Stress Analysis of Hole Orientation and Laminate Geometry Impacting on Boron/Epoxy Composites Laminates

Boron/epoxy laminates are used in aircraft and space vehicles for their high strength. Evaluation of stresses and residual strength of the laminate with square cutout are not analyzed in the literature. The present work is focused on studying the effect of hole orientation and laminate geometry on Boron/Epoxy composites laminates under in-plane loading. The analytical solution for stresses plate with different hole shapes and orientations of loading. The basic equations of failure criteria available for plain laminates are derived to calculate the residual strength of the laminates with hole using the stresses obtained from the analytical solution. The derived analytical solution is validated by reproducing exactly the same results of earlier researchers even by other formulations and also by the results of finite element analysis using ANSYS. The [0/0]s laminate is not preferred due to highest stress concentrations at the corners that range between 12 to 12.45. Similarly, [45/-45]s laminate is also not preferred due to its higher values of stress concentrations which range from 9.5 to 28. The normalized stress for [0/90]s under x-axis loading is 9.6 and for y-axis loading it is 9.5 which is almost the same. Even for equi-biaxial loading, it is 8.5 and for shear loading, it is 12.45. Except for shear loading, [0/90]s laminate seems to be a better choice for a reasonable value of stress concentration for any general case loading. The analytical solution derived in the present work is the most general and unique as it can yield the stresses around any shape of hole and laminate geometry and all types of in plane loading. This solution will be able to reproduce the results of all other solutions available in the literature by different formulations.


Introduction
Composite materials are very important in manufacturing processes because they are used widely in several areas such as aerospace industries and airframes and thanks to its remarkable strength and stiffness to weight ratios as well as its good resistance to corrosion and the fatigue [1,2]. Fiber reinforced polymeric (FRP) composites are nowadays widely used inengineering applications due to their outstanding features, such as high specific strength,specific stiffness, less/weight ratio and high structural performances, which are often theoutput of specific design and manufacturing strategies that aim to optimize the response of these composite structures to specific working conditions. Such superior mechanical properties of composite materials such as high stiffness and strength to weight ratios and its attractive replacement for metallic materials made it possible for it to be used increasingly in many areas of technology including marine, aerospace, automotive and others. This has generated an increasing interest in the study of new possible design solutions aimed to enhance the performances of composite laminate structures under prescribed rules through the appropriate choice of materials, orientation, number of stacking sequence of layers that make up composite material and the determination of the optimal fiber orientation for each FRC layer [3][4][5][6][7][8].
Boron fibers, having about six times the modulus of elasticity of glass fibers and about the same strength and density as glass fibers, have been developed for use as reinforcements in structural plastics for aerospace vehicles and other high-performance applications. They are very stiff and have a high tensile and compressive strength. Boron fibers are used to repair cracked aluminum aircraft skins, because the thermal expansion of boron is close to aluminum and there is no galvanic corrosion potential [9][10][11]. Stress concentrations around cutouts are important from design point of view because they are generally the prime cause of failure under static loads. For most materials, the failure strength is strongly notch (or hole) sensitive [12,13]. It has been experimentally validated that the net failure stress (considering the reduction in cross-sectional area due to cutouts) is generally much less than the ultimate tensile strength without any discontinuity [14].
Different cutout shapes in structural elements are needed to reduce the weight of thestructure or provide access to other parts of the structure. In some cases, structural elements are being damaged during their service life. It is well known that the presenceof a cutout or hole in a stressed member creates highly localized stresses at the vicinityof the cutout [15]. Uncoupling between extension and bending in symmetric laminatesmakes analyzing such laminates simpler. It also prevents a laminate fromtwisting due to thermal loads, such as cooling down from processing temperatures and temperature fluctuations during use such as in a space shuttle [1]. A major advantage of fibrous polymeric composites is that their anisotropy can be controlled through suitable choice of the influencing parameters. The unidirectional fiber reinforced composites provide much higher longitudinal mechanical properties compared to the transverse ones. Therefore, composite laminates are formed by stacking two or more laminas, with different fiber orientations, as to respond to complex states of stresses [1].
A major disadvantage of unidirectional composite materials is that they are highly anisotropic. One way around this is to form laminates by stacking two or more laminae' on top of each other with various orientations of the fiber direction and the mechanical properties are dependent upon the way the individual layers are stacked i.e. the stacking sequence. Predictions for cross-ply and angle-ply laminate mechanical properties are based upon classical lamination theory. Experimental data which indicates that the strength of symmetric composite laminates containing identical ply orientations can be strongly dependent on the detailed stacking sequence, i.e., the actual arrangement of the various layers [16,17]. One of the most important advantages of laminated composites is their potential to orient the laminas in different suitable directions or to choose the convenient stacking sequence configurations, such that the structural response to the complex state of stresses can be improved. It is evident that to enhance the mechanical properties of composite, controlling of the stacking sequence is very important [18,19]. The High stiffness and strength to weight ratios and its attractive replacement for metallic materials caused it to be used increasingly in many areas of technology including marine, aerospace, automotive and others. These Superior mechanical properties and extensive application of composite materials are achieved by tailoring of material properties through selective choice of orientation, number of stacking sequence of layers that make up composite material [20][21][22]. The present work addresses the determination of the stresses around square holes in symmetric boronepoxy unidirectional, cross ply and angle ply laminates under uni-axial and biaxial tensions and shear loading at infinity and to study the effect of orientation of square hole in [0/0]s, [0/90]s, and [45/-45]s boron/epoxy symmetric laminates and type of in plane loading on the residual strength of the laminate.

Theoretical Formulations
In the present work, a general solution is derived within the classical laminated plate theory to obtain two-dimensional stresses around holes in thin fiber reinforced composite laminates. The given hole problem is regarded as superposition of two subproblems corresponding to a sum of the respective complex potentials. Thus, by combining the two subproblems together will ensure the boundary conditions of no traction on the boundary of the hole while the plate is only under remote loading. Important feature of the present general solution is the introduction of generalized form of mapping function and also the arbitrary biaxial loading condition into the boundary conditions of the basic solution given by Savin [23] for in plane loading on thin anisotropic plates containing holes.

Generalized Mapping Function
In the formulation of complex variable method using conformal mapping, it required to represent the area external to a given hole in the physical z-plane by the area outside the unit circle in ] -plane using a transformation function called the mapping function. Such a mapping function is given in a generalized form as seen in equation 1: where, ݉ are the constants of the mapping function. The solution is implemented by computer through coding in MATLAB. The solution is developed to consider a maximum number of terms N equal to 1 to 19. To get the convergent results, higher number of terms in the mapping function are used. R is a constant for the size of hole and since the stress distribution depends only on the shape but not on the size of the hole, the value of R is taken as unity. For anisotropy, the equation shall be modified by introducing the complex parameters of anisotropy ‫ݏ‬ . By affine transformation, the mapping function ‫ݖ‬ in equation 2 becomes: By using the identities for sine and cosine and introducing them into Eq. (2), the mapping function in equation (2), is finally obtained in equation 3 as:

Arbitrary Biaxial Loading Condition
Arbitrary biaxial loading condition [24] is useful to consider various cases of in plane loading in to the boundary conditions. This is easily achieved by introducing the orientation angle (E ) and the biaxial loading factor (O) into the boundary conditions at infinity as seen in Figure 1.

Boundary Conditions at Infinity
The boundary conditions about the arbitrary coordinate axes x'0y' for each case of loading are given below for symmetric laminates within plane loading as seen in equation 4.
where, ߪ ஶ ௫ᇱ and ߪ ஶ ௬ᇱ are applied stresses at infinity about the axis of ‫ݔ‬Ԣ and ‫ݕ‬Ԣ axes. Using the transformation relation of axes in equation 5, it is given by: From Equations 4 and 5, the boundary conditions ߪ ஶ ௫ , ߪ ஶ ௬ and ߬ ஶ ௫௬ about x0y axes are explicitly written in equation 6 as: The boundary conditions in Equation 6 are useful to determine the stress functions of the hole free plate

(a) Scheme of Solution
The anisotropic plate with hole is applied by a remote tension , at the outer edges as shown in Figure 2(c). The edges of the hole are free from loading. To determine the stresses around the hole, the solution is considered in two stages.

(a2) Second Stage Solution
For the second stage solution, the plate with hole is applied by a negative of the boundary conditions f1, f2 on the hole boundary in the absence of remote loading as shown in Figure 2

(b) Stress Functions of First Stage Solution
By taking I'1(z1) = (B* + i C*), \'1(z2) = (B'* + i C'*) and upon integration, we get I1(z1), \1(z2). The constant C * is associated with the rotation of the infinitely distant part of the plane x0y. Since no rotation is allowed and it results in C * to be zero as seen in equation 11.

(c) Boundary Conditions from First Stage Solution
The boundary conditions on the fictitious hole are given in equation 12 as: Now the plate with hole and negative of the boundary conditions obtained in the first stage solution such as: f1 0 = f1, f2 0 = f2 are considered as shown in Figure   2 The constants, B*, B'*, C'* in Equations 11 and 13 are given in equation 14 as: Upon introducing the mapping function (3) into the boundary conditions (13), we have equations 15 and 16: where,

(d) Stress Functions of Second Stage Solution
Introducing the boundary conditions in equation 15 into the Schwarz formula Eq (4) of Savin [23], the stress functions for the second stage are obtained using the following in equation 17:  (17) where, J is the boundary of the unit circle in ]-plane O1, O2 are imaginary constants which will not affect the stresses and may be dropped further. The following results are used while evaluating the integrals in equation 18: where,

Normalised Von Misses Stresses Around Square Hole with Normal Sides
The  Table 1 for all the cases considered in Figures 3 -14.
It is noted from the results in Table 1, that a lowest stress concentration equal to 6.8 is present for [0/0]s laminate under y-axis loading, i.e., the loading is normal to the direction of fibers. On the contrary, a higher stress concentration equal to 12 is obtained for the same laminate when the loading is in the direction of fibers, i.e., along x-axis. The higher value obtained can be attributed to the prevailing discontinuity of fibers in x-direction along the vertical edges on either side of the hole. There is a shearing interface near the corners due to discontinuity of fibers in the direction of loading which results in mode-II fracture of the fiber-matrix interface which has only a weaker interaction of shear resistance to offer against the applied tension. Since the shear resistance of the interface is less, a high stress concentration is obtained near the corner.
This result is in conformation with the observation by Jong [28] that shear stress concentration is more E3S Web of Conferences 309, 01160 (2021) ICMED 2021 https://doi.org/10.1051/e3sconf/202130901160 relevant in anisotropic plates rather than the tangential or von-Mises stress. When the same [0/0]s laminate is applied by y-axis loading, the normalized stress is only 6.8 which is close to one-half of the previous value. This lower value indicates the presence of a tensile condition of mode-I fracture of fiber matrix interface near the corners due to loading in y-direction. The lower stress concentration can be attributed to the higher resistance offered by interfaces with a tensile resistance which is normally higher. Hence, there is a lower value of stress concentration for loading along y-axis.
Another feature of a shift of 5 o on either side of the corner can be noted for x-and y-axes loadings. It is at 50 o for x-axis loading while it is at 40 o for y-axis loading. This shift is oriented away from the direction of loading which can be attributed to the shearing of interface between the broken and unbroken fibers near the corners with a stretch of the fibers along the edges in the direction of loading while there will be a sliding of the broken fibers towards the empty space of the hole due to shearing action. It concludes that orientation of fibers normal to the direction of loading has reduced the stress concentration with 5 o shift from the corners in the direction of fibers which applies for the [0/0]s laminate under y-axis loading.
The normalized stress for [0/90]s under x-axis loading is 9.6 and for y-axis loading it is 9.5 which is almost the same. Even for equi-biaxial loading, it is 8.5 and for shear loading, it is 12.45. Except for shear loading, [0/90]s laminate seems to be a better choice for a reasonable value of stress concentration for any general case loading. A cursory look at the results of [45/-45]s laminates for different types of loading indicates that there is higher value of stress concentration for all cases of normal loads. The highest value is 28 for [45/-45]s laminate under equibiaxial loading, while it is 14 for both x-and y-axis loadings. This is due to the fibers being oriented at 45 o to the directions of loading and only matrix must be sharing the maximum load. This indicates a weaker resistance offered by the interfaces due to fibers oriented at an angle and not contributing much for load sharing. Hence, [45/-45]s geometry is a bad choice especially when the direction of loading is along x-, yor equi-biaxial directions. However, for shear loading, the stress concentration is 11.6 which is lower compared to the values under loading in normal directions.
Generally, shear loading has caused a higher stress concentration at the corner and the material around the hole boundary is subjected to higher severity of loading compared to other types of loading.