Recent zbMATH articles in MSC 74B20https://zbmath.org/atom/cc/74B202021-05-28T16:06:00+00:00WerkzeugOn inequalities of Korn's type.https://zbmath.org/1459.350072021-05-28T16:06:00+00:00"Chipot, Michel"https://zbmath.org/authors/?q=ai:chipot.michelSummary: The goal of this note is to identify the minimal number of linear forms which is necessary to obtain some inequalities of Korn's type. In other words we give some results allowing to estimate the \(L^2\)-norm of the gradient of a vector field in terms of a minimal sum of \(L^2\)-norms of linear forms of this gradient. We consider such inequalities either over \(\mathbb{H}_0^1(\Omega)\) or \(\mathbb{H}^1(\Omega)\).An efficient grid-based direct-volume integration BEM for 3D geometrically nonlinear elasticity.https://zbmath.org/1459.741842021-05-28T16:06:00+00:00"Deng, Yani"https://zbmath.org/authors/?q=ai:deng.yani"Rong, Junjie"https://zbmath.org/authors/?q=ai:rong.junjie"Ye, Wenjing"https://zbmath.org/authors/?q=ai:ye.wenjing"Gray, L. J."https://zbmath.org/authors/?q=ai:gray.leonard-j|gray.len-jSummary: The boundary element method is regarded as an efficient method for solving problems with complex domains. However most successful applications of the BEM have been thus far limited to linear analyses. A major difficulty with the BEM solution of a nonlinear problem is the handling of the volume integral resulting from the nonlinear terms. In this work, a grid-based direct-volume integration BEM has been developed for 3D geometrically nonlinear elastic problems. The volume integrals are evaluated using a regular Cartesian grid, and thus only the boundary discretization of the problem domain is required. The efficiency of the method is enhanced by using acceleration schemes for both surface and volume integration. Several 3-D example calculations have been performed to demonstrate the effectiveness of the three-dimensional formulas and the numerical implementation.A hybrid high-order method for nonlinear elasticity.https://zbmath.org/1459.652122021-05-28T16:06:00+00:00"Botti, Michele"https://zbmath.org/authors/?q=ai:botti.michele"Di Pietro, Daniele A."https://zbmath.org/authors/?q=ai:di-pietro.daniele-antonio"Sochala, Pierre"https://zbmath.org/authors/?q=ai:sochala.pierreThe authors propose and analyze a novel Hybrid High-Order discretization of a class of (linear and nonlinear) elasticity models in the small deformation regime. The proposed method is valid in two and three space dimensions and it can be applied using general meshes including polyhedral elements and nonmatching interfaces. Other interesting feature of the numerical method is that it can produce arbitrary approximation order and the resolution cost can be reduced by statically condensing a large subset of the unknowns for linearized versions of the problem. In addition to this, the method satisfies a local principle of virtual work inside each mesh element, with interface tractions that obey the law of action and reaction. A full analysis is carried out, and optimal error estimates are proved. Several numerical tests are presented.
Reviewer: Abdallah Bradji (Annaba)Blow-up analysis and spatial asymptotic profiles of solutions to a modified two-component hyperelastic rod system.https://zbmath.org/1459.353532021-05-28T16:06:00+00:00"Wei, Long"https://zbmath.org/authors/?q=ai:wei.long|wei.long.1"Zeng, Qi"https://zbmath.org/authors/?q=ai:zeng.qiThe authors consider the two-component hyperelastic rod problem \(
u_{t}-u_{xxt}+3uu_{x}-\gamma (2u_{x}u_{xx}+uu_{xxx})+\rho \overline{\rho }
_{x}=0\), \(\rho _{t}+(\rho u)_{x}=0\), posed in \((0,\infty )\times \mathbb{R}\). Here \(\gamma \) is a constant, \(u\) is the displacement, \(\rho
=(1-\partial _{x}^{2})(\overline{\rho }-\overline{\rho }_{0})\) is the
pointwise density, \(\overline{\rho }\) the averaged density, and \(\overline{
\rho }_{0}\) a constant. Introducing \(G(x)=\frac{1}{2}e^{-\left\vert
x\right\vert }\), \(\rho =v-v_{xx}\), and \(v=\overline{\rho }-\overline{\rho }
_{0}\), the authors rewrite the previous problem as: \(u_{t}+\gamma
uu_{x}=-\partial _{x}G\ast (\frac{3-\gamma }{2}u^{2}+\frac{\gamma }{2}
u_{x}^{2}+\frac{1}{2}v^{2}-\frac{1}{2}v_{x}^{2})\), \(v_{t}+uv_{x}=-G\ast
((u_{x}v_{x})_{x}+u_{x}v)\), posed in \((0,\infty )\times \mathbb{R}\).
Initial conditions \(z_{0}=(u_{0},v_{0})\in H^{s}(\mathbb{R})\times H^{s}(
\mathbb{R})\), \(s>\frac{3}{2}\), are added. The first main result of the paper
proves a blow-up behavior in finite time of the solution \(z=(u,v)\in
C([0,T);H^{s}\times H^{s})\cap C^{1}([0,T);H^{s-1}\times H^{s-1})\) to the
last problem. Assuming \(\gamma \in (0,4]\) and the existence of \(x_{0}\in
\mathbb{R}\) such that \(u_{0,x}(x_{0})<-\frac{C_{1}}{\sqrt{\gamma }}\), the
solution \(z=(u,v)\) blows up in finite time with an estimate of the blow-up
time \(T\) as \(0<T\leq \frac{1}{\sqrt{\gamma }C_{1}}\log (\frac{u_{0,x}(x_{0})-
\frac{C_{1}}{\sqrt{\gamma }}}{u_{0,x}(x_{0})+\frac{C_{1}}{\sqrt{\gamma }}})\)
. Here \(C_{1}\) is explicitly given in terms of \(E(0)=\int_{\mathbb{R}
}(u^{2}+u_{x}^{2}+v^{2}+v_{x}^{2})dx\). In the case where \(\gamma =1\), and
assuming that \(\rho _{0}\) does not change sign on \(\mathbb{R}\), and the
existence of a point \(x_{0}\in \mathbb{R}\) such that \(u_{0,x}(x_{0})<-\sqrt{
\frac{E(0)}{2}}\), the authors prove a blow-up behavior in finite time \(T\) of
the solution to the last problem. The blow-up rate of the solution is here
proved equal to \(u_{x}(t,x(t))\thicksim -\frac{2}{T-t}\), as \(t\rightarrow T\)
, for some \(x(t)\in \mathbb{R}\). In their last main result, the authors
describe the asymptotic profiles of the solutions to this last problem,
assuming some boundedness property of the term \(u_{0}+u_{0,x}+v_{0}+v_{0,x}\)
involving the initial conditions. The authors first prove the existence of a
solution in the above-indicated spaces and which depends continuously on the
initial data \(z_{0}\), using Kato's semigroup theory. The maximal time of
existence \(T>0\) of this solution is independent of the parameter \(s\). The
blow-up analysis is performed mainly using classical results concerning
ordinary differential equations.
Reviewer: Alain Brillard (Riedisheim)Geometry of martensite needles in shape memory alloys.https://zbmath.org/1459.741482021-05-28T16:06:00+00:00"Conti, Sergio"https://zbmath.org/authors/?q=ai:conti.sergio"Lenz, Martin"https://zbmath.org/authors/?q=ai:lenz.martin"Lüthen, Nora"https://zbmath.org/authors/?q=ai:luthen.nora"Rumpf, Martin"https://zbmath.org/authors/?q=ai:rumpf.martin"Zwicknagl, Barbara"https://zbmath.org/authors/?q=ai:zwicknagl.barbara-mariaSummary: We study the geometry of needle-shaped domains in shape-memory alloys. Needle-shaped domains are ubiquitously found in martensites around macroscopic interfaces between regions which are laminated in different directions, or close to macroscopic austenite/twinned-martensite interfaces. Their geometry results from the interplay of the local nonconvexity of the effective energy density with long-range (linear) interactions mediated by the elastic strain field, and is up to now poorly understood. We present a two-dimensional shape optimization model based on finite elasticity and discuss its numerical solution. Our results indicate that the tapering profile of the needles can be understood within finite elasticity, but not with linearized elasticity. The resulting tapering and bending reproduce the main features of experimental observations on Ni\(_{65}\)Al\(_{35}\).Stable spatially localized configurations in a simple structure -- a global symmetry-breaking approach.https://zbmath.org/1459.370782021-05-28T16:06:00+00:00"Pandurangi, Shrinidhi S."https://zbmath.org/authors/?q=ai:pandurangi.shrinidhi-s"Elliott, Ryan S."https://zbmath.org/authors/?q=ai:elliott.ryan-s"Healey, Timothy J."https://zbmath.org/authors/?q=ai:healey.timothy-j"Triantafyllidis, Nicolas"https://zbmath.org/authors/?q=ai:triantafyllidis.nicolasSummary: 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.Nonlinear and linear elastodynamic transformation cloaking.https://zbmath.org/1459.740902021-05-28T16:06:00+00:00"Yavari, Arash"https://zbmath.org/authors/?q=ai:yavari.arash"Golgoon, Ashkan"https://zbmath.org/authors/?q=ai:golgoon.ashkanSummary: In this paper we formulate the problems of nonlinear and linear elastodynamic transformation cloaking in a geometric framework. In particular, it is noted that a cloaking transformation is neither a spatial nor a referential change of frame (coordinates); a cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic body (virtual problem) to that of an anisotropic and inhomogeneous elastic body with a hole surrounded by a cloak that is to be designed (physical problem). The virtual body has a desired mechanical response while the physical body is designed to mimic the same response outside the cloak using a cloaking transformation. We show that nonlinear elastodynamic transformation cloaking is not possible while nonlinear elastostatic transformation cloaking may be possible for special deformations, e.g., radial deformations in a body with either a cylindrical or a spherical cavity. In the case of classical linear elastodynamics, in agreement with the previous observations in the literature, we show that the elastic constants in the cloak are not fully symmetric; they do not possess the minor symmetries. We prove that elastodynamic transformation cloaking is not possible regardless of the shape of the hole and the cloak. It is shown that the small-on-large theory, i.e., linearized elasticity with respect to a pre-stressed configuration, does not allow for transformation cloaking either. However, elastodynamic cloaking of a cylindrical hole is possible for in-plane deformations while it is not possible for anti-plane deformations. We next show that for a cavity of any shape elastodynamic transformation cloaking cannot be achieved for linear gradient elastic solids; similar to classical linear elasticity the balance of angular momentum is the obstruction to transformation cloaking. We finally prove that transformation cloaking is not possible for linear elastic generalized Cosserat solids in dimension two for any shape of the hole and the cloak. In particular, in dimension two transformation cloaking cannot be achieved in linear Cosserat elasticity. We show that transformation cloaking for a spherical cavity covered by a spherical cloak is not possible in the setting of linear elastic generalized Cosserat elasticity. We conjecture that this result is true for a cavity of any shape. It should be emphasized that in this paper we do not consider the so-called metamaterials [\textit{G. W. Milton}, ``New metamaterials with macroscopic behavior outside that of continuum elastodynamics'', \url{arXiv:0706.2202}; \textit{G. W. Milton} and \textit{J. R. Willis}, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 463, No. 2079, 855--880 (2007; Zbl 1347.74080)].An anisotropic hyperelastic constitutive model with bending stiffness interaction for cord-rubber composites: comparison of simulation results with experimental data.https://zbmath.org/1459.740222021-05-28T16:06:00+00:00"Sun, Shulei"https://zbmath.org/authors/?q=ai:sun.shulei"Chen, Wenguo"https://zbmath.org/authors/?q=ai:chen.wenguoSummary: Based on the invariant theory of continuum mechanics by Spencer, the strain energy depends on deformation, fiber direction, and the gradients of the fiber direction in the deformed configuration. The resulting extended theory is very complicated and brings a nonsymmetric stress and couple stress. By introducing the gradient of fiber vector in the current configuration, the strain energy function can be decomposed into volumetric, isochoric, anisotropic, and bending deformation energy. Due to the particularity of bending deformation, the reinforced material has tensile deformation and compression deformation. The bending stiffness should be taken into consideration, and it is further verified by the bending simulation.Another special case of vibrations of a rolling tire.https://zbmath.org/1459.740682021-05-28T16:06:00+00:00"Kozhevnikov, I. F."https://zbmath.org/authors/?q=ai:kozhevnikov.ivan-fSummary: We investigate a special case of vibrations of a loaded tire rolling at constant speed. A previously proposed analytical model of a radial tire is considered [the author, Nelineĭn. Din. 15, No. 1, 67--78 (2019; Zbl 1451.74103)]. The surface of the tire is a flexible tread combined with elastic sidewalls. In the undeformed state, the tread is a circular cylinder. The tread is reinforced with inextensible cords. The tread is the part of the tire that makes actual contact with the ground plane. In the undeformed state, the sidewalls are represented by parts of two tori and consist of incompressible rubber described by the Mooney-Rivlin model. The previously obtained partial differential equation which describes the tire radial in-plane vibrations about steady-state regime of rolling is investigated. Analyzing the discriminant of the quartic polynomial, which is the function of the frequency of the tenth degree and the function of the angular velocity of the sixth degree, the rare case of a root of multiplicity three is discovered. The angular velocity of rotation, the tire speed and the natural frequency, corresponding to this case, are determined analytically. The mode shape of vibration in the neighborhood of the singular point is determined analytically.Cauchy problem for the torsional vibration equation of a nonlinear-elastic rod of infinite length.https://zbmath.org/1459.740802021-05-28T16:06:00+00:00"Umarov, Kh. G."https://zbmath.org/authors/?q=ai:umarov.khasan-galsanovichThe author studies the solvability of the problem of torsional vibrations of an infinite nonlinear-elastic rod in the space of continuous functions. This is generally modeled by a Sobolev-type equation that is not resolved with respect to the time derivative of the second order
\[
Du = \beta\frac{\partial}{\partial x}\Big(\frac{\partial u}{\partial x}\Big)^3,\qquad D=\frac{\partial^2}{\partial t^2} -\frac{\partial^4}{\partial x^2\partial t^2} -\frac{\partial^2}{\partial x^2}+\alpha^2 \frac{\partial^4}{\partial x^4},
\]
which, after a suitable change of variable, becomes
\[
\frac{\partial^2 \theta }{\partial t^2} - c_1^2 \frac{\partial^2\theta}{\partial x^2} -\frac{I_\varphi}{I_0} \frac{\partial^2}{\partial x^2}\Big(\frac{\partial^2 \theta}{\partial t^2} -c_0^2 \frac{\partial^2\theta}{\partial x^2}\Big)=0.
\]
A Cauchy problem associated to this equation then requires imposing initial conditions
\[
u(x,0)=\varphi(x),\qquad u_t(x,0)=\psi(x).
\]
In the first part, the author proves that, when the initial conditions and their derivatives up to the fourth order are continuous, then the Cauchy problem is well posed, at least locally in time, and admits a solution in the strong sense.
Then, the author shows also uniqueness of the strong solution, and provides some quantitative estimates on the regularity of such solutions.
The second part is devoted to the global-in-time well-posedness of the Cauchy problem: the author shows that, when the initial data
\[
\varphi \in W^{1,4}(\mathbb{R})\cap H^2(\mathbb{R}),\qquad \psi \in W^{1,2}(\mathbb{R}),
\]
then the strong solution belongs to \(W^{1,2}(\mathbb{R})\), is globally well defined in time, and its \(C^0\) norm grows at most exponentially in time. On the other hand, if the initial data satisfies
\[
\|\varphi\|_{W^{1,2}(\mathbb{R})}^2>0,\qquad M:= \langle \varphi,\psi \rangle+\langle \varphi',\psi' \rangle>0,\qquad Z_0< M^2 \|\varphi\|_{W^{1,2}(\mathbb{R})}^{-2},
\]
then no global-in-time solution can exist, and the author estimates the maximum blow-up time.
Reviewer: Xin Yang Lu (Thunder Bay)Experimental and numerical analysis on the bending response of the geometrically gradient soft robotics actuator.https://zbmath.org/1459.700212021-05-28T16:06:00+00:00"Dilibal, S."https://zbmath.org/authors/?q=ai:dilibal.s"Sahin, H."https://zbmath.org/authors/?q=ai:sahin.hakan|sahin.hasan"Celik, Y."https://zbmath.org/authors/?q=ai:celik.yunus|celik.yildiraySummary: In this study, three different soft pneumatic actuators (SPA) are designed and directly fabricated through additive manufacturing using thermoplastic polyurethane (TPU) filaments. The equal total inner volume size is used in the three varied designs to compare their effect on the bending response. A material model is selected and implemented according to the uniaxial tensile test parameters. The experimental results obtained from three different soft pneumatic actuators are compared with numerical model results. Especially, the experimentally measured bending forces are compared with the numerical model counterparts. The highest continuous bending deformation is determined among the three different soft pneumatic actuators. Additionally, a new integrated design and manufacturing approach is presented aiming to maximize the potential bending capability of the actuator through additive manufacturing.From statistical polymer physics to nonlinear elasticity.https://zbmath.org/1459.823472021-05-28T16:06:00+00:00"Cicalese, Marco"https://zbmath.org/authors/?q=ai:cicalese.marco"Gloria, Antoine"https://zbmath.org/authors/?q=ai:gloria.antoine"Ruf, Matthias"https://zbmath.org/authors/?q=ai:ruf.matthiasSummary: A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the \(\Gamma\)-limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the \(\Gamma\)-limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis).Hybrid high-order methods for finite deformations of hyperelastic materials.https://zbmath.org/1459.740202021-05-28T16:06:00+00:00"Abbas, Mickaël"https://zbmath.org/authors/?q=ai:abbas.mickael"Ern, Alexandre"https://zbmath.org/authors/?q=ai:ern.alexandre"Pignet, Nicolas"https://zbmath.org/authors/?q=ai:pignet.nicolasSummary: We devise and evaluate numerically Hybrid High-Order (HHO) methods for hyperelastic materials undergoing finite deformations. The HHO methods use as discrete unknowns piecewise polynomials of order \(k\geq 1\) on the mesh skeleton, together with cell-based polynomials that can be eliminated locally by static condensation. The discrete problem is written as the minimization of a broken nonlinear elastic energy where a local reconstruction of the displacement gradient is used. Two HHO methods are considered: a stabilized method where the gradient is reconstructed as a tensor-valued polynomial of order \(k\) and a stabilization is added to the discrete energy functional, and an unstabilized method which reconstructs a stable higher-order gradient and circumvents the need for stabilization. Both methods satisfy the principle of virtual work locally with equilibrated tractions. We present a numerical study of the two HHO methods on test cases with known solution and on more challenging three-dimensional test cases including finite deformations with strong shear layers and cavitating voids. We assess the computational efficiency of both methods, and we compare our results to those obtained with an industrial software using conforming finite elements and to results from the literature. The two HHO methods exhibit robust behavior in the quasi-incompressible regime.Equilibrium for multiphase solids with Eulerian interfaces.https://zbmath.org/1459.741512021-05-28T16:06:00+00:00"Grandi, Diego"https://zbmath.org/authors/?q=ai:grandi.diego"Kružík, Martin"https://zbmath.org/authors/?q=ai:kruzik.martin"Mainini, Edoardo"https://zbmath.org/authors/?q=ai:mainini.edoardo"Stefanelli, Ulisse"https://zbmath.org/authors/?q=ai:stefanelli.ulisseSummary: We describe a general phase-field model for hyperelastic multiphase materials. The model features an elastic energy functional that depends on the phase-field variable and a surface energy term that depends in turn on the elastic deformation, as it measures interfaces in the deformed configuration. We prove existence of energy minimizing equilibrium states and \(\Gamma \)-convergence of diffuse-interface approximations to the sharp-interface limit.Semi-inverse method in nonlinear problems of axisymmetric shells forming.https://zbmath.org/1459.740012021-05-28T16:06:00+00:00"Yudin, Anatoly S."https://zbmath.org/authors/?q=ai:yudin.anatoly-s"Shchitov, Dmitry V."https://zbmath.org/authors/?q=ai:shchitov.dmitry-vPublisher's description: Currently, solving problems based on designing and calculating complex structures with significant nonlinearity usually require:
\begin {itemize}
\item Expensive and difficult to learn software packages
\item High-performance computers
\item Spare time as calculations take a long time to complete
\end {itemize}
The book provides an alternative method for solving problems with deep geometric and physical nonlinearity. Easily implemented on normal PCs, this method is fast and creative. The reader can use integrated packages of the MathCad variety that implement ``live mathematics''. Such packages give the reader the freedom to create programs for themselves.
In the proposed method, a function for molding pressure is constructed, which is output to a stationary value by varying the shape parameters and edge reactions. The final shape of the shell is given using analytical approximations. Applications of the method are applied to real shell structures. Forming spherical and ellipsoidal shells (flapping membranes), correcting the shape of the bottom of a container for liquid cargo, modeling the operation of a flat jack, and converting a cylindrical shell into a barrel-shape are also considered.The anelastic Ericksen problem: universal deformations and universal eigenstrains in incompressible nonlinear anelasticity.https://zbmath.org/1459.740212021-05-28T16:06:00+00:00"Goodbrake, Christian"https://zbmath.org/authors/?q=ai:goodbrake.christian"Yavari, Arash"https://zbmath.org/authors/?q=ai:yavari.arash"Goriely, Alain"https://zbmath.org/authors/?q=ai:goriely.alainSummary: Ericksen's problem consists of determining all equilibrium deformations that can be sustained solely by the application of boundary tractions for an arbitrary incompressible isotropic hyperelastic material whose stress-free configuration is geometrically flat. We generalize this by first, using a geometric formulation of this problem to show that all the known universal solutions are symmetric with respect to Lie subgroups of the special Euclidean group. Second, we extend this problem to its anelastic version, where the stress-free configuration of the body is a Riemannian manifold. Physically, this situation corresponds to the case where nontrivial finite eigenstrains are present. We characterize explicitly the universal eigenstrains that share the symmetries present in the classical problem, and show that in the presence of eigenstrains, the six known classical families of universal solutions merge into three distinct anelastic families, distinguished by their particular symmetry group. Some generic solutions of these families correspond to well-known cases of anelastic eigenstrains. Additionally, we show that some of these families possess a branch of anomalous solutions, and demonstrate the unique features of these solutions and the equilibrium stress they generate.