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) of different material models, we need to understand the behavior of each model and its corresponding mathematical expression for J(t). Let's break down each model:
a) Elastic spring with Young's modulus E:
In an elastic material, the deformation is instantaneous and fully recoverable when the load is removed. The creep compliance J(t) for the elastic spring is zero since there is no time-dependent deformation.
J(t) = 0
b) Viscous dashpot with viscosity η:
In a viscous material, the deformation is time-dependent and gradually accumulates over time. The creep compliance J(t) for the viscous dashpot is proportional to time t.
J(t) = η * t
Here, η is the viscosity of the material.
c) Maxwell model (spring and dashpot in series):
The Maxwell model combines the behavior of an elastic spring and a viscous dashpot connected in series. The strain in the spring component and the viscous component combine additively.
J(t) = E * t + η * t
Where E is the Young's modulus of the spring component and η is the viscosity of the dashpot component.
d) Voigt model (spring and dashpot in parallel):
The Voigt model combines the behavior of an elastic spring and a viscous dashpot connected in parallel. The stresses in the spring and dashpot components sum up.
J(t) = E * t + η / t
Where E is the Young's modulus of the spring component and η is the viscosity of the dashpot component.
These equations provide the mathematical expressions for the creep compliance J(t) for different material models.