Homogenized quadriaxial laminates

Introduction

Composite laminates are used in many structural applications that require geometric variations such as thickness transitions or tapered ends to achieve weight reduction, improve aerodynamics, structural efficiency, or allow for structural integration with other components. Tapering laminates, that is, reducing thickness by terminating plies in a controlled fashion, may result in change in properties and introduce stress concentrations that can affect performance and durability. Thus, staggering ply drops has been a standard practice to reduce stress concentrations and delamination. Instead of dropping off multiple plies at the same location, which creates a severe geometric and stiffness discontinuity, staggering distributes the termination points over a distance. However, the most effective way for tapering laminates is homogenization. This is achieved by stacking identical building blocks in sufficiently large number so the laminate in-plane stiffness [A*] is practically equal to the flexural stiffness [D*]—when both are normalized by the thickness [ A * ] = 1 h [ A ] and [ D * ] = 12 h 3 [ D ] and the coupling matrix [B] approaches zero. Normalized stiffnesses share consistent units, allowing direct comparison of their relative magnitudes.

Traditional quadriaxial (Quad) laminates, based on collections of 0°, ±45°, and 90° plies, have been the standard for decades. However, tapering Quad laminates can be challenging due to the complexity in their stacking sequence when the building blocks used are 8-, 10-, or higher number of plies. When the requirement for mid-plane symmetry is added, the number of plies to be dropped is doubled. There are additional plies added to meet the load-carrying requirements. In such laminates, properties are changed when plies are dropped.

Double-double (DD) laminates are a class of composite laminates formed by repeating four-ply sub-laminates with the configuration [±Φ/±Ψ], in which the ply orientations are governed by two continuous angular parameters, Φ and Ψ. In contrast to traditional composite Quad laminates that rely on a finite set of discrete ply angles, DD laminates allow for a continuous variation of ply orientation.^1–4 DD laminates with stiffnesses that exactly or approximately match those of Quad laminates can be determined. The procedure for deriving an equivalent DD laminate for a given Quad laminate is based on minimizing the differences in the thickness-normalized stiffness components A₁₁*, A₂₂*, and A₆₆*. ⁵

Double-double laminates offer a more rational and simplified approach by using two continuous angle variables (Φ and Ψ) to define the ply orientations within a repeating four-ply sub-laminate [±Φ/±Ψ]. These sub-laminates can be repeated (“stacked up”) to form the full laminate. The angles Φ, Ψ are continuous variables, that is, any value between 0 and 90° within manufacturing capability, rather than fixed at the Quad angles. As the number of repeats of the building block increases, the laminate tends toward homogenization: its behavior becomes less dependent on stacking sequence, and averaged properties (e.g., bending stiffness) approach those of symmetric laminates.^2,6–11

Despite the many attractive features of double-double laminates, traditional Quad laminates offer the advantages of the availability of extensive data and familiarity of design practices. In this study, the concept of homogenization—largely discussed in the context of DD laminates—is examined for quasi-isotropic Quad laminates. The convergence of normalized flexural stiffness to in-plane stiffness is also analyzed for Quad laminates. In addition, the concept of average strength is employed to determine ratios among carbon fiber reinforced plastic (CFRP) laminates and to compare CFRP materials for a given Quad laminate. Overall, this investigation explores opportunities for design optimization using Quad laminates.

Homogenization of quasi-isotropic laminates

An invariant-based approach that uses the trace (Tsai’s Modulus) of the plane stress stiffness matrix as a material property to describe the elastic behavior of composite plies has been described in the literature. ¹² Hence,

The traces of the in-plane and flexural normalized stiffness matrices have been shown to be constrained by the same invariant, ¹³ as expressed in equation (2). Thus,

Then, it has been demonstrated that trace-normalized stiffness parameters for unidirectional carbon fiber reinforced polymer composites are very similar, and their mean values were used to define a “master ply,” which was found to properly describe the behavior of these composites. Master ply properties given as dimensionless ratios are shown in Table 1.

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Table 1.

Master ply properties (dimensionless ratios). ⁸

E 1	E 2	ν 12	E 6
0.8796	0.0522	0.3181	0.0313
Q 11	Q 22	Q 12	Q 66	Tr [Q]
0.8849	0.0525	0.0167	0.0313	1.0

Thus, when master ply properties are considered for simulations and optimization, in-plane and flexural normalized stiffness matrices’ components can be determined by multiplying the general master ply-based parameters by the trace of the specific CFRP material. The use of master ply properties that represent all CFRPs have been proven very useful for design and optimization of CFRP laminates.^14–16

Master ply properties are used for the study of homogenization of quasi-isotropic laminates. The Quad laminate configuration comprises four plies, representing the lowest ply count possible for such a laminate. In this case, the resulting laminate is limited to have quasi-isotropic properties. There are 24 possible stacking sequences. Twelve of those are shown in Figure 1 using master ply properties. Twelve additional laminates can be obtained if the signs of [±45] are interchanged.

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Figure 1.

Normalized stiffness matrix elements of quasi-isotropic laminates with four plies (symmetric) using master ply properties.

The stiffness values A _ij * and D _ij * shown in Figure 1 are the normalized in-plane stiffness and flexural stiffness of the laminate, respectively. The listing of the 12 laminates shown is based on the descending order of the flexural stiffness component D₁₁* with positive D₁₆*. In these quasi-isotropic laminates, A₂₂* = A₁₁* and D₂₂* can be obtained from D 22 * = 1 − D 11 * − 2 D 66 * .

It can be observed, from the graph of Figure 1, that A* components are constant for all combinations of quasi-isotropic laminates. On the other hand, D* components vary depending on the Quad stacking sequences.

To homogenize this family of 4-ply Quad laminates, each building block consisting of four plies must be repeatedly stacked. The resulting normalized flexural and in-plane stiffnesses are shown in Figure 2. The rate of convergence of stiffness D₁₁* to stiffness A₁₁*—which is a characteristic of homogenized laminates—vary, and the fastest convergence shown in Figure 2 is for laminate [−45/+45/0/90], which has the same rate of convergence of laminate #6 in Figure 1, that is, [+45/−45/0/90]. In this case, the number of repeats to achieve homogenization is about 5, resulting in differences of 0.2% for D₁₁*, 7.7% for D₂₂*, and 11.4% for D₆₆*, as compared to the corresponding in-plane components A₁₁*, A₂₂*, and A₆₆*, respectively. Once a laminate is homogenized, tapering becomes much simpler.

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Figure 2.

Convergence of normalized flexural to corresponding in-plane stiffness matrix elements with increasing repeats of the 4-ply building blocks, using master ply properties.

Homogenized quad and double-double laminates

The convergences of normalized flexural to in-plane stiffness matrix elements for equivalent Quad and double-double laminates are shown in Figure 3. Three diagonal components of the trace-normalized flexural stiffness matrix of a Quad quasi-isotropic laminate [−45/+45/0/90] and its equivalent DD [±22.5/±67.5] are shown for a master ply representing carbon/epoxy systems. This stacking sequence of has the fastest convergence. At a repeated number 5, reasonable convergence is found. This stacking sequence can be obtained from the two most common building blocks: [±45] and [0/90]. It is truly a black aluminum. It is comparable to a metal that should be much easier to design than orthotropic laminates.

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Figure 3.

Convergence of normalized flexural to in-plane stiffness matrix elements of quads [−45/+45/0/90], [−45/+45/0₂/90], and [−45/+45/0₃/90] and their respective equivalent double-double [±22.5/±67.5], [±9/±62], and [0₂/±52]. Master ply properties are used.

It turned out that two hard laminates can be derived from this quasi-isotropic laminate. This is done by adding one or two [0] plies to the single [0] in this unique laminate as shown in Figure 3. The total number of plies will increase to 5 and 6, respectively. Plies are added so the stacking sequence does not change. Shear stiffness can be increased if both [+45] and [−45] are added in units of single plies. This specific ply adding procedure must be followed so that the homogenization of plies remains the same. While these hard laminates serve their purposes, their properties are limited, not suitable for general use. The quasi-isotropic case is unrestricted and can be used for all situations.

The homogenized Quad laminates, as shown in Figure 3, offer the following structural benefits found in DD:

1. Laminates built with multiple repeated stacking can become homogenized. They can be tapered and have their weight reduced by reducing the number of repeats, such as from 12 to 11 to 10 …

While Quad laminates can be homogenized as shown in Figure 3, they are limited in number of possibilities since the angles are restricted to 0°, ±45°, and 90° and there is a practical limit to increase the number of additional plies in any of the four directions. The Quad laminates proposed herein have limited scope in properties on orthotropy in only two cases of hard laminates and a quasi-isotropic. Double-double on the other hand covers the entire field from 0° to 90° for each of the 4 ply angles (±Φ and ±Ψ). In addition, its convergence from flexural to in-plane stiffness is faster than those of the three Quads shown in Figure 3. Having confidence in the design allowable of Quad laminates allows designers to use these unique Quad to learn the benefits of homogenization. This is one way to be exposed to the benefits of DD.

Failure envelopes

Prediction of laminate strengths—tensile X and compressive X’—from the unit circle last-ply failure (LPF) envelopes ¹⁷ is shown in Figure 4. Unit circle has been proposed as a strain normalized failure envelope for any carbon fiber reinforced polymer laminate. The circle is generated from the longitudinal tensile and compressive strains-to-failure of a unidirectional ply. ¹³ The envelope in stress space can be obtained using the stress-strain relations for laminate considering degraded elastic properties to account for matrix cracking prior to failure.

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Figure 4.

Unit circle failure envelopes of quad laminates and measured laminate strengths for four CFRPs. Dots represent experimentally measured tensile and compressive strengths.

Unit circle failure envelopes for four materials and three laminates are shown in Figure 4. The circular dots are the measured uniaxial tensile and compressive data for the laminates. A good correlation between the prediction and measured confirming data was observed, showing the suitability of unit circle envelopes for the prediction of strength of these Quad laminates.

Failure envelopes are then compared to show the benefits of the design, testing, and manufacturing of DD laminates. Figure 5 shows unit circle envelopes in stress space for three materials: T800/Cytec, T4708/MR60H, and T300/F35. They represent the upper bound, the average and the lower bound strength of CFRPs, respectively. For each material, three laminates were selected: Quasi-isotropic [+45/−45/0/90] and two hard Quads [+45/−45/0₂/90] and [+45/−45/0₃/90]. This is the same family of laminates shown in Figure 3, where the convergence from flexural to in-plane stiffness was the fastest among the 4-ply Quad laminates shown in Figure 2. Equally, the equivalent DD of the three Quad laminates considered in Figure 5 are the same shown in Figure 3: [±22.5/±67.5], [±9/±62], and [±0/±52], respectively related to the Quads.

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Figure 5.

Unit circle failure envelopes of Quads and equivalent DD laminates for three CFRPs: T800/Cytec, T4708/MR60H, and T300/F35. The numbers shown for each envelope indicate the radius R of a circle with equal area enclosed by the envelope.

In Figure 5, the average strength—given by the radius R of the circle with equal area of each failure envelope—is shown for each envelope. For the first two laminates of each material, Quasi-isotropic [±45/0/90] and hard Quad [+45/−45/0₂/90], the equivalent DDs, [±22.5/±67.5] and [±9/±62], result in equal failure envelopes. The difference between [+45/−45/0₂/90] and [±9/±62] results from the rounding error associated with the equivalent DD angles. In addition, their R-ratios are 1.06 (=1039/984) for T800/Cytec, 1.06 (=809/766) for T4708/MR60H, and 1.05 (=485/460) for T300/F35. Thus, the accuracy of the ratio of average strengths is within 1%. For laminate [+45/−45/0₃/90], the DD [0₂/±52] does not match exactly. They are approximate, with R values of 1037 and 875 for T800/Cytec, 807 and 680 for T4708/MR60H, and 484 and 408 for T300/F35. Nevertheless, their R-ratios over laminate [+45/−45/0₂/90] are 1.0—considering the envelope of [+45/−45/0₃/90]. Again, they show the same ratio of average strengths with 1% accuracy. If the equivalent DDs [0₂/±52] and [±9/±62] are compared, the ratio of average strengths are 0.83 for all materials (875/1048, 680/816 and 408/490).

In addition, if the relative average strengths of the materials for the same laminate are compared, the strengths of T800/Cytec are increased by 29% as compared to those of T4708/MR60H, for all three laminates. Thus, for the comparison between two materials, the same ratio holds for all laminates considered. Likewise, the R-ratio between T300/F35 and T4708/MR60H is 0.60 for all three laminates. Therefore, ratios of average laminate strengths between two materials are the same.

Overall, the correlations shown in this work are remarkable and can greatly simplify material property tests and facilitate design optimization. These results propose that materials and laminates are independent and locked in within 1% variation. Experimental variations, on the other hand, can be 3% or higher. While only three Quad laminates were offered for this unique application, it is not limited by such. Others Quad can be found. More importantly, the findings of this research open opportunities to double-double laminates.

The methodology and results presented in this investigation are based on the properties of CFRP materials. Although the concept of laminate homogenization is not limited to CFRPs, the simulations used to examine the convergence of quadriaxial laminates relied on master ply properties, which are representative of CFRP systems. Similarly, the unit circle failure criterion, together with the concept of average strength, was employed to establish strength ratios among different CFRPs and quadriaxial laminates. Consequently, the applicability of the present findings to other composite systems should first be assessed using suitably representative material properties and failure envelopes.

Conclusions

This research showed that there are opportunities for engineers to explore new design features of quadriaxial laminates that can be homogenized and tapered. Design, testing, and manufacturing strategies commonly associated with double-double laminates can also be effectively applied to specially configured quadriaxial laminates. Among quasi-isotropic configurations, the fastest convergence toward homogenization was achieved with five repeats of four-ply sub-laminates [−45/+45/0/90] and [+45/−45/0/90]. Hard laminates obtained by adding [0] plies—such as [−45/+45/0₂/90] and [−45/+45/0₃/90]—also exhibited good convergence with five repeats. In addition, the concept of average strength, defined as the radius of a circle with area equal to that of the failure envelope, was employed to establish relationships among CFRP laminates and to compare different CFRP materials for a given quadriaxial configuration. Three laminates—[+45/−45/0/90], [+45/−45/0₂/90], and [+45/−45/0₃/90]—were evaluated considering three CFRPs: T800/Cytec, T4708/MR60H, and T300/F35. The results show that, for each material, the ratios between the average strengths of different quadriaxial laminates remain constant. Likewise, for a given quadriaxial laminate, the relative average strengths among different CFRP materials are also consistent. These findings indicate that homogenized quadriaxial laminates can be designed with confidence in available data and established procedures, while retaining nearly all the simplicity and lightweight benefits of double-double laminates. Overall, the concepts of homogenized quadriaxial laminates and relative average strength discussed in this study are promising and can be effectively applied to facilitate thickness tapering and enhance the design optimization of composite structures.