1,721,036 research outputs found
On the controllability of a creasing singularity in a nonlinear elastic circular sector
No full textThe deformation of a circular sector into a full self-contacting circle can be sustained in all homogeneous, isotropic, incompressible materials by surface tractions alone. In this class of nonlinear elastic materials, this works investigates the controllability of such a peculiar mapping having uniform constant strains and a creasing singularity. By performing a perturbative analysis based on small–on–large incremental methods, we determine the critical conditions for the normal traction load to trigger a morphological transition from the circular ground state to an elliptic shape. Such predictions are given for neo-Hookean, Gent and polynomial material models to illustrate how both geometrical and physical nonlinearities concur to this elastic instability
Matched asymptotic solution for crease nucleation in soft solids
Full text linkA soft solid subjected to a large compression develops sharp self-contacting folds at its free surface, known as creases. Creasing is physically different from structural elastic instabilities, like buckling or wrinkling. Indeed, it is a fully nonlinear material instability, similar to a phase-transformation. This work provides theoretical insights of the physics behind crease nucleation. Creasing is proved to occur after a global bifurcation allowing the co-existence of an outer deformation and an inner solution with localised self-contact at the free surface. The most fundamental result here is the analytic prediction of the nucleation threshold, in excellent agreement with experiments and numerical simulations. A matched asymptotic solution is given within the intermediate region between the two co-existing states. The self-contact acts like the point-wise disturbance in the Oseen's correction for the Stokes flow past a circle. Analytic expressions of the matching solution and its range of validity are also derived
Geometric control by active mechanics of epithelial gap closure
Full text linkEpithelial wound healing is one of the most important biological processes occurring during the lifetime of an organism. It is a self-repair mechanism closing wounds or gaps within tissues to restore their functional integrity. In this work we derive a new diffuse interface approach for modelling the gap closure by means of a variational principle in the framework of non-equilibrium thermodynamics. We investigate the interplay between the crawling with lamellipodia protrusions and the supracellular tension exerted by the actomyosin cable on the closure dynamics. These active features are modeled as Korteweg forces into a generalised chemical potential. From an asymptotic analysis, we derive a pressure jump across the gap edge in the sharp interface limit. Moreover, the chemical potential diffuses as a Mullins-Sekerka system, and its interfacial value is given by a Gibbs-Thompson relation for its local potential driven by the curvature-dependent purse-string tension. The finite element simulations show an excellent quantitative agreement between the closure dynamics and the morphology of the edge with respect to existing biological experiments. The resulting force patterns are also in good qualitative agreement with existing traction force microscopy measurements. Our results shed light on the geometrical control of the gap closure dynamics resulting from the active forces that are chemically activated around the gap edge.Shedding light on the geometric control of the gap closure dynamics in epithelial wound healing through a novel diffuse interface mathematical model derived by means of a variational principle in the framework of non-equilibrium thermodynamics
Physical principles of morphogenesis in mushrooms
No full textMushroom species display distinctive morphogenetic features. For example, Amanita muscaria and Mycena chlorophos grow in a similar manner, their caps expanding outward quickly and then turning upward. However, only the latter finally develops a central depression in the cap. Here we use a mathematical approach unraveling the interplay between physics and biology driving the emergence of these two different morphologies. The proposed growth elastic model is solved analytically, mapping their shape evolution over time. Even if biological processes in both species make their caps grow turning upward, different physical factors result in different shapes. In fact, we show how for the relatively tall and big A. muscaria a central depression may be incompatible with the physical need to maintain stability against the wind. In contrast, the relatively short and small M. chlorophos is elastically stable with respect to environmental perturbations; thus, it may physically select a central depression to maximize the cap volume and the spore exposure. This work gives fully explicit analytic solutions highlighting the effect of the growth parameters on the morphological evolution, providing useful insights for novel bio-inspired material design
On the existence of elastic minimizers for initially stressed materials
No full textA soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors satisfying the boundary conditions in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed according to the symmetry group of the material at fixed initial stresses. These results allow one to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials
Elastic fingering of a bonded soft disc in traction: Interplay of geometric and physical nonlinearities
No full textThis work provides a mathematical understanding of the elastic fingering provoked by a large axial extension of a soft solid cylinder bonded between rigid plates. In this prototypical system model, a topological transition from a ground axis-symmetric meniscus is quasi-statically controlled by the applied displacement, which acts as the order parameter of a pitchfork bifurcation. Since the isotropic elastic energy becomes nonconvex under finite strains, geometric nonlinearity is of paramount importance for the loss of uniqueness of the solution of the boundary value problem. Nonetheless, physical nonlinearity in the elastic energy is found to exert an opposite stabilizing effect. It indeed penalizes the local stretching at the free boundary that would arise as a consequence of any change of its Gaussian curvature. The theoretical and numerical results are in agreement with recent experimental observations, showing that elastic fingering is strongly affected by the aspect ratio of the disc and can be even suppressed in soft materials with physical nonlinearity
Nonlinear Morphoelastic Theory of Biological Shallow Shells with Initial Stress
Full text linkShallow shells are widely encountered in biological structures, especially during embryogenesis, when they undergo significant shape variations. As a consequence of geometric frustration caused by underlying biological processes of growth and remodeling, such thin and moderately curved biological structures experience initial stress even in the absence of an imposed deformation. In this work, we perform a rigorous asymptotic expansion from three-dimensional elasticitiy to obtain a nonlinear morphoelastic theory for shallow shells accounting for both initial stress and large displacements. By application of the principle of stationary energy for admissible variation of the tangent and normal displacement fields with respect to the reference middle surface, we derive two generalised nonlinear equilibrium equations of the Marguerre-von Kármán type. We illustrate how initial stress distributions drive the emergence of spontaneous mean and Gaussian curvatures which are generally not compatible with the existence of a stress free configuration. We also show how such spontaneous curvatures influence the structural behavior in the solutions of two systems: a saddle-like and a cylindrical shallow shell
The constitutive relations of initially stressed incompressible Mooney-Rivlin materials
No full textInitial stresses originate in soft materials by the occurrence of misfits in the undeformed microstructure. Since the reference configuration is not stress-free, the effects of initial stresses on the hyperelastic behavior must be constitutively addressed. Notably, the free energy of an initially stressed material may not possess the same symmetry group as the one of the same material deforming from a naturally unstressed configuration. This work assumes that the hyperelastic strain energy density is characterized only by the deformation gradient and the initial stress tensor, using an explicit functional dependence on their independent invariants. In particular, we consider a subclass of constitutive behaviors in which the material constants do not depend on the choice of the reference configuration. Within this theoretical framework, a constitutive equation is derived for an initially stressed body that naturally behaves as an incompressible Mooney-Rivlin material. The strain energy densities for initially stressed neo-Hookean and Mooney materials are derived as special sub-cases. By assuming the existence of a virtual state that is naturally stress-free, the resulting strain energy functions are proved to fulfill the required frame-independence constraints for this special class of constitutive models. In the case of plane strain, great simplifications arise in the expression of the constitutive relations. Finally, the resulting constitutive relations prove useful guidelines for designing non-destructive methods for the quantification of the underlying initial stresses in naturally isotropic materials
Optimal surface clothing with elastic nets
Full text linkThe clothing problem aims at identifying the shape of a planar fabric for covering a target surface in the three-dimensional space. It poses significant challenges in various applications, ranging from fashion industry to digital manufacturing. Here, we propose a novel inverse design approach to the elastic clothing problem that is formulated as a constrained optimization problem. We assume that the textile behaves as an orthotropic, nonlinear elastic surface with fibers distributed along its warp and weft threads, and we enforce mechanical equilibrium as a variational problem. The target surface is frictionless, except at its boundary where the textile is pinned, imposing a unilateral obstacle condition for the reactive forces at the target surface. The constrained optimization problem also accounts for an elongation condition of the warp and weft fibers, possibly with bounded shearing angle. We numerically solve the resulting constrained optimization problem by means of a gradient descent algorithm. The numerical results are first validated against known clothing solutions for Chebyshev nets, taking the limit of inextensible fibers. We later unravel the interplay between thread and shear stiffness for driving the optimal cloth shape covering the hemisphere and the hemicatenoid. We show how the metric of these target surfaces strongly affects the resulting distribution of the reaction forces. When considering the limit of covering the full sphere, we show how clothing with elastic nets allows to avoid the onset of singularities in the corresponding Chebyshev net, by developing corners at the cloth boundary
Morphomechanical model of the Torsional c-looping in the embryonic heart
Full text linkBefore septation processes shape its four chambers, the embryonic heart is a straight tube that spontaneously bends and twists breaking the left-right symmetry. In particular, the heart tube is subjected to a cell remodeling inducing ventral bending and dextral torsion during the c-looping phase. In this work we propose a morphomechanical model for the torsion of the heart tube that behaves as a nonlinear elastic body. We hypothesize that this spontaneous looping can be modeled as a mechanical instability due to accumulation of residual stresses induced by the geometrical frustration of tissue remodeling, which mimics the cellular rearrangement within the heart tube. Thus, we perform a linear stability analysis of the resulting nonlinear elastic boundary value problem to determine the onset of c-looping as a function of the geometry of the tube and of the internal remodeling rate. We perform numerical simulations to study the fully nonlinear morphological transition, showing that the soft tube develops a realistic self-contacting looped shape in the physiological range of geometrical parameters
- …
