SIMULATION ON THE ELECTRICAL AND MECHANICAL FATIGUE DAMAGE BEHAVIOR OF CONDUCTIVE HYDROGEL INTERFACE UNDER CYCLIC LOADING
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摘要: 为了研究导电水凝胶电极贴片在疲劳载荷下因界面损伤而导致的电阻变化规律, 本文提出了一种力电学内聚力模型. 首先基于水凝胶的黏弹性力学行为, 采用Wiechert模型构建了应变率相关的内聚力牵引力-牵引位移关系方程, 并将界面电阻率定义为牵引力和损伤变量的函数, 然后通过ABAQUS的UEL子程序对黏弹性力电学内聚力模型进行了数值实现. 通过搭接剪切试验确定了模型的黏弹性参数和不同应变率下内聚力模型的损伤起始和界面断裂的能量释放率, 以及应变率下界面电阻率随牵引力的变化规律. 采用单个单元对模型的有效性进行了验证, 最后将模型应用于导电水凝胶电极贴片的在疲劳载荷下的界面电阻变化规律仿真预测, 结果表明随着电极贴片的电阻随着加载循环次数的增加呈波动上升趋势, 且随着加载速率增大, 电阻随加载次数上升的趋势逐渐减缓, 电阻的波动也越小, 有限元仿真结果与实验结果吻合良好.Abstract: In order to study the resistance change of conductive hydrogel patch electrode caused by interface damage under fatigue load, an electrical-mechanical cohesive zone model was proposed in this work. Firstly, based on the viscoelastic mechanical behavior of hydrogel, the Wiechert model was applied to construct the rate-dependent traction-displacement jump relationship of the hydrogel adhesive layer. And the interfacial resistivity was defined as a function of traction force and damage variables. Then, the viscoelastic electrical-mechanical cohesive zone model was numerically implemented by ABAQUS/UEL subroutine. The viscoelastic parameters of the model and the energy release rate of damage initiation and interface fracture of the cohesion model at different strain rates, as well as the variation law of interfacial resistivity with traction force under different loading rates, were determined by shear tests. Afterwards, the effectiveness of the proposed model was verified by a single element simulation. Finally, the model was applied to predict the interfacial resistance growth of the patch electrode adhered by conductive hydrogel under fatigue loading. The results showed that the resistance of the patch electrode presented a fluctuating increase trend with the increment of loading cycles. With the increase of the loading rate, the resistance rising rate gradually decrease, and the resistance fluctuation was smaller. The finite element simulation results were in good agreement with the experiments.
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引 言
相比于基于刚性组件的设备, 柔性电子技术将某些生理传感器与信号采集分析系统柔性化, 以满足设备的形变要求, 为医疗健康监测提供可穿戴、可移动、不受时间和地点限制的监测设备[1-2], 从而引起了人们的广泛研究. 其中的皮肤贴片式传感器因其固有的灵活性和柔韧性, 在实时监测个体生理状态方面表现出巨大的潜力[3]. 例如可用于心血管疾病预测的柔性心电贴[4], 可以检测皮肤紫外线强度的柔性UV监测贴片[5], 可检测人体体表温度的体温贴[6], 可分析表皮汗液中葡萄糖含量的柔性电化学监测贴片[7]等. 为保证电信号的稳定传递, 常见方法是在皮肤与传感器之间涂敷一层特殊的黏附胶层. 而结合导电材料和水凝胶特有优势的导电水凝胶是健康监控和电子皮肤等方面的理想材料[8-9], 并成功运用到了皮肤贴片式传感器中. 因此在实际的使用中传感器/皮肤界面事实上是由传感器、导电水凝胶电极和皮肤组成的异质层结构. 然而在服役时由于应力/应变、温度或汗液等外部载荷作用下, 水凝胶与皮肤之间的黏接界面会发生损伤, 引起传输的电信号产生漂移, 从而导致传感器检测精度下降. 以往的研究中多关注温度和湿度导致的信号漂移, 并发展了补偿电路技术对信号漂移进行校正[10-11], 但对应力应变载荷如何导致柔性贴片式传感器产生信号漂移还缺乏机理性认识, 也缺少相应模型对此问题进行描述.
针对大多数的界面破坏问题, 由于内聚力模型对于网格的节点空间坐标信息要求不高,更易于进行有限元实现, 获得了大量的研究[12]. 传统的内聚力模型自Dugdalel[13]和Barenblattl[14]提出以来已有大量研究, 如双线性内聚力模型[15]、梯形内聚力模型[16]和指数内聚力模型[17]等, 在模拟涂层剥离[18]、复合材料界面脱黏[19]、金属化合物晶间断裂[20]等领域得到了广泛应用. 在内聚力模型的疲劳损伤研究中, Nguyen等[21]将界面的黏附强度定义为随循环次数呈指数衰减规律的函数, 建立了疲劳累积损伤内聚力模型. Solana等[22]将弹塑性损伤机制嵌入双线性内聚力模型中提出了疲劳累积损伤模型, 可有效应用于复合材料的疲劳寿命预测[23]. 但是, 与传统材料不同, 水凝胶黏接层是一种典型的软材料, 具有大变形、非线性及显著的黏弹性特性[24-25]. 这些水凝胶本身所具有的特征使得传统的界面破坏理论, 在描述水凝胶黏接结构的界面破坏行为时会带来极大误差[26].
在软材料黏结的理论研究方面, 人们一般用黏附界面的断裂能量释放率G来表征材料黏接结构界面性能的强弱[27]. 软材料的断裂能量释放率G由界面本身的断裂能量释放率${G_0}$与水凝胶基体内部的变形引起的能量释放率${G_D}$两部分的和组成[28]. 软材料的黏弹性引起的能量释放率的速率相关性也已经得到了普遍的重视. Zhu等[29]研究了具有黏弹性的聚丙烯酸胶带在刚性基底上的剥离行为, 展现了黏弹性性能对剥离行为的影响. Yang等[30]研究了界面断裂能量释放率与对数形式的速度的关系, 建立了速度依赖性的能量释放率公式. 为了更好表征界面力学行为和损伤累积的速率相关性, 人们将黏弹性材料本构与内聚力模型进行结合, 建立了黏弹性内聚力模型[31-34], 可有效模拟不同加载速率下软界面损伤演化行为.
综上可知, 内聚力模型在模拟软材料界面的力学疲劳损伤行为方面得到了广泛的研究和应用. 然而, 目前内聚力模型研究中很少涉及界面的电性能衰减行为, 因此无法很好地应用于柔性电子领域模拟由于界面损伤导致的器件的性能下降. 因此, 为了研究导电水凝胶贴片在工作中的电性能衰减, 本文在黏弹性内聚力模型基础上, 引入电传导方程, 建立了力电学内聚力模型, 并通过UEL子程序嵌入ABAQUS, 应用于导电水凝胶电极贴片的界面损伤和电性能衰减行为的仿真预测, 以期为皮肤贴片式传感器在长期监测人体生理信号时由于界面疲劳损伤导致的信号漂移的校正提供技术支持.
1. 理论模型
1.1 黏弹性力电学内聚力模型
导电水凝胶电极贴片与皮肤黏结界面主要以剪切变形为主, 为了简化建模运算, 建立只含切向变形与损伤行为的内聚力模型. 首先需选用合适的黏弹性力学模型来表征水凝胶力学行为的加载速率相关性. 通过弹簧和黏壶的不同组合, 可以得到不同的黏弹性力学模型. 考虑建模方便, 本文采用如图1所示的Wiechert三元件标准固体模型表征界面的黏弹性行为. ${E_1}$和${E_2}$分别表示两弹性元件的刚度, $\eta $表示黏性元件的黏性系数, $\delta $, ${\delta _e}$和${\delta _\eta }$分别表示各自的分离位移, 且有$\delta = {\delta _e} + {\delta _\eta }$.
含损伤变量的黏结界面自由能表达式为
$$ {\psi \left( {\delta ,\alpha ,D} \right) = \left( {1 - D} \right)\left( {\frac{1}{2}{E_1}{\delta ^2} + \frac{1}{2}{E_2}{\delta _e}^2} \right)} $$ (1) 内聚力模型的牵引力表达式为
$$ {T = \frac{{\partial \psi }}{{\partial \delta }} = \left( {1 - D} \right)\left( {{E_1}\delta + {E_2}{\delta _e}} \right)} $$ (2) 以能量阈值定义界面损伤起始和断裂, 基于指数型损伤演化准则, 损伤变量可表示为
$$ {D = \left\{ \begin{split} & 0{,\qquad \left| \delta \right| \leqslant {\delta _0}{\text{ }}} \\ & {\mathop \int \nolimits_{{\delta _0}}^\delta \frac{{T{\text{d}}\delta }}{{{G_c} - {G_0}}}}{,\qquad{\delta _0} < \left| \delta \right| < {\delta _{{c}}}} \\ & 1{,\qquad\left| \delta \right| \geqslant {\delta _{{c}}}{\text{ }}} \end{split}\right.} $$ (3) 其中, ${\delta _0}$和${\delta _{{c}}}$分别表示界面损伤起始和断裂位移, ${G_0}$和${G_c}$表示界面损伤起始和断裂能量. 由于水凝胶的黏弹性行为, 界面的损伤起始和断裂的能量阈值也是加载速率相关的.
研究表明, 黏弹性材料的电学性能与力学性能一样也具有率相关性[35-37], 因此, 本文采用线性关系建立了水凝胶黏结界面的电阻率与界面牵引力关系函数[35]. 同时, 考虑损伤行为对界面电导率的影响, 可得到界面的实时电阻率与界面损伤与应力状态的函数关系
$$ \begin{array}{*{20}{c}} {\rho = \left( {{\rho _0} + kT} \right)/\left( {1 - D} \right)} \end{array} $$ (4) 其中, ${\rho _0}$为导电水凝胶的初始电阻率, $k$为界面应力对其电阻率的影响系数.
1.2 黏弹性内聚力模型的数值算法
采用有限差分法对界面的黏弹性力学行为进行求解. 根据Boltzmann叠加原理, 黏性元件的黏弹性力学响应随时间的变化可表示为
$$ \begin{split} & {T_2}\left( t \right) = {\left. {{E_2}{\delta _e}} \right|_t} = \mathop \int \nolimits_{ - \infty }^t Y\left( {t - \tau } \right)\dot \delta \left( \tau \right){\rm{d}}\tau= \\ &\qquad {E_2}\mathop \int \nolimits_{ - \infty }^t {{\rm{e}}^{ - \tfrac{{t - \tau }}{\lambda }}}\dot \delta \left( \tau \right){\rm{d}}\tau \end{split} $$ (5) 式中, $Y$为应力松弛函数, $Y\left( s \right) = {E_2}{{\rm{e}}^{ - \tfrac{s}{\lambda }}}$, 其中$\lambda $为松弛时间, 表示为$ \lambda = \eta /{E}_{2} $.
将整个加载时间段$\left[ {0,t} \right]$进行N等分, 则有$\left[ {0,t} \right] = \bigcup\limits_{n = 1}^N {\left[ {{t_n},{t_{n + 1}}} \right]} $, ${t_{n + 1}} = {t_n} + {{\Delta }}{t_n}$. 此时, 加载速率可表示为
$$ \dot \delta \left( t \right) = \frac{{\delta \left( {{t_{n + 1}}} \right) - \delta \left( {{t_n}} \right)}}{{\Delta {t_n}}} $$ (6) 将其代入方程(5)可得
$$\begin{split} & {\delta _e}\left( {{t_{n + 1}}} \right) = \mathop \int \nolimits_{ - \infty }^{{t_{n + 1}}} {{\rm{e}}^{ - \tfrac{{{t_{n + 1}} - \tau }}{\lambda }}}\dot \delta \left( \tau \right){\rm{d}}\tau = {{\rm{e}}^{ - \tfrac{{{{\Delta }}{t_n}}}{\lambda }}}{\delta _e}\left( {{t_n}} \right)+ \\ &\qquad \frac{{\left( {1 - {{\rm{e}}^{ - {{\Delta }}{t_n}/\lambda }}} \right)\left( {\delta \left( {{t_{n + 1}}} \right) - \delta \left( {{t_n}} \right)} \right)}}{{{{\Delta }}{t_n}/\lambda }} \end{split} $$ (7) 由此可通过式(2)得牵引力的计算公式
$$ {T\left( {{t_{n + 1}}} \right) = \left( {1 - D\left( {{t_{n + 1}}} \right)} \right)\left( {{E_1}\delta \left( {{t_{n + 1}}} \right) + {E_2}{\delta _e}\left( {{t_{n + 1}}} \right)} \right)} $$ (8) 1.3 模型在ABAQUS的UEL子程序中的实现
本文采用二维四节点单元来构建黏弹性力电学内聚力单元, 来实现水凝胶黏结界面的损伤和导电性能衰退行为的有限元仿真, 如图2所示.
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图 2 内聚力单元: (a)初始状态和(b)变形状态Figure 2. Cohesive element: (a) initial state and (b) deformation state内聚力单元包含4个节点, 单元节点的“位移”向量同时包含了节点的位移和电势
$$ {\boldsymbol{u}} = {\left( {{u_1},{v_1},{p_1},{u_2},{v_2},{p_2},{u_4},{v_4},{p_4},{u_3},{v_3},{p_3}} \right)^{\rm{T}}} $$ (9) 其中, ${u_i}$, ${v_i}$为节点在x和y方向上的位移, ${p_i}$为节点的电势, $i = 1,2,\cdots, 4$. 内聚力界面单元的节点相对位移为单元上下表面的位移差, 可表示为
$$ {{\boldsymbol{\delta}} = {{\boldsymbol{u}}^{{\rm{top}}}} - {{\boldsymbol{u}}^{{\rm{buttom}}}}} $$ (10) 其中, 下表面${{\boldsymbol{u}}^{{\rm{buttom}}}} = {\left( {{u_1},{v_1},{p_1},{u_2},{v_2},{p_2}} \right)^{\rm{T}}}$, 上表面${{\boldsymbol{u}}^{{\rm{top}}}} = {\left( {{u_4},{v_4},{p_4},{u_3},{v_3},{p_3}} \right)^{\rm{T}}}$. 根据节点的相对位移和插值函数可以求得单元内任意一点得相对位移为
$$ {\boldsymbol{\delta}} \left( \xi \right) = \left[ {\begin{array}{*{20}{c}} {{{\Delta }}u\left( \xi \right)} \\ {{{\Delta }}v\left( \xi \right)} \\ {{{\Delta }}p\left( \xi \right)} \end{array}} \right] = {\boldsymbol{H}}\left( \xi \right)\left( {{{\boldsymbol{u}}^{{\rm{top}}}} - {{\boldsymbol{u}}^{{\rm{buttom}}}}} \right) = {\boldsymbol{B}}\left( \xi \right){\boldsymbol{u}} \\ $$ (11) 式中, ${\boldsymbol{B}}\left( \xi \right) = \left[ { - {\boldsymbol{H}}\left( \xi \right)|{\boldsymbol{H}}\left( \xi \right)} \right]$, ${\boldsymbol{H}}\left( \xi \right)$为插值函数矩阵, 其形式为
$$ {{\boldsymbol{H}}\left( \xi \right) = \left[ {\begin{array}{*{20}{c}} {{N_1}\left( \xi \right)}&0&0&{{N_2}\left( \xi \right)}&0&0 \\ 0&{{N_1}\left( \xi \right)}&0&0&{{N_2}\left( \xi \right)}&0 \\ 0&0&{{N_1}\left( \xi \right)}&0&0&{{N_2}\left( \xi \right)} \end{array}} \right]} $$ (12) 式中, ${N_1}\left( \xi \right) = \dfrac{1}{2}\left( {1 - \xi } \right)$, ${N_2}\left( \xi \right) = \dfrac{1}{2}\left( {1 + \xi } \right)$, $ - 1 \leqslant \xi \leqslant 1$.
以上在全局坐标系下求解的相对位移是需要转换到内力单元的局部坐标系下
$$ {{\boldsymbol{\delta}} }_{{\rm{loc}}} = \left[\begin{array}{c}{\delta }_{n}\\ {\delta }_{t}\\ {\delta }_{p}\end{array}\right] = {{\boldsymbol{R}}}^{\text{T}} {\boldsymbol{\delta}} $$ (13) 其中, ${\delta _n}$, ${\delta _t}$为内聚力单元局部坐标系下的正向和切向相对位移, ${\delta _p}$为电势差, ${\boldsymbol{R}}$为转换矩阵, 表达式为
$$ {{\boldsymbol{R}} = \left[ {\begin{array}{*{20}{c}} {{\rm{cos}}\alpha }&{ - {\rm{sin}}\alpha }&0 \\ {{\rm{sin}}\alpha }&{{\rm{cos}}\alpha }&0 \\ 0&0&1 \end{array}} \right]} $$ (14) 式中$\alpha $为内聚力单元变形后的旋转角度, 如图2所示.
采用用户自定义单元子程序UEL将上述内聚力模型引入ABAQUS有限元分析软件, UEL子程序中需要定义单元的节点“力”向量${\boldsymbol{F}}$和单元刚度矩阵${\boldsymbol{K}}$, 其计算表达式如下
$$ \left.\begin{split} &{\boldsymbol{F}} = {{\displaystyle \int }}_{{S}_{c}}^{ }{{\boldsymbol{B}}}^{{\rm{T}}}{\boldsymbol{DBu}} \text{d}S = h{{\displaystyle \int }}_{-1}^{ + 1}{{\boldsymbol{B}}}^{{\rm{T}}}{{\boldsymbol{D}}}_{cz}{\boldsymbol{Bu}}\text{det}\left({\boldsymbol{J}}\right)\text{d}\xi \\ &{\boldsymbol{K}} = \frac{\partial {\boldsymbol{F}}}{\partial {\boldsymbol{u}}} = h{{\displaystyle \int }}_{-1}^{ + 1}{{\boldsymbol{B}}}^{{\rm{T}}}{{\boldsymbol{D}}}_{cz}{\boldsymbol{B}}\text{det}\left({\boldsymbol{J}}\right)\text{d}\xi \end{split}\right\} $$ (15) 式中, ${\boldsymbol{J}}$为雅可比矩阵, $h$为内聚力单元的宽度, ${{\boldsymbol{D}}_{cz}}$为单元的切线模量, 其形式为
$$ \left.\begin{split} &{{\boldsymbol{D}}}_{cz} = \left[\begin{array}{ccc}{D}_{nn}& 0& 0\\ 0& {D}_{tt}& 0\\ 0& 0& {C}_{p}\end{array}\right]\\ &{D}_{nn} = 0 \\ &{D}_{tt} = \partial T/\partial {\delta }_{t} \\ &{C}_{p} = 1/\rho \end{split}\right\} $$ (16) 其中, ${D_{nn}}$和${D_{tt}}$分别为内聚力单元在正向和切向上的切线模量, 由于只考虑内聚力切向力学行为, 因此${D_{nn}} = 0$, $ {D}_{tt} = \partial T/\partial {\delta }_{t} $, 可由式(8)进行求解; ${C_p}$为内聚力单元的电导率. 采用UEL用户自定义单元子程序将力电学内聚力模型嵌入ABAQUS进行有限元仿真运算.
2. 模型参数确定
2.1 界面搭接剪切试验
本文提出的黏弹性力电学内聚力模型需要确定的模型参数有界面的电阻率$\rho $, 黏弹性参数${E_1}$, ${E_2}$和$\eta $, 以及界面损伤起始能量${G_0}$和断裂能${G_c}$. 实验采用的水凝胶贴片为聚丙烯酸水凝胶, 具有良好的保水性和长时间工作的稳定性. 首先通过交流阻抗法测得水凝胶界面的电导率为0.17 S/m. 为了获取界面的黏弹性参数和损伤断裂参数, 对水凝胶黏结界面进行搭接剪切试验, 如图3所示, 水凝胶层厚度1.0 mm, 搭接面的尺寸为25 mm × 25 mm, 采用导电硅胶模拟人体皮肤.
如图4所示为通过剪切实验得到的水凝胶黏结界面牵引力−位移曲线, 可以把界面的力学响应大致可分为3个阶段, 首先是黏弹性力学响应阶段, 牵引力和位移呈近似线性关系; 然后进入界面损伤阶段, 伴随着界面出现局部脱黏现象, 如图4所示曲线B段; 最后是大面积脱黏和滑移阶段, 此时界面基本已经脱黏, 但由于在摩擦力和局部微小黏结力的作用下, 牵引力持续下降直至上下界面完全分离.
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图 4 水凝胶黏结界面剪切牵引力−位移曲线Figure 4. Traction-displacement curve of hydrogel adhesion interface in shear test2.2 黏弹性参数确定
为了获取不同应变率下水凝胶黏结界面的力学行为, 采用不同的加载速率进行搭接剪切, 得到的界面牵引力−位移曲线如图5所示, 可以看出随着加载速率的增大, 界面的刚度和最大牵引力有明显增大的趋势.
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图 5 不同应变率下水凝胶界面牵引力−位移曲线Figure 5. Traction-displacement curves of hydrogel interface at different strain rates采用如图1所示的三元件模型对水凝胶界面的黏弹性力学行为进行描述, 有如下关系式
$$ \qquad\qquad {\delta = {\delta _e} + {\delta _\eta } = \frac{{{T_2}}}{{{E_2}}} + {\delta _\eta }} $$ (17) $$ \qquad\qquad {T = {T_1} + {T_2} = {E_1}\delta + {T_2}} $$ (18) 将方程式(17)对时间进行求导得到
$$ {\dot \delta = \frac{{{{\dot T}_2}}}{{{E_2}}} + {{\dot \delta }_\eta } = \frac{{{{\dot T}_2}}}{{{E_2}}} + \frac{{{T_2}}}{\eta }} $$ (19) 将上式与方程式(18)进行联立, 得到模型的本构方程
$$ {\frac{{\eta \left( {{E_1} + {E_2}} \right)}}{{{E_2}}}\dot \delta + {E_1}\delta = \frac{\eta }{{{E_2}}}\dot T + T} $$ (20) 对上式两边同时求积分可得
$$ {\frac{{\eta \left( {{E_1} + {E_2}} \right)}}{{{E_2}}}\delta + {E_1}\int {\delta {\rm{d}}t} - \frac{\eta }{{{E_2}}}T = \int {T{\rm{d}}t} } $$ (21) 令$a = \dfrac{{\eta \left( {{E_1} + {E_2}} \right)}}{{{E_2}}}$, $b = {E_1}$, $c = - \dfrac{\eta }{{{E_2}}}$, 可根搭接剪切试验得到的牵引力−位移曲线, 分别取其在无损伤阶段的曲线数据, 获取$\delta $, $\displaystyle\int \delta {\text{d}}t$和T, $\displaystyle\int T{\text{d}}t$的值代入方程式(21), 并基于最小二乘法求出$a$, $b$, $c$的值, 从而可求解出${E_1}$, ${E_2}$和$\eta $. 如表1所示为根据不同应变率下的试验曲线, 取牵引位移为0 ~ 1.5 mm阶段的数据, 得到的界面的黏弹性性能参数. 可以看出, 不同应变率下求解得到的水凝胶黏结界面在剪切变形中的黏弹性力学参数是接近的. 通过求平均值, 最终得到${E_1}$ = 2.19 kPa, ${E_2}$ = 4.48 kPa, $\eta $ = 41.67 kPa·s.
表 1 不同应变率下求解得到的界面黏弹性参数Table 1. The viscoelastic parameters obtained by different strain ratesStrain rates ${E_1}$/kPa ${E_2}$/kPa $\eta $/(kPa·s) 0.01 2.11 4.29 70.46 0.02 2.03 5.23 52.09 0.03 2.04 3.07 35.82 0.04 2.23 5.14 26.25 0.05 2.56 4.68 23.75 average value 2.19 4.48 41.67 2.3 损伤断裂参数确定
如图4所示, 以A阶段终点作为界面损伤起始位移${\delta _0}$, 以B阶段的终点作为界面断裂位移${\delta _c}$. 根据不同应变率下水凝胶界面的牵引力−位移曲线数据, 结合界面损伤起始能量和断裂能的表达式${G_0} = \displaystyle\int \nolimits_0^{{\delta _0}} T{\text{d}}\delta$, ${G_c} = \displaystyle\int \nolimits_0^{{\delta _c}} T{\text{d}}\delta$, 可分别求出不同应变率下界面的${G_0}$和${G_c}$的值. 图6和图7分别为不同应变率下重复多次剪切实验得到的${G_0}$和${G_c}$随应变率的变化趋势, 可以看出${G_0}$和${G_c}$都随加载速率的增加而呈现增加的趋势. 本文采用如下公式对其规律进行拟合
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图 6 界面的损伤起始能量${G_0}$随应变率变化规律Figure 6. The variation of damage initiation energy${G_0}$versus strain rate of the interface![]()
图 7 界面的断裂能量${G_c}$随应变率变化规律Figure 7. The variation of fracture energy${G_c}$versus strain rate of the interface$$ {\left.\begin{split} & {{{ {{G_0 ^{1/2}}}}} = {{\left( {G_0^{{\rm{ref}}}} \right)}^{1/2}} + {\gamma _0}\left( {\dot \delta - {{\dot \delta }^{{\rm{ref}}}}} \right)/{{\dot \delta }^{{\rm{ref}}}}} \\ & {{{ {{G_c^{1/2}}} }} = {{\left( {G_c^{{\rm{ref}}}} \right)}^{1/2}} + {\gamma _c}\left( {\dot \delta - {{\dot \delta }^{{\rm{ref}}}}} \right)/{{\dot \delta }^{{\rm{ref}}}}} \end{split} \right\}} $$ (22) 其中, ${\gamma _0}$和${\gamma _c}$分别为${G_0}$和${G_c}$随加载速率的变化系数, $\dot \delta $为加载速率, ${\dot \delta ^{{\rm{ref}}}}$为参考加载速率, $G_0^{{\rm{ref}}}$和$G_c^{{\rm{ref}}}$分别为参考加载速率下界面的损伤起始和断裂能, 本文取${\dot \delta ^{{\rm{ref}}}} = 0.01$, 可得到$G_0^{{\rm{ref}}} = 2.6\; {\rm{J}}/{{\rm{m}}^2}$, $G_c^{{\rm{ref}}} = 53.5 \;{\rm{J}}/{{\rm{m}}^2}$. 通过式(22)对${G_0}$和${G_c}$的变化曲线进行拟合, 可得到${\gamma _0} = 0.18$, ${\gamma _c} = 0.95$.
3. 结果分析与讨论
3.1 力电学内聚力模型有效性验证
采用图8(a)所示的单个单元对构建的力电学内聚力模型的有效性进行验证, 单元尺寸为1 mm × 1 mm. 首先采用不同的应变率(0.01 ~ 0.05) s−1对单元施加剪切变形, 剪切位移为25 mm, 如图8(b)所示, 得到的内聚力单元的牵引力−牵引位移曲线如图9所示.
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图 9 不同应变率下牵引力−牵引位移曲线仿真结果Figure 9. Simulation results of traction-displacement curves under different strain rates可以看出, 不同应变率加载条件下, 内聚力单元的刚度, 最大牵引力都随着应变率的增大而增大, 这与图5所示的实验结果(剔除滑移摩擦阶段)基本是一致的, 验证了模型在描述界面黏弹性力学行为的有效性. 然后采用不同的应变率对单元施加循环剪切位移, 载荷谱如图8(c)所示, 最大位移为2 mm. 在10个循环内, 单元在不同应变率下损伤的演化随循环次数的演化规律如图10所示. 可以看出, 在循环加载下, 单元的损伤随着循环次数的增加而逐步增加, 且损伤累积随着加载速率的增大而逐渐减小的趋势.
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图 10 不同应变率循环加载下内聚力单元损伤演化曲线Figure 10. The damage evolution curve of cohesive element under cyclic loading with different strain rates同时, 为了验证模型描述界面损伤导致导电性能退化行为的有效性, 基于如图3所示的搭接剪切实验方法, 采用不同的加载速率对水凝胶黏结界面进行的循环剪切实验, 并实时监测界面的电阻变化. 循环加载过程中, 最大位移为1 mm, 水凝胶黏结界面基本处于无损伤阶段, 由此得到不同加载速率下界面的电阻变化曲线如图11所示. 可以看出, 界面的电阻随循环加载过程也呈现周期性的波动, 且波动幅度应变率的增大而减小. 式(6)描述了界面在加载过程中的电阻率与应力应变状态的关系, 如图12所示为仿真过程中采用不同$k$值得到应变率为0.01 s−1的加载速率下的内聚力单元的电阻变化曲线, 可以看出内聚力模型可有效表征循环加载过程中内聚力单元电阻的波动, 且电阻的波动幅度随着$k$值的增大而增大. 而当$k$取值为$2.0 \times {10^4}\;{{\Omega }} \cdot {\text{mm}}/{\text{MPa}}$时, 仿真结果得到电阻波动幅值与实验结果一致, 由此可以得到当前应变率下界面应力状对电阻率的影响系数$k$的取值. 通过相同的方法可以得到不同加载速率下的$k$值, 结果如图13所示, $k$值随着应变率的增加, 基本呈线性减小趋势. 通过线性拟合得到$k$值与应变率$\dot \delta $关系为$k = 2.42 - 38.5\dot \delta $, 单位为${10^4}{{\Omega }} \cdot {\text{mm}}/{\text{MPa}}$.
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图 11 不同应变率下无损伤循环加载时界面的电阻变化曲线Figure 11. The resistance variation curve of the interface under cyclic loading with different strain rates in undamaged stage![]()
图 12 应变率0.01 s−1循环加载下不同k值的仿真得到的界面电阻变化曲线与实验结果的对比Figure 12. Comparing between interface resistance variation curve obtained by simulation with different k values and experimental results under cyclic loading with strain rate of 0.01 s−13.2 界面疲劳损伤演化行为预测
基于力电学内聚力模型对水凝胶黏结界面的疲劳损伤演化行为进行仿真预测. 首先对水凝胶黏结界面的界面疲劳损伤演化行为进行试验研究, 试验方法如图14所示, 试样由导电硅胶基底, 水凝胶黏结界面和铜电极组成, 铜电极和水凝胶黏结层的尺寸为10 mm × 10 mm, 试验过程中, 试样的夹持间距为40 mm, 以50%应变对试样进行循环拉伸试验并记录试样电阻的变化.
对界面疲劳损伤的演化行为进行仿真, 建立了如图15所示的有限元模型. 模型由导电硅胶基底, 水凝胶黏结界面, 以及金属电极3部分组成. 水凝胶黏结界面采用力电学内聚力单元进行表征, 通过ABAQUS用户自定义单元子程序UEL实现. 基于稳态分析过程中热传导和电传导方程的相似性, 铜电极和导电硅胶采用常规CPE4T热力耦合单元, 其中, 铜弹性模量为120 GPa, 电导率为5.7 × 107 S/m, 导电硅胶的弹性模量为1.3 MPa, 电导率为9.5 S/m. 仿真过程中, 在金属电极上表面施加1 V电压, 导电硅胶基底下表面电势定义为0, 然后对结构进行循环拉伸加载.
如图16所示为0.05 s−1应变率下经100次循环加载的有限元仿真结果, 可知, 经循环加载后, 水凝胶黏结界面外侧出现了显著的损伤累积, 如图16(a)所示, 从而导致电极与基体之间有效接触面积减小, 使得电流的有效传输路径减少, 如图16(b)所示.
图17为不同加载速率下整体结构的电阻随循环加载次数变化曲线的仿真结果与实验结果对比. 可以看到随着循环次数的增加, 由于界面损伤不断累积, 试样的电阻都呈现出明显的上升趋势, 且随着加载速率的增大, 电阻随循环次数的上升趋势逐渐减缓. 同时单次循环造成的电阻变化令电阻产生波动, 因此电阻呈波动上升, 且应变率越大电阻波动越小. 仿真预测的电阻变化趋势和实验结果基本吻合, 验证了本文提出的力电学内聚力模型在预测由于界面损伤导致的结构电阻变化的有效性, 未来可应用于模拟和预测皮肤贴片式传感器在长期服役过程中由于界面损伤导致的信号漂移趋势, 为传感信号补偿矫正, 提升传感器服役可靠性提供技术支持.
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图 17 不同应变率下电阻随拉伸循环次数变化的仿真与实验结果Figure 17. Simulation and experimental results of resistance variation versus tensile cycles at different strain rates4. 结论
本文为研究疲劳载荷作用下导电水凝胶黏结界面的损伤演化行为, 基于水凝胶黏结层的黏弹性行为, 以及界面损伤对其电导率的影响, 建立了应变率相关的力电学内聚力模型. 基于ABAQUS的UEL子程序实现了模型的数值仿真算法, 并通过实验验证了模型的有效性. 本文研究得到的主要结论如下.
(1)导电水凝胶界面的力学行为具有明显的速率相关性, 通过搭接剪切试验获得了不同加载速率下导电水凝胶黏结界面的力学性能参数和损伤断裂参数. 结果表明, 随着加载速率的增加, 水凝胶黏结强度和断裂能量释放率都明显增大. 通过线性拟合得到了不同应变率下界面损伤断裂参数随应变率的变化规律.
(2)导电水凝胶界面的电导率与应力状态和损伤相关, 循环加载过程中水凝胶界面的电阻随着变形的加载和释放呈现上下波动的变化规律, 通过实验和仿真相结合的方法, 确定了水凝胶界面的电导率随界面牵引力的变化系数$k$. 系数$k$随着应变率的增大, 基本呈线性减小的趋势.
(3)循环加载下导电水凝胶电极贴片的电阻随着循环次数的增加而呈现波动上升趋势, 且加载速率越大, 电阻对循环次数的上升趋势越缓慢, 而且波动越小. 实验结果和有限元仿真结果基本一致.
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图 2 内聚力单元: (a)初始状态和(b)变形状态
Figure 2. Cohesive element: (a) initial state and (b) deformation state
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图 4 水凝胶黏结界面剪切牵引力−位移曲线
Figure 4. Traction-displacement curve of hydrogel adhesion interface in shear test
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图 5 不同应变率下水凝胶界面牵引力−位移曲线
Figure 5. Traction-displacement curves of hydrogel interface at different strain rates
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图 6 界面的损伤起始能量${G_0}$随应变率变化规律
Figure 6. The variation of damage initiation energy${G_0}$versus strain rate of the interface
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图 7 界面的断裂能量${G_c}$随应变率变化规律
Figure 7. The variation of fracture energy${G_c}$versus strain rate of the interface
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图 9 不同应变率下牵引力−牵引位移曲线仿真结果
Figure 9. Simulation results of traction-displacement curves under different strain rates
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图 10 不同应变率循环加载下内聚力单元损伤演化曲线
Figure 10. The damage evolution curve of cohesive element under cyclic loading with different strain rates
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图 11 不同应变率下无损伤循环加载时界面的电阻变化曲线
Figure 11. The resistance variation curve of the interface under cyclic loading with different strain rates in undamaged stage
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图 12 应变率0.01 s−1循环加载下不同k值的仿真得到的界面电阻变化曲线与实验结果的对比
Figure 12. Comparing between interface resistance variation curve obtained by simulation with different k values and experimental results under cyclic loading with strain rate of 0.01 s−1
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图 17 不同应变率下电阻随拉伸循环次数变化的仿真与实验结果
Figure 17. Simulation and experimental results of resistance variation versus tensile cycles at different strain rates
表 1 不同应变率下求解得到的界面黏弹性参数
Table 1 The viscoelastic parameters obtained by different strain rates
Strain rates ${E_1}$/kPa ${E_2}$/kPa $\eta $/(kPa·s) 0.01 2.11 4.29 70.46 0.02 2.03 5.23 52.09 0.03 2.04 3.07 35.82 0.04 2.23 5.14 26.25 0.05 2.56 4.68 23.75 average value 2.19 4.48 41.67 
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