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Journal articles on the topic 'Linear elasticty'

Author: Grafiati

Published: 20 April 2024

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1

Bakushev, S. V. "LINEAR THEORY OF ELASTICITY WITH QUADRATIC SUMMAND." STRUCTURAL MECHANICS AND ANALYSIS OF CONSTRUCTIONS 303, no. 4 (February 28, 2022): 29–36. http://dx.doi.org/10.37538/0039-2383.2022.1.29.36.

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We suggest a linear theory version based on Taylor decompositions for stresses and power-series for quadratic summand deformations. Thus, static equations of equilibrium in stresses are written in the form of the second-order partial derivatives differential equations. The resolving equations of equilibrium in displacements are represented in the form of the third order partial derivatives differential equations. The physical equations in this version of the linear theory of elasticity are written in the same way as in the classical linear theory of elasticity. Equilibrium equations, along with other parameters – physical constants of the medium – contain minor parameters dx, dy, dz, the value of which, as shown by numerical modelling, has little effect on the nature of the stress-strain state. It is suggested to use experimental data to determine them. Along with the formulating of the basic equations of the three-dimensional theory of elasticity, particular cases of the stress-strain state of elastic continuous medium are considered: uniaxial stressed state; uniaxial deformed state; flat deformation; generalized plane stress state. Determination of the stressed and deformed state of a thin elastic bar by integrating the resolving equations in stresses and displacements is considered as examples. The suggested version of the linear theory of elasticity, due to the quadratic summand in Taylor decompositions for stresses and in power-series for deformations, expands the classical linear theory of elasticity and, with an appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.

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2

Hassanpour, Soroosh, and Glenn R. Heppler. "Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations." Mathematics and Mechanics of Solids 22, no. 2 (August 5, 2016): 224–42. http://dx.doi.org/10.1177/1081286515581183.

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This paper is devoted to a review of the linear isotropic theory of micropolar elasticity and its development with a focus on the notation used to represent the micropolar elastic moduli and the experimental efforts taken to measure them. Notation, not only the selected symbols but also the approaches used for denoting the material elastic constants involved in the model, can play an important role in the micropolar elasticity theory especially in the context of investigating its relationship with the couple-stress and classical elasticity theories. Two categories of notation, one with coupled classical and micropolar elastic moduli and one with decoupled classical and micropolar elastic moduli, are examined and the consequences of each are addressed. The misleading nature of the former category is also discussed. Experimental investigations on the micropolar elasticity and material constants are also reviewed where one can note the questionable nature and limitations of the experimental results reported on the micropolar elasticity theory.

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3

KOENEMANN, FALK H. "LINEAR ELASTICITY AND POTENTIAL THEORY: A COMMENT ON GURTIN (1972)." International Journal of Modern Physics B 22, no. 28 (November 10, 2008): 5035–39. http://dx.doi.org/10.1142/s0217979208049224.

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In an exhaustive presentation of the linear theory of elasticity by Gurtin [The Linear Theory of Elasticity (Springer-Verlag, 1972)], the author included a chapter on the relation of the theory of elasticity to the theory of potentials. Potential theory distinguishes two fundamental physical categories: divergence-free and divergence-involving problems. From the criteria given in the source quoted by the author, it is evident that elastic deformation of solids falls into the latter category. It is documented in this short note that the author presented volume-constant elastic deformation as a divergence-free physical process, systematically ignoring all the information that was available to him that this is not so.

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4

Böhmer, CG, and N. Tamanini. "Rotational elasticity and couplings to linear elasticity." Mathematics and Mechanics of Solids 20, no. 8 (November 29, 2013): 959–74. http://dx.doi.org/10.1177/1081286513511093.

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5

Xiao, B., and J. Feng. "Higher order elastic tensors of crystal structure under non-linear deformation." Journal of Micromechanics and Molecular Physics 04, no. 04 (December 2019): 1950007. http://dx.doi.org/10.1142/s2424913019500073.

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The higher-order elastic tensors can be used to characterize the linear and non-linear mechanical properties of crystals at ultra-high pressures. Besides the widely studied second-order elastic constants, the third- and fourth-order elastic constants are sixth and eighth tensors, respectively. The independent tensor components of them are completely determined by the symmetry of the crystal. Using the relations between elastic constants and sound velocity in solid, the independent elastic constants can be measured experimentally. The anisotropy in elasticity of crystal structures is directly determined by the independent elastic constants.

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6

Sadegh, A. M., and S. C. Cowin. "The Proportional Anisotropic Elastic Invariants." Journal of Applied Mechanics 58, no. 1 (March 1, 1991): 50–57. http://dx.doi.org/10.1115/1.2897178.

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There are two proportional invariants for a linear isotropic material, the hydrostatic invariant, and the deviatoric invariant. The former is proportional to the trace of the tensor and the latter is proportional to the trace of the square of the associated deviatoric tensor. The hydrostatic stress and strain and the von Mises stress and strain are directly related to the hydrostatic and deviatoric proportional invariants, respectively, for an isotropic, linear elastic material. For each anisotropic linear elastic material symmetry there are up to six proportional invariants. In this paper we illustrate the six proportional invariants of an orthotropic elastic material using the elastic constants for spruce as the numerical example. The proportional elastic invariants play a role in anisotropic linear elasticity similar to the roles played by the hydrostatic stress and strain and the von Mises stress and strain in isotropic elasticity. They are the unique parameters whose contours represent both the stress and the strain distributions. They also have potential for representing failure or fracture criteria.

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7

de C. Henderson, J. C. "Introduction to linear elasticity." Applied Mathematical Modelling 9, no. 3 (June 1985): 226–27. http://dx.doi.org/10.1016/0307-904x(85)90013-7.

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8

Cudworth, C. J. "Introduction to linear elasticity." Journal of Mechanical Working Technology 12, no. 3 (February 1986): 385. http://dx.doi.org/10.1016/0378-3804(86)90008-2.

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9

Lee, KwangJin, SangRyong Lee, and Hak Yi. "Design and Control of Cylindrical Linear Series Elastic Actuator." Journal of the Korean Society for Precision Engineering 36, no. 1 (January 1, 2019): 95–98. http://dx.doi.org/10.7736/kspe.2019.36.1.95.

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10

Cowin, S. C., and M. M. Mehrabadi. "Anisotropic Symmetries of Linear Elasticity." Applied Mechanics Reviews 48, no. 5 (May 1, 1995): 247–85. http://dx.doi.org/10.1115/1.3005102.

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The objective of this paper is to present a development of the anisotropic symmetries of linear elasticity theory based on the use of a single symmetry element, the plane of mirror symmetry. In this presentation the thirteen distinct planes of mirror symmetry are catalogued. Traditional presentations of the anisotropic elastic symmetries involve all the crystallographic symmetry elements which include the center of symmetry, the n-fold rotation axis and the n-fold inversion axis as well as the plane of mirror symmetry. It is shown that the crystal system symmetry groups, as opposed to the crystal class symmetry groups, of the elastic crystallographic symmetries can be generated by the appropriate combinations of the orthogonal transformations corresponding to each of the thirteen distinct planes of mirror symmetry. It is also shown that the restrictions on the elastic coefficients appearing in Hooke’s law follow in a simple and straightforward fashion from orthogonal transformations based on a small subset of the small catalogue of planes of mirror symmetry.

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11

JURAK, MLADEN, and JOSIP TAMBAČA. "LINEAR CURVED ROD MODEL: GENERAL CURVE." Mathematical Models and Methods in Applied Sciences 11, no. 07 (October 2001): 1237–52. http://dx.doi.org/10.1142/s0218202501001318.

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A one-dimensional model of a curved rod is derived from the three-dimensional linearized elasticity. No positivity assumption on the curvature of the central line of the curved rod is made. The model is obtained by taking the limit in the equilibrium equation of the three-dimensional elastic rod when the thickness of the rod goes to zero. The appropriate convergence result is proved.

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12

Paroni, Roberto, and Giuseppe Tomassetti. "From non-linear elasticity to linear elasticity with initial stress via Γ-convergence." Continuum Mechanics and Thermodynamics 23, no. 4 (March 11, 2011): 347–61. http://dx.doi.org/10.1007/s00161-011-0184-y.

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13

He, Linjie, Yumin Chen, Caiming Zhong, and Keshou Wu. "Granular Elastic Network Regression with Stochastic Gradient Descent." Mathematics 10, no. 15 (July 27, 2022): 2628. http://dx.doi.org/10.3390/math10152628.

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Linear regression is the use of linear functions to model the relationship between a dependent variable and one or more independent variables. Linear regression models have been widely used in various fields such as finance, industry, and medicine. To address the problem that the traditional linear regression model is difficult to handle uncertain data, we propose a granule-based elastic network regression model. First we construct granules and granular vectors by granulation methods. Then, we define multiple granular operation rules so that the model can effectively handle uncertain data. Further, the granular norm and the granular vector norm are defined to design the granular loss function and construct the granular elastic network regression model. After that, we conduct the derivative of the granular loss function and design the granular elastic network gradient descent optimization algorithm. Finally, we performed experiments on the UCI datasets to verify the validity of the granular elasticity network. We found that the granular elasticity network has the advantage of good fit compared with the traditional linear regression model.

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14

Kim, Daeho, and Jaeil Kim. "Elastic Multiple Parametric Exponential Linear Units for Convolutional Neural Networks." Journal of KIISE 46, no. 5 (May 31, 2019): 469–77. http://dx.doi.org/10.5626/jok.2019.46.5.469.

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15

He, Q. C. "A Remarkable Tensor in Plane Linear Elasticity." Journal of Applied Mechanics 64, no. 3 (September 1, 1997): 704–7. http://dx.doi.org/10.1115/1.2788952.

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It is shown that any two-dimensional elastic tensor can be orthogonally and uniquely decomposed into a symmetric tensor and an antisymmetric tensor. To within a scalar multiplier, the latter turns out to be equal to the right-angle rotation on the space of two-dimensional second-order symmetric tensors. On the basis of these facts, several useful results are derived for the traction boundary value problem of plane linear elasticity.

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16

Markov, A., A. Kazakov, M. Haqberdiyev, Sh Muhitdinov, and M. Rahimova. "Calculation of tectonic stresses in the earth’s crust of SouthWestern Uzbekistan." IOP Conference Series: Earth and Environmental Science 937, no. 4 (December 1, 2021): 042087. http://dx.doi.org/10.1088/1755-1315/937/4/042087.

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Abstract In this article, based on accounting, the interaction of the Earth’s crust blocks is limited by the deep breaks in the form of three-layer panels. The analysis dependences for tectonic pressure on elasticity parameters and the Earth’s crust layers capacity were obtained using the hypothesis of linear changes of deformations on the height of panels and the elasticity for bottom layers of the Earth’s crust. This paper considers the elastic interaction of crustal blocks bounded by deep faults in the form of three-layer panels. Using the hypothesis of linear measurement of deformations along with the height of the board and the elastic limit for the lower layer of the Earth’s crust, calculated dependences for tectonic stresses on the elasticity and thickness of the layers of the Earth’s crust are obtained.

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17

Miettinen, Markku, and Uldis Raitums. "On the strong closure of strains and stresses in linear elasticity." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 5 (1999): 987–1009. http://dx.doi.org/10.1017/s0308210500031048.

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We consider the following special problem related to the optimal layout problems of materials: given two linear elastic materials, the elasticity tensors of which are C1 and C2, and a force f, find the strong closure of strains and stresses as the distribution of the materials varies, or, alternatively, find the sets of elasticity tensors which generate these strong closures. In this paper, it is shown that the local incompatibility conditions depending on C1, C2 and the local properties of strains or stresses completely characterize these sets. A connection to multiple-well problems is established.

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18

Brenner, Susanne C., and Li-Yeng Sung. "Linear finite element methods for planar linear elasticity." Mathematics of Computation 59, no. 200 (1992): 321. http://dx.doi.org/10.1090/s0025-5718-1992-1140646-2.

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19

Levitin, Michael, Peter Monk, and Virginia Selgas. "Impedance Eigenvalues in Linear Elasticity." SIAM Journal on Applied Mathematics 81, no. 6 (January 2021): 2433–56. http://dx.doi.org/10.1137/21m1412955.

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20

Cardoso, Fernando, Partially supported by CNP, q. sub, esub Brazil, Fernando Cardoso, Partially supported by CNP, q. sub, and esub Brazil. "Rayleigh Quasimodes In Linear Elasticity." Communications in Partial Differential Equations 17, no. 7 (1992): 87–100. http://dx.doi.org/10.1080/03605309208820888.

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21

Nader, J. J. "Linear Response in Finite Elasticity." Journal of Elasticity 73, no. 1-3 (December 2003): 165–72. http://dx.doi.org/10.1023/b:elas.0000029956.39597.a5.

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22

Yavari, Arash, Christian Goodbrake, and Alain Goriely. "Universal displacements in linear elasticity." Journal of the Mechanics and Physics of Solids 135 (February 2020): 103782. http://dx.doi.org/10.1016/j.jmps.2019.103782.

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23

Chu, Jeong Ho, Hi Jun Choe, and Man-Hoe Kim. "Spectral method for linear elasticity." Applied Mathematics and Computation 182, no. 1 (November 2006): 269–82. http://dx.doi.org/10.1016/j.amc.2006.01.093.

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24

LANCIA, M. R., G. VERGARA CAFFARELLI, and P. PODIO-GUIDUGLI. "NULL LAGRANGIANS IN LINEAR ELASTICITY." Mathematical Models and Methods in Applied Sciences 05, no. 04 (June 1995): 415–27. http://dx.doi.org/10.1142/s0218202595000255.

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The concept of null Lagrangian is exploited in the context of linear elasticity. In particular, it is shown that the stored energy functional can always be split into a null Lagrangian and a remainder; the null Lagrangian vanishes if and only if the elasticity tensor obeys the Cauchy relations, and is therefore determined by only 15 independent moduli (the so-called “rari-constant” theory of elasticity).

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25

Kim, Hyunseok, and Jin Keun Seo. "Identification Problems in Linear Elasticity." Journal of Mathematical Analysis and Applications 215, no. 2 (November 1997): 514–31. http://dx.doi.org/10.1006/jmaa.1997.5656.

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26

Cowin, Steven C. "Propagation of Kelvin Modes." Mathematics and Mechanics of Solids 1, no. 1 (March 1996): 25–43. http://dx.doi.org/10.1177/108128659600100103.

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The Kelvin modes of isotropic elasticity are the dilatational mode and the deviatoric mode. Kelvin modes may be defined for all the anisotropic elastic symmetries. It is shown here that any plane wave may be considered as the sum of propagating Kelvin modes, but that the only Kelvin mode that will propagate by itself as a plane wave in any anisotropic elastic material (except a material with triclinic symmetry) is a single simple shear mode. However, other Kelvin modes may propagate in special elastic materials and single simple shear modes may propagate in special triclinic materials. These results generalized the paradigm of the shear and pressure waves of linear isotropic elasticity to all plane waves in anisotropic elasticity.

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27

Ďoubal, Stanislav, Petr Klemera, Vladimír Semecký, Jiří Lamka, and Monika Kuchařová. "Non-Linear Mechanical Behavior of Visco-Elastic Biological Structures – Measurements and Models." Acta Medica (Hradec Kralove, Czech Republic) 47, no. 4 (2004): 297–300. http://dx.doi.org/10.14712/18059694.2018.110.

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Mechanical properties of biological structures affect functional ability of organism. Current knowledge is prevailingly concentrated on static characteristics. The present work analyzed dynamic mechanical responses of various biological materials. Following biological structures were measured: samples of aorta walls of human origin and from model organisms, human body surface, and samples of bones of various types and origin. Linear approximation leads in case of aortas and bones to simple Voight's model. Modules of elasticity (in tensile loading) of aortas were from 102 kPa to 103 kPa. Module of elasticity of bones were from 106 Pa to 1010 Pa. Viscous coefficients of aortas were from 102 Pa.s to 103 Pa.s. Viscous coefficients of bones were from 100 Pa.s to 102 Pa.s. Nonlinearities: We found that following types of nonlinearities are significant: strain-stress relationship, time-dependent changes in elastic as well as viscose bodies. Strain and stress is well approximated by quadratic function σ = a ε2 + b ε + c with parameters a = 1833, b = 135, c = 20.0 (porcine aorta). Time-dependence in elastic coefficient: At the beginning of responses the elastic coefficient was of 42% lower then at 0.02 s of duration of the response (porcine aortas). Analogical results follow also from experiments on other structures (skin, bones).

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28

Steinberg, Victor. "New direction and perspectives in elastic instability and turbulence in various viscoelastic flow geometries without inertia." Low Temperature Physics 48, no. 6 (June 2022): 492–507. http://dx.doi.org/10.1063/10.0010445.

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We shortly describe the main results on elastically driven instabilities and elastic turbulence in viscoelastic inertialess flows with curved streamlines. Then we describe a theory of elastic turbulence and prediction of elastic waves Re ≪ 1 and Wi ≫ 1, which speed depends on the elastic stress similar to the Alfvén waves in magneto-hydrodynamics and in a contrast to all other, which speed depends on medium elasticity. Since the established and testified mechanism of elastic instability of viscoelastic flows with curvilinear streamlines becomes ineffective at zero curvature, so parallel shear flows are proved linearly stable, similar to Newtonian parallel shear flows. However, the linear stability of parallel shear flows does not imply their global stability. Here we switch to the main subject, namely a recent development in inertialess parallel shear channel flow of polymer solutions. In such flow, we discover an elastically driven instability, elastic turbulence, elastic waves, and drag reduction down to relaminarization that contradict the linear stability prediction. In this regard, we discuss briefly normal versus non-normal bifurcations in such flows, flow resistance, velocity and pressure fluctuations, and coherent structures and spectral properties of a velocity field as a function of Wi at high elasticity number.

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29

Bakushev, S. V. "REFINED LINEAR THEORY OF ELASTICITY TAKING INTO ACCOUNT QUADRATIC TERMS." STRUCTURAL MECHANICS AND ANALYSIS OF CONSTRUCTIONS, no. 5 (October 30, 2023): 44–54. http://dx.doi.org/10.37538/0039-2383.2023.5.44.54.

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A variant of the linear theory is proposed, based on taking into account in Taylor expansions for stresses and in power series and the expansion of the radical into a binomial series for deformations of not only linear, but also quadratic terms. In this case, static equilibrium equations in stresses are written in the form of second-order partial differential equations. The law of pairing of tangential stresses is observed. The relationship between deformations and displacements is described by relations similar to Cauchy relations, but taking into account the second derivatives of displacements along spatial coordinates. The equations of continuity of deformations for this version of the theory of elasticity are written in the form of two groups of relations. The first group of relations establishes the continuity equations for the linear terms of the deformation components; The second group of relations establishes the continuity equations for the quadratic terms of the deformation components. The physical equations for this variant of the linear theory of elasticity are written in the same way as for the classical linear theory of elasticity. Equilibrium equations in stresses are represented by second-order partial differential equations. The equilibrium equations in displacements are written in the form of fourth-order partial differential equations. The equations of equilibrium, as well as the equations of continuity of deformations, along with other parameters – the physical constants of the medium – contain small parameters dx, dy, dz, the magnitude of which, as shown by numerical modeling, has little effect on the nature of the stress-strain state. To determine them, it is necessary, apparently, to use experimental data. The construction of resolving equations in both stresses and displacements for special cases of the stress-strain state of an elastic continuous medium is considered: uniaxial dressed-up state; uniaxial deformed state; plane strain; generalized plane stress state. The problem of the uniaxial deformed state of a continuous medium (solution in stresses) is numerically solved. The proposed version of the linear theory of elasticity extends the classical linear theory of elasticity and, with appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.

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30

Castro, Jorge. "Elastic potentials as yield surfaces for isotropic materials." PLOS ONE 17, no. 10 (October 26, 2022): e0275968. http://dx.doi.org/10.1371/journal.pone.0275968.

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This paper proposes that elastic potentials, which may be rigorously formulated using the negative Gibbs free energy or the complementary strain energy density, may be used as the yield surface of elasto-plastic constitutive models. Thus, the yield surface may be assumed in some materials as an elastic potential surface for a specific level of critical complementary strain energy density. Traditional approaches, such as the total strain energy criterion, only consider second order terms, i.e., the elastic potential is centred at the origin of the current stress state. Here, first order terms are considered, and consequently, the elastic potential may be translated, which allows to reproduce the desired level of tension-compression asymmetry. The proposed approach only adds two additional parameters, e.g., uniaxial compressive and tensile yield limits, to the elastic ones. For linear elasticity, the proposed approach provides elliptical yield surfaces and shows a correlation between the shape of the ellipse and the Poisson’s ratio, which agree with published experimental data for soils and metallic glasses. This elliptical yield surface also fits well experimental values of amorphous polymers and some rocks. Besides, the proposed approach automatically considers the influence of the intermediate stress. For non-linear elasticity, a wider range of elastic potentials, i.e., yield surfaces, are possible, such as distorted ellipsoids. For the case of incompressible non-linear materials, the yield surfaces are between von Mises and Tresca ones.

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31

Ike, CC. "FIRST PRINCIPLES DERIVATION OF A STRESS FUNCTION FOR AXIALLY SYMMETRIC ELASTICITY PROBLEMS, AND APPLICATION TO BOUSSINESQ PROBLEM." Nigerian Journal of Technology 36, no. 3 (June 30, 2017): 767–72. http://dx.doi.org/10.4314/njt.v36i3.15.

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In this work, a stress function is derived from first principles to describe the behaviour of three dimensional axially symmetric elasticity problems involving linear elastic, isotropic homogeneous materials. In the process, the fifteen governing partial differential equations of linear isotropic elasticity were reduced to the solution of the biharmonic problem involving the stress function. thus simplifying the solution process. The stress function derived was found to be identical with the Love stress function. The stress function was then applied to solve the axially symmetric problem of finding the stress fields, strain fields and displacement fields in the semi-infinite linear elastic, isotropic homogeneous medium subject to a point load P acting at the origin of coordinates also called the Boussinesq problem. The results obtained in this study for the stresses and displacements were exactly identical with those from literature, as obtained by Boussinesq.http://dx.doi.org/10.4314/njt.v36i3.15

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32

Haughton, D. M. "On non-linear stability in unconstrained non-linear elasticity." International Journal of Non-Linear Mechanics 39, no. 7 (September 2004): 1181–92. http://dx.doi.org/10.1016/j.ijnonlinmec.2003.07.002.

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33

Droniou, Jérôme, and Bishnu P. Lamichhane. "Gradient schemes for linear and non-linear elasticity equations." Numerische Mathematik 129, no. 2 (June 12, 2014): 251–77. http://dx.doi.org/10.1007/s00211-014-0636-y.

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34

Phan, Anh Thu, Aïmen E. Gheribi, and Patrice Chartrand. "Coherent phase equilibria of systems with large lattice mismatch." Physical Chemistry Chemical Physics 21, no. 20 (2019): 10808–22. http://dx.doi.org/10.1039/c9cp01272a.

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The elastic contribution to the Gibbs energy, representing the elastic energy stored in the coherent boundary, is formulated based on the linear elasticity theory in both the small and large deformation regimes. Several case studies have been examined in cubic systems, and the proposed formalism is showing an appropriate predictive capability.

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35

Valášek, Jan, Petr Sváček, and Jaromír Horáček. "Numerical Simulation of Interaction of Fluid Flow and Elastic Structure Modelling Vocal Fold." Applied Mechanics and Materials 821 (January 2016): 693–700. http://dx.doi.org/10.4028/www.scientific.net/amm.821.693.

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This paper deals with an interaction of the viscous incompressible fluid flow with a simplified model of the human vocal fold. In order to capture deformation of the elastic body the arbitrary Lagrangian-Euler method is used. The linear elasticity model is considered. The problem is solved by the developed finite element method based solver. For the flow approximation the crossgrid elements are used. The elastic structure motion is approximated by the piecewise linear elements. Numerical results are shown.

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36

Mañero-Sanz, Hugo, Eva M. García del Toro, Vicente Alcaraz-Carrillo de Albornoz, and Alfredo Luizaga Patiño. "Method for the Improvement of the Elasticity Module of Concrete Specimens by Active Confinement." Sustainability 11, no. 12 (June 14, 2019): 3289. http://dx.doi.org/10.3390/su11123289.

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The purpose of this work is to improve the modulus of elasticity of reinforced concrete pillars in the area where it is known with certainty that the concrete is elastic. To achieve this, an innovative method was devised to introduce an initial tension ( σ i ) resulting in an 11% increase in the working compression. Three concrete batches of five specimens each were prepared for this study. The first batch was used as a control without applying any reinforcement, the second was reinforced with a carbon fiber fabric (CF) layer in the usual way, and in the third batch, an initial tension was introduced to the CF fabric by a technique devised for this purpose. After measuring the modulus of elasticity of each of the specimens that made up each batch, it was observed that the modulus of elasticity obtained for the specimens in the third batch was 8% higher than the specimens in the first and second batches. The compression–deformation behaviour of the specimens observed throughout the study allows us to propose a stress–strain model with three different behaviours: linear elastic, parabolic elasto-plastic and linear elastic.

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37

Moskalyuk, Olga A., Andrey V. Belashov, Yaroslav M. Beltukov, Elena M. Ivan’kova, Elena N. Popova, Irina V. Semenova, Vladimir Y. Yelokhovsky, and Vladimir E. Yudin. "Polystyrene-Based Nanocomposites with Different Fillers: Fabrication and Mechanical Properties." Polymers 12, no. 11 (October 23, 2020): 2457. http://dx.doi.org/10.3390/polym12112457.

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The paper presents a comprehensive analysis of the elastic properties of polystyrene-based nanocomposites filled with different types of inclusions: small spherical particles (SiO2 and Al2O3), alumosilicates (montmorillonite, halloysite natural tubules and mica), and carbon nanofillers (carbon black and multi-walled carbon nanotubes). Block samples of composites with different filler concentrations were fabricated by melt technology, and their linear and non-linear elastic properties were studied. The introduction of more rigid particles led to a more profound increase in the elastic modulus of a composite, with the highest rise of about 80% obtained with carbon fillers. Non-linear elastic moduli of composites were shown to be more sensitive to addition of filler particles to the polymer matrix than linear ones. A non-linearity modulus βs comprising the combination of linear and non-linear elastic moduli of a material demonstrated considerable changes correlating with those of the Young’s modulus. The changes in non-linear elasticity of fabricated composites were compared with parameters of bulk non-linear strain waves propagating in them. Variations of wave velocity and decay decrement correlated with the observed enhancement of materials’ non-linearity.

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38

Itin, Yakov. "Irreducible matrix resolution for symmetry classes of elasticity tensors." Mathematics and Mechanics of Solids 25, no. 10 (April 20, 2020): 1873–95. http://dx.doi.org/10.1177/1081286520913596.

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In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

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39

Maddalena, Francesco, Danilo Percivale, and Franco Tomarelli. "Signorini problem as a variational limit of obstacle problems in nonlinear elasticity." Mathematics in Engineering 6, no. 2 (2024): 261–304. http://dx.doi.org/10.3934/mine.2024012.

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<abstract><p>An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.</p></abstract>

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40

Prado, Jose Manuel. "The Elastic Behaviour of Metal Powder Compacts." Materials Science Forum 534-536 (January 2007): 325–28. http://dx.doi.org/10.4028/www.scientific.net/msf.534-536.325.

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In this work the elastic behaviour of metallic powder compacts is studied. Cylindrical specimens with different levels of density have been submitted to uniaxial compression tests with loading and unloading cycles. The analysis of the elastic loadings shows a non linear elasticity which can be mathematically represented by means of a potential law. Results are explained by assuming that the total elastic strain is the contribution of two terms one deriving from the hertzian deformation of the contacts among particles and another that takes into account the linear elastic deformation of the powder skeleton. A simple model based in a one pore unit cell is presented to support the mathematical model.

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Korolev, Alexander Sergeevich, Anastasia Kopp, Denis Odnoburcev, Vladislav Loskov, Pavel Shimanovsky, Yulia Koroleva, and Nikolai Ivanovich Vatin. "Compressive and Tensile Elastic Properties of Concrete: Empirical Factors in Span Reinforced Structures Design." Materials 14, no. 24 (December 9, 2021): 7578. http://dx.doi.org/10.3390/ma14247578.

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Concretes with the same strength can have various deformability that influences span structures deflection. In addition, a significant factor is the non-linear deformation of concrete dependence on the load. The main deformability parameter of concrete is the instantaneous modulus of elasticity. This research aims to evaluate the relation of concrete compressive and tensile elastic properties testing. The beam samples at 80 × 140 × 1400 cm with one rod Ø8 composite or Ø10 steel reinforcement were experimentally tested. It was shown that instantaneous elastic deformations under compression are much lower than tensile. Prolonged elastic deformations under compression are close to tensile. It results in compressive elasticity modulus exceeding the tensile. The relation between these moduli is proposed. The relation provides operative elasticity modulus testing by the bending tensile method. The elasticity modulus’s evaluation for the reinforced span structures could be based only on the bending testing results. A 10% elasticity modulus increase, which seems not significant, increases at 30–40% the stress of the reinforced span structures under load and 30% increases the cracking point stress.

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Jaric, Jovo, and Dragoslav Kuzmanovic. "On damage tensor in linear anisotropic elasticity." Theoretical and Applied Mechanics 44, no. 2 (2017): 141–54. http://dx.doi.org/10.2298/tam170306018j.

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In this paper, the anisotropic linear damage mechanics is presented starting from the principle of strain equivalence. The authors have previously derived damage tensor components in terms of elastic parameters of undamaged (virgin) material in closed form solution. Here, making use of this paper, we derived elasticity tensor as a function of damage tensor also in closed form. The procedure we present here was applied for several crystal classes which are subjected to hexagonal, orthotropic, tetragonal, cubic and isotropic damage. As an example isotropic system is considered in order to present some possibility to evaluate its damage parameters.

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Kozubal, Janusz, Wojciech Puła, Marek Wyjadłowski, and Jerzy Bauer. "INFLUENCE OF VARYING SOIL PROPERTIES ON EVALUATION OF PILE RELIABILITY UNDER LATERAL LOADS." Journal of Civil Engineering and Management 19, no. 2 (April 18, 2013): 272–84. http://dx.doi.org/10.3846/13923730.2012.756426.

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A three dimensional probabilistic approach to analyzing laterally loaded piles is presented. Two typical subsurface models are used in the analyses: the first one consists of layered linear elastic soil where each layer has a random modulus of elasticity; while the second model takes the form of linear elastic soil with a random modulus of elasticity that increases with depth. Efficient step by step procedures for the reliability computation involving pile displacements are proposed. The solution is based on three-dimensional modeling by the finite element method. A series of results has been obtained for various values of elastic parameters of the soil. Next by a non-linear regression procedure a response surface is obtained. To get the final response surface allowing for a reliability analysis, an iterative algorithm based on the so-called design point concept is applied. The failure criterion is defined as the pile head displacement exceeding displacement threshold. The two cases of piles subjected to lateral load are computed. The paper illustrates the influence of the two distinct types of subsurface variability on the probabilistic analysis. A pronounced effect of the random variability of both the lateral force and the elastic modulus of the upper layer on reliability indices has been shown in results of numerical examples.

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Igumnov, Leonid, I. P. Маrkov, and A. V. Amenitsky. "A Three-Dimensional Boundary Element Approach for Transient Anisotropic Viscoelastic Problems." Key Engineering Materials 685 (February 2016): 267–71. http://dx.doi.org/10.4028/www.scientific.net/kem.685.267.

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This paper presents a three-dimensional direct boundary element approach for solving transient problems of linear anisotropic elasticity and viscoelasticity. In order to take advantage of the correspondence principle between viscoelasticity and elasticity the formulation is given in the Laplace domain. Anisotropic viscoelastic fundamental solutions are obtained using the correspondence principle and anisotropic elastic Green’s functions. The standard linear solid model is used to represent the mechanical behavior of viscoelastic material. Solution in time domain is calculated via numerical inversion by modified Durbin’s method. Numerical example is provided to validate the proposed boundary element formulation.

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Zeidi, Mahdi, and Chun IL Kim. "Mechanics of fiber composites with fibers resistant to extension and flexure." Mathematics and Mechanics of Solids 24, no. 1 (September 4, 2017): 3–17. http://dx.doi.org/10.1177/1081286517728543.

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A model of elastic solids reinforced with fibers resistant to extension and bending is formulated in finite-plane elastostatics. The linear theory of the proposed model is also derived through which a complete analytical solution is obtained. The presented model can serve as an alternative two-dimensional Cosserat theory of non-linear elasticity.

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Atchonouglo, K., G. de Saxcé, and M. Ban. "2D ELASTICITY TENSOR INVARIANTS, INVARIANTS DEFINITE POSITIVE CRITERIA." Advances in Mathematics: Scientific Journal 10, no. 8 (August 6, 2021): 2999–3012. http://dx.doi.org/10.37418/amsj.10.8.1.

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In this paper, we constructed relationships with the differents 2D elasticity tensor invariants. Indeed, let ${\bf A}$ be a 2D elasticity tensor. Rotation group action leads to a pair of Lax in linear elasticity. This pair of Lax leads to five independent invariants chosen among six. The definite positive criteria are established with the determined invariants. We believe that this approach finds interesting applications, as in the one of elastic material classification or approaches in orbit space description.

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XIA, BINGXING, and VIET HA HOANG. "BEST N-TERM GPC APPROXIMATIONS FOR A CLASS OF STOCHASTIC LINEAR ELASTICITY EQUATIONS." Mathematical Models and Methods in Applied Sciences 24, no. 03 (December 29, 2013): 513–52. http://dx.doi.org/10.1142/s0218202513500589.

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We consider a class of stochastic linear elasticity problems whose elastic moduli depend linearly on a countable set of random variables. The stochastic equation is studied via a deterministic parametric problem on an infinite-dimensional parameter space. We first study the best N-term approximation of the generalized polynomial chaos (gpc) expansion of the solution to the displacement formula by considering a Galerkin projection onto the space obtained by truncating the gpc expansion. We provide sufficient conditions on the coefficients of the elastic moduli's expansion so that a rate of convergence for this approximation holds. We then consider two classes of stochastic and parametric mixed elasticity problems. The first one is the Hellinger–Reissner formula for approximating directly the gpc expansion of the stress. For isotropic problems, the multiplying constant of the best N-term convergence rate for the displacement formula grows with the ratio of the Lamé constants. We thus consider stochastic and parametric mixed problems for nearly incompressible isotropic materials whose best N-term approximation rate is uniform with respect to the ratio of the Lamé constants.

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Craster, Richard, André Diatta, Sébastien Guenneau, and Harsha Hutridurga. "On Near-cloaking for Linear Elasticity." Multiscale Modeling & Simulation 19, no. 2 (January 2021): 633–64. http://dx.doi.org/10.1137/20m1333201.

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Boffi, Daniele, Franco Brezzi, and Michel Fortin. "Reduced symmetry elements in linear elasticity." Communications on Pure & Applied Analysis 8, no. 1 (2009): 95–121. http://dx.doi.org/10.3934/cpaa.2009.8.95.

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Cai, Zhiqiang, and Gerhard Starke. "Least-Squares Methods for Linear Elasticity." SIAM Journal on Numerical Analysis 42, no. 2 (January 2004): 826–42. http://dx.doi.org/10.1137/s0036142902418357.

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