Inter-laminar stress modeling and validation on multi-layer composite cylinders under thermal loading
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摘要: 多层复合材料筒状结构的残余应力可造成内部分层、结构失稳等，有必要结合理论和实验等研究其内部应力形成机制。基于各向异性线弹性本构和平面应力假设，建立了温度载荷作用下多层复合材料筒状结构内部应力的预测模型，并通过有限元仿真和热膨胀变形测试对模型有效性进行了校核和验证。结果表明：筒状结构环向的热膨胀变形从内层到外层逐渐增大，其中内层附近环向热膨胀系数低于面内热膨胀系数、外层附近环向热膨胀系数则高于面内热膨胀系数。在此基础上，结合模型理论分析揭示了层间应力对多层复合材料筒状结构的热膨胀变形行为的影响机制。在筒状结构形式下，温度加载引起的层间应力由一个涉及热膨胀系数和弹性参数的热力耦合项决定；由于多层复合材料面内与面外热膨胀系数存在差异，该应力耦合项不为0，从而在温度加载下形成了层间应力并影响了环向膨胀变形行为。基于上述认识，提出了调控多层复合材料筒状结构层间应力的有效措施。本文对揭示多层复合材料筒状结构的层间开裂物理机制、优化其热力匹配设计等具有重要意义。Abstract: The inter-laminar stress in multi-layer composite cylindrical structures can cause internal delamination, structural instability, etc. It is necessary to study the internal stress formation mechanism. An analytical model for prediction of the inter-laminar stress in multi-layer composite cylinders subjected to thermal loading was developed based on an anisotropic constitutive model and the plane stress assumption. The analytical model was validated by means of finite element analyses and a thermal expansion experiment performed on a composite cylinder. Using this validated model, the inter-laminar stress between layers and the thermal expansion behavior were investigated. Results show that the special geometric constraint of the cylinder itself plays an important role in the residual stress development. The thermal expansion behavior in the cylinders is much different compared to that in planer laminated composites. Due to the difference of modulus and coefficient of thermal expansion along hoop and radial direction, a significant inter-laminar tensile stress will generate during cooling down process. Furthermore, the hoop thermal expansion deformation shows an evident increasing trend from the inner layer to the outer layer. The hoop thermal expansion deformation in the inner layer is much smaller than that of the composite. With the increment of the cylinder thickness, the inter-laminar tensile stress, and the difference in coefficient of thermal expansion between inner layer and outermost layer increase. The developed model could be useful to disclose the stress-induced inter-laminar crack and optimize stress distribution in multi-layer composite cylinders.
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图 1 由N层复合材料组成的筒状结构及坐标系定义(r, θ)
Figure 1. Cylindrical coordinate system (r, θ) of an axisymmetric laminated composite cylinder containing N layers
图 2 多层复合材料结构层间应力分析有限元模型(FEM)
Figure 2. Finite element model (FEM) for analyzing the inter-laminar a multi-layer composite cylinder
图 8 多层复合材料筒状结构不同位置膨胀变形量随温度变化曲线
Figure 8. Thermal expansion behavior of a multi-layer composite cylinder for different positions
图 9 热膨胀系数与各向异性比对多层复合材料筒状结构层间应力的影响
Figure 9. Effect of coefficient of thermal expansion and orthotropic ratio on the stress in the radial direction of a multi-layer composite cylinder
图 10 厚度/外径比对多层复合材料筒状结构层间应力的影响
Figure 10. Effect of thickness on the stress in the radial direction of a multi-layer composite cylinder
表 1 材料参数
Table 1. Material properties
Property Mat-a Mat-b In-plane Young's modulus Er/MPa 2000 3000 Out-of-plane Young's modulus Eθ/MPa 2500 12000 Poisson's ratio ${\mu _{r\theta }}$ 0.2 0.075 In-plane coefficient of thermal expansion (CTE) ${\alpha _r}$/(10−6 ℃−1) 50 50 Out-of-plane coefficient of thermal expansion (CTE) ${\alpha _\theta }$/(10−6 ℃−1) 50 5 
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表 2 多层缠绕复合材料结构材料性能
Table 2. Material properties of multi-layer composite laminate
Property Value Standard In-plane Young's modulus ${{{E_r}} \mathord{\left/ {\vphantom {{{E_r}} {\rm{MPa}}}} \right. } {\rm{MPa}}}$ 4623 ASTM D3039[18] Out-of-plane Young's modulus ${{{E_\theta }} \mathord{\left/ {\vphantom {{{E_\theta }} {\rm{MPa}}}} \right. } {\rm{MPa}}}$ 3243 ASTM D3039[18] Poisson's ratio ${\mu _{r\theta }}$ 0.36 ASTM D3039[18] In-plane coefficient of thermal expansion (CTE) ${\alpha _\theta }$/(10−6 ℃−1) 38 ASTM E831[19] Out-of-plane coefficient of thermal expansion (CTE) ${\alpha _r}$/(10−6 ℃−1) 97 ASTM E831[19]
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表 3 多层复合材料筒状结构不同位置热膨胀系数模型校核结果
Table 3. Validation of the analytical model on coefficient of thermal expansion for different positions in a multi-layer composite cylinder
Position Coefficient of thermal expansion
α/(10−6 ℃−1)Experiment Prediction Error Inner-most layer (#1) 18.54 20.58 11.1% Outer-most layer (#2) 49.95 50.44 1.0%
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