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 4

Jeffrey viscoelastic medium.
              Model Constitutive equation
        1 𝜀 = 𝜂 1 𝜕 𝑡 + 1 𝐸 𝑣 2 1 1 + 𝜏 2 𝜕 𝑡 𝜎 ; the differential operator 𝜕 𝑡 ≡ 𝜕 / 𝜕 𝑡
        Viscoelastic Operator that Corresponds to the Modulus of Elasticity 𝐸
                 1 𝐸 → 1 𝜂 1 𝜕 𝑡 + 1 𝐸 𝑣 2 1 1 + 𝜏 2 𝜕 𝑡
              The modified Creep Model
          𝑢 ( 𝑡 ) ℎ 0 = ⎛ ⎜ ⎜ ⎝ 𝑏 𝜒 𝜂 1 𝜂 2 𝑃 0 𝐿 0 ( 𝜂 1 + 𝜂 2 ) ⎞ ⎟ ⎟ ⎠ 1 / ( 𝜒 + 1 ) 𝑡 1 / ( 𝜒 + 1 ) Γ ( ( 𝜒 + 2 ) / ( 𝜒 + 1 ) ) × Φ 2 ⎛ ⎜ ⎜ ⎝ − 1 𝜒 + 1 , 1 𝜒 + 1 ; 𝜒 + 2 𝜒 + 1 , − 𝑡 𝜏 2 , − 𝐸 𝑣 2 𝜂 1 + 𝜂 2 𝑡 ⎞ ⎟ ⎟ ⎠ ,
     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 !
     where ( 𝑎 ) 𝑘 is Pochhammer's notation, that is, ( 𝑎 ) 0 = 1 , ( 𝑎 ) 1 = 𝑎 , ( 𝑎 ) 𝑘 = ( 𝑎 ) 𝑘 − 1 ( 𝑎 + 𝑘 − 1 )