Recent zbMATH articles in MSC 74C https://zbmath.org/atom/cc/74C 2021-07-26T21:45:41.944397Z Werkzeug Reliable computation and local mesh adaptivity in limit analysis. https://zbmath.org/1463.49041 2021-07-26T21:45:41.944397Z "Sysala, Stanislav" https://zbmath.org/authors/?q=ai:sysala.stanislav "Haslinger, Jaroslav" https://zbmath.org/authors/?q=ai:haslinger.jaroslav "Repin, Sergey" https://zbmath.org/authors/?q=ai:repin.sergey-i This paper is concerned with limit load for a perfectly plastic model. Given a body, external forces (load) and admissible stress fields, we first find the caused stress and then determine for which factor of the external forces will the stress stop being admissible and the body would therefore collapse. The admissible stresses in this paper are tensors with bounded deviatoric part. This leads to a problem formulation with divergence-free functions. The discretization instead adds a penalty term to restrict the divergence and a sufficiently large penalization parameter is found on the initial mesh. The mesh is then refined to compute the load parameter and the upper bound for it. The method is demonstrated on an example. This is a continuation of the authors' work in [\textit{S. Repin} et al., Comput. Math. Appl. 75, No. 1, 199--217 (2018; Zbl 1458.74022)] and proofs of statements are not included in here. The contribution lies in using local mesh adaptivity. The limit load parameter for which the structure would collapse is typically determined only in a small area with the highest stress. The refinement is thus carried out on elements with the highest integral of stress function over them. We can expect that the refinement will focus almost entirely on one area. The numerical example uses P2 elements defined on isosceles right-angled triangles which are bisected upon refinement, but all reasoning behind this should also work for other elements. The computed upper bound is sharper than that on a uniform mesh. For the entire collection see [Zbl 1425.65005]. Elastic-plastic decomposition method of metallic structure based on molecule dynamics simulation https://zbmath.org/1463.74020 2021-07-26T21:45:41.944397Z "Cui, Junzhi" https://zbmath.org/authors/?q=ai:cui.junzhi "Yu, Yifan" https://zbmath.org/authors/?q=ai:yu.yifan Summary: In this paper a new elastic-plastic strain decomposition method is proposed based on Molecule Dynamics (MD) simulation for metallic structures. First the motion traces of atoms are decomposed into structural deformation component and thermal vibration, then the computational method and approximate formulae on the structural deformation are given. To the current configuration of the structure the continuous deformation functions are constructed based on the composition pattern of BCC-FCC cells and tetrahedral elements supported by 4-atoms, and the algorithm of deformation gradient is shown. By using the atomic-continuum coupled model the calculation formulae of the stress fields and elasticity tensor are developed. And then, the micro-defect forms generated by overlarge loading inside materials are analyzed, and classified into dislocations, stacking faults, twin boundaries, grain boundaries and vacancies et al. The constrained equations of rigid body motion satisfied for the stacking faults and twin boundaries during the elastic unloading process are derived, then the elastic unloading algorithm of current configuration is created by making use of minimum potential energy principle. Further, the entire elastic-plastic strain decomposition algorithm based on MD simulation is proposed. Finally, the numerical results for the tension of single crystal Cu nanowire are shown. It is shown that the elastic-plastic strain decomposition method in this paper is reasonable. The elastic-plastic decomposition method based on MD simulation presented above can be applied to the multi-scales analysis coupled with multiple models for mechanic behaviors of materials and their structures. Extension of the strip with symmetric angular notches https://zbmath.org/1463.74031 2021-07-26T21:45:41.944397Z "Chromov, A. I." https://zbmath.org/authors/?q=ai:chromov.a-i "Bukhan'ko, A. A." https://zbmath.org/authors/?q=ai:bukhanko.a-a "Patlina, O. V." https://zbmath.org/authors/?q=ai:patlina.o-v "Kocherov, E. P." https://zbmath.org/authors/?q=ai:kocherov.e-p Summary: The plastic flow of ideal rigid-plastic strip with symmetric $$V$$-notches is considered which results in a blunting of the notch under uniaxial tension. Elastic-plastic analysis of rotating solid shaft by maximum reduced stress yield criterion https://zbmath.org/1463.74032 2021-07-26T21:45:41.944397Z "Prokudin, Aleksandr Nikolaevich" https://zbmath.org/authors/?q=ai:prokudin.aleksandr-nikolaevich Summary: An elasto-plastic rotating solid cylinder under plane strain condition is investigated. The analysis is based on infinitesimal strain theory, maximum reduced stress yield criterion, its associated flow rule and perfectly plastic material behavior. It is assumed that angular velocity is monotonically increasing from 0 to the maximum value and then is monotonically reducing down to 0. In this investigation both loading and unloading phases are considered. It is assumed that angular velocity varies slowly with time, so angular acceleration can be neglected. Under above mentioned assumptions, there is only one non-trivial equilibrium equation in a cylinder. It is established that with increasing angular velocity four plastic regions appear in a cylinder. The last one forms at angular velocity which exceeds fully-plastic limit. Stresses image points of plastic regions lie on different sides and corners of yield surface. As the angular speed decreases, the whole cylinder behaves elastically again. At particular value of angular velocity secondary plastic flow may starts at the center of cylinder. Replasticization is possible only for sufficiently high maximum angular speed and the entire cylinder may be replasticized. Four secondary plastic regions may appear in the cylinder under unloading. The stresses image points in primary and secondary regions lie on opposite sides and corners of yield surface. In the present analysis it is assumed that the entire cylinder becomes replasticized just at stand-still. In this case only two secondary plastic regions emerge. Exact solutions for all stages of deformation are obtained. The systems of algebraic equations for determination of integration constants and border radii are formulated. The obtained results are illustrated by the distributions of stresses and plastic strains in the cylinder rotating at different speeds. Presented solutions are compared with known analytical solutions based on Tresca's criterion. Integro-differential equations of the second boundary value problem of linear elasticity theory. II: Inhomogeneous anisotropic body https://zbmath.org/1463.74033 2021-07-26T21:45:41.944397Z "Struzhanov, Valeriĭ Vladimirovich" https://zbmath.org/authors/?q=ai:struzhanov.valerii-vladimirovich Summary: In communication 1 [the author, Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki 21, No. 3, 496--506 (2017; Zbl 1413.74020)], the integro-differential equations of the second boundary value problem of the theory of elasticity for a homogeneous isotropic body were considered. The results obtained are extended to boundary value problems for the general case of an inhomogeneous anisotropic body. It is shown that the integro-differential equations found are also Fredholm type equations. The existence and uniqueness of their solution is proved, the conditions under which the solution can be found by the method of successive approximations are determined. An example of calculating the residual stresses in an inhomogeneous quenched cylinder is given. On the solution of one problem of deformation of rod systems that does not satisfy the Hadamard conditions by the simple iteration method https://zbmath.org/1463.74034 2021-07-26T21:45:41.944397Z "Struzhanov, Valeriĭ Vladimirovich" https://zbmath.org/authors/?q=ai:struzhanov.valerii-vladimirovich "Korkin, Aleksandr Vladimirovich" https://zbmath.org/authors/?q=ai:korkin.aleksandr-vladimirovich Summary: A rod system under the action of a quasi-statically increasing tensile tension is considered. The load is carried out according to soft and hard schemes. One of the rods of the system has the property of deformation softening, that is, its tension diagram has a branch falling to zero. As a result, the equilibrium equations do not satisfy the Hadamard conditions. The system has several equilibrium positions, including unstable ones. The application of the simple iterations method is shown to determine the parameters of all possible equilibrium positions and their stability when solving these equations that do not satisfy the Hadamard conditions. Properties of softening materials and definitive relations for uni-axial stress state https://zbmath.org/1463.74035 2021-07-26T21:45:41.944397Z "Struzhanov, V. V." https://zbmath.org/authors/?q=ai:struzhanov.valerii-vladimirovich Summary: Basing on the results obtained from the research of modified Masing mode, which allows prediction the declining branch on the diagram of material straining, we partition three classes of materials: elastic-fragile, elastic-plastic and part-plastic. For each class we obtain the definitive relations for the uni-axial stress-strain state. Modelled behaviour of granular material during loading and unloading. https://zbmath.org/1463.74038 2021-07-26T21:45:41.944397Z "Krejčí, Pavel" https://zbmath.org/authors/?q=ai:krejci.pavel "Siváková, Lenka" https://zbmath.org/authors/?q=ai:sivakova.lenka "Chleboun, Jan" https://zbmath.org/authors/?q=ai:chleboun.jan The paper is concerned with a numerical model of the behavior of granular material during loading and unloading. The standard strain-stress law is, under some conditions on the material, modified by the authors to obtain a rate-independent form. The resulting stress then admits an explicit representation. Numerical analysis of the solution of the model allows the authors to distinguish, in dependence on different parameter values, several solution categories: cyclic case, convergence to zero, divergence to infinity, and isotropic case. Strain ratcheting of granular material is typical for the isotropic case in agreement with experiments. All the solution categories are illustrated by figures. For the entire collection see [Zbl 1425.65005]. Dynamic damping -- comparison of different concepts from the point of view of their physical nature and effects on civil engineering structures. https://zbmath.org/1463.74054 2021-07-26T21:45:41.944397Z "Němec, Ivan" https://zbmath.org/authors/?q=ai:nemec.ivan "Trcala, Miroslav" https://zbmath.org/authors/?q=ai:trcala.miroslav "Vaněčková, Adéla" https://zbmath.org/authors/?q=ai:vaneckova.adela "Rek, Václav" https://zbmath.org/authors/?q=ai:rek.vaclav In their paper, the authors try to stimulate the discussion with the aim of replacing the non-justifiable and outdated Rayleigh damping model which is generally used in nonlinear dynamics. They draw attention to introducing a damping which is modelled by its real sources, using primarily (but not limited to) viscous material models. For example, the paper shows that the Rayleigh $$\beta$$ coefficient has a similar physical meaning as the viscosity $$\eta$$ of the Kelvin-Voigt material; in 1D are these coefficients identical. But the Rayleigh coefficient $$\alpha$$ is physically unjustified. Several other damping models are presented and analysed. Based on selected numerical results the authors recommend the standard linear solid'' model, unless an even more sophisticated material model is used. For the entire collection see [Zbl 1425.65005]. Modeling of viscoelastoplastic deformation of flexible shallow shells with spatial-reinforcements structures https://zbmath.org/1463.74080 2021-07-26T21:45:41.944397Z "Yankovskiĭ, Andreĭ Petrovich" https://zbmath.org/authors/?q=ai:yankovskii.andrei-petrovich Summary: Based on the procedure of time steps, a mathematical model of the viscoelastoplastic behavior of shallow shells with spatial reinforcement structures is constructed. Plastic deformation of the components of the composition is described by flow theory with isotropic hardening; viscoelastic deformation by the equations of the Maxwell-Boltzmann model. The possible weakened resistance of composite curved panels to transverse shear is taken into account in the framework of the hypotheses of Reddy's theory, and the geometric nonlinearity of the problem is taken into account in the Karman approximation. The solution of the formulated initial-boundary value problem is constructed using an explicit numerical scheme of the cross'' type. The elastoplastic and viscoelastoplastic flexural dynamic behavior of flat'' and spatially reinforced fiberglass cylindrical panels under the action of explosive loads has been investigated. Using the example of relatively thin composite structures, it is shown that, depending on which of the front surface (convex or concave), a load is applied, replacing the traditional flat'' reinforcement structure with a spatial one can lead to both an increase and a decrease in the residual deflection. However, in both cases, such a replacement can significantly reduce the intensity of residual deformations of the binder material and fibers of some families. It was demonstrated that the amplitudes of oscillations of curved composite panels in the neighborhood of the initial moment of time significantly exceed the maximum absolute values of the residual deflections. In this case, the residual deflections are rather complicated. It is shown that the calculations carried out within the framework of the elastoplastic deformation theory of the composition components do not even allow an approximate the magnitude determination of the residual deformations of the materials making up the composition. Analysis of a dynamic elasto-viscoplastic frictionless antiplan contact problem with normal compliance https://zbmath.org/1463.74095 2021-07-26T21:45:41.944397Z "Abbes, Ourahmoun" https://zbmath.org/authors/?q=ai:abbes.ourahmoun "Brahim, Bouderah" https://zbmath.org/authors/?q=ai:brahim.bouderah "Touffik, Serrar" https://zbmath.org/authors/?q=ai:touffik.serrar Summary: We consider a mathematical model which describes the dynamic evolution of a thermo elasto viscoplastic contact problem between a body and a rigid foundation. The mechanical and thermal properties of the obstacle coating material near its surface. A variational formulation of this dynamic contact phenomenon is derived in the context of general models of thermo elasto viscoplastic materials. The displacements and temperatures of the bodies in contact are governed by the coupled system consisting of a variational inequality and a parabolic differential equation. The proof is based on a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed point arguments.