Recent Advancements in Fractal Geometric-Based Nonlinear Time Series Solutions to the Micro-Quasistatic Thermoviscoelastic Creep for Rough Surfaces in Contact

Table 1

Maxwell viscoelastic medium.

Model Constitutive equation
(𝜕𝑡/𝐸+1/𝜂)𝐹=(𝜕𝑡)𝑢; the differential operator 𝜕𝑡≡𝜕/𝜕𝑡
Viscoelastic Operator Corresponds to the Modulus of Elasticity 𝐸
1𝐸→𝐸1𝜕^𝑡+1/𝜏𝜕𝑡
The modified Creep Model
𝑢(𝑡)ℎ0=𝑏𝜒𝐸𝐿𝑃^00^1/(𝜒+1)𝑒^−𝑡/𝜏×1𝐹112++𝜒𝜒;1;𝜏𝑡,^1𝐹1[2+𝜒/1+𝜒;1;𝑡/𝜏] is the confluent; hypergeometric function (called Kummer’s equation) written as: 1𝐹1[𝑐;𝑑;𝑥]=1+𝑐𝑑𝑥+𝑑𝑐((𝑐𝑑++11))𝑥2^2!+𝑐𝑑((𝑐𝑑++11))((𝑐𝑑++22))𝑥3^3!+⋯, or 1𝐹1[𝑐;𝑑;𝑥]=∑^∞𝑛=0(𝑐)𝑛(𝑑)𝑛𝑥^𝑛𝑛!