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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Keywords:
- phase field method /
- composite material /
- failure analysis /
- cohesive zone model /
- delamination
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复合材料由于其高比强度、高比刚度等优点,被广泛应用于航空航天、汽车、建筑等领域[1-3]。因复合材料具有各向异性和非均匀分布的材料属性,其失效模式和机制较复杂。对于常用的纤维增强复合材料层合板,其失效由基体断裂、纤维失效、分层等因素导致[4]。深刻揭示复合材料失效机制,进一步发展出复合材料及结构的健康监控和寿命预测技术,是工业界和学术界备受关注并且具有挑战的研究领域。然而,当涉及复杂构型及加载、边界条件时,传统的数值方法不能准确捕捉复合材料中可能发生的失效行为[5]。相场法是一种变分断裂方法,它将尖锐的裂纹定义为弥散的损伤带,避免了对裂纹产生的不连续边界的处理。相场法采用序参量——相场变量来定量地描述材料的损伤程度,并借助弥散的损伤带来实现完整材料和裂纹之间的光滑过渡,无需额外的判据来追踪裂纹的扩展路径。由于复合材料的失效由渐进损伤所致,单裂纹会导致复合材料其他区域出现裂纹萌生,而相场法可以有效地模拟材料的裂纹萌生和扩展等行为。基于这些特性,相场法在预测复合材料失效行为方面具有一定优势[6-12]。
本文将主要针对相场法模拟复合材料失效行为的研究进展进行介绍,并对未来相场法预测复合材料及结构复杂失效行为的发展方向及需要解决的问题进行探讨。
1. 相场法的控制方程
在相场断裂模型中,为了处理裂纹带来的材料不连续问题,通过弥散的损伤带来连接裂纹和完整的材料,并采用相场变量
ϕ 来定义材料的损伤:当ϕ=1 时,表示材料完全破坏;当ϕ=0 时,表示材料完整,如图1所示。其中:2l0表示损伤带的宽度;l0为长度尺度参数。图1给出了相场变量在裂纹两侧的分布情况,其分布曲面的剖面可以由指数函数
e−|x|/l0 来表达,如图2所示。该指数函数通常定义为相场变量的分布函数ϕ(x) 。显然,相场变量分布函数满足下列常微分方程:
ϕ(x)l0−l0d2ϕ(x)dx2=0 (1) 及如下条件:
ϕ(0)=1, ϕ(±∞)=0 (2) 构造函数
γ(ϕ,ϕ′) ,使得泛函∫∞−∞γ(ϕ,ϕ′)dx 取极值时的欧拉-拉格朗日方程等于式(1),即∂γ∂ϕ−ddx(∂γ∂ϕ′)=ϕ(x)l0−l0d2ϕ(x)dx2=0ϕ′=dϕdx (3) 因此,可得到如下方程:
∂γ∂ϕ=ϕ(x)l0,∂γ∂ϕ′=l0dϕ(x)dx (4) 进而得到函数
γ(ϕ,ϕ′) 的表达式:γ(ϕ,ϕ′)=12[ϕ2/ϕ2l0l0+l0(ϕ′)2] (5) 其中,
γ(ϕ,ϕ′) 为裂纹密度函数[5-7]。一维情况下,裂纹面积泛函可以定义为Γl=∫∞−∞γ(ϕ,ϕ′)dx=∫∞−∞12[ϕ2/ϕ2l0l0+l0(ϕ′)2]dx (6) 对于二维或者三维问题,只需将式(6)中
ϕ′ 替换成∇ϕ ,并将定积分转化成面积分或体积分可得到二维和三维断裂问题的裂纹面积泛函:Γl=∫Ω12[ϕ2/ϕ2l0l0+l0|∇ϕ|2]dV (7) 式中,右边第一、二项分别表示裂纹密度函数的局部项和非局部项。根据断裂表面能定义,给出其最终表达式:
Ψfracture=GcΓl = ∫ΩGc2[ϕ2/ϕ2l0l0+l0|∇ϕ|2]dV (8) 在准静态加载下,脆性相场断裂模型中的变形体总势能泛函写成
Π(u,∇u,ϕ,∇ϕ)=∫Ωg(ϕ)ψe(∇u)+Gcγ(ϕ,∇ϕ)dV−∫ΩbudV−∫∂ΩfudA (9) 其中:
u 表示位移场;g(ϕ) 表示相场退化函数,用来定量描述相场变量对材料参数的影响;ψe 为弹性应变能密度;Gc为临界应变能释放率;b 和f分别表示外体力和外面力;Ω 表示变形体占据的空间;∂Ω 表示变形体的表面。根据最小势能原理,可以分别得到关于位移场和相场变量的欧拉-拉格朗日方程:
∂Π∂u−∇∂Π∂∇u=∇σ+b=0 (10) ∂Π∂ϕ−∇∂Π∂∇ϕ=−2(1−ϕ)ψe+Gc(ϕ/ϕl0l0−l0∇2ϕ)=0 (11) 式(10)、式(11)分别为位移场和相场变量的控制方程,它们需满足以下边界条件
σn=f∇ϕn=0 (12) 其中,σ为柯西应力张量。
2. 复合材料的相场断裂模型和数值求解方法
式(10)和式(11)为两组耦合的偏微分方程,在采用相场法模拟材料的断裂问题时,需对其进行数值求解得到材料的位移场和相场。对于复合材料的相场损伤模拟,需要对原来的相场断裂模型进行修正,不仅要考虑材料本构的各向异性,还需引入各向异性的断裂能和相场参数。本节主要对复合材料层合板的断裂能模型、分层损伤模型和相场断裂模型的数值求解方法展开介绍。
2.1 各向异性的相场断裂能模型
为了能采用相场法来预测复合材料的各向异性的断裂行为,学者们在相场断裂模型中引入了各向异性的断裂能,提出了相应的相场断裂能模型。为了对三维复合材料层合板进行断裂分析,Zhang等[13]在断裂能密度中引入了一个二阶结构张量来表征断裂属性的各向异性,该相场断裂模型可以有效地模拟沿各个方向角铺层的复合材料的裂纹扩展路径,如图3所示[13-14]。Bleyer等[15]基于复合材料中的多种损伤机制,建立了双相场的断裂能模型,用两个各项同性相场断裂能来分别表示I、II型断裂的表面能。Nguyen等[16]在Bleyer等[15]的基础上,将各向同性的相场断裂能推广至各向异性,建立了各向异性的双相场模型。Zhang等[17]建立了关于纤维和基体的双相场模型来预测复合材料层合板的多种失效模式。对于双相场断裂模型,其控制方程为两组断裂能对应的相场演化方程。
高阶相场模型在二阶相场模型的基础上引入了相场变量的二阶导数项,在数值计算上具有更好的收敛性,而且不需要很精细划分的网格就可以精确捕捉到裂纹扩展路径。因此,Teichtmeister等[18]建立了四阶的各向异性相场断裂能模型,引入了四阶的结构张量,基于张量函数推导了正交各向异性、横观各向同性、立方对称这3种情况下的二阶和四阶结构张量,并采用建立的相场断裂模型模拟了含螺旋纤维结构的圆柱形管的拉伸破坏,模拟结果如图4所示。该模型的相场演化方程为四阶偏微分方程,如采用有限元方法求解,需建立C1连续的形函数。
上述相场断裂能的数学表达式,如表1所示。为了便于数值实现,表1给出了二阶和四阶结构张量的分量形式。由于相场断裂模型含有相场变量的梯度项,在采用有限元方法求解其相场演化方程时,为了较精确地对相场变量的梯度项进行数值离散,需对裂纹可能经过的区域进行网格精细划分。研究表明[19],精细划分的最小网格尺寸需要小于长度尺度参数的一半。对于复合材料相场断裂模型的有限元计算,需要确定长度尺度参数,然后对网格进行局部细分。由式(1)和式(8)可知,基于变分原理从相场变量的分布函数逆向推导可得到相场断裂能。当建立不同的复合材料相场断裂能模型时,需对长度尺度参数进行相应修正,对应的长度尺度参数如表2所示。
Model Mathematical model Ref. Second-order anisotropic model 12[1l0ϕ2+l0(∇ϕA∇ϕ)] Aij=δij+γMij Mij = NiNj [13] Double isotropic model for different crack mode ∫ΩGcI2[1l0ϕ21+l0(∇ϕ1⋅∇ϕ1)]dV+∫ΩGcII2[1l0ϕ22+l0(∇ϕ2⋅∇ϕ2)]dV [15] Double isotropic model for different component ∫ΩGf2[1l0ϕ2f+l0(∇ϕf⋅∇ϕf)]dV+∫ΩGm2[1l0ϕ2m+l0(∇ϕm⋅∇ϕm)]dV [16] Double second-order anisotropic model ∫ΩGcI2[1l0ϕ2+l0A1:(∇ϕ⊗∇ϕ)]dV+∫ΩGcII2[1l0ϕ2+l0A2:(∇ϕ⊗∇ϕ)]dV [17] Fourth-order transverse isotropic model ∫Ω12[1l0ϕ2+l02A:(∇ϕ⊗∇ϕ)+l3016(∇∇ϕ:A:∇∇ϕ)]dV,Aij=δij+γMij,Mij= NiNjAijkl=δijδkl+sym(β1MijMkl+β2δijMkl)+β32(δikMjl+Mikδjl+δilMjk+Milδjk) [18] Fourth-order orthotropic model ∫Ω12[1l0ϕ2+l02A:(∇ϕ⊗∇ϕ)+l3016(∇∇ϕ:A:∇∇ϕ)]dVAij=δij+γ1M1ij+γ2M2ij,M1ij= N1iN1j,M2ij= N2iN2jAijkl=12(δijδkl+δilδkj)+sym[2∑s=1(αs1MsijMskl+αs2δijMskl)+α7M1ijM2kl+ 2∑s=1αs32(δikMsjl+Msikδjl+δilMsjk+Msilδjk)] [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. 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. 2.2 基于相场法的分层和界面损伤模型
对复合材料层合板而言,采用相场法可以较好地模拟其层内失效行为,而对于层间失效行为则需要一些特殊的处理[5]。目前,学术界基于相场法模拟复合材料层合板损伤的主流处理方法:采用相场法来模拟层内失效行为,在相邻单层板之间插入内聚力单元来模拟层间失效行为[3, 20-24],如图5所示。为了发挥多相场复合材料断裂模型的优势,Asur Vijaya Kumar等[20]将其与内聚力模型结合,考虑了纤维失效、纤维间失效和界面脱粘的机制,基于多场变分原理导出了具有热力学一致性的计算框架。为了捕捉各向异性的损伤,Pillai等[21]为不同的复合材料损伤模式定义了裂纹驱动力和结构张量,提出了含三参数退化函数的各向异性内聚相场模型。上述的相场断裂模型仅能模拟复合材料在小应变下的断裂与损伤,对此,Carollo等[22]将其应用范围进行了拓展,在大变形分析框架下将相场法和内聚力单元法结合,模拟了非均匀材料和层合结构中的断裂现象。
上述相场断裂模型的数值求解是在隐式框架下实现,因而计算效率较低,仅能对网格数目在十万以下的复合材料层合板模型进行模拟。为了提高相场法的计算效率和解决计算收敛上的困难,Zhang等[3, 23-24]提出将显式的相场法和内聚力模型结合,来模拟复合材料的分层损伤。与隐式求解框架相比,显式求解框架不需要进行迭代及大量的矩阵储存和运算,因而计算效率较高。
2.3 微-细-介观的复合材料相场断裂模型
一些复合材料具有复杂的非均匀性和多尺度结构,在对其有限元建模时,如果仅以均匀的各向异性弹性体对其进行处理,则其网格的单元尺寸与“临界体积”的尺寸相当。然而该尺寸不一定与材料的物理性质有关,难以直接获取[25]。碳纳米管复合材料具有多尺度特性和多种特征的微结构,对其进行数值模拟具有挑战性[26]。Quinteros等[27]采用平均场均一化方法来估计纳米复合材料的力学本构和断裂属性,提出了基于细观力学的相场法来模拟碳纳米管复合材料的裂纹萌生和裂纹。
在一些复合材料中,纤维连接着两个相邻裂纹表面,这种裂纹桥接行为会推迟裂纹尖端的张开,从而提高材料断裂韧性。为了研究纤维、基体和纤维基体界面属性对裂纹桥接行为的影响,Tan等[28]建立了考虑微结构分布的相场内聚力模型,用裂纹尖端附近的嵌入单元来表示复合材料细观结构,用各向异性弹性体来表示其他部分,如图6所示。
复合材料层合板的层中横向裂纹会促进其他失效机制的发生,导致试样在加载后续阶段发生坍塌,从而对复合材料层合板的强度造成显著影响。对此,Guillén-Hernández等[29]基于相场法和双线性内聚力模型,对正交铺层的复合材料层合板进行细观力学分析,研究其在拉伸作用下的微裂纹和损伤状态的演化。混凝土作为一种典型的多相复合材料,其细观结构的不均匀性和裂纹扩展的多尺度性可能是影响其力学性能的主要因素。对此,Li等[30]、Xia等[31]分别建立了混凝土的细观和介观相场断裂模型。为了模拟多晶材料的晶粒和晶界的断裂,Paggi等[32]建立相应的各向异性相场内聚力模型,用相场法模拟穿晶断裂,用内聚力模型模拟晶间断裂。
2.4 复合材料相场断裂模型的数值求解方法
相场法的核心是求解关于位移场和相场变量的偏微分方程,这两组偏微分方程可以采用除有限元以外的数值方法求解。因此,相场法可以较方便地与其他数值方法结合,共同发挥计算和模拟上的优势。相场法需要对裂纹可能经过的区域进行网格细分,故而求解效率较低。对此,Heister等[33]采用网格自适应加密技术来实现相场控制方程的有限元求解,只需根据相场损伤的演化情况对网格进行精准的局部细分,在一定程度上提高了计算效率。除此之外,为了提高相场法数值求解中的网格自适应的便捷性和计算效率、降低了对空间场变量的连续性和光滑性要求,学者们分别采用等几何配点法[34]、无网格伽辽金法[35]、快速傅里叶变换法[36-37]和近场动力学方法[38]对相场断裂模型的两组控制方程进行数值求解,如图7所示。
3. 相场法在复合材料中的应用
近年来,随着相场法与一些力学和物理理论的结合(图8),复合材料相场断裂模型得到了不断发展,在复合材料的强度预测和失效模拟等方面得到了较广泛的应用,展示出巨大的应用潜力。
3.1 纤维增强复合材料的失效相场模拟
纤维增强复合材料通常由硬纤维和增韧环氧树脂基体等材料制成,具有较高强度和韧性。然而其纤维和基体之间及层间容易发生损伤,从而降低其整体结构的强度和耐久性[13]。Tan等[39]采用相场法和内聚力模型来模拟高分子纤维增强复合材料中的基体开裂和纤维基体脱粘行为,研究了孔隙率对纤维增强复合材料裂纹生长阻力曲线的影响。结果显示孔隙体积分数的增加,纤维增强复合材料具有更高的裂纹扩展阻力。此外,还给出了外加载荷随裂纹口张开位移(CMOD)的曲线,并与试验结果[40]进行了对比。如图9所示,相场法计算得到的曲线和裂纹扩展路径和试验结果吻合较好。
Espadas-Escalante等[41]用相场法预测了单层板和层合板的强度,结果显示当单层板承受横向拉伸时,基体中的裂纹将垂直于加载方向并围绕着纤维簇;当单层板承受横向压缩时,断裂将由剪切带诱发。静水压力-偏应力分解和谱分解是相场法中应用最广泛的两种拉压分解模型[9]。比较了基于这两种拉压分解模型的复合材料失效相场模拟的数值结果,结果显示两种拉压分解模型得到的数值结果相近。由于静水压力-偏应力分解在数值计算上具有更弱的非线性,因此在复合材料的相场模拟中收敛性更好。
在介观尺度,长纤维增强复合材料的典型失效模式可以分为层内失效、层间失效和跨层失效这几种。这些失效模式和材料各向异性的力学性能结合,使得材料力学响应的预测极为复杂。对此,Quintanas-Corominas等[42]用相场法研究了长纤维增强复合材料在准静态拉伸作用下的渐进失效行为。数值结果与连续损伤力学方法和内聚力单元法模拟的数值结果进行了比较,验证了相场法的优越性。Zhang等[3]基于显式相场模型研究纤维增强复合材料层合板开口拉伸试样的渐进失效,模拟得到的分层损伤分布与试验结果[43]进行了对比,如图10所示,−45°铺层与0°铺层、90°铺层与−45°铺层、45°铺层与90°铺层之间的分层损伤的分布特征和区域较好地复现了试验结果。此外,相场法还被应用于预测纤维增强水泥复合材料[44-45]、混杂复合材料[46-47]和变刚度复合材料[48-49]的失效。
相场法相比其他断裂和损伤数值方法,能精确捕捉复杂的裂纹扩展路径,并且结合内聚力单元法可以同时模拟纤维增强复合材料的多种损伤模式。然而,相场法需要更精细的网格划分,计算效率较低。
3.2 复合材料大变形下的断裂模拟
对高分子基复合材料的断裂行为进行数值模型具有一定难度,由于需要同时考虑超弹性本构的材料非线性、大变形下的几何非线性及材料参数退化带来的非线性。炭黑增强天然橡胶是一种复合材料,越来越多地用于工程应用。详细了解增强橡胶的成分与其断裂韧性之间的关系对于各种工程系统的分析和设计非常重要。为此,Wu等[50]采用不同的超弹性本构模型对橡胶基体和增强颗粒进行建模,然后用相场法模拟准静态载荷下炭黑增强天然橡胶的损伤成核和扩展,模拟结果如图11所示,相场变量接近1的单元都被删除。可以看出,一些椭圆形颗粒之间会产生损伤并形成空洞,随着位移载荷的增加,这些空洞合并,直到形成宏观裂纹。
在过去20年中,3D打印聚合物复合材料受到了广泛关注[51]。为了探讨夹杂物的空间分布对3D打印聚合物复合材料断裂韧性的影响,Russ等[52]通过试验手段和相场法研究了含3个硬质圆形夹杂物的3D打印聚合物复合材料的断裂行为,结果如图12所示。相场法计算得到的裂纹演化过程的特征及大变形后基体的形态与实验结果极为吻合。然而, 在Russ等[52]采用的相场断裂模型中,需要提前测得复合材料的强度来获得相场断裂模型的长度尺度参数。对于生物组织、纤维增强橡胶和仿生结构这些各向异性超弹性材料,合适的长度尺度参数的获取较困难。对此,Mandal等[53]发展了一种对长度尺度参数无依赖性的相场断裂模型,模拟了具有各向异性超弹性本构的生物组织在大变形下的断裂行为,模拟得到的位移-载荷曲线与试验结果较一致。
3.3 功能梯度材料和功能复合材料的失效模拟
功能梯度材料具有随空间位置变化的材料特性,相比传统材料可以减少应力集中、缓解残余应力和增强复合材料成分的粘结强度,被广泛应用于弹道和装甲的抗冲击结构、高温组件中的热障涂层、微电子封装中的夹层设计中[54]。然而,功能梯度材料容易发生脆性破坏,研究其损伤和失效行为具有重要的意义。Duc等[55]用相场法研究了功能梯度玻璃填充环氧树脂材料的动态断裂行为,模拟得到的裂纹扩展路径与实验结果吻合较好,研究发现该材料刚度较硬的部分更容易发生裂纹扭结。Thom等[56]用相场法分析了含裂纹功能梯度材料的热屈曲行为,研究了边界条件和材料属性对热屈曲行为的影响。基于三阶剪切板理论和相场法,Phuc等[57]研究了功能梯度板的厚长比、裂纹角度和体积分数指数对其稳定性的影响。Hirshikesh等[58]考虑了功能梯度材料随空间变化的弹性和断裂属性,采用相场法精准复现了在梯度光降解共聚物和玻璃填充环氧功能梯度材料中观察到的裂纹轨迹。
压电材料由于其良好的力电耦合性能,被广泛应用于传感器、变频器和驱动器等智能器件和结构中[59]。由于理论分析和实验在实际工程应用中存在一些局限性,数值模拟对于预测压电材料的断裂行为具有重要意义。Tan等[59]采用相场法研究了各向异性断裂韧性、极化方向和外加电场对压电材料断裂行为的影响。Dan等[60]采用相场法分析了非均质压电复合材料微结构的力电耦合响应和裂纹演化。
相场法作为一种连续性的方法,可以对材料属性在空间连续分布的功能梯度材料进行建模,并且因其变分结构可以较方便地引入新的物理场,进而建立相场、位移场和物理场之间的耦合机制,在求解功能梯度材料的断裂问题和复合材料多场耦合断裂问题上具有一定优势[8-9]。但是相场断裂模型的势能泛函具有非凸性,因此,对三场进行全耦合求解往往存在数值上的困难。
3.4 多晶材料的失效模拟
在多晶材料中,晶间断裂和穿晶断裂是其主要的两种断裂模式。多晶材料的晶格方向和晶界特性这些微观结构特征对其宏观力学断裂性能有显著影响。在计算材料领域,相场法可以模拟晶粒的生长和微结构的演化。近年来,随着相场断裂模型的发展,相场法被用来模拟多晶材料的断裂。Soukaina等[61]采用相场法研究多晶材料晶间断裂和穿晶断裂的竞争机制和相互作用。Cheng等[62]将相场法和晶体塑性理论结合,假设有限变形下多晶的裂纹驱动力由考虑材料各向异性和拉压不对称性的弹性能和由滑移系位错滑动和硬化产生的缺陷能组成,模拟了Ti-6V-4Al合金微结构的裂纹扩展,模拟结果如图13所示。Ti-6V-4Al合金微结构在较大的持续应变加载下才会出现裂纹扩展和完全断裂,这是由于塑性流动导致一部分能量耗散掉,并未参与驱动裂纹扩展。Emdadi等[63]用相场法模拟了ZrB2双晶系统在拉伸载荷下的晶间和穿晶裂纹扩展。研究了晶界错向角、相对于初始裂纹方向的晶界倾角及晶界强度对裂纹扩展路径的影响。
相比其他数值方法,相场法可以将多晶材料的晶间断裂和穿晶断裂纳入统一的损伤表征体系,能在晶体塑性的背景下模拟裂纹扩展[62],还可以和计算材料学中的相场模型结合,先后模拟晶粒的生长和断裂破坏过程[63]。
4. 总结和展望
本文就相场法模拟复合材料失效行为相关研究情况进行了系统的论述和总结,形成如下结论:
(1) 相场法结合断裂力学和损伤力学的特色,不需要预设初始裂纹和额外的损伤判据,便可以精确捕捉复合材料单层板的裂纹萌生和裂纹扩展行为;
(2) 相场法可以与内聚力模型结合有效地预测复合材料层合板的基体开裂、纤维断裂、纤维基体脱粘及分层等失效行为;
(3) 相场法控制方程的数值求解方法不限于有限元方法,任何有关偏微分方程的数值求解方法都可以与相场法结合,发挥各自的优势,模拟特定的问题;
(4) 对不同复合材料的失效行为进行相场模拟,需对相场断裂模拟进行修正,以适应真实材料本构和断裂性能的各向异性;
(5) 相场法在不同类型复合材料的强度评估上得到了广泛的应用,不仅能用来模拟复合材料宏观的断裂行为,还可以用来预测复合材料细观尺度上的损伤演化。
综合已有的研究,未来相场法模拟复合材料失效行为的研究可以从以下几个方面考虑:
(1) 建立复合材料的相场疲劳模型,并且考虑交变载荷对复合材料断裂参数的影响,给出相应的疲劳衰减函数;
(2) 建立湿热环境下复合材料的相场损伤模型,结合湿度场和温度场对材料弹性常数的退化作用,探讨相场变量对浓度扩散系数和热传导系数的衰减机制,建立湿度场、温度场、相场和位移场之间的耦合机制,研究复合材料在湿热疲劳下的损伤演化情况;
(3) 用显式相场法模拟复合材料层合板的冲击损伤,结合显式动力学和接触算法,对冲击载荷下的动态相场模型进行数值实现,开拓相场法的应用范围;
(4) 相场法需要对网格进行加密,同时需要求解相场这个自由度,因此求解效率相对较低,这成为制约相场法模拟复合材料工程结构件的重要因素。对复合材料结构件的破坏行为进行预测时,可以尝试将相场法和扩展有限元方法结合,对损伤较严重和裂纹扩展路径较复杂的区域采用相场法求解,其他区域则采用扩展有限元方法进行求解,进而提高计算效率。
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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 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. 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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