Mathematics and Mechanics of Solids

Many living structures are coated by thin films, which have distinct mechanical properties from the bulk. In particular, these thin layers may grow faster or slower than the inner core. Differential growth creates a balanced interplay between tension and compression and plays a critical role in enhancing structural rigidity. Typical examples with a compressive outer surface and a tensile inner core are the petioles of celery, caladium, or rhubarb. While plant physiologists have studied the impact of tissue tension on plant rigidity for more than a century, the fundamental theory of growing surfaces remains poorly understood. Here, we establish a theoretical and computational framework for continua with growing surfaces and demonstrate its application to classical phenomena in plant growth. To allow the surface to grow independently of the bulk, we equip it with its own potential energy and its own surface stress. We derive the governing equations for growing surfaces of zero thickness and obtain their spatial discretization using the finite-element method. To illustrate the features of our new surface growth model we simulate the effects of growth-induced longitudinal tissue tension in a stalk of rhubarb. Our results demonstrate that different growth rates create a mechanical environment of axial tissue tension and residual stress, which can be released by peeling off the outer layer. Our novel framework for continua with growing surfaces has immediate biomedical applications beyond these classical model problems in botany: it can be easily extended to model and predict surface growth in asthma, gastritis, obstructive sleep apnoea, brain development, and tumor invasion. Beyond biology and medicine, surface growth models are valuable tools for material scientists when designing functionalized surfaces with distinct user-defined properties.

We model cardiac muscle contractions in the framework of finite elasticity with large distortions and couple a mechanical model with reaction–diffusion equations representing electrophysiological activity. Both models are implemented using anisotropic constitutive relations: we use stress–strain relations for fiber-reinforced materials, and anisotropic diffusion tensors for both the membrane potential and calcium ions. The effects of these choices on the electromechanical behavior are presented and discussed.

At any point in space the material properties of the myocardium are characterized as orthotropic, that is, there are three mutually orthogonal axes along which both electrical and mechanical parameters differ. To investigate the role of spatial structural heterogeneity in an orthotropic material, electro-mechanically coupled models of the left ventricle (LV) were used. The implemented models differed in their arrangement of fibers and sheets in the myocardium, but were identical otherwise: (i) a generic homogeneous model, where a rule-based method was applied to assign fiber and sheet orientations, and (ii) a heterogeneous model, where the assignment of the orthotropic tissue structure was based on experimentally obtained fiber/sheet orientations. While both models resulted in pressure–volume loops and metrics of global mechanical function that were qualitatively and quantitatively similar and matched well with experimental data, the predicted deformations were strikingly different between these models, particularly with regard to torsion. Thus, the simulation results strongly suggest that heterogeneous structure properties play an important nonnegligible role in LV mechanics and, consequently, should be accounted for in computational models.

The application of nonlinear elasticity concepts to the mechanical modeling of soft biomaterials is currently the subject of intense investigation. For fibrous soft biomaterials, some specific strain-energy density models for anisotropic hyperelastic materials have been proposed in the literature that are particularly useful as they reflect the typical J-shaped stress–stretch stiffening response due to collagen fibers that is observed experimentally. These models have the feature of incorporating the increased stiffness of the collagen fibers with deformation and involve a maximum fiber stretch (or locking stretch). Here we apply such models to the analysis of the fracture or tearing of fibrous soft biomaterials. Attention is focused on a particular fracture test namely the trousers test where two legs of a cut specimen are pulled horizontally apart out of the plane of the test piece. It is shown that, in general, the location of the cut in the specimen plays a key role in the fracture analysis, and that the effect of the cut position depends crucially on the degree of strain–stiffening. This dependence is characterized explicitly for the specific strain–stiffening constitutive models considered. The effects of anisotropy and strain–stiffening on the fracture toughness (resistance to tearing) are also examined.

Quasi-static motions are motions for which inertial effects can be neglected, to the first order of approximation. It is crucial to be able to identify the quasi-static regime in order to efficiently formulate constitutive models from standard material characterization test data. A simple non-dimensionalization of the equations of motion for continuous bodies yields non-dimensional parameters which indicate the balance between inertial and material effects. It will be shown that these parameters depend on whether the characterization test is strain- or stress-controlled and on the constitutive model assumed. A rigorous definition of quasi-static behaviour for both strain- and stress-controlled experiments is obtained for elastic solids and a simple form of a viscoelastic solid. Adding a rate dependence to a constitutive model introduces internal time-scales and this complicates the identification of the quasi-static regime. This is especially relevant for biological soft tissue as this tissue is typically modelled as being a non-linearly viscoelastic solid. The results obtained here are applied to some problems in cardiac mechanics and to data obtained from simple shear experiments on porcine brain tissue at high strain rates.

Fiber alignment in biological tissues is created and maintained by the cells, which respond to mechanical stimuli arising from properties of the surrounding material. This coupling between mechanical anisotropy and tissue remodeling can be modeled in nonlinear elasticity by a fiber-reinforced hyperelastic material where remodeling is represented as the change in fiber orientation. Here, we study analytically a simple model of fiber reorientation in a rectangular elastic tissue reinforced by two symmetrically arranged families of fibers subject to constant external loads. In this model, the fiber direction tends to align with the maximum principal stretch. We characterize the global behaviour of the system for all material parameters and applied loads, and show that provided the fibers are tensile initially, the system converges to a stable equilibrium, which corresponds to either complete or intermediate fiber alignment.

We develop an elastic–isotropic rod model for superhelical DNA structures where the helical angle is varying as a function of the arc-length. Our motivation for a variable helical angle comes from some experiments and simulations on DNA braids where complex superhelical structures have been observed. The helical solutions are minimizers of a free energy consisting of elastic, entropic and electrostatic terms. These minimizers are obtained within a variational framework where the end-points of the helices are allowed to be variable so that the length of the superhelix is computed as part of the solution. Considering variable curvature solutions brings up the possibility of finding more complex DNA structures because for two (or more) interwound helices there is a geometrical lock-up helical angle which puts a limit on the length of a superhelix. We perform calculations with different ionic concentrations and study the effects of lock up for braided structures. We also extend the variable curvature model to study the formation of plectonemes in the presence of multivalent salts where the supercoiling radius can be regarded as a constant prescribed by the balance of attractive and repulsive forces in DNA–DNA interactions, and provide analytical solutions in terms of elliptic functions for the supercoil parameters.