Conjugate stress of strain \(E^{(3)}= \frac 13 (U^3-I)\) (Q1572719)

In continuum mechanics, a conjugate stress tensor is defined as a symmetric second-order tensor whose trace of its product with the time rate of a chosen strain measure yields the stress power. In this work the authors evaluate the conjugate stress tensor corresponding to the third-order Seth strain measure in terms of either Jaumann stress tensor or Piola-Kirchhoff stress tensor of the second kind. In order to achieve this task, they employ the well-known solution of matrix equation \(UX+XU=C\), where \(U\) and \(C\) are given \(3\times 3\) matrices, and \(U\) is symmetric.