We develop a viscoelastic generalization of the elastic Eshelby inclusion solution, where the inclusion and surrounding matrix are two different viscoelastic solids and the inclusion’s eigenstrain is a time-periodic oscillatory input. The solution exploits the Correspondence Principle of Linear Viscoelasticity and a Discrete Fourier Transform to efficiently capture the steady-state oscillatory behavior of the 3-D mechanical fields. The approach is illustrated here in the context of the recently-developed in vitro Cell-in-Gel system, where an isolated live cardiomyocyte (the inclusion) is paced to contract periodically within a soft hydrogel (the matrix), for the purpose of studying the effect of mechanical load on biochemical signals that regulate contractility. The addition of viscoelasticity improves the fidelity of our previous elastic Eshelby inclusion analysis of the Cell-in-Gel system by accounting for the time-varying fields and the resulting hysteresis and dissipated mechanical energy. This mathematical model is used to study the parametric sensitivities of the relative stiffness of the inclusion, the inclusion’s aspect ratio (slenderness), and the cross-link density of the hydrogel matrix.

我们开发了弹性 Eshelby 夹杂物解的粘弹性推广，其中夹杂物和周围基质是两种不同的粘弹性固体，夹杂物的本征应变是时间周期性振荡输入。该解决方案利用线性粘弹性的对应原理和离散傅里叶变换来有效地捕捉 3-D 机械场的稳态振荡行为。该方法在最近开发的体外Cell-in-Gel 系统，其中分离的活心肌细胞（内含物）在软水凝胶（基质）内定期收缩，目的是研究机械负荷对调节收缩性的生化信号的影响。通过考虑时变场和由此产生的滞后和耗散的机械能，粘弹性的增加提高了我们之前的凝胶中细胞系统的弹性 Eshelby 包含分析的保真度。该数学模型用于研究夹杂物的相对刚度、夹杂物的纵横比（细长）和水凝胶基质的交联密度的参数敏感性。