Page 243 - Mechanics of Asphalt Microstructure and Micromechanics

Models for Asphalt Concrete 235

Where W i = dissipated energy per cycle; W n = dissipated energy at cycle n; and
n
∑ W = total sum of dissipated energy up to cycle n. Then fatigue life, based on the DER
i
i=1
criterion, can be represented as:
N = K ( 1 ) K 1 (7-100)
f 2 W
i
Where K 1 and K 2 are the slope and the Y intercept, respectively, of the fitted fatigue
curves (W i versus N p ) for a given asphalt binder. The slope and the Y intercept from fa-
tigue curves obtained from different types of fatigue testing were used in the equation
to determine the number of cycles to failure (N f ).
The major limitations for all the fatigue models include the criteria for fatigue life;
the testing boundary conditions; and the lack of fundamental mechanism at micro-
scopic levels.

7.11 Cohesive Zone Models for Numerical Simulations

7.11.1 General Ideas
The idea to describe fracture as a material separation across a surface was started by
Barenblatt (1962). Generally, it is called a cohesive zone model. The cohesive zone is a
surface in a bulk material where displacement discontinuities occur. Continuum is ex-
tended with discontinuities in the form of displacement jump which requires addi-
tional constitutive description. Equations relating normal and tangential displacement
jumps across the cohesive surfaces with the proper tractions define a specific cohesive
zone model.
Based on the elementary functions used in cohesive zone models, they can be clas-
sified as (1) multilinear, (2) polynomial, (3) trigonometric, and (4) exponential. Almost
all these models are constructed as follows: tractions increase, reach a maximum, and
then approach zero with increasing separation. Needleman (1987) introduced the cohe-
sive zone models in computational practice. Since then, cohesive zone models are used
widely in finite element simulations of crack tip plasticity and creep; crazing in poly-
mers; adhesively bonded joints; interface cracks in bimaterials; delamination in com-
posites and multilayered materials; fast crack propagation in polymers, and so on.
Four typical cohesive zone models are briefly introduced here. They present rela-
tionships between surface tractions T and displacement jumps Δ across the cohesive
zone. Maximum surface traction T max is the cohesive strength. The corresponding dis-
placement jump is Δ max . Dimensionless parameters used are:

σ = T (7-101)
T
max
Δ
δ = Δ (7-102)
max
The work of separation is:
J = ∫ TdΔ (7-103)

Or dimensionless:
φ = J (7-104)
T Δ
max max