Page 50 - Engineering Plastics Handbook

24 Introduction

Maxwell spring-and-dashpot model
The Maxwell model integrates the elastic and viscous behavior of a ther-
moplastic by combining the spring and dashpot, providing a simple model
for viscoelastic polymers. The spring and dashpot are in series [10].

Stress = σ = E ε
spring s
Stress = σ = η d ε
dashpot d
dt
where E = Young’s modulus, MPa (psi)
η = viscosity of newtonian fluid (oil) in dashpot, Pa⋅s
Total strain ε= strain in the spring ε + strain in the dashpot ε d
s
The Maxwell model represents elastic creep, elastic recovery, and per-
manent set. In general, the Maxwell model provides reasonable expres-
sions for stress relaxation, and the Voigt-Kelvin model provides reasonable
expressions for creep behavior. Creep occurs when a constant stress is
instantly or rapidly applied over a period of time, and a corresponding
strain increases over that time. The instant or rapid application of stress
to a Maxwell model causes instantaneous stretching of the spring, which
reaches an equilibrium strain ε. Under constant stress the dashpot con-
tinues to extend with time. When the stress is released, the spring imme-
diately contracts to an amount equal to its extension, which is elastic
recovery. The dashpot does not recover, leaving a permanent set, which is
the amount that the dashpot extended during the time the constant stress
was applied. This model may not be easy to fit to experimental quantita-
tive data, but it demonstrates some viscoelastic characteristics [11, 12].

Voigt spring-and-dashpot model
The Voigt model spring and dashpot are parallel. The model is a con-
ventional concept for understanding stress and strain relationships
when load is applied to a viscoelastic material [10].

Voigt-Kelvin spring-and-dashpot models
Spring-and-dashpot models are extended by the Voigt-Kelvin (V-K)
model, which broadens linear viscoelastic concepts. The spring and
dashpot are always in parallel. The V-K spring-and-dashpot models are
useful for understanding creep behavior [11].
+σ
σ=σ s d
where σ= total stress acting on a V-K spring-and-dashpot model,
MPa (psi)