Buckling stability of variable stiffness composite cylinder shell based on automated fiber placement technology

Automated fiber placement technology (AFP) is the key technology to realize variable-stiffness design that have fiber orientation variation across its in-plane. However, defects such as fiber angle deviation and overlap are inevitable during AFP manufacturing. A finite element model has been developed for variable-stiffness cylinder shell that can investigate the effect of the overlap areas and fiber angle deviation on the buckling stability of variable stiffness cylinder shell. Furthermore, the influence of the design parameters of the curved fiber path and the process parameters such as the tow width, the number of tow feeding, and the overlapping strategies on the buckling performance of the structure was studied by parametric analysis. The results showed under the condition of compression-bending coupling, the buckling performance is improved by the variable stiffness design in which the fiber angle varies along the circumferential direction, the maximum critical buckling load factor of optimal variable-stiffness design [0 /〈30|80〉/ 90 /−〈30|80〉] s is 10771, which is 21.1% higher than that of the optimal normal stiffness design. It is found that increasing the course width reduces the buckling performance of the variable stiffness structure, when the tow width is kept constant. When the course width is kept constant, the buckling performance of the variable stiffness structure increases with the increase of the tow width.


Introduction
Automated fiber placement (AFP) technology is an advanced manufacturing technology that can be used to fabricate structures with complex geometries such as double-curvature profiles.In addition, the AFP process can precisely control the fiber orientation, and the variable stiffness laminate can be manufactured by laying fibers along curved fiber paths.The variable stiffness design can provide a more effective load transfer method, so as to achieve higher structural efficiency and lower weight that cannot be achieved by traditional straight-fiber counterparts.
Yau and Chou [1] increase the strength of the open-hole laminate by inserting metal into the braid before curing.It is the first attempt of variable stiffness design of curvilinear fibers.Hyer and Charette [2]   effectively improved the tensile strength of the laminate by designing the orientation of the fibers to align with the principal stress directions, but there was no significant improvement on the buckling load.
Blom [3,4] successfully applied geodesic and equal curvature fiber trajectories to cylindrical and conical surfaces, and discussed the influence of the different fiber trajectories on the mechanical performance.
Gürdal [5] proposed a fiber angle definition method that fiber angle linearly varies along the specified direction, using a small amount of parameters to define the change trend of the fiber angle.Several researches [6][7][8][9][10] studied on the advantage of using curvilinear fibers to improve mechanical responses of a composite laminate have demonstrated the potential of the variable stiffness design for weight reduction.
However, the above works are often based on an ideal model that ignores the specific inherent defect such as fiber angle deviation, overlaps and gaps induced by the fiber steering.Although the cut/restart technology of the AFP process can change the course width through adding or dropping tows, small defect still widespread in laminates.The coverage strategies determine the defect form between adjacent courses.0% coverage strategy creates small triangular areas without fibers.100% coverage strategy creates small triangular overlap areas.The effects of these AFP process-induced defects on the buckling behavior of variable stiffness composite structure are not yet well understood.
Hence, the purpose of this paper is to study the effects of path design parameters and process parameters on the buckling behavior of variable stiffness cylindrical shells.In this work, the define method of the curved fiber path is developed, including the generation of the reference path and the densification of the remaining path.Then, a high-precision AFP variable-stiffness finite element model algorithm based on automated fiber placement technology is proposed, which can effectively simulate the local features in a variable stiffness structure, including the fiber angle deviation within the courses and the overlaps or gaps between adjacent courses boundary.Finally, the mechanism of the curved fiber variable stiffness design to improve the buckling performance is analyzed, and the influence of fiber path design parameters and process parameters on the buckling performance of the structure are studied.

Variable stiffness composite cylindrical shell
Automatic fiber placement is a process in which the fiber placement head lays fiber tows along a predetermined path, and the fiber path will directly affect the mechanical properties and forming quality of components.Gürdal and Olmedo proposed a simple fiber path define method based on a linear function, which defines a continuous curvilinear fiber path with a linear variation of the fiber angle along a specified direction.Fiber angle θ varies linearly along the reference coordinate direction x as shown in Equation (1).
Where 0 T is initial fiber angle, 1 T is end fiber angle, d is characteristic length.
A single layer with this fiber path definition is represented by 01 TT .When the fiber orientation angle varies linearly along the axis direction, the tow-steered fiber orientation angle varies continuously from 0 T on the shell middle to 1 T on the shell both ends.When the fiber orientation angle varies linearly along the circumferential direction, the tow-steered fiber orientation angles varies continuously from 0 T on the crown and keel to 1 T on the shell sides as shown in Fig. 1.
(a) (b) Fig. 1 The fiber angle on the cylindrical shell varies linearly along the circumference (a) or axial (b) direction.
As shown in Fig. 2, by expanding the cylinder to a reference plane, the transformation of the fiber path between the cylindrical surface and the reference plane is realized by establishing the mapping relationship between the cylindrical coordinates and the cartesian coordinates.The path with linear variation of the fiber angle along the circumferential and axial directions of the cylinder is generated in the reference plane.According to the coordinate transformation, the equation of the corresponding curve on the cylindrical surface is obtained.

Fig.2 Transformation of fiber trajectories on cylinder and reference plane
When the fiber orientation angle varies linearly along the cylinder axis, the slope at any point on the fiber path can be expressed as shown: Substituting equation( 1) into equation( 2), the function of the fiber path can be obtained through integration as: In which, 10 k 2(T T ) L  , 0 bT  .When the fiber orientation angle varies linearly along the cylinder circumferential direction, the slope at any point on the fiber path can be expressed as shown: Substituting equation(1) into equation( 4), the function of the fiber path can be obtained through integration as: In which,

Ideal variable stiffness finite element model
The ideal variable stiffness finite element model, which assumes that the width of the tow is small enough to be ignored, is the translation of the reference fiber path along the specified direction, without any overlap or gap in the entire model.Therefore, the ideal variable stiffness finite element model only needs to consider the discretization of the reference fiber path, so that the fiber direction of each element is along the tangent direction of the fiber path.The variable stiffness finite element modelling procedure is schematically summarized in Fig. 4. First step is to map the cylindrical surface to the reference plane, and establish the correspondence of elements and nodes between the cylindrical surface and the reference plane.Then, the following steps are to establish affiliation between elements and nodes in the reference plane, calculate the coordinates of the element center point according to the coordinates of the subordinate nodes of each element, and use the center point coordinates to calculate the fiber angle of each element.Finally, according to the corresponding relationship between the reference plane and the cylindrical surface, the stacking sequence of each element in the reference plane is inversely mapped to the cylindrical surface to obtain the finite element model of ideal variable stiffness cylindrical shell as shown in Fig. 5.

Variable stiffness finite element model based on automated placement technology
In the actual production process, the width of the course is determined by the number of tows fed by the laying head, and the tows in the course are parallel to each other.Therefore, the fiber direction in the course is deflected along the normal direction of the reference path, which makes the fiber angle deviate from the ideal state.The maximum deviation occurs at the edge of the course, and the wider the course, the greater the fiber angle deviation.
In addition, the effective distance of the course in the translation direction is different due to the change of the curvature of the curvilinear fiber path, resulting in overlap or gap on the course boundary, and the translation distance determines the defect form at the boundary.In order to ensure that the course boundary does not produce gaps, the translation distance meets w cos(min ) s T    .At this time, a large amount of overlap area will be generated at the course boundary.The cut/restart function can be used to cut the tow when the tow overlaps, leaving a small wedge-shaped area on the boundary of the course, effectively reducing the area of the overlap.The coverage strategies determine the state of the wedge-shaped area.
In order to accurately simulate the fiber angle deviation in the course and the overlap or gap on the course boundary during the placement process, a high-precision variable stiffness finite element model algorithm based on the automatic placement process needs to be developed.The high-precision variable stiffness finite element model algorithm contains three modules, as shown in Fig. 6.The finite element model of AFP technology is established by the above algorithm can effectively simulate the angle deviation within the courses and the overlap on the course boundary as shown in Fig. 7 and 8.

Linear buckling analyses
Cylindrical shells are often used as the bearing structure, experiencing complex load conditions such as axial pressure, bending, torsion loads during the working process.The cylinder shell has a length L of 300 mm and a radius r of 100 mm.Both shells are built with the 8-ply symmetric  layup subjected to unit compression-bending coupling conditions.Each course consists of 8 tows with 6.35mm wide M40J/4211 high modulus carbon fiber/epoxy prepreg tows.All courses use 100% coverage strategy and one-sided tow drop approach.
The above method is used to establish the variable stiffness cylindrical shell with the fiber angle varies along the circumferential and axial directions respectively.Fiber path design parameters 0 T and 1 T are set at intervals of 10 for parametric buckling analysis.Fig. 9 shows the buckling performance of ideal variable stiffness cylindrical shells with different design parameters under compression-bending coupling conditions without considering the tow width.When the fiber angle varies in the circumferential direction, the 0 / 20 80 / 90 / 20 80 s      design yields the highest buckling load factors of 10144.When the fiber angle varies in the axial direction, the 0 / 60 50 / 90 / 60 50 s      design yields the highest buckling load factors of 8917.6.Hence, the critical buckling load of the fiber angle varies along the circumferential direction is larger than the axial direction, which is 14.3% higher than the optimal constant stiffness design (   0 / 30 / 90 / 30 s  ) that has a critical buckling load factors of 8876.1.Under the condition of compression-bending coupling loads, one side of the cylindrical shell is compressed and the other is tensed.By vary the fiber angle along the circumferential direction, the stiffness distribution of the structure in the circumferential direction is changed, thereby improving the distribution of the load, reducing the compressive load on the compression side, and effectively improving the buckling performance of the structure.Fig. 10 shows the buckling performance of the AFP variable stiffness cylindrical shell with different design parameters under compression-bending coupling conditions.When the fiber angle varies along the circumferential direction, the design yields the highest buckling load factors of 10771.When the fiber angle varies along the axial direction, the 0 / 60 40 / 90 / 60 40 design yields the highest buckling load factors of 9360.It is not difficult to find that the variation trends of the buckling characteristics of variable stiffness cylindrical shell of the AFP and that of the ideal cylindrical shell are similar.It means that the fiber angle deviation in the course and the small triangular overlap on the course boundary are not significantly changing the variation trend of the buckling performance with the design parameters.The small overlap acts as stiffeners.The critical buckling load of the structure is increased, and the optimal buckling performance of the variable stiffness cylindrical shell is 21.3% higher than that of the optimal constant stiffness design.The tow width and the number of tows within a course are process parameters that have a significant influence on the buckling behaviour of variable stiffness structure.The typically widths of the tow are 3.175mm, 6.35mm and 12.7mm.The course width depends on the number of tows.The variation of the course width affects the deviation of the fiber angle and the overlap area, which in turn affects the buckling performance of the structure.When the tow width is constant, the buckling performance of the variable stiffness cylindrical shell decreases with the increase of the course width.When the course width is constant, the buckling performance of the variable stiffness cylindrical shell increases with the increase of the tow width, as shown in

Conclusion
In this text, the buckling performance of the variable stiffness composite cylinder shell under the compression-bending coupling condition is investigated.By establishing a high-precision variable stiffness finite element model based on automated fiber placement technology, the influences of fiber path design parameters and process parameters on the buckling performance of the structure are studied, the conclusions are obtained as below: (1) The AFP variable-stiffness finite element modeling method is developed to accurately simulate the fiber angle deviation in the course and the overlap on the course boundary.
(2) Under the compression-bending coupling condition, the 21.3% improvement on buckling performance of the variable stiffness cylinder compares with the optimal constant stiffness design by vary the fiber angle along the circumferential direction.It is attributed to stiffness varies along the circumferential direction release the compressive load and transfer it to the tension side so that a better load distribution and more area is involved in carrying the compressive load.
(3) It is seen that the course widths and buckling performance are negatively correlated, and tow widths are positively correlated with buckling performance, so that the smallest course width and the largest tow width yields the maximum buckling performance of the variable-stiffness cylinder.

10 k 2 (
T T ) π  , 0 bT  , 1 cT  .After determining the reference path, the translate method was used to construct the variablestiffness laminate.Subsequent courses were placed by equidistant translation along the direction perpendicular to the fiber angle change until the entire surface was covered.Due to the change of the curvature of the curved fiber path, the overlap or gap is inevitable, as shown in Figure3.(a) (b) (c) Fig.3 Methods for densifying paths with overlap (a), gap (b), or both overlap and gap (c).

Fig. 4
Fig.4 Finite element modelling procedure of ideal variable stiffness cylindrical shell

Fig. 9
Buckling characteristic of ideal variable stiffness cylindrical shells with fiber angle varies circumferentially (a) and axially (b) (a) (b) Fig.10 Buckling characteristic of AFP variable stiffness cylindrical shells with fiber angle varies circumferentially (a) and axially (b) Fig.11.(a) (b) Fig.11The effect of course width (a) and tow width (b) process parameters on structural buckling performance