Determine the creep compliance J(t) of a material that responds like:
a) an elastic spring with Young's modulus E
J(t):
b) a viscous dashpot with viscosity η as a function of time t)
J(t):
c) the Maxwell model (E for spring, and η for dashpot as a function of time t)
J(t):
d) the Voigt model (E for spring, and η for dashpot as a function of time t)
J(t):
To determine the creep compliance J(t) for different material models, we need to understand the behavior of each model and how it relates to time.
a) Elastic spring:
An elastic spring obeys Hooke's Law, which states that the deformation is directly proportional to the applied force and follows a linear relationship. The creep compliance of an elastic spring is given by J(t) = 1 / E, where E represents the Young's modulus of the material.
b) Viscous dashpot:
A viscous dashpot exhibits viscous behavior, meaning that the deformation is directly proportional to the rate of loading. The creep compliance for a viscous dashpot changes with time and follows a logarithmic relationship. The equation for the creep compliance of a viscous dashpot is J(t) = (1 / η) * ln(t), where η represents the viscosity of the material.
c) Maxwell model:
The Maxwell model combines the behavior of an elastic spring and a viscous dashpot in series. This means that both the spring and dashpot experience the same applied stress but respond differently over time. The creep compliance of the Maxwell model is the sum of the creep compliance of the spring and dashpot. Therefore, J(t) = (1 / E) + (1 / η) * ln(t).
d) Voigt model:
The Voigt model combines the behavior of an elastic spring and a viscous dashpot in parallel. This means that both the spring and dashpot experience the same deformation but respond differently to applied stress over time. The creep compliance of the Voigt model is the sum of the creep compliance of the spring and dashpot. Therefore, J(t) = (1 / E) + (1 / η) * t.
So, to determine the creep compliance J(t) for each material model, use the respective equations provided above based on the material properties (Young's modulus for the spring, and viscosity for the dashpot), along with the time variable t.