The problem on constructing new time-effective characteristics of a linear viscoelastic body is considered. The initial medium is modeled as a viscoelastic composite. One its part has properties determined by time-effective moduli of the Lagrangian type, but the other part has properties determined by moduli of the Castigliano type. This model makes it possible to employ the methods of mechanics of composite materials, for example, to formulate the effective Voigt and Reuss moduli and to construct iterative transformations narrowing the Voigt and Reuss fork. These transformations are constructed in such a way that, at each iteration, the inequalities following from the minimum total potential energy principle and the theorem of complementary work are satisfied for the effective moduli. It is shown that, at each moment of time t, the sequences of iteratively transformed Voigt and Reuss moduli converge to the same limit, equal to the geometric mean of their initial values. By the example of the problem on bending of a viscoelastic plate, the approximate solutions obtained on the basis of the new time-effective characteristics found, are compared with an analytical solution. Their good agreement points to a high accuracy of the approximate solutions.

中文翻译：

---

粘弹性体时效模量的修正

考虑了构建线性粘弹性体的新时效特性的问题。初始介质被建模为粘弹性复合材料。其一部分具有由拉格朗日型时效模量确定的特性，但另一部分具有由 Castigliano 型模量确定的特性。该模型使得采用复合材料力学方法成为可能，例如，制定有效的 Voigt 和 Reuss 模以及构建缩小 Voigt 和 Reuss 叉的迭代变换。这些变换的构造方式使得在每次迭代时，有效模量满足最小总势能原理和互补功定理的不等式。可以看出，在每个时刻 t，迭代变换的 Voigt 和 Reuss 模数序列收敛到相同的极限，等于它们初始值的几何平均值。以粘弹性板弯曲问题为例，将根据新的时效特性获得的近似解与解析解进行比较。它们的良好一致性表明近似解的精度很高。