Chapter 3

Constitutive Models $–$ Relations between Stress and Strain

3.13 Constitutive models for contacting surfaces and interfaces in solids

Many practical problems involve two contacting surfaces that roll or slide against one another: examples include machine elements such as gears and bearings; machining and metal forming processes; or slip along a geological fault.  In addition, models of deformation and failure in materials must often account for the nucleation and growth of cracks in the solid.   In these applications, constitutive equations must be used to specify the forces transmitted across the interface or contacting surface as a function of their relative motion.

The simplest and most familiar such constitutive law is Coulomb friction, which relates the normal and tangential tractions acting across a contacting surface.  More complex constitutive laws are also available, which can model very complex interactions between surfaces.   In this section, we outline two general classes of interface law: (i) Cohesive zone models, which are used to model interfaces in materials or adhesion between very clean (often nanoscale) surfaces; and (ii) Models that are intended to model contact and friction between two sliding surfaces.

3.13.1 Cohesive zone models of interfaces

Cohesive zone models are usually used to model the nucleation and propagation of cracks along an interface between two solids, and to model adhesion between two contacting surfaces.  The figure illustrates the problem to be solved.  We assume that

Two solids meet at a surface S

In the undeformed configuration, the interface is free of traction, and there is no overlap or separation between the solids along S

When the solid is loaded, forces are transmitted across the interface, while the two solids may separate, slide, or overlap at the interface.  The notion that two solids may interpenetrate can be disturbing at first sight.  However, the surface S where the solids meet does not represent a plane of atoms $–$ it merely characterizes the equilibrium separation between the two solids when the interface is stress free.  If the two solids overlap, this means that the atomic or material planes just adjacent to the interface move closer together.

We shall assume that the relative displacement of the two solids across S is small compared with any characteristic dimension of the solid; and also that the two contacting solids themselves experience small shape changes.

A “cohesive zone law” relates the relative motion of the two solids adjacent to S to the tractions transmitted across S.  A large number of such constitutive equations have been developed, but there are two general classes: (i) reversible force-displacement laws, in which the traction is simply a function of the relative distance between the two surfaces, and independent of the history of loading.  These are often used to model nucleation and growth of a crack on an interface that is subjected to monotonically increasing loading, where irreversibility plays no role; and are also used to model interaction between surfaces of nanoscale structures, whose dimension can be comparable to the distance of action of long-range interatomic forces.  (ii) Irreversible force-displacement laws, which model failure processes that lead to the creation of new free surface in the solid.   These could include separation of atomic planes due to cleavage, or more complex processes such as rupture by void nucleation and coalescence, or fatigue.

Kinematics:  The relative motion of the two solids is characterized as follows:

1.      Let n denote a unit vector normal to the interface.  The sense of n is arbitrary (i.e. it can point up or down, as you prefer).  Once n has been chosen, however, we designate the two material surfaces adjacent to $S$ by ${S}^{+}$ and ${S}^{-}$, with n the outward normal to ${S}^{-}$.

2.      Introduce two mutually perpendicular unit vectors ${e}_{\alpha }$ that are tangent to the interface.

3.      Let ${u}^{+}\left(x\right)$ and ${u}^{-}\left(x\right)$ denote the displacement of two material points that are just adjacent to a point $x$ on S  in the undeformed solid.

4.      Let $\Delta ={u}^{+}-{u}^{-}$ denote the relative displacement of two initially coincident points. To specify constitutive equations, it is convenient to characterize the relative displacement using the three scalar Cartesian components $\left({\Delta }_{n},{\Delta }_{1},{\Delta }_{2}\right)$ of $\Delta$ in the basis $\left\{n,{e}_{1},{e}_{2}\right\}$.  If the interface is isotropic (i.e. its response is independent of the direction of the relative tangential displacement between the surfaces), the behavior of the interface depends only on ${\Delta }_{n}$ and ${\Delta }_{t}=\sqrt{{\Delta }_{1}^{2}+{\Delta }_{2}^{2}}$.

Kinetics: The forces acting between the two surfaces are characterized as follows:

1.      Two points that are initially coincident in the undeformed interface are assumed to exert equal and opposite tractions on one another.  Since the relative displacements of ${S}^{+}$ and ${S}^{-}$ are assumed to be small, and both solids are assumed to experience small shape changes, there is no need to distinguish between forces acting on the deformed and undeformed solids.  Let ${t}^{-}\left(x\right)$ and ${t}^{+}\left(x\right)$ denote the force per unit area acting on ${S}^{-}$ and ${S}^{+}$, respectively.

2.      Since ${t}^{+}\left(x\right)=-{t}^{-}\left(x\right)$, the tractions can be characterized by the three scalar components $\left({T}_{n},{T}_{1},{T}_{2}\right)$ of ${t}^{-}$ in the basis $\left\{n,{e}_{1},{e}_{2}\right\}$.

The constitutive equations for the interface must relate $\left({T}_{n},{T}_{1},{T}_{2}\right)$ to $\left({\Delta }_{n},{\Delta }_{1},{\Delta }_{2}\right)$

Constitutive equations representing reversible separation between interfaces are the simplest cohesive zone laws.  For these models, the tractions are a function only of the relative displacement of the material planes adjacent to the interface, and are independent of the history or rate of loading. This means that the traction-displacement relation for the interface is reversible.  The interface will heal if the two surfaces are brought back into contact after separation.

The constitutive equations relating $\left({T}_{n},{T}_{1},{T}_{2}\right)$ to $\left({\Delta }_{n},{\Delta }_{1},{\Delta }_{2}\right)$ for a reversible interface are constructed as follows:

1.      The traction-displacement relation is most conveniently characterized by a scalar inter-planar potential $\Phi \left({\Delta }_{n},{\Delta }_{1},{\Delta }_{2}\right)$ by setting

${T}_{n}=\frac{\partial \Phi }{\partial {\Delta }_{n}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{1}=\frac{\partial \Phi }{\partial {\Delta }_{1}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{2}=\frac{\partial \Phi }{\partial {\Delta }_{2}}$

The value of $\Phi$ represents  the work done per unit area in separating the interface by $\Delta$.

2.      A number of different functions are used to approximate $\Phi$.  Here, we will just give one example (a simplified version of a potential suggested by Xu and Needleman, (1995), J Mech. Phys. Solids 42 p. 1397)

$\Phi \left({\Delta }_{n},{\Delta }_{t}\right)={\varphi }_{n}-{\varphi }_{n}\left(1+\frac{{\Delta }_{n}}{{\delta }_{n}}\right)\mathrm{exp}\left(-\frac{{\Delta }_{n}}{{\delta }_{n}}\right)\mathrm{exp}\left(-\frac{{\beta }^{2}{\Delta }_{t}^{2}}{{\delta }_{n}^{2}}\right)$

Here, ${\Delta }_{t}^{2}={\Delta }_{1}^{2}+{\Delta }_{2}^{2}$, while ${\varphi }_{n},{\delta }_{n},\beta$ are material properties. Their physical significance is discussed below.

3.      Formulas relating $\left({T}_{n},{T}_{1},{T}_{2}\right)$ to $\left({\Delta }_{n},{\Delta }_{1},{\Delta }_{2}\right)$ can be calculated by differentiating the potential.  The result is

$\begin{array}{l}{T}_{n}={\sigma }_{\mathrm{max}}\frac{{\Delta }_{n}}{{\delta }_{n}}\mathrm{exp}\left(1-\frac{{\Delta }_{n}}{{\delta }_{n}}\right)\\ {T}_{\alpha }=2{\sigma }_{\mathrm{max}}\left(\frac{{\beta }^{2}{\Delta }_{\alpha }}{{\delta }_{n}}\right)\left(1+\frac{{\Delta }_{n}}{{\delta }_{n}}\right)\mathrm{exp}\left(1-\frac{{\Delta }_{n}}{{\delta }_{n}}\right)\mathrm{exp}\left(-\frac{{\beta }^{2}{\Delta }_{t}^{2}}{{\delta }_{n}^{2}}\right)\end{array}$

where ${\sigma }_{\mathrm{max}}={\varphi }_{n}/{\delta }_{n}\mathrm{exp}\left(1\right)$.  The traction-displacement relations are plotted in the figure.  Under purely normal tensile loading, the interface has work of separation ${\varphi }_{n}$, and the noremal traction reaches a value of ${\sigma }_{\mathrm{max}}$ at an interface separation ${\Delta }_{n}={\delta }_{n}$.  Under purely shear loading, the tangential traction has a maximum value ${\tau }_{\mathrm{max}}=\beta {\varphi }_{n}\sqrt{2}/\left({\delta }_{n}\sqrt{\mathrm{exp}\left(1\right)}\right)$ at a tangential shear displacement ${\Delta }_{t}={\delta }_{n}/\sqrt{2}$.

Constitutive equations modeling irreversible separation between interfaces. Most interfaces do not heal when brought back into contact after separation.   In applications where interfaces are subjected to cyclic loading, more complicated constitutive equations must be used to account for this irreversible behavior.   Again, a very large number of such constitutive equations have been developed: we will  illustrate their general features using a model adapted from Ortiz and Pandolfi, Int J. Numerical Methods in Engineering, 44 1276 (1999) as a representative example.

The behavior of the interface can be illustrated using its response to a purely normal tensile traction, shown in the traction-separation law on the right.   The interface initially responds elastically, with a constant stiffness ${k}_{0}={\sigma }_{\mathrm{max}}/{d}_{1}$, so that ${T}_{n}={k}_{0}{\Delta }_{n}$.   As long as ${\Delta }_{n}\le {d}_{1}$ the interface is reversible and undamaged.   If the displacement exceeds  ${\Delta }_{n}={d}_{1}$, the interface begins to accumulate irreversible damage, which causes the stress to drop.  At the same time, the damage reduces the stiffness of the interface, so that during unloading the traction-displacement relation remains linear, but with a reduced slope. Note that the total work of separation for the interface under tensile loading is ${\varphi }_{0}={\sigma }_{\mathrm{max}}\left({d}_{1}+{d}_{2}\right)/2$. The constitutive equation is constructed as follows:

1. The material surfaces ${S}^{+}$ and ${S}^{-}$ are completely prevented from overlapping, by enforcing the constraint ${\Delta }_{n}\ge 0$
2. The magnitude of the relative displacement between ${S}^{+}$ and ${S}^{-}$ is quantified by a scalar parameter $\lambda =\sqrt{{\Delta }_{n}^{2}+{\beta }^{2}\left({\Delta }_{1}^{2}+{\Delta }_{2}^{2}\right)}$, where $\beta$ is a material property, which controls the relative stiffness and strength of the interface to normal and shear loading.
3. Similarly, the magnitude of the traction can be quantified by an effective stress $\tau =\sqrt{{T}_{n}^{2}+\left({T}_{1}^{2}+{T}_{2}^{2}\right)/{\beta }^{2}}$
4. The tractions acting between ${S}^{+}$ and ${S}^{-}$ are related to the relative displacement by an elastic potential  $\Phi \left({\Delta }_{n},{\Delta }_{1},{\Delta }_{2},D\right)$ by setting

${T}_{n}=\frac{\partial \Phi }{\partial {\Delta }_{n}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{1}=\frac{\partial \Phi }{\partial {\Delta }_{1}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{2}=\frac{\partial \Phi }{\partial {\Delta }_{2}}$

Here, $0\le D\le 1$ is a scalar parameter that quantifies the irreversible damage accumulated by the interface.

1. A linear traction-displacement relation is constructed by making $\Phi$ a quadratic function of $\lambda$, as follows

$\Phi ={k}_{0}\left(1-D\right){\lambda }^{2}/2$

Here, ${k}_{0}$ is a material property that corresponds to the slope of the traction-displacement relation for the undamaged interface.  It follows that the tractions are related to the displacements by

${T}_{n}={k}_{0}\left(1-D\right){\Delta }_{n}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{1}={\beta }^{2}{k}_{0}\left(1-D\right){\Delta }_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{2}={\beta }^{2}{k}_{0}\left(1-D\right){\Delta }_{2}$

1. The constitutive law is completed by devising an appropriate equation governing the evolution of D.   $D$ remains constant if the traction on the interface is less than its current strength; if the interface is unloaded, or if D reaches 1.  Otherwise, D must evolve so that the strength of the interface decreases linearly from its initial value ${\sigma }_{\mathrm{max}}$ to zero as the effective displacement $\lambda$ increases from $\lambda ={d}_{1}$ to $\lambda ={d}_{2}$. This requires

Representative values for properties of cohesive zones.

The two constitutive laws contain the following parameters:

1. The reversible interface can be conveniently characterized by its strength, ${\sigma }_{\mathrm{max}}$; the total work of tensile separation ${\varphi }_{0}$, and the parameter $\beta$ that controls the ratio of shear to normal strength.
2. The irreversible interface can be characterized by its strength, ${\sigma }_{\mathrm{max}}$; the total work of tensile separation ${\varphi }_{0}$, the parameter $\beta$ and the displacement ${d}_{1}$ at the instant of maximum stress.

It is difficult to give precise values for these material properties.  This is partly because the constitutive equations are used to model a variety of physical processes that lead to failure, and partly because there is no simple way to measure the values of the parameters.   The following guidelines are usually followed:

1. If the cohesive zone is used to model atomic-scale cleavage in a brittle elastic material, or adhesion between two elastic solids, then ${\varphi }_{0}$ is set equal to the fracture toughness of the interface (typically ${\varphi }_{0}\approx 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{Jm}}^{\text{-2}}$ ), and the peak strength of the material ${\sigma }_{\mathrm{max}}\approx E/100$ where E is the Young’s modulus.  Available data suggests that interfaces are stronger in shear than in tension so $\beta$ is usually taken to be slightly less than 1 ( $\beta \approx 0.7$ ).   Computations are usually not strongly sensitive to the shape of the cohesive zone, so ${d}_{1}/{d}_{2}$ can be taken to be approximately 1 in the irreversible model.
2. If the cohesive zone is intended to model both the plastic zone and the failure process at the tip of a crack in an otherwise elastic solid, then ${\varphi }_{0}$ is set equal to the fracture toughness of the solid (fracture toughness values are tabulated in Section 9.3.6), while ${\sigma }_{\mathrm{max}}$ is taken to be roughly three times the yield stress of the solid in uniaxial tension (yield stress values are tabulated in Section 3.6.9).  Again, $\beta \approx 0.7$, while ${d}_{1}/{d}_{2}\approx 1$ in the irreversible model.
3. Cohesive zones are sometimes used to model material separation at the tip of a crack in a plastic solid, together with an elastic-plastic constitutive equation for the two solids adjacent to the cohesive interface.   In this case it is not usually clear what process the cohesive zone represents.  Experience shows that if the strength ${\sigma }_{\mathrm{max}}$ of the cohesive zone is taken to be too high (greater than approximately three times the yield stress of the plastic material) the crack will never propagate.  If the strength of the cohesive zone is less than the yield stress, there is no plasticity.  Consequently $3Y<{\sigma }_{\mathrm{max}}.  It is difficult to interpret the meaning of ${\varphi }_{0}$ in these models, but fortunately simulations tend to be relatively insensitive to ${\varphi }_{0}$.  A value for ${\varphi }_{0}$ is usually estimated by choosing a sensible characteristic length (between $1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu \text{m}$ and  $10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu \text{m}$ ) for ${\delta }_{n}$ or ${d}_{1}$ and setting ${\varphi }_{n}={\sigma }_{\mathrm{max}}{\delta }_{n}\mathrm{exp}\left(1\right)$ for the reversible model; or ${\varphi }_{0}={\sigma }_{\mathrm{max}}\left({d}_{1}+{d}_{2}\right)/2$ for the irreversible version.

3.13.2 Models of contact and friction between surfaces

Experimental observations of friction between contacting surfaces

A friction experiment is conceptually very simple: two surfaces are pressed into contact by a controlled normal pressure p, and the specimens are loaded so as to induce a state of uniform shear traction $T$ acting between the contacting surfaces.   The experiment seeks to answer the following questions:

1.      What is the critical combination of normal and tangential forces cause the surfaces to start to slide?

2.      If the two surfaces do start to slip, what tangential force is required to keep them sliding?

3.      If the surfaces are slipping, how does the tangential force vary with sliding velocity and normal pressure, and how does the surface respond to changes in sliding velocity and pressure?

4.      How does friction depend on the contact area, the properties of the two contacting surfaces; surface roughness; environment; lubricant films, etc?

The results of these experiments show that, for most engineering surfaces which make contact over a nominal area exceeding $100\mu {\text{m}}^{\text{2}}$ or so:

The critical tangential traction required to initiate sliding between two surfaces is proportional to the normal pressure.  If the normal force is zero, the contact can’t support any tangential force.  Doubling the normal force will double the critical tangential force that initiates slip.

Surface roughness has a very modest effect on friction.  Doubling the surface roughness might change the friction force by a few percent.

Contaminants or lubricant on the two surfaces has a big effect on friction.  Even a little moisture on the surfaces can reduce friction by 20-30%.  If there is a thin layer of grease on the surfaces it can cut friction by a factor of 10.  If the contaminants are removed,  friction forces can be huge, and the two surfaces can seize together completely.

Friction forces depend quite strongly on what the two surfaces are made from.  Some materials like to bond with each other (metals generally bond well to other metals, for example) and so have high friction forces.  Some materials (e.g. Teflon) don’t bond well to other materials.  In this case friction forces will be smaller.

If the surfaces start to slide, the tangential force often (but not always) drops slightly.  Thus, kinetic friction forces are often a little lower than static friction forces.  Otherwise, kinetic friction forces behave just like static friction $–$ they are proportional to the normal force, etc.

The steady-state kinetic friction force usually (but not always) decreases slightly as the sliding speed increases.  Increasing sliding speed by a factor of 10 might drop the friction force by a few percent.

 The transient response to changes in sliding speed has been extensively studied in geological materials, motivated by the need to understand earthquakes. In these materials, increasing the sliding speed causes an instantaneous increase in shear traction, which then gradually decays to a lower steady-state value, as illustrated in the figure.   If the sliding speed is reduced, there is an instantaneous drop in the friction force, which subsequently increases towards a steady state.  The transients occur over a sliding distance of order 10-50 microns.   This behavior has been observed in other materials (including metals) as well, but is not universal $–$ for example Gearing et al In J. Plasticity 17 p. 237, (2001) observe an increase in steady state friction forces with sliding speed in sliding of Al against steel. The transient response to a change in contact pressure has not been studied as extensively as the response to changes in sliding speed.  The data of Prakash (Journal of Tribology, 120, 1998 p. 27) indicates that when the contact pressure is suddenly increased, the shear traction is initially unchanged, and subsequently asymptotes towards a value proportional to the new contact pressure as the relative distance of sliding between the two surfaces increases, as illustrated in the figure.

These trends can be attributed to the effects of surface roughness.   All surfaces are rough, and when brought into contact meet only at highest points on the two surfaces.   The true area of contact between the two surfaces is much less than the nominal contact area, and increases roughly in proportion to the nominal contact pressure acting between the surfaces.   The nominal tangential traction is proportional to the product of the true contact area and the shear strength of the contacting surfaces, and is therefore approximately proportional to the nominal contact pressure.

There are some situations where the true area of contact approaches the nominal contact area. Examples include (i) the tip of an atomic force microscope, which has roughness comparable to atomic scale dimensions; (ii) friction between the tool and workpiece in metal forming applications.  In these situations the traction acting tangent to the surface is relatively insensitive to the contact pressure.

Kinematics:  Constitutive laws for friction must account for large relative motion between the contacting surfaces. Consequently, the contact is best characterized by the relative position and motion of the two surfaces in the deformed configuration.

1.      One of the two surfaces is arbitrarily designated the master’ surface, and labeled ${S}^{-}$.  The other surface is designated the ‘slave’, and is labeled ${S}^{+}$.  Note that in some friction models (e.g. the plasticity model described below) exchanging the master and slave surface will have a small influence on the behavior of the interface.

2.      At a representative point ${y}^{-}$ on ${S}^{-}$, we let n denote a unit vector normal to ${S}^{-}$ and introduce two mutually perpendicular unit vectors ${e}_{\alpha }$ that are tangent to ${S}^{-}$.  We take ${e}_{1}$ to point along a characteristic material direction in ${S}^{-}$, i.e. if ${m}_{1}$ is a unit vector tangent to ${S}^{-}$ in the undeformed slave surface, ${e}_{1}=\left(F\cdot {m}_{1}\right)/|F\cdot {m}_{1}|$ in the deformed surface.

3.      The gap between the two surfaces is characterized by the points on the two surfaces that lie along n, i.e. ${y}^{+}={y}^{-}+{\Delta }_{n}n$

4.      The relative velocity of the two surfaces is defined as

$v=\frac{d}{dt}\left({y}^{+}-{y}^{-}\right)\text{\hspace{0.17em}}-{\Delta }_{n}\frac{dn}{dt}$

It is convenient to separate the relative velocity into components normal and tangent to the surface ${v}_{n}=d{\Delta }_{n}/dt=v\cdot n$${v}_{1}=v\cdot {e}_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{2}=v\cdot {e}_{2}$.

5.      In finite element computations it is sometimes convenient to introduce a small elastic compliance for the interface.   In this case, the relative velocity of the surfaces is divided into a reversible elastic part and an irreversible (plastic) part by defining

$v={v}^{e}+{v}^{p}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{n}={v}_{n}^{e}+{v}_{n}^{p}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{\alpha }={v}_{\alpha }^{e}+{v}_{\alpha }^{p}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =1,2$

Kinetics: The forces acting between the two surfaces are characterized as follows:

1.      The points on the two surfaces at position ${y}^{+}$ and ${y}^{-}$ are assumed to exert equal and opposite tractions on one another. We let ${t}^{+}\left({y}^{+}\right)$, ${t}^{-}\left({y}^{-}\right)$ denote the tractions on ${S}^{+}$ and ${S}^{-}$, respectively.

2.      Since ${t}^{+}\left({y}^{+}\right)=-{t}^{-}\left({y}^{-}\right)$, the tractions can be characterized by the three scalar components $\left({T}_{n},{T}_{1},{T}_{2}\right)$ of ${t}^{-}$ in the basis $\left\{n,{e}_{1},{e}_{2}\right\}$.

Constitutive equations for sliding friction must specify relationships between $\left({\Delta }_{n},{v}_{i}\right)$ and $\left({T}_{n},{T}_{1},{T}_{2}\right)$.  Various alternatives are summarized briefly below.

Coulomb Friction:  This is the most familiar friction law.  For this model

(i) The interface separates, with an indeterminate ${\Delta }_{n}$ if ${T}_{n}>0$;

(ii) The surfaces are prevented from inter-penetrating ${\Delta }_{n}=0$ if ${T}_{n}<0$;

(iii) No slip occurs between the surfaces ${v}_{i}=0$ if $\sqrt{{T}_{1}^{2}+{T}_{2}^{2}}<\mu |{T}_{n}|$, where $\mu$ is the coefficient of friction;

(iv) The tangential traction is proportional to the normal pressure and opposes the direction of slip if the two surfaces slide ${T}_{1}=\mu |{T}_{n}|{v}_{1}/\sqrt{{v}_{1}^{2}+{v}_{2}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{2}=\mu |{T}_{n}|{v}_{2}/\sqrt{{v}_{1}^{2}+{v}_{2}^{2}}$.

The table below (taken from Engineering Materials’ by Ashby and Jones, Pergammon, 1980) lists rough values for friction coefficients for various material pairs.  These are rough guides only $–$ friction coefficients for a given material can by highly variable (for example, friction for a steel/steel contact can vary anywhere between 0.001 to 3), and can even vary significantly with time or sliding distance during an experiment.

 Material Approx friction coefficient Clean metals in air 0.8-2 Clean metals in wet air 0.5-1.5 Steel on soft metal (lead, bronze, etc) 0.1-0.5 Steel on ceramics (sapphire, diamond, ice) 0.1-0.5 Ceramics on ceramics (eg carbides on carbides) 0.05-0.5 Polymers on polymers 0.05-1.0 Metals and ceramics on polymers (PE, PTFE, PVC) 0.04-0.5 Boundary lubricated metals (thin layer of grease) 0.05-0.2 High temperature lubricants (eg graphite) 0.05-0.2 Hydrodynamically lubricated surfaces (full oil film) 0.0001-0.0005

HEALTH WARNING: Descriptions of Coulomb friction in elementary mechanics and physics texts often distinguish between kinetic and static friction coefficients.  It is not advisable to do adopt this approach when posing a boundary value problem in continuum mechanics, as it is likely to make the problem ill-posed (with either no solution, or an infinite number of solutions).  In fact, even with a single friction coefficient the Coulomb friction model can be ill-posed and should be used with caution.

Coulomb Friction with a shear cutoff: In metal forming applications contacting surfaces can be subjected to extremely high pressure, with the result that the true area of contact approaches the nominal area.  Under these conditions, the shear traction is no longer proportional to the contact pressure.  Behavior at high pressure is often approximated by truncating the shear traction at a critical value (usually taken to be somewhat lower than the shear yield strength of the softer of the two contacting surfaces).  The modified friction has the following constitutive equations:

(i) The interface separates, with an indeterminate ${\Delta }_{n}$ if ${T}_{n}>0$;

(ii) The surfaces are prevented from inter-penetrating ${\Delta }_{n}=0$ if ${T}_{n}<0$;

(iii) We introduce the shear resistance of the interface ${\tau }_{0}$ defined as

$\tau =\left\{\begin{array}{c}\mu |{T}_{n}|\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu |{T}_{n}|\le {\tau }_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ {\tau }_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu |{T}_{n}|\ge {\tau }_{0}\end{array}$

(iii) No slip occurs between the surfaces ${v}_{i}=0$ if $\sqrt{{T}_{1}^{2}+{T}_{2}^{2}}<\tau$, where $\mu$ is the coefficient of friction;

(iv) The tangential traction is proportional to the normal pressure and opposes the direction of slip if the two surfaces slide ${T}_{1}=\tau {v}_{1}/\sqrt{{v}_{1}^{2}+{v}_{2}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{2}=\tau {v}_{2}/\sqrt{{v}_{1}^{2}+{v}_{2}^{2}}$.

Rate and state variable models of friction: The variation of friction with sliding velocity and transient behavior following changes in contact pressure play an important role in controlling the stability of sliding on an interface.   Several friction laws have been developed to describe this behavior, and are widely used in geophysics applications.  As a representative example, we outline a constitutive law based loosely on work of Dieterich, J.H.,. J. Geophys. Res. 84, 2161 (1979), Ruina, A.L.,. J. Geophys. Res. 88, 10359 (1983) and Prakash, V., J. Tribol. 120, 97 (1998).

The transient behavior of sliding friction is modeled by introducing two state variables’ ${\omega }^{±},{p}^{±}$ for each material point on ${S}^{±}$.  The state variables evolve according to

$\frac{d\omega }{dt}=\left({v}_{s}-\omega \right)\left(\frac{{v}_{s}}{{L}_{v}}+\frac{1}{{t}_{v}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{dp}{dt}=-\left({T}_{n}+p\right)\left(\frac{{v}_{s}}{{L}_{p}}+\frac{1}{{t}_{p}}\right)$

where ${v}_{s}=\sqrt{{\left({v}_{1}^{p}\right)}^{2}+{\left({v}_{2}^{p}\right)}^{2}}$, $\left({L}_{v},{L}_{p}\right)$ are material properties with units of length, and $\left({t}_{v},{t}_{p}\right)$ are material properties with units of time.  The two surfaces ${S}^{+}$ and ${S}^{-}$ may have different properties. To interpret these equations, note that

1.      Both $\omega$ and $p$ evolve with time and the sliding distance.

2.      If the surfaces slide at constant speed, then $\omega \to {v}_{s}$ in the steady state, while if the surfaces are subjected to a time independent normal traction $p\to -{T}_{n}$.

3.      The two constants  $\left({t}_{v},{t}_{p}\right)$ control the time-scale associated with this evolution for a static contact; while $\left({L}_{v},{L}_{p}\right)$ control the distance required for $\omega$ and $p$ to reach their steady state values under a rapidly sliding contact.

HEALTH WARNING: Note that (i) state variables must be introduced to characterize both contacting surfaces, because coincident points on the two surfaces experience different histories of contact pressure and slip velocity.  To see this, note that as you slide your finger over the surface of a table, a point on your finger sees a constant contact pressure, while a point on the table experiences a cycle of loading.  (ii) the time derivatives of the state variables should be interpreted as the rate of change experienced by an observer traveling with a particular material particle in each surface.

The variation of steady-state friction coefficient with sliding velocity is modeled by introducing a friction coefficient that is a function of the state variables ${\omega }^{±}$

$\mu \left(\overline{\omega }\right)={\mu }_{k}+\left({\mu }_{s}-{\mu }_{k}\right)\mathrm{exp}\left[-{\left(\overline{\omega }/{V}_{1}\right)}^{n}\right]$

where $\overline{\omega }=\left({\omega }^{+}+{\omega }^{-}\right)/2$, and ${\mu }_{k},{\mu }_{s},{V}_{1},n$ are all material properties.  The constant ${\mu }_{k}$ represents the limiting value of the friction coefficient as sliding velocity approaches infinity (it can be interpreted as the kinetic friction coefficient), while ${\mu }_{s}$ is the steady-state value of friction coefficient for a static contact (and can be interpreted as the static friction coefficient).  The two constants ${V}_{1},n$ control the rate at which the friction transitions from one value to the other.

The friction law can be conveniently expressed as a relationship between the tractions and the relative velocity of the contact, as follows.  For ${\Delta }_{n}>0$ the surfaces are traction free ${T}_{n}={T}_{1}={T}_{2}=0$.  For ${\Delta }_{n}\le 0$,

1.      The elastic part of the relative velocity is related to the traction components by

${v}_{n}^{e}=\frac{1}{{k}_{n}}\frac{d{T}_{n}}{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{1}^{e}=\frac{1}{{k}_{t}}\frac{d{T}_{1}}{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{2}^{e}=\frac{1}{{k}_{t}}\frac{d{T}_{2}}{dt}$

where ${k}_{n}$ and ${k}_{t}$ are two elastic stiffnesses.  Note that these equations assume that the elastic distorsion of the interface occurs on the slave surface (the time derivatives correspond to the traction rate experienced by an observer fixed to the slave surface).

2.      The irreversible part of the normal component of velocity ${v}_{n}^{p}=0$

3.      The irreversible part of the tangential component of velocity is calculated from

${v}_{\alpha }^{p}=\left\{\begin{array}{c}{V}_{0}\left[\left(\frac{\overline{\omega }}{{V}_{0}}+1\right){\left(\frac{{T}_{t}}{\mu \overline{p}}\right)}^{m}-1\right]\frac{{T}_{\alpha }}{{T}_{t}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{t}\ge \mu \overline{p}\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{t}<\mu \overline{p}\end{array}$

where ${T}_{t}=\sqrt{{T}_{1}^{2}+{T}_{2}^{2}}$, $\overline{\omega }=\left({\omega }^{+}+{\omega }^{-}\right)/2$, $\overline{p}=\left({p}^{+}+{p}^{-}\right)/2$ and ${V}_{0},m$ are two constants that control the variation of shear stress to a step change in sliding velocity.

To interpret this equation, suppose that the interface is subjected to a constant (i.e. time independent) pressure, and is constrained to slip at a rate ${v}_{s}$.  The magnitude of the shear traction follows as

${T}_{t}=\mu \left(\overline{\omega }\right)\overline{p}{\left(\frac{{v}_{s}/{V}_{0}+1}{\overline{\omega }/{V}_{0}+1}\right)}^{1/m}$

The steady state value is ${T}_{t}=\mu \left({v}_{s}\right)\overline{p}$.   Following an instantaneous increase in sliding speed, the shear traction first jumps to a new, higher value, then progressively decreases to a lower steady-state value as $\overline{\omega }$ approaches the new value of sliding speed.  Similarly, if the sliding speed is suddenly reduced, the shear stress first drops to a lower value, and subsequently increases gradually to a higher steady-state.

Representative values of material properties for state variable model of friction.  The subtle features of friction captured by this constitutive equation are very sensitive to the materials involved, the surface finish, and the environment.  Extensive tests are required to characterize a particular contacting pair.  As a rough guide to the orders of magnitudes of the various material parameters, the table below lists rough estimates for parameters based on a discussion by Coker, Needleman and Rosakis (J. Mech. Phys. solids, 53 p.884 (2005)) of transient friction in Homalite.

 Representative values of parameters for a simple state variable friction law (the two contacting surfaces are assumed to have identical properties) ${k}_{n}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{GPa/m)}$ ${k}_{t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{GPa/m)}$ ${\mu }_{k}$ ${\mu }_{s}$ ${V}_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{(m/s)}$ n 300 100 0.5 0.6 26 1.2 ${L}_{p}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{(}\mu \text{m)}$ ${t}_{p}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(s\right)$ ${L}_{v}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{(}\mu \text{m)}$ ${t}_{v}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(s\right)$ ${V}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{(m/s)}$ m 20 $\infty$ 20 $\infty$ 100 5

Friction laws based on plasticity theory: The general framework of viscoplasticity can easily be adapted to construct friction laws that approximate the variation of friction with sliding speed and the evolution of friction with slip.   Laws of this kind are often used in metal forming simulations.   Several such models exist and will not be described in detail here.  Instead, we will illustrate the general idea by adapting the critical state theory of plasticity outlined in Section 3.10, together with the viscoplasticity law described in Section 3.7 to construct a friction law that captures the transient behavior of a sliding interface, as follows:

1.      The normal traction must satisfy ${T}_{n}\le 0$: if ${T}_{n}=0$ the surfaces separate and ${T}_{1}={T}_{2}=0$

2.      The relative velocity of the two surfaces is divided into elastic and plastic parts

${v}_{n}={v}_{n}^{e}+{v}_{n}^{p}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{\alpha }={v}_{\alpha }^{e}+{v}_{\alpha }^{p}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =1,2$

3.      The elastic part of the relative velocity is related to the traction components by

${v}_{n}^{e}=\frac{1}{{k}_{n}}\frac{d{T}_{n}}{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{1}^{e}=\frac{1}{{k}_{t}}\frac{d{T}_{1}}{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{2}^{e}=\frac{1}{{k}_{t}}\frac{d{T}_{2}}{dt}$

where ${k}_{n}$ and ${k}_{t}$ are two elastic stiffnesses.  Note that these equations assume that the elastic distorsion occurs on the slave surface (the time derivatives correspond to the traction rate experienced by an observer fixed to the slave surface).

4.      Three state variables ${p}^{±},{\omega }^{±},{s}^{±}$ are introduced to track the history of contact pressure, sliding speed and sliding distance on each of the two contacting surfaces.   The state variables evolve according to

$\frac{dp}{dt}=-cp{v}_{n}^{p}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{ds}{dt}={v}_{s}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{d\omega }{dt}=\left({v}_{s}-\omega \right)\left(\frac{{v}_{s}}{{L}_{v}}+\frac{1}{{t}_{v}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

where ${v}_{s}=\sqrt{{v}_{1}^{2}+{v}_{2}^{2}}$, and $c,{L}_{v},{t}_{v}$ are material properties.  Naturally, the two surfaces may have different values of $c,{L}_{v},{t}_{v}$.  The governing equations for the evolution of the state variables have been designed so that $p\to -{T}_{n}$ and $\omega \to {v}_{s}$ under conditions of steady sliding.

5.      The variation of relative velocity between the surfaces with traction is approximated using a slip potential (similar to the viscoplastic potential described in 3.7), defined as

$g\left({T}_{n},{T}_{t}\right)=\left\{\begin{array}{c}\frac{{V}_{0}{\tau }_{0}}{m+1}{\left[{\left(1+\frac{{T}_{n}}{\overline{p}}\right)}^{2}+{\left(\frac{{T}_{t}}{{\tau }_{0}}\right)}^{2}-1\right]}^{\left(m+1\right)/2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(1+\frac{{T}_{n}}{\overline{p}}\right)}^{2}+{\left(\frac{{T}_{t}}{{\tau }_{0}}\right)}^{2}-1\ge 0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(1+\frac{{T}_{n}}{\overline{p}}\right)}^{2}+{\left(\frac{{T}_{t}}{{\tau }_{0}}\right)}^{2}-1\le 0\end{array}$

where ${V}_{0},m$ are material properties which control the response of the interface to an instantaneous change in traction, $\overline{p}=\left({p}^{+}+{p}^{-}\right)/2$, and ${\tau }_{0}$ is a representative shear strength which may be a function of one or more of the state variables, as discussed further below.   The state variable $\overline{p}$ plays the role of a in the critical state soil model outlined in Section 3.10.

6.      The plastic part of the relative velocity between the surfaces is related to g through an associated flow law

${v}_{n}^{p}=\frac{\partial g}{\partial {T}_{n}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{\alpha }^{p}=\frac{\partial g}{\partial {T}_{\alpha }}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =1,2$

Evaluating the derivatives gives

${v}_{n}^{p}=\left\{\begin{array}{c}{V}_{0}\frac{{\tau }_{0}}{\overline{p}}\left(1+\frac{{T}_{n}}{\overline{p}}\right){\left[{\left(1+\frac{{T}_{n}}{\overline{p}}\right)}^{2}+{\left(\frac{{T}_{t}}{{\tau }_{0}}\right)}^{2}-1\right]}^{\left(m-1\right)/2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(1+\frac{{T}_{n}}{\overline{p}}\right)}^{2}+{\left(\frac{{T}_{t}}{{\tau }_{0}}\right)}^{2}-1\ge 0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(1+\frac{{T}_{n}}{\overline{p}}\right)}^{2}+{\left(\frac{{T}_{t}}{{\tau }_{0}}\right)}^{2}-1\le 0\end{array}$

${v}_{\alpha }^{p}=\left\{\begin{array}{c}{V}_{0}\frac{{T}_{\alpha }}{{\tau }_{0}}{\left[{\left(1+\frac{{T}_{n}}{\overline{p}}\right)}^{2}+{\left(\frac{{T}_{t}}{{\tau }_{0}}\right)}^{2}-1\right]}^{\left(m-1\right)/2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(1+\frac{{T}_{n}}{\overline{p}}\right)}^{2}+{\left(\frac{{T}_{t}}{{\tau }_{0}}\right)}^{2}-1\ge 0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(1+\frac{{T}_{n}}{\overline{p}}\right)}^{2}+{\left(\frac{{T}_{t}}{{\tau }_{0}}\right)}^{2}-1\le 0\end{array}$

7.      Finally, the variation of the tangential force with contact pressure, sliding speed, and slip distance must be specified by an appropriate equation for ${\tau }_{0}\left(\overline{p},\overline{\omega },\overline{s}\right)$.  Any sensible function can be chosen, depending on the behavior that you would like to approximate.   For example,

Setting ${\tau }_{0}=\mu \overline{p}$ will produce Coulomb friction like behavior, with a delayed response to changes in contact pressure.   To see this, note that the model behaves like the critical state soil model discussed in Section 3.10, with volumetric strain’ replaced by the normal separation between the surfaces, and $\mu \equiv M$.

Setting ${\tau }_{0}=\left\{{\mu }_{k}+\left({\mu }_{s}-{\mu }_{k}\right)\mathrm{exp}\left[-{\left(\overline{\omega }/{V}_{1}\right)}^{n}\right]\right\}\overline{p}/f\left(\overline{\omega }/{V}_{0}\right)$, where $f\left(y\right)$ is the root of the equation $y=x{\left({x}^{2}-1\right)}^{\left(m-1\right)/2}$ will give a Coulomb-like friction law with a velocity dependent friction coefficient similar to the rate-and state-variable model outlined earlier.

There is very little to distinguish the rate- and state-variable model from the plasticity based model.  The plasticity model has some advantages for numerical simulations, because (a) the transition from stick to slip is gradual; (b) The plasticity model has a `soft’ relationship between the normal displacement of the surfaces and the normal pressure; (c) the plasticity model has an associated flow rule.  All these tend to stabilize numerical computations.