Hierarchical modelling of a polymer matrix composite

Abstract

A hierarchical modelling scheme to predict the properties of a polymer matrix composite is introduced. The stress–strain curves of amine-cured tetraglycidyl 4,4′-diaminodiphenylmethane (TGDDM) cured have been predicted using group interaction modelling (GIM). The GIM method, originally applied primarily to linear polymers, has been significantly extended to give accurate, consistent results for TGDDM, a highly crosslinked two-component matrix. The model predicts a complete range of temperature-dependent properties, from fundamental energy contributions, through engineering moduli to full stress–strain curves through yield. The predicted properties compare very well with experiment. Using the GIM-predicted TGDDM stress–strain curve, a 3D finite element model is used to obtain strain concentration factors (SCF) of fibres adjacent to a fibre break in a unidirectional (UD) composite. The strain distribution among the intact neighbouring fibres is clearly affected by the yielding mechanism in the resin matrix. A Monte Carlo simulation is carried out to predict the tensile failure strain of a single composite layer with the thickness equal to the fibre ineffective length. The effect of matrix shear yielding is introduced to the model through the SCF of surviving fibres adjacent to the fibre-break. The tensile failure strain of the composite is then predicted using a statistical model of a chain of composite layers.

Glossary

A :

Loss factor

B :

Bulk modulus

C _p :

Heat capacity

E :

Young’s modulus in tension

E _total :

Total GIM energy

E _coh :

Cohesive energy

E_coh (0 K):

Cohesive energy at 0 K

f :

Characteristic vibrational frequency of the polymer chain

H _T :

Thermal energy

ΔH_β :

Activation energy of the beta transition

h:

Planck’s constant

k:

Boltzmann’s constant

L :

Length of the mer unit

M :

Molecular weight of the mer unit

N :

Degrees of freedom

ΔN :

Degrees of freedom change for a single beta event

R:

Gas constant

r :

Strain rate

T :

Temperature

T _g :

Glass transition temperature

T _β :

Beta transition temperature

ΔT :

Temperature change for a single beta event

tanΔ_β :

Cumulative loss tangent through the beta transition

tanδ :

Local total loss tangent

V :

Volume of the mer unit

V _w :

van der Waal’s volume of the mer unit

z _i :

Normal random numbers

z _l :

Probability of layer failure

α _l :

Linear thermal expansion coefficient

α _v :

Volumetric thermal expansion coefficient

ε :

Full strain

ε _e :

Elastic strain contribution

$\bar \varepsilon _{{{\text{composite}}}}$ :

Average failure strain of a size of composite

$\bar \varepsilon _{{{\text{if}}}}$ :

Average failure strain of composite layer

μ:

Mean fibre failure strain

ν :

Poisson’s ratio

θ ₁ :

Debye temperature normal to polymer chain axis

ρ :

Density

σ :

Standard fibre failure strain

σ _c :

Compressive stress

σ _l :

Standard deviation of layer failure strain

σ _y :

Compressive yield stress