Revisiting the Persson theory of elastoplastic contact: A simpler closed-form
solution and a rigorous proof of boundary conditions

Yang Xua111Corresponding author<EMAIL_ADDRESS>Xiaobao Lib, Daniel M.
Mulvihillc
aSchool of Mechanical Engineering, Hefei University of Technology, Hefei,
230009, China

bSchool of Civil Engineering, Hefei University of Technology, Hefei, 230009,
China

cMaterials and Manufacturing Research Group, James Watt School of Engineering,
University of Glasgow, Glasgow, G12 8QQ, UK

Abstract

Persson’s theory of contact is extensively used in the study of the purely
normal interaction between a nominally flat rough surface and a rigid flat. In
the literature, Persson’s theory was successfully applied to the elastoplastic
contact problem with a scale-independent hardness $H$. However, it yields a
closed-form solution, $P(p,\xi)$, in terms of an infinite sum of sines. In
this study, $P(p,\xi)$ is found to have a simpler form which is a
superposition of three Gaussian functions. A rigorous proof of the boundary
condition $P(p=0,\xi)=P(p=H,\xi)=0$ is given based on the new solution.

Keywords: Persson’s theory of contact; Rough surface; Elastoplastic contact;
Hardness model

##  1 Introduction

In the study of the purely normal contact between a nominally flat rough
surface and a rigid flat, there are two major theoretical approaches, namely,
the multi-asperity contact model (e.g., the Greenwood and Williamson (GW)
model[1]) and Persson’s theory of contact [2]. In the GW model, as well as
other related multi-asperity contact models, the physical basis for the rough
surface contact is clear, i.e., solid interaction occurs at the summits of
higher asperities. Therefore, multi-asperity contact models rely on the
analytical asperity models (e.g., Hertzian contact theory [3]) or empirical
models (e.g., Jackson and Green model [4]). One major assumption adopted in
the multi-asperity contact models is that each contacting asperity can be
studied individually without considering its interaction with nearby
asperities. However, this assumption restricts the application of the multi-
asperity contact models in the low load range where asperity interaction and
coalescence are largely avoided. So far, several attempts have been made to
overcome this limitation [5, 6, 7, 8].

A major new development in this field arrived some two decades ago with the
publication of Persson’s theory of contact[2]. Since then, it has gradually
become one of the dominant theoretical approaches in the area of rough surface
contact. Here, unlike multi-asperity contact models, the rough surface contact
problem is solved by Persson’s theory in the probability domain. Assuming the
contact pressure, $p$, within the real area of contact is a stochastic
variable, Persson derived a diffusion equation of a scale-dependent
probability density function (PDF) $P(p,\xi)$ where the scale
$\xi=\kappa_{s}/\kappa_{l}$ is the ratio of the upper cut-off frequency
$\kappa_{s}$ to the lower cut-off frequency $\kappa_{l}$ associated with the
power spectrum density of the rough surface topography. As the scale $\xi$
increases, higher frequency components are added to the roughness topography.
As a matter of fact, the corresponding probability density function $P(p,\xi)$
is broadened and evolves following the diffusion equation [2]. Compared with
the multi-asperity contact models, the asperity interaction is effectively
included in Persson’s theory of contact. Moreover, the theory covers the
entire load range of rough surface contact from the first touch to nearly
complete contact.

Persson’s theory of contact was first applied by Persson [2, 9] to the
elastoplastic contact problem. To account for yielding, the contact pressure
is truncated at $p=H$ where $H$ is a scale-independent hardness. Therefore,
the elastoplastic material model used in Persson’s theory [2, 9] is indeed a
constant hardness model. It should not be confused with the elastoplastic
models (e.g., elastic-perfectly plastic model) used in the solid mechanics
community. The probability of the elastic portion of the contact pressure is
assumed to satisfy the diffusion equation. Persson’s theory of contact yields
an analytical solution of $P(p,\xi)$ in a Fourier series (in terms of an
infinite sum of sines). An attempt to include the size-dependent plasticity
has been made by Persson [10] using a scale-dependent hardness $H(\xi)$.
Venugopalan _et al._ [11] found good agreement between Persson’s theory and
discrete dislocation plasticity modelling. The agreement deteriorates as root
mean square roughness increases. This disagreement probably arises for two
possible reasons [11]: (1) Persson’s theory does not account for the strain
hardening, and (2) an inaccurate estimation of the hardness using
$H=3\sigma_{Y}$ where $\sigma_{Y}$ is the initial yield strength.

In the present study, Persson’s theory of elastoplastic contact with a scale-
independent hardness is revisited. Inspired by the mirror Gaussian solution
[12, 13] of the linear elastic contact problem, a simpler analytical solution
of $P(p,\xi)$ is found which is a superposition of three Gaussian functions.

##  2 Elastoplastic Contact

Figure 1: Schematic of the deformed interface, the elastoplastic contact
pressure and the corresponding PDF $P(p,\xi)$.

Consider a purely normal contact between a nominally flat rough surface and a
rigid flat, see Fig. 1. The contact pair is subjected to a uniform normal
traction $\bar{p}$ applied remotely. The scale-dependent PDF of the contact
pressure, $P(p,\xi)$, is discontinuous at $p=0$ and $p=H$, i.e.,
$P(p=0,\xi)\neq P(p\to 0^{+},\xi)$ and $P(p=H,\xi)\neq P(p\to H^{-},\xi)$.
This is because the non-contact area and plastically deformed area is
associated with $p=0$ and $p=H$, respectively. Inspired by a similar form
proposed by Persson [14] for an adhesive contact problem, a discontinuous PDF
$P(p,\xi)$ can be formulated as follows

$P(p,\xi)=\delta(p)A_{nc}^{*}(\xi)+\delta(p-H)A_{pl}^{*}(\xi)+\left[H(p)-H(p-H)\right]P_{0}(p,\xi),$
(1)

where $\delta(p)$ and $H(p)$ are _Dirac_ and _Heaviside_ functions,
respectively. The first two terms in the right hand side of Eq. (1) represent
the PDF of contact pressure $p=0$ and $p=H$, respectively. The last term is
associated with the PDF of the elastic contact pressure $p\in(0,H)$ which is
denoted by $P_{0}(p,\xi)$. Outside $p\in(0,H)$, $P_{0}(p,\xi)=0$.
$A_{nc}^{*}(\xi)$, $A_{el}^{*}(\xi)=\int_{0^{+}}^{H^{-}}P_{0}(p,\xi)dp$ and
$A_{pl}^{*}(\xi)$ are the ratios of the non-contact, elastic and plastic
contact areas to the nominal contact area, respectively. The PDF $P(p,V)$ is
automatically vanishing outside the nominal range $p\in[0,H]$. A schematic
plot of $P(p,V)$ can be found in Fig. 1.

Based on Persson’s theory of elastoplastic contact [2, 9], the PDF of elastic
contact pressure $P_{0}(p,\xi)$ satisfies the following diffusion equation:

$\frac{\partial}{\partial\xi}P_{0}(p,\xi)=\frac{1}{2}\frac{dV}{d\xi}\frac{\partial^{2}}{\partial
p^{2}}P_{0}(p,\xi)~{}~{}~{}p\in(0,H)~{}~{}~{}\xi\geq 1,$ (2)

where $V=\langle p_{c}^{2}\rangle$ is the variance of the elastic contact
pressure $p_{c}$ at the stage of complete contact [12]. For a self-affine
isotropic roughness with a deterministic power spectrum density [15, 16],
$V(\xi)$ monotonically increases with the scale $\xi$. Neglecting the finite
size effect (i.e., $\kappa_{l}\to 0$), roughness vanishes so that
$V(\xi=1)=0$.

Assuming $V(\xi)$ monotonically increases with $\xi$ with $V(\xi=1)=0$, then
$V(\xi)$ is a one to one mapping from $\xi$ to $V$, and we can replace $\xi$
with $V$ in Eq. (2), i.e., $P(p,\xi)=P(p,V)$. The corresponding diffusion
equation is exactly what Manners and Greenwood derived [12]

$\frac{\partial}{\partial V}P_{0}(p,V)=\frac{1}{2}\frac{\partial^{2}}{\partial
p^{2}}P_{0}(p,V)~{}~{}~{}p\in(0,H)~{}~{}~{}V\geq 0.$ (3)

Since the finite size effect is neglected, we can expect that the contact
surface becomes perfectly flat with an infinite size as $V(\xi=1)=0$ and solid
contact occurs over the entire nominal contact area with a uniform pressure of
magnitude $\bar{p}$. Therefore, the initial condition of Eq. (3) is simply a
Dirac function [9]

$P_{0}(p,V=0)=\delta(p-\bar{p}).$ (4)

The corresponding boundary conditions at $p=0^{+}$ and $p=H^{-}$ was given by
Persson with a less rigorous proof [9, 10]:

$P_{0}(p=0^{+},V)=P_{0}(p=H^{-},V)=0.$ (5)

Eqs. (3-5) are solved below by two approaches: the former one is the Fourier
series method proposed by Persson [9], and the latter one is a new method we
propose here based on the superposition of three Gaussian distributions. A
rigorous proof of the boundary conditions in Eq. (5) is given in Section 3.

###  2.1 Fourier series method

A general solution of Eq. (3) maybe given in Fourier series form as follows:

$P_{0}(p,V)=\sum_{n=1}^{\infty}B_{n}(V)\sin\left(\frac{n\pi p}{H}\right),$ (6)

which automatically satisfies the boundary conditions in Eq. (5). Substituting
Eq. (6) into Eqs. (3) and (4), we can have an explicit form of $P_{0}(p,V)$ in
terms of an infinite sum of sines[2, 9]:

$P_{0}(p,V)=\frac{2}{H}\sum_{n=1}^{\infty}\sin\left(\frac{n\pi\bar{p}}{H}\right)\sin\left(\frac{n\pi
p}{H}\right)\exp\left[-\frac{1}{2}\left(\frac{n\pi}{H}\right)^{2}V\right].$
(7)

###  2.2 Superposition method

The superposition method relies on a specific solution $\tilde{P}_{0}(p,V)$ of
the diffusion equation [12]

$\tilde{P}_{0}(p,V)=\frac{1}{\sqrt{2\pi
V}}\exp\left(-\frac{p^{2}}{2V}\right),$ (8)

which is a Gaussian function. $P_{0}(p,V)$ can be built using a superposition
of three $\tilde{P}_{0}(p,V)$ with two unknown coefficients:

$P_{0}(p,V)=\tilde{P}_{0}(p-\bar{p},V)-a\tilde{P}_{0}(p+\bar{p},V)-b\tilde{P}_{0}(p-2H+\bar{p},V).$
(9)

This idea is inspired by the mirror Guassian solution of the linear elastic
contact [12, 13]. It is obvious that the diffusion equation in Eq. (3) is
automatically satisfied by the solution in Eq. (9). The boundary conditions in
Eq. (5) is enforced in this solution using two constants $a$ and $b$. An
illustration of the superposition method can be found in Fig. 2.

Figure 2: Schematic illustration of $P_{0}(p,V)$ and its three Gaussian
functions where $\bar{p}/\sqrt{V}=1.8$, $H/\sqrt{V}=3$ and $V=1$

Asymptotic solutions of $a$ and $b$ can be found when $\bar{p}\to 0^{+}$ and
$\bar{p}\to H^{-}$. As $\bar{p}\to 0$,
$P_{0}(p,V)=(1-a)\tilde{P}_{0}(p,V)-b\tilde{P}_{0}(p-2H,V)=0$, and we can get
$a=1$ and $b=0$. Similarly, as $\bar{p}\to H$,
$P_{0}(p,V)=(1-b)\tilde{P}_{0}(p-H,V)-a\tilde{P}_{0}(p+H,V)=0$, and we have
$a=0$ and $b=1$. Notice that this asymptotic analysis is independent of the
boundary conditions.

As $V\to 0^{+}$, Eq. (9) deduces to

$P_{0}(p,V\to
0^{+})=\delta(p-\bar{p})-\delta(p+\bar{p})-\delta(p-2H+\bar{p}).$

Since $\bar{p},p\in[0,H)$, $\delta(p+\bar{p})$ and $\delta(p-2H+\bar{p})$ are
zero. Therefore, the initial condition is satisfied.

Enforcing boundary conditions in Eq. (5) results in two linear equations

$\displaystyle(1-a)\tilde{P}_{0}(\bar{p},V)-b\tilde{P}_{0}(-2H+\bar{p},V)$
$\displaystyle=0,$ (10)
$\displaystyle(1-b)\tilde{P}_{0}(H-\bar{p},V)-a\tilde{P}_{0}(H+\bar{p},V)$
$\displaystyle=0.$ (11)

After solving the above two linear equations, explicit forms of $a$ and $b$
are derived

$\displaystyle a=$
$\displaystyle\frac{\tilde{P}_{0}(H-\bar{p},V)\left[\tilde{P}_{0}(\bar{p},V)-\tilde{P}_{0}(2H-\bar{p},V)\right]}{\tilde{P}_{0}(H-\bar{p},V)\tilde{P}_{0}(\bar{p},V)-\tilde{P}_{0}(H+\bar{p},V)\tilde{P}_{0}(2H-\bar{p},V)},$
(12) $\displaystyle b=$
$\displaystyle\frac{\tilde{P}_{0}(\bar{p},V)\left[\tilde{P}_{0}(H-\bar{p},V)-\tilde{P}_{0}(H+\bar{p},V)\right]}{\tilde{P}_{0}(H-\bar{p},V)\tilde{P}_{0}(\bar{p},V)-\tilde{P}_{0}(H+\bar{p},V)\tilde{P}_{0}(2H-\bar{p},V)}.$
(13)

$a$ and $b$ monotonically decreases and increases, respectively, with
$\bar{p}/\sqrt{V}$ for a given hardness $H$, see Fig. 3.

Figure 3: Numerical solutions of $a$ and $b$ where $H/\sqrt{V}=3$

The final form of $P_{0}(p,V)$ is given below

$P_{0}(p,V)=\frac{1}{\sqrt{2\pi
V}}\left\\{\exp\left[-\frac{(p-\bar{p})^{2}}{2V}\right]-a\exp\left[-\frac{(p+\bar{p})^{2}}{2V}\right]-b\exp\left[-\frac{(p-2H+\bar{p})^{2}}{2V}\right]\right\\},$
(14)

where $a$ and $b$ are given in Eqs. (12) and (13), respectively. As
$H\to\infty$, the third Gaussian term in Eq. (14) vanishes, and Eq. (12)
results in $a=1$. Therefore, Eq. (14) deduces to a mirror Gaussian solution of
linear elastic contact [10, 12]

$P_{0}(p,V)=\frac{1}{\sqrt{2\pi
V}}\left\\{\exp\left[-\frac{(p-\bar{p})^{2}}{2V}\right]-\exp\left[-\frac{(p+\bar{p})^{2}}{2V}\right]\right\\}.$
(15)

Note that it is not straightforward to deduce the elastic solution when
$P_{0}(p,V)$ is in the form of Fourier series, see Appendix B in [2].

###  2.3 $A_{el}^{*}$, $A_{pl}^{*}$ and $A_{nc}^{*}$

Normalization of $P(p,V)$ results in the following equality

$A_{nc}^{*}(V)+A_{el}^{*}(V)+A_{pl}^{*}(V)=1.$ (16)

Integrating both sides of Eq. (3) over $p\in[0,H]$ and resorting to Eq. (16),
we can have explicit forms of $A_{nc}^{*}$ and $A_{pl}^{*}$ [9]:

$\displaystyle A_{nc}^{*}(V)$
$\displaystyle=~{}~{}\frac{1}{2}\int_{0}^{V}\left[\frac{\partial}{\partial
p}P_{0}(p,V^{\prime})\right]\bigg{|}_{p=0}dV^{\prime},$ (17) $\displaystyle
A_{pl}^{*}(V)$
$\displaystyle=-\frac{1}{2}\int_{0}^{V}\left[\frac{\partial}{\partial
p}P_{0}(p,V^{\prime})\right]\bigg{|}_{p=H}dV^{\prime}.$ (18)

Substituting Eq. (7) into Eqs. (16-18), the Fourier series method results in
the following closed-form ratios [9]

$\displaystyle A_{el}^{*}(V)$
$\displaystyle=\sum_{n=1}^{\infty}\left[1-(-1)^{n}\right]\frac{2}{n\pi}\sin\left(\frac{n\pi\bar{p}}{H}\right)\exp\left[-\frac{1}{2}\left(\frac{n\pi}{H}\right)^{2}V\right],$
(19) $\displaystyle A_{pl}^{*}(V)$
$\displaystyle=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{n\pi}\sin\left(\frac{n\pi\bar{p}}{H}\right)\left\\{1-\exp\left[-\frac{1}{2}\left(\frac{n\pi}{H}\right)^{2}V\right]\right\\},$
(20) $\displaystyle A_{nc}^{*}(V)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{2}{n\pi}\sin\left(\frac{n\pi\bar{p}}{H}\right)\left\\{1-\exp\left[-\frac{1}{2}\left(\frac{n\pi}{H}\right)^{2}V\right]\right\\}.$
(21)

The superposition method gives a simpler forms as follows

$\displaystyle A_{el}^{*}(V)$
$\displaystyle=\frac{1+b}{2}\text{erf}\left(\frac{H-\bar{p}}{\sqrt{2V}}\right)+\frac{1+a}{2}\text{erf}\left(\frac{\bar{p}}{\sqrt{2V}}\right)-\frac{a}{2}\text{erf}\left(\frac{H+\bar{p}}{\sqrt{2V}}\right)+\frac{b}{2}\text{erf}\left(\frac{\bar{p}-2H}{\sqrt{2V}}\right),$
(22) $\displaystyle A_{pl}^{*}(V)$
$\displaystyle=-\frac{1+b}{2}\text{erf}\left(\frac{H-\bar{p}}{\sqrt{2V}}\right)+\frac{a}{2}\text{erf}\left(\frac{H+\bar{p}}{\sqrt{2V}}\right)+\frac{1}{2}(1-a+b),$
(23) $\displaystyle A_{nc}^{*}(V)$
$\displaystyle=-\frac{(1+a)}{2}\text{erf}\left(\frac{\bar{p}}{\sqrt{2V}}\right)-\frac{b}{2}\text{erf}\left(\frac{\bar{p}-2H}{\sqrt{2V}}\right)+\frac{1}{2}(1+a-b).$
(24)

It is easy to check that Eq. (16) is satisfied by Eqs. (22–24). The
satisfaction of Eq. (16) by the Fourier series solutions, Eqs. (19–21), is
also not straightforward. As $H\to\infty$, $a=1$, $b=0$, and the corresponding
real contact area can be obtained directly from Eq. (22)

$A_{el}^{*}(V)=\text{erf}\left(\frac{\bar{p}}{\sqrt{2V}}\right).$ (25)

##  3 Results and discussion

Figure 4: (a) PDF of elastic contact pressure, $P_{0}(p,V)$ where
$\bar{p}/\sqrt{V}=0.5,1.5$, and $2.5$, $H/\sqrt{V}=3$; (b) Variation of
$A_{el}^{*}$, $A_{pl}^{*}$ and $A_{nc}^{*}$ with $\bar{p}/\sqrt{V}$ where
$H/\sqrt{V}=3$

In this section, we first compare the results of the Fourier series method
with that of the superposition method. Consider the case where
$\bar{p}/\sqrt{V}=0.5,1.5$, and $2.5$, $H/\sqrt{V}=3$. The corresponding
$P_{0}(p,V)$ are shown in Fig. 4(a). Excellent agreement can be found between
$P_{0}(p,V)$ predicted by Eq. (7) (Fourier series method) and Eq. (14)
(superposition method). Only $10$ sine terms are used in the Fourier series
method in Fig. 4(a). Similar agreement can also be observed in Fig. 4(b) for
the variation of $A_{el}^{*}$, $A_{pl}^{*}$ and $A_{nc}^{*}$ with
$\bar{p}/\sqrt{V}$ predicted by the Fourier series method, Eqs. (19–21), and
the superposition method, Eqs. (22–24). An insufficient number of sine terms
(say $10$) would result in the Fourier series solutions, $A_{pl}^{*}$ and
$A_{nc}^{*}$, oscillating about the corresponding solutions of the
superposition method. A larger amount of sine terms (say $1000$ in Fig. 4(b))
are needed. Fig. 4 also shows that $P_{0}(p,V)|_{\bar{p}=p_{0}}$ is exactly
the reflection of $P_{0}(p,V)|_{\bar{p}=H-p_{0}}$ about the axis $p=H/2$ where
$p_{0}\in(0,H)$.

In Section 2, the boundary conditions in Eq. (5) was given without a rigorous
proof. In this discussion, a rigorous proof of the boundary conditions, Eq.
(5), will be given with the aid of the new superposition solution given in Eq.
(9).

The load equilibrium of elastoplastic contact can be formulated as

$\int_{0^{+}}^{H^{-}}pP_{0}(p,V)dp+HA_{pl}^{*}(V)=\bar{p}.$ (26)

Since $H$ and $\bar{p}$ are scale-independent, we can get the following
equation by differentiating both sides of Eq. (26) with respect to $V$:

$\int_{0}^{H}p\frac{\partial}{\partial
V}P_{0}(p,V)dp+H\frac{d}{dV}A_{pl}^{*}(V)=0.$ (27)

Using Eq. (3) and (18), we can replace $\displaystyle\frac{\partial}{\partial
V}P_{0}(p,V)$ and $\displaystyle\frac{d}{dV}A_{pl}^{*}(V)$ with
$\displaystyle\frac{1}{2}\frac{\partial^{2}}{\partial p^{2}}P_{0}(p,V)$ and
$\displaystyle-\frac{1}{2}\left[\frac{\partial}{\partial
p}P_{0}(p,V)\right]\bigg{|}_{p=H}$, respectively. Using integration by parts,
Eq. (27) eventually becomes

$P_{0}(p=0^{+},V)=P_{0}(p=H^{-},V)=C(V),$ (28)

where $C$ is a function of $V$ only.

Persson [10] argued that $P_{0}(p=0^{+},V)=0$ which proved in linear elastic
contact problem is still held in the elastoplastic problem. Thus, $C(V)=0$ and
eventually, boundary condition in Eq. (28) is deduced to Eq. (5). This is not
a rigorous proof since $P_{0}(p=0^{+},V)=0$ is only proved to be correct when
$H\to\infty$. A rigorous proof of $P_{0}(p=0^{+},V)=P_{0}(p=H^{-},V)=0$
becomes much easier using the new solution.

###### Proof.

Since PDF has non-negative value, $C(V)\geq 0$ according to Eq. (28). Assuming
$C(V)>0$, then we can rewrite Eq. (10) with $\bar{p}=H$ as

$(1-a-b)\tilde{P}_{0}(H,V)=C(V)>0.$ (29)

Given a finite value of $V$, $\tilde{P}_{0}(H,V)>0$, then $1-a-b>0$. According
to the previous asymptotic analysis, as $\bar{p}=H$, $a=0$ and $b=1$. This
contradicts with $1-a-b>0$. Therefore, $C(V)$ must be zero, and the boundary
condition, $P_{0}(p=0^{+},V)=P_{0}(p=H^{-},V)=0$. ∎

The constant hardness assumption is adopted to account for surface plasticity
in many elastoplastic rough surface contact models which have been applied to
various tribological problems [17, 18, 2]. The hardness assumption stems from
the work of Tabor [19] where he found the mean contact pressure on a spherical
indenter does not exceed 3$\sigma_{Y}$. Therefore, the plastic flow on the
surface is approximately accounted for in many models by an extra constraint
where the contact pressure cannot exceed 3$\sigma_{Y}$. In the last two
decades, the accuracy of the constant hardness model has been challenged by
several counter examples [21, 20]. Additionally, volume conservation cannot be
strictly satisfied under the constant hardness assumption [22]. Recently,
Frérot et al. [23] studied the effect of different types of plasticity model
on the prediction of the real contact area and PDF of the contact pressure.
Given the same normal load, the more realistic J2 plasticity model with a
linear isotropic hardening results in a larger real contact area [23]. A sharp
peak is introduced by the constant hardness assumption in the PDF of the
contact pressure, and does not exist for J2 plasticity model. The PDF of the
contact pressure whose value is less than the hardness is significantly
increased when the constant hardness assumption is replaced by J2 plasticity
model. In summary, care should be taken when applying the present Persson
theory to elastoplastic rough surface contact. It may underestimate both the
real contact area and the PDF of the contact pressure if the material follows
J2 plasticity model with strain hardening and the plastic deformation on the
interface is dominant [11]. It remains a future goal to find an effective way
to extend the Persson’s model to cover elastoplastic contact with strain
hardening.

In the above derivation, Persson’s theory of elastoplastic contact has only
been applied to the contact scheme where a rigid flat is in normal contact
with a nominally flat rough surface. For a linear elastic contact, it was
proved by Barber [24] that the roughness with surface height $h$ in contact
with a rigid flat is equivalent to a roughness with surface height $h_{1}$ in
contact with a roughness with surface height $h_{2}$ using the plane strain
modulus $E^{*}$ and $h=h_{1}+h_{2}$, as long as root mean square surface
gradients $\langle\nabla h_{1}\rangle\ll 1$ and $\langle\nabla h_{2}\rangle\ll
1$. This equivalent formulation was introduced to Persson’s theory of contact
by Scaraggi and Persson [25]. For elastoplastic contact, no relevant work has
been conducted so far to prove the validity of this equivalent formulation.
The closest one is the recent work of Shi and Zou [26]. They found that, (1)
if the material model follows J2 plasticity with a linear isotropic hardening,
the real area of contact between two rough surfaces predicted by the finite
element model is underestimated using the equivalent roughness vs rigid flat
scheme (see Fig. 7(a) of [26]) and (2) if the material is elastic-perfectly
plastic, both schemes result in the same real area of contact (see Fig. 7(b)
of [26]). Since both plasticity models are solid mechanics models, rather than
a constant hardness model, it remains unclear whether the equivalent
formulation of Scaraggi and Persson can be applied to Persson’s theory of
elastoplastic contact.

##  4 Conclusion

In this study, Persson’s theory of elastoplastic contact with a scale-
independent hardness is revisited. The previous solution of $P(p,\xi)$ is in
the form of a Fourier series (in terms of an infinite sum of sines). Here we
find a simpler form of $P(p,\xi)$ using the superposition of three Gaussian
functions. Excellent agreement is found between both solutions. A rigorous
proof of the boundary condition is given based on the new solution.

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## Declarations

### Funding

This work was supported by the Leverhulme Trust through Project Grant
“Fundamental mechanical behavior of nano and micro structured interfaces”
(RPG-2017-353), the National Natural Science Foundation of China (No.
52105179), the Fundamental Research Funds for the Central Universities of
China (No. PA2021KCPY0029) and Jiangsu Key Laboratory of Engineering
Mechanics, Southeast University (No. LEM21A03).

### Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

### Author Contributions

Yang Xu: Methodology, Conceptualization, Software, Validation, Formal
analysis, Visualization, Investigation, Writing – original draft, Funding
acquisition, Supervision, Writing - review and editing. Xiaobao Li: Writing -
review and editing, Funding acquisition. Daniel M. Mulvihill: Methodology,
Writing - review and editing, Funding acquisition.