A fibril-based structural constitutive theory reveals the dominant role of network characteristics on the mechanical behavior of fibroblast-compacted collagen gels

In this paper, we present a general, fibril-based structural constitutive theory which accounts for three material aspects of crosslinked filamentous materials: the single fibrillar force response, the fibrillar network model, and the effects of alterations to the fibrillar network. In the case of the single fibrillar response, we develop a formula that covers the entropic and enthalpic deformation regions, and introduce the relaxation phase to explain the observed force decay after crosslink breakage. For the filamentous network model, we characterize the constituent element of the fibrillar network in terms its end-to-end distance vector and its contour length, then decompose the vector orientation into an isotropic random term and a specific alignment, paving the way for an expanded formalism from principal deformation to general 3D deformation; and, more important, we define a critical core quantity over which macroscale mechanical characteristics can be integrated: the ratio of the initial end-to-end distance to the contour length (and its probability function). For network alterations, we quantitatively treat changes in constituent elements and relate these changes to the alteration of network characteristics. Singular in its physical rigor and clarity, this constitutive theory can reproduce and predict a wide range of nonlinear mechanical behavior in materials composed of a crosslinked filamentous network, including: stress relaxation (with dual relaxation coefficients as typically observed in soft tissues); hysteresis with decreasing maximum stress under serial cyclic loading; strain-stiffening under uniaxial tension; the rupture point of the structure as a whole; various effects of biaxial tensile loading; strain-stiffening under simple shearing; the so-called "negative normal stress" phenomenon; and enthalpic elastic behaviors of the constituent element. Applied to compacted collagen gels, the theory demonstrates that collagen fibrils behave as enthalpic elasticas with linear elasticity within the gels, and that the macroscale nonlinearity of the gels originates from the curved fibrillar network. Meanwhile, the underlying factors that determine the mechanical properties of the gels are clarified. Finally, the implications of this study on the enhancement of the mechanical properties of compacted collagen gels and on the cellular mechanics with this model tissue are discussed.