Page 51 - Engineering Plastics Handbook

Products and Design 25

σ = stress acting on spring element, MPa (psi)
s
σ = stress acting on dashpot element, MPa (psi)
d
Total strain ε in a Voigt-Kelvin spring-and-dashpot model [11] is
ε=ε +ε d
s
where ε = strain in spring element, cm/cm (in/in) %
s
ε = strain in dashpot element, cm/cm (in/in) %
d
Three-, four-, and five-parameter models
Spring-and-dashpot models developed for three-, four-, and five-param-
eter models evolve from the linear elastic (spring) and viscous (dashpot)
elements in Maxwell and Voigt-Kelvin models. A larger number of
hookean spring and newtonian dashpot elements may be needed to
obtain a more accurate description of viscoelastic responses. An exam-
ple of a three-element standard linear solid model (SLSM) has a second
hookean spring element added parallel to the conventional hookean
spring and in series with the newtonian dashpot element. The three-
element model depicts a more realistic stress relaxation time and creep
behavior with addition of the second spring [11–13].
To calculate the stress relaxation time for the standard linear solid
model with a second spring added parallel to the initial spring, use

 1 1 
τ* = η +
m   E* E  
m
where τ* = stress relaxation time
= viscosity of Maxwell dashpot oil, Pa⋅s (cP)
η m
E* = modulus of additional spring, MPa (psi)
E = modulus of Maxwell spring, MPa (psi)
m
To calculate the stress relaxation time for the Maxwell model alone, use
η
τ* = m
E
m
To calculate the shear modulus of elasticity G at any given time t, use

τ η
G = =
γ t

where τ= shear stress
γ= shear strain
η = newtonian shear viscosity