# Optimization of Hybrid Laminates with Extension-Shear Coupling

Authors: Da Cui; Daokui Li
Journal: International Journal of Aerospace Engineering
 (2018)

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## Abstract

The introduction of hybrid composites into the structure with coupling effect can greatly reduce the cost of materials. The expressions of stiffness coefficient, thermal stress, and thermal moment for hybrid laminates are derived based on the geometrical factors of laminates, and the necessary and sufficient conditions for the hybrid extension-shear-coupled laminates with immunity to hygrothermal shear distortion (HTSD) are further derived. The extension-shear-coupled effect of hybrid laminates is optimized with improved differential evolution algorithm. Results are presented for the hybrid laminates that consist of carbon fiber and glass fiber composite materials. The hygrothermal effect and extension-shear-coupled effect are simulated and verified, meanwhile the robustness of hybrid laminates is analyzed by Monte Carlo method.


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## Body

## 1. Introduction

Laminated composites are playing an important and irreplaceable role in designing structures with coupling effect. For example, the bending-twisting-coupled wing structure can be designed by using composite extension-shear-coupled laminates [1] and the bending-twisting-coupled wind turbine blades structure can be designed by using composite extension-twisting-coupled laminates [2]. However, with the large-scale use of these bending-twisting coupling structures, the common glass fiber composites have difficulties in meeting the requirements of structural reliability. On the other hand, the carbon fiber composites with good comprehensive properties [3, 4] cost about ten times more than glass fiber composites, which restricts its wide range of applications. Therefore, it is necessary to introduce hybrid fiber composites into the design of the bending-twisting-coupled structure to achieve the purpose that greatly reduces material cost under meeting the structural reliability requirements.According to different hybrid modes, hybrid fiber composites are mainly divided into two types of composites, the in-layer composites and between-layer composites [5]. The in-layer hybrid composites consist of two or more fibers which uniformly dispersed in the same matrix of the lamina; the between-layer hybrid composites are composed of two or more different laminae which consist of different single-fiber composites. In this paper, the bending-twisting-coupled structure is designed by between-layer hybrid laminates.At present, the widely used materials in the study of the bending-twisting coupled structure are the single-fiber composite laminates [6–10], whose important design parameters contain paving angles merely. However, the important design parameters for hybrid laminates also include the paving materials [11]. Once the paving materials become variable, the design of laminates becomes more complicated and the number of optimization constraints even multiplies at the same time.J. Li and D. Li [2] have designed a kind of single-fiber composite laminates with immunity to hygrothermal shear distortion (HTSD) with only extension-shear-coupled effect—the AFB0DS laminates (refer to the nomenclature of references [1, 2])—and the sequential quadratic programming (SQP) is used to optimize its coupled effect. However, we found that feasible solutions cannot be found when using the SQP algorithm to optimize the hybrid AFB0DS laminates with immunity to HTSD, for this algorithm cannot meet the strong constraints of hybrid laminates. Therefore, this paper takes the improved differential evolution algorithm DE_CMSBHS to optimize this problem.The DE_CMSBHS algorithm is an efficient global optimization algorithm, which has the characteristics of simple structure, easy realization, fast convergence, and strong robustness, and can effectively solve the single-objective optimization problems such as integer problems, real problems, and mixed integer-real problems [12]. The single-objective nonlinear optimization problems are able to be solved well by combining the penalty function’s ability to handle constraints with the optimal performance of DE_CMSBHS algorithm [13, 14]. The optimization problem of hybrid AFB0DS laminates with immunity to HTSD is a typical nonlinear mixed integer-real single-objective optimization problem.In this paper, the model of hybrid laminates is firstly established. Then the expressions of stiffness coefficient, thermal stress, and thermal moment for laminates are derived with important parameter geometric factor [15, 16]. Secondly, the necessary and sufficient conditions for the hybrid extension-shear-coupled laminates with immunity to HTSD are derived. Thirdly, the DE_CMSBHS algorithm combined with the penalty function is used to optimize the extension-shear-coupled effect of hybrid laminates with immunity to HTSD. Finally, the mechanical properties of optimized laminates are verified.


## 2. Stiffness Coefficient, Thermal Stress, and Thermal Moment of Hybrid Laminates

Introducing the geometrical factors into the design of laminates can effectively improve the efficiency of this progress. In this section, the model of hybrid laminates will be established. Based on the geometric factors, the expressions of stiffness coefficient, thermal stress, and thermal moment of hybrid laminates are derived.The research object is set as a kind of hybrid laminates which are composed of two different types of lamina, and each lamina has the same thickness, as shown in Figure1. Wherein, zk is the position of the k-ply in the entire laminates, n is the number of plies of the laminates, and H is the entire thickness of the laminates. ① and ② represent two kinds of laminae with different material properties, respectively, and the number and layer order of each kind of lamina are variables.Figure 1
Model of hybrid laminates.


### 2.1. Stiffness Coefficient

The off-axis stiffness coefficients of thek-ply in the entire hybrid laminates can be defined as
(1)Q¯11k=U1q+U2qcos2θk+U3qcos4θk,Q¯12k=−U3qcos4θk+U4q,Q¯16k=U2q2sin2θk+U3qsin4θk,Q¯22k=U1q−U2qcos2θk+U3qcos4θk,Q¯26k=U2q2sin2θk−U3qsin4θk,Q¯66k=−U3qcos4θk+U5q.Wherein,θk is the paving angle of the k-ply of the laminates. The superscript “q” is defined to mean which kind of lamina corresponds to: q=① implies that the lamina corresponds to the type ① lamina and q=② implies that the lamina corresponds to the type ② lamina. Moreover, Uiqi=1,2,…,5 are the material constants of hybrid laminates, which are only related to the material parameters of the laminae, as shown in
(2)U1q=3Q11q+3Q22q+2Q12q+4Q66q8,U2q=Q11q−Q22q2,U3q=Q11q+Q22q−2Q12q−4Q66q8,U4q=Q11q+Q22q+6Q12q−4Q66q8,U5q=Q11q+Q22q−2Q12q+4Q66q8=12U1q−U4q.In which,Qijq are the stiffness coefficients of two types of lamina. The geometric factors ξjqj=1,2,…,15,q=①,② of hybrid laminates, which are convenient to express the stiffness matrices, are defined as
(3)ξ1qξ2qξ3qξ4q=∑k=qcos2θkcos4θksin2θksin4θkzk−zk−1,ξ5qξ6qξ7qξ8q=∑k=qcos2θkcos4θksin2θksin4θkzk2−zk−12,ξ9qξ10qξ11qξ12q=∑k=qcos2θkcos4θksin2θksin4θkzk3−zk−13,ξ13qξ14qξ15q=∑k=qzk−zk−1zk2−zk−12zk3−zk−13,where k=q means the sum of all the type ① or ② laminae on the right side of the equation. Obviously, the geometric factors are only related to the paving angle and location of each lamina.


#### 2.1.1. Extension Stiffness MatrixA

According to the classical theory of laminates, the extension stiffness matrix of laminates can be expressed as(4)Aij=∑k=1nQ¯ijkzk−zk−1,i,j=1,2,6.Substituting (1), (2), and (3) into (4) can be used to obtain the expressions of the extension stiffness coefficients, which are only related to the material constants and geometric factors, as shown in
(5)A11=∑k=1nU1q+U2qcos2θk+U3qcos4θkzk−zk−1=U1①ξ13①+U2①ξ1①+U3①ξ2①+U1②ξ13②+U2②ξ1②+U3②ξ2②,A12=A21=∑k=1n−cos4θkU3q+U4qzk−zk−1=−U3①ξ2①+U4①ξ13①−U3②ξ2②+U4②ξ13②,A16=A61=∑k=1nU2qsin2θk2+U3qsin4θkzk−zk−1=12U2①ξ3①+U3①ξ4①+12U2②ξ3②+U3②ξ4②,A22=∑k=1nU1q−U2qcos2θk+U3qcos4θkzk−zk−1=U1①ξ13①−U2①ξ1①+U3①ξ2①+U1②ξ13②−U2②ξ1②+U3②ξ2②,A26=A62=∑k=1nU2qsin2θk2−U3qsin4θkzk−zk−1=12U2①ξ3①−U3①ξ4①+12U2②ξ3②−U3②ξ4②,A66=∑k=1n−U3qcos4θk+U5qzk−zk−1=−U3①ξ2①+U5①ξ13①−U3②ξ2②+U5②ξ13②.


#### 2.1.2. Coupling Stiffness MatrixB

Similarly, the coupling stiffness matrix of the hybrid laminates can be expressed as(6)Bij=∑k=1nQ¯ijkzk2−zk−12,i,j=1,2,6.Substituting (1), (2), and (3) into (6) can be used to obtain the expression of the coupling stiffness coefficients, which are only related to the material constants and geometric factors, as shown in
(7)B11=12∑k=1nU1q+U2qcos2θk+U3qcos4θkzk2−zk−12=12U1①ξ14①+12U2①ξ5①+12U3①ξ6①+12U1②ξ14②+12U2②ξ5②+12U3②ξ6②,B12=B21=12∑k=1n−cos4θkU3q+U4qzk2−zk−12=−12U3①ξ6①+12U4①ξ14①−12U3②ξ6②+12U4②ξ14②,B22=12∑k=1nU1q−U2qcos2θk+U3qcos4θkzk2−zk−12=12U1①ξ14①−12U2①ξ5①+12U3①ξ6①+12U1②ξ14②−12U2②ξ5②+12U3②ξ6②,B16=B61=12∑k=1nU2qsin2θk2+U3qsin4θkzk2−zk−12=14U2①ξ7①+12U3①ξ8①+14U2②ξ7②+12U3②ξ8②,B26=B62=12∑k=1nU2qsin2θk2−U3qsin4θkzk2−zk−12=14U2①ξ7①−12U3①ξ8①+14U2②ξ7②−12U3②ξ8②,B66=12∑k=1n−U3qcos4θk+U5qzk2−zk−12=−12U3①ξ6①+12U5①ξ14①−12U3②ξ6②+12U5②ξ14②.


#### 2.1.3. Bending Stiffness MatrixD

The bending stiffness matrix of the hybrid laminates can be expressed as(8)Dij=∑k=1nQ¯ijkzk3−zk−13,i,j=1,2,6.Substituting (1), (2), and (3) into (8) can be used to obtain the expressions of the bending stiffness coefficients, which are only related to the material constants and geometric factors, as shown in
(9)D11=13∑k=1nU1q+U2qcos2θk+U3qcos4θkzk3−zk−13=13U1①ξ15①+13U2①ξ9①+13U3①ξ10①+13U1②ξ15②+13U2②ξ9②+13U3②ξ10②,D12=D21=13∑k=1n−cos4θkU3q+U4qzk3−zk−13=−13U3①ξ10①+13U4①ξ15①−13U3②ξ10②+13U4②ξ15②,D16=D61=13∑k=1nU2qsin2θk2+U3qsin4θkzk3−zk−13=16U2①ξ11①+13U3①ξ12①+16U2②ξ11②+13U3②ξ12②,D22=13∑k=1nU1q−U2qcos2θk+U3qcos4θkzk3−zk−13=13U1①ξ15①−13U2①ξ9①+13U3①ξ10①+13U1②ξ15②−13U2②ξ9②+13U3②ξ10②,D26=D62=13∑k=1nU2qsin2θk2−U3qsin4θkzk3−zk−13=16U2①ξ11①−13U3①ξ12①+16U2②ξ11②−13U3②ξ12②,D66=13∑k=1n−U3qcos4θk+U5qzk3−zk−13=−13U3①ξ10①+13U5①ξ15①−13U3②ξ10②+13U5②ξ15②.To sum up, the stiffness coefficients of the hybrid laminates, which are only related to the material constants and geometric factors, can be expressed as(10)A11A12A16A22A26A66=∑q=①,②ξ13qξ1qξ2q0000−ξ2qξ13q00ξ3q2ξ4q00ξ13q−ξ1qξ2q000ξ3q2−ξ4q0000−ξ2q0ξ13①U1qU2qU3qU4qU5q,(11)B11B12B16B22B26B66=12∑q=①,②ξ14qξ5qξ6q0000−ξ6qξ14q00ξ7q2ξ8q00ξ14q−ξ5qξ6q000ξ7q2−ξ8q0000−ξ6q0ξ14qU1qU2qU3qU4qU5q,(12)D11D12D16D22D26D66=13∑q=①,②ξ15qξ9qξ10q0000−ξ10qξ15q00ξ11q2ξ12q00ξ15q−ξ9qξ10q000ξ11q2−ξ12q0000−ξ10q0ξ15qU1qU2qU3qU4qU5q.


### 2.2. Thermal Stress and Thermal Moment

The thermal expansion coefficients of thek-ply in the entire hybrid laminates are
(13)αxk=α1qcos2θk+α2qsin2θk,αyk=α1qsin2θk+α2qcos2θk,αxyk=α1q−α2q2sinθkcosθk,wherein α1q and α2qq=①,② are the thermal expansion coefficients of two different types of lamina. The variation of temperature can be expressed by ΔT, and the thermal stress and thermal moment of laminates are expressed as
(14)NxT=∑k=1nΔTQ¯11kαxk+Q¯12kαyk+Q¯16kαxykzk−zk−1,NyT=∑k=1nΔTQ¯12kαxk+Q¯22kαyk+Q¯26kαxykzk−zk−1,NxyT=∑k=1nΔTQ¯16kαxk+Q¯26kαyk+Q¯66kαxykzk−zk−1,(15)MxT=12∑k=1nΔTQ¯11kαxk+Q¯12kαyk+Q¯16kαxykzk2−zk−12,MyT=12∑k=1nΔTQ¯12kαxk+Q¯22kαyk+Q¯26kαxykzk2−zk−12,MxyT=12∑k=1nΔTQ¯16kαxk+Q¯26kαyk+Q¯66kαxykzk2−zk−12.Substituting (1) and (13) into (14) can be used to obtain the expressions of thermal stress, which are only related to the material constants and geometric factors, as shown in
(16)NxT=∑k=1nΔT12U1Tq+12U2Tqcos2θkzk−zk−1=ΔT2U1T①ξ13①+U2T①ξ1①+U1T②ξ13②+U2T②ξ1②,NyT=∑k=1nΔT12U1Tq−12U2Tqcos2θkzk−zk−1=ΔT2U1T①ξ13①−U2T①ξ1①+U1T②ξ13②−U2T②ξ1②,NxyT=ΔT2∑k=1nU2Tqsin2θkzk−zk−1=ΔT2ξ3①U2T①+ξ3②U2T②.In which,U1Tq and U2Tq are defined as the thermal invariants of the hybrid laminates, which can be calculated from the thermal expansion coefficients and the invariants of laminates, as shown in
(17)U1Tq=α1q+α2qU1q+U4q+α1q−α2qU2q,U2Tq=α1q+α2qU2q+α1q−α2qU1q+2U3q−U4q.Substituting (1) and (13) into (15) can be used to obtain the expressions of the thermal moment, which are only related to the material constants and geometric factors, as shown in
(18)MxT=12∑k=1nΔT12U1Tq+12U2Tqcos2θkzk2−zk−12=ΔT4U1T①ξ14①+U2T①ξ5①+U1T②ξ14②+U2T②ξ5②,MyT=12∑k=1nΔT12U1Tq−12U2Tqcos2θkzk2−zk−12=ΔT4U1T①ξ14①−U2T①ξ5①+U1T②ξ14②−U2T②ξ5②,MxyT=ΔT4∑k=1nU2Tqsin2θkzk−zk−1=ΔT4ξ7①U2T①+ξ7②U2T②.To sum up, the thermal stress and thermal moment of the hybrid laminates, which are only related to the thermal invariants and geometric factors, can be expressed as(19)NxTNyTNxyT=ΔT2∑q=①,②U1Tqξ13q+U2Tqξ1qU1Tqξ13q−U2Tqξ1qU2Tqξ3q,(20)MxTMyTMxyT=ΔT4∑q=①,②U1Tqξ14q+U2Tqξ5qU1Tqξ14q−U2Tqξ5qU2Tqξ7q.


## 2.1. Stiffness Coefficient

The off-axis stiffness coefficients of thek-ply in the entire hybrid laminates can be defined as
(1)Q¯11k=U1q+U2qcos2θk+U3qcos4θk,Q¯12k=−U3qcos4θk+U4q,Q¯16k=U2q2sin2θk+U3qsin4θk,Q¯22k=U1q−U2qcos2θk+U3qcos4θk,Q¯26k=U2q2sin2θk−U3qsin4θk,Q¯66k=−U3qcos4θk+U5q.Wherein,θk is the paving angle of the k-ply of the laminates. The superscript “q” is defined to mean which kind of lamina corresponds to: q=① implies that the lamina corresponds to the type ① lamina and q=② implies that the lamina corresponds to the type ② lamina. Moreover, Uiqi=1,2,…,5 are the material constants of hybrid laminates, which are only related to the material parameters of the laminae, as shown in
(2)U1q=3Q11q+3Q22q+2Q12q+4Q66q8,U2q=Q11q−Q22q2,U3q=Q11q+Q22q−2Q12q−4Q66q8,U4q=Q11q+Q22q+6Q12q−4Q66q8,U5q=Q11q+Q22q−2Q12q+4Q66q8=12U1q−U4q.In which,Qijq are the stiffness coefficients of two types of lamina. The geometric factors ξjqj=1,2,…,15,q=①,② of hybrid laminates, which are convenient to express the stiffness matrices, are defined as
(3)ξ1qξ2qξ3qξ4q=∑k=qcos2θkcos4θksin2θksin4θkzk−zk−1,ξ5qξ6qξ7qξ8q=∑k=qcos2θkcos4θksin2θksin4θkzk2−zk−12,ξ9qξ10qξ11qξ12q=∑k=qcos2θkcos4θksin2θksin4θkzk3−zk−13,ξ13qξ14qξ15q=∑k=qzk−zk−1zk2−zk−12zk3−zk−13,where k=q means the sum of all the type ① or ② laminae on the right side of the equation. Obviously, the geometric factors are only related to the paving angle and location of each lamina.


### 2.1.1. Extension Stiffness MatrixA

According to the classical theory of laminates, the extension stiffness matrix of laminates can be expressed as(4)Aij=∑k=1nQ¯ijkzk−zk−1,i,j=1,2,6.Substituting (1), (2), and (3) into (4) can be used to obtain the expressions of the extension stiffness coefficients, which are only related to the material constants and geometric factors, as shown in
(5)A11=∑k=1nU1q+U2qcos2θk+U3qcos4θkzk−zk−1=U1①ξ13①+U2①ξ1①+U3①ξ2①+U1②ξ13②+U2②ξ1②+U3②ξ2②,A12=A21=∑k=1n−cos4θkU3q+U4qzk−zk−1=−U3①ξ2①+U4①ξ13①−U3②ξ2②+U4②ξ13②,A16=A61=∑k=1nU2qsin2θk2+U3qsin4θkzk−zk−1=12U2①ξ3①+U3①ξ4①+12U2②ξ3②+U3②ξ4②,A22=∑k=1nU1q−U2qcos2θk+U3qcos4θkzk−zk−1=U1①ξ13①−U2①ξ1①+U3①ξ2①+U1②ξ13②−U2②ξ1②+U3②ξ2②,A26=A62=∑k=1nU2qsin2θk2−U3qsin4θkzk−zk−1=12U2①ξ3①−U3①ξ4①+12U2②ξ3②−U3②ξ4②,A66=∑k=1n−U3qcos4θk+U5qzk−zk−1=−U3①ξ2①+U5①ξ13①−U3②ξ2②+U5②ξ13②.


### 2.1.2. Coupling Stiffness MatrixB

Similarly, the coupling stiffness matrix of the hybrid laminates can be expressed as(6)Bij=∑k=1nQ¯ijkzk2−zk−12,i,j=1,2,6.Substituting (1), (2), and (3) into (6) can be used to obtain the expression of the coupling stiffness coefficients, which are only related to the material constants and geometric factors, as shown in
(7)B11=12∑k=1nU1q+U2qcos2θk+U3qcos4θkzk2−zk−12=12U1①ξ14①+12U2①ξ5①+12U3①ξ6①+12U1②ξ14②+12U2②ξ5②+12U3②ξ6②,B12=B21=12∑k=1n−cos4θkU3q+U4qzk2−zk−12=−12U3①ξ6①+12U4①ξ14①−12U3②ξ6②+12U4②ξ14②,B22=12∑k=1nU1q−U2qcos2θk+U3qcos4θkzk2−zk−12=12U1①ξ14①−12U2①ξ5①+12U3①ξ6①+12U1②ξ14②−12U2②ξ5②+12U3②ξ6②,B16=B61=12∑k=1nU2qsin2θk2+U3qsin4θkzk2−zk−12=14U2①ξ7①+12U3①ξ8①+14U2②ξ7②+12U3②ξ8②,B26=B62=12∑k=1nU2qsin2θk2−U3qsin4θkzk2−zk−12=14U2①ξ7①−12U3①ξ8①+14U2②ξ7②−12U3②ξ8②,B66=12∑k=1n−U3qcos4θk+U5qzk2−zk−12=−12U3①ξ6①+12U5①ξ14①−12U3②ξ6②+12U5②ξ14②.


### 2.1.3. Bending Stiffness MatrixD

The bending stiffness matrix of the hybrid laminates can be expressed as(8)Dij=∑k=1nQ¯ijkzk3−zk−13,i,j=1,2,6.Substituting (1), (2), and (3) into (8) can be used to obtain the expressions of the bending stiffness coefficients, which are only related to the material constants and geometric factors, as shown in
(9)D11=13∑k=1nU1q+U2qcos2θk+U3qcos4θkzk3−zk−13=13U1①ξ15①+13U2①ξ9①+13U3①ξ10①+13U1②ξ15②+13U2②ξ9②+13U3②ξ10②,D12=D21=13∑k=1n−cos4θkU3q+U4qzk3−zk−13=−13U3①ξ10①+13U4①ξ15①−13U3②ξ10②+13U4②ξ15②,D16=D61=13∑k=1nU2qsin2θk2+U3qsin4θkzk3−zk−13=16U2①ξ11①+13U3①ξ12①+16U2②ξ11②+13U3②ξ12②,D22=13∑k=1nU1q−U2qcos2θk+U3qcos4θkzk3−zk−13=13U1①ξ15①−13U2①ξ9①+13U3①ξ10①+13U1②ξ15②−13U2②ξ9②+13U3②ξ10②,D26=D62=13∑k=1nU2qsin2θk2−U3qsin4θkzk3−zk−13=16U2①ξ11①−13U3①ξ12①+16U2②ξ11②−13U3②ξ12②,D66=13∑k=1n−U3qcos4θk+U5qzk3−zk−13=−13U3①ξ10①+13U5①ξ15①−13U3②ξ10②+13U5②ξ15②.To sum up, the stiffness coefficients of the hybrid laminates, which are only related to the material constants and geometric factors, can be expressed as(10)A11A12A16A22A26A66=∑q=①,②ξ13qξ1qξ2q0000−ξ2qξ13q00ξ3q2ξ4q00ξ13q−ξ1qξ2q000ξ3q2−ξ4q0000−ξ2q0ξ13①U1qU2qU3qU4qU5q,(11)B11B12B16B22B26B66=12∑q=①,②ξ14qξ5qξ6q0000−ξ6qξ14q00ξ7q2ξ8q00ξ14q−ξ5qξ6q000ξ7q2−ξ8q0000−ξ6q0ξ14qU1qU2qU3qU4qU5q,(12)D11D12D16D22D26D66=13∑q=①,②ξ15qξ9qξ10q0000−ξ10qξ15q00ξ11q2ξ12q00ξ15q−ξ9qξ10q000ξ11q2−ξ12q0000−ξ10q0ξ15qU1qU2qU3qU4qU5q.


## 2.1.1. Extension Stiffness MatrixA

According to the classical theory of laminates, the extension stiffness matrix of laminates can be expressed as(4)Aij=∑k=1nQ¯ijkzk−zk−1,i,j=1,2,6.Substituting (1), (2), and (3) into (4) can be used to obtain the expressions of the extension stiffness coefficients, which are only related to the material constants and geometric factors, as shown in
(5)A11=∑k=1nU1q+U2qcos2θk+U3qcos4θkzk−zk−1=U1①ξ13①+U2①ξ1①+U3①ξ2①+U1②ξ13②+U2②ξ1②+U3②ξ2②,A12=A21=∑k=1n−cos4θkU3q+U4qzk−zk−1=−U3①ξ2①+U4①ξ13①−U3②ξ2②+U4②ξ13②,A16=A61=∑k=1nU2qsin2θk2+U3qsin4θkzk−zk−1=12U2①ξ3①+U3①ξ4①+12U2②ξ3②+U3②ξ4②,A22=∑k=1nU1q−U2qcos2θk+U3qcos4θkzk−zk−1=U1①ξ13①−U2①ξ1①+U3①ξ2①+U1②ξ13②−U2②ξ1②+U3②ξ2②,A26=A62=∑k=1nU2qsin2θk2−U3qsin4θkzk−zk−1=12U2①ξ3①−U3①ξ4①+12U2②ξ3②−U3②ξ4②,A66=∑k=1n−U3qcos4θk+U5qzk−zk−1=−U3①ξ2①+U5①ξ13①−U3②ξ2②+U5②ξ13②.


## 2.1.2. Coupling Stiffness MatrixB

Similarly, the coupling stiffness matrix of the hybrid laminates can be expressed as(6)Bij=∑k=1nQ¯ijkzk2−zk−12,i,j=1,2,6.Substituting (1), (2), and (3) into (6) can be used to obtain the expression of the coupling stiffness coefficients, which are only related to the material constants and geometric factors, as shown in
(7)B11=12∑k=1nU1q+U2qcos2θk+U3qcos4θkzk2−zk−12=12U1①ξ14①+12U2①ξ5①+12U3①ξ6①+12U1②ξ14②+12U2②ξ5②+12U3②ξ6②,B12=B21=12∑k=1n−cos4θkU3q+U4qzk2−zk−12=−12U3①ξ6①+12U4①ξ14①−12U3②ξ6②+12U4②ξ14②,B22=12∑k=1nU1q−U2qcos2θk+U3qcos4θkzk2−zk−12=12U1①ξ14①−12U2①ξ5①+12U3①ξ6①+12U1②ξ14②−12U2②ξ5②+12U3②ξ6②,B16=B61=12∑k=1nU2qsin2θk2+U3qsin4θkzk2−zk−12=14U2①ξ7①+12U3①ξ8①+14U2②ξ7②+12U3②ξ8②,B26=B62=12∑k=1nU2qsin2θk2−U3qsin4θkzk2−zk−12=14U2①ξ7①−12U3①ξ8①+14U2②ξ7②−12U3②ξ8②,B66=12∑k=1n−U3qcos4θk+U5qzk2−zk−12=−12U3①ξ6①+12U5①ξ14①−12U3②ξ6②+12U5②ξ14②.


## 2.1.3. Bending Stiffness MatrixD

The bending stiffness matrix of the hybrid laminates can be expressed as(8)Dij=∑k=1nQ¯ijkzk3−zk−13,i,j=1,2,6.Substituting (1), (2), and (3) into (8) can be used to obtain the expressions of the bending stiffness coefficients, which are only related to the material constants and geometric factors, as shown in
(9)D11=13∑k=1nU1q+U2qcos2θk+U3qcos4θkzk3−zk−13=13U1①ξ15①+13U2①ξ9①+13U3①ξ10①+13U1②ξ15②+13U2②ξ9②+13U3②ξ10②,D12=D21=13∑k=1n−cos4θkU3q+U4qzk3−zk−13=−13U3①ξ10①+13U4①ξ15①−13U3②ξ10②+13U4②ξ15②,D16=D61=13∑k=1nU2qsin2θk2+U3qsin4θkzk3−zk−13=16U2①ξ11①+13U3①ξ12①+16U2②ξ11②+13U3②ξ12②,D22=13∑k=1nU1q−U2qcos2θk+U3qcos4θkzk3−zk−13=13U1①ξ15①−13U2①ξ9①+13U3①ξ10①+13U1②ξ15②−13U2②ξ9②+13U3②ξ10②,D26=D62=13∑k=1nU2qsin2θk2−U3qsin4θkzk3−zk−13=16U2①ξ11①−13U3①ξ12①+16U2②ξ11②−13U3②ξ12②,D66=13∑k=1n−U3qcos4θk+U5qzk3−zk−13=−13U3①ξ10①+13U5①ξ15①−13U3②ξ10②+13U5②ξ15②.To sum up, the stiffness coefficients of the hybrid laminates, which are only related to the material constants and geometric factors, can be expressed as(10)A11A12A16A22A26A66=∑q=①,②ξ13qξ1qξ2q0000−ξ2qξ13q00ξ3q2ξ4q00ξ13q−ξ1qξ2q000ξ3q2−ξ4q0000−ξ2q0ξ13①U1qU2qU3qU4qU5q,(11)B11B12B16B22B26B66=12∑q=①,②ξ14qξ5qξ6q0000−ξ6qξ14q00ξ7q2ξ8q00ξ14q−ξ5qξ6q000ξ7q2−ξ8q0000−ξ6q0ξ14qU1qU2qU3qU4qU5q,(12)D11D12D16D22D26D66=13∑q=①,②ξ15qξ9qξ10q0000−ξ10qξ15q00ξ11q2ξ12q00ξ15q−ξ9qξ10q000ξ11q2−ξ12q0000−ξ10q0ξ15qU1qU2qU3qU4qU5q.


## 2.2. Thermal Stress and Thermal Moment

The thermal expansion coefficients of thek-ply in the entire hybrid laminates are
(13)αxk=α1qcos2θk+α2qsin2θk,αyk=α1qsin2θk+α2qcos2θk,αxyk=α1q−α2q2sinθkcosθk,wherein α1q and α2qq=①,② are the thermal expansion coefficients of two different types of lamina. The variation of temperature can be expressed by ΔT, and the thermal stress and thermal moment of laminates are expressed as
(14)NxT=∑k=1nΔTQ¯11kαxk+Q¯12kαyk+Q¯16kαxykzk−zk−1,NyT=∑k=1nΔTQ¯12kαxk+Q¯22kαyk+Q¯26kαxykzk−zk−1,NxyT=∑k=1nΔTQ¯16kαxk+Q¯26kαyk+Q¯66kαxykzk−zk−1,(15)MxT=12∑k=1nΔTQ¯11kαxk+Q¯12kαyk+Q¯16kαxykzk2−zk−12,MyT=12∑k=1nΔTQ¯12kαxk+Q¯22kαyk+Q¯26kαxykzk2−zk−12,MxyT=12∑k=1nΔTQ¯16kαxk+Q¯26kαyk+Q¯66kαxykzk2−zk−12.Substituting (1) and (13) into (14) can be used to obtain the expressions of thermal stress, which are only related to the material constants and geometric factors, as shown in
(16)NxT=∑k=1nΔT12U1Tq+12U2Tqcos2θkzk−zk−1=ΔT2U1T①ξ13①+U2T①ξ1①+U1T②ξ13②+U2T②ξ1②,NyT=∑k=1nΔT12U1Tq−12U2Tqcos2θkzk−zk−1=ΔT2U1T①ξ13①−U2T①ξ1①+U1T②ξ13②−U2T②ξ1②,NxyT=ΔT2∑k=1nU2Tqsin2θkzk−zk−1=ΔT2ξ3①U2T①+ξ3②U2T②.In which,U1Tq and U2Tq are defined as the thermal invariants of the hybrid laminates, which can be calculated from the thermal expansion coefficients and the invariants of laminates, as shown in
(17)U1Tq=α1q+α2qU1q+U4q+α1q−α2qU2q,U2Tq=α1q+α2qU2q+α1q−α2qU1q+2U3q−U4q.Substituting (1) and (13) into (15) can be used to obtain the expressions of the thermal moment, which are only related to the material constants and geometric factors, as shown in
(18)MxT=12∑k=1nΔT12U1Tq+12U2Tqcos2θkzk2−zk−12=ΔT4U1T①ξ14①+U2T①ξ5①+U1T②ξ14②+U2T②ξ5②,MyT=12∑k=1nΔT12U1Tq−12U2Tqcos2θkzk2−zk−12=ΔT4U1T①ξ14①−U2T①ξ5①+U1T②ξ14②−U2T②ξ5②,MxyT=ΔT4∑k=1nU2Tqsin2θkzk−zk−1=ΔT4ξ7①U2T①+ξ7②U2T②.To sum up, the thermal stress and thermal moment of the hybrid laminates, which are only related to the thermal invariants and geometric factors, can be expressed as(19)NxTNyTNxyT=ΔT2∑q=①,②U1Tqξ13q+U2Tqξ1qU1Tqξ13q−U2Tqξ1qU2Tqξ3q,(20)MxTMyTMxyT=ΔT4∑q=①,②U1Tqξ14q+U2Tqξ5qU1Tqξ14q−U2Tqξ5qU2Tqξ7q.


## 3. The HybridAFB0DS Laminates with Immunity to HTSD

### 3.1. Necessary and Sufficient Conditions

From (10), (11), and (12), for matrices A, B, and D, respectively, can we see that the stiffness coefficients of laminates are only related to geometrical factors and material constants. In order to make the designed laminates suitable for all materials, the necessary and sufficient conditions of geometric factors for the hybrid AFB0DS laminates with immunity to HTSD will be derived in this section. The stiffness coefficients of hybrid AFB0DS laminates should meet the following relationships [2].
(21)B11=B12=B16=B22=B26=B66=0,D16=D26=0.Substituting (11) and (12) into Eq. (21) can be used to obtain the necessary and sufficient conditions of geometric factors for the hybrid AFB0DS laminates.
(22)ξ5①=ξ6①=ξ7①=ξ8①=ξ11①=ξ12①=ξ14①=0,ξ5②=ξ6②=ξ7②=ξ8②=ξ11②=ξ12②=ξ14②=0.In order to ensure that the hybridAFB0DS laminates will not cause hygrothermal shear distortion, its thermal shear strain should meet following relationship.
(23)γxyT=0.Moreover, the connection between thermal stressNT and thermal shear strain γxyT of AFB0DS hybrid laminates is
(24)εxTεyTγxyT=A11A12A16A12A22A26A16A26A66−1NxTNyTNxyT=A∗ANxTNyTNxyT,where
(25)A∗=A22A66−A262A16A26−A12A66A12A26−A16A22A16A26−A12A66A11A66−A162A16A16−A11A26A12A26−A16A22A16A16−A11A26A11A22−A122.Substituting (25) into (24) can be used to obtain the expression of thermal shear strain of the hybrid AFB0DS laminates.
(26)γxyT=1AA12A26−A16A22NxT+A12A16−A11A26NyT+A11A22−A122NxyT.Inserting (26) into (23) gives
(27)A12A26−A16A22NxT+A12A16−A11A26NyT+A11A22−A122NxyT=0.Inserting (19) into (27) and simplifying gives
(28)A12A26−A16A22+A12A16−A11A26∑q=①,②U1Tqξ13q+A12A26−A16A22−A12A16−A11A26∑q=①,②U2Tqξ1q+A11A22−A122∑q=①,②U2Tqξ3q=0.If (28) is constantly established, the previous coefficient of thermal material constant U1Tq and U2Tqq=①,② must be zero.
(29)A12A26−A16A22+A12A16−A11A26ξ13q=0q=①,②,(30)A12A26−A16A22−A12A16−A11A26ξ1q+A11A22−A122ξ3q=0q=①,②.According to (30) can we find out that
(31)ξ1q=ξ3q=0q=①,②.Inserting (31) into (10) gives
(32)A11A12A16A22A26A66=∑q=①,②ξ13q0ξ2q0000−ξ2qξ13q000ξ4q00ξ13q0ξ2q0000−ξ4q0000−ξ2q0ξ13qU1qU2qU3qU4qU5q.Then inserting (32) into (29) further gives
(33)A12A26−A16A22+A12A16−A11A26=A26+A16A12−A11.Furthermore, because ofA26+A16=0 and ξ13q≠0, (29) is also established. Considering that if ξ4q=0, A16 and A26 of laminates are both zero and the laminates will not have extension-shear-coupled effect. Therefore, geometric factor ξ4q cannot be zero; thus, the necessary and sufficient conditions for “with immunity to HTSD” can be expressed as
(34)ξ1①=ξ3①=ξ1②=ξ3②=0,ξ4①≠0,ξ4②≠0.In summary, the necessary and sufficient conditions of the hybridAFB0DS laminates with immunity to HTSD can be expressed as
(35)ξ1①=ξ3①=ξ1②=ξ3②=0,ξ4①≠0,ξ4②≠0,ξ5①=ξ6①=ξ7①=ξ8①=ξ11①=ξ12①=ξ14①=0,ξ5②=ξ6②=ξ7②=ξ8②=ξ11②=ξ12②=ξ14②=0.


### 3.2. Curing Deformation

Considering that the curing deformation phenomenon will occur during the process of molding, which may bring about deformation difference between the expected design shape and the free shape after taking off the model at room temperature, now, the thermal strain of the hybridAFB0DS laminates with immunity to HTSD will be derived by the necessary and sufficient conditions during the process of curing deformation.Due to the similarity of influence between humidity changes and temperature changes on composite materials, it only needs to replace the thermal expansion coefficients by humidity expansion coefficients. In order to simplify the analysis and derivation process, the analysis for hygrothermal distortion contains the thermal effect merely.Substituting (25) into (24) can be used to obtain the expressions of thermal strain of the hybrid AFB0DS laminates with immunity to HTSD.
(36)εxT=1AA22A66−A262NxT+A16A26−A12A66NyT+A12A26−A16A22NxyT,εyT=1AA16A26−A12A66NxT+A11A66−A162NyT+A16A16−A11A26NxyT.Substituting (35) into (19) can give the expressions of thermal stress of the hybrid AFB0DS laminates with immunity to HTSD.
(37)NxTNyTNxyT=ΔT2U1T①ξ13①U1T①ξ13①0+ΔT2U1T②ξ13②U1T②ξ13②0.Inserting (10), (35), and (37) into (36) gives
(38)εxT=εyT=ΔT2A16A26−A16+A66A11−A12U1T①ξ13①+U1T②ξ13②.Therefore, the thermal strains of two main directions of the hybridAFB0DS laminates with immunity to HTSD are equal to each other and the value of the thermal strain is not only related to the material constants and temperature variation but also changing with geometric factors ξ2q, ξ4q, and ξ13q.


## 3.1. Necessary and Sufficient Conditions

From (10), (11), and (12), for matrices A, B, and D, respectively, can we see that the stiffness coefficients of laminates are only related to geometrical factors and material constants. In order to make the designed laminates suitable for all materials, the necessary and sufficient conditions of geometric factors for the hybrid AFB0DS laminates with immunity to HTSD will be derived in this section. The stiffness coefficients of hybrid AFB0DS laminates should meet the following relationships [2].
(21)B11=B12=B16=B22=B26=B66=0,D16=D26=0.Substituting (11) and (12) into Eq. (21) can be used to obtain the necessary and sufficient conditions of geometric factors for the hybrid AFB0DS laminates.
(22)ξ5①=ξ6①=ξ7①=ξ8①=ξ11①=ξ12①=ξ14①=0,ξ5②=ξ6②=ξ7②=ξ8②=ξ11②=ξ12②=ξ14②=0.In order to ensure that the hybridAFB0DS laminates will not cause hygrothermal shear distortion, its thermal shear strain should meet following relationship.
(23)γxyT=0.Moreover, the connection between thermal stressNT and thermal shear strain γxyT of AFB0DS hybrid laminates is
(24)εxTεyTγxyT=A11A12A16A12A22A26A16A26A66−1NxTNyTNxyT=A∗ANxTNyTNxyT,where
(25)A∗=A22A66−A262A16A26−A12A66A12A26−A16A22A16A26−A12A66A11A66−A162A16A16−A11A26A12A26−A16A22A16A16−A11A26A11A22−A122.Substituting (25) into (24) can be used to obtain the expression of thermal shear strain of the hybrid AFB0DS laminates.
(26)γxyT=1AA12A26−A16A22NxT+A12A16−A11A26NyT+A11A22−A122NxyT.Inserting (26) into (23) gives
(27)A12A26−A16A22NxT+A12A16−A11A26NyT+A11A22−A122NxyT=0.Inserting (19) into (27) and simplifying gives
(28)A12A26−A16A22+A12A16−A11A26∑q=①,②U1Tqξ13q+A12A26−A16A22−A12A16−A11A26∑q=①,②U2Tqξ1q+A11A22−A122∑q=①,②U2Tqξ3q=0.If (28) is constantly established, the previous coefficient of thermal material constant U1Tq and U2Tqq=①,② must be zero.
(29)A12A26−A16A22+A12A16−A11A26ξ13q=0q=①,②,(30)A12A26−A16A22−A12A16−A11A26ξ1q+A11A22−A122ξ3q=0q=①,②.According to (30) can we find out that
(31)ξ1q=ξ3q=0q=①,②.Inserting (31) into (10) gives
(32)A11A12A16A22A26A66=∑q=①,②ξ13q0ξ2q0000−ξ2qξ13q000ξ4q00ξ13q0ξ2q0000−ξ4q0000−ξ2q0ξ13qU1qU2qU3qU4qU5q.Then inserting (32) into (29) further gives
(33)A12A26−A16A22+A12A16−A11A26=A26+A16A12−A11.Furthermore, because ofA26+A16=0 and ξ13q≠0, (29) is also established. Considering that if ξ4q=0, A16 and A26 of laminates are both zero and the laminates will not have extension-shear-coupled effect. Therefore, geometric factor ξ4q cannot be zero; thus, the necessary and sufficient conditions for “with immunity to HTSD” can be expressed as
(34)ξ1①=ξ3①=ξ1②=ξ3②=0,ξ4①≠0,ξ4②≠0.In summary, the necessary and sufficient conditions of the hybridAFB0DS laminates with immunity to HTSD can be expressed as
(35)ξ1①=ξ3①=ξ1②=ξ3②=0,ξ4①≠0,ξ4②≠0,ξ5①=ξ6①=ξ7①=ξ8①=ξ11①=ξ12①=ξ14①=0,ξ5②=ξ6②=ξ7②=ξ8②=ξ11②=ξ12②=ξ14②=0.


## 3.2. Curing Deformation

Considering that the curing deformation phenomenon will occur during the process of molding, which may bring about deformation difference between the expected design shape and the free shape after taking off the model at room temperature, now, the thermal strain of the hybridAFB0DS laminates with immunity to HTSD will be derived by the necessary and sufficient conditions during the process of curing deformation.Due to the similarity of influence between humidity changes and temperature changes on composite materials, it only needs to replace the thermal expansion coefficients by humidity expansion coefficients. In order to simplify the analysis and derivation process, the analysis for hygrothermal distortion contains the thermal effect merely.Substituting (25) into (24) can be used to obtain the expressions of thermal strain of the hybrid AFB0DS laminates with immunity to HTSD.
(36)εxT=1AA22A66−A262NxT+A16A26−A12A66NyT+A12A26−A16A22NxyT,εyT=1AA16A26−A12A66NxT+A11A66−A162NyT+A16A16−A11A26NxyT.Substituting (35) into (19) can give the expressions of thermal stress of the hybrid AFB0DS laminates with immunity to HTSD.
(37)NxTNyTNxyT=ΔT2U1T①ξ13①U1T①ξ13①0+ΔT2U1T②ξ13②U1T②ξ13②0.Inserting (10), (35), and (37) into (36) gives
(38)εxT=εyT=ΔT2A16A26−A16+A66A11−A12U1T①ξ13①+U1T②ξ13②.Therefore, the thermal strains of two main directions of the hybridAFB0DS laminates with immunity to HTSD are equal to each other and the value of the thermal strain is not only related to the material constants and temperature variation but also changing with geometric factors ξ2q, ξ4q, and ξ13q.


## 4. Optimized Design of Laminates

Considering that the extension-shear-coupled effect of hybridAFB0DS laminates with immunity to HTSD is the main performance index, the maximum extension-shear-coupled effect is therefore the major objective. Regarding the ply materials and ply angles of each lamina as variables, take the ply materials and ply angles as the optimized design variables. The optimized constraint conditions can be obtained by (35), which ensures that the final optimized laminates are hybrid AFB0DS laminates with immunity to HTSD. Take the flexibility coefficient a16 as the parameter to test the extension-shear-coupled effect of laminates, which can be obtained by inverting the stiffness matrix of laminates, as shown in (39).
(39)a11a12a16b11b12b16a12a22a26b12b22b26a16a26a66b16b26b66b11b12b16d11d12d16b12b22b26d12d22d26b16b26b66d16d26d66=A11A12A16B11B12B16A12A22A26B12B22B26A16A26A66B16B26B66B11B12B16D11D12D16B12B22B26D12D22D26B16B26B66D16D26s66−1.However, taking into account the actual engineering demands, which should be combined with the good comprehensive properties of carbon fiber composites and the low cost of glass fiber composites, the proportion of two kinds of material must have a minimum requirement. On the one hand, if the proportion of carbon fiber composites is too small, the structural stability of the hybrid laminates cannot be guaranteed. On the other hand, if the proportion of glass fiber composites is too small, the cost reduction effect is not obvious. Therefore, in this paper, the minimum proportions of two types of material are both set as 30% and the mathematical model of optimization problem can be formulated as(40)minFq1,q2,…,qn,θ1,θ2,…,θn=−a16,s.t.1.3n≤∑i=1nqk≤1.7n,−90°<θk≤90°,k=1,2,…,n,∑qk=1cos2θkzk−zk−1=∑qk=2cos2θkzk−zk−1=0,∑qk=1sin2θkzk−zk−1=∑qk=2sin2θkzk−zk−1=0,∑qk=1cos2θkzk2−zk−12=∑qk=2cos2θkzk2−zk−12=0,∑qk=1cos4θkzk2−zk−12=∑qk=2cos4θkzk2−zk−12=0,∑qk=1sin2θkzk2−zk−12=∑qk=2sin2θkzk2−zk−12=0,∑qk=1sin4θkzk2−zk−12=∑qk=2sin4θkzk2−zk−12=0,∑qk=1sin2θkzk3−zk−13=∑qk=2sin2θkzk3−zk−13=0,∑qk=1sin4θkzk3−zk−13=∑qk=2sin4θkzk3−zk−13=0,∑qk=1zk2−zk−12=∑qk=2zk2−zk−12=0,∑qk=1sin4θkzk−zk−1≠0,∑qk=2sin4θkzk−zk−1≠0.Wherein,qk=1,2k=1,2,…,n represents the values of “q” for the k-ply, qk=1 shows that the lamina corresponds to the type ① lamina, and qk=2 shows that the lamina corresponds to the type ② lamina. If ∑k=1nqk=n, it means that all lamina are made of type ① material, and similarly, if ∑k=1nqk=2n, it means that all lamina are made of type ② material. θk represents the paving angle of the k-ply, whose initial value is selected randomly from the range of [−90°, 90°].The improved differential evolution algorithm DE_CMSBHS is used to optimize this problem. Because of the strong constraints of optimization problems, which have 18 equality constraints and 2 inequality constraints, the penalty function is applied to deal with the constraints. The type① lamina and type ② lamina are made up of carbon fiber composites and glass fiber composites, respectively, and the corresponding material parameters are shown in Table 1.Table 1
Material properties of carbon fiber lamina and glass fiber lamina.
Performance parameters
Carbon fiber lamina
Glass fiber lamina
Elastic modulus (GPa)
E1
181.0
38.6
E2
10.2
8.3
Shear modulus (GPa)
G12
7.2
4.14
Poisson’s ratio
ν12
0.28
0.26
Thickness (mm)
t
0.1
0.1
Thermal expansion coefficient (μ/°C)
α1
−0.1
8.6
α2
25.6
22.1Table2 shows the 12–20-ply hybrid AFB0DS laminates with immunity to HTSD which are optimized by using DE_CMSBHS algorithm to maximize the extension-shear-coupled effect. The subscripts “c” and “gl” in the table indicate that the lamina is made of carbon fiber composites and glass fiber composites, respectively. As can be seen from the table, (1) there is no hybrid AFB0DS laminates with immunity to HTSD for 1–11-ply laminates; (2) for 12–20-ply hybrid AFB0DS laminates with immunity to HTSD, the proportions of two kinds of lamina are both not less than 30%; and (3) for optimized laminates with the maximum extension-shear-coupled effect, paving materials are symmetrical about the geometric middle plane and have a relatively uniform distribution, which is the one in the middle and the other on both sides.Table 2
HybridAFB0DS laminates with immunity to HTSD.
Number of plies
Stacking sequence/°
a16/m⋅N−1
1–11
No feasible solution
12
[1.2623c/−88.1563c/86.2219c/−4.7557c/90gl/0gl/0gl/90gl/85.2442c/−3.7780c/1.8436c/−88.7376c]T
2.28 × 10−9
13
[78.6287c/2.5518c/−62.5528c/−4.4571c/0gl/90gl/57.6811c/90gl/0gl/−22.1785c/74.0623c/−77.5022c/5.4030c]T
2.15 × 10−9
14
[−82.7030c/3.4955c/−9.5277c/68.6342c/−87.1720gl/−4.8666gl/44.7270gl/−44.7270gl/4.8666gl/87.1720gl/−86.6975c/−18.9236c/84.4406c/9.1474c]T
2.62 × 10−9
15
[13.7114c/79.6085c/−25.7171c/−74.2581c/85.3581gl/7.4729gl/−44.4060gl/15.8105c/44.4060gl/−7.4729gl/−85.3581gl/−88.3194c/−87.3471c/24.6320c/−19.9766c]T
1.68 × 10−9
16
[−82.2580c/3.4541c/−7.8035c/71.9029c/0.0178gl/−0.0360gl/−89.9848gl/89.9417gl/−89.9861gl/89.9911gl/−0.0421gl/0.0222gl/−86.5215c/−16.3448c/84.9501c/9.0614c]T
2.10 × 10−9
17
[−86.7602c/86.2949c/−3.1492c/3.7823c/−4.3788gl/35.9403gl/−50.2104gl/89.9014gl/−72.6480gl/41.0040gl/−73.7900gl/27.9322gl/−16.0242gl/−3.7823c/3.1492c/−86.2949c/86.7602c]T
8.26 × 10−10
18
[−0.0130gl/−89.9908gl/0.0171gl/89.9876gl/−80.5781c/79.3162c/7.2684c/−20.7364c/13.9886c/12.0891c/−86.7465c/−66.9711c/0.1373c/80.6754c/−89.9869gl/−0.0163gl/89.9904gl/0.0125gl]T
1.93 × 10−9
19
[−3.8195c/−1.6516c/−87.2690c/−86.3382c/39.6427c/−81.7483c/−0.0004gl/−89.9971gl/89.9928gl/−14.8789c/−0.0071gl/0.0028gl/89.9995gl/25.4403c/−72.0969c/−83.9994c/−0.8820c/79.5202c/−2.8331c]T
1.93 × 10−9
20
[88.5760gl/38.9195gl/−76.4222gl/−17.7876gl/−8.6194gl/12.7407c/−21.3605c/−87.3614c/64.3117c/−30.3418c/−87.4610c/48.9897c/89.9981c/−24.1090c/7.9710c/14.9679gl/69.9391gl/0.3258gl/−50.8020gl/86.2064gl]T
2.20 × 10−9In order to reflect the influence of the hybrid form on the extension-shear-coupled effect of laminates more intuitively, this paper further optimizes the extension-shear-coupled effect of single-material laminates and laminates with different proportions of two materials. Table3 shows the optimization results of two kinds of single-material laminates. Taking 16-ply laminates and 17-ply laminates as an example, the optimized results of laminates with different proportions of two materials are exhibited in Figure 2.Table 3
Maximum extension-shear-coupled effect of two single-composite laminates.
Number of plies
12
13
14
15
16
17
18
19
20
Carbon fiber laminates (10−9 m·N−1)
2.80
2.71
2.36
2.10
3.01
0.809
1.93
2.19
1.26
Glass fiber laminates (10−9 m·N−1)
12.3
13.0
11.0
8.76
11.0
10.4
9.40
8.53
7.59Figure 2
Maximal extension-shear-coupled effect of hybrid laminates with different proportions of two materials.It can be obtained from Table3 that the maximum extension-shear-coupled effect of glass fiber composite laminates is greater than that of carbon fiber composite laminates in the case of the same number of plies. In the process of optimization by DE_CMSBHS algorithm, if the minimum proportion limits of the two materials are removed, which means the paving material is free, the final optimal results are totally the same as those of single-glass fiber composite laminates of Table 3. Furthermore, the data in Tables 2 and 3 suggests that the maximum extension-shear-coupled effects of the 14-ply, 17-ply, and 20-ply hybrid laminates are greater than those of single-carbon fiber composite laminates, which indicates that the introduction of glass fiber composite materials into carbon fiber composite laminates can improve the extension-shear-coupled effect of partial numbers of plies laminates.The abscissa “ngl” in Figure 2 represents the number of glass fiber lamina. As can be seen from Figure 2, (1) for hybrid laminates with different proportions of two materials, there is no feasible solution when ngl takes some values, for which the geometrical factors of those kinds of laminates cannot satisfy the necessary and sufficient condition of (35) and (2) compared with single-carbon material composite laminates, some hybrid laminates can not only significantly reduce costs but also increase the maximum extension-shear-coupled effect.


## 5. Verification of Mechanical Properties of Laminates

### 5.1. Verification of Hygrothermal Effect

The finite element method is used to verify the distortion caused by the temperature change of the hybridAFB0DS laminates with immunity to HTSD. In this paper, the hygrothermal effect of laminates in Table 2 is verified by the example of 16-ply and 17-ply laminates and the conclusion of other laminates is the same as that of these laminates.Based on the finite element software MSC.Patran, the 10 m × 1 m finite element model is established and 360 shell units are divided. In order to simulate the displacement boundary condition of the composite laminates, the geometric center of the finite element model is fixed, as shown in Figure3. The typical temperature difference of the high-temperature curing process is −180°C to this finite element model. Then the finite element software MSC.Nastran is used to compute with the linear statics calculation function.Figure 3
Finite element model of the laminates for hygrothermal effect verification.The calculated results of two kinds of hybridAFB0DS laminates with immunity to HTSD are shown in Table 4. In which, εxT and εyT are the thermal strain of two main directions, γxyT is the thermal shear strain of the laminates, and κxT, κyT, and κxyT are expressed as the surface curvature and distortion of the laminates caused by temperature changes. It can be seen from the table that the shear strain of these laminates are all zero during the high-temperature curing process, which means two kinds of laminates will not cause hygrothermal distortion. The thermal strains of two directions are equal to each other, but this regulation is not suitable for laminates with different numbers of plies. Furthermore, their bending curvature and twist rate are all zero, which indicates that two kinds of laminates will not cause hygrothermal warping distortion.Table 4
Simulation results of hybridAFB0DS laminates on thermal distortion.
Type of laminates
εxT
εyT
γxyT
κxT
κyT
κxyT
16-ply laminates
−6.54 × 10−4
−6.54 × 10−4
0
0
0
0
17-ply laminates
−6.90 × 10−4
−6.90 × 10−4
0
0
0
0


### 5.2. Verification of Extension-Shear-Coupled Effect

In order to verify the extension-shear-coupled effect of two kinds of hybridAFB0DS laminates in Table 2, the finite element method is also used based on the finite element software MSC.Patran and the finite element model of a rectangular plate with 12 m × 1 m is established. A total of 800 shell units and 891 nodes (six degrees of freedom) are divided, and a multipoint constraint element (RBE2) is used to connect the nodes in the 1.2 m × 1 m region at both ends of the model. The axial tension F=1000 N is applied to the multipoint confinement unit to ensure that the intermediate 9.6 m × 1 m area bears a uniform load. The geometric center of the finite element model is fixed, as shown in Figure 4. The finite element software MSC.Nastran is used to compute with the linear statics calculation function.Figure 4
Finite element model of the laminates for extension-shear-coupled effect verification.The calculated displacement nephogram of two kinds of hybridAFB0DS laminates with immunity to HTSD under axial extension force is shown in Figure 5, and the concrete results of distortion are shown in Table 5. It can be seen from the table that under the axial extension force of 1000 N, the two kinds of hybrid AFB0DS laminates not only have axial distortion but also shear distortion and the extension-shear-coupled effect of 16-ply laminates is obviously more than that of 17-ply laminates.Figure 5
Deformation of two kinds of hybridAFB0DS laminates due to extension load.
(a)
16-ply laminates
(b)
17-ply laminatesTable 5
Simulation results of hybridAFB0DS laminates on extension-shear distortion.
Type of laminates
Simulation resultεx
Theoretical resultεx
Error ofεx
Simulation resultγxy
Theoretical result γxy
Error of γxy
16-ply laminates
1.08 × 10−5
1.10 × 10−5
1.82%
2.23 × 10−6
2.27 × 10−6
1.76%
17-ply laminates
1.04 × 10−5
1.06 × 10−5
1.89%
8.20 × 10−7
8.26 × 10−7
0.73%Table5 also shows the theoretical calculation results of two kinds of hybrid AFB0DS laminates subjected to axial extension distortion, and through comparison, it is found that the results of finite element analysis agree well with the theoretical results. The error is controlled within 2%, and the extension-shear-coupled effect of laminates is validated. The reason for the error is that the finite element simulation is loaded on the unit node and cannot be completely equivalent to the linear loading method.


### 5.3. Robustness Analysis

Considering that there may be human error and equipment error in the actual process of paving the laminates, which may have a negative effect on the extension-shear-coupled effect, in order to ensure the practicability of the composite laminates, the slight angle deviation should not have an obvious influence on the extension-shear-coupled effect of the hybridAFB0DS laminates with immunity to HTSD. Now, the extension-shear-coupled effect of laminates is analyzed under the ply angle existing deviation, which is based on the Monte Carlo composite laminates robustness analysis method.Assume that the paving angle of thek-ply for the hybrid AFB0DS laminates with immunity to HTSD is θk±Δθk, in which θk is the theoretical paving angle of the k-ply and Δθk is the angle deviation and with the value of Δθk=2° [17]. Figure 6 shows the error distribution of the extension-shear-coupled effect for the 16-ply laminates and the 17-ply laminates in Table 2 under 10,000 random samples. The figure suggests that when there exists a random error in the paving angle of the laminates, the error of extension-shear-coupled effect is in accordance with the normal distribution law and can be controlled within 2%.Figure 6
Robustness analysis of hybridAFB0DS laminates on extension-shear distortion.
(a)
16-ply laminates
(b)
17-ply laminates


## 5.1. Verification of Hygrothermal Effect

The finite element method is used to verify the distortion caused by the temperature change of the hybridAFB0DS laminates with immunity to HTSD. In this paper, the hygrothermal effect of laminates in Table 2 is verified by the example of 16-ply and 17-ply laminates and the conclusion of other laminates is the same as that of these laminates.Based on the finite element software MSC.Patran, the 10 m × 1 m finite element model is established and 360 shell units are divided. In order to simulate the displacement boundary condition of the composite laminates, the geometric center of the finite element model is fixed, as shown in Figure3. The typical temperature difference of the high-temperature curing process is −180°C to this finite element model. Then the finite element software MSC.Nastran is used to compute with the linear statics calculation function.Figure 3
Finite element model of the laminates for hygrothermal effect verification.The calculated results of two kinds of hybridAFB0DS laminates with immunity to HTSD are shown in Table 4. In which, εxT and εyT are the thermal strain of two main directions, γxyT is the thermal shear strain of the laminates, and κxT, κyT, and κxyT are expressed as the surface curvature and distortion of the laminates caused by temperature changes. It can be seen from the table that the shear strain of these laminates are all zero during the high-temperature curing process, which means two kinds of laminates will not cause hygrothermal distortion. The thermal strains of two directions are equal to each other, but this regulation is not suitable for laminates with different numbers of plies. Furthermore, their bending curvature and twist rate are all zero, which indicates that two kinds of laminates will not cause hygrothermal warping distortion.Table 4
Simulation results of hybridAFB0DS laminates on thermal distortion.
Type of laminates
εxT
εyT
γxyT
κxT
κyT
κxyT
16-ply laminates
−6.54 × 10−4
−6.54 × 10−4
0
0
0
0
17-ply laminates
−6.90 × 10−4
−6.90 × 10−4
0
0
0
0


## 5.2. Verification of Extension-Shear-Coupled Effect

In order to verify the extension-shear-coupled effect of two kinds of hybridAFB0DS laminates in Table 2, the finite element method is also used based on the finite element software MSC.Patran and the finite element model of a rectangular plate with 12 m × 1 m is established. A total of 800 shell units and 891 nodes (six degrees of freedom) are divided, and a multipoint constraint element (RBE2) is used to connect the nodes in the 1.2 m × 1 m region at both ends of the model. The axial tension F=1000 N is applied to the multipoint confinement unit to ensure that the intermediate 9.6 m × 1 m area bears a uniform load. The geometric center of the finite element model is fixed, as shown in Figure 4. The finite element software MSC.Nastran is used to compute with the linear statics calculation function.Figure 4
Finite element model of the laminates for extension-shear-coupled effect verification.The calculated displacement nephogram of two kinds of hybridAFB0DS laminates with immunity to HTSD under axial extension force is shown in Figure 5, and the concrete results of distortion are shown in Table 5. It can be seen from the table that under the axial extension force of 1000 N, the two kinds of hybrid AFB0DS laminates not only have axial distortion but also shear distortion and the extension-shear-coupled effect of 16-ply laminates is obviously more than that of 17-ply laminates.Figure 5
Deformation of two kinds of hybridAFB0DS laminates due to extension load.
(a)
16-ply laminates
(b)
17-ply laminatesTable 5
Simulation results of hybridAFB0DS laminates on extension-shear distortion.
Type of laminates
Simulation resultεx
Theoretical resultεx
Error ofεx
Simulation resultγxy
Theoretical result γxy
Error of γxy
16-ply laminates
1.08 × 10−5
1.10 × 10−5
1.82%
2.23 × 10−6
2.27 × 10−6
1.76%
17-ply laminates
1.04 × 10−5
1.06 × 10−5
1.89%
8.20 × 10−7
8.26 × 10−7
0.73%Table5 also shows the theoretical calculation results of two kinds of hybrid AFB0DS laminates subjected to axial extension distortion, and through comparison, it is found that the results of finite element analysis agree well with the theoretical results. The error is controlled within 2%, and the extension-shear-coupled effect of laminates is validated. The reason for the error is that the finite element simulation is loaded on the unit node and cannot be completely equivalent to the linear loading method.


## 5.3. Robustness Analysis

Considering that there may be human error and equipment error in the actual process of paving the laminates, which may have a negative effect on the extension-shear-coupled effect, in order to ensure the practicability of the composite laminates, the slight angle deviation should not have an obvious influence on the extension-shear-coupled effect of the hybridAFB0DS laminates with immunity to HTSD. Now, the extension-shear-coupled effect of laminates is analyzed under the ply angle existing deviation, which is based on the Monte Carlo composite laminates robustness analysis method.Assume that the paving angle of thek-ply for the hybrid AFB0DS laminates with immunity to HTSD is θk±Δθk, in which θk is the theoretical paving angle of the k-ply and Δθk is the angle deviation and with the value of Δθk=2° [17]. Figure 6 shows the error distribution of the extension-shear-coupled effect for the 16-ply laminates and the 17-ply laminates in Table 2 under 10,000 random samples. The figure suggests that when there exists a random error in the paving angle of the laminates, the error of extension-shear-coupled effect is in accordance with the normal distribution law and can be controlled within 2%.Figure 6
Robustness analysis of hybridAFB0DS laminates on extension-shear distortion.
(a)
16-ply laminates
(b)
17-ply laminates


## 6. Summary

In this paper, the model of hybrid laminates is established and the expressions of stiffness coefficient, thermal stress, and thermal moment of laminates are derived based on the geometric factor and the necessary and sufficient conditions for hybrid extension-shear-coupled laminates with immunity to HTSD are further derived. The method of combing the improved differential evolution algorithm DE_CMSBHS with the penalty function is used to optimize the extension-shear-coupled effect of the hybridAFB0DS laminates with immunity to HTSD. The hygrothermal effect, extension-shear-coupled effect, and robustness of laminates are verified. Some conclusions of optimization and simulation can be obtained as follows.
(1)
Although the optimized constraints is enhanced, which is mainly due to the introduction of variable parameter paving materials, the feasible solutions of the hybridAFB0DS laminates with immunity to HTSD can still be obtained by using the DE_CMSBHS algorithm combined with the penalty function. The optimal solution can be obtained according to the different requirements of different proportions of two materials, which achieved the synchronous optimization of paving angle and paving materials.
(2)
The maximum extension-shear-coupled effect of the single-glass fiber composite laminates is greater than that of single-carbon fiber composite laminates in the case of the same number of plies, and the introduction of glass fiber composite material into single-carbon fiber composite laminates can improve the extension-shear-coupled effect of the partial number of plies laminates.
(3)
The optimized laminates are able to meet the conditions of no hygrothermal shearing distortion and no hygrothermal warping distortion. When there exists a random error in the paving angle of the laminates, the error of extension-shear-coupled effect is in accordance with the normal distribution law and can be controlled within 2%.The expressions for parameters of the hybrid laminates can also be applied to laminates with other coupling effects. The DE_CMSBHS algorithm combined with the penalty function can also optimize the different objectives of other types of laminates.



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*Source: 9869105-2018-05-27.xml*