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1
Slaughter, WS, and J. Petrolito. "Linearized Theory of Elasticity." Applied Mechanics Reviews 55, no. 5 (September 1, 2002): B90—B91. http://dx.doi.org/10.1115/1.1497478.
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Javili, Ali, Niels Saabye Ottosen, Matti Ristinmaa, and Jörn Mosler. "Aspects of interface elasticity theory." Mathematics and Mechanics of Solids 23, no. 7 (April 10, 2017): 1004–24. http://dx.doi.org/10.1177/1081286517699041.
Повний текст джерелаАнотація:
Interfaces significantly influence the overall material response especially when the area-to-volume ratio is large, for instance in nanocrystalline solids. A well-established and frequently applied framework suitable for modeling interfaces dates back to the pioneering work by Gurtin and Murdoch on surface elasticity theory and its generalization to interface elasticity theory. In this contribution, interface elasticity theory is revisited and different aspects of this theory are carefully examined. Two alternative formulations based on stress vectors and stress tensors are given to unify various existing approaches in this context. Focus is on the hyper-elastic mechanical behavior of such interfaces. Interface elasticity theory at finite deformation is critically reanalyzed and several subtle conclusions are highlighted. Finally, a consistent linearized interface elasticity theory is established. We propose an energetically consistent interface linear elasticity theory together with its appropriate stress measures.
3
Freddi, Francesco, and Gianni Royer-Carfagni. "From Non-Linear Elasticity to Linearized Theory: Examples Defying Intuition." Journal of Elasticity 96, no. 1 (February 20, 2009): 1–26. http://dx.doi.org/10.1007/s10659-009-9191-7.
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4
Sini, Mourad. "Absence of Positive Eigenvalues for the Linearized Elasticity System." Integral Equations and Operator Theory 49, no. 2 (June 1, 2004): 255–77. http://dx.doi.org/10.1007/s00020-002-1265-x.
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Ciarlet, Philippe G., and Cristinel Mardare. "The intrinsic theory of linearly elastic plates." Mathematics and Mechanics of Solids 24, no. 4 (May 28, 2018): 1182–203. http://dx.doi.org/10.1177/1081286518776047.
Повний текст джерелаАнотація:
In an intrinsic approach to a problem in elasticity, the only unknown is a tensor field representing an appropriate ‘measure of strain’, instead of the displacement vector field in the classical approach. The objective of this paper is to study the displacement traction problem in the special case where the elastic body is a linearly elastic plate of constant thickness, clamped over a portion of its lateral face. In this respect, we first explicitly compute the intrinsic three-dimensional boundary condition of place in terms of the Cartesian components of the linearized strain tensor field, thus avoiding the recourse to covariant components in curvilinear coordinates and providing an interesting example of actual computation of an intrinsic boundary condition of place in three-dimensional elasticity. Second, we perform a rigorous asymptotic analysis of the three-dimensional equations as the thickness of the plate, considered as a parameter, approaches zero. As a result, we identify the intrinsic two-dimensional equations of a linearly elastic plate modelled by the Kirchhoff–Love theory, with the linearized change of metric and change of curvature tensor fields of the middle surface of the plate as the new unknowns, instead of the displacement field of the middle surface in the classical approach.
6
CIARLET, PHILIPPE G., GIUSEPPE GEYMONAT, and FRANÇOISE KRASUCKI. "A NEW DUALITY APPROACH TO ELASTICITY." Mathematical Models and Methods in Applied Sciences 22, no. 01 (January 2012): 1150003. http://dx.doi.org/10.1142/s0218202512005861.
Повний текст джерелаАнотація:
The displacement-traction problem of three-dimensional linearized elasticity can be posed as three different minimization problems, depending on whether the displacement vector field, or the stress tensor field, or the strain tensor field, is the unknown. The objective of this paper is to put these three different formulations of the same problem in a new perspective, by means of Legendre–Fenchel duality theory. More specifically, we show that both the displacement and strain formulations can be viewed as Legendre–Fenchel dual problems to the stress formulation. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new duality approach to elasticity.
7
CIARLET, PHILIPPE G., LILIANA GRATIE, CRISTINEL MARDARE, and MING SHEN. "SAINT VENANT COMPATIBILITY EQUATIONS ON A SURFACE APPLICATION TO INTRINSIC SHELL THEORY." Mathematical Models and Methods in Applied Sciences 18, no. 02 (February 2008): 165–94. http://dx.doi.org/10.1142/s0218202508002644.
Повний текст джерелаАнотація:
We first establish that the linearized change of metric and change of curvature tensors, with components in L2 and H-1 respectively, associated with a displacement field, with components in H1, of a surface S immersed in ℝ3 must satisfy in the distributional sense compatibility conditions that may be viewed as the linear version of the Gauss and Codazzi-Mainardi equations. These compatibility conditions, which are analogous to the familiar Saint Venant equations in three-dimensional elasticity, constitute the Saint Venant equations on the surface S. We next show that these compatibility conditions are also sufficient, i.e. that they in fact characterize the linearized change of metric and the linearized change of curvature tensors in the following sense: If two symmetric matrix fields of order two defined over a simply-connected surface S ⊂ ℝ3 satisfy the above compatibility conditions, then they are the linearized change of metric and linearized change of curvature tensors associated with a displacement field of the surface S, a field whose existence is thus established. The proof provides an explicit algorithm for recovering such a displacement field from the linearized change of metric and linearized change of curvature tensors. This algorithm may be viewed as the linear counterpart of the reconstruction of a surface from its first and second fundamental forms. Finally, we show how these results can be applied to the "intrinsic theory" of linearly elastic shells, where the linearized change of metric and change of curvature tensors are the new unknowns. These new unknowns solve a quadratic minimization problem over a space of tensor fields whose components, which are only in L2, satisfy the Saint Venant compatibility conditions on a surface in the sense of distributions.
8
Jesenko, Martin, and Bernd Schmidt. "Geometric linearization of theories for incompressible elastic materials and applications." Mathematical Models and Methods in Applied Sciences 31, no. 04 (March 22, 2021): 829–60. http://dx.doi.org/10.1142/s0218202521500202.
Повний текст джерелаАнотація:
We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.
9
Froiio, Francesco, and Antonis Zervos. "Second-grade elasticity revisited." Mathematics and Mechanics of Solids 24, no. 3 (April 24, 2018): 748–77. http://dx.doi.org/10.1177/1081286518754616.
Повний текст джерелаАнотація:
We present a compact, linearized theory for the quasi-static deformation of elastic materials whose stored energy depends on the first two gradients of the displacement (second-grade elastic materials). The theory targets two main issues: (1) the mechanical interpretation of the boundary conditions and (2) the analytical form and physical interpretation of the relevant stress fields in the sense of Cauchy. Since the pioneering works of Toupin and Mindlin et al. in the 1960’s, a major difficulty has been the lack of a convincing mechanical interpretation of the boundary conditions, causing second-grade theories to be viewed as ‘perturbations’ of constitutive laws for simple (first-grade) materials. The first main contribution of this work is the provision of such an interpretation based on the concept of ortho-fiber. This approach enables us to circumvent some difficulties of a well-known ‘reduction’ of second-grade materials to continua with microstructure (in the sense of Mindlin) with internal constraints. A second main contribution is the deduction of the form of the linear and angular-momentum balance laws, and related stress fields in the sense of Cauchy, as they should appear in a consistent Newtonian formulation. The viewpoint expressed in this work is substantially different from the one in a well known and influential paper by Mindlin and Eshel in 1968, while affinities can be found with recent studies by dell’Isola et al. The merits of the new formulation and the associated numerical approach are demonstrated by stating and solving three example boundary value problems in isotropic elasticity. A general finite element discretization of the governing equations is presented, using C1-continuous interpolation, while the numerical results show excellent convergence even for relatively coarse meshes.
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Giovine, Pasquale. "A Multiscale Approximation Method to Describe Diatomic Crystalline Systems: Constitutive Equations." Journal of Multiscale Modelling 09, no. 03 (September 2018): 1840001. http://dx.doi.org/10.1142/s1756973718400012.
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We model the mechanical behavior of diatomic crystals in the light of mixture theory. Use is made of an approximation method similar to one proposed by Signorini within the theory of elasticity, by supposing that the relative motion between phases is infinitesimal. The constitutive equations for a mixture of elastic bodies in the absence of diffusion are adapted to the partially linearized case considered here, and the representation theorems for constitutive fields are applied to obtain the final expression of dynamical equations in the form which appears in theories of continua with vectorial microstructure. Comparisons are made with results of lattice theories.
11
Goloveshkina, Evgeniya V. "Influence of Distributed Dislocations on Stability of Hollow Nonlinearly Elastic Sphere." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 4 (208) (December 23, 2020): 17–21. http://dx.doi.org/10.18522/1026-2237-2020-4-17-21.
Повний текст джерелаАнотація:
The phenomenon of stability loss of a hollow elastic sphere containing distributed dislocations and loaded with external hydrostatic pressure is studied. The study was carried out in the framework of the nonlinear elasticity theory and the continuum theory of continuously distributed dislocations. An exact statement and solution of the stability problem for a three-dimensional elastic body with distributed dislocations are given. The static problem of nonlinear elasticity theory for a body with distributed dislocations is reduced to a system of equations consisting of equilibrium equations, incompatibility equations with a given dislocation density tensor, and constitutive equations of the material. The unperturbed state is caused by external pressure and a spherically symmet-ric distribution of dislocations. For distributed edge dislocations in the framework of a harmonic (semi-linear) mate-rial model, the unperturbed state is defined as an exact spherically symmetric solution to a nonlinear boundary value problem. This solution is valid for any function that characterizes the density of edge dislocations. The perturbed equilibrium state is described by a boundary value problem linearized in the neighborhood of the equilibrium. The analysis of the axisymmetric buckling of the sphere was performed using the bifurcation method. It consists in determining the equilibrium positions of an elastic body, which differ little from the unperturbed state. By solving the linearized problem, the value of the external pressure at which the sphere first loses stability is found. The effect of dislocations on the buckling of thin and thick spherical shells is analyzed.
12
Vala, Jiri. "On a Computational Smeared Damage Approach to the Analysis of Strength of Quasi-Brittle Materials." WSEAS TRANSACTIONS ON APPLIED AND THEORETICAL MECHANICS 16 (December 23, 2021): 283–92. http://dx.doi.org/10.37394/232011.2021.16.31.
Повний текст джерелаАнотація:
Computational analysis of strength of quasi-brittle materials, crucial for the durability of building structures and industrial components, needs typically a smeared damage approach, referring to the Eringen theory of nonlocal elasticity. Unfortunately its ad hoc constitutive relations cannot avoid potential divergence of sequences of approximate solutions, exploiting some extended finite element techniques, as well as questionable or missing existence results for corresponding boundary value problems. Introducing a simple static partially linearized model problem of such type, this article demonstrates some relevant remedies and their limitations, with numerous references to desirable generalizations
13
Azimloo, Hadi, Ghader Rezazadeh, and Rasoul Shabani. "Bifurcation Analysis of an Electro-Statically Actuated Nano-beam Based on the Nonlocal Theory considering Centrifugal Forces." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 3-4 (May 26, 2020): 303–18. http://dx.doi.org/10.1515/ijnsns-2017-0230.
Повний текст джерелаАнотація:
AbstractA nonlocal elasticity theory is a popular growing technique for mechanical analysis of the micro- and nanoscale structures which captures the small-size effects. In this paper, a comprehensive study was carried out to investigate the influence of the nonlocal parameter on the bifurcation behavior of a capacitive clamped-clamped nano-beam in the presence of the electrostatic and centrifugal forces. By using Eringen’s nonlocal elasticity theory, the nonlocal equation of the dynamic motion for a nano-beam has been derived using Euler–Bernoulli beam assumptions. The governing static equation of motion has been linearized using step by step linearization method; then, a Galerkin based reduced order model have been used to solve the linearized equation. In order to study the bifurcation behavior of the nano-beam, the static non-linear equation is changed to a one degree of freedom model using a one term Galerkin weighted residual method. So, by using a direct method, the equilibrium points of the system, including stable center points, unstable saddle points and singular points have been obtained. The stability of the fixed points has been investigated drawing motion trajectories in phase portraits and basins of attraction set and repulsion have been illustrated. The obtained results have been verified using the results of the prior studies for some cases and a good agreement has been observed. Moreover, the effects of the different values of the nonlocal parameter, angular velocity and van der Waals force on the fixed points have been studied using the phase portraits of the system for different initial conditions. Also, the influence of the nonlocal beam theory and centrifugal forces on the dynamic pull-in behavior have been investigated using time histories and phase portraits for different values of the nonlocal parameter.
14
Wu, Mengjun, Quan Yuan, Honglin Li, Bin Wu, Lin Fang, and Mengyang Huang. "Effect of Curvature-Dependent Surface Elasticity on the Flexural Properties of Nanowire." Advances in Civil Engineering 2021 (July 26, 2021): 1–5. http://dx.doi.org/10.1155/2021/6726685.
Повний текст джерелаАнотація:
Surface elasticity and residual stress strongly influence the flexural properties of nanowire due to the excessively large ratio of surface area to volume. In this work, we adopt linearized surface elasticity theory, which was proposed by Chhapadia et al., to capture the influence of surface curvature on the flexural rigidity of nanowire with rectangular cross section. Additionally, we have tried to study the bending deformation of circular nanowire. All stresses and strains are measured relative to the relaxed state in which the difference in surface residual stress between the upper and lower faces of rectangular nanowire with no external load induces additional bending. The bending curvature of nanowire in the reference and relaxed states is obtained. We find that flexural rigidity is composed of three parts. The first term is defined by the precept of continuum mechanics, and the last two terms are defined by surface elasticity. The normalized curvature increases with the decrease in height, thereby stiffening the nanowire. We also find that not only sizes but also surface curvature induced by surface residual stress influence the bending rigidity of nanowire.
15
Daşdemir, Ahmet. "A Mathematical Model for Forced Vibration of Pre-Stressed Piezoelectric Plate-Strip Resting On Rigid Foundation." MATEMATIKA 34, no. 2 (December 2, 2018): 419–31. http://dx.doi.org/10.11113/matematika.v34.n2.988.
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A mathematical model to investigate the dynamic response of a piezoelectric plate-strip with initial stress under the action of a time-harmonic force resting on a rigid foundation is presented within the scope of the three-dimensional linearized theory of electro-elasticity waves in initially stressed bodies (TLTEEWISB). The governing system of equations of motion is solved by employing the Finite Element Method (FEM). The numerical results illustrating the dependencies of different problem parameters are investigated. In particular, the influence of a change in the value of the initial stress parameter on the dynamic response of the plate-strip is discussed.
16
Shuvalov, Gleb M., and Sergey A. Kostyrko. "Stability analysis of a nanopatterned bimaterial interface." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 17, no. 1 (2021): 97–104. http://dx.doi.org/10.21638/11701/spbu10.2021.109.
Повний текст джерелаАнотація:
In the article it is shown that the nanopatterned interface of bimaterial is unstable due to the diffusion atom flux along the interface. The main goal of the research is to analyze the conditions of interface stability. The authors developed a model coupling thermodynamics and solid mechanics frameworks. In accordance with the Gurtin—Murdoch theory of surface/interface elasticity, the interphase between two materials is considered as a negligibly thin layer with the elastic properties differing from those of the bulk materials. The growth rate of interface roughness depends on the variation of the chemical potential at the curved interface, which is a function of interface and bulk stresses. The stress distribution along the interface is found from the solution of plane elasticity problem taking into account plane strain conditions. Following this, the linearized evolution equation is derived, which describes the amplitude change of interface perturbation with time.
17
HODA, NAZISH, MIHAILO R. JOVANOVIĆ, and SATISH KUMAR. "Energy amplification in channel flows of viscoelastic fluids." Journal of Fluid Mechanics 601 (April 25, 2008): 407–24. http://dx.doi.org/10.1017/s0022112008000633.
Повний текст джерелаАнотація:
Energy amplification in channel flows of Oldroyd-B fluids is studied from an input–output point of view by analysing the ensemble-average energy density associated with the velocity field of the linearized governing equations. The inputs consist of spatially distributed and temporally varying body forces that are harmonic in the streamwise and spanwise directions and stochastic in the wall-normal direction and in time. Such inputs enable the use of powerful tools from linear systems theory that have recently been applied to analyse Newtonian fluid flows. It is found that the energy density increases with a decrease in viscosity ratio (ratio of solvent viscosity to total viscosity) and an increase in Reynolds number and elasticity number. In most of the cases, streamwise-constant perturbations are most amplified and the location of maximum energy density shifts to higher spanwise wavenumbers with an increase in Reynolds number and elasticity number and a decrease in viscosity ratio. For similar parameter values, the maximum in the energy density occurs at a higher spanwise wavenumber for Poiseuille flow, whereas the maximum energy density achieves larger maxima for Couette flow. At low Reynolds numbers, the energy density decreases monotonically when the elasticity number is sufficiently small, but shows a maximum when the elasticity number becomes sufficiently large, suggesting that elasticity can amplify disturbances even when inertial effects are weak.
18
Korenskii, S. A. "The formulation of linearized boundary integral equations of the anisotropic theory of elasticity and their application in geometrical inverse problems." Journal of Applied Mathematics and Mechanics 62, no. 3 (January 1998): 435–42. http://dx.doi.org/10.1016/s0021-8928(98)00055-0.
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Babich, S. Yu, and N. O. Yaretska. "Contact interaction of prestressed annular punch and half-space." Reports of the National Academy of Sciences of Ukraine, no. 11 (2020): 24–30. http://dx.doi.org/10.15407/dopovidi2020.11.024.
Повний текст джерелаАнотація:
The article is devoted to the task of contact interaction of the pressure of a pre-stressed cylindrical annular punch on the half-space with initial (residual) stresses without friction. It is solved for the case of unequal roots of the characteristic equation. In general, the research was carried out for the theory of great initial (ultimate) deformations and two variants of the theory of small initial ones within the framework of linearized theory of elasticity with the elastic potential having any structure. It is assumed that the initial states of the elastic annular stamp and the elastic half-space remain homogeneous and equal. The study is carried out in the coordinates of the initial deformed state, which are interrelated with Lagrange coordinates (natural state). In addition, the influence of the annular stamp causes small perturbations of the basic elastic deformed state. It is assumed that the elastic annular stamp and the elastic half-space are made of different isotropic, transversal-isotropic or composite materials.
20
Hechmer, J. L., and G. L. Hollinger. "The ASME Code and 3D Stress Evaluation." Journal of Pressure Vessel Technology 113, no. 4 (November 1, 1991): 481–87. http://dx.doi.org/10.1115/1.2928784.
Повний текст джерелаАнотація:
The ASME Code [1] identifies the modes of failure that must be addressed to ensure acceptable pressure vessel designs. The failure modes addressed in this paper are precluded by limits on the primary and primary plus secondary stress. Both involve the transition from elasticity to plasticity. Their evaluation requires the computation of membrane and bending stresses (the linearized stresses). The original techniques for evaluating the limits were based on beam and shell theory. Since beam and shell theory were the basis of the then-current tools, the transition from analysis results to failure assessment was straightforward. With the advent of finite elements (FE), the transition from the stress distribution to the failure modes requires a different path. For three-dimensional finite element (3D FE), the path is obscure. Since the development of FE, the ASME Code has made no additions to clarify the correlations between FE stress distributions and the failure modes. The authors believe that the Code should provide guidance in this area.
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Dikhtyaruk, N. N., and E. A. Poplavskaya. "A flat contact problem the interaction two prestressed stripes with an infinite stringer." Problems of Tribology 24, no. 94/4 (December 19, 2019): 40–48. http://dx.doi.org/10.31891/10.31891/2079-1372-2019-94-4-40-48.
Повний текст джерелаАнотація:
The article is devoted to the research of problems of contact interaction of infinite elastic stringer with two identical clamped along one edge of pre-stressed strips. In general, the research was carried out for the theory of great initial and different variants of the theory of small initial deformations within the framework of linearized theory of elasticity with the elastic potential having arbitrary structure. The integral integer-differential equations are obtained using the integral Fourier transform. Their solution is represented in the form of quasiregular infinite systems of algebraic equations. In the article alsaw was investigated the influence of the initial (residual) stresses in strips on the law of distribution of contact stresses along the line of contact with an infinite stringer. The system is solved in a closed forms using transformation of Fourier. Expressions of stresses are represented by Fourier integrals with a simple enough structure. Influence of initial stress on the distribution of contact stresses is study and discovered the mechanical effects under the influence of concentrated loads
22
Bagno, O. M. "On the influence of finite initial deformations on the surface instability of the incompressible elastic layer interacting with the half-space of an ideal fluid." Reports of the National Academy of Sciences of Ukraine, no. 1 (February 2021): 24–32. http://dx.doi.org/10.15407/dopovidi2021.01.024.
Повний текст джерелаАнотація:
The problem of the propagation of quasi-Lamb waves in a pre-deformed incompressible elastic layer that interacts with the half-space of an ideal compressible fluid is considered. The study is conducted on the basis of the three-dimensional linearized equations of elasticity theory of finite deformations for the incompressible elastic layer and on the basis of the three-dimensional linearized Euler equations for the half-space of an ideal compressible fluid. The problem is formulated, and the approach based on the utilization of representations of the general solutions of the linearized equations for an elastic solid and a fluid is developed. Applying the Fourier method, we arrive at two eigenvalue problems for the equations of motion of the elastic body and the fluid. Solving them, we find the eigenfunctions. Substituting the general solutions into the boundary conditions, we obtain a homogeneous system of linear algebraic equations for the arbitrary constants. From the condition for the existence of a nontrivial solution, we derive the dispersion equation. A dispersion equation, which describes the propagation of normal waves in the hydroelastic system, is obtained. The dispersion curves for quasi-Lamb waves over a wide range of frequencies are constructed. The effect of the finite initial deformations in an elastic layer, the thickness of the elastic layer, and the half-space of an ideal compressible fluid on the phase velocities and dispersion of quasi-Lamb modes are analyzed. It follows from the graphical material presented above that, in the case of compression with 0.54, i.e., with a 46 percent’s reduction in the length of the highly elastic incompressible body, the phase velocities of the surface waves (Stoneley waves and Rayleigh waves) vanish. This indicates that surface instability develops at 0.54 for a highly elastic incompressible non-Hookean body initially in a plane stress-strain state. We should point out that these figures agree with results obtained earlier in the theory of stability and correspond to the critical value of the contraction parameter. In the case of highly elastic incompressible bodies, the linearized wave theory makes it possible to study not only general and several specific wave processes, but also the conditions under which the surface instability begins in elastic bodies and hydroelastic systems. It also follows from the graphs that the ideal fluid slightly affects the surface instability of hydroelastic systems. The numerical results are presented in the form of graphs, and their analysis is given.
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Habrusiev, Hryhorii, and Iryna Habrusieva. "Contact interaction of a predeformed plate which lies without friction on rigid base with a parabolic indenter." Scientific journal of the Ternopil national technical university 102, no. 2 (2021): 87–95. http://dx.doi.org/10.33108/visnyk_tntu2021.02.087.
Повний текст джерелаАнотація:
Within the framework of linearized formulation of a problem of the elasticity theory, the stress-strain state of a predeformed plate, which is modeled by a prestressed layer, is analyzed in the case of its smooth contact interaction with a rigid axisymmetric parabolic indenter. The dual integral equations of the problem are solved by representing the quested-for functions in the form of a partial series sum by the Bessel functions with unknown coefficients. Finite systems of linear algebraic equations are obtained for determination of these coefficients. The influence of the initial strains on the magnitude and features of the contact stresses and vertical displacements on the surface of the plate is analyzed for the case of compressible and incompressible solids. In order to illustrate the results, the cases of the Bartenev – Khazanovich and the harmonic-type potentials are addressed.
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Nguyen, Thai Binh, J. N. Reddy, Jaroon Rungamornrat, Jintara Lawongkerd, Teerapong Senjuntichai, and Van Hai Luong. "Nonlinear Analysis for Bending, Buckling and Post-buckling of Nano-Beams with Nonlocal and Surface Energy Effects." International Journal of Structural Stability and Dynamics 19, no. 11 (October 23, 2019): 1950130. http://dx.doi.org/10.1142/s021945541950130x.
Повний текст джерелаАнотація:
The modeling and analysis for mechanical response of nano-scale beams undergoing large displacements and rotations are presented. The beam element is modeled as a composite consisting of the bulk material and the surface material layer. Both Eringen nonlocal elasticity theory and Gurtin–Murdoch surface elasticity theory are adopted to formulate the moment–curvature relationship of the beam. In the formulation, the pre-existing residual stress within the bulk material, induced by the residual surface tension in the material layer, is also taken into account. The resulting moment-curvature relationship is then utilized together with Euler–Bernoulli beam theory and the elliptic integral technique to establish a set of exact algebraic equations governing the displacements and rotations at the ends of the beam. The linearized version of those equations is also established and used in the derivation of a closed-form solution of the buckling load of nano-beams under various end conditions. A discretization-free solution procedure based mainly upon Newton iterative scheme and a selected numerical quadrature is developed to solve a system of fully coupled nonlinear equations. It is demonstrated that the proposed technique yields highly accurate results comparable to the benchmark analytical solutions. In addition, the nonlocal and surface energy effects play a significant role on the predicted buckling load, post-buckling and bending responses of the nano-beam. In particular, the presence of those effects remarkably alters the overall stiffness of the beam and predicted solutions exhibit strong size-dependence when the characteristic length of the beam is comparable to the intrinsic length scale of the material surface.
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FEDOROV, A. V., N. D. MALMUTH, and V. G. SOUDAKOV. "Supersonic scattering of a wing-induced incident shock by a slender body of revolution." Journal of Fluid Mechanics 585 (August 7, 2007): 305–22. http://dx.doi.org/10.1017/s0022112007006714.
Повний текст джерелаАнотація:
The lift force acting on a slender body of revolution that separates from a thin wing in supersonic flow is analysed using Prandtl–Glauert linearized theory, scattering theory and asymptotic methods. It is shown that this lift is associated with multi-scattering of the wing-induced shock wave by the body surface. The local and global lift coefficients are obtained in simple analytical forms. It is shown that the total lift is mainly induced by the first scattering. Contributions from second, third and higher scatterings are zero in the leading-order approximation. This greatly simplifies calculations of the lift force. The theoretical solution for the flow field is compared with numerical solutions of three-dimensional Euler equations and experimental data at free-stream Mach number 2. There is agreement between the theory and the computations for a wide range of shock-wave strength, demonstrating high elasticity of the leading-order asymptotic approximation. Theoretical and experimental distributions of the cross-sectional normal force coefficient agree satisfactorily, showing robustness of the analytical solution. This solution can be applied to the moderate supersonic (Mach numbers from 1.2 to 3) multi-body interaction problem for crosschecking with other computational or engineering methods.
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Dikhtyaruk, N., and E. Poplavskaya. "Plane contact problem on the interaction of a pre-stressed strip with an infinite inhomogeneous stringer." Problems of tribology 97, no. 3 (September 28, 2020): 55–63. http://dx.doi.org/10.31891/2079-1372-2020-97-3-55-63.
Повний текст джерелаАнотація:
The article is devoted to the study of problems of contact interaction of an infinite elastic inhomogeneous stringer with a prestressed strip clamped along one edge. As a result of the research, we have obtained a resolving system of recurrent systems of integro-differential equations. In general, the studies were carried out for the theory of large initial and various versions of the theory of small initial deformations within the framework of the linearized theory of elasticity with an elastic potential of an arbitrary structure. Integral integer differential equations are obtained using the integral Fourier transform. Their solution is presented in the form of quasiregular infinite systems of algebraic equations. The article also investigates the influence of the initial (residual) stresses in strips on the distribution law of contact stresses along the line of contact with an infinite stringer. The system is solved in a closed form using the Fourier transform. The stress expressions are represented by Fourier integrals with a fairly simple structure. The influence of the initial stress on the distribution of contact stresses has been studied and mechanical effects have been found under the action of concentrated loads.
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Akbarov, Surkay D., and Masoud Negin. "Near-surface waves in a system consisting of a covering layer and a half-space with imperfect interface under two-axial initial stresses." Journal of Vibration and Control 23, no. 1 (August 9, 2016): 55–68. http://dx.doi.org/10.1177/1077546315575466.
Повний текст джерелаАнотація:
The influence of shear-spring + normal-spring type imperfect interface conditions on the dispersion of the generalized Rayleigh waves in a system consisting of a covering layer and a half-space with two-axial homogeneous initial stresses is investigated. The three-dimensional linearized theory of elastic waves in initially stressed bodies is employed and the plane-strain state is considered. The elasticity relations of the materials of the constituents are described through the Murnaghan potential and the influence of the third order elastic constants which enter the expression of this potential is taken into consideration. The corresponding dispersion equation is derived and an algorithm is developed for numerical solution to this equation. Numerical results on the action of the parameters, which enter the formulation of the imperfect contact conditions, on the wave dispersion curves are presented and discussed. The results of these investigations can be successfully used for estimation of the degree of the bonded defects between the covering layer and the half-space.
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Daşdemir, Ahmet. "Modeling the effect of the inclination angle on the dynamic response of a biaxially pre-stressed plate." Transactions of the Canadian Society for Mechanical Engineering 46, no. 1 (March 1, 2022): 153–64. http://dx.doi.org/10.1139/tcsme-2020-0111.
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In this study, I report on an investigation of the forced vibrations procured by an arbitrary angled time-harmonic loading from a plate based on a rigid foundation. The study was formulated according to the three-dimensional linearized theory of elasticity for solids under initial stress (TLTESIS). It was assumed throughout the investigation that there is a rigid clamped state between the system and the rigid ground; further, it was assumed that the plate was exposed to biaxially static initial stresses. Given this, a mathematical model was developed, and then solved using a three-dimensional finite element method (3D-FEM). Presented are numerical investigations that illustrate the influence of changes in the inclination of the force, as well as other important factors such as dimensionless frequency parameters, on the dynamic behavior of the system. In particular, the results indicate that the effect the initial stresses have on the dynamic stress distribution character increases with the aspect ratio but decreases with the thickness ratio.
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Emel’yanov, V. I. "The 3D Kuramoto-Sivashinsky Equation for Nonequilibrium Defects Interacting through Self-Consisting Strain and Nanostructuring of Solids." ISRN Nanomaterials 2013 (October 21, 2013): 1–6. http://dx.doi.org/10.1155/2013/981616.
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It is shown that the bulk defect-deformational (DD) nanostructuring of isotropic solids can be described by a closed three-dimensional (3D) nonlinear DD equation of the Kuramoto-Sivashinsry (KS) type for the nonequilibrium defect concentration, derived here in the framework of the nonlocal elasticity theory (NET). The solution to the linearized DDKS equation describes the threshold appearance of the periodic self-consistent strain modulation accompanied by the simultaneous formation of defect piles at extremes of the strain. The period and growth rate of DD nanostructure are determined. Based on the obtained results, a novel mechanism of nanostructuring of solids under the severe plastic deformation (SPD), stressing the role of defects generation and selforganization, described by the DDKS, is proposed. Theoretical dependencies of nanograin size on temperature and shear strain reproduce well corresponding critical dependencies obtained in experiments on nanostructuring of metals under the SPD, including the effect of saturation of nanofragmentation. The scaling parameter of the NET is estimated and shown to determine the limiting small grain size.
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Khaled, A. R. A., and K. Vafai. "Cooling Enhancements in Thin Films Supported by Flexible Complex Seals in the Presence of Ultrafine Suspensions." Journal of Heat Transfer 125, no. 5 (September 23, 2003): 916–25. http://dx.doi.org/10.1115/1.1597612.
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Flow and heat transfer inside thin films supported by flexible soft seals having voids of a stagnant fluid possessing a large coefficient of volumetric thermal expansion βT are studied in the presence of suspended ultrafine particles. The study is conducted under periodically varying thermal load conditions. The governing continuity, momentum and energy equations are non-dimensionalized and reduced to simpler forms. The deformation of the seal is related to the internal pressure and lower plate’s temperature based on the theory of linear elasticity and a linearized model for thermal expansion. It is found that enhancements in the cooling are achieved by an increase in the volumetric thermal expansion coefficient, thermal load, thermal dispersion effects, softness of the supporting seals and the thermal capacitance of the coolant fluid. Further, thermal dispersion effects are found to increase the stability of the thin film. The noise in the thermal load is found to affect the amplitude of the thin film thickness, Nusselt number and the lower plate temperature however it has a negligible effect on their mean values.
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BABUŠKA, IVO, EUGENE G. PODNOS, and GREGORY J. RODIN. "NEW FICTITIOUS DOMAIN METHODS: FORMULATION AND ANALYSIS." Mathematical Models and Methods in Applied Sciences 15, no. 10 (October 2005): 1575–94. http://dx.doi.org/10.1142/s0218202505000893.
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Two closely-related fictitious domain methods for solving problems involving multiple interfaces are introduced. Like other fictitious domain methods, the proposed methods simplify the task of finite element mesh generation and provide access to solvers that can take advantage of uniform structured grids. The proposed methods do not involve the Lagrange multipliers, which makes them quite different from existing fictitious domain methods. This difference leads to an advantageous form of the inf–sup condition, and allows one to avoid time-consuming integration over curvilinear surfaces. In principle, the proposed methods have the same rate of convergence as existing fictitious domain methods. Nevertheless it is shown that, at the cost of introducing additional unknowns, one can improve the quality of the solution near the interfaces. The methods are presented using a two-dimensional model problem formulated in the context of linearized theory of elasticity. The model problem is sufficient for presenting method details and mathematical foundations. Although the model problem is formulated in two dimensions and involves only one interface, there are no apparent conceptual difficulties to extending the methods to three dimensions and multiple interfaces. Further, it is possible to extend the methods to nonlinear problems involving multiple interfaces.
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Ozisik, Muslum. "Dispersion of generalized Rayleigh waves in the half-plane covered with pre-stretched two layers under complete contact." Thermal Science 25, Spec. issue 2 (2021): 247–53. http://dx.doi.org/10.2298/tsci21s2247o.
Повний текст джерелаАнотація:
In this paper the dispersion of the generalized Rayleigh wave propagation in the non-prestressed half-plane covered with pre-stretched two layers under complete contact conditions is investigated by 3-D linearized theory of elasticity. The layers and the half-plane are assumed that elastic, homogeneous, isotropic, and the complete contact conditions are existed. The inter phase zone between the upper layer and half-plane is modeled by this second layer. The purpose of the investigation is the determination on the effect of the existence of the second layer to the considered generalized Rayleigh wave propagation velocity. For this purpose, firstly the same materials were selected for both layers and the results obtained in previous studies for a single layer in the literature were verified, the accuracy of the modeling was shown, and then the effect of the second layer on the considered problem was shown by selecting the different materials and applying different initial pre-stresses. Consequently, the present study can be considered as the investigation of the existence of the inter phase zone which is characteristic one for the composite materials to the dispersion of the generalized Rayleigh wave propagation. Numerical results obtained and discussed.
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Jafarsadeghi-Pournaki, Ilgar, Ghader Rezazadeh, and Rasool Shabani. "Nonlinear Instability Modeling of a Nonlocal Strain Gradient Functionally Graded Capacitive Nano-Bridge in Thermal Environment." International Journal of Applied Mechanics 10, no. 08 (September 2018): 1850083. http://dx.doi.org/10.1142/s1758825118500837.
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This paper develops a theoretical model directed towards investigation of the static pull-in instability of a functionally graded (FG) electrostatically actuated nano-bridge via nonlocal strain gradient theory (NLSGT) of elasticity and Euler–Bernoulli beam theory in thermal environment. The nano-beam is under the influence of electrostatic and van der Waals (vdW) forces. In addition to the nonlinear nature of the electrostatic force, the other type of nonlinearity namely geometric nonlinearity resulting from the mid-plane stretching is considered. Material properties of FG nano-beam are assumed to vary gradually along the thickness direction according to simple power-law form. With the purpose of eliminating the coupling between the stretching and bending due to the asymmetrical material variation along the thickness, a new surface reference is introduced. The nonlinear integro-differential governing equation is derived utilizing minimum total potential energy principle, linearized by means of the step-by-step linearization method (SSLM) and solved by Galerkin-based weighted residual method. The numerical investigations are performed while the emphasis is placed on studying the effect of various parameters including: nonlocal parameter, material characteristic length scale, material gradient index, thermal effect and intermolecular force on the static pull-in instability of FG nano-beam. To establish the validity of the present formulation, a comparison is conducted with experimental and numerical results reported in previous studies.
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Abdolali, Ali, and James T. Kirby. "TSUNAMI PHASE SPEED REDUCTION DUE TO WATER COMPRESSIBILTY." Coastal Engineering Proceedings, no. 36 (December 30, 2018): 9. http://dx.doi.org/10.9753/icce.v36.currents.9.
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Most existing tsunami propagation models consider the ocean to be an incompressible, homogenous medium. Recently, it has been shown that a number of physical features can slow the propagation speed of tsunami waves, including wave frequency dispersion, ocean bottom elasticity, water compressibility and thermal or salinity stratification. These physical effects are secondary to the leading order, shallow water or long wave behavior, but still play a quantifiable role in tsunami arrival time, especially at far distant locations. In this work, we have performed analytical and numerical investigations and have shown that consideration of those effects can actually improve the prediction of arrival time at distant stations, compared to incompressible forms of wave equations. We derive a modified Mild Slope Equation for Weakly Compressible fluid following the method proposed by Sammarco et al. (2013) and Abdolali et al. (2015) using linearized wave theory, and then describe comparable extensions to the Boussinesq model of Kirby et al. (2013). Both models account for water compressibility and compression of static water column to simulate tsunami waves. The mild slope model is formulated in plane Cartesian coordinates and is thus limited to medium propagation distances, while the Boussinesq model is formulated in spherical polar coordinates and is suitable for ocean scale simulations.
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Volokh, K. Y., and H. Gao. "On the Modified Virtual Internal Bond Method." Journal of Applied Mechanics 72, no. 6 (April 5, 2005): 969–71. http://dx.doi.org/10.1115/1.2047628.
Повний текст джерелаАнотація:
The virtual internal bond (VIB) method was developed for the numerical simulation of fracture processes. In contrast to the traditional approach of fracture mechanics where stress analysis is separated from a description of the actual process of material failure, the VIB method naturally allows for crack nucleation, branching, kinking, and arrest. The idea of the method is to use atomic-like bond potentials in combination with the Cauchy-Born rule for establishing continuum constitutive equations which allow for the material separation–strain localization. While the conventional VIB formulation stimulated successful computational studies with applications to structural and biological materials, it suffers from the following theoretical inconsistency. When the constitutive relations of the VIB model are linearized for an isotropic homogeneous material, the Poisson ratio is found equal to 1∕4 so that there is only one independent elastic constant—Young’s modulus. Such restriction is not suitable for many materials. In this paper, we propose a modified VIB (MVIB) formulation, which allows for two independent linear elastic constants. It is also argued that the discrepancy of the conventional formulation is a result of using only two-body interaction potentials in the microstructural setting of the VIB method. When many-body interactions in “bond bending” are accounted for, as in the MVIB approach, the resulting formulation becomes consistent with the classical theory of isotropic linear elasticity.
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Cazeaux, Paul, and Céline Grandmont. "Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs." Mathematical Models and Methods in Applied Sciences 25, no. 06 (March 24, 2015): 1125–77. http://dx.doi.org/10.1142/s0218202515500293.
Повний текст джерелаАнотація:
We are interested in the mathematical modeling of the deformation of the human lung tissue, called the lung parenchyma, during the respiration process. The parenchyma is a foam-like elastic material containing millions of air-filled alveoli connected by a tree-shaped network of airways. In this study, the parenchyma is governed by the linearized elasticity equations and the air movement in the tree by the Poiseuille law in each airway. The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0. We use the two-scale convergence theory to study the asymptotic behavior as ε goes to zero. The effect of the network of airways is described by a nonlocal operator and we propose a simple geometrical setting for which we show that this operator converges as ε goes to zero. We identify in the limit the equations modeling the homogenized behavior under an abstract convergence condition on this nonlocal operator. We derive some mechanical properties of the limit material by studying the homogenized equations: the limit model is nonlocal both in space and time if the parenchyma material is considered compressible, but only in space if it is incompressible. Finally, we propose a numerical method to solve the homogenized equations and we study numerically a few properties of the homogenized parenchyma model.
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Mirzaei, M. "Lord–Shulman Nonlinear Generalized Thermoviscoelasticity of a Strip." International Journal of Structural Stability and Dynamics 20, no. 02 (December 5, 2019): 2050017. http://dx.doi.org/10.1142/s0219455420500170.
Повний текст джерелаАнотація:
In this paper, the response of a bounded one-dimensional medium (strip) subjected to a thermal shock is investigated. The strip is made of a linear visco-elastic material, of which the time-dependency of the elastic modulus is described by the simple Kelvin–Voigt model. To obtain the displacement, stress and temperature within the strip, the Lord and Shulman theory of generalized thermo-elasticity containing a single relaxation time is used. Three coupled equations, namely, the equation of motion, the modified Fourier law and the second law of thermodynamics, are established in terms of the displacement, temperature and heat flux. It is worth noting that the second law of thermodynamics is not linearized and is kept in the nonlinear form. The equations derived are first represented in a dimensionless presentation. Then, they are discretized using the well-known generalized differential quadrature. To obtain the response of the strip in time, the Newmark time marching scheme is implemented. It should be mentioned that due to the nonlinear nature of the governing equations, the successive Picard algorithm is used. The results of the present study are compared with those available in the literature for an elastic strip. Besides, numerical results are given for a strip made of Kelvin–Voigt visco-elastic material. The effects of visco-elastic parameter, coupling parameter, thermal relaxation time and nonlinearity are discussed in numerical examples.
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Bustamante, R., and K. R. Rajagopal. "Solutions of some boundary value problems for a new class of elastic bodies. Comparison with predictions of the classical theory of linearized elasticity: Part II. A problem with spherical symmetry." Acta Mechanica 226, no. 6 (December 27, 2014): 1807–13. http://dx.doi.org/10.1007/s00707-014-1289-8.
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Dobrokhotov, S. Yu, V. E. Nazaikinskii, and A. I. Shafarevich. "Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations." Russian Mathematical Surveys 76, no. 5 (October 1, 2021): 745–819. http://dx.doi.org/10.1070/rm9973.
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Abstract We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.
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Irschik, Hans, and Franz Ziegler. "Dynamic Processes in Structural Thermo-Viscoplasticity." Applied Mechanics Reviews 48, no. 6 (June 1, 1995): 301–16. http://dx.doi.org/10.1115/1.3005104.
Повний текст джерелаАнотація:
A multiple field theory dates back to developments in linear thermo-elasticity where the thermal strain is considered to be imposed on the linear elastic background structure in an isothermal state. Load strain and thermal strain fields are coupled by the boundary conditions or exhibit volume coupling. Taking into account a proper auxiliary problem, eg, the structure loaded by a unit force under isothermal conditions, results in Maysel’s formula and thus represents the solution in an optimal manner. That approach was generalized to dynamic problems and its efficiency was enhanced by splitting the solution in the quasi-static part and in the dynamic part thus considering the inertia of mass in the latter portion. This paper reviews recent extensions of such a multiple field theory to nonlinear problems of structural thermo-viscoplasticity. Their basis is given in formulations of micromechanics. In an incremental setting, the inelastic strains enter the background material as a second imposed strain field. Considering the rate form of the generalized Hooke’s law such a three-field representation is identified by inspection. Geometric nonlinearity is not fully taken into account, only approximations based on a second order theory, and, for beams and plates with immovable boundaries, in Berger’s approximation, render a third “strain field” to be imposed on the linearized background structure. A novel derivation of the dynamic generalization of Maysel’s formula is given in the paper. An elastic-viscoplastic semi-infinite impacted rod serves as the illustrative example of the fully dynamic analysis in the multiple field approach. Ductile damage is taken into account and the dissipated energy is deposited under adiabatic conditions. Material parameters are considered temperature dependent. An extension to a finite element discretization of the background structure seems to be possible, when preserving the solely linear elastic solution technique. Quite naturally, the domain integral boundary element method of solution results from such a multiple field approach.
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Achenbach, Jan D. "Reciprocity and Related Topics in Elastodynamics." Applied Mechanics Reviews 59, no. 1 (January 1, 2006): 13–32. http://dx.doi.org/10.1115/1.2110262.
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Reciprocity theorems in elasticity theory were discovered in the second half of the 19th century. For elastodynamics they provide interesting relations between two elastodynamic states, say states A and B. This paper will primarily review applications of reciprocity relations for time-harmonic elastodynamic states. The paper starts with a brief introduction to provide some historical and general background, and then proceeds in Sec. 2 to a brief discussion of static reciprocity for an elastic body. General comments on waves in solids are offered in Sec. 3, while Sec. 4 provides a brief summary of linearized elastodynamics. Reciprocity theorems are stated in Sec. 5. For some simple examples the concept of virtual waves is introduced in Sec. 6. A virtual wave is a wave motion that satisfies appropriate conditions on the boundaries and is a solution of the elastodynamic equations. It is shown that combining the desired solution as state A with a virtual wave as state B provides explicit results for state A. Basic elastodynamic states are discussed in Sec. 7. These states play an important role in the formulation of integral representations and integral equations, as shown in Sec. 8. Reciprocity in 1-D and full-space elastodynamics are discussed in Secs. 910, respectively. Applications to a half-space and a layer are reviewed in Secs. 1112. Section 13 is concerned with reciprocity of coupled acousto-elastic systems. The paper is completed with a brief discussion of reciprocity for piezoelectric systems. There are 61 references cited in this review article.
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Bustamante, R., and K. R. Rajagopal. "Solutions of some boundary value problems for a new class of elastic bodies undergoing small strains. Comparison with the predictions of the classical theory of linearized elasticity: Part I. Problems with cylindrical symmetry." Acta Mechanica 226, no. 6 (December 27, 2014): 1815–38. http://dx.doi.org/10.1007/s00707-014-1293-z.
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Yavari, Arash, and Arkadas Ozakin. "Covariance in linearized elasticity." Zeitschrift für angewandte Mathematik und Physik 59, no. 6 (March 26, 2008): 1081–110. http://dx.doi.org/10.1007/s00033-007-7127-2.
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Ponnusamy, Palaniyandi. "Stress wave analysis of thermo-piezoelectric solid bar of polygonal cross-sections immersed in fluid." Multidiscipline Modeling in Materials and Structures 10, no. 4 (November 4, 2014): 537–61. http://dx.doi.org/10.1108/mmms-12-2013-0076.
Повний текст джерелаАнотація:
Purpose – The purpose of this paper is to study the problem of wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of polygonal (triangle, square, pentagon and hexagon) cross-section immersed in fluid is using Fourier expansion collocation method, with in the frame work of linearized, three-dimensional theory of thermo-piezoelectricity. Design/methodology/approach – A mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of polygonal cross-sections immersed in fluid is studied using the three-dimensional theory of elasticity. Three displacement potential functions are introduced, to uncouple the equations of motion and the heat and electric conductions. The frequency equations are obtained for longitudinal and flexural (symmetric and antisymmetric) modes of vibration and are studied numerically for triangular, square, pentagonal and hexagonal cross-sectional bar immersed in fluid. Since the boundary is irregular in shape; it is difficult to satisfy the boundary conditions along the curved surface of the polygonal bar directly. Hence, the Fourier expansion collocation method is applied along the boundary to satisfy the boundary conditions. The roots of the frequency equations are obtained by using the secant method, applicable for complex roots. Findings – From the literature survey, it is clear that the free vibration of an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of polygonal cross-sectional bar immersed in fluid have not been analyzed by any of the researchers, also the previous investigations in the vibration problems of transversely isotropic thermo-piezoelectric solid bar of circular cross-sections only. So, in this paper, the wave propagation in thermo-piezoelectric cylindrical bar of polygonal cross-sections immersed in fluid are studied using the Fourier expansion collocation method. The computed non-dimensional frequencies are plotted in the form of dispersion curves and its characteristics are discussed, also a comparison is made between non-dimensional wave numbers for longitudinal and flexural modes piezoelectric, thermo-piezoelectric and thermo-piezoelectric polygonal cross-sectional bars immersed in fluid. Research limitations/implications – Wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of polygonal cross-sectional bar immersed in fluid have not been analyzed by any of the researchers, also the previous investigations in the vibration problems of transversely isotropic thermo-piezoelectric solid bar of circular cross-sections only. So, in this paper, the wave propagation in thermo-piezoelectric cylindrical bar of polygonal cross-sections immersed in fluid are studied using the Fourier expansion collocation method. The computed non-dimensional frequencies are plotted in the form of dispersion curves and its characteristics are discussed, also a comparison is made between non-dimensional wave numbers for longitudinal and flexural modes of piezoelectric, thermo-piezoelectric and thermo-piezoelectric polygonal cross-sectional bars immersed in fluid. Originality/value – The researchers have discussed the wave propagation in thermo-piezoelectric circular cylinders using three-dimensional theory of thermo-piezoelectricity, but, the researchers did not analyzed the wave propagation in an arbitrary/polygonal cross-sectional bar immersed in fluid. So, the author has studied the free vibration analysis of thermo-piezoelectric polygonal (triangle, square, pentagon and hexagon) cross-sectional bar immersed in fluid using three-dimensional theory elasticity. The problem may be extended to any kinds of cross-sections by using the proper geometrical relations.
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Krokos, Vasilis, Viet Bui Xuan, Stéphane P. A. Bordas, Philippe Young, and Pierre Kerfriden. "A Bayesian multiscale CNN framework to predict local stress fields in structures with microscale features." Computational Mechanics 69, no. 3 (November 27, 2021): 733–66. http://dx.doi.org/10.1007/s00466-021-02112-3.
Повний текст джерелаАнотація:
AbstractMultiscale computational modelling is challenging due to the high computational cost of direct numerical simulation by finite elements. To address this issue, concurrent multiscale methods use the solution of cheaper macroscale surrogates as boundary conditions to microscale sliding windows. The microscale problems remain a numerically challenging operation both in terms of implementation and cost. In this work we propose to replace the local microscale solution by an Encoder-Decoder Convolutional Neural Network that will generate fine-scale stress corrections to coarse predictions around unresolved microscale features, without prior parametrisation of local microscale problems. We deploy a Bayesian approach providing credible intervals to evaluate the uncertainty of the predictions, which is then used to investigate the merits of a selective learning framework. We will demonstrate the capability of the approach to predict equivalent stress fields in porous structures using linearised and finite strain elasticity theories.
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Bociu, Lorena, Steven Derochers, and Daniel Toundykov. "Linearized hydro-elasticity: A numerical study." Evolution Equations and Control Theory 5, no. 4 (October 2016): 533–59. http://dx.doi.org/10.3934/eect.2016018.
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CIARLET, PHILIPPE G., PATRICK CIARLET, OANA IOSIFESCU, STEFAN SAUTER, and JUN ZOU. "LAGRANGE MULTIPLIERS IN INTRINSIC ELASTICITY." Mathematical Models and Methods in Applied Sciences 21, no. 04 (April 2011): 651–66. http://dx.doi.org/10.1142/s0218202511005167.
Повний текст джерелаАнотація:
In an intrinsic approach to three-dimensional linearized elasticity, the unknown is the linearized strain tensor field (or equivalently the stress tensor field by means of the constitutive equation), instead of the displacement vector field in the classical approach. We consider here the pure traction problem and the pure displacement problem and we show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations (the form of which depends on the type of boundary conditions considered). Using the Babuška-Brezzi inf-sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints. Such results have potential applications to the numerical analysis and simulation of the intrinsic approach to three-dimensional linearized elasticity.
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Ciarlet, Philippe G., and Cristinel Mardare. "Intrinsic formulation of the displacement-traction problem in linearized elasticity." Mathematical Models and Methods in Applied Sciences 24, no. 06 (March 28, 2014): 1197–216. http://dx.doi.org/10.1142/s0218202513500814.
Повний текст джерелаАнотація:
The displacement-traction problem of linearized elasticity is a system of partial differential equations and boundary conditions whose unknown is the displacement field inside a linearly elastic body. We explicitly identify here the corresponding boundary conditions satisfied by the linearized strain tensor field associated with such a displacement field. Using this identification, we are then able to provide an intrinsic formulation of the displacement-traction problem of linearized elasticity, by showing how it can be recast into a boundary value problem whose unknown is the linearized strain tensor field.
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Serrano, Hélia. "Homogenization of kinetic laminates in linearized elasticity." Mathematical Methods in the Applied Sciences 41, no. 1 (October 5, 2017): 270–80. http://dx.doi.org/10.1002/mma.4611.
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Ciarlet, Philippe G., Patrick Ciarlet, Oana Iosifescu, Stefan Sauter, and Jun Zou. "A Lagrangian approach to intrinsic linearized elasticity." Comptes Rendus Mathematique 348, no. 9-10 (May 2010): 587–92. http://dx.doi.org/10.1016/j.crma.2010.04.011.
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