Microdamage Modeling in Laminates

Abstract

During service life structures made of laminated composites are subjected to complex combinations of thermo-mechanical and environmental loads. The final macroscopic failure of composite laminate is preceded by initiation and evolution of several microdamage modes in layers. This is because the transverse tensile strain to failure of unidirectional composites is lower than other failure strain components. Therefore transverse cracking of layers with off-axis orientation with respect to the main load direction, caused by combined action of transverse tensile stress and shear stress, is usually the first mode of damage (Parvizi and Bailey, J. Mater. Sci. 13:2131–2136, 1978 [1]; Jamison et al., ASTM STP 836:21–55, 1984 [2]).

Appendix: Expressions for COD and CSD

The COD, $u_{2an}^{0}$ of non - interactive crack is considered in a coordinate system where the cracked layer has 90-orientation with respect to x-axis. In other words x-direction is direction 2 for the layer with crack. Index k denoting the layer is omitted in expressions below. A distinction has to be made between cracks in surface layers and cracks in inside layers. Obviously the normalized average COD of surface cracks is larger because the cracked layer is supported only from one side. The fitting expressions are presented for symmetric case where the bottom support layer has equal properties, orientation and geometry as the top support layer. The expression for $u_{2an}^{0}$ is

$$u_{2an}^{0} = A + B\left( {\frac{{E_{2} }}{{E_{x}^{S} }}} \right)^{n}$$

(6.45)

In (6.45) $E_{x}^{S}$ is the Young’s modulus of the support layer measured in the x-direction which is the transverse direction for the cracked layer. For a crack in internal layer

$$\begin{aligned} A & = 0.52\quad B = 0.3075 + 0.1652\left( {\frac{{t_{90} }}{{2t_{s} }} - 1} \right) \\ n & = 0.030667\left( {\frac{{t_{90} }}{{2t_{s} }}} \right)^{2} - 0.0626\frac{{t_{90} }}{{2t_{s} }} + 0.7037 \\ \end{aligned}$$

(6.46)

In (6.46) $t_{s}$ is thickness of the adjacent support layer and $t_{90}$ is thickness of the cracked layer.

For a crack in surface layer

$$\begin{aligned} A & = 1.2\quad B = 0.5942 + 0.1901\left( {2\frac{{t_{90} }}{{t_{s} }} - 1} \right) \\ n & = - 0.52292\left( {\frac{{t_{90} }}{{t_{s} }}} \right)^{2} + 0.8874\frac{{t_{90} }}{{t_{s} }} + 0.2576 \\ \end{aligned}$$

(6.47)

Suggestions for calculations in more realistic cases when the support layers from different sides are different are given in [42].

Crack face sliding displacements (CSD), $u_{1an}^{0}$, see [20] for details, also follows a power law

$$u_{1an}^{0} = A + B\left( {\frac{{G_{12} }}{{G_{xy}^{S} }}} \right)^{n}$$

(6.48)

In (6.48) $G_{xy}^{S}$ is the in-plane shear modulus of the support layer.

For cracks in internal layer

$$A = 0.3\quad B = 0.066 + 0.054\frac{{t_{90} }}{{2t_{s} }}\quad n = 0.82$$

(6.49)

For cracks in surface layer

$$A = 0.6 \quad B = 0.134 + 0.105\frac{{t_{90} }}{{t_{s} /2}}\quad n = 0.82$$

(6.50)

Expressions (6.45)–(6.47) and (6.48)–(6.50) show that the normalized average COD and CSD are larger for less stiff surrounding layers and approach to certain asymptotic value with increasing support layer and cracked layer stiffness ratio. For thicker support layers the COD and CSD is smaller. This effect of neighbouring layers on the crack face displacements is called “constraint effect”.

Due to nonlinear shear stress-shear strain response the secant shear modulus of the layer will change with increasing laminate strain and will affect the value of $u_{1an}^{0}$ calculated according to (6.48).

When the distance between cracks decreases (high dimensionless crack density) the stress perturbation regions of individual cracks overlap and the normalized average COD and CSD start to decrease. The $u_{2an}^{k}$ has been related to COD of non-interactive cracks, $u_{2an}^{0k}$ by relationship [40]

$$u_{2an}^{k} = \lambda_{k} \left( {\rho_{kn} } \right)u_{2an}^{0k}$$

(6.51)

The crack interaction function $\lambda$ is a function of the crack density in the layer and generally speaking it depends on material and geometrical parameters of the cracked layer and surrounding layers. For non-interactive cracks $\lambda = 1$.

Detailed analysis of the effect of different parameters on interaction function was performed in [40] using FEM. Weak interaction (2–5 %) is observable at normalized spacing $2l_{90} /t_{90} = 2.5$. Further decrease of spacing leads to dramatic drop of the values of the interaction function to 0.3. The interaction of cracks in Glass fiber/epoxy laminates is stronger than in Carbon fiber/epoxy laminates. In the latter at high stiffness ratio the interaction function is not sensitive to layer thickness ratio. In the former with lower layer stiffness ratio the interaction is stronger if the support layer is thicker.

The calculated values of the interaction function were fitted by an empirical relationship with an origin in a simple shear lag model. The interaction function according to the shear lag model is

$$\lambda_{k} = { \tan }h\left( {\frac{k}{{\rho_{kn} }}} \right)$$

(6.52)

The shape function (6.52) was used to obtain the k value from the best fit. The best fit with this function to data corresponding to CF laminates (k _CF  = 1.12) and for GF laminate (k _GF  = 0.84). The interaction effect on $u_{2an}^{k}$ for cracks in surface layer was analyzed in [41] where also more accurate interaction functions for internal cracks are presented. The effect of nonuniform crack distribution on COD was analyzed in [21].