A micromechanics-based interface mesomodel for virtual testing of laminated composites

The damage mechanisms on the microscale

Four scenarios can be distinguished. The first two mechanisms have been studied for laminated composites by the micromechanics community (see the reviews [1, 21–23]). Matrix microcracking (Scenario 1) is driven by the ply’s microstructure: usually, matrix microcracks originate perpendicular to the plane of the ply, then run throughout the ply’s thickness, and finally grow parallel to the fibers’ direction. Moreover, the microcracking pattern can be considered to be locally periodic: thus the amount of microcracking can be quantified by the microcracking rate ρ=H/L, where H is the ply thickness and L the distance between two cracks (see Figure 1). Local delamination (Scenario 2) generally occurs after the saturation of matrix microcracking: it is caused by the stress concentrations at the tips of the intraply cracks. This mechanism is quantified by the local delamination ratio τ=e/H, where e is the length of the delaminated zone (see Figure 1). Diffuse intra- and interply damage mechanisms (Scenarios 3 and 4) were introduced into the damage mechanics of laminates a long time ago, but they are usually not taken into account in micromechanics. As they occur at the fiber’s scale, they can be homogenized and they are introduced directly through damage variables and evolution laws at the ply’s scale.

In order to handle these mechanisms, a computational micromodel was introduced in [1, 23] and developed in [24, 29]. This micromodel reproduces the key points observed in the micromechanics of laminates [1, 23] quite well.

The bridge between micromechanics and mesomechanics

where π is the projection operator onto the plane and Γ is an arbitrary section of the unit cell perpendicular to vector N₃ (see Figure 1). Thus, there are two basic problems, one associated with in-plane loading and the other associated with out-of-plane loading.

The problem associated with out-of-plane loading, which defines the mesodescription of the interface, is summarized in Figure 1. Considering an interface Γ _j (in this case, a 3D matrix layer of thickness $\frac{H_e}{20}$, where H _e is the thickness of the elementary ply) between two cracked plies S _i and S_i+1, the upper part $S_{+}^{\mathrm{\prime \prime }}$ and the lower part $S_{-}^{\mathrm{\prime \prime }}$ of the laminate are homogenized. Periodic boundary conditions are defined. Uniform elementary loadings are introduced on the cracked surfaces: this residual problem can be superposed to an uncracked problem in order to obtain the full solution of the cracked cell under elementary loadings. More details on the definition of the interface problem can be found in [29].

Using the finite element method, the 3D reference problem on the microscale was solved for different sets of parameters (thickness, stiffness, ρ∈[0,0.7], τ∈[0,0.4]) which are likely to be encountered in practice, leading to a set of mesodamage indicators associated to the preferential directions of the interface (${\bar{\mathbf{N}}}_1$, ${\bar{\mathbf{N}}}_2$) defined in Figure 1. It was shown that the mesodamage of the interface depends only on the interface itself and on the microcracking rates of the adjacent plies [29].

Properties of the basic 2D interface problem

For the interfaces, which are considered to be thin 3D matrix layers made of isotropic material, the material properties are: E=2.4 GPa, ν=0.33, h=H _e /20.

The problem to be solved is elastic and follows the generalized plane strain assumption (i.e. the displacement in direction N₁ is constant). It has been proven that the mesobehavior of interface Γ _j depends only on interface Γ _j and ply S _i , i.e. on parameters λ=2τ ρ, ρ and on the ply thickness [29].

and the calculated points were τ=(0.1, 0.2) and ρ=(0.2, 0.4, 0.6, 0.8).

Now, let us introduce approximations for coefficients c₃₃, c₁₃ and c₂₃, which depend on λ=2τ ρ and ρ. These approximations are derived from the analysis of the extreme cases: small ρ, large ρ and λ equal to 0 or 1.

As shown in Figure 2, the following approximations work quite well:

$\begin{array}{ll}{\bar{d}}_{33,i}=\lambda ,{\bar{d}}_{13,i}=\lambda ,\frac{{\bar{d}}_{23,i}}{1-{\bar{d}}_{23,i}}=& \frac{\lambda }{1-\lambda }+A\left(\rho \right),A\left(\rho \right)=\frac{a\left(\rho \right)}{1-a\left(\rho \right)}\end{array}$

(7)

with the material function a(ρ) assumed to be linear (a(ρ)=0.5ρ for the material studied).

where $\lambda =2\mathrm{\tau \rho }=\frac{e}{L}$, λ^′=2τ^′ρ^′ and $\left(1-\bar{\lambda }\right)=\left(1-{\lambda }^{\prime }\right)\left(1-\lambda \right)$.

The microcracking/stiffness interaction of the interface mesomodel

The new interface mesomodel - stiffness and damage

with the coupling term ω written as ω=2 sinα cosα(A−A^′).

In previous papers [11, 12], a simplified expression was considered, based on $\bar{\rho }=\frac{\rho +{\rho }^{\prime }}{2}$. This expression is equivalent to (13) for ρ=ρ^′, τ small (i.e. d₃₃→0) and α∼ 45°.

It is remarkable that this energy depends only on ρ, ρ^′ and $\bar{\lambda }$. As mentioned previously, a(ρ) is a material function which can be identified from the basic 2D interface problem. In the present work, we used a linear law.

Computation of the dissipation

where $\dot{\omega }$ depends on $\dot{\rho }$ and ${\dot{\rho }}^{\prime }$.

Since $\dot{A}=\frac{\mathrm{d}A}{\mathrm{d}\rho }\dot{\rho }$ is positive, it follows that $\dot{D}\ge 0$; thus, the interface mesomodel is compatible with the principles of thermodynamics.

Delamination criterion for out-of-plane loading

where ${\left.\bar{Y}\right|}_t={\text{sup}}_{\tau \le t}{\left.Y\right|}_{\tau }$ and k, n, Y _c , Y₀ are material constants which can be identified using standard delamination tests. Let us note that the interface mesomodel is independent of the angle 2α between the fiber directions of the adjacent plies.

Delamination criteria for in-plane loading

In order to analyze the microdelamination due to in-plane loading, let us review the modeling of transverse microcracking going back to the basic 2D interface problem.

The modeling of microdelamination

With ρ constant, the energy release rates related to microdelamination can be calculated as λ-derivatives. For τ=0, they are equal to zero. Let us use finite fracture mechanics again and consider the τ values:

$\tau =(0.05,0.1,0.15,0.2);\mathrm{\Delta \tau }=0.05$

The curves giving the unit energy release rates are shown in Figure 3. It follows that the initiation criterion can be defined as:

(23)

where $Q_{22}^u$ and $Q_{12}^u$ are the unit energy release rates associated to ρ, τ=0.05, Δ τ=0.05. Q ^c and γ₁₂ are critical material values. One has:

$\begin{array}{l}\left\{\begin{array}{c}\dot{\tau }\ge 0g(\tilde{\sigma },\rho )\le 1\\ \dot{\tau }\left[g(\tilde{\sigma },\rho )-1\right]=0\end{array}\right.\end{array}$

(24)

Here, the out-of-plane effective stress ${\tilde{\sigma }}_{23}$ is not considered. Indeed, it is negligible except in high-gradient zones (e.g. because of edge effects), in which case it is taken into account by the interface model.

The curves of Figure 3 are either increasing or flat and show that in most cases the microdelamination mechanism is unstable. When it is activated, one can consider that the interface has been completely fractured; thus, Equations (23)-(24) can be viewed as a mesodelamination criterion.

A remark on the identification of the mesodelamination criterion

The criterion given in Equations (23)-(24) depends on two material constants Q ^c and γ₁₂ which can be identified by taking advantage of available experimental results related to microcracking saturation.

Let us consider the case ${\tilde{\sigma }}_{22}\ne 0$, ${\tilde{\sigma }}_{12}=0$. From [0 _m /90 _n ] _s tensile tests, one can identify the material constant ρ _s which represents the microcracking density at saturation [5]. From an energy point of view, this saturation is associated to the decrease in the microcracking strain energy release rate and the corresponding nearly constant strain energy release rate for microdelamination (see Figure 4). This quantity is associated with the onset of microdelamination. It follows that Q ^c can be identified as:

$\begin{array}{l}{Q}^c=\frac{Q_{22}^u\left({\rho }_s\right)}{G_I^u\left({\rho }_s\right)}{G}_I^c\left(\text{ply}\right)\end{array}$

(25)

where Q ^u and $G_I^u$ are evaluated for ρ=ρ _s . In the case ${\tilde{\sigma }}_{12}\ne 0$, ${\tilde{\sigma }}_{22}=0$, a saturation value seems to exist, but it may be different from that observed in mode I.

Remark

The constant γ₁₂ can be identified from a tensile test of a [+45/−45] _ns stacking sequence or a tensile test of a holed specimen, in which shear plays an important role. Otherwise, one can take the value related to the interface model.

Tension test on [0/90 ₄ ] _s : a first validation of the proposed interface model

Experimental results are well-known and analysed in many papers such as [21]. The top part of Figure 5 gives an overview of the sequence of damage mechanisms. Three dimensional finite element calculations are performed with a very refined mesh, a small initial defect being introduced at the center of the plate. The elastic material properties used in the simulation are the same as the ones given in Section Properties of the basic 2D interface problem. As for the parameters associated to fiber breaking and diffuse damage, they are taken as typical values for carbon/epoxy composites. The energy release rates associated with transverse cracking are: $G_I^c=200{\text{J/m}}^2$ and G II c=800 J/m². The ones related to the interfaces are assumed to be the same. Finally, the values of the parameters introduced are taken from the curves shown in the previous section. The enhanced interface model is used combined with the ply mesomodel [5, 11, 12].

Figure 5 shows that the simulation reproduces correctly the damage physics. Until (1), transverse microcracking development is observed. Diffuse damage remains weak and is not shown in the damage charts. From (1) to (2), delamination develops very quickly and, in the end, the specimen fails by fiber failure.

For this test case, a finite element calculation carried out with a classical cohesive interface, would not reproduce correctly the interface damage physics. Indeed, in this type of model, the delamination is activated by out-of-plane stresses which are really small in these cases and would not be sufficient to activate the damage mechanism.

Moreover, the enhanced interface model proposed in this paper bring a real improvement in the damage prediction compared to the former model used previously as in [12]. Indeed, this former model uses the mean value of the microcracking densities in the two adjacent plies of the interface to trigger delamination. Then, in this particular case where only one adjacent ply of the interface is damaged, the former model fails in predicting the interface breaking.

Open-hole tensile test: need of the coupling introduction

The test case used hereafter is part of Wisnom and Hallet’s experimental campaign on open-hole tensile tests [31]. Series of tests were carried out on quasi-isotropic IM7/8552 carbon/epoxy specimens with a [45 _m /90 _m /−45 _m /0 _m ] _ns lay-up and the geometry described in Figure 6.

The lay-up of the specimen chosen for this illustration is [45₄/90₄/−45₄/0₄] _s with a ply thickness h=0.5 mm, the hole diameter is D=6.35 mm and the ratio W/D=5. Experimental results reported in [31] show that this specimen experiences a delamination-dominated failure: the spread of transverse cracking in the plies, and the important amount of delamination associated lead to the coupon failure. Hence, the failure relies on the interaction between the transverse cracking in the plies and the delamination of the interface.

Concerning the damage evolution, the experiments show that the transverse cracking first develops in the upper 45° ply, resulting in damage in the 45/90 interface. Then, transverse cracking reaches the 90° plies. Damage goes through plies and interfaces until the degradation of the −45/0 interface on the whole width of the coupon, which corresponds to the failure.

Because a large amount of subcritical damage occurs, the stress-strain curve experiences a slope change before the final breakdown.

In order to highlight the influence of the interface models on the damage evolution prediction, the test case is simulated using the enhanced interface model and a more classical one which does not include the intra-interlaminar coupling.

Simulation results: global behavior

The stress-strain curves given by the two simulations are presented in Figure 7.

The two simulations show a slope change for a imposed strain ε=0.38%. This corresponds to the development of subcritical damage in the coupon which matchs the experimental observations.

The model including coupling predicts a failure stress close to the experimental one: σ _max =280 MPa for the simulation versus σ _max =285 MPa for the experimental value. The second one, that does not include coupling, predict a failure stress higher than the experimental one (σ _max =315 MPa vs σ _max =285 MPa).

In the following, the damage evolution predicted by the two models are compared. The study focuses on transverse cracking in the plies (represented in the damage charts by the variable ρ) and delamination in the interfaces (represented by the variable d _I ) as they are the main mechanisms concerned by the interface model.

The models are compared at four strain level:

1. 1.

   ε=0.38%: transverse cracking appears in the plies

    
2. 2.

   ε=0.42%: all plies experience transverse cracking

    
3. 3.

   ε=0.52%: transverse cracking has spread all over the width of the upper 45° ply

    
4. 4.

   ε=0.58%: specimen has failed

    

Damage prediction comparison: need of the intra-interlaminar coupling

For ε=0.38% (Figure 8), the two models give similar results in terms of transverse cracking. It appears in the upper ply and goes through plies and interfaces as described in [31].

For ε=0.42% (Figure 9) and ε=0.52% (Figure 10), the two models go on predicting similar behavior in terms of transverse cracking. However, whereas the model including coupling predicts a spread of delamination in the interfaces, the model without it does not predict any degradation of the interfaces.

For ε=0.58% (Figure 11), the model including coupling leads to a delamination-dominated failure, as reported in [31], whereas the second model yields a fiber breaking dominated failure.

To resume, the two simulations predict similar behaviors for the transverse cracking, which match experimental observations. However, the enhanced model predicts a spread of delamination in the different interfaces almost as soon as transverse cracking appears, whereas the second model do not predict any delamination until an equivalent strain of ε=0.5%. This difference of behavior leads to different failure mode: the new model predicts a delamination dominated failure matching the experimental observations, the second model predicts a delayed failure due to fiber breaking.

These results highlight the need for introducing intra-/interlaminar’s behavior coupling in order to accurately predict the damage evolution and failure stress and mode. More, the comparison with the experimental results illustrates the good capabilities of the enhanced interface model to predict the damage evolution and the failure pattern in the case of structural test cases such as open-hole tensile tests. Let us note that for this case the former version of our interface model gives similar results to the enhanced present one [12].