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4. Contour lines for the bulk potential function ( left) and phase space diagram ( right) χ = 1000.0.
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In this article, we construct and analyze models of anisotropic
crosslinked polymers employing tools from the theory of liquid crystal
elastomers. The anisotropy of these systems stems from the presence of
rigid-rod molecular units in the network. We study minimization of the
energy for incompressible as well as compressible materials, combining
me...
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Stiffness variable materials have been applied in a variety of engineering fields that require adaptation, automatic modulation, and morphing because of their unique property to switch between a rigid, load‐bearing state and a soft, compliant state. Stiffness variable polymers comprising phase‐changing side‐chains (s‐SVPs) have densely grafted, hig...
Citations
... The drawback of this generalization is an existence theorem for models of magnetic elastomers, liquid crystals and magnetoelasticity, see, e.g., [7,11,34]. The existence theorems were proved in the scale of Sobolev spaces with > − 1 in [8] and extended to our Sobolev-Orlicz class in [29]. ...
We extend the global invertibility result [28] to a class of orientation-preserving Orlicz-Sobolev maps with an integrability just above − 1, whose traces on the boundary are also Orlicz-Sobolev and which do not present cavitation in the interior or at the boundary. As an application, we prove the existence of a.e. injective minimizers within this class for functionals in nonlinear elasticity.
... for a certain α > 0 and some polyconvex energy function W . Functionals with a similar structure appear also in models describing the nematic mesogens with the Landau-de Gennes theory, and in magnetoelasticity and plasticity, see, e.g., [5,6,12,18,28]. The major difficulties are that I depends on the composition of the two unknowns and that the nematic director n is defined in the domain u(Ω ) which is also determined only as a part of the solution of the variational problem. The analysis is based on the inverse function theorem for Sobolev maps due to Fonseca & Gangbo [18], which is valid for W 1,p maps from a domain in R n to R n when p > n. ...
... It is necessary to take into account that if a map is differentiable at a given point then the condition of regular approximate differentiability, used in [5], is automatically satisfied. Also, the proof uses The other main conclusions in [5] are the lower semicontinuity for Div-quasiconvex integrals (under the constraint of incompressibility) of Proposition 7.6; the lower semicontinuity for the model for plasticity of [12,18]; the existence of minimizers in Theorem 8.6 for the Landau-de Gennes model for nematic elastomers of [6]; and Theorem 8.9 for the magnetostriction model of [28] where minimizers (u, m) are sought for ∫ Ω W (Du(x), m(u(x)))dx + All of these results (not only the existence of minimizers for (1.1), stated in Theorem 5.1) can be proved under the milder coercivity condition (2.8) considered in this paper, using the results of Sections 3 and 4. ...
We extend the existence theorems in Barchiesi et al. (2017), for models of nematic elastomers and magnetoelasticity, to a larger class in the scale of Orlicz spaces. These models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration, and a functional corresponding to the nematic energy (or the exchange and magnetostatic energies in magnetoelasticity) where the energy density is integrated over the deformed configuration. In order to obtain the desired compactness and lower semicontinuity, we show that the regularity requirement that maps create no new surface can still be imposed when the gradients are in an Orlicz class with an integrability just above the space dimension minus one. We prove that the fine properties of orientation-preserving maps satisfying that regularity requirement (namely, being weakly 1-pseudomonotone, H ¹ -continuous, a.e. differentiable, and a.e. locally invertible) are still valid in the Orlicz–Sobolev setting.
... where Ω ⊂ R 3 , u ∈ W 1,p (Ω, R 3 ) for some p > 3, n ∈ H 1 (u(Ω), R 2 ), and W mec (F, n) = W (α −1 n ⊗ n + √ α(I − n ⊗ n))F (1.2) for a certain α > 0 and some polyconvex energy function W . Functionals with a similar structure appear also in models describing the nematic mesogens with the Landau-de Gennes theory, and in magnetoelasticity and plasticity, see, e.g., [6,12,18,28,5]. The major difficulties are that I depends on the composition of the two unknowns and that the nematic director n is defined in the domain u(Ω) which is also determined only as a part of the solution of the variational problem. The analysis is based on the inverse function theorem for Sobolev maps due to Fonseca & Gangbo [18], which is valid for W 1,p maps from a domain in R n to R n when p > n. ...
... 2.6 and Lemma 2.24], which have to be replaced by Proposition 4.5 and Lemma 2.11 (their Orlicz counterparts). The other main conclusions in [5] are the lower semicontinuity for Div-quasiconvex integrals (under the constraint of incompressibility) of Proposition 7.6; the lower semicontinuity for the model for plasticity of [12,18]; the existence of minimizers in Theorem 8.6 for the Landau-de Gennes model for nematic elastomers of [6]; and Theorem 8.9 for the magnetostriction model of [28] where minimizers (u, m) are sought for All of these results (not only the existence of minimizers for (1.1), stated in Theorem 5.1) can be proved under the milder coercivity condition (2.8) considered in this paper, using the results of Sections 3 and 4. ...
We extend the existence theorems in [Barchiesi, Henao \& Mora-Corral; ARMA 224], for models of nematic elastomers and magnetoelasticity, to a larger class in the scale of Orlicz spaces. These models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration, and a functional corresponding to the nematic energy (or the exchange and magnetostatic energies in magnetoelasticity) where the energy density is integrated over the deformed configuration. In order to obtain the desired compactness and lower semicontinuity, we show that the regularity requirement that maps create no new surface can still be imposed when the gradients are in an Orlicz class with an integrability just above the space dimension minus one. We prove that the fine properties of orientation-preserving maps satisfying that regularity requirement (namely, being weakly 1-pseudomonotone, $\mathcal H^1$-continuous, a.e.\ differentiable, and a.e.\ locally invertible) are still valid in the Orlicz-Sobolev setting.
... In recent years, considerable attention has been devoted to fully nonlinear mechanical models describing the coupling between elasticity and nematic order (see [16] [1] [2] [6] [7] [10] and references therein). The Frank term, however, has been typically evaluated in the reference configuration while, when large deformations are in order, it is more natural to consider spatial variations in the deformed configuration. ...
We discuss the well-posedness of a new nonlinear model for nematic elastomers. The main
novelty in our work is that the Frank energy penalizes spatial variations of the nematic
director in the deformed, rather than in the reference configuration, as
it is natural in the case of large deformations.
... The nature of the connections between polymer chains and rigid monomer units plays a main role in determining the behavior of liquid crystal elastomers [43]. From a different perspective, studies of actin and cytoskeletal networks ([11], [28], [42]) show lyotropic rigid rod systems crosslinked into networks by elastomer chains or linkers that present qualitative properties of liquid crystal elastomoers. A main feature of these networks is the average number of connections between rods and the location of these connections in the rod. ...
... The study of a lyotropic elastomer with Lagrangian liquid crystal energy was carried out in previous work where we also analyzed liquid crystal phase transitions triggered by change in rod density [11]. In this work, we assume that the anisotropic behavior of the rigid units is represented by a liquid-like, Eulerian liquid crystal energy. ...
... Motivated by the theory of existence of minimizers of isotropic nonlinear elasticity ([3]), in his PhD, thesis [32], Luo generalizes W BTW to polyconvex stored energy density functionsˆw (G(x)); that is, ˆ w(G) = Ψ(G, adj G, det G) is a convex function of (G, adj G, det G). This approach was used later in [11]) to model phase transitions in rod networks. In order to recover the limiting deformation gradient F * from the minimizing sequences {G k } k≥1 , it is necessary that the minimizing sequences {L k } yield a nonsingular limit. ...
In this article, we study minimization of the Landau-de Gennes energy for
liquid crystal elastomer.The total energy, is of the sum of the Lagrangian
elastic stored energy function of the elastomer and the Eulerian Landau-de
Gennes energy of the liquid crystal. Our model consider two sources of
anisotropy represented by the traceless nematic order tensor $Q$ (rigid units),
and the positive definite step-length tensor $L$ (network). This work is
motivated by the study of cytoskeletal networks which can be regarded as
consisting of rigid rod units crosslinked into a polymeric-type network.
Due to the mixed Eulerian-Lagrangian structure of the energy, it is essential
that the deformation maps $\varphi$ be invertible. We require sufficient
regularity of the fields $(\varphi, Q)$ of the problem, and that the
deformation map satisfies the Ciarlet-Necas condition. These, in turn,
determine what boundary conditions are admissible, which include the case of
Dirichlet conditions on both fields. The approach of including the
Rapini-Papoular surface energy for the pull-back tensor $\tilde Q$ is also
discussed. The regularity requirements lead us to consider powers of the
gradient of the order tensor $Q$ higher than quadratic in the energy.
We assume polyconvexity of the stored energy function with respect to the
effective deformation tensor and apply methods from isotropic nonlinear
elasticity. We formulate a necessary and sufficient condition to guarantee this
invertibility property in terms of the growth to infinity of the bulk liquid
crystal energy $f(Q)$, as the minimum eigenvalue of $Q$ approaches the singular
limit of $-\frac{1}{3}$. $L$ becomes singular as the minimum eigenvalue of $Q$
reaches $-\frac{1}{3}$. Lower bounds on the eigenvalues of $Q$ are needed to
ensure compatibility between the theories of Landau-de Gennes and Maier-Saupe
of nematics (see Ball 2010).
We extend the global invertibility result (Henao et al., Adv Calculus Var 14(2):207–230, 2021) to a class of orientation-preserving Orlicz–Sobolev maps with an integrability just above n − 1, whose traces on the boundary are also Orlicz–Sobolev and which do not present cavitation in the interior or at the boundary. As an application, we prove the existence of a.e. injective minimizers within this class for functionals in nonlinear elasticity.
MSC2020 Classifications:
49J9546E3074B20
We study the shear flow of active filaments confined in a thin channel for extensile and contractile fibers. We apply the Ericksen-Leslie equations of liquid crystal flow with an activity source term. The dimensionless form of this system includes the Ericksen, activity, and Reynolds numbers, together with the aspect ratio of the channel, as the main driving parameters. We perform a normal mode stability analysis of the base shear flow. For both types of fibers, we arrive at a comprehensive description of the stability properties and their dependence on the parameters of the system. The transition to unstable frequencies in extensile fibers occurs at a positive threshold value of the activity parameter, whereas for contractile ones a complex behavior is found at low absolute value of the activity number. The latter might be an indication of the biologically relevant plasticity and phase transition issues. In contrast with extensile fibers, flows of contractile ones are also found to be highly sensitive to the Reynolds number. The work on extensile fibers is guided by experiments on active filaments in confined channels and aims at quantifying their findings in the prechaotic regime.
In this article, we study shear flow of active extensile filaments confined in a narrow channel. They behave as nematic liquid crystals that we assumed are governed by the Ericksen–Leslie equations of balance of linear and angular momentum. The addition of an activity source term in the Leslie stress captures the role of the biofuel prompting the dynamics. The dimensionless form of the governing system includes the Ericksen, activity, and Reynolds numbers together with the aspect ratio of the channel as the main driving parameters affecting the stability of the system. The active system that guides our analysis is composed of microtubules concentrated in bundles, hundreds of microns long, placed in a narrow channel domain, of aspect ratios in the range between 10 − 2 and 10 − 3 dimensionless units, which are able to align due to the combination of adenosine triphosphate-supplied energy and confinement effects. Specifically, this work aims at studying the role of confinement on the behavior of active matter. It is experimentally observed that, at an appropriately low activity and channel width, the active flow is laminar, with the linear velocity profile and the angle of alignment analogous to those in passive shear, developing defects and becoming chaotic, at a large activity and a channel aspect ratio. The present work addresses the laminar regime, where defect formation does not play a role. We perform a normal mode stability analysis of the base shear flow. A comprehensive description of the stability properties is obtained in terms of the driving parameters of the system. Our main finding, in addition to the geometry and magnitude of the flow profiles, and also consistent with the experimental observations, is that the transition to instability of the uniformly aligned shear flow occurs at a threshold value of the activity parameter, with the transition also being affected by the channel aspect ratio. The role of the parameters on the vorticity and angular profiles of the perturbing flow is also analyzed and found to agree with the experimentally observed transition to turbulent regimes. A spectral method based on Chebyshev polynomials is used to solve the generalized eigenvalue problems arising in the stability analysis.
We compute the relaxation of the total energy related to a variational model for nematic elastomers, involving a nonlinear elastic mechanical energy depending on the orientation of the molecules of the nematic elastomer, and a nematic Oseen--Frank energy in the deformed configuration. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation belongs to an Orlicz-Sobolev space with an integrability just above the space dimension minus one, and does not present cavitation. We benefit from the fine properties of orientation-preserving maps satisfying that regularity requirement proven in \cite{HS} and extend the result of \cite{MCOl} to Orlicz spaces with a suitable growth condition at infinity.
We compute the relaxation of the total energy related to a variational model for nematic elastomers, involving a nonlinear elastic mechanical energy depending on the orientation of the molecules of the nematic elastomer, and a nematic Oseen-Frank energy in the deformed configuration. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation belongs to an Orlicz-Sobolev space with an integrability just above the space dimension minus one, and does not present cavitation. We benefit from the fine properties of orientation-preserving maps satisfying that regularity requirement proven in [26] and extend the result of [34] to Orlicz spaces with a suitable growth condition at infinity.
