In the present analysis several simulation analysis were carried out to analyze the structural response of CF / PEEK laminated composites showing great potential for application in the biomedical field.**Background**, in a laminated composite the structural behavior depends on the arrangement of the individual layers, in these analyzes it is investigated how the different layouts influence the response of some structural elements;

**Results**, different simulation campaigns show the variation of the stiffness and of the coupling conditions between the load conditions and the structural response to the variation of the layout and the grain angle of the foil;**Conclusions**, structural responses are highly variable, from simpler behaviors to predict to more complex responses. An analysis that takes into account the layout of the entire laminate, starting from the composition of the single layer, allows for precise results to understand the structural behavior.

The following report is extracted from a project of the Computational Mechanics of Tissues and Biomaterials course, held for Medical Engineering at the University of Rome Tor Vergata.

**Table of Contents**

**Introduction**

The purpose of the following analysis is to investigate the behavior of different layouts of a laminated composite. A CF / PEEK laminate composite is analyzed, which is a composite with a thermoplastic polymer matrix reinforced with carbon fibers that shows great potential for biomedical applications.

Two fundamental structural elements are considered, a cylinder and a plate, and it is studied how the variation of the layout leads to a different mechanical response. An in-depth study is also carried out by analyzing the distribution of stresses at the level of the individual layers.

The knowledge and the possibility of predicting the** structural behavior** starting from the arrangement of the layers is fundamental both for the correct simulation and for the production of devices and objects with laminated composites.

**CF/PEEK laminates**

Recent advances in composite materials, in particular with a PEEK matrix, or a polymeric matrix of poly-ether-ether ketone, have greatly expanded the application scenarios by making these materials fall into the class of **technopolymers**. Given its properties, this polymer ranks among the most important polymers in the engineering field. It has excellent properties such as high mechanical resistance, excellent thermal stability, chemical resistance and stable behavior even in chemically hostile environments. It also has an anticorrosive nature and good resistance to degradation, properties that make it excellent for applications in the biomedical field.

In these applications, some surface properties have also been improved by combining PEEK with bioactive particles such as hydroxyapatite. This has made it possible to address some problems such as the limited effectiveness in making cells adhere and in integrating with bone. Clearly the integration of reinforcing fibers has also allowed the improvement of the mechanical properties.

One of the most important reinforcements is that of unidirectional carbon fibers. Indicated as CF / PEEK, it was introduced around 1980 and in recent years it has proved to be an excellent biomaterial with great potential for application in biomedical systems [1]. Generic PEEK find great applications both in customized plants made with 2D printing and in orthodontics. Furthermore, CF / PEEK finds great applications in orthopedics. It should also be noted that continuous fiber CF / PEEK exhibits better mechanical properties than generic or short fiber reinforced PEEK but requires more advanced manufacturing methods.

**Biomedical applications**

Typically in orthopedics, metal implants are used with a stiffness of about an order of magnitude higher than that of physiological bone and this can lead to problems related to bone resorption. In addition, there may be problems related to CT or X-ray diagnostic imaging [2].

The use of composite materials can provide several advantages such as the presence of anisotropic properties, radio-transparency, high fatigue strength and a **high strength / weight ratio**. Currently, there are still very few laminated composites actually used in orthopedic applications. An example of this material is PEEK reinforced with continuous carbon fiber, known as PEEK-OPTIMUM Ultra-Reinforced, and has been approved for implantation in humans by the American Food and Drug Administration [3].

The following simulation campaign refers to a composite laminate with CF / PEEK layers with a fiber volumetric fraction of 62% and considers different laminated layouts by analyzing the mechanical performance and the structural response.

**Background**

Typically in orthopedics, metal implants are used with a stiffness of about an order of magnitude higher than that of physiological bone and this can lead to problems related to bone resorption. In addition, there may be problems related to CT or X-ray diagnostic imaging [2].

Modern composites are based on the use of a **reinforcement phase**, typically in the form of fiber, immersed in a matrix. Usually a polymeric matrix is used and glass, carbon, aramid and others can be used as reinforcing fibers. There are several methods of arranging the fibers and their distribution influences the overall behavior.

By putting together several layers of composite material, a package is obtained which is called a laminated composite.

The problem can be divided on three different scales of magnitude: the behavior of the final structure (macroscale) which depends on the laminate which in turn is influenced by the single laminae (mesoscale). In turn, the behavior is determined by the individual constituents and their composition (microscale). A schematic representation is given in fig. 1.

The following simulation campaign refers to a composite laminate with CF/PEEK layers with a fiber volumetric fraction of 62% and considers different laminated layouts by analyzing the mechanical performance and the structural response.

**Mixtures rule**

The single layer is composed of a matrix and reinforcing fibers with a precise volume fraction. The first step is to **homogenize **the material properties of the single layer. We consider a grain such as to ensure transversely isotropic symmetry and the axis of the fiber with isotropic direction.

CF | PEEK |
---|---|

Ef =236 GPa | Em =4 GPa |

νf =0.2 | νm =0.36 |

Gf = 27.6 GPa | Gm = 1.47 GPa |

**Tab. 1.**Material parameters [8]

To get to the homogenized properties of the single layer it is possible to apply the rule of blends [6].

Considering a global volume of the composite given by the sum of the volume of the matrix and that of the reinforcement and assuming perfect adhesion between the two, it is possible to analytically calculate the resulting modules. The volumetric fractions apply:

\begin{equation} v_{m}+f_{f}=1 \end{equation}

And it is possible to express the resulting material properties as a function of the volumetric fraction of fibers (ff) and the material properties of the individual constituents, as expressed in eqs. (2) to (6).

\begin{equation} E_{1}=f_{f} E_{f}+\left(1-f_{f}\right) E_{m} \end{equation}

\begin{equation} E_{2}=E_{3}=\left(\frac{1-f_{f}}{E_{m}}+\frac{f_{f}}{E_{f}}\right)^{-1} \end{equation}

\begin{equation} G_{12}=\left(\frac{v_{m}}{G_{m}}+\frac{v_{f}}{G_{f}}\right)^{-1} \end{equation}

\begin{equation} v_{12}=v_{f} v_{f}+v_{m} v_{m} \end{equation}

\begin{equation} v_{23}=\frac{E_{2}}{2 G_{23}}-1=\frac{E_{2}}{2 G_{12}}-1 \end{equation}

It is possible to consider a compliance matrix of the type in eq. (7)

\begin{equation} \left[\begin{array}{cccccc} \frac{1}{E_{1}} & -\frac{v_{21}}{E_{2}} & -\frac{v_{31}}{E_{3}} & 0 & 0 & 0 \\ -\frac{v_{12}}{E_{1}} & \frac{1}{E_{2}} & -\frac{v_{32}}{E_{3}} & 0 & 0 & 0 \\ -\frac{v_{13}}{E_{1}} & -\frac{v_{23}}{E_{2}} & \frac{1}{E_{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{23}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{12}} \end{array}\right] \end{equation}

In particular, it is possible to observe the trend of the Young-type modules in fig. 2. The linear trend of E1 is evident. All the modules increase as the ff increases as the contribution of carbon fibers that have higher modules than PEEK increases. The following analysis considers an ff set at 62%.

**Thin laminate theory**

**Thin laminate theory**

After having homogenized the single layer, the theory of thin laminates [7] is used, treating the single layer as a homogeneous thin sheet. Each lamina is subjected to a plane state of tension and the stiffness matrix in the single layer can be expressed with respect to a global reference system.

\begin{equation} [\bar{Q}]=\left[T_{\sigma}\right]^{-1}[Q]\left[T_{\epsilon}\right] \end{equation}

Considering the deformations in the plane of the mean surface and the curvature of the mean surface we arrive at the generalized constitutive laws, eq. (9).

\begin{equation} \left[\begin{array}{c} N_{x} \\ N_{y} \\ N_{x y} \\ M_{x} \\ M_{y} \\ M_{x y} \end{array}\right]=\left[\begin{array}{llllll} A_{11} & A_{12} & A_{16} & B_{11} & B_{12} & B_{16} \\ A_{21} & A_{22} & A_{26} & B_{21} & B_{22} & B_{26} \\ A_{61} & A_{62} & A_{66} & B_{61} & B_{62} & B_{66} \\ B_{11} & B_{12} & B_{16} & D_{11} & D_{12} & D_{16} \\ B_{21} & B_{22} & B_{26} & D_{21} & D_{22} & D_{26} \\ B_{61} & B_{62} & B_{66} & D_{61} & D_{62} & D_{66} \end{array}\right]\left[\begin{array}{c} \varepsilon_{x}^{0} \\ \varepsilon_{y}^{0} \\ \gamma_{x y}^{0} \\ \kappa_{x}^{0} \\ \kappa_{y}^{0} \\ \kappa_{x y}^{0} \end{array}\right] \end{equation}

Where the three blocks of the matrix have a precise **physical meaning**. Block A, eq. (10), describes the mebranal stiffness of the single lamina. Block B, eq. (11), links the coupling between the membrane and flexural effects. Block D, eq. (12), represents the flexural stiffness of the laminate.

\begin{equation} A_{i j}=\sum_{k=1}^{N} \bar{Q}_{i j}^{k}\left(z_{k}-z_{k-1}\right) \end{equation}

\begin{equation} B_{i j}=\frac{1}{2} \sum_{k=1}^{N} \bar{Q}_{i j}^{k}\left(z_{k}^{2}-z_{k-1}^{2}\right) \end{equation}

\begin{equation} D_{i j}=\frac{1}{3} \sum_{k=1}^{N} \bar{Q}_{i j}^{k}\left(z_{k}^{3}-z_{k-1}^{3}\right) \end{equation}

This description will allow to predict the couplings and the structural behavior starting from the structure of the three blocks.

**Properties of constituents**

Starting from the single layer, a PEEK polymer matrix is considered which has a relatively low elastic modulus of 4 GPa with Poisson’s ratio ν = 0.36. The matrix is reinforced with carbon fibers which instead have a high stiffness, elastic modulus of 236 GPa and a Poisson’s ratio of 0.2. More information in tab 1.

The results of the mixture rule considering a volumetric fraction of fibers of 62% are shown in table 2. Starting from these numerical results, the stiffness matrix of the single layer is calculated.

We then proceed to consider different laminate configurations. Every single layer is considered with the fibers aligned along a main direction, with the exception of the ± 45f layers which are considered as a weave of fibers. The individual layers are then rotated with respect to a global reference and put together to form the composite laminate. Then it is possible to calculate the generalized constitutive bond of the laminate.

ff | 0.62 |

E1 | 147.96 GPa |

E2 | 10.83 GPa |

ν12 | 0.26 |

G12 | 4.24 GPa |

**Tab. 2.**Results by applying the rule of mixtures

**Laminate layout**

The variations of the laminate layout are infinite, it is possible to vary the number of layers and the grain angle. In the following analysis, 4 different cases are addressed:

\begin{aligned} (a)& \:\left[\alpha /-\alpha / 30 /-30 / 0_{2}\right]_{\mathrm{s}} \operatorname{con} \alpha \in\left[0^{\circ} ; 90^{\circ}\right] \\ (b)&\: \left[-45 / 45 /-45 / 45\right]\\ (c)&\:\left[\pm 45^{f} / \pm 45^{f}\right]\\ (d)& \:\left[-30 /-45 /-30 /-45\right] \end{aligned}

A representative illustration of the different layouts is shown in fig. 3. Two different structural elements are considered:

- Plate like element
- Cylinder like element

For the analysis of the structural response, different types of load are considered. On the plate type element, either an axial load, or a transverse load or both is considered. For the cylinder-type element, a uniform internal pressure is instead considered.

**Plate element**

For the plate-like element, the four different laminate layouts are analyzed.

*Case(a)*

*Case(a)*

The first analysis deals with a parametric study (parameter α) on a layout of the type:

\begin{equation} \left[\alpha /-\alpha / 30 /-30 / 0_{2}\right]_{S} \end{equation}

Three different load situations are considered: axial load only, transverse load only and both loads. The results are reported in figs. 5 to 7. As the parameter α varies between 0 ° and 90 ° different structural responses are observed.

Being a symmetrical laminate, a coupling between extensional and bending effects is not expected. This is confirmed by the comparison between the three cases. It is evident that by applying only an axial load, the flexural effect is much reduced compared to the other load configurations. In the case α = 0◦ the flexion is the minimum possible. Asα grows, the flexion effect increases. This does not happen with only the axial load where the bending effect is very low and practically zero both for α = 90◦ and for α = 0◦. In the latter cases, the coupling between in-plane and out-plane effects is therefore also null.

##### Simulation results

With the exception of the cases α = 90◦ and α = 0◦ there is always a torsional effect even if very limited.

It is evident that the arrangement of a greater number of layers aligned along the axis provides less flexural stiffness and this is reflected in a more pronounced flexural effect. In order to reduce the flexural effect it is good to arrange the parametric layers with the fibers orthogonal to the axis of the laminate. This also cancels the torsional effect and the in-plane out-plane coupling. In addition, it also provides greater membrane stiffness. Instead, if you want to have a more pronounced flexural effect, you can arrange the largest number of layers with the fibers aligned along the axis.

This leads to a shift in the free terminal side of the plate up to 5 times higher but makes the coupling with the torsional effects more evident. It is interesting to see that with only the transverse load, depending on the configuration of the laminate, it is possible to have such a bending as to move the final end both upwards and downwards (fig. 7 (iii)). This effect is not present in the cases α = 0 ° or α = 90 °.

Intermediate cases provide intermediate structural behaviors between the two borderline cases described. The coupling between the axial load and the torsional effect is maximum for the layout with α = 20 ° and α = 10 °.

*Case (b)*

*Case (b)*

A balanced laminate with layout is analyzed:

\begin{equation} [-45 / 45 /-45 / 45] \end{equation}

Again the three different load configurations are considered.

Being a balanced laminate, a shear-extension fit is not expected. From the results obtained, figs. 11 to 13, several couplings are observed. A coupling is observed between the axial load and a bending effect. A coupling is also observed between the axial load and the torsional effect while a torsional effect is not observed with only the transverse load.

The behavior of this laminate is very similar to the behavior of the case (§4.1.1) with the parameter α = 90 ° but with a much lower bending stiffness due to the smaller number of layers.

*Case (c)*

*Case (c)*

A layout is analyzed:

\begin{equation} \left[\pm 45^{f} / \pm 45^{f}\right] \end{equation}

The single layer is made up of an intertwined layer, that is a particular configuration where the fibers are crossed with a direction of ± 45◦. In the case of woven fabrics, the definition of the stiffness matrix of the layer changes:

\begin{equation} Q_{i j}^{\text {woven }}=\frac{1}{2}\left[\left(\bar{Q}_{i j}\right)_{45}+\left(\bar{Q}_{i j}\right)_{-45}\right] \end{equation}

There is no torsional effect in this layout. There is also no coupling between axial loads and bending effect. The laminate responds without showing torsional or extensional-flexural couplings also due to symmetry and balance.

*Case (d)*

*Case (d)*

The following layout is analyzed:

\begin{equation} [-30 /-45 /-30 /-45] \end{equation}

The numerical results are presented in figs. 11 to 13 compared with the case §4.1.2.

This layout is completely unbalanced and asymmetrical with the grain direction showing negative angles with respect to the laminate axis. This is reflected in the coupling of the torsional effect with all load conditions.

There is also the in-plane out-plane coupling as can be seen from the bending induced by the axial load alone (fig. 13 (iii)). Extensional-extensional coupling is also very marked.

**Data analysis**

It is interesting to see the effect of the transverse load only (fig. 12) in which, due to the imbalance of the grain directions and torsion, there is an overall effect whereby the free end of the plate tends to have a displacement in the opposite direction to that of the load. This effect, in the case in which both load conditions are present, results in an initial raising and then flexing according to the direction of the transverse load (fig. 11 (iii)).

The complete deformation is also shown in fig. 14. This coupling results in a bending effect by applying only the longitudinal load (fig. 13 (iii)).

**Cylinder like element**

A cylinder-type structural element with radius R and length L is then considered. A load configuration expressed by a uniform internal pressure on the cylinder wall is considered. The four different laminate layouts are analyzed again. The results are read on the edge of the cylinder in the four segments obtainable at x = ± R and y = ± R, so as to be able to understand if the widening is uniform or not along the axis and if there is a torsional effect.

*Case (a)*

A parametric simulation campaign is carried out and the structural response to the variation of the parameter α is analyzed. The results are reported in fig. 15. The layout of the laminate is of the type:

\begin{equation} \left[\alpha /-\alpha / 30 /-30 / 0_{2}\right]_{S} \quad \operatorname{con} \alpha \in\left[0^{\circ} ; 90^{\circ}\right] \end{equation}

It is evident that as α increases, the cylinder becomes less and less compliant, that is, with the same load it expands less and less. The expansion is uniform and symmetrical, both along y ^ and along z ^. The greater number of fibers oriented parallel to the load direction (α → 90◦) makes the material more rigid and therefore reduces the extensional effect. Furthermore, being a symmetrical and balanced layout, no twisting effects are observed either.

*Case (b)*

A balanced laminate with the following layout is analyzed:

\begin{equation} [-45 / 45 /-45 / 45] \end{equation}

In this layout, a uniform and symmetrical expansion of the cylinder is observed. The circular section widens and remains constant for the entire length of the cylinder. Furthermore, there are no torsional effects. The results are reported in fig. 16

*Case (c)*

*Case (c)*

The following layout is analyzed:

\begin{equation} \left[\pm 45^{f} / \pm 45^{f}\right] \end{equation}

The results are shown in fig. 17.

The intertwined layout provides a symmetrical and balanced layout overall.

A uniform and constant dilation is observed as in the previous cases. Numerically, a slight difference is obtained by measuring the vertical displacement (along z) of the upper and lower edges, but it is negligible and less than 10^{–5}. A lower stiffness than in the case §4.2.2 is also observed due to the smaller number of overall layers.

**C***ase (d)*

*ase (d)*

The following layout is analyzed:

\begin{equation} [-30 /-45 /-30 /-45] \end{equation}

The results are shown in fig. 16 and a three-dimensional deformation is shown in fig. 4 (iv) The layout is highly unbalanced and although the dilation is uniform, a torsional effect is observed. Overall, the cylinder tends to screw.

This can be observed by reading the horizontal displacement (along *y*) of the lower and upper edge of the cylinder (* z *= ± *R *). As reported in fig. 18 an increasing linear displacement is observed along the axis of the cylinder. It is positive for the bottom edge and negative for the top edge.

**Plate thickness analysis**

In the theory of **thin laminates** the strain field is reduced to a stain field of the average surface linked to sliding and expansion in the plane. Added to this are out-of-plane effects related to the deflection of the average surface and any torsion contributions. This implies that the deformation field is assumed to be constant and identical in all layers.

Let’s consider a further case with 12 layers having a layout like:

\begin{equation} \left[10 /-10 / 30 /-30 / 0_{2}\right]_{\mathrm{s}} \end{equation}

A schematic representation is shown in fig. 19 (i).

The only thing that distinguishes one layer from the other, according to the considered theory, is the only coordinate *z *. So we can consider the reduced bond in the plane where each layer will have its constitutive aspect while the deformations will be related to the *z* coordinate.

\begin{equation} \left[\begin{array}{c} \sigma_{x} \\ \sigma_{y} \\ \tau_{x y} \end{array}\right]_{k}=[\bar{Q}]_{k}\left[\begin{array}{c} \varepsilon_{x} \\ \varepsilon_{y} \\ \gamma_{x y} \end{array}\right] \end{equation}

That is, the state of tension will be continuous in the layer but discontinuous at the interface and at most it will vary linearly along the thickness due to the linear dependence of the deformation fig. 22.

**Stress**

A certain fixed axial load is applied while the transverse load scaled by a multiplier, eq. (24), so as to be able to analyze the variation of internal stresses as the transverse load varies.

\begin{equation} \lambda F_{\perp} \quad \operatorname{con} \lambda \in\{1 ; 2 ; 5 ; 10 ; 20 ; 30\} \end{equation}

It is evident that the displacement is proportional to the applied load, as the load multiplier increases, the deflection increases. There are no torsional effects being a symmetrical and balanced laminate.

There are no extension effects and the shift along *y* of the plate edge is less than 5 · 10–4. Going to see the tensions along the thickness, or rather when the coordinate is launched

\begin{equation*} z \in\left[-\frac{h}{2} ; \frac{h}{2}\right] \end{equation*}

It is possible to see the linear trend at times. The **linear trend** is given by the deformations that multiply a different stiffness coefficient layer by layer (see fig. 22). The stress also increase, in modulus, in proportion to the load multiplier.

**Fig. 19. **Representation of the layout considered in section §5

**Conclusions**

Knowing the response of structural elements can be useful for correctly modeling a given composite laminate structure and for carrying out detailed damage studies by analyzing internal stresses.

Some cases can be as simple to predict as the alignment of all the fibers along the same direction. More complex cases, such as unbalanced layouts, can lead to structural behavior that is not always easy to predict.

Implementing an analysis of the laminated composite that takes into account the configuration of the individual layers provides useful results for understanding the structural behavior. In addition, it also allows you to take into account the internal stresses in the individual layers that present a dis-continuity at the layer-layer interface.

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