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46 卷 第 1力 学 学 报 Vol. 46No. 1
2014 1Chinese Journal of Theoretical and Applied Mechanics Jan.2014
研 究 论 文
复合材料层合板面内渐进损伤分析的 CDM 模型1)
吴义韬 姚卫,2) 吴富强
(飞行器先进设计技术国防重点学科实验室,南京航空航天大学,南京 210016)
(机械结构力学及控制国家重点实验室,南京航空航天大学,南京 210016)
摘要 基于连续介质损伤力学,提出了一个预测复合材料层合板面内渐进损伤分析的模型,它包括损伤表征、
损伤判定和损伤演化 3部分.模型能够区分纤维拉伸断裂、纤维压缩断裂、纤维间拉伸损伤和纤维间压缩损伤
4种损伤模式,定义了与 4个损伤模式对应的损伤状态变量,导出了材料主轴系下损伤前后材料本构之间的关
.损伤起始采用 Puck 准则判定,损伤演化由特征长度内应变能释放密度控制.假定材料服从线性应变软化行
为,建立了损伤状态变量关于断裂面上等效应变的渐进损伤演化法则.模型涵盖了复合材料面内损伤起始、
化直至最终失效的全过程.完成了含孔 [45/0/45/90]2S 层合板在拉伸和压缩载荷下失效分析,结果表明该模
型能合理进行层合板的强度预测和损伤失效分析.
关键词 复合材料层合板,材料本构,渐进损伤演化,Puck 准则
中图分类号:TB33 文献标识码:A doi10.6052/0459-1879-13-106
引 言
纤维增强复合材料因比强度高、比模量大、可设
计性强等优点在新一代客机上得到广泛应用 [1].
维增强复合材料的各向异性,叠层铺设灵活性,以及
使用过程中损伤模式多样性和损伤机理复杂性等特
点给复合材料应力分析和结构强度预测带来很大的
挑战.因而,在深入理解失效机理的基础上建立合理
的分析模型是复合材料失效研究和强度预测的关键.
复合材料层合板的渐进损伤失效分析是当前复
合材料强度研究的热点之一.分析方法大致可分为
断裂力学方法、连续介质损伤力学 (continuum dam-
age mechanics, CDM) 方法和唯象分析法.断裂力学
方法主要是通过研究纤维间裂纹增生机制来预测层
合板剩余刚度和剩余强度.该方法能考虑铺层顺序
和铺层厚度的影响,一般只适用于单一损伤模式、
简单叠层层合板失效分析 [2-4].象分析法基于应
-- 应变分析和失效判定,失效发生后根据失效模
式对材料属性进行直接折减 [5-9] 或建立折减系数关
于单层应力 -- 应变的函数关系 [10-12].该方法不受铺
层顺序的约束,但需要根据具体材料和结构形式选
取合适的折减系数,而这些系数的确定需要通过大
量的试验校核得出.近年来,连续介质损伤力学分析
方法在复合材料渐进损伤失效研究中被越来越多学
者所采用.该方法包括损伤表征、损伤判定和损伤
演化 3部分.损伤用损伤状态变量表征,文献 [13-
25] 中研究者都采用多个损伤状态变量对应复合材
料的不同损伤模式,通过损伤状态变量的引入建立
了损伤材料与完好材料本构之间的关系.损伤是否发
生用强度准则判定,最常见的失效准则是 Hashin
则,该准则能区分损伤模式,且考虑了材料主轴应力
之间的交互作用影响,但无法解释了横向压缩对剪
切破坏的抑制效应 [26]也无法确定真实材料损伤断
裂面方向 [10].损伤出现后其演化规律是研究难点,
目前研究者提出了不少损伤演化模型. Matzenmiller
[13] 认为损伤的演化与损伤断裂面上的有效应力
分量有关,提出损伤状态变量的指数型表达公式.
Maimi [14-15] 从材料的 Gibbs 自由能密度出发,
理论推导出含面内损伤的复合材料二维损伤本构,
损伤演化法则是热力学参量的函数. Lopes [16]
Maimi 的二维模型发展为三维模型,对复合材料层
合板低速冲击损伤进行了有效的数值模拟. Lapczyk
[17] Falzon [18-19] 基于应变能释放分析,分别
提出了材料损伤线性软化模型.其中,Lapczyk 的线
2013–04–03 收到第 1稿,2013–07–01 收到修改稿.
1) 国家自然科学基金 (11202098) 和江苏高校优势学科建设工程资助项目.
2) 姚卫星,教授,主要研究方向:飞行器综合设计和结构设计理论. E-mailwxyao@nuaa.edu.cn
1 吴义韬等:复合材料层合板面内渐进损伤分析的 CDM 模型 95
性软化模型中损伤的增加采用等效位移控制,Fal-
zon 的线性软化模型中损伤的增加采用应变控制.
Linde [20] 分别给出了对应纤维和基体损伤模式的
指数型演化法则,该演化法则中还引入了单元特征
长度,在一定程度上降低了损伤演化阶段网格的敏
感性.姚辽军等 [21-23] 基于 CDM 方法对复合材料插
层补强、补片补强和含开口层合板拉伸进行了强度
预测,损伤演化采用 Linde 的指数型演化法则.庞宝
君等 [24] 基于平面应力假定和不可逆热力学分析,
立了忽略塑性应变和考虑塑性应变的 2种面内剪切
连续损伤模型.模型中广义力的确定需结合宏观面
内剪切拉伸试验拟合得到.方国东 [25] 建立了与组分
材料断裂能、局部应力应变场以及单元尺度相关的
损伤演化模型,用于三维四向编织复合材料代表体
积单胞 (RVC) 模型的渐进损伤失效研究.
基于复合材料层合板的损伤机理分析,本文
认为:(1) 纤维间损伤断裂面一般与横向主轴应力
作用面不重合,即损伤主轴与材料主轴一般不重
合;(2) 损伤发生后直接造成损伤断裂面而非材料
主轴应力作用面的承载能力下降,材料主轴系下
损伤因子应由损伤主轴系下损伤因子经坐标转换
得到.据此分析,采用连续介质损伤力学方法开展
了复合材料的本构、损伤的起始和演化的研究,
立了一个新的渐进损伤分析模型.模型同时还通过
单元特征长度的引入考虑了网格的依赖性.编写本
文模型ABAQUS 用户子程序 UMAT完成了典
算例分析.
1本构模型
一般认为,复合材料面内损伤包含纤维断裂
(fiber fracture, FF)基体开裂和基纤剪切破坏 3种形
.基体开裂和基纤剪切破坏都发生在纤维间,
不容易区分,统称为纤维间损伤 (inter fiber fracture,
IFF). FF 损伤断裂面为纵向主应力作用面,IFF
伤断裂面与横向主轴应力作用面一般不重合,夹角
IFF 损伤断裂角 [27]. IFF 损伤断裂面上法向应力为
拉伸时促进纤维间裂纹开裂,法向应力为压缩时抑
制纤维间剪切滑移.区分断裂面上法向载荷拉压形
式,复合材料面内损伤可细分为纤维拉伸断裂 (fiber
tension fracture, FFT)纤维压缩断裂 (fiber compression
fracture, FFC)以及纤维间拉伸损伤 (inter fiber tension
fracture, IFFT)纤维间压缩损伤 (inter fiber compres-
sion fracture, IFFC) 4 种模式.
1.1 应力分析
Mohr--Coulomb 断裂理论认为:材料断裂与否取
决于断裂面上的应力.于层合板中的某一子层,
空间应力状态如图 1所示,图中 (x1,x2,x3)为材料主
轴系,(x1,xn,xt)为损伤主轴系 (2). 垂直于 x1
面为 FF 损伤断裂面,垂直于 xn的面为 IFF 损伤断
裂面.材料主轴系和损伤主轴系下,材料本构关系分
别表示为
σ=C0ε(1)
σ0=C0ε0(2)
式中,σ=[σ1σ2σ3σ23 σ31 σ12], ε=
[ε1ε2ε3ε23 ε31 ε12], C0为材料主轴系下应力、
应变及刚度矩阵;σ0=[σ1σnσtσnt σt1 σ1n]
ε0=[ε1εnεtεnt εt1 ε1n]C0为损伤主轴系下应力、
应变及刚度矩阵.
损伤主轴系下应力、应变可由材料主轴系下应
力、应变通过坐标转换得到,
σ0=T1σ(3)
ε0=TTε(4)
式中,T为坐标转换矩阵.
1子层应力状态和断裂角
Fig. 1 Intralaminar stress and fracture angle
将式 (3) 和式 (4) 代入式 (2)结合式 (1)则材料
刚度矩阵、柔度矩阵在损伤主轴系与材料主轴系之
间相互表出如下
C0=T1C0TT(5)
S0=TTS0T(6)
1.2 本构模型
对于正交各向异性材料,出现 IFF 损伤后,引入
损伤因子张量,建立损伤主轴系下有效应力 ˆ
σ0与表
观应力 σ0之间关系
ˆ
σ0=M0(D)σ0(7)
96 力 学 学 报 2014 年 第 46
式中,M0(D)为损伤因子张量,其损伤主轴系下矩阵
形式可表示为 [28]
M0(D)=
1
1d10 0 0 0 0
01
1dn0 0 0 0
0 0 1
1dt0 0 0
0 0 0 1
1dnt 0 0
0 0 0 0 1
1dt1 0
0 0 0 0 0 1
1d1n
(8)
式中,di(i=1, n, t) 分别表征材料点在损伤主轴
系下 x1轴向、xn轴向和 xt轴向损伤状态变量.
中,x1FF 损伤主轴,xn轴为 IFF 损伤主轴,
d1,dn分别对应 FF 损伤模式和 IFF 损伤模式.xt
轴垂直于 x1轴和 xn轴,即始终垂直于 FF
断裂面和 IFF 损伤断裂面,因而,xt轴垂直的
面上不会出现 FF 损伤和 IFF 损伤dt恒为 0.
dij =1(1 di)(1 dj) (i,j=1, n, t)[28]
剪切模量 Gnt,Gt1,G1n 的损伤状态变量.考虑到载
荷反向刚度回[18]d1=max hσ1i
σ1dFFT,dFFC!
dn=max hσni
σndIFFT,dIFFC!.h·i Macauley 符号,
义为 xR时,hxi=(x+|x|)/2.
损伤主轴系下,材料损伤本构用表观应力 σ0
有效应力 ˆ
σ0分别表示为
ε0=S0dσ0(9)
ε0=S0ˆ
σ0(10)
将式 (7) 代入式 (10)结合式 (9)Lemaitre
变等效性假设可得
S0d=M0(D)S0(11)
上式中为了保证损伤主轴系下柔度矩阵的对称
性,泊松比 νij 退化方式与弹性模量 Ei相同,即损伤
前后 νij
Ei=νji
Ej(i,j=1, n, t) 依然满足.
将式 (6) 代入式 (11) 中,可推导出材料坐标系下
损伤柔度矩阵
Sd=TTM0(D)TTS0(12)
由上式,材料坐标系下损伤张量可定义如下
M(D)=TTM0(D)TT(13)
(13) 中,IFF 损伤 dn,0IFF 损伤断裂
θ,0时,M(D)为非对角矩阵,所以式 (12) Sd
不对称.为避免有限元计算中柔度矩阵奇异,作对称
化处理,Sd=Sd+SdT
2.
IFF 损伤出现前材料一般都有剪切非线性行
为,引入 Hashin Tsai [29] 提出的剪切非线性表达
(14)将材料柔度矩阵 S0中纵向剪切模量用其
割线模量形式代替,有式 (15). IFF 损伤出现后,
纵向剪切模量的退化建立在损伤起始发生前的割线
剪切模量上进行.
ε12 =1
G12 σ12 +ασ3
12 (14)
C66 =1
1/G12 +3ασ2
12
C55 =1
1/G13 +3ασ2
13
C44 =1
1/G23 +3ασ2
23
(15)
式中,α为材料剪切非线性系数.
(8) 获得了损伤主轴系下损伤因子张量,代入
(13) 中可计算材料主轴系下损伤因子张量,再通
过式 (12) 可建立材料主轴系下损伤前后本构之间的
关系.(8) 中损伤状态变量由后面的式 (21) 确定.
2损伤判据
考虑横向泊松效应影响,且区分纵向拉伸和压
缩,Knops 建立以损伤参量 fE表征的 FFT 损伤准则
FFC 损伤准则 [30] 如下
fE,FFT =1
RT
k"ˆσ1 ν12 ν12fmσfE1
E1f !(ˆ
σ2+ˆσ3)#
ˆσ1>0
(16)
fE,FFC =1
Rc
k"ˆσ1 ν12 ν12fmσfE1
E1f !(ˆ
σ2+ˆσ3)#
ˆσ1<0
(17)
式中,fE为当前应力水平下材料危险程度度量的指
标,其物理意义当前有效应力与对应断裂包络面上
破坏应力的比值.fE介于 0 之间,小于 1表示
无损伤发生,大于等于 1表示出现损伤.RT
kRc
k
别为单向板纵向拉伸和压缩强度;E1E1f 分别为单
1 吴义韬等:复合材料层合板面内渐进损伤分析的 CDM 模型 97
向板纵向拉伸模量和纤维拉伸模量;ν12ν12f 分别为
单向板面内主泊松比和纤维材料泊松比;mσf
大因子,与纤维和基体弹性模量比有关,GFRP
mσf1.3CFRPmσf1.1[31].
Puck IFF 作用面断裂假设认为 [30]IFF 作用
面上法向拉伸时,法向应力和面上剪应力共同作用
促进 IFF 损伤发生;法向压缩时,法向应力抑制 IFF
损伤发生.据此,作者建立了以 fE表征的 IFFT 损伤
准则和 IFFC 损伤准则 [10,30] 如下
fE,IFFT (θ)=pT
ψ
RA
ψ
σn(θ)+
v
u
t 1
RT
pT
ψ
RA
ψ!σn(θ)
2
+ σnt (θ)
RA
⊥⊥ !2
+ σn1 (θ)
R⊥k !2(18)
σn>0
fE,IFFC (θ)=pc
ψ
RA
ψ
σn(θ)+
v
t σnt (θ)
RA
⊥⊥ !2
+ σn1 (θ)
R⊥k !2
+ pc
ψ
RA
ψ
σn(θ)!2(19)
σn<0
式中,σn(θ)σnt(θ)σn1 (θ)由式 (2) 结合式 (4) 和式
(5) 计算得到
pT
ψ
RA
ψ
=pT
⊥⊥
RA
⊥⊥
cos2ψ+pT
⊥k
R⊥k
sin2ψ
pc
ψ
RA
ψ
=pc
⊥⊥
RA
⊥⊥
cos2ψ+pc
⊥k
R⊥k
sin2ψ
RA
⊥⊥ =Rc
21+pc
⊥⊥
sin2ψ=σ2
n1
σ2
n1 +σ2
nt
,cos2ψ=σ2
nt
σ2
n1 +σ2
nt
其中,RT
,Rc
R⊥k 分别为单向板横向拉伸强度、
横向压缩强度和纵向剪切强度;pT
⊥⊥,pc
⊥⊥ 为横向斜
率参数,pT
⊥k,pc
⊥k 为纵向斜率参数 (见表 1)Tc
别对应拉伸和压缩状态.pT
⊥⊥ pT
⊥k 表征法向拉伸对
IFF 损伤的促进效应,pc
⊥⊥ pc
⊥k 表征法向压缩IFF
损伤的抑制效应.
1典型 FRP 材料斜率参数 [27]
Table 1 Inclination parameters for typical FRP[27]
ϕ=60% pT
⊥k[-] pc
⊥k[-] pT
⊥⊥[-] pc
⊥⊥[-]
GFRP/Epoxy 0.30 0.25 0.20 0.25 0.20 0.25
CFRP/Epoxy 0.35 0.30 0.25 0.30 0.25 0.30
文献 [30] 中提出了一种确定材料 IFF 损伤断裂
面的求解方法.首先将子层绕纵轴 x1180划分
(2)得到 180 IFF 作用面.然后由式 (2) 可计算
得到某一加载时刻各个作用面上的应力 σn(θ)σn1(θ)
σnt(θ)再由式 (18) 和式 (19) 计算各个作用面上的
fE(θ). 比较 180 次计算所得的 fE(θ)值,确定最大的
fE值,此时对应的面即为当前加载时刻该材料点
”IFF 断裂面,对应的夹角 θ即为当前加载时刻该
材料点 潜在”IFF 断裂角.当载荷增加到某一时刻
时,潜在断裂面上 fE(θ)>1时,表明出现 IFF
伤,此时 潜在”IFF 断裂面为该材料点IFF 损伤断
裂面,潜在”IFF 断裂角为该材料点的 IFF 损伤断裂
θfp.
2 IFF 断裂面求解
Fig. 2 Search for the IFF fracture plane
3损伤演化
损伤演化过程实际上是应变能释放的过程.
材料点应变能释放密度 (面积 SOAC)等于其断裂能
密度 (面积 SOAB)时,意味着该材料点完全失效.
应变能释放过程中,材料会软化,宏观表现为弹性
模量的退化和承载能力的下降.常见分析中,材料软
化形式有线性 [17-19,25] 和指数 [20-23] 形式,本文模型
中,假定损伤起始发生后材料满足线性等效应变--
化行为,如图 3所示.3A点对应材料损伤起始
点;B点对应材料完全失效点;C点对应损伤过程中
当前时刻.
3.1 网格依赖性
在有限元模拟材料失效过程中,若尺度不同的
单元材料遵循相同的本构关系,则损伤演化过程
中能量释放率正比于单元尺度,而不是断裂面的面
[17].采用裂纹带模型,引入单元特征长度 l和修
改单元材料本构,可以降低有限元分析过程中能量
98 力 学 学 报 2014 年 第 46
3材料本构关系
Fig. 3 Material constitutive relation
释放率对网格的依赖性 [17-18].若材料服从线性应
--软化行为 (3)修改后的单元材料本构中失效
应变 εf不再维持不变,定义为
εf=2Gc
Rl=2gc
R(20)
式中,Gc为断裂韧度,gc为弥散分布在特征长度内
的断裂能密度.
值得注意的是,单元材料本构中失效应变按式
(20) 定义时,损伤造成的总应变能释放与网格粗细
无关,但损伤的区域依然依赖于网格尺度.因而,
元特征长度的引入在一定程度上降低了材料软化阶
段对网格的敏感性.
3.2 损伤状态变量
如前面所述,起始损伤发生后,伴随应变能释
放,料出现软化行为.假定材料服从线性等效应
-- 软化行为,从开始加载到材料完全失效的整个
过程中,建立损伤状态变量关于等效应变的损伤演
化法则如下
dI(τ)=max
τ
0,min
d,
εf
eq,Iεeq,I(¯τ)ε0
eq,I
εeq,I(¯τ)εf
eq,Iε0
eq,I
ε0
eq,I6εeq,I6εf
eq,I,I(FFT,FFC,IFFT,IFFC)
(21)
式中,¯τ6τdI(τ)表示为有限元仿真中某 τ时刻
4种损伤模式相对应的损伤状态状量,其值表征
的是材料点可能出现的 4种面内损伤模式的损伤程
.采用 max 函数表示 dI表明损伤演化不可逆.εeq,I
为当前等效应变,ε0
eq,Iεf
eq,I分别为起始损伤等效
应变和完全失效等效应变.损伤起始发生前 dI=0
表明材料完好;完全失效时 dI=d理论上 d=1
为避免有限元计算中刚度矩阵奇异,本文取 0.99.
区分损伤模式,(21) ε0
eq,Iεeq,Iεf
eq,I的计
算分述如下.
(1) εeq ε0
eq 的计算
如前面所分析,材料损伤与否是基于损伤断裂
面上的应力分析和判定的.因而,材料出现初始
损伤前,εeq 损伤断裂面上应变的组合形
式,材料出现初始损伤后,εeq 是损伤断裂面上应变
的组合形式.对于 FF 损伤模式,断裂面上应力有
σ1, τ13, τ12其中,τ13, τ12 很小且不足以剪断纤维造
FF 损伤,εeq 可直接由应变 ε1表达.IFF
损伤模式,损伤断裂面上应力有 σn, τnt, τn1εeq
由应变 εn, εnt, εn1 的组合形式来表达.
基于以上分析,区分 FF IFF 损伤模式,当前
等效应变可表达为
εeq,I=qε2
1,I,I(FFT,FFC) (22)
εeq,I=qhεn,Ii2+(εnt,I)2+(εn1,I)2,I(IFFT,IFFC)
(23)
起始损伤应变 ε0
I由起始损伤判据确定,对应式
(16)(19) fE,I=1时的应变值,可由当前应变
εI经缩放得到,即有
ε0
I=εI/fE,I,I(FFT,FFC,IFFT,IFFC) (24)
将式 (24) 代入式 (22) 和式 (23) 中,则对应不同
损伤模式的起始损伤等效应变分别为
ε0
eq,I=ε0
1,I=εeq,I/fE,I,I(FFT,FFC) (25)
ε0
eq,I=qhε0
n,Ii2+(ε0
nt,I)2+(ε0
n1,I)2=εeq,I/fE,I
I(IFFT,IFFC) (26)
(2) εf
eq 的计算
εf
eq,I的大小能反映损伤演化速率,由基于应变
能释放密度的损伤演化准则确定.当特征长度内应
变能释放密度等于断裂能密度时,材料完全失效,
此时损伤断裂面上的等效应变即为 εf
eq,I.由于纤维固
有弹脆性,FF 损伤发生后能量瞬间释放,εf
eq 非常
接近 ε0
eq.因而 FF 损伤发生后,dI直接由 0增加到
d.
IFF 损伤发生后,建立能量混合模式损伤演化判
gn
Gk
2c/l!ζ
+ gn1
G12c/l!ζ
+ gnt
G23c/l!ζ
=1 (27)
1 吴义韬等:复合材料层合板面内渐进损伤分析的 CDM 模型 99
式中,gn,gn1,gnt 分别为 IFF 损伤演化时与 σn, σn1, σnt
对应的应变能释放密度;Gk
2c,G12c,G23c 分别为横向
拉伸 (压缩)面内剪切和横向剪切临界能量释放率.
k=T,c表示拉伸和压缩.ζ为材料参数,本文选值
2.
IFF 损伤完全失效时,
gf
j=Zεf
j
0σjdεj1
2σ0
jεf
j=1
2σ0
jβjεf
eq ,
j=n,n1,nt (28)
式中,σ0
jIFF 起始损伤时断裂面上各应力分量,
将式 (24) 代入式 (2) 计算得到.βj为模式混合率,
βn=hεni
εeq βnt =εnt
εeq βn1 =εn1
εeq .
将式 (28) 代入式 (27) 中,IFF 损伤发生后的
完全失效应变 εf
eq,I
εf
eq,I=2
l σ0
n,Iβn
Gk
2c !ζ
+ σ0
n1,Iβn1
G12c !ζ
+ σ0
nt,Iβnt
G23c !ζ
1
,
I(IFFT,IFFC) (29)
4算例
4.1 分析对象
选用文献 [32] 中含孔试验件为分析对象.试验
件为 [45/0/45/90]2S 准各向同性层合板,单层厚
度为 0.13mm. 中心圆孔孔径 D分别为:0 (不含孔),
2.00, 3.81, 6.35 9.55见图 4. 材料为 AS4/3501-
6工程弹性常数和强度列于表 2其他弹性常数按
E3=E2,G13 =G12, ν13 =ν12 给出.由于断裂模式相
同,横向拉伸临界能量释放率 GT
2c 和面内剪切临界
能量释放率 G12c 可用实测 I型断裂韧度 GT
Ic II
断裂韧度 GIIc 代替.横向剪切临界能量释放率 G23c
在本文中取值与 G12c 相同.能量参数列于表 3.
4试验件几何尺寸 (mm)
Fig. 4 Dimension of the specimen (mm)
2材料 AS4/3501-6 工程弹性常数和强度 [32-33]
Table 2 AS4/3501-6 material engineering properties and strength [32-33]
E1E2G12 G23 E1f ν12 ν23 ν12f
145.5GPa 10.3GPa 7.2GPa 1.57GPa 225GPa 0.27 0.34 0.20
XT/RT
kXc/Rc
kYT/RT
Yc/RT
S/R⊥k α
2280 MPa 1 507MPa 57 MPa 228 MPa 71MPa 5.33×108
3AS4/3501-6 断裂韧度 [33-34]
Table 3 AS4/3501-6 material fracture
toughness[33-34]
GT
2c G12c,G23c
0.22N/mm 0.65 N/mm
4.2 有限元模型
本文模型的有限元分析基于商用有限元软件
ABAQUS/Standard 模块,通过自编用户子程序 UMAT
实现. UMAT 子程序实现了单元积分点应力分析、
伤判定、损伤状态变量计算、损伤本构建立,以及损
伤状态信息的反馈.建立 3D 有限元模型,采用 8
点六面体线性非协调模式单元 C3D8I 划分网格.
降低计算规模同时兼顾层间效应模拟,每一层厚度
方向划分一个单元.由于孔边周围应力集中,且是
损伤起始发生的地方,因此对孔边周围进行局部网
格细化,单层 1/4孔边网格数为 20. 利用结构的对
称性,有限元分析模型仅取 1/8试验件结构.分别在
3个对称面上施加 1个位移约束和 2个转动约束,
位移约束限制面外方向位移运动,转动约束限制面
外转动.加载由位移控制,施加在自由端面外参考点
上,建立参考点和自由端面之间加载方向位移一致
约束.见图 5.
5有限元分析模型
Fig. 5 FEM model of 1/8 specimen
4.3 结果分析
6和图 7分别给出了拉伸和压缩下载荷 --
移曲线.从图中可以看出,孔径越大,层合板拉伸和
100 力 学 学 报 2014 年 第 46
6层合板拉伸载荷--位移曲线
Fig.6 Predicted laminates load--displacement curve for tensile loading
7层合板压缩载荷--位移曲线
Fig. 7 Predicted laminates load--displacement curve for
compressive loading
压缩承载能力越低.不同孔径层合板的载荷--位移曲
线在损伤出现之前为线性且一致,损伤出现后出现
非线性.各层合板在断裂前载--位移曲线变化
趋势一致,表明本文中孔径不同的同一叠层层合板
损伤演化模式基本相同.
4给出了在拉伸和压缩载荷下净截面破坏应
力值,可以看到预测结果与试验值十分接近,拉伸
载荷下最大误差为 5.31%压缩载荷下最大误差为
6.53%说明本文模型对不同孔径和不同载荷形式
的复合材料层合板的极限强度均有较好预测能力.
5给出了不同网格尺寸下有限元预测的净截
面破坏应力值.作为对比,1/4孔边分别布置了 20
网格和 40 个网格,前者模型网格划分如4后者模
型网格尺寸为前者的 1/2. 从表中数据看出,两种网
格尺寸下预测结果非常相近 (差别大约 1% 左右)
说明在本文的损伤演化模型中通过单元特征长度的
引入在一定程度上降低了计算结果对网格的依赖性.
相比拉伸载荷,压缩载荷下有限元计算结果对网格
依赖性要大些,这与压缩载荷下材料损伤演化过程
不如拉伸载荷下平缓有关.
4试件净截面破坏应力预测结果与对比 [32]
Table 4 Comparison of the numerical and experimental net-section failure stress of specimens[32]
D/mm Tension Compression
numerical/experimental/error/numerical/experimental/error/
MPa MPa % MPa MPa %
0 714.8 702.9 1.70 621.6 665.0 6.53
2.00 528.4 558.0 5.31 440.8 456.8 3.50
3.81 481.2 494.9 2.77 424.8 427.6 0.66
6.35 455.3 472.4 3.62 417.3 398.2 4.80
9.55 440.7 447.9 1.61 408.9 389.0 5.12
5不同网格尺度下净截面破坏应力预测值
Table 5 Predicted net-section failure stress for dierent
meshed scale
D/mm Tension/MPa Compression/MPa
n=20 n=40 n=20 n=40
2.00 528.4 530.1 440.8 433.2
3.81 481.2 484.2 424.8 419.7
6.35 455.3 457.3 417.3 414.1
9.55 440.7 444.5 408.9 405.4
4.4 损伤演化分析
由于本文分析的含孔层合板的圆孔尺寸对其损
伤演化模式没有明显影响,故以孔径 D=6.35 mm
层合板为例分析损伤演化过程.8和图 9模拟了
单向拉伸和单向压缩载荷下各子层损伤演化.拉伸
载荷下层合板损伤模式为 FFT IFFT压缩载荷下
损伤模式为 FFC IFFC. 从图中可知,拉伸和压缩
下,只有 0层中分别出现 FFT 损伤和 FFC 损伤,
他层不出现;IFFT 损伤在拉伸下非 0中都出现,
90层中最为严重;IFFC 损伤在压缩下非 0层中
都出现,而各层中损伤程度大体相当.
1 吴义韬等:复合材料层合板面内渐进损伤分析的 CDM 模型 101
8拉伸载荷下子层损伤演化过程
Fig.8 Intralaminar damage evolution process under tensile loading
单向拉伸载荷下,由于基体拉伸强度较弱和孔
边应力集中,8中可以看出,48.1% 峰值载荷时
90层中已出现 IFFT 损伤.90IFFT 诱发,
0层中出现了少量 IFFT 损伤.受孔边应力集中
影响,78.6% 峰值载荷时,FFT 损伤后于 IFFT 损伤
0层开始出现,此时 90层中自由侧边处已出现
102 力 学 学 报 2014 年 第 46
IFFT 损伤.92.4% 峰值载荷前,FFT 损伤在孔
边区域渐进演化,90层中 IFFT 损伤已经弥散覆
盖了整个板宽.之后 FFT 损伤在 0层内快速演化.
FFT 损伤加速了孔边横截面处应变局部化,100%
峰值载荷前,孔边区域非 0层基本失效.最终由于
0层内孔边区域大面积的 FFT 损伤导致了层合板的
整体断裂.
单向压缩载荷下,损伤演化过程不同于单向拉
伸载荷.9中可以看出,0层内孔边区域出现
纤维压缩断裂损伤 (FFC) 之前,由于基体抗压缩性
能较强,0层没有 IFFC 损伤出现.93.3%
值载荷前 0层内 FFC 损伤在孔边区域缓慢扩展,
0层内 IFFC 损伤受 0层内 FFC 损伤影响扩展也
缓慢.之后 0层内 FFC 损伤向自由侧边迅速扩展,
100% 峰值载荷时,FFC 损伤已扩展到孔边整个板
宽,导致整个层合板整体倒塌失效.
9压缩载荷下子层损伤演化过程
Fig. 9 Intralaminar damage evolution process under compressive loading
以上研究表明载荷形式不同,[45/0–45/90]2S
合板损伤演化规律不同.单向拉伸下,首先在 90
出现 IFFT 损伤,之后诱发 ±45层中 IFFT 损伤和
0层中 FFT 损伤,最终因 0层内孔边区域大面积
FFT 损伤导致了层合板整体拉伸破坏.单向压缩
下,首先在 0层内出现 FFC 损伤,之后诱发其他层
1 吴义韬等:复合材料层合板面内渐进损伤分析的 CDM 模型 103
IFFC 损伤出现,最终因 0层内整个板宽的 FFC
损伤导致了层合板整体倒塌破坏.
5结 论
本文建立了一个预测复合材料层合板面内渐进
损伤的分析模型.
(1) 考虑到材料主轴一般与损伤主轴不重合,
于连续介质损伤力学,推导了材料主轴坐标系下损
伤前后材料本构之间的联系;
(2) 损伤演化由特征长度内应变能释放密度控
制,建立了损伤状态变量关于断裂面上等效应变的
渐进损伤演化法则;
(3) 通过单元特征长度的引入在一定程度上解决
了材料损伤阶段能量释放对网格的依赖性;
(4) 对材料为 AS4/3501-6 的中心含孔和不含孔
[45/0–45/90]2S 层合板拉伸和压缩失效分析,结果
表明本文模型预测结果与实验值吻合较好,拉伸载
荷下预测强度的最大误差为 5.31%压缩载荷下为
最大误差为 6.53%.
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(责任编辑:周冬冬)
CDM MODEL FOR INTRALAMINAR PROGRESSIVE DAMAGE ANALYSIS OF
COMPOSITE LAMINATES 1)
Wu YitaoYao Weixing,2) Wu Fuqiang
(Key Laboratory of Fundamental Science for National Defense-Advanced Design Technology of Flight VehicleNanjing University of Aeronautics
and AstronauticsNanjing 210016China)
(State Key Laboratory of Mechanics and Control of Mechanical StructuresNanjing University of Aeronautics and Astronautics
Nanjing 210016China)
Abstract Based on continuum damage mechanics (CDM), a model was proposed for predicting intralaminar progressive
damage of composite laminates, including damage description, damage model judgment and damage evolution. Four
dierent damage modes existent within lamina, namely, fiber tension fracture (FFT), fiber compression fracture (FFC),
inter fiber tension fracture (IFFT) and inter fiber compression fracture (IFFC), were considered, and damage state variables
corresponding to these four modes were also defined. Material constitutive relationship being in the damaged states in the
material principal axes was derived compared to that of undamaged states. The onset of the damage was estimated with
Puck’s criteria and the evolution of the damage was controlled by the strain energy release density within the characteristic
length. Assuming that the material was exhibited linear strain-softening behavior, a new damage evolution law associating
damage state variables with equivalent strain on fracture plane was established in this paper. The proposed model can
predict damage initiation, damage evolution and final failure of composite laminates. Failure analyses of [45/0/–45/90]2S
notched laminates under tension and compression were performed with the present model and showed that it is capable
to predict the strength of the composite laminate and analyze the failure process.
Key words composite laminate, material constitutive relation, progressive damage evolution, puck criteria
Received 3 April 2013, revised 1 July 2013.
1) The project was supported by the National Natural Science Foundation of China(11202098) and the Priority Academic Program Development of
Jiangsu Higher Education Institutions.
2) Yao Weixing, professor, research interests: aircraft integrated design and structural design theory. E-mail: wxyao@nuaa.edu.cn
... The board edge strain distribution of longitudinal plates on X-axis in different groups. [36]. 574 ...
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