基于剪切非线性三维损伤本构模型的复合材料层合板失效强度预测

杨凤祥; 陈静芬; 陈善富; 刘志明

doi:10.13801/j.cnki.fhclxb.20200110.002

基于剪切非线性三维损伤本构模型的复合材料层合板失效强度预测

doi: 10.13801/j.cnki.fhclxb.20200110.002

暨南大学　力学与建筑工程学院　 “重大工程灾害与控制”教育部重点实验室，广州 510632

基金项目: 国家自然科学基金青年项目(11502095)；广东省自然科学基金博士启动项目(2015A030310306)；中央高校基本科研业务费专项资金(暨南大学科研培育与创新基金)(21615306)

详细信息

通讯作者:
陈静芬，博士，副研究员，硕士生导师，研究方向为复合材料及其结构静动力非线性分析　E-mail：tjingfen.chen@jnu.edu.cn

中图分类号: TB33
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出版历程
- 收稿日期: 2019-10-30
- 录用日期: 2020-01-02
- 网络出版日期: 2020-01-10
- 刊出日期: 2020-09-15

Failure strength prediction of composite laminates using 3D damage constitutive model with nonlinear shear effects

MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou 510632, China

摘要

摘要: 基于连续损伤力学，建立了同时考虑复合材料剪切非线性效应和损伤累积导致材料属性退化的三维损伤本构模型。模型能够区分纤维损伤、基体损伤和分层损伤不同的失效模式，并定义了相应损伤模式的损伤变量。复合材料层合板层内纤维初始损伤采用最大应力准则判定，基体初始损伤采用三维Puck准则中的基体失效准则判定，分层初始损伤采用三维Hou准则中的分层破坏准则判定，为了计算Puck失效理论中的基体失效断裂面角度，本文提出了分区抛物线法，通过Matlab软件编写计算程序并进行分析。结果表明，与Puck遍历法和分区黄金分割法对比，本文提出的分区抛物线法有效地降低了求解断裂面角度的计算次数，提高了计算效率和计算精度。推导了本构模型的应变驱动显式积分算法以更新应力和解答相关的状态变量，开发了包含数值积分算法的用户自定义子程序VUMAT，并嵌于有限元程序Abaqus v6.14中。通过对力学行为展现显著非线性效应的AS4碳纤维/3501-6环氧树脂复合材料层合板进行渐进失效分析，验证了本文提出的材料本构模型的有效性。结果显示，已提出的模型能够较准确地预测此类复合材料层合板的力学行为及其失效强度，为复合材料构件及其结构设计提供一种有效的分析方法。
- 复合材料本构模型 /
- 渐进失效分析 /
- 剪切非线性 /
- 三维Puck准则 /
- 分区抛物线法 /
- 刚度退化
Abstract: Based on the continuum damage mechanics, a 3D damage constitutive model which takes into account the nonlinear shear behavior of composites and material properties degradation due to damage development was proposed. The model differentiates between different failure modes, such as fiber failure mode, matrix failure mode and delamination. The damage variables corresponding to each failure mode were defined. The onsets of fiber damage, matrix damage and delamination of composite laminates were predicted using maximum stress failure criteria, Puck’s matrix failure criteria and Hou’s delamination criteria, respectively. In order to predict the angle of fracture surface in Puck’s matrix fracture failure theory, a selective parabola algorithm was proposed and coded using Matlab procedure. Compared with the Puck’s algorithm and the selective range golden section search algorithm, it shows that the selective parabola algorithm effectively reduces the number of calculations and improves the calculation efficiency and accuracy. A strain-driven explicit integration algorithm for the proposed material constitutive model was developed to update stresses and solution dependent state variables. The user-defined material subroutine VUMAT containing the numerical integration algorithm was coded and implemented in the finite element procedure Abaqus v6.14. The efficiency of the material constitutive model was demonstrated through progressive failure analyses of AS4 carbon fiber/3501-6 epoxy composite laminates, the mechanical behavior of which demonstrates significant nonlinear shear effects. The numerical results show that the proposed model is able to predict the mechanical behavior and failure strength of composites with sufficient accuracy. The proposed approach provides an efficient method for the design of composite components and structures.
- composite constitutive model /
- progressive failure analysis /
- nonlinear shear behavior /
- Puck’s 3D failure criteria /
- selective parabola algorithm /
- stiffness degradation

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图 1 层合板子层应力状态与潜在断裂面的定义

Figure 1. Definition of stress state and potential fracture surface for composite layer

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图 2 基体应力危险系数含局部极大值点的搜索区间

Figure 2. Search region with local maximum value of matrix stress exposure

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图 3 三种应力状态下应力危险系数随旋转角度的变化曲线及Case 3的应力和应力危险系数随旋转角度的变化曲线

Figure 3. Variation of matrix stress exposure with rotation angle in three stress states and variation of stresses and matrix stress exposure for case 3

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图 4 材料属性退化模式

Figure 4. Material property degradation modes

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图 5 剪切损伤演化方案

Figure 5. Shear damage evolution program

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图 6 AS4碳纤维/3501-6环氧树脂复合材料偏轴压缩试件的几何形状及尺寸

Figure 6. Geometry and dimensions of AS4 carbon fiber/3501-6 epoxy composite specimen under off-axis compression

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图 7 准静态荷载下不同偏轴角度AS4碳纤维/3501-6环氧树脂复合材料层合板的试验和预测应力-应变曲线

Figure 7. Experimental and predicted stress-strain curves of off-axis AS4 carbon fiber/3501-6 epoxy unidirectional composite laminates under quasi-static load

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图 8 含缺口AS4碳纤维/3501-6环氧树脂复合材料层合板的几何尺寸

Figure 8. Geometry of notched ASAS4 carbon fiber/3501-6 epoxy composite

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图 9 [0°/90°/±45°]_2S层合板达到破坏荷载时0°、45°和−45°单层的纤维断裂损伤变量${d_{\rm{f}}}$预测分布图(D=12.7 mm)

Figure 9. Predicted patterns of damage variables ${d_{\rm{f}}}$ in 0°, 45° and−45° plies of [0°/90°/±45°]_2S laminates at maximum load(D=12.7 mm)

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图 10 [0°/90°/±45°]_2S层合板达到破坏荷载时0°、90°、45°和−45°单层的基体断裂损伤变量${d_{\rm{m}}}$预测分布图(D=12.7 mm)

Figure 10. Predicted patterns of damage variables ${d_{\rm{m}}}$ in 0°, 90°, 45° and −45°plies of [0°/90°/±45°]_2S laminates at maximum load(D=12.7 mm)

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图 11 [0°/90°/±45°]_2S层合板达到破坏荷载时0°、90°、45°和−45°单层的分层损伤变量${d_{\rm{z}}}$预测分布图(D=12.7 mm)

Figure 11. Predicted patterns of damage variables ${d_{\rm{z}}}$ in 0°, 90°, 45° and −45° plies of [0°/90°/±45°]_2S laminates at maximum load(D=12.7 mm)

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图 12 [0°/90°/±45°]_2S层合板达到破坏荷载时0°/90°、90°/45°、45°/−45°层间粘结单元分层损伤预测分布图(D=12.7 mm)

Figure 12. Predicted patterns of delamination in 0°/90°, 90°/45°, 45°/ −45° interfaces of [0°/90°/±45°]_2S laminates at maximum load using cohesive element method (D=12.7 mm)

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图 13 [±45°]_4S层合板达到破坏荷载时，45°和−45°单层的基体断裂损伤变量${d_{\rm{m}}}$预测分布图(D=12.7 mm)

Figure 13. Predicted patterns of damage variables ${d_{\rm{m}}}$ in 45° and −45° plies of [±45°]_4S laminates at maximum load(D=12.7 mm)

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图 14 [0°/90°/±45°]_2S及[±45°]_4S带孔层合板拉伸应力-应变预测曲线

Figure 14. Predicted tensile stress-strain curves of [0°/90°/±45°]_2S and [±45°]_4S laminates with holes

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表 1 三种典型的三维应力状态及材料参数

Table 1. Typical 3D stress states and material properties

Material property		Stress state	${\sigma _2}$/ MPa	${\sigma _3}$/ MPa	${\tau _{12}}$/ MPa	${\tau _{13}}$/ MPa	${\tau _{23}}$/ MPa
$R_ {\bot} ^{\rm{t}}/{\rm{MPa}}$	25	Case 1	0	0	0	0	59.1
$R_ {\bot} ^{\rm{c}}/{\rm{MPa}}$	120	Case 2	−231.2	0	0	0	0
$R_{ \bot \parallel }^{\rm{A}}/{\rm{MPa}}$	70	Case 3	34.0	−87.0	22.0	46.0	29.0
Notes: $R_ {\bot} ^{\rm{t}}$, $R_ {\bot} ^{\rm{c}}$—Tensile strength and compressive strength in the transverse direction，respectively; $R_{ \bot \parallel }^{\rm{A}}$—Strength of the fracture plane against failure due to longitudinal shear stress; ${\sigma _2}$, ${\sigma _3}$—Normal stresses in the transverse and through-thickness directions, respectively; ${\tau _{12}}$, ${\tau _{13}}$, ${\tau _{23}}$—Shear stresses for the 1-2, 1-3, 2-3 plane, respectively.

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表 2 三种典型三维应力状态下断裂角计算精度和计算效率对比

Table 2. Comparison of accuracy and efficiency for fracture angle calculation under three typical 3D stress states

Stress state	Precision/(°)	Theoretical solution/(°)	Puck’s algorithm^[11]			SRGSS^[16]			Proposed method
Stress state	Precision/(°)	Theoretical solution/(°)	θ/(°)	N	T/s	θ/(°)	N	T/s	θ/(°)	N	T/s
Case 1	1	45.000	45.0	180	0.437	44.854	24	0.107	45.000	19	0.101
Case 1	0.1	45.000	45.0	1 800	4.160	44.985	29	0.148	45.000	19	0.101
Case 2	1	±51.137	51.0	180	0.428	51.245	31	0.109	51.044	21	0.104
Case 2	0.1	±51.137	51.1	1 800	4.162	51.145	41	0.151	51.044	21	0.104
Case 3	1	14.901	15.0	180	0.432	14.854	38	0.112	14.974	23	0.107
Case 3	0.1	14.901	14.9	1 800	4.187	14.916	53	0.156	14.974	23	0.107
Notes: θ—Fracture angle; N—Number of state points; T—Calculated time by different search methods; SRGSS—Selective range golden section search algorithm.

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表 3 AS4碳纤维/3501-6环氧树脂复合材料单向板材料属性

Table 3. Material properties of unidirectional AS4 carbon fiber/3501-6 epoxy composite laminate

$\begin{aligned}&{E_1}/{\rm{GPa}}\end{aligned}$	$\begin{aligned}&{E_2} = {E_3}/{\rm{GPa}}\end{aligned}$	$\begin{aligned}&{S_{{\rm{fc}}}}/{\rm{MPa}}\end{aligned}$	$\begin{aligned}&{S_{{\rm{ft}}}}/{\rm{MPa}}\end{aligned}$	$\begin{aligned}&{G_{12}} = {G_{13}}/{\rm{GPa}}\end{aligned}$	$\begin{aligned}&{G_{23}}/{\rm{GPa}}\end{aligned}$	$\begin{aligned}{\nu _{23}}\end{aligned}$
138	10.1	1690	1450	7.0	1.57	0.34
$\begin{aligned}{\nu _{12}} = {\nu _{13}}\end{aligned}$	$\begin{aligned}R_ {\bot} ^{\rm{t}}/{\rm{MPa}}\end{aligned}$	$\begin{aligned}R_ {\bot} ^{\rm{c}}/{\rm{MPa}}\end{aligned}$	$\begin{aligned}{S_{{\rm{zt}}}}/{\rm{MPa}}\end{aligned}$	$\begin{aligned}{S_{{\rm{zc}}}}/{\rm{MPa}}\end{aligned}$	$\begin{aligned}{S_{13}} = {S_{23}}/{\rm{MPa}}\end{aligned}$	$\begin{aligned}A\end{aligned}$
0.29	65	285	65	285	81	1.3
$\begin{aligned}R_{ \bot \parallel }^{\rm{A}}/{\rm{MPa}}\end{aligned}$	$\begin{aligned}{G_{{\rm{ft}}}}\end{aligned}$/(N·mm⁻¹)	$\begin{aligned}{G_{{\rm{fc}}}}\end{aligned}$/(N·mm⁻¹)	$\begin{aligned}{G_{ {\simfont\text{Ⅰ} }\!\!\!{\rm{C} } } }\end{aligned}$/(N·mm⁻¹)	$\begin{aligned}{G_{ {\simfont\text{Ⅱ} }\!\!{\rm{C} } } }\end{aligned}$/(N·mm⁻¹)	${G_{ {\simfont\text{Ⅲ} } {\rm{C} } } }$/(N·mm⁻¹)
81	220.6	220.6	0.22	0.65	0.65
Notes: $\begin{aligned}{E_1}\end{aligned}$, $\begin{aligned}{E_2}\end{aligned}$, $\begin{aligned}{E_3}\end{aligned}$—Elastic moduli in the fiber, transverse and through-thickness directions; $\begin{aligned}{S_{{\rm{fc}}}}\end{aligned}$, $\begin{aligned}{S_{{\rm{ft}}}}\end{aligned}$—Tensile strength and compressive strength in the fiber direction; $\begin{aligned}{S_{{\rm{zt}}}}\end{aligned}$, $\begin{aligned}{S_{{\rm{zc}}}}\end{aligned}$— Tensile and compressive strengths in the through-thickness directions; $\begin{aligned}{\nu _{ij}}\end{aligned}$, $\begin{aligned}{G_{ij}}\end{aligned}$(ij=12, 13, 23)—Poisson’s ratios and shear moduli for 1-2, 1-3, 2-3 plane, respectively; $\begin{aligned}{S_{13}}\end{aligned}$, $\begin{aligned}{S_{23}}\end{aligned}$—Shear strengths for 1-3 and 2-3 planes; $\begin{aligned}{G_{{\rm{ft}}}}\end{aligned}$ and $\begin{aligned}{G_{{\rm{fc}}}}\end{aligned}$—Tensile and compressive fracture energies in the fiber direction; $\begin{aligned}{G_{ {\simfont\text{Ⅰ} }\!\!\!{\rm{C} } } }\end{aligned}$, $\begin{aligned}{G_{ {\simfont\text{Ⅱ} }\!\!{\rm{C} } } }\end{aligned}$, $\begin{aligned}{G_{ {\simfont\text{Ⅲ} }{\rm{C} } } }\end{aligned}$—Fracture energies for mode Ⅰ, Ⅱ and Ⅲ in the through-thickness direction, respectively; A—Parameter in the Soutis’ formula related to the composite material.

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表 4 [0°/90°/±45°]_2S及[±45°]_4S带孔层合板拉伸破坏强度预测值与试验结果对比

Table 4. Comparison between predicted and experimental tensile failure strengths of [0°/90°/±45°]_2S and [±45°]_4S composite laminates

Lay-up	D/ mm	Test value/ MPa	Present model		Li model^[9]		Abaqus built-in model
Lay-up	D/ mm	Test value/ MPa	Predicted value/MPa	Error/ %	Predicted value/MPa	Error/ %	Predicted value/MPa	Error/ %
	6.35	420±25	416	−1.0	426	1.4	629	49.8
[0°/90°/±45°]_2S	12.70	340±13	338	−0.6	352	3.5	557	63.8
	25.40	262±3	253	−3.4	256	−2.3	458	74.8
	6.35	159±7	163	2.5	174	9.4	78	−50.9
[±45°]_4S	12.70	153±7	150	−2.0	148	−3.2	74	−51.6
	25.40	126±2	118	−6.3	130	3.2	63	−50.0

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