Failure analysis of composite materials based on phase field method: A review
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摘要: 预测复合材料的失效行为，对复合材料结构设计具有重要意义。由于其失效模式和失效机制较复杂，传统的计算断裂力学方法和基于损伤力学的数值方法在对其进行失效分析存在一定困难。相场法结合了断裂力学和损伤力学的优点，无需额外的判据便可精确捕捉裂纹的萌生、扩展和扭结行为，近年来被广泛地应用于复合材料的失效分析。本文首先简要介绍相场法的基本理论，给出了基本的断裂能模型和控制方程。然后着重介绍了基于相场法的复合材料失效分析的研究进展，梳理了相场法在复合材料领域的应用范围。最后，对相场法模拟复合材料在疲劳、疲劳湿热环境下和冲击下的损伤进行了展望。Abstract: Predicting the failure behavior of composite materials is of great significance to the design of composite structures. Due to the complexity of its failure mode and failure mechanism, the traditional computational fracture mechanics method and the numerical method based on damage mechanics are difficulty to model modeling its failure behavior. The phase field method combines the advantages of fracture mechanics and damage mechanics. It can accurately capture the crack initiation, propagation and kink behavior without additional criteria. Recently, it has been widely used in the failure analysis of composite materials. In this paper, the basic theory of phase field method was briefly introduced, and the fundamental fracture energy model and governing equations were given. Following that, the review focused on the research progress of composite failure analysis based on phase field method. The application ranges of phase field method on composite material field were reviewed. Finally, the damage simulations of composites under fatigue, hygrothermal environment and impact by using phase field method were discussed.
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Key words:
- phase field method /
- composite material /
- failure analysis /
- cohesive zone model /
- delamination
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图 1 断裂面的光滑过渡
l0—Length; $ \phi $—Phase field variable
Figure 1. Smooth transition of fracture surface
图 6 纤维桥联模型的机制和桥联区长度[28]
L—Length; h—Height; at—Crack length; ae—Effective crack length; lc—Fracture process zone length; lb—Bridging zone length; ${\delta _{\rm{n}}} $—Opening displacement of the bridging ligament; $\delta _{\rm{n}}^{\rm{f}} $—Opening displacement of the cracked fiber
Figure 6. Sketch of the fibre bridging mechanism and the bridging zone length[28]
图 7 相场法与其他数值方法的结合
Figure 7. Combination of phase field method and other numerical methods
图 9 模拟[39]和测量[40]得到的纤维增强复合材料的载荷-裂纹口张开位移(CMOD)曲线(a)和裂纹扩展路径(b)
FE—Finite element simulation; Exp—Experimental results; P—Loading; Δ—Crack mouth opening displacement
Figure 9. Predicted[39] and measured[40] of load-crack opening displacement (CMOD) curves (a) and crack propagation path (b) for fibre-reinforced composites
图 12 3D打印聚合物复合材料在不同伸长下的裂纹萌生序列[52]：((a), (c), (e), (g), (i)) 数值结果；((b), (d), (f), (h), (j))实验结果
Figure 12. Crack initiation sequence of 3D-printed hyperelastic composites at different values of global stretch[52]: ((a), (c), (e), (g), (i)) Numerical results; ((b), (d), (f), (h), (j)) Experimental results
表 1 复合材料的相场断裂能模型
Table 1. Phase field fracture energy model of composite material
Model Mathematical model Ref. Second-order anisotropic model $\dfrac{1}{2}\left[ {\dfrac{1}{ { {l_0} } }{\phi ^2} + {l_0}\left( {\nabla \phi {{A} } \nabla \phi } \right)} \right]$ ${A_{ij}} = {\delta _{ij}} + \gamma {M_{ij}}$ ${M_{ij}}{\text{ = }}{N_i}{N_j}$ [13] Double isotropic model for different crack mode $\displaystyle\int_\varOmega {\dfrac{{{G_{{\rm{cI}}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}\phi _1^2 + {l_0}\left( {\nabla {\phi _1} \cdot \nabla {\phi _1}} \right)} \right]} {\rm{d}}V + \displaystyle\int_\varOmega {\dfrac{{{G_{{\rm{cII}}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}\phi _2^2 + {l_0}\left( {\nabla {\phi _2} \cdot \nabla {\phi _2}} \right)} \right]} {\rm{d}}V$ [15] Double isotropic model for different component $\displaystyle\int_\varOmega {\dfrac{{{G_{\rm{f}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}\phi _{\rm{f}}^2 + {l_0}\left( {\nabla {\phi _{\rm{f}}} \cdot \nabla {\phi _{\rm{f}}}} \right)} \right]} {\rm{d}}V + \displaystyle\int_\varOmega {\dfrac{{{G_{\rm{m}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}\phi _{\rm{m}}^2 + {l_0}\left( {\nabla {\phi _{\rm{m}}} \cdot \nabla {\phi _{\rm{m}}}} \right)} \right]} {\rm{d}}V$ [16] Double second-order anisotropic model $ \displaystyle\int_\varOmega {\dfrac{{{G_{{\rm{cI}}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}{\phi ^2} + {l_0}{{{A}}_1}:\left( {\nabla \phi \otimes \nabla \phi } \right)} \right]} {\rm{d}}V + \displaystyle\int_\varOmega {\dfrac{{{G_{{\rm{cII}}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}{\phi ^2} + {l_0}{{{A}}_2}:\left( {\nabla \phi \otimes \nabla \phi } \right)} \right]} {\rm{d}}V $ [17] Fourth-order transverse isotropic model $ \begin{array}{l}{\displaystyle {\int }_{\varOmega }\dfrac{1}{2}\left[\dfrac{1}{{l}_{0}}{\phi }^{2}+\dfrac{{l}_{0}}{2}A:\left(\nabla \phi \otimes \nabla \phi \right)+\dfrac{{l}_{0}^{3}}{16}\left(\nabla \nabla \phi :\mathbb{A}:\nabla \nabla \phi \right)\right]}{\rm{d}}V，{A}_{ij}={\delta }_{ij}+\gamma {M}_{ij}，{M}_{ij}\text{= }{N}_{i}{N}_{j}\\ {\mathbb{A}}_{ijkl}={\delta }_{ij}{\delta }_{kl}+{\rm{sym}}\left({\beta }_{1}{M}_{ij}{M}_{kl}+{\beta }_{2}{\delta }_{ij}{M}_{kl}\right)+\dfrac{{\beta }_{3}}{2}\left({\delta }_{ik}{M}_{jl}+{M}_{ik}{\delta }_{jl}+{\delta }_{il}{M}_{jk}+{M}_{il}{\delta }_{jk}\right)\end{array} $ [18] Fourth-order orthotropic model $ \begin{array}{l}{\displaystyle {\int }_{\varOmega }\dfrac{1}{2}\left[\dfrac{1}{{l}_{0}}{\phi }^{2}+\dfrac{{l}_{0}}{2}A:\left(\nabla \phi \otimes \nabla \phi \right)+\dfrac{{l}_{0}^{3}}{16}\left(\nabla \nabla \phi :\mathbb{A}:\nabla \nabla \phi \right)\right]}{\rm{d}}V\\ {A}_{ij}={\delta }_{ij}+{\gamma }_{1}{M}_{ij}^{1}+{\gamma }_{2}{M}_{ij}^{2}，{M}_{ij}^{1}\text{= }{N}_{i}^{1}{N}_{j}^{1}，{M}_{ij}^{2}\text{= }{N}_{i}^{2}{N}_{j}^{2}\\ {\mathbb{A}}_{ijkl}=\dfrac{1}{2}\left({\delta }_{ij}{\delta }_{kl}+{\delta }_{il}{\delta }_{kj}\right)+{\rm{sym}}[{\displaystyle \sum _{s=1}^{2}\left({\alpha }_{1}^{s}{M}_{ij}^{s}{M}_{kl}^{s}+{\alpha }_{2}^{s}{\delta }_{ij}{M}_{kl}^{s}\right)}+{\alpha }_{7}{M}_{ij}^{1}{M}_{kl}^{2}+\\ \text{ }{\displaystyle \sum _{s=1}^{2}\dfrac{{\alpha }_{3}^{s}}{2}\left({\delta }_{ik}{M}_{jl}^{s}+{M}_{ik}^{s}{\delta }_{jl}+{\delta }_{il}{M}_{jk}^{s}+{M}_{il}^{s}{\delta }_{jk}\right)}]\end{array} $ [18] Fourth-order cubic symmetric model $ \begin{array}{l}{\displaystyle {\int }_{\varOmega }\dfrac{1}{2}\left[\dfrac{1}{{l}_{0}}{\phi }^{2}+\dfrac{{l}_{0}}{2}\left(\nabla \phi \cdot \nabla \phi \right)+\dfrac{{l}_{0}^{3}}{16}\left(\nabla \nabla \phi :\mathbb{A}:\nabla \nabla \phi \right)\right]}{\rm{d}}V\\ {\mathbb{A}}_{ijkl}=\dfrac{1}{2}\left({\delta }_{ij}{\delta }_{kl}+{\delta }_{il}{\delta }_{kj}\right)+{\displaystyle \sum _{m=1}^{2}{\displaystyle \sum _{n=1}^{2}\left[\alpha {\delta }_{mn}+\dfrac{\beta }{2}\left(1-{\delta }_{mn}\right)\right]}}{M}_{ij}^{m}{M}_{kl}^{m}\end{array} $ [18] Notes: A and ${A_{ij}}$—Second-order structure tensor and its component form; ${\delta _{ij}}$—Component of the second-order indentiy tensor; $\gamma $—Penalty parameter;${N_i}$—Component of unit vector along the fiber direction; ${G_{{\rm{cI}}}}$ and ${G_{{\rm{cII}}}}$—Crtical energy release rate for mode I and mode II crack; ${\phi _1}$ and ${\phi _2}$—Phase field variables for mode I and mode II crack ; ${G_{\rm{f}}}$ and ${G_{\rm{m}}}$—Crtical energy release rate of fiber and matrix; ${\phi _{\rm{f}}}$ and ${\phi _{\rm{m}}}$—Phase field variables of fiber and matrix; $ {{{A}}_1} $ and $ {{{A}}_2} $—Second-order structure tensor for mode I and mode II crack ; $ \mathbb{A} $—Fourth-order structure tensor; $ N_i^1 $and $ N_i^2 $—Components of two orthonomal basis;$ {\beta _1} $, $ {\beta _2} $, $ {\beta _3} $, $ {\gamma _1} $, $ {\gamma _2} $, $ \alpha _1^s $, $ \alpha _2^s $, $ \alpha _3^s $, $ {\alpha _7} $, $ \alpha $and $ \beta $—Material parameters. 
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表 2 复合材料的长度尺度参数模型
Table 2. Length scale parameter model of composite material
Model Mathematical model Ref. Zhang's model $l_0^{{\rm{ani}}}\left( {{\varphi _1}} \right) = l_0^{{\rm{iso}}}\sqrt {1 + \beta {{\cos }^2}{\varphi _1}} $ [13] Transverse isotropic model $l_0^{{\rm{tran}}}\left( {\varphi ,\theta } \right) = {l_0}\left[ {1 + \alpha {{\sin }^2}\left( {\varphi - \theta } \right)} \right]$ [18] Orthotropic model $l_0^{{\rm{orth}}}\left( {\varphi ,\theta } \right) = {l_0}\left[ {1 + {\alpha ^1}{{\sin }^2}\left( {\varphi - \theta } \right) + {\alpha ^2}{{\cos }^2}\left( {\varphi - \theta } \right)} \right]$ [18] Cubic symmetric model $\begin{gathered} l_0^{\rm cubic}\left( {\varphi ,\theta } \right) = \gamma {\left\{ {1 + \eta \cos \left[ {4\left( {\varphi - \theta } \right)} \right]} \right\}^{1/3}} \\ \gamma = {l_0}{\left( {\frac{{8 + 6\alpha + \beta }}{8}} \right)^{1/3}},\quad \eta = \left( {\frac{{8 + 6\alpha + \beta }}{{2\alpha - \beta }}} \right) \\ \end{gathered} $ [18] Notes: ${\varphi _1}$—Angle between the direction of gradient of phase field and the weak failure direction; ${l_0}$—Length scale parameter of isotropic phase field fracture model; $\varphi $—Angle between the horizontal axis and the tangent of the crack at position; $\theta $—Angle between the horizontal axis and the direction of the fiber; $\alpha $, ${\alpha ^1}$, ${\alpha ^2}$, $\gamma $, $\eta $, $\beta $—Material parameter.
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