1. Introduction

Carbon-fibre composite structures are being used for an increasing variety of applications. Carbon-fibre reinforced polymer (CFRP) has the desirable properties of low density, high stiffness and high strength. Interest is growing in the use of CFRP for the manufacture of optical devices; a successful implementation of CFRP technology providing the potential for large, lightweight and robust optics for any application requiring such optics, but of particular interest is for space-based devices and large ground based telescopes. CFRP is already used for satellite dishes, space and ground-based telescopes operating at longer wavelengths (e.g. far-infra-red/sub-mm), examples include the space-based Planck telescope [1] and the ground-based ARO SMT telescope [2].

The aim is to improve the manufacturing procedure to provide optics suitable for visible wavelength applications. The primary concerns in regard to CFRP optics for visible wavelength use are: surface quality (i.e. micro surface roughness) and form error. For active or adaptive optics material homogeneity also needs to be considered in terms of the influence function of the actuators and the linear elastic behaviour of the material due to deformation of the optical material by the actuators. The nature of CFRP unidirectional fibre material, which is commonly used for the applications mentioned, is that the material has orthotropic mechanical and thermal properties and inhomogeneities which arise due to imperfect placement of fibres in the resin matrix. These characteristics will result in a finished material of non-perfect form (both fibre print-through on the micro-scale and large-scale deformations). These effects must be minimised to an acceptable level. In optics the level of acceptable error is related to the wavelength (e.g. λ/20 RMS form error and ~λ/200 RMS micro surface roughness), and so the task is more challenging for shorter wavelength applications.

This paper investigates the issue of form error with regard to the composite ply layup sequence. Other CFRP manufacturing processes can also have an impact on the form error, however these are not discussed here. Discussion is also given to the choice of finite element model used to represent the composite structure. This is particularly important where the best estimate of a design value (e.g. surface form deformation) is required rather than a comparative or trend-based study. This paper builds on certain aspects of work presented in a previous study [3,4]. The paper by Thompson et al [3] considered the general manufacturing of nickel coated CFRP mirrors and presented a worst-case example of the effect of a single ply misalignment on the mirror form. The work presented here includes a more in-depth analysis to account for the real-life situation of randomly distributed errors in ply-placement across all plies simultaneously; this is then compared with the error margin of ± 0.5 degrees that is generally used in high accuracy layups. A recent study by Arao et al [5] investigated the case of deformations in composites due to moisture absorption (post-curing) within a 24-ply composite containing random ply misalignments for 3 different layups (0/90, 0/60/120 and 0/+/−45/90). The work presented here is also pertinent for the moisture expansion issue discussed in that paper, since any environmental change that results in an orthotropic expansion/contraction of the ply material will follow a similar trend. Compared to other similar studies [3–6], this paper has investigated a greater variety of ply layup arrangements, utilizing angular increments smaller than 45 degrees, which will be of interest for applications requiring a higher degree of material isotropy.

2. Layup sequences

The composite mirrors modelled in this work are composed of stacked layers (plies) that are arranged in a certain sequence (a layup). It is usual for layups to be balanced and symmetric. Balanced means that for every ply at angle θ, the laminate contains another at -θ. Symmetric is such that each ply is mirrored about the centre plane. An unbalanced layup naturally deforms when taken out of the mould following curing.

There are several parameters that define a ply: thickness, fibre and matrix types and orientation. In this study, only mirrors with the same ply thickness and material for all plies have been compared in order to focus on orientation arrangements.

A ply stacking sequence is defined by 3 parameters.

Table 1 lists all the sequences that were investigated using the finite element model. Angles inferior to 9° have not been considered in this paper because they require a large number of plies to form a layup pattern. A thick mirror, containing more plies, will use more material and increase the layup time which increases cost. As the composite thickness increases above 5mm there is also an increased risk of a run-away exotherm [7] during the cure cycle which would render the material useless. The cure cycle of thicker materials has to be more strictly controlled. We are interested in thin and lightweight CFRP mirrors and so have limited the thickness of the mirrors to the range 1-4 mm. If extra stiffness is required this is best achieved by adding structures such as reinforcing ribs on the rear surface of the mirror or sandwich constructs.

Table 1. The list of layup sequences that were modelled in this study. For a given number of plies in the layup and a given increment angle the number indicates how many symmetric sequences can be obtained in a clockwise stacking arrangement. A * indicates whether the sequence can also be obtained using the perpendicular-pair arrangement.

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3. Finite element method

The finite element method (FEM) has been used to model the effect that errors in the angular positioning of plies in the layup has on the form of a CFRP mirror due to resin cure shrinkage during manufacture. FEM is a mathematical tool for solving problems by dividing-up a continuum into a number of elements and solving the relevant partial differential equations for each element simultaneously. A finite element model must contain suitable boundary conditions to properly condition the solution; these are discussed in Section 3.2.

3.1 Modelling the deformations due to resin shrinkage

A 200 mm diameter, flat circular mirror has been modelled using a FEM software package: namely, Abaqus/CAE in conjunction with the Abaqus/Implicit solver. The thickness of the mirror depends on the sequence which varies from 20 to 40 plies, with each ply being 0.1 mm thick. This is a typical thickness for unidirectional pre-pregnated material that would be used for a mirror or satellite dish application. Thirty-two different layup sequences have been created by varying the 3 parameters: arrangement (clockwise/perpendicular pairs), ply number and increment angle.

The general material properties have been set to those of the pre-pregnated unidirectional material LTM123/M55J [7,8]. It consists of high modulus carbon fibres in a cyanate-ester resin matrix. A 60% fibre volume fraction has been chosen giving an average density = 1638 kgm⁻³, E₁ = 313.4 GPa and E₂ = 7.47 GPa, where direction 1 is along the fibre axis and direction 2 is perpendicular to this. To model the effect of the resin cure shrinkage using the finite element package, the material has been assigned dummy coefficients of thermal expansion (CTE). Minimal shrinkage occurs in the fibre direction, so the dummy CTE has been set to zero (α₁₁ = 0); for the other two directions a dummy CTE value is entered which when combined with a suitable temperature change simulates a 0.1% material cure shrinkage in those directions (e.g. α₂₂ = α₃₃ = 1e-5 K⁻¹ for ΔT = −100 K).

In this work both 3D continuum and shell geometries have been used to formulate the models. It is a less complicated task and more memory efficient to define the model for surface shell geometry. The geometry for both models is straightforward but the layup definition, for instance, is simplified with surface shells in Abaqus/CAE and allows the user to import and save layups directly from comma-separated values (CSV) text files.

3.2 Modelling considerations

In creating a finite element model there are generally two problems that have to be considered: solver accuracy and solver run time.

In this case, as the number of runs is substantial (> 10,000), run times are critical. Both solver run time and accuracy are affected by the choice of element type and number of elements. As such, special care has been taken to optimize the models.

For both the model types investigated, surface shell and 3D continuum, careful consideration of the boundary conditions has been made so that they reflect the real system. The boundary conditions should not constrain any movement the mirror may make to deform and they should prevent free-body motions caused by a lack of constraint. This is straightforward for shell elements as their nodes have, in addition to translational degrees of freedom (dof), rotational dof which provide a means of constraining rotation at the node. All free body movements in this model type can be prevented by fixing all six dof at one node. Nodes on 3D elements only have translational dof. Therefore, the rotations have to be prevented by constraining an off-axis translation. Figure 1 illustrates two sets of boundary conditions that block all 6 dof to stop free body movements, with the right-hand side (Boundary Condition Set 2) diagram representing the models used in this work. By way of example, the left-hand side (Boundary Condition Set 1) diagram is shown with boundary conditions which over-constrain.

 

Fig. 1 Examples of boundary conditions: 0 indicates a blocked displacement; U indicates the point is unconstrained and the number in which axes. Set 1 is an over-constrained model. Set 2 is used for the main body of work in this study.

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Boundary condition Set 1 prevents tangential displacements along 4 linear supports on the edge of the mirror: A, B, C and D. This blocks all 3 rotations and 2 translations along directions 1 and 3. The last translation along the mirror axis (direction 2) is constrained at point O. The first case (BC Set 1) is presented to illustrate how over-defined boundary conditions can affect the model solution. The problem with this definition is that it constrains nodes tangentially to the mirror’s edge where in reality they would displace around the circumference of the mirror, as can be seen in Fig. 2.

 

Fig. 2 Close-up of the mirror’s edge showing an example of deformation.

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In BC Set 2, point O is fixed in 3 directions. The unwanted free-body rotations are then constrained: the tip/tilt of the mirror is prevented by fixing the translations in two directions at point O’ and the in-plane rotation is blocked at point A.

There are generally two main factors that have an influence on model running times: mesh size and element type. Abaqus allows the user to choose between different element types [9]. To test and verify the choice of element used for these simulations a series of the composite mirror models were run to compare the results and computational time.

Figure 3 illustrates the different geometries that define the element types. A 3-D continuum model uses 6 and 8-noded linear element types (denoted as C3D6 and C3D8 with full integration), whereas a 2-D definition (shell geometry) uses a 4-noded surface element (ABAQUS’s S4R). When available, reduced integration (R) and incompatible modes (I) were tested (denoted C3DnI orC3DnR, where n = number of nodes per element).

 

Fig. 3 Mesh element geometries.

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In order to control the meshing process in ABAQUS/CAE the mirror was partitioned into 4 quarters. This is more important for hexahedral than for triangular meshes as it is easier to fit triangles than rectangles over a circular shape; other similar techniques can also be used for this purpose.

The results of convergence studies using different element types are shown in Fig. 4.This figure shows surface deformation peak-to-valley (PV) values against the number of nodes on the top surface of the mirror for a range of element types. It can be seen that there is a significant variation in the results depending on the number and type of element used. The results also compare two sets of models using either boundary condition set 1 or set 2.

 

Fig. 4 Comparison of the results obtained with different element types and meshes for a 24-ply layup, with 5° error on the ply immediately below the surface ply.

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As shown in Fig. 4, the reduced integration 3D continuum ‘brick’ or ‘hex’ elements contrast most markedly with results from other models with the same boundary conditions. This is especially the case for BC Set 1. With increasing elements the BC Set 2 results may eventually converge with each other, however this is very unlikely for the BC Set 1 results. The element type C3D8R (breaks trend for BC Set2 results) is not appropriate for simulations which involve bending due to a phenomenon known as hour-glassing since the bending effects are not appropriately represented. Therefore, this type of element should be avoided. The Set 1 BCs constrain the astigmatic nodal displacements and creates bending in the mirror which manifests as errors when using the reduced integration elements. Prisms, or wedge elements (C3D6) also tend to give inaccurate results as the geometry gives them increased stiffness. These elements are primarily designed for hex-dominated meshes. It should be noted that again in the Set 1 BC case these elements under-estimate the PV result, but at a high element density they converge with the Set 2 BC’s. Both the full integration hex (C3D8) and the incompatible nodes (C3D8I) element offer very good performance in this type of problem, with the latter offering very good convergence characteristics. Given the very small difference in result and that the computational time was also an important consideration a surface element (S4R) was chosen for speed and memory efficiency. As seen on Fig. 4, this element gives very good results with a limited number of nodes and as such is used in the analyses which are to follow.

3.3 Simulating the fabrication errors

In order to compare the different layup sequences, a certain number of models with a normal distribution of random angular alignment errors were generated for each layup sequence. Each of these samples was constructed with the characteristics of a given sequence with perfect ply orientations and was given a set of ply alignment errors, generated randomly, that was added to the base orientation.

The indicator that has been used to compare the layup sequences is the surface PV averaged over all the results for a given sequence. For the results to be consistent, a minimum number of models was required per sequence. In order to determine this number, a convergence plot, shown in Fig. 5, was calculated by running a number of models from the same sequence and observing the evolution of the surface PV average as the number of samples increases.

 

Fig. 5 Plot showing the running average of the surface PV deformation against the number of sample models generated. Where the average PV stabilises was deemed a suitable number of samples for the Monte Carlo simulations.

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After approximately 300 samples, the average reaches a value that oscillates within an interval of ± 1µm. Other layup sequences showed a similar behaviour, but as the exact number of samples that are necessary to get such a convergence depends on the exact sequence defined, a ‘margin’ has been employed with 500 samples used per layup sequence tested. In this way, an accuracy of 2µm on the mean surface PV for a layup sequence can be achieved which is represented by the two red (dashed) lines on Fig. 5.

4. Simulation results

Models were created to investigate the effect of layup sequence/type on the PV deformation of a circular flat mirror due to random errors in the angular placement of plies in the layup sequence (i.e. the layup is no longer perfectly balanced creating a residual strain in the material due to resin cure shrinkage). The effect of varying the standard deviation of the angular errors and the size (radius) of the mirrors was also investigated.

4.1 Layup sequences

The plots in Fig. 6 show the surface PV against the increment angle for the 32 different layup sequences as detailed in Table 1. The mirror in this simulation has a diameter of 200 mm. These data were obtained by averaging each set of (500) model results for every sequence. A standard deviation of 1° was chosen for the random alignment error added to each ply orientation.

 

Fig. 6 Plots of surface PV versus increment angle for 32 different layup sequences; all plies contain angular alignment errors with a 1 degree standard deviation. Each point corresponds to a specific layup sequence, showing the mean surface PV of the 500 samples for that sequence; the error bars indicate the standard deviation of those results. The sequences are grouped by a fixed number of plies per graph.

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As can be observed in Fig. 6, one of the most interesting findings is that for a given number of plies, having one pattern repetition inside the sequence reduces the deformations considerably. The relative difference between the surface PV of a sequence without any repeat (using the smallest increment angle possible) and the adjacent sequence (i.e. a doubling in the angular increment angle) which has one pattern repetition is 26-28%. Yet, having two or more repetitions does not reduce the form error by any significant further amount. This is also reflected in the standard deviation of the results: going from no pattern repeat to one or more repeats significantly reduces the standard deviation of the form error, but there is no significant reduction in the spread of results by having more than one repeat.

Another observation is that clockwise sequences deform slightly less than perpendicular pair sequences. The exceptions to this rule are where the difference is less than the error on the result. In these cases the difference between the two layup types can be considered to be insignificant.

The increment angle has a greater influence on deformation than the layup arrangement type. For example, the largest difference between a clockwise and a perpendicular pair sequence that have the same increment angle and number of plies is 0.9 µm whereas the largest difference between sequences that have the same number of plies, but a different increment angle is 5 µm.

As expected, the PV deformation decreases with increasing number of plies. It can be explained by the fact that as the thickness increases the stiffness of the mirror increases more rapidly than the size of the residual strain and so there is a smaller force-to-stiffness ratio.

Smaller angles are preferred as they increase the isotropy of material. Hence for this purpose, we can conclude that for a given number of plies, the optimum increment angle to minimize the curing deformations is the smallest angle that allows a pattern repetition within the thickness.

4.2 The size of the angular error

We also investigated the effect of varying σ, the standard deviation of the applied angular errors. A high accuracy layup is generally considered to be ± 0.5°, achieved by a manual placement of the plies over the mandrel. The recent study by Arao et al [5] measured the standard deviation of the ply misalignment to be 0.4°, resulting from a combination of ply placement errors and intrinsic material properties. This intrinsic error is due to the manufacturing process used to create the ply material and random ‘waviness’ of the carbon-fibre placement in the resin matrix. Assuming an intrinsic minimum angular error for unidirectional CFRP material to lie within a range of 0.1 to 0.25° suggests that even with a near perfect placement of the plies within the layup sequence there is a random intrinsic magnitude of angular error of the material that will limit the best form error that can be achieved.

Figure 7 shows the result for 32-ply sequences; a similar trend is observed for other mirror thicknesses. Again, one can observe that the sequences with no pattern repeat show PV deformations ~30% greater than the others. The cross-over of some lines in Fig. 7 are due to the method that is used to obtain them; the results represent mean values of 500 virtual mirrors made with randomly-distributed errors. The separation of those lines are smaller than the associated errors ( ± 1 μm) determining them.

 

Fig. 7 The change in PV form error with different applied angular error distributions for 32-ply sequences. The 3-part nomenclature in the key, e.g. 32_1125_clock, denotes the number of plies (32), the angular separation (1125 = 11.25°, 225 = 22.5°, 45 = 45°) and the sequence arrangement (clock = clockwise, perp = perpendicular pairs). The 32_225_clock sequence has 2 additional results at σ = 0.1 and 0.25 degrees; error bars have been shown only on this sequence for clarity and are representative of the trend in the standard deviation of the PV.

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A composite mirror with perfectly-aligned plies except for one (the ply immediately below the surface ply) that is misaligned by 5° produces a 9µm form error. To get an equivalent deformation for the same sequence with a random error on every ply, Fig. 7 shows the standard deviation of error required to lie in the range 1.2° to 1.7°. This highlights the importance of maintaining a low consistent error throughout the layup, as any large errors on one ply have a significant effect on surface form. This is more important for the outer plies as the moment of the forces increases with the distance from the centre plane of the composite.

4.3 The effect of mirror diameter

Figure 8 presents the relationship between PV and mirror surface area for 3 different mirror sizes (diameter = 200, 600 and 1000 mm). The deformation is proportional to the mirror surface area and the gradient depends on the layup sequence and the errors within that layup sequence. The sensitivity of the mirror to ply misalignments is also proportional to mirror area, i.e. the standard deviation of the PV form errors increases proportionally with mirror area. A negligible difference between layup sequences on a small mirror can become significant as the size of the mirror increases.

 

Fig. 8 The deformation of the mirror is proportional to the surface area of the mirror. The random error applied to the plies for all sequences (plotted with a solid line) is sigma = 1°; the sequence with a dashed line shows the effect of reducing the ply placement error and has a sigma = 0.5°. Two sequences: 32_225_clock and 32_45_clock are very similar and overlap on this graph. The key nomenclature is the same as used previously.

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It can also be seen from the figure that halving the error in ply placement approximately halves the PV deformation. The distribution of these form errors on a 1-metre diameter mirror is plotted in Fig. 9.

 

Fig. 9 The surface deformations due to ply misalignments are of an astigmatic form. This 1000 mm diameter, 32 ply (3.2 mm thickness) mirror has a PV form error of ~50 μm for a 0.5° sigma ply error.

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5. Conclusions

This study investigated the effect of angular errors in ply placement in uni-directional fibre-polymer composites on the overall deformation of the structure post-cure. The metric used to quantify this was the surface PV and the composite structures considered were flat-form circular plates; our application area of interest being in lightweight composite optics and their support structures (e.g. backing plates for active optics).

Most high-accuracy layups call for a tolerance of ± 0.5° where the plies are aligned manually using guides on the edge of the mirror, like a protractor. This is relatively easy to achieve for large mirrors but is more difficult for smaller mirrors given that the arc-size tolerance is r × Δθ at the edge (where r is the radius of the mirror). In addition there is an intrinsic angular error within the material due to imperfect placement of the fibres in the polymer matrix. Intrinsic error is likely to be slightly higher in larger mirrors. This implies that there is a minimum average form error that can be expected for a mirror of a given size, layup sequence and angular error. It should be anticipated that a certain number of mirrors will have to be manufactured so that there is a reasonable chance that the desired form error is achieved. The likelihood of an unfavourable form error can be reduced by ensuring that there is at least one pattern repeat in the layup sequence. In reality the statistics for this are more favourable than those presented since the ply placement and intrinsic error are not truly random, with the requested tolerance (e.g. ± 0.5°) being close to an upper clipped limit (i.e. the 0.5° could be considered to be at the 3σ error level).

Within the context of an optical quality CFRP mirror operating in the visible wavelengths, the simulations presented here do not meet the required PV surface error. As an example, using the material properties in these simulations, a 1m diameter CFRP mirror with a random angular error of sigma = 0.1° and the 22.5° clockwise increment 32-ply layup (3.2 mm in thickness) has an average form error of 14 ± 7 μm; the 200 mm diameter case has an average PV of 0.6 ± 0.3 μm. A mirror such as this would require form correction, which can be achieved using active or adaptive support structures.

The form error could be reduced further by using a ply material that has lower overall cure shrinkage (< 0.1%). This can be achieved one of two ways: use a higher fibre volume fraction (V_f) or select a matrix (resin) that has a lower cure shrinkage. We used a V_f = 60% in the models presented. The theoretical V_f limit is 90.7% (for hexagonal fibre packing) or 78.5% (for square packing), however there is a practical limit of around 70% V_f. This is due to a minimum amount of resin that is required to form a fibre-polymer ply to ensure adequate bonding and the fact that perfect packing cannot be achieved. Additionally, selection of the optimum resin in a composite is not based on cure shrinkage alone. The environment in which it is to be used will determine other properties such as micro-cracking resistance, thermal performance, moisture absorption expansion and UV ageing characteristics. Use of a higher modulus fibre (a higher E_fibre/E_matrix ratio) will also reduce the impact of strains within the matrix and improve the overall stiffness of the mirror to further reduce deformations.

The principal findings drawn from this work are summarised as follows:

Taking note of these points will aid the designer in producing an optimal construction for a given diameter and thickness requirement. The tools developed for this study could be applied to any size of mirror (or similarly constructed artefact) to predict the deformations that can be expected from the ply misalignment.