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

Table 2

Kelvin-Voigt viscoelastic medium.

Model Constitutive equation
𝜀=𝜎𝐸𝑒+𝜂𝜕𝑡; the differential operator 𝜕𝑡≡𝜕/𝜕𝑡
Viscoelastic Operator that Corresponds to the Modulus of Elasticity 𝐸
1𝐸→𝐸1𝑒11+𝜏𝜕𝑡
The modified Creep Model
𝑢(𝑡)ℎ0=1Γ((2+𝜒)/(1+𝜒))𝜏𝑏𝐸𝜒𝑒𝐹0𝐿0^1/(𝜒+1)𝜏^(2+𝜒)/2(1+𝜒)×𝑡^−𝜒/2(1+𝜒)𝑒^−𝑡/2𝜏𝑀−𝜒/2(1+𝜒),1/2(1+𝜒)[𝑡/𝜏],; where Γ is the gamma function, 𝑀𝜇,𝜈(𝑥) is the Whittaker’s hypergeometric function and equals to: 𝑀𝜇,𝜈(𝑥)=𝑥^1/2+𝜈𝑒^−𝑥/2×1𝐹112+𝜈−𝜇;2𝜈+1;𝑥 and 1𝐹1[𝑐;𝑑;𝑥] is the Kummer’s confluent; hypergeometric function which could be expressed as 1𝐹1[𝑐;𝑑;𝑥]=∑^∞𝑛=0(𝑐)𝑛(𝑑)𝑛𝑥^𝑛𝑛!, or: 1𝐹1[𝑐;𝑑;𝑥]=1+𝑐𝑑𝑥+𝑑𝑐((𝑐𝑑++11))𝑥2^2!+𝑑𝑐((𝑑𝑐++11))((𝑐𝑑++22))𝑥3^3!+⋯
The modified Thermal Creep Model
𝑢(𝑡)ℎ0=𝑏𝜒𝜏𝐸𝑒𝑃0𝐿0^1/(𝜒+1)𝑡^1/(1+𝜒)Γ((2+𝜒)/(1+𝜒))^1𝐹11+1𝜒;21++𝜒𝜒;−𝜏𝑡×exp𝑄𝑅𝑇1−𝑇10