Problems In Continuum Mechanics Research Articles

TESTING OF ROCK SAMPLES FROM OIL AND GAS FIELDS TO DETERMINE THEIR PHYSICAL AND MECHANICAL CHARACTERISTICS

Determination of the physical and mechanical properties of anisotropic limestone rocks is an important and urgent task, since carbonate deposits are very often found in oil and gas fields of the Mangistau region. The data show that limestone creates from 70% to 90% of all problems related to the stability of wells, which is due to the peculiarities of their physical and mechanical properties. Experimental studies of the physical and mechanical properties of limestone have features both in terms of monolith selection and in terms of preparation of samples for research, and in the methods of conducting and processing experiments. These features are due to extremely low permeability, anisotropy of elastic and strength properties due to their layered structure. The experience of testing core samples of layered limestone is considered, tests are carried out on equipment under the program of consolidated-undrained loading in accordance with the standards. As a result of experiments on core samples cut at an angle, along and across the layering, the anisotropic elastic properties of rocks, as well as their dependencies on geophysical parameters, were studied. The anisotropy of elastic properties has a significant effect both on the stress-strain state when solving specific problems of continuum mechanics and on the design values of the initial stress field as a whole. Taking into account the anisotropy parameters obtained in this paper will make it possible to solve Geomechanics problems in relation to layered limestones of the Sarmatian stage.

To the Development and Perspective of Some Linear and Non-linear Problems of Continuum Mechanics

To the Development and Perspective of Some Linear and Non-linear Problems of Continuum Mechanics

Об идентификации характеристик неоднородных вязкоупругих тел в рамках модели дробного порядка

Nowadays, the development of models of viscoelastic materials with complex inhomogeneous structure is one of the hottest problems of continuum mechanics. Along with classical models, fractional-differential models of viscoelasticity have become increasingly popular. In this paper, we present a model for describing steady-state oscillations of inhomogeneous viscoelastic bodies in terms of fractional-order differential operators. Taking into account the fractional order of the model operators, the corresponding form of the complex module describing the material properties is constructed. The module includes four characteristics: instantaneous and long-term elastic moduli (in the case of material inhomogeneity, they are the functions of coordinates), relaxation time and fractionality parameter. The properties of the complex module are studied, and the ranges of the model parameters, at which the rheological properties are most pronounced, are identified. A general formulation of the inverse problem (IP) on the identification of function-parameters of the model based on the acoustic sounding data is proposed. In the framework of this formulation, the inverse problems for specific objects , namely, an inhomogeneous rod and a round plate, are considered. In both model problems, the influence of the fractionalization parameter on the amplitude-frequency characteristics is analyzed. It has been found that in the vicinity of viscoelastic resonances the fractionality parameter affects the parameters of the oscillatory process most significantly, which is typical for such problems. A solution to the nonlinear IPs under consideration is constructed based on the linearization method, which also serve as basis for the iterative processes supplemented with the elements of the projection approach. This allows one to determine corrections to the required functions in the specified classes of functions using regularization. For both IPs, a series of computational experiments were carried out. The analysis of the experimental results made it possible to formulate recommendations for the selection of optimal sensing modes.

Разработка модуля для управления моделированием системы двух сжимаемых жидких фаз с одной дисперсной фазой на базе платформы OpenFOAM

The purpose of the presented research is to refine the initial release of the graphical shell for the OpenFOAM package by designing and connecting an additional module focused on numerical experiments using the twoPhaseEulerFoam solver in the field of modeling problems in continuum mechanics. This module, unlike existing analogue applications, has the status of an open source software product, does not require the purchase of maintenance services, and has a Russian-language interface. In the presented software, to simplify further support and modification, the source code of the external part of the application is separated from the code that provides the operating logic. The key original approaches proposed by the author also include a subsystem for serializing design parameters, which allows you to convert the parameters of a design case into json and csv objects and perform the reverse process. This allows the user to switch between different parameter sets for one design case. In addition, it is worth emphasizing the presence in the created software module of a mechanism for checking the completeness of the design case before starting the numerical experiment. Some features of the solver and the principles of its use in preparing calculation cases are considered. The purpose of the study was determined and a list of required tasks was compiled. The selected technology stack is described, as well as development aids. A process diagram is provided to demonstrate how the application works, along with a description of each step. The results of the study were tested using the example of one of the fundamental problems of continuum mechanics and are presented in the form of an updated version of the graphical shell, publicly available on the GitHub resource. Based on the results of the study, the effectiveness of the selected technology stack for achieving development goals was confirmed, and the completed tasks were noted. The practical significance of the results is formulated, expressed in the potential saving of working time for engineers and researchers, minimizing modeling errors and simplifying the process of preparing a design case.

Forced vibrations of a cantilever beam using radial point interpolation methods: A comparison study

Meshless methods are a type of numerical method used to simulate continuum mechanics problems. These methods have been applied to several types of problems, there are a few works using meshless method focused on dynamic problems, but most works study static loading conditions. The current work aims at using two different meshless methods, the Radial Point Interpolation Method (RPIM) and the Natural Neighbours RPIM (NNRPIM), in a dynamic problem, specifically forced vibrations. This problem requires time integration, therefore three different time integration methods have also been tested, namely: the Central Difference Method (CDM), the Wilson method, and the Newmark method. The CDM is an explicit method, while the other two are implicit. A discretization study was performed to assess the ideal nodal discretization before the numerical and time integration methods are validated. For the implicit methods, different time step lengths were also tested. In the final example damping was introduced. The results prove the validity of the two meshless methods by having similar results to the Finite Element Method (FEM) using the three distinct time integration methods, different loading conditions, and damping.

Interface Development of a Module for the interFoam Solver of the OpenFOAM Package

The results of a study on expanding the list of tasks solved using the graphical shell of the OpenFOAM platform by creating and connecting an additional module for the interFoam solver to the basic version of the graphical shell are presented. The relevance of the problem is formulated, the goal and the list of tasks necessary to achieve it. The features of this solver are studied and described in relation to experiments on numerical modeling of problems in continuum mechanics. The possibility of using open technologies to supplement the source code of the graphical shell with new modules covering a larger range of types of problems in continuum mechanics is analyzed. Graphic materials have been prepared that describe the structure and logic of the module being developed. The technology stack used is presented, and the results of testing the updated version of the graphical shell using the example of one of the classic problems of continuum mechanics are presented. The results of the research were summed up, the tasks completed were noted, the expected practical value of the development and possible ways for further research in this direction were indicated, the repository with the source code of the graphical shell on the GitHub service was updated.

Selection of shaped charges parameters for producing aluminum particles with velocities in the range of 2.5–16 km/s

Testing rocket and space technology objects in ground conditions for resistance to the impact of meteoroids and fragments of space debris can be carried out using shaped charges. To substantiate the design parameters of shaped charges that ensure the formation of aluminum particles in a wide velocity range (from 2.5 to 16 km/s), numerical modeling of the formation process was carried out within the framework of a two-dimensional axisymmetric problem of continuum mechanics using three different computing codes to increase the reliability of the results. The calculations consider shaped charges with a diameter of 20–100 mm with aluminum liners of various shapes. It is shown that the formation of particles with velocities close to the lower limit of the considered range is ensured by gently sloping segmental liners of degressive thickness. To form higher-velocity particles with velocities over 5 km/s, it is proposed to use combined liners, the jet-forming part of which has the shape of a hemisphere of constant thickness or the shape of a semi-ellipsoid or semi-superellipsoid of rotation of degressive thickness.

Approximated decompositions for computational continuum mechanics

Solid and fluid mechanics problems often involve the computation of eigenvalues and eigenvectors. One has short algebraic expressions for the Euler equations and the perfect gas equation of state. However, algebraic expressions for solid mechanics models can become substantial. Computing the eigenvalues and eigenvectors algebraically for different models can become cumbersome. Also, larger formulae may result in increased overall computational time. Internal energy plays a significant role in modeling the medium, and thermodynamic consistency is indispensable. The models should correctly capture heat transfer, mechanical response, and fluid flow, e.g., in mechanocaloric and bone remodeling (BR) problems. In mechanocaloric problems, temperature variation occurs because of load and unload stresses. In BR problems, several mathematical models were developed for modeling its behavior. Flexibility for testing different models is an essential requirement. Hence, our goal is to propose and test approximated eigensystem and singular value (SVD) decompositions for continuum mechanics problems. We tested the resolution and accuracy of our propositions in benchmarking CFD problems. For the solid mechanics, we solved a few classical problems, verified the accuracy, and solved mechanocaloric tests and bar under tension. Although more expensive, the approximated eigensystem decomposition is more flexible. It is high-order and high-resolution, can capture heat transfer, fluid flow, discontinuities, and mechanical response, and can model real-world applications. The SVD is a viable possibility and can be used to adjust the computational time of severe situations, but it needs to be improved.

Structure‐preserving invariant interpolation schemes for invertible second‐order tensors

AbstractTensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non‐symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data , followed by an individual interpolation of the two rotation tensors and and the positive definite diagonal eigenvalue tensor resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group , either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, and remain orthogonal tensors and remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non‐symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log‐Euclidean, Cholesky and Log‐Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first‐ and second‐order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor‐valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second‐order square tensors independent of the specific application.

Temetos Software Platform and Its Applications in Problems of Continuum Mechanics

Temetos Software Platform and Its Applications in Problems of Continuum Mechanics

Korn’s inequality in anisotropic Sobolev spaces

Abstract Korn’s inequality has been at the heart of much exciting research since its first appearance in the beginning of the 20th century. Many are the applications of this inequality to the analysis and construction of discretizations of a large variety of problems in continuum mechanics. In this paper, we prove that the classical Korn inequality holds true in anisotropic Sobolev spaces. We also prove that an extension of Korn’s inequality, involving non-linear continuous maps, is valid in such spaces. Finally, we point out that another classical inequality, namely Poincaré’s inequality, also holds in anisotropic Sobolev spaces.

Black-Box Solver for Numerical Simulations and Mathematical Modelling in Engineering Physics

This article presents a two-grid approach for developing a black-box iterative solver for a large class of real-life problems in continuum mechanics (heat and mass transfer, fluid dynamics, elasticity, electromagnetism, and others). The main requirements on this (non-)linear black-box solver are: (1) robustness (the lowest number of problem-dependent components), (2) efficiency (close-to-optimal algorithmic complexity), and (3) parallelism (a parallel robust algorithm should be faster than the fastest sequential one). The basic idea is to use the auxiliary structured grid for more computational work, where (non-)linear problems are simpler to solve and to parallelize, i.e., to combine the advantages of unstructured and structured grids: simplicity of generation in complex domain geometry and opportunity to solve (non-)linear (initial-)boundary value problems by using the Robust Multigrid Technique. Topics covered include the description of the two-grid algorithm and estimation of their robustness, convergence, algorithmic complexity, and parallelism. Further development of modern software for solving real-life problems justifies relevance of the research. The proposed two-grid algorithm can be used in black-box parallel software for the reduction in the execution time in solving (initial-)boundary value problems.

Review on Design and Analysis in Ultimate State of Bridge for Class 70R IRC Loading

Abstract: Due to its increased carrying capacity, pre-pressurized concrete bridge flooring have recently seen growth in the field of bridge engineering. IRC codal keywords were used for the analysis. One of the largest varieties of cast in situ concrete decks with a concrete slab attached to belts are T-beam bridge decks. The standard structural analysis method FEM (standard feature approach) in STAAD Pro rates the continuum mechanics problem. This study analyses a single-span t-beam bridge by varying the lengths by 25, 30, 35, and 40 metres while keeping the width constant. To ascertain the curve time and shear strength, models of bridges of the IRC AA category and the IRC loading system of section 70Rtracked were used. According to the investigation, there are no discernible differences between the Courbon method and the limited feature method as the span increases.

Review on Design and Analysis in Ultimate State of Bridge for Class 70R IRC Loading

Abstract: Due to its increased carrying capacity, pre-pressurized concrete bridge flooring have recently seen growth in the field of bridge engineering. IRC codal keywords were used for the analysis. One of the largest varieties of cast in situ concrete decks with a concrete slab attached to belts are T-beam bridge decks. The standard structural analysis method FEM (standard feature approach) in STAAD Pro rates the continuum mechanics problem. This study analyses a single-span t-beam bridge by varying the lengths by 25, 30, 35, and 40 metres while keeping the width constant. To ascertain the curve time and shear strength, models of bridges of the IRC AA category and the IRC loading system of section 70Rtracked were used. According to the investigation, there are no discernible differences between the Courbon method and the limited feature method as the span increases.

Smooth particle hydrodynamics: a meshless approach for structural mechanics

This article provides an overview of the smooth particle hydrodynamics (SPH) approach and its mathematical modeling. SPH is a numerical technique based on a mesh-free Lagrangian scheme for evaluating the continuum mechanics problems. This method is suitable in the case of continuum objects undergoing large deformation, as conventional finite element methods are unreliable due to mesh failure and convergence issues. It is a widely used approach in the field of astrophysics, fluid mechanics, structural mechanics, soil mechanics, automobiles, and so on. A numerical example is also considered in this research paper to demonstrate the applicability of the method. The simulation process was achieved using LS-Dyna explicit solver software, and plots related to cutting and thrust forces, von Mises stress, plastic strain, temperature distribution, and so on, were obtained. Also, the effect of Time-Scaling Factor (TSSFAC) on SPH simulations was observed in this research.

MODELING OF DIRECTIONAL SOLIDIFICATION/MELTING BY THE ENTHALPY-POROSITY METHOD

The research is focused on the development of mathematical models and software based on them to simulate complex processes of structural-phase transformations for new-generation materials, such as materials with phase transitions (PCM), biomedical materials, materials for additive manufacturing, and materials for the space industry. The mathematical description of the enthalpy-porosity model is performed in this work. The equations of viscous fluid hydrodynamics are used to describe fluid motion in time and space. The analysis of necessary restrictions and assumptions in the model related to consideration of laminar flows and Newtonian fluid model is performed. The computational problem is formulated in terms of the finite volume method and the computational domain and hydrodynamic equations are discretized. The OpenFOAM software, an open integrated platform for numerical simulation of continuum mechanics problems, was used for the computations. The computational algorithm OpenFOAM was developed to analyze the physical state of the system taking into account the initial and boundary conditions in the case of conductive and convective heat transfer. The simulations of gallium melting are performed and the model is verified for the conductive and convective cases. It is shown that in the conductive case the material melting occurs uniformly along the heat sources, while different velocities of convection flows have a significant influence on the formation of the melting boundary. The mathematical models developed in the study, as well as the analytical dependences and the computer simulations are applied to describe real experimental data on crystal growth in supersaturated solutions and supercooled melts.

Spatio-Trajectorial Optical Flow for Higher-Order Deformation Analysis in Solid Experimental Mechanics.

Material models are required to solve continuum mechanical problems. These models contain parameters that are usually determined by application-specific test setups. In general, the theoretically developed models and, thus, the parameters to be determined become increasingly complex, e.g., incorporating higher-order motion derivatives, such as the strain or strain rate. Therefore, the strain rate behaviour needs to be extracted from experimental data. Using image data, the most-common way in solid experimental mechanics to do so is digital image correlation. Alternatively, optical flow methods, which allow an adaption to the underlying motion estimation problem, can be applied. In order to robustly estimate the strain rate fields, an optical flow approach implementing higher-order spatial and trajectorial regularisation is proposed. Compared to using a purely spatial variational approach of higher order, the proposed approach is capable of calculating more accurate displacement and strain rate fields. The procedure is finally demonstrated on experimental data of a shear cutting experiment, which exhibited complex deformation patterns under difficult optical conditions.

THE EXACT SOLUTION OF THE WIENER–HOPF EQUATION ON THE SEGMENT FOR CONTACT PROBLEMS AND PROBLEMS OF THE THEORY OF CRACKS IN A LAYERED MEDIUM

This paper presents an approach that allows for the first time to construct an exact solution of the Wiener–Hopf integral equations on a finite segment for the case of meromorphic functions in Fourier transforms of the kernel. The Wiener–Hopf integral equation is traditionally considered set on a semi-infinite segment. However, in applications, there are often cases of their application specified on a finite segment. For these purposes, approximate methods of applying these integral equations have been developed. However, when considering the Wiener–Hopf integral equations generated by mixed problems of continuum mechanics and mathematical physics in a multilayer medium of finite thickness, it turned out that these integral equations are solved exactly both on semi-infinite and finite segments. The approach is based on a new modeling method in differential equations and in some types of integral equations. It allows the reduction of Wiener–Hopf integral equations to infinite systems of linear algebraic equations that are solved exactly. The obtained result opens up the possibility of constructing exact solutions to boundary value problems for deformable stamps and cracks of a new type in bounded bodies.

Analysis and mitigation of spatial integration errors for the material point method

SummaryThe material point method (MPM) is a robust numerical simulation approach for continuum mechanics problems involving large material deformations coupled to changing surface topographies. These types of problems are present in many different engineering contexts, from understanding the failure processes of earthen slopes to predicting the strengths and failure mechanisms of body armor to modeling the impact forces of waves in fluid tanks. By using a set of persistent material point tracers to follow the motion and deformation of the continuum material while solving the equations of motion on a static simulation grid, the MPM avoids several shortcomings of more traditional numerical approaches including blurring of material surfaces — as in Eulerian finite element or finite volume methods (FEMs or FVMs) — and mesh tangling — as in Lagrangian FEMs. Despite its robustness, MPM is known to develop significant numerical errors: namely, (i) the particle ringing instability and (ii) solution dependent discretization and integration errors. In this work, we present an analysis of local‐in‐time, spatial integration errors in the MPM and several techniques designed to mitigate these errors. Error mitigation approaches previously described in the literature are compared to a new method we propose for problems involving very large material deformations. The proposed method is shown to offer substantial improvement over standard MPM for simulations of fluid‐like materials without requiring significant augmentation of existing MPM frameworks.

A review on the progress and research directions of ocean engineering

This paper reviews research in ocean engineering over the last 50+ years with the aim to (I) understand the technological challenges and evolution in the field, (II) investigate whether ocean engineering studies meet present global demands, (III) explore new scientific/engineering tools that may suggest pragmatic solutions to problems, and (IV) identify research and management gaps, and the way forward. Six major research divisions are identified, namely (I) Ocean Hydrodynamics, (II) Risk Assessment and Safety, (III) Ocean Climate and Geophysics: Data and Models, (IV) Control and Automation in the Ocean, (V) Structural Engineering and Manufacturing for the Ocean, and (VI) Ocean Renewable Energy. As much as practically possible research sub-divisions of the field are also identified. It is highlighted that research topics dealing with ocean renewable energy, control and path tracking of ships, as well as computational modelling of wave-induced motions are growing. Updating and forecasting energy resources, developing computational methods for wave generation, and introducing novel methods for the optimised control of energy converters are highlighted as the potential research opportunities. Ongoing studies follow the global needs for environmentally friendly renewable energies, though engineering-based studies often tend to overlook the longer-term potential influence of climate change. Development and exploitation of computational engineering methods with focus on continuum mechanics problems remain relevant. Notwithstanding this, machine learning methods are attracting the attention of researchers. Analysis of COVID-19 transmission onboard is rarely conducted, and 3D printing-based studies still need more attention from researchers.