In an elastic medium, it was proved that the stiffness tensor is symmetric with respect to the exchange of the first pair of indices and the second pair of indices, but the proof does not apply to a viscoelastic medium. In this paper, we thus derive the representation theorem for viscoelastic waves in a medium with a non-symmetric stiffness matrix. The representation theorem expresses the wave field at a receiver, situated inside a subset of the definition volume of the viscoelastodynamic equation, in terms of the volume integral over the subset and the surface integral over the boundary of the subset. For the given medium, we define the complementary medium corresponding to the transposed stiffness matrix. We define the frequency-domain complementary Green function as the frequency-domain Green function in the complementary medium. We then derive the provisional representation theorem as the relation between the frequency-domain wave field in the given medium and the frequency-domain complementary Green function. This provisional representation theorem yields the reciprocity relation between the frequency-domain Green function and the frequency-domain complementary Green function. The final version of the representation theorem is then obtained by inserting the reciprocity relation into the provisional representation theorem.

在弹性介质中，已证明刚度张量相对于第一对指标和第二对指标的交换是对称的，但是该证明不适用于粘弹性介质。因此，在本文中，我们推导了具有非对称刚度矩阵的介质中粘弹性波的表示定理。表示定理根据子集上的体积积分和子集边界上的表面积分，表达了位于粘弹动力学方程定义体积子集内的接收器处的波场。对于给定的介质，我们定义对应于转置刚度矩阵的互补介质。我们将频域互补格林函数定义为互补介质中的频域格林函数。然后，我们将临时表示定理推导为给定介质中的频域波场与频域互补格林函数之间的关系。该临时表示定理产生了频域格林函数与频域互补格林函数之间的互易关系。然后通过将互易关系插入到临时表示定理中来获得表示定理的最终版本。