70.5. A FRICTION CONTACT PROBLEM 2413

|σT |= F((un−g)+)µ(∣∣u̇T − U̇T

∣∣) implies u̇T − U̇T =−λσT (u, u̇). (70.5.21)

Here Cn is a positive function in L∞(ΓC), which we take equal to 1 to simplify notation, U̇Tis the velocity of the foundation, λ is non negative, and µ is a bounded positive functionhaving a bounded continuous derivative. We could also let µ depend on x ∈ ΓC to modelthe roughness of the contact surface but we will suppress this dependence in the interestof simpler notation. Also, n is the unit outward normal to ∂U and un,uT ,σT , and σn aredefined by the following.

un = u ·n

uT = u−(u ·n)n

σn = σ i jn jni

σTi = σ i jn j−σnni,

written more simply,σT = σn−σnn

Systems like the above model dynamic friction contact problems [93], [51] [46]. Thefunction g represents the gap between the contact surface of U , ΓC, and a foundation whichis sliding tangent to ΓC with tangential velocity U̇T .

The new ingredient in this paper is that we allow

g = g(t,x,ω)

where ω ∈ (Ω,F ) and we assume (t,x,ω)→ g(x,ω) is B ([0,T ]×ΓC)×F measurable.Also, we make the reasonable assumption that

0≤ g(t,x,ω)≤ l < ∞

for all (t,x,ω). We also assume that the given motion of the foundation U̇T is a stochasticprocess

U̇T = U̇T (t,x,ω)

and is B ([0,T ]×ΓC)×F measurable. Here B ([0,T ]×ΓC) denotes the Borel sets of[0,T ]×ΓC. We make the reasonable assumption that U̇T (t,x,ω) is uniformly bounded. Inthe interest of notation, we will often suppress the dependence on t,x, and ω .

The condition 70.5.18 is the contact condition. It says the normal component of thetraction force density is dependent on the normal penetration of the body into the foun-dation surface. Conditions 70.5.19 -70.5.21 model friction. They say that the tangentialpart of the traction force density is bounded by a function determined by the normal forceor penetration. No sliding takes place until |σT | reaches this bound, F((un− g)+)µ (0),70.5.20. When this occurs, the tangential force density has a direction opposite the relativetangential velocity 70.5.21. The dependence of the friction coefficient on the magnitudeof the slip velocity,

∣∣u̇T − U̇T∣∣ may be experimentally verified and so it has been included.

The new feature in this model is the assumption that the gap is a random variable for eachx ∈ ΓC and we want to consider measurability of the solutions. Thus for a fixed ω, we havea standard friction problem and it is the measurability which is of interest here.