arXiv subjects

Timothy J. Healey

Publications and source records attributed to Timothy J. Healey.

18 recordsLinked to original sources

arXiv2024

Sharp-interface limits for brittle fracture via the inverse-deformation formulation

We derive sharp-interface models for one-dimensional brittle fracture via the inverse-deformation approach. Methods of Gamma-convergence are employed to obtain the singular limits of previously proposed models. The latter feature a local, non-convex stored energy of inverse strain, augmented by small interfacial energy, formulated in terms of the inverse-strain gradient. They predict spontaneous fracture with exact crack-opening discontinuities, without the use of damage (phase) fields or pre-existing cracks; crack faces are endowed with a thin layer of surface energy. The models obtained herewith inherit the same properties, except that surface energy is now concentrated at the crack faces. Accordingly, we construct energy-minimizing configurations. For a composite bar with a breakable layer, our results predict a pattern of equally spaced cracks whose number is given as an increasing function of applied load.

math.AP↗

arXiv2023

Nonlinearly Elastic Maps: Energy Minimizing Configurations of Membranes on Prescribed Surfaces

We propose a model for nonlinearly elastic membranes undergoing finite deformations while confined to a regular frictionless surface in $\mathbb{R}^3$. This is a physically correct model of the analogy sometimes given to motivate harmonic maps between manifolds. The proposed energy density function is convex in the strain pair comprising the deformation gradient and the local area ratio. If the target surface is a plane, the problem reduces to 2-dimensional, polyconvex nonlinear elasticity addressed by J.M. Ball. On the other hand, the energy density is not rank-one convex for unconstrained deformations into $\mathbb{R}^3$. We show that the problem admits an energy-minimizing configuration when constrained to lie on the given surface. For a class of Dirichlet problems, we demonstrate that the minimizing deformation is a homeomorphism onto its image on the given surface and establish the weak Eulerian form of the equilibrium equations.

math.AP↗

arXiv2023

An Existence Theorem for a Class of Wrinkling Models for Highly Stretched Elastic Sheets

We consider a class of models motivated by previous numerical studies of wrinkling in highly stretched, thin rectangular elastomer sheets. The model used is characterized by a finite-strain hyperelastic membrane energy perturbed by small bending energy. Without bending energy, the stored-energy density is not rank-one convex for general spatial deformations but reduces to a polyconvex function when restricted to the plane, i.e., two-dimensional hyperelasticity. In addition, it grows unbounded as the local area ratio approaches zero. The small-bending component of the model is the same as that in the classical von K\'arm\'an model. Here we prove the existence of energy minima for a general class of such models.

math.AP↗

arXiv2022

Energy Minimizing Configurations for Single-Director Cosserat Shells

We consider a class of single-director Cosserat shell models accounting for both curvature and finite mid-plane strains. We assume a polyconvexity condition for the stored-energy function that reduces to a physically correct membrane model in the absence of bending. With appropriate growth conditions, we establish the existence of energy minimizers. The local orientation of a minimizing configuration is maintained via the blowup of the stored energy as a version of the local volume ratio approaches zero. Finally, we specialize our results to three constrained versions of the theory commonly employed in the subject.

math.AP↗

arXiv2022

Nucleation and development of multiple cracks in thin composite fibers

We study the nucleation and development of crack patterns in thin composite fibers under tension in this work. A fiber comprises an elastic core and an outer layer of a weaker brittle material. In recent tensile experiments on such composites, multiple cracks were observed to develop simultaneously on the outer layer. We propose here a simple one-dimensional model to predict such phenomenon. We idealize the problem as two axially loaded rods coupled by a linear interfacial condition. The latter can be regarded as an adhesive that resists slip between the two materials. One rod is modeled as a brittle material, and the other a linearly elastic material, both undergoing finite deformations.

math.NA↗

arXiv2022

A group-theoretic approach to the bifurcation analysis of elastic frameworks with symmetry

We present a general approach to the bifurcation analysis of elastic frameworks with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of elastic frameworks is not straightforward. We consider frames comprising assemblages of Cosserat rods, and the main difficulty arises from the nonlinear configuration space, due to the presence of (cross-sectional) rotation fields. We avoid this via a single-rod formulation, developed earlier by one of the authors, whereby the governing equations are embedded in a linear space. The field equations comprise the assembly of all rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the symmetry-group action, and the implementation of group-theoretic methods is now natural within the linear-space context. All potential generic, symmetry-breaking bifurcations are predicted apriori. We then employ an open-source path-following code, which can detect and compute simple, onedimensional bifurcations; multiple bifurcation points are beyond its capabilities. For the latter, we construct symmetry-reduced problems implemented by appropriate substructures. Multiple bifurcations are rendered simple, and the path-following code is again applicable. We first analyze a simple tripod framework, providing all details of our methodology. We then treat a hexagonal space frame via the same approach. The tripod and the hexagonal dome both exhibit simple and double bifurcation points.

physics.app-ph↗

arXiv2021

Existence of Weak Solutions for Non-Simple Elastic Surface Models

We consider a class of models for nonlinearly elastic surfaces in this work. We have in mind thin, highly deformable structures modeled directly as two-dimensional nonlinearly elastic continua, accounting for finite membrane and bending strains and thickness change. We assume that the stored-energy density is polyconvex with respect to the second gradient of the deformation, and we require that it grow unboundedly as the local area ratio approaches zero. For sufficiently fast growth, we show that the latter is uniformly bounded away from zero at an energy minimizer. With this in hand, we rigorously derive the weak form of the Euler-Lagrange equilibrium equations.

math.AP↗

arXiv2020

Stable Spatially Localized Configurations in a Simple Structure -- A Global Symmetry-Breaking Approach

We revisit the classic stability problem of the buckling of an inextensible, axially compressed beam on a nonlinear elastic foundation with a semi-analytical approach to understand how spatially localized deformation solutions emerge in many applications in mechanics. Instead of a numerical search for such solutions using arbitrary imperfections, we propose a systematic search using branch-following and bifurcation techniques along with group-theoretic methods to find all the bifurcated solution orbits (primary, secondary, etc.) of the system and to examine their stability and hence their observability. Unlike previously proposed methods that use multi-scale perturbation techniques near the critical load, we show that to obtain a spatially localized deformation equilibrium path for the perfect structure, one has to consider the secondary bifurcating path with the longest wavelength and follow it far away from the critical load. The novel use of group-theoretic methods here illustrates a general methodology for the systematic analysis of structures with a high degree of symmetry.

nlin.PS↗

arXiv2020

The inverse-deformation approach to fracture

We propose a one-dimensional, nonconvex elastic constitutive model with higher gradients that can predict spontaneous fracture at a critical load via a bifurcation analysis. It overcomes the problem of discontinuous deformations without additional field variables, such as damage or phase-field variables, and without a priori specified surface energy. Our main tool is the use of the inverse deformation, which can be extended to be a piecewise smooth mapping even when the original deformation has discontinuities describing cracks opening. We exploit this via the inverse-deformation formulation of finite elasticity due to Shield and Carlson, including higher gradients in the energy. The problem is amenable to a rigorous global bifurcation analysis in the presence of a unilateral constraint. Fracture under hard loading occurs on a bifurcating solution branch at a critical applied stretch level and fractured solutions are found to have surface energy arising from higher gradient effects.

math.AP↗

arXiv2019

Hidden Asymptotic Symmetry in Long Elastic Beams on Softening Foundations

Transverse wrinkles are known to appear in thin rectangular elastic sheets when stretched in the long direction. Numerically computed bifurcation diagrams for extremely thin, highly stretched films indicate entire orbits of wrinkling solutions, cf. Healey, et. al. [J. Nonlinear Sci., 23 (2013), pp.~777--805]. These correspond to arbitrary phase shifts of the wrinkled pattern in the transverse direction. While such behavior is normally associated with problems in the presence of a continuous symmetry group, an unloaded rectangular sheet possesses only a finite symmetry group. In order to understand this phenomenon, we consider a simpler problem more amenable to analysis -- a finite-length beam on a nonlinear softening foundation under axial compression. We first obtain asymptotic results via amplitude equations, that are valid as a certain non-dimensional beam length becomes sufficiently large. We deduce that any two phase-shifts of a solution differ from one another by exponentially small terms in that length. We validate this observation with numerical computations, indicating the presence of solution orbits for sufficiently long beams. We refer to this as "hidden asymptotic symmetry".

cond-mat.soft↗

arXiv2018

Existence of Global Symmetry-Breaking Solutions in an Elastic Phase-Field Model for Lipid Bilayer Vesicles

We consider a well known model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field with finite curvature elasticity. We prove the existence of a plethora of equilibria, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations, via global bifurcation from the spherical state. To the best of our knowledge, this constitutes the first rigorous existence results for this class of problems. We overcome several difficulties in carrying this out. Due to inherent in-plane fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. Instead we adopt a singularity-free radial-map description that effectively eliminates the grossly underdetermined in-plane fluid deformation. The resulting governing equations comprise a quasi-linear elliptic system with lower-order nonlinear constraints. We then show the equivalence of our problem to that of finding the zeros of compact vector field. The latter is not routine; we obtain certain spectral estimates and then shift the principle part of the operator. With this in hand, we combine well known group-theoretic ideas for symmetry-breaking with global bifurcation theory to obtain our results.

math.AP↗

arXiv2018

Classical Injective Solutions in the Large in Incompressible Nonlinear Elasticity

We consider a general class of parametrized displacement boundary value problems in incompressible nonlinear elasticity. We prove the existence of an unbounded solution branch of classical injective solutions emanating from the unforced stress-free reference configuration, which constitutes a sharp global implicit function theorem. The ramifications of this are: There exists at least one solution for any given applied loading, and or there exists solutions of arbitrarily large norm for some finite loading. We refer to this as solutions in the large. The nonlinear constraint equation enforcing incompressibility obviates the direct use of either the Leray-Schauder degree or an oriented degree previously developed for compressible problems. Instead we employ the nonlinear Fredholm degree for proper maps developed by Fitzpatrick, et. al. We establish the requisite admissibility properties in the presence of the constraint equation before proving the main results.

math.AP↗

arXiv2018

The Mullins effect in the wrinkling behavior of highly stretched thin films

Recent work demonstrates that finite-deformation nonlinear elasticity is essential in the accurate modeling of wrinkling in highly stretched thin films. Geometrically exact models predict an isola-center bifurcation, indicating that for a bounded interval of aspect ratios only, stable wrinkles appear and then disappear as the macroscopic strain is increased. This phenomenon has been verified in experiments. In addition, recent experiments revealed the following striking phenomenon: For certain aspect ratios for which no wrinkling occurred upon the first loading, wrinkles appeared during the first unloading and again during all subsequent cyclic loading. Our goal here is to present a simple pseudo-elastic model, capturing the stress softening and residual strain observed in the experiments, that accurately predicts wrinkling behavior on the first loading that differs from that under subsequent cyclic loading. In particular for specific aspect ratios, the model correctly predicts the scenario of no wrinkling during first loading with wrinkling occurring during unloading and for all subsequent cyclic loading.

cond-mat.soft↗

arXiv2016

The Shape of a Möbius Strip via Elastic Rod Theory Revisited

In 1993 Mahadevan and Keller used the Kirchhoff rod theory to predict the shape of a Möbius band. Starting from the solution for a square cross-section (isotropic), they employ numerical continuation in the cross-sectional aspect ratio in order to approach the solution for a thin strip. Certain smoothly varying configurations are obtained. More recently in 2007, Starostin and van der Heijden pointed out that an actual Möbius band "localizes" into a nearly flat triangular configuration as the ratio of the strip width to center-line circumference is no longer small. Accordingly they return to the developable, thin-plate model of Sadowsky and Wunderlich, obtaining such localized shapes. In this work we strike a middle ground between these two approaches. We employ the standard two-director (special) Cosserat model, and we also use the cross-sectional aspect ratio as a numerical continuation parameter. Nonetheless, we are able to capture both the smoothly varying shapes and the localized, nearly triangular configurations. Our point of departure is that we do not always employ the strength-of-materials formula for the torsional stiffness - a standard feature of the Kirchhoff theory. Our key observation comes from the usual Cosserat ansatz in light of developability, suggesting an increasing torsional stiffness ratio (as well as one increasing bending stiffness ratio) as the thickness becomes small. We demonstrate that the closed-loop configurations we obtain are stable with respect to small perturbations.

cond-mat.soft↗

arXiv2016

Computation of Unconstrained Elastic Equilibria of Complete Möbius Bands and their Stability

Determining the equilibrium configuration of an elastic Möbius band is a challenging problem. In recent years numerical results have been obtained by other investigators, employing first the Kirchhoff theory of rods and later the developable, ruled-surface model of Wunderlich. In particular, the strategy employed previously for the latter does not deliver an unconstrained equilibrium configuration for the complete strip. Here we present our own systematic approach to the same problem for each of these models, with the ultimate goal of assessing the stability of flip-symmetric configurations. The presence of pointwise constraints considerably complicates the latter step. We obtain the first stability results for the problem, numerically demonstrating that such equilibria render the total potential energy a local minimum. Along the way we introduce a novel regularization for the for the singular Wunderlich model that delivers unconstrained equilibria for the complete strip, which can then be tested for stability.

cond-mat.soft↗

arXiv2015

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations, via global bifurcation from the spherical state. To the best of our knowledge, this constitutes the first rigorous existence results for this class of problems. We overcome several difficulties in carrying this out. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly underdetermined in-plane deformation. The resulting governing equations comprise a quasi-linear elliptic system with nonlinear constraints. We show the equivalence of the problem to that of finding the zeros of compact vector field. The latter is not routine. Using spectral and a-priori estimates together with Fredholm properties, we demonstrate that the principal part of the quasi-linear mapping defines an operator with compact resolvent. We then combine well known group-theoretic ideas for symmetry breaking with global bifurcation theory to obtain our results.

math.AP↗

arXiv2015

Stability boundaries for wrinkling in highly stretched elastic sheets

We determine stability boundaries for the wrinkling of highly uni-directionally stretched, finely thin, rectangular elastic sheets. For a given fine thickness and length, a stability boundary here is a curve in the parameter plane, aspect ratio vs. the macroscopic strain; the values on one side of the boundary are associated with a flat, unwrinkled state, while wrinkled configurations correspond to all values on the other. In our recent work we demonstrated the importance of finite elasticity in the membrane part of such a model in order to capture the correct phenomena. Here we present and compare results for four distinct models:(i) the popular Föppl-von Kármán plate model (FvK), (ii) a correction of the latter, used in our earlier work, in which the approximate 2D Föppl strain tensor is replaced by the exact Green strain tensor, (iii) and (iv): effective 2D finite-elasticity membrane models based on 3D incompressible neo-Hookean and Mooney-Rivlin materials, respectively. In particular, (iii) and (iv) are superior models for elastomers. The 2D nonlinear, hyperelastic models (ii)-(iv) all incorporate the same quadratic bending energy used in FvK. Our results illuminate serious shortcomings of the latter in this problem, while also pointing to inaccuracies of model (ii), in spite of yielding the correct qualitative phenomena in our earlier work. In each of these, the shortcoming is a due to a deficiency of the membrane part of the model.

cond-mat.soft↗

arXiv2014

On The Equivalence of Local and Global Area-constraint Formulations for Lipid Bilayer Vesicles

Lipid bilayer membranes are commonly modeled as area-preserving fluid surfaces that resist bending. There appear to be two schools of thought in the literature concerning the actual area constraint. In some works the total or global area (GA) of the vesicle is a prescribed constant, while in others the local area ratio is assigned to unity. In this work we demonstrate the equivalence of these ostensibly distinct approaches in the specific case when the equilibrium configuration is a smooth, closed surface of genus zero. We accomplish this in the context of the Euler-Lagrange equilibrium equations, constraint equations and the second-variation with admissibility conditions, for a broad class of models - including the phase-field type.

physics.bio-ph↗