arXiv subjects

Carlos Mora-Corral

Publications and source records attributed to Carlos Mora-Corral.

At least 19 recordsLinked to original sources

arXiv2026

Stress Asymmetry in Hard Magnetic Soft Materials

Hard magnetic soft materials -- soft polymers embedded with hard magnetic particles -- are modeled using continuum magnetomechanical formulations in which the deformation and the magnetization field are the primary kinematic variables. A recent question in such formulations is whether the Cauchy stress is symmetric, which is directly related to frame invariance and angular momentum balance. This note discusses energetically equivalent formulations, related by a change of variables between referential and current descriptions of the magnetization, and shows that they generally yield different Cauchy stresses, including a change in their symmetry. Specifically, the formulation based on a referential magnetization produces a symmetric Cauchy stress, while that based on a current magnetization generally yields an asymmetric Cauchy stress. We highlight that when the internal variable (magnetization field) is at the energy-minimizing equilibrium configuration, the divergences of these stresses are the same, and both stresses are symmetric.

cond-mat.mtrl-sci↗

arXiv2026

A Lavrentiev phenomenon in the neo-Hookean model

We exhibit a Lavrentiev gap phenomenon for the neo-Hookean energy in three-dimensional nonlinear elasticity. More precisely, we construct boundary data for which the infimum of the neo-Hookean energy over deformations satisfying a natural regularity and invertibility condition is strictly larger than the infimum over the weak $H^1$-closure of that class. The mechanism underlying the gap is a deformation with a dipole-type singularity.

math.AP↗

arXiv2025

Crack Face Contact Modeling is Essential to Predict Crack-Parallel Stresses

Phase-field fracture models provide a powerful approach to modeling fracture, potentially enabling the unguided prediction of crack growth in complex patterns. To ensure that only tensile stresses and not compressive stresses drive crack growth, several models have been proposed that aim to distinguish between compressive and tensile loads. However, these models have a critical shortcoming: they do not account for the crack direction, and hence they cannot distinguish between crack-normal tensile stresses that drive crack growth and crack-parallel stresses that do not. In this study, we apply a phase-field fracture model, developed in our earlier work, that uses the crack direction to distinguish crack-parallel stresses from crack-normal stresses. This provides a transparent energetic formulation that drives cracks to grow in when crack faces open or slide past each other, while the cracks respond like the intact solid when the crack faces contact under normal compressive loads. We compare our approach against two widely used approaches, Spectral splitting and the Volumetric-Deviatoric splitting, and find that these predict unphysical crack growth and unphysical stress concentrations under loading conditions in which these should not occur. Specifically, we show that the splitting models predict spurious crack growth and stress concentration under pure crack-parallel normal stresses. However, our formulation, which resolves the crack-parallel stresses from the crack-normal stresses, predicts these correctly.

cond-mat.mtrl-sci↗

arXiv2025

Invertibility of Sobolev maps through approximate invertibility at the boundary and tangential polyconvexity

We work in a class of Sobolev $W^{1,p}$ maps, with $p > d-1$, from a bounded open set $Ω\subset \mathbb{R}^{d}$ to $\mathbb{R}^{d}$ that do not exhibit cavitation and whose trace on $\partial Ω$ is also $W^{1,p}$. Under the assumptions that the Jacobian is positive and the deformation can be approximated on the boundary by injective maps, we show that the deformation is injective. We prove the existence of minimizers in this class for functionals accounting for a nonlinear elastic energy and a boundary energy. The energy density in $Ω$ is assumed to be polyconvex, while the energy density in $\partial Ω$ is assumed to be tangentially polyconvex, a new type of polyconvexity on $\partial Ω$.

math.AP↗

arXiv2024

Phase-Field Modeling of Fracture under Compression and Confinement in Anisotropic Geomaterials

Strongly anisotropic geomaterials undergo fracture under compressive loading. This paper applies a phase-field fracture model to study this fracture process. While phase-field fracture models have several advantages, they provide unphysical predictions when the stress state is complex and includes compression that can cause crack faces to contact. Building on a phase-field model that accounts for compressive traction across the crack face, this paper extends the model to anisotropic fracture. The key features include: (1) a homogenized anisotropic elastic response and strongly-anisotropic model for the work to fracture; (2) an effective damage response that accounts consistently for compressive traction across the crack face, that is derived from the anisotropic elastic response; (3) a regularized crack normal field that overcomes the shortcomings of the isotropic setting, and enables the correct crack response, both across and transverse to the crack face. To test the model, we first compare the predictions to phase-field fracture evolution calculations in a fully-resolved layered specimen with spatial inhomogeneity, and show that it captures the overall patterns of crack growth. We then apply the model to previously-reported experimental observations of fracture evolution in laboratory specimens of shales under compression with confinement, and find that it predicts well the observed crack patterns in a broad range of loading conditions. We further apply the model to predict the growth of wing cracks under compression and confinement. The effective crack response model enables us to treat the initial crack simply as a non-singular damaged zone within the computational domain, thereby allowing for easy and general computations.

cond-mat.mtrl-sci↗

arXiv2024

A relaxation approach to the minimisation of the neo-Hookean energy in 3D

Despite its high significance in nonlinear elasticity, the neo-Hookean energy is still not known to admit minimisers in some appropriate admissible class. Using ideas from relaxation theory, we propose a larger minimisation space and a modified functional that coincides with the neo-Hookean energy on the original space. This modified energy is the sum of the neo-Hookean energy and a term penalising the singularities of the inverse deformation. The new functional attains its minimum in the larger space, so the initial question of existence of minimisers of the neo-Hookean energy is thus transformed into a question of regularity of minimisers of this new energy.

math.AP↗

arXiv2024

Tests for almost stochastic dominance

We introduce a 2-dimensional stochastic dominance (2DSD) index to characterize both strict and almost stochastic dominance. Based on this index, we derive an estimator for the minimum violation ratio (MVR), also known as the critical parameter, of the almost stochastic ordering condition between two variables. We determine the asymptotic properties of the empirical 2DSD index and MVR for the most frequently used stochastic orders. We also provide conditions under which the bootstrap estimators of these quantities are strongly consistent. As an application, we develop consistent bootstrap testing procedures for almost stochastic dominance. The performance of the tests is checked via simulations and the analysis of real data.

econ.EM↗

arXiv2024

Nonlocal gradients: Fundamental theorem of calculus, Poincaré inequalities and embeddings

We address the study of nonlocal gradients defined through general radial kernels $ρ$. Our investigation focuses on the properties of the associated function spaces, which depend on the characteristics of the kernel function. Specifically, even with minimal assumptions on $ρ$, we establish Poincaré inequalities and compact embeddings into Lebesgue spaces. Additionally, we present a fundamental theorem of calculus that enables us to recover a function from its nonlocal gradient through a convolution. This is used to demonstrate embeddings into Orlicz spaces and spaces of continuous functions that mirror the well-known Sobolev and Morrey inequalities for classical gradients. Finally, we establish conditions for inclusions and equality of spaces associated to different kernels.

math.AP↗

arXiv2024

Variational models with Eulerian-Lagrangian formulation allowing for material failure

We investigate the existence of minimizers of variational models with Eulerian-Lagrangian formulations. We consider energy functionals depending on the deformation of a body, defined on its reference configuration, and an Eulerian map defined on the unknown deformed configuration in the actual space. Our existence theory moves beyond the purely elastic setting and accounts for material failure by addressing free-discontinuity problems where both deformations and Eulerian fields are allowed to jump. To do so, we build upon the work of Henao and Mora-Corral regarding the variational modeling of cavitation and fracture in nonlinear elasticity. Two main settings are considered by modeling deformations as Sobolev and SBV-maps, respectively. The regularity of Eulerian maps is specified in each of these two settings according to the geometric and topological properties of the deformed configuration. We present some applications to specific models of liquid crystals, phase transitions, and ferromagnetic elastomers. Effectiveness and limitations of the theory are illustrated by means of explicit examples.

math.AP↗

arXiv2024

On the lack of compactness in the axisymmetric neo-Hookean model

We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti-- De Lellis is generic in some sense. On this map we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of $\mathbb{S}^2$-valued harmonic maps.

math.AP↗

arXiv2023

Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity

We consider the problem of minimizing the neo-Hookean energy in $3D$. The difficulty of this problem is that the space of maps without cavitation is not compact, as shown by Conti \& De Lellis with a pathological example involving a dipole. In order to rule out this behaviour we consider the relaxation of the neo-Hookean energy in the space of axisymmetric maps without cavitation. We propose a minimization space and a new explicit energy penalizing the creation of dipoles. This new energy, which is a lower bound of the relaxation of the original energy, bears strong similarities with the relaxed energy of Bethuel-Brezis-Hélein in the context of harmonic maps into the sphere.

math.AP↗

arXiv2022

Nonlocal gradients in bounded domains motivated by Continuum Mechanics: Fundamental Theorem of Calculus and embeddings

In this paper we develop a new set of results based on a nonlocal gradient jointly inspired by the Riesz s-fractional gradient and Peridynamics, in the sense that its integration domain depends on a ball of radius delta > 0 (horizon of interaction among particles, in the terminology of Peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for nonlocal models in Calculus of Variations and partial differential equations. Our motivation is to develop the proper functional analysis framework in order to tackle nonlocal models in Continuum Mechanics, which requires working with bounded domains, while retaining the good mathematical properties of Riesz s-fractional gradients. This functional space is defined consistently with Sobolev and Bessel fractional ones: we consider the closure of smooth functions under the natural norm obtained as the sum of the Lp norms of the function and its nonlocal gradient. Among the results showed in this investigation we highlight a nonlocal version of the Fundamental Theorem of Calculus (namely, a representation formula where a function can be recovered from its nonlocal gradient), which allows us to prove inequalities in the spirit of Poincaré, Morrey, Trudinger and Hardy as well as the corresponding compact embeddings. These results are enough to show the existence of minimizers of general energy functionals under the assumption of convexity. Equilibrium conditions in this nonlocal situation are also established, and those can be viewed as a new class of nonlocal partial differential equations in bounded domains.

math.AP↗

arXiv2022

Minimizers of Nonlocal Polyconvex Energies in Nonlocal Hyperelasticity

We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz' fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola's identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient. Contrary to classical elasticity, this existence result is compatible with cavitation and fracture.

math.AP↗

arXiv2022

Phase-Field Finite Deformation Fracture with an Effective Energy for Regularized Crack Face Contact

Phase-field models are a leading approach for realistic fracture problems. They treat the crack as a second phase and use gradient terms to smear out the crack faces, enabling the use of standard numerical methods for simulations. This regularization causes cracks to occupy a finite volume in the reference, and leads to the inability to appropriately model the closing or contacting -- without healing -- of crack faces. Specifically, the classical idealized crack face tractions are that the shear component is zero, and that the normal component is zero when the crack opens and identical to the intact material when the crack closes. Phase-field fracture models do not replicate this behavior. This work addresses this shortcoming by introducing an effective crack energy density that endows the regularized (finite volume) phase-field crack with the effective properties of an idealized sharp crack. The approach is based on applying the QR (upper triangular) decomposition of the deformation gradient tensor in the basis of the crack, enabling a transparent identification of the crack deformation modes. By then relaxing over those modes that do not cost energy, an effective energy is obtained that has the intact response when the crack faces close and zero energy when the crack faces are open. A highlight of this approach is that it lies completely in the setting of finite deformation, enabling potential application to soft materials and other settings with large deformation or rotations. The model is applied to numerically study representative complex loadings, including (1) cyclic loading on a cavity in a soft solid that shows the growth and closing of cracks in complex stress states; and (2) cyclic shear that shows a complex pattern of crack branching driven by the closure of cracks.

math.AP↗

arXiv2021

Extremal points of Lorenz curves and applications to inequality analysis

We find the set of extremal points of Lorenz curves with fixed Gini index and compute the maximal $L^1$-distance between Lorenz curves with given values of their Gini coefficients. As an application we introduce a bidimensional index that simultaneously measures relative inequality and dissimilarity between two populations. This proposal employs the Gini indices of the variables and an $L^1$-distance between their Lorenz curves. The index takes values in a right-angled triangle, two of whose sides characterize perfect relative inequality-expressed by the Lorenz ordering between the underlying distributions. Further, the hypotenuse represents maximal distance between the two distributions. As a consequence, we construct a chart to, graphically, either see the evolution of (relative) inequality and distance between two income distributions over time or to compare the distribution of income of a specific population between a fixed time point and a range of years. We prove the mathematical results behind the above claims and provide a full description of the asymptotic properties of the plug-in estimator of this index. Finally, we apply the proposed bidimensional index to several real EU-SILC income datasets to illustrate its performance in practice.

econ.EM↗

arXiv2020

$Γ$-convergence of polyconvex functionals involving s-fractional gradients to their local counterparts

In this paper we study localization properties of the Riesz $s$-fractional gradient $D^s u$ of a vectorial function $u$ as $s \nearrow 1$. The natural space to work with $s$-fractional gradients is the Bessel space $H^{s,p}$ for $0 < s < 1$ and $1 < p < \infty$. This space converges, in a precise sense, to the Sobolev space $W^{1,p}$ when $s \nearrow 1$. We prove that the $s$-fractional gradient $D^s u$ of a function $u$ in $W^{1,p}$ converges strongly to the classical gradient $Du$. We also show a weak compactness result in $W^{1,p}$ for sequences of functions $u_s$ with bounded $L^p$ norm of $D^s u_s$ as $s \nearrow 1$. Moreover, the weak convergence of $D^s u_s$ in $L^p$ implies the weak continuity of its minors, which allows us to prove a semicontinuity result of polyconvex functionals involving $s$-fractional gradients defined in $H^{s,p}$ to their local counterparts defined in $W^{1,p}$. The full $Γ$-convergence of the functionals is achieved only for the case $p>n$.

math.AP↗

arXiv2020

Fractional Piola identity and polyconvexity in fractional spaces

In this paper we address nonlocal vector variational principles obtained by substitution of the classical gradient by the Riesz fractional gradient. We show the existence of minimizers in Bessel fractional spaces under the main assumption of polyconvexity of the energy density, and, as a consequence, the existence of solutions to the associated Euler--Lagrange system of nonlinear fractional PDE. The main ingredient is the fractional Piola identity, which establishes that the fractional divergence of the cofactor matrix of the fractional gradient vanishes. This identity implies the weak convergence of the determinant of the fractional gradient, and, in turn, the existence of minimizers of the nonlocal energy. Contrary to local problems in nonlinear elasticity, this existence result is compatible with solutions presenting discontinuities at points and along hypersurfaces.

math.AP↗

arXiv2017

Quasistatic elastoplasticity via Peridynamics: existence and localization

Peridynamics is a nonlocal continuum-mechanical theory based on minimal regularity on the deformations. Its key trait is that of replacing local constitutive relations featuring spacial differential operators with integrals over differences of displacement fields over a suitable positive interaction range. The advantage of such perspective is that of directly including nonregular situations, in which discontinuities in the displacement field may occur. In the linearized elastic setting, the mechanical foundation of the theory and its mathematical amenability have been thoroughly analyzed in the last years. We present here the extension of Peridynamics to linearized elastoplasticity. This calls for considering the time evolution of elastic and plastic variables, as the effect of a combination of elastic energy storage and plastic energy dissipation mechanisms. The quasistatic evolution problem is variationally reformulated and solved by time discretization. In addition, by a rigorous evolutive $Γ$-convergence argument we prove that the nonlocal peridynamic model converges to classic local elastoplasticity as the interaction range goes to zero.

math.AP↗