Self-Similar Mode of Ice Surface Softening During Friction

Time series analysis of friction force at self-affine mode of ice surface softening

“…From the system (1.2)-(1.4), the Langevin equation is acquired within the framework of the adiabatic approximation 𝜏 𝜎 , 𝜏 𝑇 𝜏 𝜀 in [20] 𝜏…”

“…The subsequent analysis consists in a numerical solution of the Langevin stochastic differential equation (2.1) by the Euler method [7,20,26,27]. With this aim, the Stratonovich's stochastic differential equation (2.1) is transformed into the Ito's one:…”

“…In figure 1, the time dependencies of frictional force are depicted, which are built on the basis of approximation 𝐹 (𝑡) = 𝐴𝐺 𝜀 |𝜀(𝑡)| in accordance with [18,20]. Therein, we have shown that 𝐹 (𝑡) corresponds to the experiments described in [11][12][13][14][15][16][17] for the following quantities: 1) the strain relaxation time 𝜏 𝜀 ≈ 0.1 − 5 s; 2) the contact area 𝐴 ≈ 10 −6 − 10 −1 m 2 that is typical of friction of a polymer (poly(methyl methacrylate) PMMA), rubber and steel on ice [11,16,17] as well as for polycrystalline freshwater and saline ice over itself [12][13][14][15]; 3) the shear modulus of ice 𝐺 𝜀 ≈ 0.1 − 10 MPa, i.e., yield stress [11].…”

“…Moreover, in the study [19] it was shown that the intermittent regime is also self-affine for most natural and practically important surfaces. A new stochastic model has helped to find out the reasons for the appearance of this mode and its features [20]. We used the measure units…”

“…for magnitudes 𝜀, 𝜎, 𝑇, 𝐼 𝜀 , 𝐼 𝜎 , and 𝐼 𝑇 , respectively, where 𝐺 𝜀 ≡ 𝜂 𝜀 /𝜏 𝜀 ≡ 𝐺 (𝜔)| 𝜔→0 is the relaxed value of the ice shear modulus (𝜔 is the circular frequency of a periodic external influence), 𝜂 𝜀 is the effective value of shear viscosity 𝜂, 𝜏 𝜀 is the relaxation time of ice strain, 𝑐 𝑝 is the heat capacity, 𝑇 𝑐 is the characteristic temperature, 𝜏 𝑇 ≡ 𝑙 2 𝑐 𝑝 /𝜅 is the time of heat conductivity, 𝑙 is the distance into which heat penetrates the ice, 𝜅 is the thermal conductivity. The nonlinear relaxation term of strain and feedbacks were considered in the governing equations with the fractional exponent 0 < 𝑎 < 1 [20]:…”