Designing curing conditions in order to analyse the influence of process-induced stresses upon some mechanical properties of carbon / epoxy laminates at constant T_g and degree of cure

Abstract

In the aim to study of the effects of process-induced stresses upon some mechanical properties of simple stacking sequence composite laminates, three different cure cycles were designed. This was achieved on the basis of a cure kinetics modelling. A minimum number of curing parameters were modified in order to restrain changes in material physical properties. Those latter were determined from the molecular network of the thermosetting matrix to the fibres volume fraction and thermal expansion of the laminated plates. The level of process-induced stresses generated by the three cure cycles was assessed by measuring the out-of-plane deflection on unsymmetrical laminated strips. Manufactured laminates were submitted to a set of tensile tests in order to determine the changes in their mechanical properties due to the different curing conditions. All the mechanical tests were followed by acoustic emission. This has enabled to distinguish the occurrence of the various damaging processes with respect to curing conditions and therefore process-induced stresses. Lastly, process-induced stresses on a micromechanical level were theoretically determined. This determination has enabled a first analysis of the effects of process-induced stresses upon mechanical properties to be settled.

Appendices

Appendix 1: Relations for the calculation of process-induced stresses on a micromechanical level

Stresses in the fibre (f) are given by:

$$ \left\{ {\begin{array}{*{20}{c}} {{\sigma_{zz}} = 2 \cdot {C_{11}} \cdot {\chi_f} + \frac{{{C_{33}} \cdot {\omega_f}}}{{{C_{13}} + {G_{13}}}}} \\{{\sigma_{rr}} = {\sigma_{\theta \theta }} = \left( {{C_{11}} + {C_{12}}} \right) \cdot {\chi_f} + \frac{{{C_{13}} \cdot {\omega_f}}}{{{C_{13}} + {G_{13}}}}} \\{{\sigma_{r\theta }} = {\sigma_{rz}} = {\sigma_{\theta z}} = 0} \\\end{array} } \right. $$

(A-1)

Stresses in the matrix (m) cylinder are given by:

$$ \left\{ {\begin{array}{*{20}{c}} {{\sigma_{zz}} = 2 \cdot {\lambda_m} \cdot {\chi_m} + \frac{{\left( {{\lambda_m} + 2{\mu_m}} \right) \cdot {\omega_m}}}{{{\lambda_m} + {\mu_m}}}} \\{{\sigma_{rr}} = 2 \cdot \left( {{\lambda_m} + {\mu_m}} \right) \cdot {\chi_m} + \frac{{{\lambda_m} \cdot {\omega_m}}}{{\left( {{\lambda_m} + {\mu_m}} \right)}} - \frac{{2 \cdot {\mu_m} \cdot {\varsigma_m}}}{{R_m^2}}} \\{{\sigma_{\theta \theta }} = 2 \cdot \left( {{\lambda_m} + {\mu_m}} \right) \cdot {\chi_m} + \frac{{{\lambda_m} \cdot {\omega_m}}}{{\left( {{\lambda_m} + {\mu_m}} \right)}} + \frac{{2 \cdot {\mu_m} \cdot {\varsigma_m}}}{{R_m^2}}} \\\end{array} } \right. $$

(A-2)

With: χ _f , ω _f , constants (fibre); C _ij , G _ij , stiffness coefficients of carbon fibre; λ _m , μ _m , Lamé coefficients of matrix and χ _m , ω _m , ζ _m , constants (matrix).

For matrix: σ_rθ  = σ_θz  = σ _rz  = 0

Appendix 2: Inverse determination of carbon fibres transverse coefficient of thermal expansion (CTE)

In order to determine the transverse coefficient of thermal expansion of a carbon fibre, the unidirectional ply was divided into 2 materials. As shown in Fig. 16, its thickness h, was separated into an isotropic material made of matrix (thickness h_m) and a transversely isotropic material corresponding to carbon fibre (thickness h_f). The following equations (Eqs. A-3 to A-XX) show how aft can be thus determined, provided that Vf% is known:

Fig. 16

The unidirectional composite ply (a) and its equivalent multi-material made of a layer of matrix and a layer of carbon fibre material (b)

$$ h = {h_m} + {h_f} $$

(A-3)

Where:

$$ {h_f} = {V_f} \cdot h $$

(A-4)

And:

$$ {h_m} = \left( {1 - {V_f}} \right) \cdot h = {V_m} \cdot h $$

(A-5)

The changes in thicknesses Δh due to temperature differences are given by:

$$ \Delta h = \Delta {h_m} + \Delta {h_f} $$

(A-6)

With:

$$ \left\{ {\begin{array}{*{20}{c}} {\Delta h = {\alpha_2} \cdot h \cdot \Delta T} \\{\Delta {h_m} = {\alpha_m} \cdot {h_m} \cdot \Delta T} \\{\Delta {h_f} = \alpha_f^t \cdot {h_f} \cdot \Delta T} \\\end{array} } \right. $$

(A-7)

In Eq. (A-7): α₂, transverse CTE of unidirectional ply (cf. Table 1); α_m, CTE of matrix, α ^t_f , transverse CTE of carbon fibre.

$$ {\alpha_2} \cdot h \cdot \Delta T = {\alpha_m} \cdot {h_m} \cdot \Delta T + \alpha_f^t \cdot {h_f} \cdot \Delta T $$

(A-8)

$$ \Rightarrow \alpha_f^t = \frac{{{\alpha_2} - {\alpha_m} \cdot {V_m}}}{{{V_f}}} $$

(A-9)

Lastly, if not available, the CTE of matrix α_m can be obtained from various modelling such as the one proposed by C.C. Chamis In: Simplified composite micromechanics equations for Hygral, Thermal and Mechanical properties. SAMPE Quarterly, 15(3): 14–23 (April 1984):

$$ {\alpha_1} = \frac{{E_1^f \cdot \alpha_1^f \cdot {V_f} + {E_m} \cdot {\alpha_m} \cdot {V_m}}}{{E_1^f \cdot {V_f} + {E_m} \cdot {V_m}}} $$

(A-10)

$$ {\alpha_2} = \alpha_2^f \cdot \sqrt {{{V_f}}} + \left( {1 - \sqrt {{{V_f}}} } \right) \cdot \left( {1 + \frac{{{V_f} \cdot {\nu_m} \cdot E_1^f}}{{{E_1}}}} \right) \cdot {\alpha_m} $$

(A-11)

Where V_f is the composite fibres volume fraction, E ^f_l , the fibres longitudinal (// to axis) tensile modulus and E_l the unidirectional ply tensile modulus.

Appendix 3: Thermoviscoleastic behaviour of unreinforced matrix

The unreinforced epoxy matrix samples were cured according to mrcc (120 min. at 180°C). After curing they were submitted to an accelerated thermoviscoelastic characterisation procedure in order to get the parameters of behaviour. The matrix was considered to be isotropic and the visoelastic behaviour to be linear. This procedure has consisted in dynamical thermal mechanical analyses carried out on a Polymer Lab MKII DMTA between 30 and 280°C (ramp at 3°C/min) at 8 different frequencies: 0.03, 0.1, 0.3, 1, 2, 5 10 and 50 Hz under a uniaxial tensile loading. Static and dynamic forces were respectively set at 1 N and 0.3 N. This enables to get a master curve describing the changes in matrix modulus E(T,t) as a function of Log(a_T) (Fig. 17). Log (a_T) is itself a function of temperature T (Fig. 18). The matrix master curve was obtained by applying the time-temperature superposition principle (Eq. A-12) (see refs. [27, 70]) with 180°C taken as the reference temperature (T_R) since the largest changes in matrix storage modulus E’ were recorded at 180°C. In the process towards the building of a master curve a_T is the horizontal shift factor either given by an Arrhenius-type Eq. (A-13) or a WLF-type Eq. (A-14) depending on considered temperature:

Fig. 17

Changes in –Log(aT) as a function of temperature

Fig. 18

Master curve of the unreinforced epoxy matrix. Tensile modulus as a function of Log(a_T)

$$ E\prime \left( {{t_1},{T_1}} \right) = E\prime \left( {{t_2},{T_R}} \right) $$

(A-12)

$$ \log {a_T} = \frac{{\Delta {H_a}}}{{2,303 \cdot R}} \cdot \left( {\frac{1}{T} - \frac{1}{{{T_R}}}} \right)\quad when\quad T < {T_R} $$

(A-13)

$$ log\;{a_T} = \frac{{ - c_1^R \cdot \left( {T - {T_R}} \right)}}{{c_2^R + \left( {T - {T_R}} \right)}}\quad when\quad {\hbox{T}} > {{\hbox{T}}_{\rm{R}}} $$

(A-14)

In Eqs. (A-13) and (A-14): $\Delta {H_a}$ is activation energy, R is the universal gas constant and $c_1^R$ and $c_2^R$ are constants relatives to T _R . They can be determined from Fig. 18 experimental results.