Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 109

idea for Hooke's law is that of a spring. If a force displaces the end of a spring, there
is a relationship between the force and the resulting displacement. Thus, to develop
Hooke's law, the stress T at a particle X is expressed as a function of the variation in
the displacement field u ( X , t ) in the neighborhood of X , N ( X ),

T

¼ T

all X in N

ð u ð X ; t Þ; X Þ; ð X Þ: (5.1H)

The constitutive idea for the Newtonian law of viscosity is that of the dashpot or
damper, namely that the force is proportional to the rate at which the deformation is
accomplished rather than to the size of the deformation itself. The total stress in a
viscous fluid is the sum of the viscous stresses T v plus the fluid pressure
p , T

T v . The constitutive idea for the Newtonian law of viscosity is
that the stress T v , due to the viscous effects at a particle X , is expressed as a function
of the variation in the velocity field v ( X , t ) in the neighborhood of X , N ( X ). The
expression for the total stress in a fluid is the pressure plus the viscous stresses.

¼ p1 þ

T ¼

pU þ T v ðvðX;

all X in N

t Þ; XÞ; ðXÞ: (5.1N)

Recall that U is the six-dimensional vector with components {1, 1, 1, 0, 0, 0}; it is
the image of the three-dimensional unit tensor 1 in six dimensions. Each of the four
constitutive ideas described yields the value of a flux or stress at time t due to the
variation in a field (temperature, pressure, displacement, velocity) at the particle X
at time t . The constitutive idea for viscoelasticity is different in that the stress at
time t is assumed to depend upon the entire history of a field, the displacement field.
Thus, while the first three constitutive ideas are expressed as functions, the consti-
tutive idea for viscoelasticity is expressed as a functional of the history of the
displacement field. A functional is like a function, but rather than being evaluated at
a particular value of its independent variables like a function, it requires an entire
function to be evaluated; a functional is a function of function(s). An example of a
functional is the value of an integral in which the integrand is a variable function.
The constitutive idea for a viscoelastic material is that the stress T at a particle X is a
function of the variation in the history of the velocity field v ( X , t ) in the neighbor-
hood of X , N ( X ),

1

T T

for all X in N ¼ ð v ð X ; t s Þ ,s, X Þ d s ; ð X Þ; (5.1V)

s¼ 0

where s is a backward running time variable that is 0 at the present instant and
increases with events more distant in the past. Thus the stress T at a particle X is a
function of the entire history of the displacement of the particle; to evaluate the
stress, knowledge of the entire history is required. In the sections that follow this one
these four constitutive ideas will be developed into linear constitutive equations.