Mathematical Problems in Engineering

Research Article

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

Table 3

Standard Linear Solid viscoelastic medium.
              Model Constitutive equation
        1 𝜀 = 𝐸 𝑣 1 + 1 𝐸 𝑣 2 1 1 + 𝜏 𝜕 𝑡 𝜎 ; the differential operator 𝜕 𝑡 ≡ 𝜕 / 𝜕 𝑡
        Viscoelastic Operator that Corresponds to the Modulus of Elasticity 𝐸
                 1 𝐸 → 1 𝐸 𝑣 1 + 1 𝐸 𝑣 2 1 1 + 𝜏 𝜕 𝑡
              The modified Creep Model
         𝑢 ( 𝑡 ) ℎ 0 = 𝑏 𝜒 𝐸 𝑣 1 𝑃 0 𝐿 0 1 / ( 𝜒 + 1 ) Φ 2 − 1 , 1 𝜒 + 1 𝐸 𝜒 + 1 , 1 , − 𝑣 1 + 𝐸 𝑣 2 𝜂 2 𝑡 𝑡 , − 𝜏
             The modified Thermal Creep Model
     𝑢 ( 𝑡 ) ℎ 0 = 𝑏 𝜒 𝐸 𝑣 1 𝑃 0 𝐿 0 1 / ( 𝜒 + 1 ) Φ 2 − 1 , 1 𝜒 + 1 𝐸 𝜒 + 1 , 1 , − 𝑣 1 + 𝐸 𝑣 2 𝜂 2 𝑡 𝑡 , − 𝜏 𝑄 × e x p 𝑅 1 𝑇 − 1 𝑇 0 ,
where Φ 2 is the confluent hypergeometric function of two variables (Appell function)
              which is defined by:
       Φ 2 ∑ ( 𝛼 , 𝛽 , 𝛾 , 𝑥 , 𝑦 ) = ∞ 𝑚 = 0 ∑ ∞ 𝑛 = 0 ( 𝛼 ) 𝑚 ( 𝛽 ) 𝑛 ( 𝛾 ) 𝑚 + 𝑛 𝑥 𝑚 𝑦 𝑛 𝑚 ! 𝑛 ! , and ( 𝑥 ) 𝑛 = ( 𝑥 + 𝑛 − 1 ) ! ( 𝑥 − 1 ) !