Page 205 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 197

brittle damage, ductile damage, creep damage, low-cycle fatigue damage, and high-
cycle fatigue damage. Brittle damage refers to the cases where plastic strain is smaller
than elastic strain. Ductile damage refers to the cases where plastic strains can be
equal or even larger than elastic strains. Creep damage refers to the cases when sig-
nificant creep strains occur (for example at an elevated temperature). Cyclic loading
also causes damage. At relatively higher stresses, the number of loading repetitions to
rupture could be smaller, for example, smaller than 10,000. This process is referred to
as low-cycle fatigue damage. If the number of loading repetitions to rupture is larger
than 10,000, it is called high-cycle fatigue damage.
All materials have inherent defects or induced damage and microstructure. The
representative material properties should be defined on a representative volume ele-
ment (RVE). Depending on the material structure, the distribution of inherent defects,
and induced damage, the sizes of the representative volume elements are different.
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3
Typically this size is (0.1 mm) for metals and ceramics; (1 mm) for polymers and most
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composites; (10 mm) for wood; and (100 mm) for concrete. Damage parameters and
the governing equations are valid for materials of sizes larger that the RVE.
6.5.2 Damage Parameter
Considering a section through an RVE at location M and in the n direction, the damage
is defined as:
δ S
DM n x) = Dx (6-164)
,
,
(
δ S
Where dS Dx and dS are the areas that cannot take stresses and the overall cross sec-
tion area; x is the location along the n direction. Typically, damage thus defined will
vary with x. The maximum value of the damage variable is then defined as the damage
value for location M in the orientation n.
[
D = Max D ]
,
( Mn) ( Mn x)
,
,
x ()
δ S
D ( Mn , ) = D (6-165)
δ S
Clearly, this damage variable is dependent on orientations as well. It can be usually
represented as a damage tensor or a damage effect tensor.
In the 1-D and homogeneous case, it reduces to a scalar D = S D . Clearly, this D may
S
vary between 0 and 1.
0 D 1
With D = 0, no damage; D = 1, complete damage. In reality, material will fail at D
much smaller than 1.

6.5.3 Effective Stress
In the same 1-D and homogeneous case for defining the damage parameter, the effec-
tive stress can be defined. In the simplest case where the force on the surface is in the
normal direction, the normal stress is defined as:
F
σ =
S (6-166)