Page 196 - MODELING OF ASPHALT CONCRETE

174 Cha pte r Se v e n

quantified by a set of parameters often referred to as internal state variables or damage
parameters in the context of thermodynamics of irreversible processes. The growth of
damage is governed by an appropriate damage (or internal state) evolution law. The
stiffness of the material, which varies with the extent of damage, is determined as a
function of the internal state variables by fitting the theoretical model to available
experimental data.
The mechanical behavior of an elastic medium with constant material properties
(i.e., without damage growth) can usually be described using an appropriate
thermodynamic potential (e.g., the Helmholtz free energy for isothermal processes or the
Gibbs free energy for isentropic processes). These potentials are point functions of
thermodynamic state variables. When thermal effects are not considered, both the
Helmholtz free energy and the Gibbs free energy potentials are identified with the so-
called strain energy and represent the energy stored in the system which is algebraically
equal to the work done on the system by external loading. However, when damage occurs
due to external loading, the work done on the body is not entirely stored as strain energy;
part of it is consumed in causing damage to the body. The amount of energy required to
produce a given extent of damage is expressed as a function of internal state variables. The
total work input to the body during the processes in which damage occurs depends, in
general, on the path of loading. However, it has been observed that, for certain processes
in which damage occurs, the work input is independent of the path of loading (Schapery
1987a; Lamborn and Schapery 1988, 1993).
Schapery (1990) applied the method of thermodynamics of irreversible processes
and the observed phenomenon of path independence of work in damage-inducing
processes to develop the work potential theory so that it may be applicable to describing
the mechanical behavior of elastic media with growing damage and other structural
changes. The theory is general enough to allow for strong nonlinearities and coupling
between the internal state variables and to describe a variety of mechanisms including
micro- and macrocrack growth in monolithic and composite materials. Sicking (1992)
applied the theory to model the damage-related material nonlinearity in graphite-epoxy
laminates, and Lamborn and Schapery (1993) showed the existence of a work potential
for suitably limited deformation paths using experimental data from axial and torsional
deformation tests on angle-ply fiber-reinforced plastic laminates. The elements of work
potential theory in terms of a strain energy formulation may be represented as follows:

Strain energy density function: W = W(,ε ij S ) (7-18)
m
∂ W
Stress-strain relationships: σ = ε ∂ (7-19)
ij
ij
∂W ∂W
Damage evolution laws: − = S (7-20)
∂S ∂S
m m
where s = stresses
ij
e = strains
ij
S = internal state variables (or damage parameters)
m
W = W (S ) = dissipated energy due to damage growth
S S m
The internal state variables, S (m = 1, 2,..., M), account for the effects of damage,
m
and the number of internal state variables (i.e., M) is typically determined by the number