力学学报, 2020, 52(6): 1690-1699 DOI: 10.6052/0459-1879-20-165

固体力学

低速冲击激励下嵌入黏弹性阻尼芯层的纤维金属混杂层合板动态响应预测模型1)

李则霖*,, 李晖,*,,**,2), 王东升*,, 任朝晖*,, 祖旭东††,***, 周晋††,†††, 官忠伟††, 王相平**

*东北大学机械工程与自动化学院,沈阳 110819

东北大学航空动力装备振动及控制教育部重点实验室,沈阳 110819

**中国航发沈阳发动机研究所,沈阳 110015

††利物浦大学工程学院,英国利物浦 L693GQ

***南京理工大学机械工程学院,南京 210094

†††西安交通大学机械工程学院,西安 710054

A DYNAMIC RESPONSE PREDICTION MODEL OF FIBER-METAL HYBRID LAMINATED PLATES EMBEDDED WITH VISCOELASTIC DAMPING CORE UNDER LOW-VELOCITY IMPACT EXCITATION 1)

Li Zelin*,, Li Hui,*,,**,2), Wang Dongsheng*,, Ren Chaohui*,, Zu Xudong††,***, Zhou Jin††,†††, Guan Zhongwei††, Wang Xiangping**

*School of Mechanical Engineering & Automation,Northeastern University,Shenyang 110819,China

Key Laboratory of Vibration and Control of Aeronautical Power Equipment of the Ministry of Education, Northeastern University,Shenyang 110819, China

**AECC Shenyang Engine Research Institute,Shenyang 110015, China

††School of Engineering,University of Liverpool,Liverpool L693GQ,UK

***School of Mechanical Engineering,Nanjing University of Science and Technology,Nanjing 210094,China

†††School of Mechanical Engineering,Xi'an Jiaotong University,Xi'an 710054,China

通讯作者: 2) 李晖,副教授,研究方向:复合材料结构动力学. E-mail:lh200300206@163.com

收稿日期: 2020-05-16   接受日期: 2020-08-4   网络出版日期: 2020-11-18

基金资助: 1) 国家自然科学基金.  51505070
国家自然科学基金.  51970530
国家自然科学基金.  U1708257
中央高校基本科研业务费专项资金.  N160313002
中央高校基本科研业务费专项资金.  N160312001
中央高校基本科研业务费专项资金.  N170302001
中央高校基本科研业务费专项资金.  N180302004
中央高校基本科研业务费专项资金.  N180703018
中央高校基本科研业务费专项资金.  N180312012
中央高校基本科研业务费专项资金.  N180313006
装备预研重点实验室基金.  6142905192512

Received: 2020-05-16   Accepted: 2020-08-4   Online: 2020-11-18

作者简介 About authors

摘要

本文首次从解析角度建立了低速冲击激励下嵌入黏弹性阻尼芯层的纤维金属混杂层合板动态响应预测模型. 首先,结合经典层合板理论和冯$\cdot$卡门假设,建立了嵌入黏弹性芯层的纤维金属混杂层合板弹性损伤本构关系. 然后,将层合板受冲击时的变形分成接触和拉伸两个区域,在接触区域内,对金属层采用 Von Mises 失效准则,纤维层采用 Tsai-Hill 失效准则和对黏弹性层采用指数 Drucker-Prager 失效准则判断层合板损伤情况. 考虑不同材料层对冲击动态响应的贡献来修正两个变形区域的位移公式,进而计算结构因弹性变形产生的应变能,以及接触区域因塑性变形消耗的能量,实现每次失效事件发生后各层材料的能量、位移和冲击接触力的理论求解,并给出了结构动态响应分析的具体流程图. 最后,以嵌入 Zn33 黏弹性芯层的 TA2 钛合金混杂 T300 碳纤维/树脂层合板为研究对象,开展落锤冲击实验. 验证结果表明,理论预测与测试获得的冲击接触力、位移响应以及冲击载荷-位移曲线吻合较好,且关注的峰值点计算误差最大不超过 9%,进而验证了所提出的理论模型的有效性.

关键词: 低速冲击 ; 黏弹性阻尼芯层 ; 纤维金属层合板 ; 动态响应 ; 失效准则

Abstract

A dynamic response prediction model of fiber-metal hybrid laminated plates embedded with a viscoelastic damping core under low-velocity impact excitation is established analytically for the first time in this research. Firstly, based on the classical laminates theory and von Kármán theory, the constitutive relation of elastic damage of fiber-metal hybrid laminated plates embedded with a viscoelastic damping core is established. Then, the deformation of laminated plates under impact is divided into contact and stretching areas. Within the contact areas, Von Mises failure criteria are used for metal layers, Tsay-Hill failure criteria for fiber layers and Drucker-Prager failure criteria for viscoelastic layer to determine the damage of laminated plates. Considering the contribution of different material layers to the dynamic response subjected to the impact load for modifying the displacement formula, the theoretical solutions of energy, displacement and impact contact force in each layer of such laminated plates are obtained after each failure event occurs, and gives the flow chart of structure dynamic response analysis of concrete. Finally, a TA2 titanium alloy and T300 fiber/epoxy hybrid plate embedded with the Zn33 viscoelastic core is taken as the research object to carry out the drop-weight impact test. The theoretical prediction results of the impact contact force, displacement response, and impact load-displacement curve are found to agree well with the measured ones. Besides, the maximum calculation errors of the concerned peaks are less than 9%. Thus, the effectiveness of the proposed theoretical model has been verified.

Keywords: low-velocity impact ; viscoelastic damping core ; fiber-metal laminates ; dynamic response ; failure criteria

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本文引用格式

李则霖, 李晖, 王东升, 任朝晖, 祖旭东, 周晋, 官忠伟, 王相平. 低速冲击激励下嵌入黏弹性阻尼芯层的纤维金属混杂层合板动态响应预测模型1). 力学学报[J], 2020, 52(6): 1690-1699 DOI:10.6052/0459-1879-20-165

Li Zelin, Li Hui, Wang Dongsheng, Ren Chaohui, Zu Xudong, Zhou Jin, Guan Zhongwei, Wang Xiangping. A DYNAMIC RESPONSE PREDICTION MODEL OF FIBER-METAL HYBRID LAMINATED PLATES EMBEDDED WITH VISCOELASTIC DAMPING CORE UNDER LOW-VELOCITY IMPACT EXCITATION 1). Chinese Journal of Theoretical and Applied Mechanics[J], 2020, 52(6): 1690-1699 DOI:10.6052/0459-1879-20-165

引言

纤维金属层合板 (fiber metal laminated plates,FMLPs) 是一种由金属和纤维增强复合材料组成的混杂复合材料板,具有轻质、高强、阻燃、耐冲击、抗疲劳等诸多优点[1-3],已逐步在航空、航天、轨道交通、兵器工业等领域得到应用[4-8].目前,FMLPs的抗低速冲击动力学性能研究一直备受关注.

在 20 世纪 90 年代,荷兰学者 Vlot[9] 就对FMLPs 的冲击特性进行了研究,并第一个建立了该类型层合结构的冲击解析模型.之后,Vlot 研究团队[10-12]对分析方法不断完善,深入研究了低速、高速冲击激励下的动态力、位移响应、损伤面积、裂纹长度等问题.Davies 等[13]建立了FMLPs的有限元模型,并预测了冲击激励下结构的动响应和动应变.Payeganeh 等[14]通过两自由度弹簧-质量系统,建立了 FMLPs 在低速冲击激励下的解析模型,并分析了不同冲击速度对冲击接触力、响应等参数的影响.傅衣铭等[15-16]基于多种失效准则,综合运用正交配点法、Newmark 法和迭代法对复合材料板壳的冲击动态响应进行求解.Starikov[17]对不同材料组成的 FMLPs 进行了一系列低速冲击试验,推导获得了层合板结构、性能与压痕冲击响应之间的关系.陈勇[18]基于 Hashin 失效准则建立了冲击载荷下 FMLPs 的二维和三维损伤模型,预测了结构的损伤行为及动态响应,并通过落锤实验进行了模型验证.Jones[19]采用刚性与塑性材料近似法,对 FMLPs 的实际横截面进行了修正,得到了简单的计算公式来预估低速冲击后结构的最大永久横向位移.

黏弹性材料可有效地提升结构阻尼性能[20-21],若能够将该类型材料填充到纤维金属层合板中,则可极大地提升 FMLPs 的抗冲击性、耐疲劳性及服役可靠性. 然而,目前对嵌入黏弹性芯层 (viscoelastic core,VC) 的 FMLPs 结构 (简称为 VC-FMLPs) 冲击特性的研究报道很少,仅有零星文献对嵌入黏弹性芯层的三明治结构的低速冲击问题进行了初步的研究.例如 Malekzadeh 等[22]提出了一种改进的动态高阶冲击理论,并求解获得了嵌入软芯层的三明治结构的低速冲击响应,但是该模型选用的冲击能量较低,所以没有考虑损伤对冲击响应的影响.Shariyat 和 Hosseini[23]提出了一种双叠加幂指数全局-局部理论,在改进赫兹接触理论的同时,分析了带黏弹性芯层的三明治层合板的低速冲击特性. 该模型虽然有损伤产生,但是建模时也没有考虑其影响.

针对上述研究不足,本文结合经典层合板理论,冯$\cdot$卡门假设和能量守恒定律,首次从解析角度提出了带黏弹性芯层的纤维金属混杂层合板在低速冲击激励下的动态响应预测模型. 通过对金属层采用 VonMises 失效准则,对纤维层采用 Tsai-Hill 失效准则和对黏弹性层采用指数 Drucker-Prager 失效准则,考虑不同材料层对冲击动态响应的影响,推导获得了每次失效事件发生后的结构位移、能量和冲击接触力的表达式,还提出了结构动态响应分析的具体流程图.最后,基于自行设计的落锤试验系统开展了一系列测试,验证了所提出的理论预测模型的有效性.本文所采用的分析方法和模型可为带黏弹性芯层的纤维金属层合板的冲击问题研究,提供一种新思路和新模型.

1 理论模型

1.1 模型概述

所建立的 VC-FMLPs 结构的冲击动力学模型如图1 所示,其长、宽、厚分别为 $a$,$b$,$h$. 其中,金属层厚度为 $h_{m}$,黏弹性层厚度为 $h_{v}$,每个纤维层厚度均相同且为 $h_{f}$. 首先,以结构受到球头型圆柱杆冲击激励时的中心位置为原点,以该类型层合板的中面为参考平面,并沿着厚度方向,建立 $o$-$xyz$ 坐标系. 图中的 1 代表纤维纵向,2 代表纤维横向,3 代表垂直于 1-2 平面的方向. 且纤维方向与整体坐标系 $x$ 轴的夹角为 $\theta$. 假设纤维增强复合薄板平行纤维方向的弹性模量为 $E_1 $,垂直纤维方向的弹性模量为 $E_2 $,1-2 平面内的剪切弹性模量为 $G_{12} $,1 方向作用应力引起 1, 2 方向应变的泊松比为 $\nu _{12} $,2 方向作用应力引起 1,2 方向应变的泊松比为 $\nu _{21} $. 金属层和黏弹性层的弹性模量分别 $E_{M} $ 和 $E_{V}$,剪切模量分别为 $G_{M} $ 和 $G_{V} $,泊松比分别为 $\nu _{M}$ 和 $\nu _{V}$. 另外,$V$ 为球头型圆柱杆在接触 VC-FMLPs 结构瞬间的速度,$\xi $ 和 $\zeta $ 为冲击位置参数,其取值范围分别为 $0 \leqslant \xi \leqslant a$, $0 \leqslant \zeta \leqslant b $.

图1

图1   嵌入黏弹性芯层的纤维金属混杂复合薄板理论模型

Fig. 1   The theoretical model of VC-FMLPs


1.2 本构关系

在冲击过程中,由于随着 VC-FMLPs 结构的逐步破坏,其参考平面不断变化. 因此,VC-FMLPs 结构的参考平面 $z_0$ 可表示为[24]

$z_0 = \dfrac{\sum_{k = 1}^n {E_k h_k h_d } }{\sum_{k = 1}^n {E_k h_k } }$

其中,$E_k $ 为第 $k$ 层弹性模量,$h_k $ 为第 $k$ 层厚度,$h_d$ 为第 $k$ 层与球头型圆柱杆之间未被破坏层的总厚度.

由于冲击变形沿 $z$ 轴方向的位移 $w$ 远远大于沿 $x$ 轴和 $y$ 轴方向的位移 $u$ 和 $v$,因此忽略位移 $u$ 和 $v$. 根据经典层合板理论,正应变 $\varepsilon _z $ 和剪应变 $\gamma _{yz} $, $\gamma_{xz} $ 都为 0,即 $\varepsilon _z =\gamma _{yz} =\gamma _{xz} =0$. 结合冯$\cdot$卡门假设,用中面位移 $w_0 $ 表示 VC-FMLPs 结构任意点的应变[25], 可写为

$\left.\begin{array}{l} \varepsilon _x = \dfrac{1}{2}\left( {\dfrac{\partial w_0 }{\partial x}} \right)^2 - z\dfrac{\partial ^2w_0 }{\partial x^2} \\ \varepsilon _y = \dfrac{1}{2}\left( {\dfrac{\partial w_0 }{\partial y}} \right)^2 - z\dfrac{\partial ^2w_0 }{\partial y^2} \\ \gamma _{xy} = \dfrac{\partial w_0 }{\partial y}\dfrac{\partial w_0 }{\partial x} - 2z\dfrac{\partial ^2w_0 }{\partial x\partial y} \end{array}\!\!\right\}$

忽略应变率的影响[2,7-10],则材料在第 $k$ 层主轴方向的应力-应变关系为

$\left\{ \!\! \begin{array}{c} {{\sigma }'_1 } \\ {{\sigma }'_2 } \\ {{\sigma }'_{12} } \end{array} \!\! \right\}^{(k)} = \left[\!\! \begin{array}{ccc} {Q_{11} } & {Q_{12} } & 0 \\ {Q_{21} } & {Q_{22} } & 0 \\ 0 & 0 & {Q_{66} } \end{array} \!\! \right]^{(k)}\left\{ \!\! \begin{array}{c} {\varepsilon _1 } \\ {\varepsilon _2 } \\ {\gamma _{12} } \end{array} \!\! \right\}$

对于黏弹性层,式中的各个元素可表示为

$ Q_{11} = Q_{22} = \dfrac{E_{V} }{1 - \nu _{V}^2 } , \ Q_{66} = \dfrac{E_{V} }{2\left( {1 + \nu _{V} } \right)} $

$ Q_{12} = Q_{21} = \dfrac{\nu _{V} E_{V} }{1 - \nu _{V}^2 } $

对于金属层,式中的各个元素可表示为

$ Q_{11} = Q_{22} = \dfrac{E_{M} }{1 - \nu _{M}^2 } , \ Q_{66} = \dfrac{E_{M}}{2\left( {1 + \nu _{M} } \right)} $

$ Q_{12} = Q_{21} = \dfrac{\nu _{M} E_{M} }{1 - \nu _{M}^2 } $

对于纤维层,式中的各个元素可表示为

$\left.\begin{array}{c} Q_{11} = \dfrac{E_1 }{1 - \nu _{12} \nu _{21} } , \ Q_{12} = \dfrac{\nu _{12} E_2 }{1 - \nu _{12} \nu _{21} } \\ Q_{22} = \dfrac{E_2 }{1 - \nu _{12} \nu _{21} } , \ Q_{66} = G_{12} ,\quad \nu _{21} = \nu _{12} \dfrac{E_2 }{E_1 } \end{array}\right\}$

当某一层失效后,失效层面内 1 方向,2 方向以及 3 方向上的应力 $\sigma _1^{f} $, $\sigma _2^{f}$,$\sigma _{12}^{f}$ 会分配到剩余层.根据各层刚度的不同,第 $k$ 层面内 1 方向,2 方向以及 3 方向分配得的应力 $\bar{\sigma }_1 $, $\bar{\sigma }_2 $, $\bar {\sigma }_{12} $ 为

$ \left\{ \!\!\begin{array}{c} {\bar {\sigma }_1 } \\ {\bar {\sigma }_2 } \\ {\bar {\sigma }_{12} } \end{array} \!\! \right\}^{(k)} = \left\{ \!\! \begin{array}{c} {\sigma _1^{f} \dfrac{Q_{11}^{(k)} h_k }{K_1 h}} \\ {\sigma _2^{f} \dfrac{Q_{22}^{(k)} h_k }{K_2 h}} \\ {\sigma _{12}^{f} \dfrac{Q_{66}^{(k)} h_k }{K_{12} h}} \end{array}\!\! \right\} $

其中, $K_1 $, $K_2 $ 和 $K_{12}$ 为 VC-FMLPs 结构的等效刚度,其表达式为

$K_1 = \dfrac{\bar {E}_1 }{1 - \bar {\nu }_{12}^{ 2} \dfrac{{\mathop{E}\limits^{ \_}}_2 }{{\mathop{E}\limits^{ \_}}_1 }} , K_2 = K_1 \dfrac{\bar {E}_2 }{{\mathop{E}\limits^{ \_}}_1 } , K_{12} = \bar {G}_{12}$

式中,$\bar {E}_1 $,$\bar {E}_2 $ 和 $\bar {G}_{12}$ 为 VC-FMLPs 结构的等效弹性模量,其表达式为

$\bar {E}_1 = \dfrac{A_{11} - \dfrac{A_{12}^2 }{A_{22} }}{h} , \ \ \bar {E}_2 = \bar {E}_1 \dfrac{A_{22} }{A_{11} } , \ \ \bar {G}_{12} = \dfrac{A_{66} }{h}$

相应的等效泊松比是

$ \bar {\nu }_{12} = \dfrac{A_{12} }{A_{22} } , \ \ \bar {\nu }_{21} = \bar {\nu }_{12} \dfrac{\bar {E}_2 }{{\mathop{E}\limits^{ \_}}_1 }$

式中,$A_{ij}, B_{ij}, D_{ij} $ 分别为拉伸系数、拉伸弯曲耦合系数和弯曲系数, 可写为

$ \left[ {A_{ij} , B_{ij} , D_{ij} } \right]^{T} = \int_{ - h / 2}^{h / 2} {\bar {Q}_{ij}^k \left[ {1,z,z^2} \right]^{T}} \text{d}z$

式中,$\bar {Q}_{ij}^k $ 为转轴刚度系数,用来表示考虑纤维层的材料主轴与 $ {x}$ 轴的夹角 $\theta $ 影响下的第 $ {k}$ 层板在整体坐标系下的应力-应变关系[26-27], 可写为

$ \left\{ \!\! \begin{array}{c} {\sigma _x } \\ {\sigma _y } \\ {\sigma _{xy} } \end{array}\!\! \right\}^{(k)} = \left[\!\! \begin{array}{ccc} {\bar {Q}_{11} } & {\bar {Q}_{12} } & {\bar {Q}_{16} } \\ {\bar {Q}_{12} } & {\bar {Q}_{22} } & {\bar {Q}_{26} } \\ {\bar {Q}_{16} } & {\bar {Q}_{26} } & {\bar {Q}_{66} } \end{array}\!\! \right]^{(k)}\left\{\!\! \begin{array}{c} {\varepsilon _x } \\ {\varepsilon _y } \\ {\gamma _{xy} } \end{array}\!\! \right\}$

其中

$ \left. \!\!\begin{array}{c} \bar {Q}_{11} = Q_{11} \cos ^4\theta _k + 2(Q_{12} + 2Q_{66} )\sin ^2\theta _k \cos ^2\theta _k +\\ Q_{22} \sin ^4\theta _k \\ \bar {Q}_{12} = \left( {Q_{11} + Q_{22} - 4Q_{66} } \right)\sin ^2\theta _k \cos ^2\theta _k +\\ Q_{12} \left( {\sin ^4\theta _k + \cos ^4\theta _k } \right) \\ \bar {Q}_{22} = Q_{11} \sin ^4\theta _k + 2\left( {Q_{12} + 2Q_{66} } \right)\sin ^2\theta _k \cos ^2\theta _k +\\ Q_{22} \cos ^4\theta _k \\ \bar {Q}_{16} = \left( {Q_{11} - Q_{12} - 2Q_{66} } \right)\sin \theta _k \cos ^3\theta _k +\\ \left( {Q_{12} - Q_{22} + 2Q_{66} } \right)\sin ^3\theta _k \cos \theta _k \\ \bar {Q}_{26} = \left( {Q_{11} - Q_{12} - 2Q_{66} } \right)\sin ^3\theta _k \cos \theta _k +\\ \left( {Q_{12} - Q_{22} + 2Q_{66} } \right)\sin \theta _k \cos ^3\theta _k \\ \bar {Q}_{66} = \left( {Q_{11} + Q_{22} - 2Q_{12} - 2Q_{66} } \right)\sin ^2\theta _k \cos ^2\theta _k +\\ Q_{66} \left( {\sin ^4\theta _k + \cos ^4\theta _k } \right) \end{array}\!\!\right\}$

式中,$\theta$ 为第 ${k}$ 层板的纤维方向与整体坐标系 ${x}$ 轴的夹角.

结合式 (3) 和式 (5),就可以得到第 $k$ 层面内 1 方向,2 方向以及 3 方向实际应力 $\sigma_1 $, $\sigma _2 $, $\sigma _{12} $ 为

$ \left\{ \!\! \begin{array}{c} {\sigma _1 } \\ {\sigma _2 } \\ {\sigma _{12} } \end{array} \!\! \right\}^{(k)} = \left\{ \!\! \begin{array}{c} {{\sigma }'_1 } \\ {{\sigma }'_2 } \\ {{\sigma }'_{12} } \end{array}\!\! \right\}^{(k)} + \left\{ \!\! \begin{array}{c} {\bar {\sigma }_1 } \\ {\bar {\sigma }_2 } \\ {\bar {\sigma }_{12} } \end{array}\!\! \right\}^{(k)}$

1.3 失效准则

在文献[28,29,30] 提出的低速冲击下层合板结构损伤失效准则的基础上,利用逐渐累积损伤分析方法,对于金属层,可应用 Von Mises 等效应力失效准则[28]

$ \dfrac{\sigma _1^2 + \sigma _2^2 - \sigma _1 \sigma _2 + 3\sigma _{12}^2 }{\sigma _{M}^{e} } = 1$

其中,$\sigma _{M}^{\rm e}$ 为金属层材料的等效屈服应力,低速冲击时认为金属层达到屈服抗拉强度,就已经失去承载能力,不考虑之后塑性变形对结构的影响.

对于纤维层,应用 Tsai-Hill 应力失效准则[29]

$ \left( {\dfrac{\sigma _1 }{X_{T} }} \right)^2 + \left( {\dfrac{\sigma _2 }{Y_{T} }} \right)^2 - \dfrac{\sigma _1 \sigma _2 }{X_{T} ^2} + \left( {\dfrac{\sigma _{12} }{S_{f} }} \right)^2 = 1$

其中,$X_{T}$ 为纤维纵向拉伸强度,$Y_{T}$ 为纤维横向拉伸强度,$S_{f}$ 为纤维的剪切强度.

对于黏弹性层,应用指数 Drucker-Prager 应力失效准则[30]

$ \dfrac{ (\sigma_1-\sigma_2)^2+(\sigma_1-\sigma_{12})^2+(\sigma_2-\sigma_{12})^2 }{2(\sigma^{e}_{\rm v})^2} =1$

其中,$\sigma^{e}_{v}$ 为黏弹性层材料的等效屈服应力,可表示为

$ \sigma^{e}_{v} = \left[ {\lambda \left( {\sigma _{v}^{T} } \right)^2 - 3\left( {\lambda - 1} \right)\sigma _{v}^{T} \sigma _{v}^{A} } \right]^{1 / 2}$

式中,$\sigma _{v}^{T}$ 为黏弹性层的屈服抗拉强度,$\sigma _{v}^{A}$ 为主应力表示的流体静应力,$\lambda $ 为流体静应力敏感系数. $\lambda$ 的值由两种或两种以上不同应力状态下的屈服应力确定,低速冲击状态下,主要考虑屈服抗拉强度 $\sigma _{v}^{T}$ 和屈服剪切强度 $\sigma _{v}^{s}$. 其各自表达式为

$\sigma_{v}^{A}= \dfrac{1}{3}(\sigma _1 + \sigma _2 + \sigma _{12} )$
$\lambda = 3(\sigma_{v}^{s} )^2 / (\sigma_{v}^{T})^2$

1.4 位移方程

在与球头型圆柱杆冲击接触过程中,VC-FMLPs 结构将形成两个变形的区域,分别称其为接触区域与拉伸区域,如图2 所示. 假设冲击过程中,损伤仅发生在接触区域内. 在此区域内,需计算结构因塑性变形 (及其损伤) 消耗的能量以及因弹性变形产生的应变能;在拉伸区域内,只需计算因弹性变形产生的应变能.

图2

图2   VC-FMLPs 结构变形区域

Fig. 2   The deformation region of FMLPs


对于接触区域,其主要由球头型圆柱杆形状决定,其范围可看作由冲击中心位置到接触半径,即 $0 \leqslant x \leqslant R$, $0 \leqslant y \leqslant R $. 考虑到接触半径相对于该类型层合板的尺寸非常小,可将该接触区域近似看成一个平面. 在这个范围内,中面位移 $w_0 (x,y)$ 的表达式为

$w_0 (x,y) = w_{\max }$

其中,$w_{\max } $为冲击中心处的最大位移.

对于拉伸区域,其范围可看作由接触半径到结构约束边界组成,即 $R \leqslant x\leqslant \max \left\{ {\xi , a - \xi } \right\}$,$R \leqslant y \leqslant\max \left\{ {\zeta , b - \zeta } \right\}$. 对于图1 所示的四边固支边界下的 VC-FMLPs 结构,$w_0 (x,y)$ 有如下关系式

$w_0 = 0 , \ \ \dfrac{\partial w_0 }{\partial x} \approx 0 , \ \ \dfrac{\partial w_0 }{\partial y} \approx 0$

为了解释不同材料层对 VC-FMLPs 结构中面位移的贡献,改进文献[31] 中的位移公式,引入金属体积分数 $v_{\rm m}$,纤维层体积分数 $v_{f}$,黏弹性层体积分数 $v_{v}$,并使其满足式 (20) 中的边界条件

$ w_0 (x,y) = w_{M} v_{m} \left( {1 - \dfrac{x}{\xi }} \right)^2\left( {1 - \dfrac{y}{\zeta }} \right)^2 + \\ w_{f} v_{f} \left[ {1 - \left( {\dfrac{x}{\xi }} \right)^2} \right]\left[ {1 - \left( {\dfrac{y}{\zeta }} \right)^2} \right] + \\ w_{v} v_{\rm v} \left( {1 - \dfrac{x}{\xi }} \right)^2\left( {1 - \dfrac{y}{\zeta }} \right)^2$

其中

$w_{M} = \dfrac{w_{\max } }{\left( {1 - \dfrac{R}{\xi }} \right)^2\left( {1 - \dfrac{R}{\zeta }} \right)^2}$
$w_{f} = \dfrac{w_{\max } }{\left[ {1 - \left( {\dfrac{R}{\xi }} \right)^2} \right]\left[ {1 - \left( {\dfrac{R}{\zeta }} \right)^2} \right]}$
$w_{v} = \dfrac{w_{\max } }{\left( {1 - \dfrac{R}{\xi }} \right)^2\left( {1 - \dfrac{R}{\zeta }} \right)^2}$

在利用模型计算获得了中面位移后,将其代入式 (2)、式 (3)、式 (5) 和式 (12) 来获得变形最大位置处 (在冲击中心位置处,近似取 $x=R$, $y=R$ 处) 应力,并进一步应用 1.3 部分对应的失效准则来判别某层是否发生失效. 如果式 (13) 或式 (14) 或式 (15) 对应的左侧表达式大于 1,则认为该层发生了损伤 (此时,可认为满足了失效准则要求,并可将对应的接触区域内刚度设置为零);如果没有发生失效,则将中面位移对应的 $w_{\max } $ 进行更新,并在获得新的中面位移基础上,重复利用式 (13) 或式 (14) 或式 (15) 进行判别,直到满足失效准则的要求为止.

2 冲击响应求解

2.1 冲击接触力的计算

由球头型圆柱杆冲击产生的总动能 $T$ 为

$T=\dfrac 12 MV^2_0$

VC-FMLPs 结构的应变能为

$U = \dfrac{1}{2}\int_x {\int_y {\int_z {\sigma _x \varepsilon _x + \sigma _y \varepsilon _y + \sigma _{xy} \gamma _{xy} } } } \text{d} x \text{d} y \text{d}z$

将式 (2)、式 (9) 和式 (10) 代入式 (26) 中,可得到 VC-FMLPs 结构由拉伸变形产生的应变能 $U_{m}$

$\begin{array}{c} U_{m} = \int_0^\xi {\int_0^\zeta { \Bigg [\dfrac{1}{8}A_{11} } } \left( {\dfrac{\partial w}{\partial x}} \right)^4 + \dfrac{1}{4}A_{12} \left( {\dfrac{\partial w}{\partial x}} \right)^2\left( {\dfrac{\partial w}{\partial y}} \right)^2+\\ \dfrac{1}{8}A_{22} \left( {\dfrac{\partial w}{\partial y}} \right)^2 + \dfrac{1}{2}\left[ {A_{16} \left( {\dfrac{\partial w}{\partial x}} \right)^2 + A_{26} \left( {\dfrac{\partial w}{\partial y}} \right)^2} \right]\dfrac{\partial w}{\partial x}\dfrac{\partial w}{\partial y} +\\ \dfrac{1}{2}A_{66} \left( {\dfrac{\partial w}{\partial x}\dfrac{\partial w}{\partial y}} \right)^2 \Bigg ] \text{d} x \text{d} y\end{array}$

VC-FMLPs 结构拉伸弯曲耦合应变能 $U_{c}$ 为

$\begin{array}{c} U_{c} = - \int_0^\xi {\int_0^\zeta { \Bigg [\dfrac{1}{4}B_{11} } } \left( {\dfrac{\partial w}{\partial x}} \right)^2\dfrac{\partial ^2w}{\partial x^2} +\\ \dfrac{1}{2}B_{12} \left[ {\left( {\dfrac{\partial w}{\partial y}} \right)^2\dfrac{\partial ^2w}{\partial x^2} + \left( {\dfrac{\partial w}{\partial x}} \right)^2\dfrac{\partial ^2w}{\partial y^2}} \right] +\\ \dfrac{1}{4}B_{22} \left( {\dfrac{\partial w}{\partial y}} \right)^2\dfrac{\partial ^2w}{\partial y^2} + \\ B_{16} \left[ {\dfrac{\partial ^2w}{\partial x^2}\dfrac{\partial w}{\partial x}\dfrac{\partial w}{\partial y} + \left( {\dfrac{\partial w}{\partial x}} \right)^2\dfrac{\partial ^2w}{\partial x\partial y}} \right] +\\ B_{26} \left[ {\dfrac{\partial ^2w}{\partial y^2}\dfrac{\partial w}{\partial x}\dfrac{\partial w}{\partial y} + \left( {\dfrac{\partial w}{\partial y}} \right)^2\dfrac{\partial ^2w}{\partial x\partial y}} \right] +\\ 2B_{66} \dfrac{\partial ^2w}{\partial x\partial y}\dfrac{\partial w}{\partial x}\dfrac{\partial w}{\partial y} \Bigg ] \text{d} x \text{d} y\end{array}$

VC-FMLPs 结构弯曲产生的应变能 $U_{b} $ 为

$\begin{array}{c} U_{b} = \int_0^\xi {\int_0^\zeta { \Bigg [\dfrac{1}{2}D_{11} } } \left( {\dfrac{\partial ^2w}{\partial x^2}} \right)^4 + D_{12} \dfrac{\partial ^2w}{\partial x^2}\dfrac{\partial ^2w}{\partial y^2}+\\ \dfrac{1}{2}D_{22} \left( {\dfrac{\partial ^2w}{\partial y^2}} \right)^2 + 2\left( {D_{16} \dfrac{\partial ^2w}{\partial x^2} + D_{26} \dfrac{\partial ^2w}{\partial y^2}} \right)\dfrac{\partial ^2w}{\partial x\partial y}+\\ \dfrac{1}{2}D_{66} \dfrac{\partial ^2w}{\partial x\partial y}^2 \Bigg ] \text{d} x \text{d} y \end{array}$

这里,需要说明的是,式 (27),式 (28) 和式 (29) 只是对 VC-FMLPs 结构在 ${xoy}$ 面内的第一象限积分后的应变能结果,结构的总应变能需要对 4 个象限要分别积分,并进行叠加才可获得.

冲击接触力 $F$ 所做的功为

$W = Fw_{\max }$

VC-FMLPs 结构总势能 $U_{s}$ 表达式为

$U_{s} = U_{m} + U_{c} + U_{b} -W$

当 $\partial U_{s} / \partial w_{\max } = 0$ 时,$U_{s }$ 会获得最大值,此时,可计算获得冲击接触力 $F$

$F = \partial U_{m} / \partial w_{\max } + \partial U_{c} / \partial w_{\max } + \partial U_{b} / \partial w_{\max }$

2.2 失效模式

首先,为了求解方便,引入冲击"失效事件"的概念[32],假设该事件是由冲击造成的 VC-FMLPs 结构在接触区域的分层损伤和断裂效应引发的,则每个失效事件 $j$ 吸收的总应变能 $U_{j}$ 可表示为

$U_j = (U_{m}^j + U_{c}^j + U_{b}^j ) - (U_{m}^{j - 1} + U_{c}^{j - 1} + U_{b}^{j - 1} )$

其中,$U_{m}^j , U_{c}^j , U_{b}^j$ 代表第 $j$ 个失效事件发生时的拉伸应变能,拉伸弯曲耦合应变能和弯曲应变能,$U_{m}^{j - 1} , U_{c}^{j - 1} , U_{b}^{j - 1} $ 为前一个失效事件对应的拉伸应变能,拉伸弯曲耦合应变能和弯曲应变能.

参考文献[32],可获得 $x=R_{j}$, $y =0$ 处变形协调条件,并得到下列等式条件

$ \sqrt {R_{\max }^2 - R_j^2 } + w_{\max } - R_{\max } = w_{M} v_{m} \left( {1 - \dfrac{R_j }{\xi }} \right)^2 +\\ w_{f} v_{f} \left[ {1 - \left( {\dfrac{R_j }{\xi }} \right)^2} \right] + w_{v} v_{v} \left( {1 - \dfrac{R_j }{\xi }} \right)^2 \\ $

其中,$R_{j}$ 为第 $j$ 个失效事件时的接触半径,$R_{\max}$ 为球头型圆柱杆球头半径. 这样,$R_{j}$ 就通过式 (34) 求解获得.

第 $j$ 次失效事件中,因分层损伤消耗的能量 $T_{d}^j $ 为

$T_{d}^j = \dfrac{\pi \bar {E}_1^j hG_{II}^2 }{9\left[ {1 - (\bar {\nu }_{12}^j )^2} \right]\left( {\sigma _{IL} } \right)^2} + \dfrac{\pi \bar {E}_2^j hG_{II}^2 }{9\left[ {1 - (\bar {\nu }_{21}^j )^2} \right]\left( {\sigma _{IL} } \right)^2}$

其中,$G_{II} $ 为 VC-FMLPs 结构的第二类层间断裂能量释放率[33],$\sigma_{IL} $ 为层间剪切强度. $\bar {E}_1^j $,$\bar {E}_2^j$ 为失效事件 $j$ 发生时 1 和 2 方向对应的等效弹性模量,$\bar {\nu }_{12}^j $,$\bar {\nu }_{21}^j $ 为等效泊松比.

第 $j$ 次失效事件中,各层材料由于断裂效应产生的能量 $T_{f}^j $ 为

$T_{f}^j = w_{\max }^j \pi e_t R_j^2 / 3$

其中,$e_t $ 为某层能量密度. 对于纤维层、金属层以及黏弹性层,其能量密度 $e_t^{f} $, $e_t^{m}$ 和 $e_t^{v} $ 分别为

$ \left. \begin{array}{l} e_t^{f} = \dfrac{1}{2}\sigma _{f}^u \varepsilon _{f}^u \\ e_t^{m} = \dfrac{1}{2}\sigma _{m}^y \varepsilon _{m}^y + \left( {\sigma _{m}^u + \sigma _{m}^y } \right)\dfrac{\varepsilon _{m}^u - \varepsilon _{m}^y }{2} \\ e_t^{v} = \dfrac{1}{2}\sigma _{v}^y \varepsilon _{v}^y + \left( {\sigma _{v}^u + \sigma _{v}^y } \right)\dfrac{\varepsilon _{v}^u - \varepsilon _{v}^y }{2} \end{array}\!\!\right\}$

其中,$\sigma _{f}^u $ 和 $\varepsilon _{f}^u $ 为纤维层应力极限和纵向极限应变;$\sigma _{m}^y$和$\sigma _{m}^u $ 为金属层屈服抗拉强度和极限抗拉强度,其对应的应变为 $\varepsilon _{m}^y$ 和 $\varepsilon _{m}^u $;$\sigma _{v}^y $ 和 $\sigma _{v}^u$ 为黏弹性层屈服抗拉强度和极限抗拉强度,其对应的应变为 $\varepsilon _{v}^y $ 和 $\varepsilon _{\rm v}^u $.

这样,可获得每个失效事件 $j$ 消耗的总能量 $T_{a}^j $

$T_{a}^j = U_j + T_{d}^j + T_{f}^j$

进一步,由能量守恒定律可获得每次失效事件 $j$ 发生后的冲击速度的表达式

$V_{j + 1} = \sqrt { MV_j^2 - 2T_{a}^j / M }$

另外,根据参考文献[31],采用一个单自由度弹簧-质量-阻尼模型来描述 VC-FMLPs 结构在球头型圆柱杆冲击下的整个过程. 则第$j$次失效事件中的接触力 $F_j$ 可由动力学方程表示

$F_j = (M + m_{e}^j )\dfrac{\partial ^2w_{\max }^j }{\partial t^2} + C_{s}^j \dfrac{\partial w_{\max }^j }{\partial t} + K_{s}^j w_{\max }^j$

其中,$M$ 为球头型圆柱杆质量,$m_{e}^j $ 和 $w_{\max }^j $ 为第 $j$ 次失效事件时 VC-FMLPs 结构受冲击作用影响的的等效质量和冲击中心处最大位移,$K_{s}^j $, $C_{s}^j $ 为分别为结构整体剪切刚度和黏滞阻尼系数.

进一步可得第 $j$ 次失效事件对应的动力学方程

$F_j - K_{s}^j w_{\max }^j - C_{s}^j \dot {w}_{\max }^j = (M + m_{e}^j )\dot {V}_j$

其中

$ \dot {V}_j = \dfrac{\left| {V_j - V_{j - 1} } \right|}{t_j - t_{j - 1} } , \ \ \dot {w}_{\max }^j = \dfrac{w_{\max }^j - w_{\max }^{j - 1} }{t_j - t_{j - 1} } \\ K_{s}^j = K_{12}^j , \ \ C_{s}^j = 2\varsigma _j \sqrt {K_{s}^j (M + m_{s}^j )} $

$K_{12}^j$ 为失效事件 $j$ 发生时结构整体剪切刚度,$\varsigma _j $ 第失效事件 $j$ 发生时的等效阻尼比.$V_{j}$, $V_{j - 1}$, $w_{\max }^j $, $w_{\max }^{j - 1} $ 以及 $t_{j}$ 和 $t_{j -1}$ 分别代表了第 $j$ 次失效事件和第 $j -1$ 次失效事件中的速度,冲击中心处最大位移和时间.

第 $j$ 次失效事件对应的动能 $T_{c}^j $ 可表示为

$ T_{c}^j = 2\mathop{\displaystyle\iint}\limits_{A - A_j } \Bigg[\dfrac{\rho _{a} (\dot {w}_{\max }^j )^2} {\left( {1 - \dfrac{2R_j }{a}} \right)^2\left( {1 - \dfrac{2R_j }{b}} \right)^2}\cdot\\ \left( {1 - \dfrac{2x}{a}} \right)^2\left( {1 - \dfrac{2y}{b}} \right)^2 \Bigg ] \text{d}A + \dfrac{1}{2}\pi R_j^2 \rho _{a} (\dot {w}_{\max }^j )^2 $

其中,$\rho _{a}$ 为 VC-FMLPs 结构的面密度,$A$ 为其整体积分区域,$A_{j}$ 为接触区域.

由于可近似认为 $T_{c}^j = m_{e}^j (\dot {w}_{\max }^j )^2 / 2$,则 $m_{e}^j $ 可表示为

$m_{e}^j = \pi R_j^2 \rho _{a} + \dfrac{1}{9}(a - 2R_j )(b - 2R_j )\rho _{a}$

如此,可根据式 (41),确定失效事件 $j$ 发生的时间 $t_j $

$t_j = \dfrac{(M + m_{e}^j )\left| {V_j - V_{j - 1} } \right| + C_{s}^j (w_{\max}^j - w_{\max }^{j - 1} )}{F_j - K_{s}^j w_{\max }^j } + t_{j - 1}$

至此,就实现了 VC-FMLPs 结构冲击动态响应的理论求解,其详细的分析流程如图3 所示. 当失效事件 $j$ 发生时,通过式 (33) 求解结构系统吸收的总应变能 $U_j$,通过式 (35) 和式 (36) 求解接触区域,因分层损伤消耗的能量 $T_{d}^j$ 和塑性变形消耗的能量 $T_{f}^j $,进而求解冲击接触力 $F_j $、位移 $w_{\max }^j$ 和失效事件 $j$ 发生的时间 $t_j $,直至球头型圆柱杆冲击能量消耗为零,输出计算结果.

图3

图3   低速冲击激励下 VC-FMLPs 结构动态响应分析流程图

Fig. 3   Flow chart of dynamic response analysis of VC-FMLPs under low-velocity impact excitation


3 实验验证

图4给出了自行设计的 VC-FMLPs 结构落锤冲击试验系统,落锤质量为 1 kg,刚性冲头的顶端被设计成半球形状 (直径 8.0 mm). 在冲击测试过程中,通过磁力表座固定在冲头上的 LSM 位移传感器和嵌入式动态力传感器 (型号为联能 CL-YD-305),并利用 LMS 数据采集仪和笔记本工作站,可实时采集并记录结构在低速冲击下的冲击接触力和响应数据. 试验系统经过 "江苏联能电子技术有限公司" 和 "KISTLER" 专业技术员的调试和校准,并在正式实验前进行多次预实验,保证了实验结果的可重复性.

图4

图4   落锤低速冲击试验设备

Fig. 4   Drop weight impact tester


以嵌入 Zn33 黏弹性芯层的 TA2 钛合金混杂T300 碳纤维/树脂基层合板为对象,通过夹具夹紧其四边,夹持后长、宽、厚为 170 mm $\times $ 160 mm $\times$ 2.6 mm,如图5 所示. 其中金属外层厚度为 0.3 mm,$E_{M} =$ 108 GPa,$\nu _{M} =0.3 $,$\rho _{M} = 4150$ kg/m$^3$,$\sigma _{M}^{e} = 600$ MPa;纤维层为对称正交铺设 $[0^ \circ / 90^ \circ / 0^ \circ / 90^ \circ ]$ 的 TC300 碳纤维/E21 环氧树脂,共有 9 层,$E_{1} = 136$ GPa, $E_{2} =$ $7.92$ GPa, $G_{12} =3.39$ GPa,$\nu _{12} = 0.32$,$\rho _{f} = 1780$ kg/m$^3$, $X_{T}=2210$ MPa,$ Y_{\rm T}=49$ MPa,$S_{f} = 135$ MPa;黏弹性层厚度为 1 mm,$E_{V} = 5$ MPa,$\nu _{V} = 0.3$,$\rho_{V} = 4510$ kg/m$^3$,$\sigma _{v}^{T} = 7$ MPa.

图5

图5   VC-FMLPs 结构试验件及夹具

Fig. 5   Test piece and fixtures of VC-FMLPs


在落锤高度为 0.6 m 和 1 m 处 (即冲击能量为5.9 J 和 9.8 J 时),分别开展冲击实验,图6 给出了测试获得的 VC-FMLPs 结构的冲击接触力、位移响应和冲击载荷-位移曲线.为了方便与实验数据进行对比,将利用第 1 部分建立的理论模型计算获得的分析结果也一并绘制在图7 上,还以图中的峰值结果作为统计,给出了相应的计算相对误差.另外,以冲击能量达到 9.8 J 为例,图7 给出了结构在冲击面和非冲击面产生的损伤照片.

图6

图6   理论和测试获得的低速冲击下 VC-FMLPs 结构的冲击接触力时程曲线、位移响应曲线和冲击载荷-位移曲线

Fig. 6   Impact contact force time-history curve, displacement response curve and load-displacement curve of VC-FMLPs under low-velocity impact obtained by theory and test


图7

图7   冲击能量为 9.8 J 时 VC-FMLPs 结构的冲击损伤

Fig. 7   Impact damage of VC-FMLPs under the impact energy of 9.8 J


通过对上述结果进行分析可知:利用该理论模型,计算获得的 VC-FMLPs 结构冲击下的 冲击接触力时程曲线、位移响应以及冲击载荷-位移曲线与实验结果的变化趋势吻合较好,且关注的曲线上峰值点的计算误差最大不超过 9% (详见图6),进而验证了所建立的动态响应预测模型及其分析方法的正确性,可以利用该模型对 VC-FMLPs 结构的冲击响应进行较为可靠的预测和分析. 另外,需要说明的是,利用该模型完成上述计算的时间在3 min以内,相对于商用软件 ANSYS 的 LS-DYNA 模块,大大提高了计算效率 (其通常花费 3$\sim $5 h).

但仍有必要对模型的局限性问题进行分析: (1)建模时忽略了已失效层在冲击过程中的持续损伤累积,这也是计算误差的来源之一;(2)认为每个失效事件发生时,将接触区域内刚度整体清零,在后续的研究中,应在考虑损伤区域的剩余刚度;(3)由于该模型没有考虑应变率等因素的影响,因而不适合预测高速冲击问题;(4)由于模型需要进行足够次数的迭代,才能进行高精度预测,如果是太低速的冲击,则会因迭代次数过少而影响计算精度.因此,参考相关文献的假设条件[2],推荐该模型适用的冲击速度范围为2$\sim $10 m/s.

4 结论

本文建立了低速冲击激励下嵌入黏弹性阻尼芯层的纤维金属混杂层合板动态响应预测模型,并对其分析获得的冲击接触力、位移响应以及冲击载荷-位移曲线进行了测试验证. 结果表明,理论计算与获得的上述冲击参数的变化趋势与实验吻合较好,且关注的曲线上峰值点的计算误差最大不超过 9%,进而验证了理论模型和分析方法的正确性. 另外,相对于 ANSYS 的 LS-DYNA 模块,该模型大大提高了计算效率,可为复杂层合板结构动态冲击问题的高效求解提供一种新思路和新手段.

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