## 留言板 引用本文: 彭帆, 马玉娥, 黄玮, 等. 基于相场法的复合材料失效分析研究进展[J]. 复合材料学报, 2023, 40(5): 2495-2506 PENG Fan, MA Yu'e, HUANG Wei, CHEN Pengcheng, MA Weili. Failure analysis of composite materials based on phase field method: A review[J]. Acta Materiae Compositae Sinica, 2023, 40(5): 2495-2506. doi: 10.13801/j.cnki.fhclxb.20220818.001
 Citation: PENG Fan, MA Yu'e, HUANG Wei, CHEN Pengcheng, MA Weili. Failure analysis of composite materials based on phase field method: A review[J]. Acta Materiae Compositae Sinica, 2023, 40(5): 2495-2506. • 中图分类号: TB332

## Failure analysis of composite materials based on phase field method: A review

Funds: National Natural Science Foundation of China (91860128)
• 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。

• 图  1  断裂面的光滑过渡

l0—Length; $\phi$—Phase field variable

Figure  1.  Smooth transition of fracture surface

图  2  相场变量的分布函数

Figure  2.  Distribution function of phase field variable

图  3  铺层角α=30° (a), α=45° (b), α=60° (c)和α=90° (d)铺层复合材料层合板的数值结果(i)和试验结果(ii)

Figure  3.  Numerically results (i) and experimentally (ii) obtained crack patterns for ply angle α=30° (a), α=45° (b), α=60° (c) and α=90° (d)

图  4  含螺旋状纤维的圆柱形管在拉伸下的断裂行为： (a) 缠绕角θ=0°；(b) 缠绕角θ=60°

Figure  4.  Fracture in a cylindrical tube with helix-type fiber structure under tension crack patterns for helix angles: (a) Wrap angle θ=0°; (b) Wrap angle θ=60°

图  5  复合材料层合板失效行为的模拟策略

θ1 and θ2—Ply angle; d—Phase field

Figure  5.  Modelling strategy about failure composite laminates

图  6  纤维桥联模型的机制和桥联区长度

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

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  8  相场法与其他理论的结合

Figure  8.  Combination of phase field method and other theories

图  9  模拟和测量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)

Figure  9.  Predicted and measured of load-crack opening displacement (CMOD) curves (a) and crack propagation path (b) for fibre-reinforced composites

图  10  纤维增强复合材料：((a)~(c)) 损伤演化阶段不同铺层之间的分层 ；((d)~(f)) 试验结果的X光扫描的分层图像

Figure  10.  Fibre reinforced composites: ((a)-(c)) Delamination between different plies at the damage evolution stage; ((d)-(f)) Experimental X-ray images of delamination

图  11  大变形拉伸导致的炭黑增强天然橡胶复杂裂纹扩展拓扑

Figure  11.  Snapshots showing the complex crack propagation topology of natural rubber filled with stiff/very stiff carbon black particles due to large deformation stretching

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：((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: ((a), (c), (e), (g), (i)) Numerical results; ((b), (d), (f), (h), (j)) Experimental results

图  13  Ti-6V-4Al合金微结构韧性裂纹扩展在7.6%体积平均真实应变加载(a)和13.0%体积平均真实应变加载下(b)的相场云图

Figure  13.  Contour plots of the phase field for Ti-6V-4Al microstructure in the deformed configuration from ductile crack propagation for 7.6% volume-averaged true strain (a) and 13.0% volume-averaged true strain (b)

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 $\dfrac{1}{2}\left[ {\dfrac{1}{ { {l_0} } }{\phi ^2} + {l_0}\left( {\nabla \phi {{A} } \nabla \phi } \right)} \right]$ ${A_{ij}} = {\delta _{ij}} + \gamma {M_{ij}}$ ${M_{ij}}{\text{ = }}{N_i}{N_j}$  Double isotropic model for different crack mode $\displaystyle\int_\varOmega {\dfrac{{{G_{{\rm{cI}}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}\phi _1^2 + {l_0}\left( {\nabla {\phi _1} \cdot \nabla {\phi _1}} \right)} \right]} {\rm{d}}V + \displaystyle\int_\varOmega {\dfrac{{{G_{{\rm{cII}}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}\phi _2^2 + {l_0}\left( {\nabla {\phi _2} \cdot \nabla {\phi _2}} \right)} \right]} {\rm{d}}V$  Double isotropic model for different component $\displaystyle\int_\varOmega {\dfrac{{{G_{\rm{f}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}\phi _{\rm{f}}^2 + {l_0}\left( {\nabla {\phi _{\rm{f}}} \cdot \nabla {\phi _{\rm{f}}}} \right)} \right]} {\rm{d}}V + \displaystyle\int_\varOmega {\dfrac{{{G_{\rm{m}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}\phi _{\rm{m}}^2 + {l_0}\left( {\nabla {\phi _{\rm{m}}} \cdot \nabla {\phi _{\rm{m}}}} \right)} \right]} {\rm{d}}V$  Double second-order anisotropic model $\displaystyle\int_\varOmega {\dfrac{{{G_{{\rm{cI}}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}{\phi ^2} + {l_0}{{{A}}_1}:\left( {\nabla \phi \otimes \nabla \phi } \right)} \right]} {\rm{d}}V + \displaystyle\int_\varOmega {\dfrac{{{G_{{\rm{cII}}}}}}{2}\left[ {\dfrac{1}{{{l_0}}}{\phi ^2} + {l_0}{{{A}}_2}:\left( {\nabla \phi \otimes \nabla \phi } \right)} \right]} {\rm{d}}V$  Fourth-order transverse isotropic model $\begin{array}{l}{\int }_{\varOmega }\dfrac{1}{2}\left[\dfrac{1}{{l}_{0}}{\phi }^{2}+\dfrac{{l}_{0}}{2}A:\left(\nabla \phi \otimes \nabla \phi \right)+\dfrac{{l}_{0}^{3}}{16}\left(\nabla \nabla \phi :\mathbb{A}:\nabla \nabla \phi \right)\right]}{\rm{d}}V，{A}_{ij}={\delta }_{ij}+\gamma {M}_{ij}，{M}_{ij}\text{= }{N}_{i}{N}_{j}\\ {\mathbb{A}}_{ijkl}={\delta }_{ij}{\delta }_{kl}+{\rm{sym}}\left({\beta }_{1}{M}_{ij}{M}_{kl}+{\beta }_{2}{\delta }_{ij}{M}_{kl}\right)+\dfrac{{\beta }_{3}}{2}\left({\delta }_{ik}{M}_{jl}+{M}_{ik}{\delta }_{jl}+{\delta }_{il}{M}_{jk}+{M}_{il}{\delta }_{jk}\right)\end{array}$  Fourth-order orthotropic model $\begin{array}{l}{\int }_{\varOmega }\dfrac{1}{2}\left[\dfrac{1}{{l}_{0}}{\phi }^{2}+\dfrac{{l}_{0}}{2}A:\left(\nabla \phi \otimes \nabla \phi \right)+\dfrac{{l}_{0}^{3}}{16}\left(\nabla \nabla \phi :\mathbb{A}:\nabla \nabla \phi \right)\right]}{\rm{d}}V\\ {A}_{ij}={\delta }_{ij}+{\gamma }_{1}{M}_{ij}^{1}+{\gamma }_{2}{M}_{ij}^{2}，{M}_{ij}^{1}\text{= }{N}_{i}^{1}{N}_{j}^{1}，{M}_{ij}^{2}\text{= }{N}_{i}^{2}{N}_{j}^{2}\\ {\mathbb{A}}_{ijkl}=\dfrac{1}{2}\left({\delta }_{ij}{\delta }_{kl}+{\delta }_{il}{\delta }_{kj}\right)+{\rm{sym}}[\sum _{s=1}^{2}\left({\alpha }_{1}^{s}{M}_{ij}^{s}{M}_{kl}^{s}+{\alpha }_{2}^{s}{\delta }_{ij}{M}_{kl}^{s}\right)}+{\alpha }_{7}{M}_{ij}^{1}{M}_{kl}^{2}+\\ \text{ }\sum _{s=1}^{2}\dfrac{{\alpha }_{3}^{s}}{2}\left({\delta }_{ik}{M}_{jl}^{s}+{M}_{ik}^{s}{\delta }_{jl}+{\delta }_{il}{M}_{jk}^{s}+{M}_{il}^{s}{\delta }_{jk}\right)}]\end{array}$  Fourth-order cubic symmetric model $\begin{array}{l}{\int }_{\varOmega }\dfrac{1}{2}\left[\dfrac{1}{{l}_{0}}{\phi }^{2}+\dfrac{{l}_{0}}{2}\left(\nabla \phi \cdot \nabla \phi \right)+\dfrac{{l}_{0}^{3}}{16}\left(\nabla \nabla \phi :\mathbb{A}:\nabla \nabla \phi \right)\right]}{\rm{d}}V\\ {\mathbb{A}}_{ijkl}=\dfrac{1}{2}\left({\delta }_{ij}{\delta }_{kl}+{\delta }_{il}{\delta }_{kj}\right)+\sum _{m=1}^{2}\sum _{n=1}^{2}\left[\alpha {\delta }_{mn}+\dfrac{\beta }{2}\left(1-{\delta }_{mn}\right)\right]}}{M}_{ij}^{m}{M}_{kl}^{m}\end{array}$  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.

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 $l_0^{{\rm{ani}}}\left( {{\varphi _1}} \right) = l_0^{{\rm{iso}}}\sqrt {1 + \beta {{\cos }^2}{\varphi _1}}$  Transverse isotropic model $l_0^{{\rm{tran}}}\left( {\varphi ,\theta } \right) = {l_0}\left[ {1 + \alpha {{\sin }^2}\left( {\varphi - \theta } \right)} \right]$  Orthotropic model $l_0^{{\rm{orth}}}\left( {\varphi ,\theta } \right) = {l_0}\left[ {1 + {\alpha ^1}{{\sin }^2}\left( {\varphi - \theta } \right) + {\alpha ^2}{{\cos }^2}\left( {\varphi - \theta } \right)} \right]$  Cubic symmetric model $\begin{gathered} l_0^{\rm cubic}\left( {\varphi ,\theta } \right) = \gamma {\left\{ {1 + \eta \cos \left[ {4\left( {\varphi - \theta } \right)} \right]} \right\}^{1/3}} \\ \gamma = {l_0}{\left( {\frac{{8 + 6\alpha + \beta }}{8}} \right)^{1/3}},\quad \eta = \left( {\frac{{8 + 6\alpha + \beta }}{{2\alpha - \beta }}} \right) \\ \end{gathered}$  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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##### 出版历程
• 收稿日期:  2022-06-22
• 修回日期:  2022-07-23
• 录用日期:  2022-08-04
• 网络出版日期:  2022-08-18
• 刊出日期:  2023-05-15

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