2.1.4.2. Serial-Parallel rule of mixtures

In the classical lamination theory presented in section 5.1.5 individual composite plies are treated as a continuum homogenized material. The laminate stiffness matrix is further evaluated by using the classical lamination theory (CLT). To all effects, the laminated shell is finally treated as an homogeneous orthotropic material.

RamSeries also provides an alternative formulation for FRP laminates in which a serial-parallel (SP) continuum approach is considered [Rastellini_2007], [Martínez_2008]. In the SP formulation, the composite material components (namely fibres and matrix) behave as parallel materials in the direction of the fibres alignment and as serial materials in the orthogonal directions. At the same time, each material component is treated taking into account its own constitutive law so that the micro-mechanics of unidirectional composite plies can be assessed. The Serial-Parallel model is further complemented with the application of the rule of mixtures at the level of the laminate to describe the mechanical behaviour of multilayered composite materials [Rastellini_2007].

The aim of using the SP model is to assess the mechanical behaviour of the composite as a whole but retaining the individual behaviour of its components (i.e. matrix and fibres). The non-linear behaviour of composite structures due to material degradation is also assessed with the SP model as far as the non-linear behaviour of the constituents is considered in their respective constitutive models. This is not the case in the classic lamination theory in section 5.1.5 where the material is considered to be linear elastic.

The SP-RoM formulation can be described as follows. First, the heterogeneous composite material is characterized by a composite material domain (c) that can be decomposed into two non-overlapping subdomains associated to the matrix (m) and fibre (f) components respectively:

(2.134)\[\begin{aligned} {^c}\Omega = {^m}\Omega \cup {^f}\Omega \end{aligned}\]

The matrix and fibre volumetric fractions are denoted by \(k_m\) and \(k_f\) respectively so that:

(2.135)\[\begin{aligned} {^m}k + {^f}k = 1 \end{aligned}\]

Then, average stress and strain fields can be defined as:

(2.136)\[\begin{aligned} {^c}\varepsilon = \frac{\int_{{^c}\Omega} \varepsilon \, dV}{\int_{{^c}\Omega} dV} , \quad {^m}\varepsilon = \frac{\int_{{^m}\Omega} \varepsilon \, dV}{\int_{{^m}\Omega} dV} , \quad {^f}\varepsilon = \frac{\int_{{^f}\Omega} \varepsilon \, dV}{\int_{{^f}\Omega} dV} \end{aligned}\]

(2.137)\[\begin{aligned} {^c}\sigma = \frac{\int_{{^c}\Omega} \sigma \, dV}{\int_{{^c}\Omega} dV} , \quad {^m}\sigma = \frac{\int_{{^m}\Omega} \sigma \, dV}{\int_{{^m}\Omega} dV} , \quad {^f}\sigma = \frac{\int_{{^f}\Omega} \sigma \, dV}{\int_{{^f}\Omega} dV} \end{aligned}\]

And by virtue of the rule of mixtures we can decompose the strain and stress fields as:

(2.138)\[\begin{aligned} {^c}\varepsilon_c = {^f}\varepsilon \cdot {^f}k + {^m}\varepsilon \cdot {^m}k \end{aligned}\]

(2.139)\[\begin{aligned} {^c}\sigma = {^f}\sigma \cdot {^f}k + {^m}\sigma \cdot {^m}k \end{aligned}\]

Regarding the stress as a dependent variable, the material’s constitutive law is given by the following set of differential equations:

(2.140)\[\begin{aligned} \dot{{^i}\sigma} = \dot{g} ({^i}\varepsilon,{^i}\beta,\dot{{^i}\varepsilon}) \end{aligned}\]

(2.141)\[\begin{aligned} \dot{{^i}\beta} = \dot{h} ({^i}\varepsilon,{^i}\beta,\dot{{^i}\varepsilon}) \end{aligned}\]

where \(i=m,f\) and \(\beta\) denotes a vector of internal variables.

Now, the stress and strain fields need to be decomposed in their serial and parallel components. To this aim, appropriate projector tensors are constructed as follows:

(2.142)\[\begin{aligned} N_{11} = e_1 \otimes e_1 \end{aligned}\]

(2.143)\[\begin{aligned} P_P = N_{11} \otimes N_{11} \end{aligned}\]

(2.144)\[\begin{aligned} P_S = I-P_P \end{aligned}\]

Where \(e_1\) is the unit vector parallel to the fibre direction, :\(N_{11}\) is the projector second order tensor corresponding to \(e_1\), \(P_P\) is the parallel fourth order projector tensor and \(P_S\) is the serial counterpart.

Hence, the following decomposition can be applied to the composite’s stress and strain fields:

(2.145)\[\begin{aligned} \varepsilon_P = P_P : \varepsilon \end{aligned}\]

(2.146)\[\begin{aligned} \varepsilon_S = P_S : \varepsilon \end{aligned}\]

(2.147)\[\begin{aligned} \varepsilon = \varepsilon_P + \varepsilon_S \end{aligned}\]

(2.148)\[\begin{aligned} \sigma_P = P_P : \sigma \end{aligned}\]

(2.149)\[\begin{aligned} \sigma_S = P_S : \sigma \end{aligned}\]

(2.150)\[\begin{aligned} \sigma = \sigma_P + \sigma_S \end{aligned}\]

The above decomposition, which is applied to the composite stress and strain fields, are analogously extended to matrix and fibre materials.

Finally, an appropriate closure equation must be devised for long fibre reinforced polymer composites. In this case, it is assumed that an iso-strain condition is applied in the parallel direction between composite and its constituent materials. On the other hand, an iso-stress condition is applied in the serial direction between composite and its constituent materials. This results in what is known as the basic serial parallel (BSP) closure equations that read as follows:

(2.151)\[\begin{aligned} {^c}\varepsilon_P = {^m}\varepsilon_P = {^f}\varepsilon_P \end{aligned}\]

(2.152)\[\begin{aligned} {^c}\sigma_S = {^m}\sigma_S = {^f}\sigma_S \end{aligned}\]

Hence, the governing equations of the serial-parallel problem are the constitutive laws of matrix and fibre materials in Eq. (2.140), (2.141), the equations that relate average strain and stresses in Eq. (2.138), (2.139) and the closure equations Eq. (2.151), (2.152).

The Serial/Parallel algorithm implemented is introduced next. The algorithm is defined as a strain-driven problem, given that the serial strain component of the matrix is the unknown to be predicted. On the other hand, the current prediction of the composite strain is the input variable of the algorithm. Basically, the algorithm has to find a solution for the strain and stress components of each constituent material that satisfies the compatibility and equilibrium equations, (2.151), (2.152), taking into account the constitutive law of each compounding material, (2.140), (2.141).

Some clarifications have to be done in order to correctly follow the algorithm steps. The composite algorithm iteration is denoted as \(K\), and the iteration for the FE algorithm is denoted as \(n\).

The serial-parallel composite solver algorithm implemented in RamSeries as a constitutive law consists in the following steps:

First, the tangent constitutive tensors, \({^i}[\mathbb{C}]^t\), for each constituent materials are evaluated for the converged step. Next, the current composite strain increment, \([{^c} \Delta \varepsilon]\), is split in its serial and parallel components by using the projector tensors.

(2.153)\[\begin{aligned} {^c}\Delta \varepsilon_P = P_P : [{^c} \Delta \varepsilon] \qquad {^c}\Delta \varepsilon_S = P_S : [{^c} \Delta \varepsilon] \end{aligned}\]

Then, the initial prediction of the serial matrix strain increment for the first algorithm iteration, \(K\), is conducted by the following expressions, in terms of the known variables:

(2.154)\[\begin{aligned} {^m}\Delta \varepsilon_S^0 = \mathbb{A} : {^f} \mathbb{C}_{SS} : {^c}\Delta \varepsilon_S + {^f}k \cdot ({^f}\mathbb{C}_{SP} - {^m}\mathbb{C}_{SP}) : {^c}\Delta \varepsilon_P \end{aligned}\]

Where, \(\mathbb{A} = ({^m}k \cdot {^f} \mathbb{C}_{SS} + {^f}k \cdot {^m} \mathbb{C}_{SS})^{-1}\).
The serial matrix strain tensor is the sum of the increment and the previous serial strain converged, \(n-1\):

(2.155)\[\begin{aligned} \ [{^m}\varepsilon_S]^0 = [{^m}\varepsilon_S]^{n-1} + {^m}\Delta \varepsilon_S^0 \end{aligned}\]

And the total strain tensor of the matrix is the sum of the parallel and serial contributions:

(2.156)\[\begin{aligned} \ [{^m}\varepsilon]^0 = {^m}\varepsilon_P + [{^m}\varepsilon_S]^0 \end{aligned}\]

The parallel fibre strain is the same than the parallel matrix strain, \({^c}\varepsilon_P = {^f}\varepsilon_P = {^m}\varepsilon_P\). An the serial term is obtained by:

(2.157)\[\begin{aligned} \ [{^f}\varepsilon_S]^K = \frac{1}{{^f}k} \cdot [{^c}\varepsilon_S]^n - \frac{{^m}k}{{^f}k} \cdot [{^m}\varepsilon_S]^K \end{aligned}\]

At this point, the strain tensor for fibre is obtained in a similar manner than Eq. (2.156). Therefore, both constitutive equations for each compounding material can be evaluated, obtaining the stress state for each one.

Recalling the iso-strain condition in the serial direction of the laminate, we can verify if the equilibrium has been reached in the algorithm by obtaining the residue term, \([\Delta \sigma_S]_K = [{^m} \sigma_S]_K - [{^f} \sigma_S]_K\). The convergence of the algorithm is obtained if the residue is lower than a tolerance, such as \([\Delta \sigma_S]_K \leq tol\).
If the equilibrium is reached, then the strain tensors predicted are correct for fibre and matrix and all their variables, these are stress tensor and internal variables. Hence, all the internal variables are updated in the FE algorithm. Hence, the composite stress can be computed by Eq. (2.139).
In case that the convergence is not reached, a new prediction of the serial matrix strain tensor is required. The new prediction of the unknown is conducted by applying a Newton-Rapshon scheme, being the residue of the serial stresses the term to be zeroed. The new prediction of the serial matrix strain is obtained by correcting the initial prediction, using the Jacobean of the residual forces. The Jacobean is obtained by deriving the residue function with respect the unknown, such as:

(2.158)\[\begin{aligned} \mathbb{J} = [{^m}\mathbb{C_{SS}}]^K + \frac{{^m}k}{{^f}k} \cdot [{^f}\mathbb{C_{SS}}]^K \end{aligned}\]

And the expression to correct the prediction of the unknown is:

(2.159)\[\begin{aligned} \ [{^m}\varepsilon_S]^{K+1} = [{^m}\varepsilon_S]^K - \mathbb{J}^{-1} : [\Delta \sigma_S]_K \end{aligned}\]