Mechanical Modeling of Material Behavior - Continuum Mechanics of Anisotropic Materials - page 16

Fig. 1.9 Lumped two-
parameter models.
( a ) The Maxwell element;
( b ) the Voigt element; and
( c ) the consolidation or
Terzaghi element
Each of the three lumped parameter models described has been characterized by
a single constitutive parameter, namely the spring constant, the damping constant,
and the permeability constant. The next higher level of lumped parameter models is
characterized by combinations of these elementary models with two constitutive
parameters, and the level after that by three-parameter models. There are models
with more than three parameters, but they are less useful. The three two-parameter
models of interest, the Maxwell model, the Voigt model, and the consolidation
model are illustrated in Fig. 1.9 . James Clerk Maxwell (1831-1879) was a Scottish
natural philosopher who first formulated the basic equations of electromagnetism
(“The Maxwell Equations”), and Woldemar Voigt (1850-1919) was a German
theoretical physicist who wrote a classic volume on crystal physics. The Maxwell
model is a combination of a spring and a dashpot in series. When a force applied to
a Maxwell model is changed from 0 to a finite value at an instant of time and held
constant thereafter, there is an instantaneous initial elastic extension and then there
is a continued deformation forever as the damper in the dashpot is drawn through
the dashpot cylinder. Thus a Maxwell model exhibits the characteristics of a fluid
with an initial elastic response.
Neither the Maxwell model nor the Voigt models are considered to be particu-
larly good models of the force-deformation-time behavior of real materials.
The Terzaghi consolidation model element is a special lumped two-parameter
model peculiar to soil mechanics (see, for example (Terzaghi 1943 )) that has
applications in geomechanics and biomechanics to interstitial water flow in both
hard and soft tissues. The model is constructed by combining a spring element and a
Darcy or permeability element in parallel (Fig. 1.9c ). It was developed by Karl
Terzaghi (1883-1963) about 1923 to explain the settlement or consolidation of the
soils under the foundations of buildings built on porous, water-saturated soils.
A sponge easily illustrates the mechanical principles involved. If the sponge is
waterlogged, then its compression under loading can only proceed as fast as the
water can drain from the sponge. When a load is initially placed on the piston in