Senior Laboratory -- Composites

Constitutive Equations

The following relationships describe the behavior of a composite cure and are used within the TGCure simulation. Equations referenced with a page number are taken from Travis Bogetti's PhD thesis, CCM Report 89-32.

Boundary Conditions

a*dTs/dn + b*Ts + c*T(t) = 0 (page 20)

T(t) is the temperature outside the system (water or air temp) as a function of time, which defines the cure cycle
Ts is the temperature at the surface of the composite
n is the normal direction (z for through thickness)
a, b, c are coefficients defined below

Coefficients are derived from the assumption that the heat flow entering the composite is equal to that traveling through the mold.

q/A = -kc*dTs/dn = ho*(Ts - T(t))

kc*dTs/dn + ho*Ts - ho*T(t) = 0

dTs/dn + ho/kc*Ts - ho/kc*T(t) = 0

kc is the thermal conductivity of the composite
ho is the overall heat transfer coefficient through the mold from T(t) to Ts

Hence the coefficients for the boundary conditions are:
a=1, b=ho/kc, c=-ho/kc for convective conditions
a=1, b=0, c=0 for insulating conditions
a=0, b=1, c=-1 for prescribed (given) conditions

Heat Transfer

rho*cp*dT/dt = kzz*d^2T/dz^2 + dq/dt (page 18)

rho is the density of the composite
cp is the heat capacity of the composite
kzz is the thermal conductivity of the composite in the z dimension (through thickness)
dq/dt is the heat rate resulting from the cure reaction

dq/dt = rho*mr*Hrxn*deoc/dt (page 25)

mr is the mass fraction of the resin in the composite
Hrxn is the heat of reaction of the cure
deoc/dt is the cure rate calculated from the kinetics

Kinetic Relationships

Autocatalytic Model
deoc/dt = k*(eoc^m)*(eocmax - eoc)^n (page 48)

Arrhenius Relationship
k = A * exp(-Ea/RT) (page 48)

Maximum Extent of Cure
eocmax = a*T + b

eoc is the extent of cure of the resin, sometimes shown as alpha
eocmax is the maximum extent of cure for a particular temperature
k is the reaction rate constant
m and n are model exponents, usually m + n = 2
A is the pre-exponential factor
Ea is the activation energy
R is the ideal gas constant

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