Recent zbMATH articles in MSC 74A https://zbmath.org/atom/cc/74A 2022-07-25T18:03:43.254055Z Werkzeug Continuum model and numerical method for dislocation structure and energy of grain boundaries https://zbmath.org/1487.35363 2022-07-25T18:03:43.254055Z "Qin, Xiaoxue" https://zbmath.org/authors/?q=ai:qin.xiaoxue "Gu, Yejun" https://zbmath.org/authors/?q=ai:gu.yejun "Zhang, Luchan" https://zbmath.org/authors/?q=ai:zhang.luchan "Xiang, Yang" https://zbmath.org/authors/?q=ai:xiang.yang Computable constants for Korn's inequalities on Riemannian manifolds https://zbmath.org/1487.53057 2022-07-25T18:03:43.254055Z "Knops, R. J." https://zbmath.org/authors/?q=ai:knops.robin-j Summary: A method is presented for the explicit construction of the non-dimensional constant occurring in Korn's inequalities for a bounded two-dimensional Riemannian differentiable simply connected manifold subject to Dirichlet boundary conditions. The method is illustrated by application to the spherical cap and minimal surface. Hamilton's principle in continuum mechanics https://zbmath.org/1487.74002 2022-07-25T18:03:43.254055Z "Bedford, Anthony" https://zbmath.org/authors/?q=ai:bedford.anthony Publisher's description: This revised, updated edition provides a comprehensive and rigorous description of the application of Hamilton's principle to continuous media. To introduce terminology and initial concepts, it begins with what is called the first problem of the calculus of variations. For both historical and pedagogical reasons, it first discusses the application of the principle to systems of particles, including conservative and non-conservative systems and systems with constraints. The foundations of mechanics of continua are introduced in the context of inner product spaces. With this basis, the application of Hamilton's principle to the classical theories of fluid and solid mechanics are covered. Then recent developments are described, including materials with microstructure, mixtures, and continua with singular surfaces. \begin{itemize} \item Presents a comprehensive, rigorous description of the application of Hamilton's principle to continuous media; \item Includes recent applications of the principle to continua with microstructure, mixtures, and media with surfaces of discontinuity; \item Discusses foundations of continuum mechanics and variational methods therein in the context of linear vector spaces. \end{itemize} The effect of superheat on the nucleation undercooling of metallic melts https://zbmath.org/1487.74008 2022-07-25T18:03:43.254055Z "Xu, Junfeng" https://zbmath.org/authors/?q=ai:xu.junfeng "Fan, Dandan" https://zbmath.org/authors/?q=ai:fan.dandan "Zhang, Tao" https://zbmath.org/authors/?q=ai:zhang.tao.2|zhang.tao.1|zhang.tao.4|zhang.tao.6|zhang.tao.5 Summary: The influences of the superheating temperature $$(T_s)$$ on the nucleation undercooling $$(\Delta T)$$ of metallic melts were investigated by using molecular dynamics simulations based on the embedded atom model (EAM) potential function. The results agree with the intuitive expectation that extremely high heating rates followed by short equilibration time lead to a superheating and partial melting of the solid phase. The fraction of the remained crystalline clusters in the superheated phase depends on the superheating temperature $$T_s$$ and the equilibration time, as long as $$T_s$$ is below the maximal superheating. A subsequent fast cooling facilitates a substantial undercooling of the molten phase. The achieved undercooling $$\Delta T$$ below the steady-state melting temperature $$T_m$$ depends on the size and the concentration of the crystalline clusters remained in the liquid phase, and thus on the initial superheating temperature $$T_s$$. Based on the simulated results, a model was proposed for describing the relationship of $$\Delta T$$ and $$T_s$$, with which simulated data are well fitted and the maximal undercooling for metals can be predicted. Mathematical modeling of the stress-strain state in metallic media based on the concept of force lines https://zbmath.org/1487.74009 2022-07-25T18:03:43.254055Z "Chukanov, Aleksandr Nikolaevich" https://zbmath.org/authors/?q=ai:chukanov.aleksandr-nikolaevich "Terëshin, Valeriĭ Alekseevich" https://zbmath.org/authors/?q=ai:tereshin.valerii-alekseevich "Tsoĭ, Evgeniĭ Vladimirovich" https://zbmath.org/authors/?q=ai:tsoi.evgenii-vladimirovich Summary: In this article, based on the classical works of G. Kirsch, K. Inglis, G. V. Kolosov, and N. I. Muskhelishvili, we continue to develop a mathematical apparatus that allows us to obtain solutions to a number of three-dimensional problems of fracture mechanics in a hardened metal medium. Based on the work of G. R. Irwin, G. I. Barenblatt, Westergaard, L. D. Landau, and E. M. Livshits, the authors performed mathematical modeling of the stress-strain state in the volume of a loaded steel sample in the vicinity of pores of various morphologies resulting from operational loads and aggressive environmental influences. An algorithm for determining the components of the stress tensor near concentrators in the form of pores of various shapes is proposed for understanding the force lines of the stress field in a metallic medium. A stationary case with a fixed ratio of external stress and yield strength was considered. Identifying processes governing damage evolution in quasi-static elasticity. I: Analysis https://zbmath.org/1487.74010 2022-07-25T18:03:43.254055Z "Grützner, Simon" https://zbmath.org/authors/?q=ai:grutzner.simon "Muntean, Adrian" https://zbmath.org/authors/?q=ai:muntean.adrian The authors consider a quasi-static elasticity model with damage written as $$\sigma =(1-d)\mathbb{E}\varepsilon (u)$$, $$-div(\sigma )=f$$, $$d^{\prime }=(1-d)^{-\alpha }g(\nabla u)$$, and posed in $$S\times \Omega$$, where $$S=(0,T)$$ and $$\Omega$$ is a bounded and Lipschitz domain in $$\mathbb{R }^{N}$$, $$N=1,2,3$$. The boundary $$\partial \Omega$$ of $$\Omega$$ may be decomposed as $$\Gamma _{0}\cup \Gamma _{1}$$ where $$\Gamma _{0}$$ and $$\Gamma _{1}$$ are disjoint, closed, have positive surface measures and are the union of connected components of $$\partial \Omega$$. The authors define the set $$\mathcal{G}$$ of admissible damage process as $$\mathcal{G}=\{g\in L^{\infty }(S;L^{\infty }(\Omega ;C^{1,1}(\overline{Y})));$$ $$\forall y\in \overline{Y} :0\leq g(\cdot ,\cdot ,y)\leq T^{-1}(\omega _{1}-\omega _{0})(1-\omega _{1})^{\alpha }$$ a.e. in $$S\times \Omega \}$$, where $$\overline{Y}=\overline{B }(0,\overline{y})\subset \mathbb{R}^{N^{2}}$$. They prove that for every $$f\in L^{\infty }(S;L^{\infty }(\Omega ))^{N^{2}}$$ an admissible damage process $$g\in \mathcal{G}$$ generates a Lipschitz-continuous Nemytskii operator $$G:L^{\infty }(S;L^{\infty }(\Omega ))^{N^{2}}\rightarrow L^{\infty }(S;L^{\infty }(\Omega ))$$ through $$G(f)(t,x)=g(t,x,f(t,x))$$. They define the set of admissible tractions $$\tau$$ as $$L^{\infty }(S;W^{\ast })$$ and they introduce the function spaces $$V=\{v\in W^{1,2}(\Omega )^{N};$$ $$v=0$$ on $$\Gamma _{0}\}$$, $$W=W^{1/2,2}(\Gamma _{1})^{N}$$, their duals $$V^{\ast }$$ and $$W^{\ast }$$ and the family of operators $$A_{d}(t):V\rightarrow V^{\ast }$$ for a.e. $$t\in S$$ and its realization $$\mathcal{A}(d):L^{2}(S;V)\rightarrow L^{2}(S;V^{\ast })$$ through $$\left\langle (\mathcal{A}(d)u)(t),v\right \rangle _{L^{2}(\Omega )}=\left\langle (A_{d}(t)u(t)),v\right\rangle _{L^{2}(\Omega )}=\int_{\Omega }(1-d(t))\mathbb{E}\varepsilon (u(t)):\varepsilon (v)dx$$. They finally define the traction-driven problem as: For given $$f\in L^{\infty }(S;V^{\ast })$$, $$\tau \in L^{\infty }(S;W^{\ast })$$, $$d_{0}\in D_{0}=\{d_{0}\in L^{\infty }(\Omega );$$ $$0\leq d_{0}(x)\leq \omega _{0}$$ a.e. in $$\Omega \}$$, and $$g\in \mathcal{G}$$, find $$u\in L^{\infty }(S;V)$$ such that $$\mathcal{A}(d)u=f+\tau$$ in $$L^{\infty }(S;V^{\ast })$$, $$d^{\prime }=(1-d)^{-\alpha }G(\nabla ^{\mu }u)$$ in $$L^{\infty }(S;L^{\infty }(\Omega ))$$, and $$d(0)=d_{0}$$ in $$L^{\infty }(\Omega )$$. They also define the forward operator $$\Phi :\mathcal{G} \rightarrow L^{\infty }(S\times \Omega )^{N}$$ through $$\Phi =\pi _{1}\circ F$$, where $$\pi _{1}$$ is the projection on the first component of $$F$$. The first result proves that the traction-driven problem has a unique solution which is Lipschitz continuous with respect to the data. The proof is obtained decoupling the momentum balance and the damage evolution equation. The solution to the coupled problem is obtained applying Banach's fixed point theorem. The authors also prove that the above-defined Nemytskii operator is Fréchet-differentiable and they compute its Fréchet-differential. They characterize the adjoint of the linearized forward operator. In the last part of their paper, the authors analyze the ill-posedness of the inverse problem: Find $$g$$ such that $$\Phi (g)=u^{\delta }\in L^{2}(S\times \Omega )^{N}$$, whose linearization is $$\partial \Phi (g)h=u^{\delta }$$ in $$L^{2}(S\times \Omega )^{N}$$. They prove that the inverse problem is locally and globally ill-posed. They use a tangential condition they establish. Reviewer: Alain Brillard (Riedisheim) An intrinsic geometric formulation of hyper-elasticity, pressure potential and non-holonomic constraints https://zbmath.org/1487.74012 2022-07-25T18:03:43.254055Z "Kolev, B." https://zbmath.org/authors/?q=ai:kolev.boris "Desmorat, R." https://zbmath.org/authors/?q=ai:desmorat.rodrigue Summary: Isotropic hyper-elasticity, altogether with the equilibrium equations and the usual boundary conditions, are formulated directly on the body $$\mathcal{B}$$, a three-dimensional compact and orientable manifold with boundary equipped with a mass measure. Pearson-Sewell-Beatty pressure potential on the boundary is recovered, using the Poincaré formula. The existence of such a potential requires conditions, which are formulated as non-holonomic constraints on the configuration space. On the incompressible behavior in weakly nonlinear elasticity https://zbmath.org/1487.74013 2022-07-25T18:03:43.254055Z "Kube, Christopher M." https://zbmath.org/authors/?q=ai:kube.christopher-m Summary: This article considers the influence of incompressibility on the compliance and stiffness constants that appear in the weakly nonlinear theory of elasticity. The formulation first considers the incompressibility constraint applied to compliances, which gives explicit finite limits for the second-, third-, and fourth-order compliance constants. The stiffness/compliance relationships for each order are derived and used to determine the incompressible behavior of the second-, third-, and fourth-order stiffness constants. Unlike the compressible case, the fourth-order compliances are not found to be dependent on the fourth-order stiffnesses. Applied questions of Il'yushin theory of elastoplastic processes https://zbmath.org/1487.74014 2022-07-25T18:03:43.254055Z "Molodtsov, I. N." https://zbmath.org/authors/?q=ai:molodtsov.i-n.1 Summary: The experimental results of the processes of complex loading along helical strain trajectories are used to find out that the response to the helical strain trajectory following the simple loading after exhaustion of some trace takes a certain shape of the limit mode, that is, there is a correspondence between the deformation trajectory geometry and the form of response. A new variant of constitutive equations for describing complex loading processes with strain trajectories of arbitrary geometry and dimension is considered. The vector constitutive equations and the system of differential equations for the four angles from the Frenet decomposition are obtained. It is proved that the stress vector is represented in the form of sum of three terms: rapidly decaying plastic traces of elastic states, instantaneous responses to the deformation process, and irreversible stresses accumulated along the deformation trajectory. A new method for mathematical modeling of five-dimensional processes of complex loading is constructed and tested on two- and three-dimensional processes. A fractional derivative-based numerical approach to rate-dependent stress-strain relationship for viscoelastic materials https://zbmath.org/1487.74018 2022-07-25T18:03:43.254055Z "Su, Teng" https://zbmath.org/authors/?q=ai:su.teng "Zhou, Hongwei" https://zbmath.org/authors/?q=ai:zhou.hongwei "Zhao, Jiawei" https://zbmath.org/authors/?q=ai:zhao.jiawei "Liu, Zelin" https://zbmath.org/authors/?q=ai:liu.zelin "Dias, Daniel" https://zbmath.org/authors/?q=ai:dias.daniel-m|dias.daniel-a|dias.daniel-b Summary: Strain/stress-controlled loading, loading\textbf{-}unloading, loading-relaxation (or creep), and corresponding cyclic tests are essential for characterizing the viscoelastic materials' rate-dependent stress-strain relationship. A three-parameter model is proposed based on the basic definition of fractional derivative viscoelasticity and time-varying viscosity. This model is applied to many complex loading conditions. The solutions for three monocyclic loading conditions are given and then further generalized to arbitrary linear loading conditions, which are assumed to be first-order functions of time. The generalized solution for the arbitrary linear loading path is validated by modelling the mechanical response of cyclic loading-unloading and loading-relaxation (or creep) tests. Four sets of experimental data for polymer materials are employed to demonstrate the proposed fractional derivative model's efficiency. The results show that it can accurately model strain/ stress-controlled response under various loading conditions using only three parameters. The model is then implemented in numerical software to explore its capacity further, and the simulation results show that it also succeeds in simulating cyclic loading-unloading tests. Free energies for nonlinear materials with memory https://zbmath.org/1487.74019 2022-07-25T18:03:43.254055Z "Golden, J. M." https://zbmath.org/authors/?q=ai:golden.john-murrough Summary: An exploration of representations of free energies and associated rates of dissipation for a broad class of nonlinear viscoelastic materials is presented in this work. Also included are expressions for the stress functions and work functions derivable from such free energies. For simplicity, only the scalar case is considered. Certain standard formulae are generalized to include higher power terms. It is shown that the correct initial procedure in this context is to specify the rate of dissipation as a positive semi-definite functional and then to determine the free energy from this, rather than the other way around, which would be the traditional approach. Particularly detailed versions of these formulae are given for the model with two memory contributions in the free energy, the first being the well-known quadratic functional leading to constitutive relations with linear history terms, while the second is a quartic functional yielding a cubic term for the stress function memory dependence. Also, the discrete spectrum model, for which each memory kernel is a sum of exponentials, is generalized from the quadratic functional representation for the free energy to that with the quartic functional included. Finally, a model is considered, allowing functional power series with an infinite number of terms for the free energy, rate of dissipation and stress function. Prismatic dislocation loops in crystalline materials with empty and coated channels https://zbmath.org/1487.74020 2022-07-25T18:03:43.254055Z "Kolesnikova, Anna L." https://zbmath.org/authors/?q=ai:kolesnikova.anna-l "Chernakov, Anton P." https://zbmath.org/authors/?q=ai:chernakov.anton-p "Gutkin, Mikhail Yu." https://zbmath.org/authors/?q=ai:gutkin.mikhail-yu "Romanov, Alexey E." https://zbmath.org/authors/?q=ai:romanov.alexey-e Summary: This paper presents for the first time an analytical solution to the boundary-value problem in the theory of elasticity for a circular prismatic dislocation loop (PDL) coaxial to a hollow cylindrical channel in an elastically isotropic infinite matrix. The stress fields and energy of the PDL are calculated and analyzed in detail. Based on the solution, a theoretical model for the misfit stress relaxation through the formation of a misfit PDL around a misfitting nanotube embedded in an infinite matrix is suggested. The critical radii of the embedded nanotube are found and discussed. It is shown that, for thin nanotubes prepared by nanolayer growth on the initial channel surface, there are two critical inner radii of the nanotube, between which the formation of a misfit PDL is energetically favorable. Fiber-reinforced composites: nonlinear elasticity and beyond https://zbmath.org/1487.74023 2022-07-25T18:03:43.254055Z "Wineman, A." https://zbmath.org/authors/?q=ai:wineman.alan-s "Pence, Thomas J." https://zbmath.org/authors/?q=ai:pence.thomas-j Summary: A fiber-reinforced material comprised of a soft polymeric matrix reinforced with polymeric filaments is often modeled as an equivalent anisotropic nonlinearly elastic solid. Although the response of a single constituent polymeric material can be modeled by nonlinear thermo-elasticity over a large range of deformations and temperatures, there can be conditions requiring a theory that extends the range of application to account for other features, such as nonlinear viscoelasticity and an evolving microstructure due to a combination of mechanical and nonmechanical factors. In a multi-constituent fiber-reinforced material these effects can be expected to occur with different initial triggering and ongoing potency in the separate polymer matrix and fiber constituents. This paper summarizes a number of constitutive models for fiber-reinforced materials that include these features, discusses the connection of these models to a nonlinearly elastic scaffold, provides a framework for the incorporation of these features into the constitutive theory for an equivalent general simple solid, and shows how certain terms in the mathematical structure can be associated with the matrix constituent while other terms can associated with the fibrous constituent. Some BVP in the plane theory of thermodynamics with microtemperatures https://zbmath.org/1487.74026 2022-07-25T18:03:43.254055Z "Gulua, B." https://zbmath.org/authors/?q=ai:gulua.bakur "Janjgava, R." https://zbmath.org/authors/?q=ai:janjgava.roman "Kasrashvili, T." https://zbmath.org/authors/?q=ai:kasrashvili.tamar "Narmania, M." https://zbmath.org/authors/?q=ai:narmania.miranda Summary: In this work we consider the two-dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. The Dirichlet BVP is solved for a circle. Irreversible deformation under thermomechanical loading of solids https://zbmath.org/1487.74027 2022-07-25T18:03:43.254055Z "Kikvidze, O. G." https://zbmath.org/authors/?q=ai:kikvidze.o-g Summary: The article considers irreversible deformation of solid under thermomechanical loading, using the phenomenological approach. It is assumed that the strains are small. On the basis of the dilatometric curves and the stress-strain curves, the condition was formulated for the stability of material, and the major inequality and constitutive equations for the irreversible strains under thermo-mechanical loading were obtained. These equations describe the pattern of inelastic deformation of a wide class of metallic materials in the temperature ranges of the phase transformations. Fluid mechanics of viscoplasticity https://zbmath.org/1487.76001 2022-07-25T18:03:43.254055Z "Huilgol, Raja R." https://zbmath.org/authors/?q=ai:huilgol.raja-r "Georgiou, Georgios C." https://zbmath.org/authors/?q=ai:georgiou.georgios-c Publisher's description: This book considers the kinematics and dynamics of the flows of fluids exhibiting a yield stress. Continuum mechanics governing the fluid mechanics is described. Two chapters are dedicated to analytical solutions to several steady and unsteady flows of viscoplastic fluids, including flows with pressure-dependent rheological parameters. Perturbation methods, variational inequalities to solve fluid flow problems, and the use of energy methods are discussed. Numerical modeling using augmented Lagrangian, operator splitting, finite difference, and lattice Boltzmann methods are employed. The second edition provides new sections on flows of yield stress fluids with pressure-dependent rheological parameters, on flows with wall slip, and on deriving the fundamental equations for Boltzmann lattice materials. Furthermore new material on the lubrication approximation and applications of finite differences has been added. See the review of the first edition in [Zbl 1328.76001]. Effect of magnetic field and non-uniform surface on squeeze film lubrication https://zbmath.org/1487.78005 2022-07-25T18:03:43.254055Z "Muthu, P." https://zbmath.org/authors/?q=ai:muthu.p "Pujitha, V." https://zbmath.org/authors/?q=ai:pujitha.v Summary: In the present paper, the combined effect of magnetic field and nonuniform shape of the surface on squeeze film characteristics is investigated. The non-uniform squeeze film thickness is calculated using Lagrange interpolation technique. Numerical integration procedure is used to obtain the solution for pressure, load carrying capacity. The effects of field parameters on squeeze film characteristics are discussed and are presented graphically. It is observed that externally applied magnetic field and non-uniform shape of the bearing surface enhance the squeeze film lubrication. An analytical model for effective thermal conductivity of the media embedded with fracture networks of power law length distributions https://zbmath.org/1487.80010 2022-07-25T18:03:43.254055Z "Miao, Tongjun" https://zbmath.org/authors/?q=ai:miao.tongjun "Chen, Aimin" https://zbmath.org/authors/?q=ai:chen.aimin "Jiang, Lijuan" https://zbmath.org/authors/?q=ai:jiang.lijuan "Zhang, Huajie" https://zbmath.org/authors/?q=ai:zhang.huajie "Liu, Junfeng" https://zbmath.org/authors/?q=ai:liu.junfeng "Yu, Boming" https://zbmath.org/authors/?q=ai:yu.boming Evaporation of black holes in flat space entangled with an auxiliary universe https://zbmath.org/1487.83101 2022-07-25T18:03:43.254055Z "Miyata, Akihiro" https://zbmath.org/authors/?q=ai:miyata.akihiro "Ugajin, Tomonori" https://zbmath.org/authors/?q=ai:ugajin.tomonori (no abstract)