The nonlinear mechanics of continual damage and its application to problems of creep and fatigue

The results of investigations of the mechanics of continual damage of materials and structural elements are reviewed. The focus is on nonlinea models of damage accumulation for problems of creep and high-cyclic fatigue. Nonlinearity criteria are formulated, and methods of construction of nonlinear models are analyzed. Approaches to the solution of problems of prediction of long-term strength and lifetime under creep and interaction of creep and fatigue are considered. S. P. Timoshenko Institute of Mechanics, national Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 31–66, March, 2000.     

On applications of the microlocal parameter method in modelling of temperature distributions in composite cylinders

Summary The problems of transient heat conduction in a periodically stratified medium consisting of a large number of alternating concentric cylinders of two homogeneous isotropic rigid materials and in a rotationally periodic cylinder consisting of a large number of circular homogeneous isotropic rigid sectors are considered. The equations of the homogenized models with microlocal parameters are derived by using the homogenization procedure given in [17]. The obtained models take into account certain microlocal effects connected with the microperiodic structure of the considered composites. Some examples of the application of the presented models to the problems of temperature distributions in composite cylinders are detailed. Received 10 March 1997; accepted for publication 23 October 1997     

Numerical simulation of solidification processes in enclosures

The present work deals with the development and application of numerical models for the simulation of solidification problems liquid/solid taking diffusion and convection into account. For the calculation of the thermal coupled flow process the finite element method is applied. In order to improve the numerical stability of the free convection problems, the streamline-upwind/Petrov–Galerkin method is used. Solidification processes are moving boundary problems. Three different models are set up which consider latent heat at the solidification front respectively in the mixed zone during the phase transition. Moreover, numerical methods are investigated in order to describe the behaviour of the flow at the boundary of the moving phase. Three examples serve illustrations; the technical example – casting of a transport and storage container – was provided by the company Siempelkamp Gießerei GmbH.     

Some results for approximate strain and rotation tensor formulations in geometrically non-linear Reissner-Mindlin plate theory

Finite element deflection and stress results are presented for four flat plate configurations and are computed using kinematically approximate (rotation tensor, strain tensor or both) non-linear Reissner-Mindlin plate models. The finite element model is based on a mixed variational principle and has both displacement and force field variables. High order interpolation of the field variables is possible through p-type discretization. Results for some of the higher order approximate models are given for what appears to be the first time. It is found that for the class of example problems examined, exact strain tensor but approximate rotation tensor theories can significantly improve the solution over approximate strain tensor models such as the von Kármán and moderate rotation models when moderate deflections/rotations are present. However, for each of the problems examined (with the exception of a postbuckling problem) the von Kármán and moderate rotation model results compared favorably with the higher order models for deflection magnitudes which could be reasonably expected in typical aeroelastic configurations.     

Solutions of some simple boundary value problems within the context of a new class of elastic materials

Some simple boundary value problems are studied, for a new class of elastic materials, wherein deformations are expressed as non-linear functions of the stresses. Problems involving ‘homogeneous’ stress distributions and one-dimensional stress distributions are considered. For such problems, deformations are calculated corresponding to the assumed stress distributions. In some of the situations, it is found that non-unique solutions are possible. Interestingly, non-monotonic response of the deformation is possible corresponding to monotonic increase in loading. For a subclass of models, the strain-stress relationship leads to a pronounced strain-gradient concentration domain in the body in that the strains increase tremendously with the stress for small range of the stress (or put differently, the gradient of the strain with respect to the stress is very large in a narrow domain), and they remain practically constant as the stress increases further. Most importantly, we find that for a large subclass of the models considered, the strain remains bounded as the stresses become arbitrarily large, an impossibility in the case of the classical linearized elastic model. This last result has relevance to important problems in which singularities in stresses develop, such as fracture mechanics and other problems involving the application of concentrated loads.     

Mathematical models for the study of interactions in the system lake baikal-atmosphere of the region

The structure of mathematical models for studying the processes of thermodynamics and transfer of pollutants in a climatic system involving the atmosphere of an industrial region and a lake is presented. These models are used to solve the problems of climatic and ecological monitoring and prediction. The problems of constructing numerical schemes and simulation methods are discussed. An example for estimating the effect of pollutants from sources located in the northern hemisphere of the Earth on the Baikal region is given. Institute of Computational Mathematics and Mathematical Geophysics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 137–147, March–April, 1999.     

Identification and nonlinearity compensation of hysteresis using NARX models

This paper deals with two problems: the identification and compensation of hysteresis nonlinearity in dynamical systems using nonlinear polynomial autoregressive models with exogenous inputs (NARX). First, based on gray-box identification techniques, some constraints on the structure and parameters of NARX models are proposed to ensure that the identified models display a key feature of hysteresis. In addition, a more general framework is developed to explain how hysteresis occurs in such models. Second, two strategies to design hysteresis compensators are presented. In one strategy, the compensation law is obtained through simple algebraic manipulations performed on the identified models. In the second strategy, the compensation law is directly identified from the data. Both numerical and experimental results are presented to illustrate the efficiency of the proposed procedures. Also, it has been found that the compensators based on gray-box models outperform the cases with models identified using black-box techniques.

     

Variational Evolution Problems and Nonlocal Geometric Motion

We consider two variational evolution problems related to Monge-Kantorovich mass transfer. These problems provide models for collapsing sandpiles and for compression molding. We prove the following connection between these problems and nonlocal geometric curvature motion: The distance functions to surfaces moving according to certain nonlocal geometric laws are solutions of the variational evolution problems. Thus we do the first step of the proof of heuristics developed in earlier works. The main techniques we use are differential‐equation methods in the Monge‐Kantorovich theory. (Accepted May 19, 1998)     

A Finite Element Analysis of a Viscoplastic Plate Subjected to Rapid Heating

Abstract

This paper models the response of a thin metallic plate that is subjected to a rapid heat input. In order to accurately model plate response, both the dynamic mechanical and transient heat transfer problems must be solved. The solution is complicated by nonlinearities due to radiation boundary conditions and material inelasticity. Furthermore, the viscoplastic constitutive equations that model the mechanical material behavior are numerically stiff. Nonlinear finite element algorithms are developed for both heat transfer and mechanical analyses. The algorithms are both stable and efficient for solving the problems considered herein. Example problems presented in the paper demonstrate the importance of including material nonlinearity in the model     

Residual‐based turbulence models for moving boundary flows: hierarchical application of variational multiscale method and three‐level scale separation

This paper presents residual‐based turbulence models for problems with moving boundaries and interfaces. The method is developed via a hierarchical application of variational multiscale ideas and the models are cast in an arbitrary Lagrangian–Eulerian (ALE) frame to accommodate the deformation of domain boundaries. An overlapping additive decomposition of velocity and pressure fields into coarse and fine scale components leads to coarse and fine scale mixed‐field problems. The problem governing fine scales is subjected to a further decomposition of the fine scale velocity into overlapping components termed as fine scales level I and level II. In turn, in the bottom‐up integration of scales, the model for level II fine scales serves to stabilize the problem governing level I fine scales, and model for level I fields yields the turbulence models. From the computational perspective, the coarse scales are represented in terms of the standard Lagrange shape functions, whereas level I and level II scales are represented via quadratic and fourth order polynomial bubbles, respectively. Because of the bubble functions approach employed in the consistently derived fine scale models, the resulting method is free of any embedded or tunable parameters. The proposed turbulence models share a common feature with the LES models in that the largest scales in the flow are numerically resolved, whereas the subgrid scales are modeled. The method is applied to flow around a plunging airfoil at Re = 40,000, and results are compared with experimental and numerical data published in the literature. Also presented are the results for the plunging airfoil at Re = 60,000 to show the robustness and range of applicability of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

A new approach in modeling of constant temperature boundary condition in thermal lattice‐Boltzmann method

The lattice‐Boltzmann method is being applied to a diversity of fluid flow and heat transfer problems nowadays. Because of its microscale nature, strict attention should be paid when introducing macroscopic inputs to the model. One of the challenging issues dealing with macroscale and microscale treatment is the implementation of boundary conditions. In this regard constant‐temperature boundaries are frequently used in energy transfer problems. Such boundaries are simply modeled in Navier–Stokes based solvers, but they are not so harnessed in lattice‐Boltzmann models. One of the problems is that the calculated tangential heat flux is not zero along such boundaries in most of the previous models. In the present paper, a model has been developed, which has the capability of controlling tangential heat flux along the constant‐temperature boundaries. It aims to set the heat flux nearly zero along the boundary in midplane grid schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

Normal Compliance Contact Models with Finite Interpenetration

We study contact problems with contact models of normal compliance type, where the compliance function tends to infinity for a given finite interpenetration. Such models are physically more realistic than standard normal compliance models, where the interpenetration is not restricted, because the interpenetration is typically justified by deformations of microscopic asperities of the surface; therefore it should not exceed a certain value that corresponds to a complete flattening of the asperities. The model can be interpreted as intermediate between the usual normal compliance models and the unilateral contact condition of Signorini type. For the static problem without friction, we prove the existence and uniqueness of solutions and establish the equivalence to an optimization problem. For the static problem with Coulomb friction, we show the existence of a solution. The analysis is based on an approximation of the problems by standard normal compliance models, the derivation of regularity results for these auxiliary problems in Sobolev spaces of fractional order by a special translation technique, and suitable limit procedures.     

A convergence study of two ocean models applied to a density driven flow

A three‐dimensional, baroclinic and rotational benchmark for hydrostatic coastal ocean models is suggested. The computational domain is a quadratic basin 200×200 km horizontally and 200 m deep. The lateral boundaries are closed. Solutions to the problems are estimated with two different sigma co‐ordinate models, both for the diagnostic and the prognostic case. Grid‐ and time‐converged results are presented. For the transports, convergence within two significant digits is reported. Copyright © 2001 John Wiley & Sons, Ltd.     

Basic principles and concepts of model analysis

The techniques and the problems associated with the design and use of models are covered. Particular emphasis is placed on the development of modeling laws by means of dimensional analysis. Special problems associated with the use of true and distorted models are illustrated by means of selected examples.     

Finite Element Analysis of Elasto-plastic Plate Bending Problems using Transition Rectangular Plate Elements

In this work, the finite element analysis of the elasto-plastic plate bending problems is carried out using transition rectangular plate elements. The shape functions of the transition plate elements are derived based on a practical rule. The transition plate elements are all quadrilateral and can be used to obtain efficient finite element models using minimum number of elements. The mesh convergence rates of the models including the transition elements are compared with the regular element models. To verify the developed elements, simple tests are demonstrated and various elasto-plastic problems are solved. Their results are compared with ANSYS results.The English text was polished by Keren Wang.     

Quasi-one-dimensional approximation in two-dimensional problems of gas dynamics

A useful means of constructing approximate flow models is the hydraulic (for two-dimensional problems quasi-one-dimensional) approach, based on averaging the initial nonuniform flows over some direction or cross section [1]. In this case, at the expense of a rougher model it is possible to reduce the dimensionality of the problem. Here, this approach is extended to unsteady two-dimensional gas-dynamic processes; certain problems (flow around a cone or a blunt body, jet flows) are considered in the framework of the quasi-one-dimensional model obtained, and results are compared with the solutions of the corresponding two-dimensional problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 136–143, March–April, 1989.     

Applied Problems in the Mechanics of Strain Hardening of Structural Metallic Materials

Some applied problems of the mechanics of strain-hardening processes in metallic materials are considered. To solve these problems, the concept of loading surface, which separates the elastic and elastoplastic domains in the stress space, is used. Strain-hardening models are analyzed. For a wide range of steels and alloys, the most commonly used hypothesis of isotropic-and-kinematic hardening is experimentally justified __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 10, pp. 12–51, October 2005.     

SOME VALIDATION OF STANDARD,MODIFIED AND NON-LINEAR k–ε TURBULENCE MODELS

Standard, modified and non-linear k–ε: turbulence models are validated against three axisymmetric flow problems—flow through a pipe expansion, flow through a pipe constriction and an impinging jet problem—to underpin knowledge about the solution quality obtained from two-equation turbulence models. The extended models improve the prediction of turbulence as a flow approaches a stagnation point and the non-linear model allows for the prediction of anisotropic turbulence. Significantly different values for the non-linear model coefficients are proposed in comparison with values found in the literature. Nevertheless, current turbulence models are still unable to accurately predict the spreading rate of shear layers. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids, 24: 965–986, 1997.     

Free and forced vibration of repetitive structures

In this paper, the vibration problems of some repetitive structures, including symmetric, cyclic periodic, linear periodic, chain, and axi-symmetric structures is investigated. Eigen-value problems derived from the vibration equations of these structures are established based on their continuous models. The special properties of the structural modes of these structures are deduced. Applying these properties can provide effective reduction approach to solving the natural and forced vibration problems of these structures by either numerical or experimental methods. Furthermore, these properties can be applied in other aspects such as evaluating the reasonableness of the discrete models of these repetitive structures.