Abstract
AbstractThere is growing experimental evidence for non‐affine deformations occurring in different types of fibrous soft tissues; meaning that the fiber orientations do not follow the macroscopic deformation gradient. Suitable mathematical modeling of this phenomenon is an open challenge, which we here tackle in the framework of continuum micromechanics. From a rate‐based analogon of Eshelby's inhomogeneity problem, we derive strain and spin concentration tensors relating macroscopic strain rate tensors applied to the boundaries of a Representative Volume Element (RVE), to strain rates and spins within the tissue microstructure, in particular those associated with fiber rotations due to external mechanical loading. After presenting suitable algorithms for integrating the resulting rate‐type governing equations, a first relevance check of the novel modeling approach is undertaken, by comparison of model results to recent experiments performed on the adventitia layer of rabbit carotid tissue.
Highlights
Sound continuum mechanics models for soft tissues, as reviewed and developed over the last decades by Holzapfel and co-workers,[20,30,40] have resulted in a thriving research community with increasing impact in biomedical engineering and biomedicine, and so in cardiology.[32,33,35,47]In this context, the importance of tissue anisotropy has been more and more understood and considered, thereby aiming at a more and more resolved representation of the collagen and elastin fiber morphologies evolving during tissue deformation
These interesting observations have motivated the theoretical developments described in the present paper: Here, the continuum micromechanics of fiber-matrix composites and polycrystals, which turned out as a versatile tool for predicting the material
The paper is organized as follows: In Section 2, we extend the classical notions of continuum micromechanics, from strains to strain rates, providing the basis for a continuum micromechanics theory applicable to microheterogeneous materials undergoing large deformations and largeconfigurational changes
Summary
Sound continuum mechanics models for soft tissues, as reviewed and developed over the last decades by Holzapfel and co-workers,[20,30,40] have resulted in a thriving research community with increasing impact in biomedical engineering and biomedicine, and so in cardiology.[32,33,35,47]. While this concept frequently provided very satisfactory results, in particular so in the context of mitral valve leaflet modeling,[39] we note several experimental observations where the fibers do not follow such a deformation pattern These observations concern the adventitia layer of carotid arteries,[36,37] and tissues beyond the cardiac realm, such as the human liver capsule and murine skin.[34] Typically, the aforementioned deformation patterns are associated with large shear strains in the soft matrix being situated in-between the fibers; and such discrepancies between macroscopic and microscopic strains, being incompatible with affine deformation characteristics, have been reported for tendon fascicles.[23,24,55] These interesting observations have motivated the theoretical developments described in the present paper: Here, the continuum micromechanics of fiber-matrix composites and polycrystals, which turned out as a versatile tool for predicting the material. ∂Ω ω ωf ib,r ωI ωnf ib,r ωlog ωnf,ilbo,gr constant axial deformation rate time increment eigenstrain rate tensor co-latitudinal angle co-latitudinal angle of the r-th fiber phase evaluated at time instant tn circumferential stretch axial stretch shear modulus of phase r coefficients related to the definition of nlog hypoelastic Poisson's ratio of the matrix phase of the Eshelby problem hypoelastic Poisson's ratio of the matrix phase current mass density initial mass density microscopic Cauchy stress macroscopic Cauchy stress longitudinal angle longitudinal angle of the r-th fiber phase, evaluated at time instant tn Helmholtz free energy per unit mass volume of RVE surface of RVE Eulerian spin tensor Eulerian spin tensor averaged over the r-th fiber phase homogeneous Eulerian spin tensor in the inclusion I Eulerian spin tensor averaged over the r-th fiber phase, evaluated at time instant tn logarithmic material rotation tensor logarithmic material rotation tensor in the r-th fiber phase, evaluated at time instant tn
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