First, let us study the change in the stiffness of the interface mesomodel due to microcracking in the adjacent plies in the general case of different microcracking rates. To obtain these stiffness changes, the basic interface problem must be solved under out-of-plane loading. With only a limited loss of accuracy, one can consider the solution to be the superposition of the solutions of two 2D problems (one of which is depicted in Figure 1), which are associated with the fiber directions of Ply *S*_{
i
} and Ply *S*_{i+1}[29].

### Properties of the basic 2D interface problem

The basic 2D interface problem is defined in Figure 1. Since the results are quasi-independent of the stacking sequence, a sequence of [90/0/90_{2}] with *x*≡**N**_{1} was chosen. *h* denotes the interface’s thickness; the main parameters are the microcracking rate *ρ* and the delamination ratio *τ*. The typical properties of carbon/epoxy unidirectional plies are considered:

\begin{array}{ll}{E}_{1}& =148\phantom{\rule{2.77626pt}{0ex}}\text{GPa},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{E}_{2,3}=10.8\phantom{\rule{2.77626pt}{0ex}}\text{GPa},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{\nu}_{12,13}=0.3,\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{\nu}_{23}=0.4,\phantom{\rule{2em}{0ex}}\\ {G}_{12,13}& =5.8\phantom{\rule{2.77626pt}{0ex}}\text{GPa},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{G}_{23}=\frac{{E}_{2}}{2(1+{\nu}_{23})},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}H={H}_{e}=0.125\xb71{0}^{-3}\phantom{\rule{2.77626pt}{0ex}}\mathrm{m}.\phantom{\rule{2em}{0ex}}\end{array}

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 **N**_{1} 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*=2*τ* *ρ*, *ρ* and on the ply thickness [29].

The cell was analyzed for unit values of stresses *σ*_{33}, *σ*_{23}, *σ*_{13}, *σ*_{22} and *σ*_{12} using a relatively refined FE mesh, leading to a residual energy expressed as a surface energy:

\begin{array}{ll}\mathrm{\Delta e}=& \phantom{\rule{2.77626pt}{0ex}}{c}_{33}{\left({\sigma}_{33}\right)}^{2}+{c}_{23}{\left({\sigma}_{23}\right)}^{2}+{c}_{13}{\left({\sigma}_{13}\right)}^{2}+{c}_{22}{\left({\sigma}_{22}\right)}^{2}+{c}_{12}{\left({\sigma}_{12}\right)}^{2}\phantom{\rule{2em}{0ex}}\\ +{c}_{3313}{\sigma}_{33}{\sigma}_{13}+{c}_{3323}{\sigma}_{33}{\sigma}_{23}+{c}_{1323}{\sigma}_{13}{\sigma}_{23}+{c}_{2212}{\sigma}_{22}{\sigma}_{12}+{c}_{2233}{\sigma}_{22}{\sigma}_{33}\phantom{\rule{2em}{0ex}}\\ +{c}_{2213}{\sigma}_{22}{\sigma}_{13}+{c}_{2223}{\sigma}_{22}{\sigma}_{23}+{c}_{1233}{\sigma}_{12}{\sigma}_{33}+{c}_{1213}{\sigma}_{12}{\sigma}_{13}+{c}_{1223}{\sigma}_{12}{\sigma}_{23}\phantom{\rule{2em}{0ex}}\end{array}

(2)

The values of the coupling coefficients have been computed

\begin{array}{l}{\alpha}_{\mathit{\text{ijkl}}}=\frac{{c}_{\mathit{\text{ijkl}}}}{{\left({c}_{\mathit{\text{ij}}}{c}_{\mathit{\text{kl}}}\right)}^{\frac{1}{2}}},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{\alpha}}_{\mathit{\text{ijkl}}}=\text{max}{|}_{\text{calculated points}}\left|{\alpha}_{\mathit{\text{ijkl}}}\right|\end{array}

(3)

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

Except for *c*_{2233}, these coupling coefficients are negligible, the maximum being around 6.1·10^{−13}. Thus, *Δ* *e* can be taken as:

\begin{array}{l}\mathrm{\Delta e}={c}_{33}{\left({\sigma}_{33}\right)}^{2}+{c}_{23}{\left({\sigma}_{23}\right)}^{2}+{c}_{13}{\left({\sigma}_{13}\right)}^{2}+{c}_{22}{\left({\sigma}_{22}\right)}^{2}+{c}_{12}{\left({\sigma}_{12}\right)}^{2}+{c}_{2233}{\sigma}_{22}{\sigma}_{33}\end{array}

(4)

Moreover, the last three terms, which are proportional to *h*, are small compared to the ply’s residual energy, which is proportional to *H*, so they, too, are negligible. Consequently, the interface’s residual energy can be taken as:

\begin{array}{l}\mathrm{\Delta e}={c}_{33}{\left({\sigma}_{33}\right)}^{2}+{c}_{23}{\left({\sigma}_{23}\right)}^{2}+{c}_{13}{\left({\sigma}_{13}\right)}^{2}\end{array}

(5)

Now, let us introduce approximations for coefficients *c*_{33}, *c*_{13} and *c*_{23}, which depend on *λ*=2*τ* *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.

Let us introduce the damage parameters *d*_{33,i}, *d*_{13,i} and *d*_{23,i} associated to the 2D basic interface problem involving Ply *S*_{
i
}:

\begin{array}{l}\frac{{d}_{33,i}}{1-{d}_{33,i}}={c}_{33}\frac{2E}{h},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\frac{{d}_{13,i}}{1-{d}_{13,i}}={c}_{13}\frac{2G}{h},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\frac{{d}_{23,i}}{1-{d}_{23,i}}={c}_{23}\frac{2G}{h}\end{array}

(6)

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

\begin{array}{ll}{\stackrel{\u0304}{d}}_{33,i}=\lambda ,\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{d}}_{13,i}=\lambda ,\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\frac{{\stackrel{\u0304}{d}}_{23,i}}{1-{\stackrel{\u0304}{d}}_{23,i}}=& \frac{\lambda}{1-\lambda}+A\left(\rho \right),\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}A\left(\rho \right)=\frac{a\left(\rho \right)}{1-a\left(\rho \right)}\phantom{\rule{2em}{0ex}}\end{array}

(7)

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

Taking into account the delaminated area in the other direction *N*^{′}_{1}, one gets:

\mathrm{\Delta e}\sim \frac{h}{2}\left[\frac{\lambda}{1-\stackrel{\u0304}{\lambda}}\left(\frac{{\sigma}_{13}^{2}}{G}+\frac{{\sigma}_{33}^{2}}{E}\right)+\frac{{\sigma}_{23}^{2}}{G}\left(\frac{\lambda}{1-\stackrel{\u0304}{\lambda}}+A\left(\rho \right)\right)\right]

(8)

where \lambda =2\mathrm{\tau \rho}=\frac{e}{L}, *λ*^{′}=2*τ*^{′}*ρ*^{′} and \left(1-\stackrel{\u0304}{\lambda}\right)=\left(1-{\lambda}^{\prime}\right)\left(1-\lambda \right).

### The microcracking/stiffness interaction of the interface mesomodel

One must add up the residual energies of the two basic 2D interface problems:

\begin{array}{ll}\phantom{\rule{-10.0pt}{0ex}}\mathrm{\Delta e}=& \phantom{\rule{0.3em}{0ex}}\frac{h}{2}\left[\frac{\lambda +{\lambda}^{\prime}}{1-\stackrel{\u0304}{\lambda}}\frac{{\sigma}_{33}^{2}}{E}+\frac{{\sigma}_{13}^{2}+{\sigma}_{23}^{2}}{G}\frac{\lambda}{1-\stackrel{\u0304}{\lambda}}+\frac{{\sigma}_{{1}^{\prime}3}^{2}+{\sigma}_{{2}^{\prime}3}^{2}}{G}\frac{{\lambda}^{\prime}}{1-\stackrel{\u0304}{\lambda}}+\frac{{\sigma}_{23}^{2}}{G}A\left(\rho \right)+\frac{{\sigma}_{{2}^{\prime}3}^{2}}{G}A\left({\rho}^{\prime}\right)\right]\phantom{\rule{2em}{0ex}}\end{array}

(9)

Let {\stackrel{\u0304}{\sigma}}_{33}, {\stackrel{\u0304}{\sigma}}_{13} and {\stackrel{\u0304}{\sigma}}_{23} be the mesostress components written in the interface’s basis ({\stackrel{\u0304}{\mathbf{N}}}_{1},{\stackrel{\u0304}{\mathbf{N}}}_{2}) and let 2*α* be the angle between the fiber directions of the adjacent plies. One has:

\begin{array}{ll}{\sigma}_{13}& =cos\alpha \phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{\sigma}}_{13}-sin\alpha \phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{\sigma}}_{23},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{\sigma}_{23}=sin\alpha \phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{\sigma}}_{13}+cos\alpha \phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{\sigma}}_{23}\phantom{\rule{2em}{0ex}}\\ {\sigma}_{{1}^{\prime}3}& =cos\alpha \phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{\sigma}}_{13}+sin\alpha \phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{\sigma}}_{23},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{\sigma}_{{2}^{\prime}3}=-sin\alpha \phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{\sigma}}_{13}+cos\alpha \phantom{\rule{2.77626pt}{0ex}}{\stackrel{\u0304}{\sigma}}_{23}\phantom{\rule{2em}{0ex}}\end{array}

(10)

Neglecting the term *λ* *λ*^{′} which is very small compared to 1, one easily obtains:

\begin{array}{ll}\phantom{\rule{-18.0pt}{0ex}}\mathrm{\Delta e}& =\frac{h}{2}\left\{\frac{\stackrel{\u0304}{\lambda}}{1-\stackrel{\u0304}{\lambda}}\frac{{\stackrel{\u0304}{\sigma}}_{33}^{2}}{E}+\frac{{\stackrel{\u0304}{\sigma}}_{13}^{2}}{G}\left[\frac{\stackrel{\u0304}{\lambda}}{1-\stackrel{\u0304}{\lambda}}+\left(A\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{A}^{\prime}\right)\stackrel{2}{sin}\alpha \right]+\frac{{\stackrel{\u0304}{\sigma}}_{23}^{2}}{G}\left[\frac{\stackrel{\u0304}{\lambda}}{1-\stackrel{\u0304}{\lambda}}+\left(A\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{A}^{\prime}\right)\stackrel{2}{cos}\alpha \right]\right.\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\left(\right)close="\}">+\phantom{\rule{0.3em}{0ex}}\frac{2{\stackrel{\u0304}{\sigma}}_{13}{\stackrel{\u0304}{\sigma}}_{23}sin\alpha cos\alpha}{G}\left(A-{A}^{\prime}\right)& \phantom{\rule{2em}{0ex}}\end{array}\n

(11)

### The new interface mesomodel - stiffness and damage

Let us note that the interface mesomodel is described as a cohesive interface with a very small ‘thickness’ compared to the cell’s dimensions. The contributions due to microcracking should be viewed as relatively long-wavelength contributions. Thus, the energy of the interface mesomodel is:

\begin{array}{l}\u0113=\frac{h}{2}\left[\frac{{\left.\right)"\; close=">">-{\sigma}_{33}}^{}2}{}E+\frac{{\left.\right)"\; close=">">{\sigma}_{33}}^{}2}{}E\left(1-{d}_{33}\right)\right]+\frac{{\stackrel{\u0304}{\sigma}}_{13}^{2}}{G\left(1-{d}_{13}\right)}+\frac{{\stackrel{\u0304}{\sigma}}_{23}^{2}}{G\left(1-{d}_{23}\right)}+\frac{\omega}{G}{\stackrel{\u0304}{\sigma}}_{13}{\stackrel{\u0304}{\sigma}}_{23}\end{array}\n

(12)

where the purpose of the positive part <∙> is to account for crack opening and crack closure. The usual damage variables, deduced from the micro-meso energy equivalence, are:

\begin{array}{l}{d}_{33},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{d}_{13}=\frac{{d}_{33}+\left(1-{d}_{33}\right)\left(A+{A}^{\prime}\right)\stackrel{2}{sin}\alpha}{1+\left(1-{d}_{33}\right)\left(A+{A}^{\prime}\right)\stackrel{2}{sin}\alpha},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{d}_{23}=\frac{{d}_{33}+\left(1-{d}_{33}\right)\left(A+{A}^{\prime}\right)\stackrel{2}{cos}\alpha}{1+\left(1-{d}_{33}\right)\left(A+{A}^{\prime}\right)\stackrel{2}{cos}\alpha}\end{array}

(13)

with the coupling term *ω* written as *ω*=2 sin*α* cos*α*(*A*−*A*^{′}).

In previous papers [11, 12], a simplified expression was considered, based on \stackrel{\u0304}{\rho}=\frac{\rho +{\rho}^{\prime}}{2}. This expression is equivalent to (13) for *ρ*=*ρ*^{′}, *τ* small (*i.e.* *d*_{33}→0) and *α*∼ 45°.

It is remarkable that this energy depends only on *ρ*, *ρ*^{′} and \stackrel{\u0304}{\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

The dissipation work associated with the new interface model is:

\begin{array}{l}\stackrel{\u0307}{D}=\mathrm{\Delta \u0117}={Y}_{I}{\stackrel{\u0307}{d}}_{33}+{Y}_{\mathit{\text{II}}}{\stackrel{\u0307}{d}}_{13}+{Y}_{\mathit{\text{III}}}{\stackrel{\u0307}{d}}_{23}+\frac{h}{2G}\stackrel{\u0307}{\omega}{\stackrel{\u0304}{\sigma}}_{13}{\stackrel{\u0304}{\sigma}}_{23}\end{array}

(14)

where \stackrel{\u0307}{\omega} depends on \stackrel{\u0307}{\rho} and {\stackrel{\u0307}{\rho}}^{\prime}.

One can easily see that \stackrel{\u0307}{D}\ge 0, as {\stackrel{\u0307}{d}}_{33}, \stackrel{\u0307}{\rho} and {\stackrel{\u0307}{\rho}}^{\prime} are positive or equal to zero. Using (11) and (7):

\begin{array}{ll}\stackrel{\u0307}{D}& =\frac{h}{2}\left[\frac{{\stackrel{\u0307}{d}}_{33}}{{\left(1-{d}_{33}\right)}^{2}}\left(\frac{{\stackrel{\u0304}{\sigma}}_{33}^{2}}{E}+\frac{{\stackrel{\u0304}{\sigma}}_{13}^{2}+{\stackrel{\u0304}{\sigma}}_{23}^{2}}{G}\right)+{\left({\stackrel{\u0304}{\sigma}}_{13}sin\alpha +{\stackrel{\u0304}{\sigma}}_{23}cos\alpha \right)}^{2}\frac{\stackrel{\u0307}{A}}{G}\right.\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\left(\right)close="]">{\left({\stackrel{\u0304}{\sigma}}_{13}sin\alpha -{\stackrel{\u0304}{\sigma}}_{23}cos\alpha \right)}^{2}\frac{{\stackrel{\u0307}{A}}^{\prime}}{G}& \phantom{\rule{2em}{0ex}}\end{array}\n

(15)

Since \stackrel{\u0307}{A}=\frac{\mathrm{d}A}{\mathrm{d}\rho}\stackrel{\u0307}{\rho} is positive, it follows that \stackrel{\u0307}{D}\ge 0; thus, the interface mesomodel is compatible with the principles of thermodynamics.