Mathematical modeling of the human heart and its function can expand our understanding of various cardiac diseases, which remain the most common cause of death in the developed world. Like other ...physiological systems, the heart can be understood as a complex multiscale system involving interacting phenomena at the molecular, cellular, tissue, and organ levels. This article addresses the numerical modeling of many aspects of heart function, including the interaction of the cardiac electrophysiology system with contractile muscle tissue, the sub-cellular activation–contraction mechanisms, as well as the hemodynamics inside the heart chambers. Resolution of each of these sub-systems requires separate mathematical analysis and specially developed numerical algorithms, which we review in detail. By using specific sub-systems as examples, we also look at systemic stability, and explain for example how physiological concepts such as microscopic force generation in cardiac muscle cells, translate to coupled systems of differential equations, and how their stability properties influence the choice of numerical coupling algorithms. Several numerical examples illustrate three fundamental challenges of developing multiphysics and multiscale numerical models for simulating heart function, namely: (i) the correct upscaling from single-cell models to the entire cardiac muscle, (ii) the proper coupling of electrophysiology and tissue mechanics to simulate electromechanical feedback, and (iii) the stable simulation of ventricular hemodynamics during rapid valve opening and closure.
We discuss connections between the strong ellipticity condition and the infinitesimal instability within the nonlinear strain gradient elasticity. The strong ellipticity (SE) condition describes the ...property of equations of statics whereas the infinitesimal stability is introduced as the positive definiteness of the second variation of an energy functional. Here we establish few implications which simplify the further analysis of stability using formulated SE conditions. The results could be useful for the analysis of solutions of homogenized models of beam-lattice materials at different scales.
•Nonlinear strain gradient elasticity is studied.•First- and second-order strong ellipticity conditions (1st and 2nd SE) are formulated.•Relations between SE and infinitesimal stability is clarified.•Infinitesimal stability implies the weak form of 2nd SE.•1st and 2nd SE imply infinitesimal stability of affine deformations.•Infinitesimal stability could be proven even without SE.
•The proposed Peidynamic model is validated by comparing the fracture toughness and the fracture forms of multilayer graphene sheets with existing experiments.•Asynchronous crack propagation with ...independent path observed in multilayer graphene sheets is a unique mechanism for strengthening the fracture property.•The Peridynamic theory is extended for the first time to investigate the in-plane fracture of large-sized nano multilayer graphene sheets.
Understanding the fracture properties of graphene sheets is a crucial step towards their practical applications. However, due to the limitations of experimental operations and all-atom (AA) methods, investigating the fracture of large-sized nano graphene sheets remains a formidable challenge. Especially, study on the layer-by-layer fracture of multi-layer graphene sheets (MLGS) is nearly impossible. To overcome this challenge, a peridynamic (PD) model is proposed in this study, which comprises the intra-layer part and the inter-layer part. The proposed PD model is validated by comparing the fracture toughness and the fracture forms of MLGS with existing experiments. It is found that the uniaxial tensile stress-strain curve of pre-cracked MLGS is closely related to the number of graphene layers in MLGS. The fracture property of MLGS can be enhanced by increasing the number of graphene layers, reducing the pre-crack length and blunting the pre-crack tip. Notably, asynchronous crack propagation with independent path observed in MLGS is a unique mechanism for strengthening the fracture property, which is distinct from monolayer graphene sheet. In this work, the PD theory is extended for the first time to investigate the in-plane fracture of large-sized nano MLGS.
Animal cells in tissues are supported by biopolymer matrices, which typically exhibit highly nonlinear mechanical properties. While the linear elasticity of the matrix can significantly impact cell ...mechanics and functionality, it remains largely unknown how cells, in turn, affect the nonlinear mechanics of their surrounding matrix. Here, we show that living contractile cells are able to generate a massive stiffness gradient in three distinct 3D extracellular matrix model systems: collagen, fibrin, and Matrigel. We decipher this remarkable behavior by introducing nonlinear stress inference microscopy (NSIM), a technique to infer stress fields in a 3D matrix from nonlinear microrheology measurements with optical tweezers. Using NSIM and simulations, we reveal large long-ranged cell-generated stresses capable of buckling filaments in the matrix. These stresses give rise to the large spatial extent of the observed cell-induced matrix stiffness gradient, which can provide a mechanism for mechanical communication between cells.
A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that ...the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.
In this paper, a consistent finite-strain plate theory for incompressible hyperelastic materials is formulated. Within the framework of nonlinear elasticity and through a variational approach, the ...three-dimensional (3D) governing system is derived. Series expansions of the independent variables in the governing system are taken about the bottom surface of the plate, which, together with some further manipulations, yield a 2D vector plate equation. Suitable position and traction boundary conditions on the edge are also proposed. The 2D plate system ensures that each term in the variations of the generalized potential energy functional attains the required asymptotic order. The associated weak formulation of the plate model is also derived, and can be simplified to accommodate distinct types of practical edge conditions. To demonstrate the validity of the derived 2D vector plate system, the pure finite-bending of a plate made of an incompressible neo-Hookean material is studied. Both the exact solutions and the plate solutions of the problem are obtained. Through some comparisons, it is found that the plate theory can provide second-order correct results.
Growth-induced instabilities are ubiquitous in biological systems and lead to diverse morphologies in the form of wrinkling, folding, and creasing. The current work focusses on the mechanics behind ...growth-induced wrinkling instabilities in an incompressible annular hyperelastic plate. The governing differential equations for a two-dimensional plate system are derived using a variational principle with no apriori kinematic assumptions in the thickness direction. A linear bifurcation analysis is performed to investigate the stability behaviour of the growing hyperelastic annular plate by considering both axisymmetric and asymmetric perturbations. The resulting differential equations are then solved numerically using the compound matrix method to evaluate the critical growth factor that leads to wrinkling. The effect of boundary constraints, thickness, and radius ratio of the annular plate on the critical growth factor is studied. For most of the considered cases, an asymmetric bifurcation is the preferred mode of instability for an annular plate. Our results are useful to model the physics of wrinkling phenomena in growing planar soft tissues, swelling hydrogels, and pattern transition in two-dimensional films growing on an elastic substrate.
Display omitted
•A computational framework for growing nonlinear plates is developed.•Growth under boundary constraints causes instability leading to wrinkling patterns.•Number and shape profile of wrinkles can be tuned by the annular plate dimensions.
PDF
