The stress–strain behaviors of viscoelastic materials are often simulated using a model composed of various combinations of springs and dampers. With the increase in the number of springs and dampers, the viscoelastic characteristics of the model will approach those of the actual material. This study discusses how to obtain the differential constitutive equation of a viscoelastic model composed of any number of springs and dampers. First, the general viscoelastic model is regarded as the combination of various Kelvin units. The viscoelastic model is then transformed into a digraph. Based on the relationships between the independent path of the digraph and the strain equation of the viscoelastic model and between the closed enclosure and the stress equation, the derivation of the constitutive equation is transformed into operations involving the incidence matrix of the digraph. Finally, the coefficients of the linear differential operator of the constitutive equation of the viscoelastic model can be obtained by block matrix operations. This method is suitable for computer programming and has a certain significance for accurately constructing viscoelastic models of engineering materials.

粘弹性材料的应力-应变行为通常使用由弹簧和阻尼器的各种组合组成的模型来模拟。随着弹簧和阻尼器数量的增加，模型的粘弹性特性将接近实际材料的粘弹性特性。本研究讨论如何获得由任意数量的弹簧和阻尼器组成的粘弹性模型的微分本构方程。首先，一般粘弹性模型被视为各种开尔文单位的组合。然后将粘弹性模型转换为有向图。根据有向图独立路径与粘弹性模型应变方程、封闭外壳与应力方程之间的关系，将本构方程的推导转化为有向图关联矩阵的运算。最后，通过分块矩阵运算即可得到粘弹性模型本构方程的线性微分算子的系数。该方法适合计算机编程，对于准确构建工程材料的粘弹性模型具有一定的意义。