Recent zbMATH articles in MSC 74Ghttps://www.zbmath.org/atom/cc/74G2021-02-27T13:50:00+00:00WerkzeugEquilibrium of a linearly elastic body under generalized boundary data.https://www.zbmath.org/1453.740112021-02-27T13:50:00+00:00"Starita, Giulio"https://www.zbmath.org/authors/?q=ai:starita.giulio"Tartaglione, Alfonsina"https://www.zbmath.org/authors/?q=ai:tartaglione.alfonsinaSummary: We consider the interior Dirichlet, Neumann and Robin problems associated to the differential system of linear elastostatics with singular data. We prove that if the assigned displacement field \(a\) on the \(C^2\) boundary \(S\) of the reference configuration of the elastic body belongs to \(W^{-1/2,2}(S) \), then there exists a unique solution to the equilibrium problem which takes the boundary datum \(a\) in a well-defined sense; similar results hold if we assign the traction or a linear combination of displacement and traction on the boundary. Moreover, natural estimates controlling the norms of the solutions with the norms of the data hold and analogous results are obtained for the exterior problems requiring the displacement vanishes at infinity.
For the entire collection see [Zbl 1445.34003].Generalized Eshelby problem in the gradient theory of elasticity.https://www.zbmath.org/1453.740132021-02-27T13:50:00+00:00"Volkov-Bogorodskiy, D. B."https://www.zbmath.org/authors/?q=ai:volkov-bogorodsky.dmitrij-b"Moiseev, E. I."https://www.zbmath.org/authors/?q=ai:moiseev.evgeny-ivanovichSummary: A generalized Eshelby problem of arbitrary order in the gradient elasticity for a multilayer inclusions of spherical shape with a polynomial strain field at infinity is considered. For this problem we propose a constructive method of solution in a closed finite form, using generalized Papkovich-Neuber representation and the system of canonical potentials based on harmonic polynomials. We use also the Gauss theorem on the decomposition of an arbitrary homogeneous polynomials. The solutions of the generalized Eshelby problem are applied in the method of asymptotic homogenization of the gradient elasticity to accurately calculation of the effective characteristics of composite materials with scale effects.Weak solutions within the gradient-incomplete strain-gradient elasticity.https://www.zbmath.org/1453.740122021-02-27T13:50:00+00:00"Eremeyev, V. A."https://www.zbmath.org/authors/?q=ai:eremeyev.victor-a"dell'Isola, F."https://www.zbmath.org/authors/?q=ai:dellisola.francescoSummary: In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the strain-gradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first- and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.On the coupling of plastic slip and deformation-induced twinning in magnesium: A variationally consistent approach based on energy minimization.https://www.zbmath.org/1453.740172021-02-27T13:50:00+00:00"Homayonifar, M."https://www.zbmath.org/authors/?q=ai:homayonifar.m"Mosler, J."https://www.zbmath.org/authors/?q=ai:mosler.jornSummary: The present paper is concerned with the analysis of the deformation systems in single crystal magnesium at the micro-scale and with the resulting texture evolution in a polycrystal representing the macroscopic mechanical response. For that purpose, a variationally consistent approach based on energy minimization is proposed. It is suitable for the modeling of crystal plasticity at finite strains including the phase transition associated with deformation-induced twinning. The method relies strongly on the variational structure of crystal plasticity theory, i.e., an incremental minimization principle can be derived which allows to determine the unknown slip rates by computing the stationarity conditions of a (pseudo) potential. Phase transition associated with twinning is modeled in a similar fashion. More precisely, a solid-solid phase transition corresponding to twinning is assumed, if this is energetically favorable. Mathematically speaking, the aforementioned transition can be interpreted as a certain rank-one convexification. Since such a scheme is computationally very expensive and thus, it cannot be applied to the analysis of a polycrystal, a computationally more efficient approximation is elaborated. Within this approximation, the deformation induced by twinning is decomposed into the reorientation of the crystal lattice and simple shear. The latter is assumed to be governed by means of a standard Schmid-type plasticity law (pseudo-dislocation), while the reorientation of the crystal lattice is considered, when the respective plastic shear strain reaches a certain threshold value. The underlying idea is in line with experimental observations, where dislocation slip within the twinned domain is most frequently seen, if the twin laminate reaches a critical volume. The resulting model predicts a stress-strain response in good agreement with that of a rank-one convexification method, while showing the same numerical efficiency as a classical Taylor-type approximation. Consequently, it combines the advantages of both limiting cases. The model is calibrated for single crystal magnesium by means of the channel die test and finally applied to the analysis of texture evolution in a polycrystal. Comparisons of the predicted numerical results to their experimental counterparts show that the novel model is able to capture the characteristic mechanical response of magnesium very well.Less is often more: applied inverse problems using \(hp\)-forward models.https://www.zbmath.org/1453.740802021-02-27T13:50:00+00:00"Smyl, Danny"https://www.zbmath.org/authors/?q=ai:smyl.danny"Liu, Dong"https://www.zbmath.org/authors/?q=ai:liu.dongSummary: To solve an applied inverse problem, a numerical forward model for the problem's physics is required. Commonly, the finite element method is employed with discretizations consisting of elements with variable size \(h\) and polynomial degree \(p\). Solutions to \(hp\)-forward models are known to converge exponentially by simultaneously decreasing \(h\) and increasing \(p\). On the other hand, applied inverse problems are often ill-posed and their minimization rate exhibits uncertainty. Presently, the behavior of applied inverse problems incorporating \(hp\) elements of differing \(p, h\), and geometry is not fully understood. Nonetheless, recent research suggests that employing increasingly higher-order \(hp\)-forward models (increasing mesh density and \(p)\) decreases reconstruction errors compared to inverse regimes using lower-order \(hp\)-forward models (coarser meshes and lower \(p)\). However, an affirmative or negative answer to following question has not been provided, ``Does the use of higher order \(hp\)-forward models in applied inverse problems always result in lower error reconstructions than approaches using lower order \(hp\)-forward models?'' In this article we aim to reduce the current knowledge gap and answer the open question by conducting extensive numerical investigations in the context of two contemporary applied inverse problems: elasticity imaging and hydraulic tomography -- nonlinear inverse problems with a PDE describing the underlying physics. Our results support a \textit{negative} answer to the question -- i.e. decreasing \(h\) (increasing mesh density), increasing \(p\), or simultaneously decreasing \(h\) and increasing \(p\) does not guarantee lower error reconstructions in applied inverse problems. Rather, there is complex balance between the accuracy of the \(hp\)-forward model, noise, prior knowledge (regularization), Jacobian accuracy, and ill-conditioning of the Jacobian matrix which ultimately contribute to reconstruction errors. As demonstrated herein, it is often more advantageous to use lower-order \(hp\)-forward models than higher-order \(hp\)-forward models in applied inverse problems. These realizations and other counterintuitive behavior of applied inverse problems using \(hp\)-forward models are described in detail herein.Interactions of anisotropic inclusions on a fluid membrane.https://www.zbmath.org/1453.740532021-02-27T13:50:00+00:00"Kwiecinski, James A."https://www.zbmath.org/authors/?q=ai:kwiecinski.james-a"Goriely, Alain"https://www.zbmath.org/authors/?q=ai:goriely.alain"Chapman, S. Jon"https://www.zbmath.org/authors/?q=ai:chapman.stephen-jonathanOn the geometric rigidity interpolation estimate in thin bi-Lipschitz domains.https://www.zbmath.org/1453.740552021-02-27T13:50:00+00:00"Harutyunyan, Davit"https://www.zbmath.org/authors/?q=ai:harutyunyan.davitSummary: This work is concerned with developing asymptotically sharp geometric rigidity estimates in thin domains. A thin domain \(\Omega\) in space is roughly speaking a shell with non-constant thickness around a regular enough two dimensional compact surface. We prove a sharp geometric rigidity interpolation inequality that permits one to bound the \(L^p\) distance of the gradient of a \(\mathbf{u}\in W^{1,p}\) field from any constant proper rotation \(\mathbf{R} \), in terms of the average \(L^p\) distance (nonlinear strain) of the gradient from the rotation group, and the average \(L^p\) distance of the field itself from the set of rigid motions corresponding to the rotation \(\mathbf{R}\). The constants in the estimate are sharp in terms of the domain thickness scaling. If the domain mid-surface has a constant sign Gaussian curvature then the inequality reduces the problem of estimating the gradient \(\nabla \mathbf{u}\) in terms of the nonlinear strain \(\int_\Omega \text{dist}^p(\nabla \mathbf{u}(x),SO(3))\text{d}x\) to the easier problem of estimating only the vector field \(\mathbf{u}\) in terms of the nonlinear strain with no asymptotic loss in the constants. This being said, the new interpolation inequality reduces the problem of proving ``any'' geometric one well rigidity problem in thin domains to estimating the vector field itself instead of the gradient, thus reducing the complexity of the problem.Wavelet-based numerical techniques for 1D peristatic problems in infinite domain.https://www.zbmath.org/1453.740822021-02-27T13:50:00+00:00"Singh, Debabrata"https://www.zbmath.org/authors/?q=ai:singh.debabrata"Panja, Madan Mohan"https://www.zbmath.org/authors/?q=ai:panja.madan-mohanSummary: Peristatics is an important branch of continuum mechanics dealing with nonlocal effects in solid structures. Some 1D peristatic problems with four different micromodulus functions have been studied here. A wavelet-based collocation method has been developed to find multiscale approximate solutions of the peristatic problems that avoid cumbersome evaluation of singular type integrals present in analytical solution available in literature. The obtained results offer important insights into applications and simulations of peristatic models and seem to be useful for investigation in the field of peridynamics in one or higher dimensions.Phase transitions in two-phase media with the same moduli of elasticity.https://www.zbmath.org/1453.740712021-02-27T13:50:00+00:00"Osmolovskii, V. G."https://www.zbmath.org/authors/?q=ai:osmolovskii.viktor-g|osmolovskij.v-gSummary: For a variational problem of the theory of phase transitions in continuum mechanics with the same moduli of elasticity we obtain explicit formulas for the phase transition temperatures and equilibrium energy. The existence of equilibrium states is studied in some particular cases.