  We review continuum elastic models for the transmission of both external
forces and internal active cellular forces in biopolymer gels, and relate them
to recent experiments. Rather than being exhaustive, we focus on continuum
elastic models for small affine deformations and intend to provide a systematic
continuum method and some analytical perspectives to the study of force
transmissions in biopolymer gels. We start from a very brief review of the
nonlinear mechanics of individual biopolymers and a summary of constitutive
models for the nonlinear elasticity of biopolymer gels. We next show that the
simple 3-chain model can give predictions that well fit the shear experiments
of some biopolymer gels, including the effects of strain-stiffening and
negative normal stress. We then review continuum models for the transmission of
internal active forces that are induced by a spherically contracting cell
embedded in a three-dimensional biopolymer gel. Various scaling regimes for the
decay of cell-induced displacements are identified for linear isotropic and
anisotropic materials, and for biopolymer gels with nonlinear
compressive-softening and strain-stiffening elasticity, respectively. After
that, we present (using an energetic approach) the generic and unified
continuum theory proposed in [Ben-Yaakov, Soft Matter, 2015, 11, 1412] about
how the transmission of forces in the biogel matrix can mediate long-range
interactions between cells with mechanical homeostasis. We show the predictions
of the theory in a special hexagonal multicellular array, and relate them to
recent experiments. Finally, we conclude this paper with comments on the
limitations and outlook of continuum modeling, and highlight the needs of
complementary theoretical approaches such as discrete network simulations to
force transmissions in biopolymer gels and phenomenological active gel theories
for multicellular systems.
