij}}}={m}_{{w}_{\mathrm{ij}}}\cdot {p}_{\mathrm{ij}}\)

where \({p}_{\mathrm{ij}}\) is the percentage of elements in class \(\mathrm{ij}\).

This leads to an overall wear intensity factor

\(w=\frac{{(\sum {P}_{{w}_{\mathrm{ij}}})}^{2}}{\sum {P}_{{h}_{\mathrm{ij}}}+\sum {P}_{{w}_{\mathrm{ij}}}}\)

The global intensity factor \(w\) is used as a correction factor for the wear coefficient in the sense of the law of ARCHARD according to the expression:

\({K}_{\mathrm{KWU}}={k}_{r}\cdot w/{w}_{r}\)

\(V={K}_{\mathrm{KWU}}\cdot P\cdot t\)

where \({k}_{r}\):	is the reference wear coefficient obtained experimentally for conventional oscillating slip test conditions,
and \({w}_{r}\)	is the overall intensity factor evaluated for this same test.

2.3. Law of usury “EDF_MZ”

It is currently being developed for the sole case of control clusters.

The feedback shows that the kinetics of wear slows down over time \(t\); one way to take into account the observations is to express the worn volume in the form:

\(V=(\frac{{S}_{0}-S}{n})\cdot (1-{e}^{-\mathrm{nt}})+S\cdot t\)

where \({S}_{0}\) is the initial speed and \(S\) is the asymptotic wear rate (see below),

\(n\) is a model parameter.

The values of \(n\) and \(S\) are deduced from the feedback.

Tests on simulators, of short duration compared to that of a reactor operating cycle, show that the initial wear rate \({S}_{0}\) follows a law of the type:

\({S}_{0}=A\cdot {({P}_{0})}^{b}\)

where \({P}_{0}\) is the initial wear power

\(A\) and \(b\) are coefficients determined by tests on simulators [bib4]

The feedback shows that the wear rate reaches an asymptotic value \(S\). The previous relationship, observed on a simulator, is assumed to be valid for all moments of the wear phenomenon. This assumes a wear power \(P\) that makes it possible to reach \(S=A\cdot {(P)}^{b}\), for high values of time \(t\) (typically, one or more operating cycles).

The corresponding evolution of the worn volume as a function of time is of the form:

The used volume \(V\) calculated using the POST_USURE operator is written as:

\(V=(\frac{A\cdot {({P}_{0})}^{b}-S}{n})\cdot (1-{e}^{\mathrm{nt}})+S\cdot t\)

where	\(V\)	: wear volume,
	\({P}_{0}\)	: wear power calculated by the Code_Aster ®,
	\(A,b,S,n\)	: model coefficients defined above.

This model is described in detail by the reference [bib4].