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Step 3: Prediction of thermal stresses
Using viscoelastic transformation theory, the compliance, D(t), can be related to the
relaxation modulus, Er, of the asphalt mix. Knowledge of this parameter, coupled with
the temperature data obtained from the EICM model, allows for the prediction of the
thermal stress at any given depth and time within the asphalt layer.
The relaxation modulus function is obtained by transforming the creep compliance
function. The relaxation modulus is represented by a generalized Maxwell model and
expressed by a Prony series relationship:
E( ) = E e- / i (3.3.63)
i
N+1
i=1
ξ ∑ ξ λ
where:
E(ξ) = Relaxation modulus at reduced time ξ.
Ei,λI = Prony series parameters for master relaxation modulus curve
(spring constants or moduli and relaxation times for the Maxwell
elements).
3.3.92
The knowledge of the relaxation modulus function allows for the computation of the
thermal stresses in the pavement according to the following constitutive equation:
ξ
ξ
ε
σ ξ ξ ξ
ξ
′
′
∫ ′ d
d
( ) = E( - ) d
0
(3.3.64)
where:
σ(ξ) = Stress at reduced time ξ.
E(ξ-ξ′) = Relaxation modulus at reduced time ξ-ξ′.
ε = Strain at reduced time ξ (= α (T(ξ′ ) - T0)).
α = Linear coefficient of thermal contraction.
T(ξ′) = Pavement temperature at reduced time ξ′.
T0 = Pavement temperature when σ = 0.
ξ′ = Variable of integration.
Step 4: Growth of the thermal crack length computation
Fracture mechanics (Paris’ Law) is used to compute the growth of the thermal crack
length within the asphalt layer. This is accomplished by knowledge of the stress intensity
factor, K, as well as the A and n fracture parameters obtained from the creep compliance
and strength of the mixture.
TCMODEL is used to predict the amount of transverse cracking expected in the
pavement system. As previously noted, the climatic input and viscoelastic properties
(compliance-relaxation modulus) allow for the computation of the thermal stress, at any
given time and location within the asphalt layer. Once this is accomplished, fracture
mechanics, based upon Paris’ Law, is used to compute the stress intensity and fracture
properties of the material.
The stress intensity parameter, K, has been formulated by developing a simplified
equation based upon theoretical FEM studies and results. From this analysis, it was
determined that K could be estimated from:
K = (0.45 + 1.99C0.56 ) (3.3.65)
σ o
where:
K = Stress intensity factor.
σ = Far-field stress from pavement response model at depth of crack tip.
Co = Current crack length, feet.
The crack propagation model used in the thermal fracture model is:
ΔC = AΔK n (3.3.66)
where:
ΔC = Change in the crack depth due to a cooling cycle.
ΔK = Change in the stress intensity factor due to a cooling cycle.