  We use the normal mode formalism to study the shear rheology of the vertex
model for epithelial tissue mechanics in the overdamped linear response regime.
We consider systems with external (e.g., cell-substrate) and internal (e.g.,
cell-cell) dissipation mechanisms, and derive expressions for stresses on cells
due to mechanical and dissipative forces. The semi-analytical method developed
here is, however, general and can be directly applied to study the linear
response of a broad class of soft matter systems with internal and external
dissipation. It involves normal mode decomposition to calculate linear loss and
storage moduli of the system. Specifically, displacements along each normal
mode produce stresses due to elastic deformation and internal dissipation,
which are in force balance with loads due to external dissipation. Each normal
mode responds with a characteristic relaxation timescale, and its rheological
behavior can be described as a combination of a standard linear solid element
due to elastic stresses and a Jeffreys model element due to the internal
dissipative stresses. The total response of the system is then fully determined
by connecting in parallel all the viscoelastic elements corresponding to
individual normal modes. This allows full characterization of the potentially
complex linear rheological response of the system at all driving frequencies
and identification of collective excitations. We show that internal and
external dissipation mechanisms lead to qualitatively different rheological
behaviors due to the presence or absence of Jeffreys elements, which is
particularly pronounced at high driving frequencies. Our findings, therefore,
underscore the importance of microscopic dissipation mechanisms in
understanding the rheological behavior of soft materials and tissues, in
particular.
