Relevant bibliographies by topics / Elasticity - Cauchy's Problem

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Elasticity - Cauchy's Problem'

Author: Grafiati
Published: 4 June 2021
Last updated: 6 September 2023

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Journal articles on the topic "Elasticity - Cauchy's Problem"

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Makhmudov, O. I., and I. E. Niyozov. "Cauchy Problem for Dynamic Elasticity Equations." Differential Equations 56, no. 9 (September 2020): 1130–39. http://dx.doi.org/10.1134/s0012266120090037.

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Ang, Dang Dinh, and Nguyen Dung. "A Cauchy like problem in plane elasticity." Vietnam Journal of Mechanics 29, no. 3 (September 22, 2007): 245–48. http://dx.doi.org/10.15625/0866-7136/29/3/5536.

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Let \(\Omega \) be a bounded domain in the plane, representing an elastic body. Let \(\Gamma_0 \) be a portion of the boundary \(\Gamma\) of \(\Omega \), \(\Gamma_0 \) being assumed to be paralled to the \(x\)-axis. It is proposed to determine the stress field in \(\Omega \) from the displacements and surface stresses given on \(\Gamma_0 \). Under the assumption of plane stress, it is shown that \(u_x + u_y\) is a harmonic function. An Airy stress function is introduced, from which the stress field is computed.
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Koenemann, Falk H. "Unorthodox Thoughts about Deformation, Elasticity, and Stress." Zeitschrift für Naturforschung A 56, no. 12 (December 1, 2001): 794–808. http://dx.doi.org/10.1515/zna-2001-1202.

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Abstract The nature of elastic deformation is examined in the light of the potential theory. The concepts and mathematical treatment of elasticity and the choice of equilibrium conditions are adopted from the mechanics of discrete bodies, e. g., celestial mechanics; they are not applicable to a change of state. By nature, elastic deformation is energetically a Poisson problem since the buildup of an elastic potential implies a change of the energetic state in the sense of thermodynamics. In the Euler-Cauchy theory, elasticity is treated as a Laplace problem, implying that no change of state occurs, and there is no clue in the Euler-Cauchy approach that it was ever considered as one. The Euler-Cauchy theory of stress is incompatible with the potential theory and with the nature of the problem; it is therefore wrong. The key point in the understanding of elasticity is the elastic potential.
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Nezam, Ali. "Numerical Results of Sub-Cauchy Problem for Linear Elasticity." International Journal for Research in Applied Science and Engineering Technology 7, no. 6 (June 30, 2019): 1693–99. http://dx.doi.org/10.22214/ijraset.2019.6284.

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Makhmudov, O. I., and I. E. Niezov. "A cauchy problem for the system of elasticity equations." Differential Equations 36, no. 5 (May 2000): 749–54. http://dx.doi.org/10.1007/bf02754234.

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Marin, L., L. Elliott, D. B. Ingham, and D. Lesnic. "Boundary element method for the Cauchy problem in linear elasticity." Engineering Analysis with Boundary Elements 25, no. 9 (October 2001): 783–93. http://dx.doi.org/10.1016/s0955-7997(01)00062-5.

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Baranger, Thouraya N., and Stéphane Andrieux. "An optimization approach for the Cauchy problem in linear elasticity." Structural and Multidisciplinary Optimization 35, no. 2 (May 3, 2007): 141–52. http://dx.doi.org/10.1007/s00158-007-0123-5.

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Cheng, Jin, Victor Isakov, Masahiro Yamamoto, and Qi Zhou. "Lipschitz stability in the lateral Cauchy problem for elasticity system." Journal of Mathematics of Kyoto University 43, no. 3 (2003): 475–501. http://dx.doi.org/10.1215/kjm/1250283691.

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Joumaa, Hady, and Martin Ostoja-Starzewski. "Stress and couple-stress invariance in non-centrosymmetric micropolar planar elasticity." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2134 (May 11, 2011): 2896–911. http://dx.doi.org/10.1098/rspa.2010.0660.

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The stress-invariance problem for a chiral (non-centrosymmetric) micropolar material model is explored in two different planar problems: the in-plane and the anti-plane problems. This material model grasps direct coupling between the Cauchy-type and Cosserat-type (or micropolar) effects in Hooke's law. An identical strategy of invariance is set for both problems, leading to a remarkable similarity in their results. For both problems, the planar components of stress and couple-stress undergo strong invariance, while their out-of-plane counterparts can only attain weak invariance, which restricts all compliance moduli transformations to a linear type. As an application, when heterogeneous (composite) materials are subjected to weak invariance, their effective (volume-averaged) compliance moduli undergo the same linear shift as that of the moduli of the local phases forming the material, independently of the microstructure, geometry and phase distribution. These analytical results constitute a valuable means to validate computational procedures that handle this particular type of material model.
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Niyozov, I. E., and O. I. Makhmudov. "The Cauchy problem of the moment elasticity theory in R m." Russian Mathematics 58, no. 2 (February 2014): 24–30. http://dx.doi.org/10.3103/s1066369x14020042.

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Dissertations / Theses on the topic "Elasticity - Cauchy's Problem"

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O, Makhmudov, I. Niyozov, and Nicolai Tarkhanov. "The cauchy problem of couple-stress elasticity." Universität Potsdam, 2006. http://opus.kobv.de/ubp/volltexte/2009/3007/.

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We study the Cauchy problem for the oscillation equation of the couple-stress theory of elasticity in a bounded domain in R3. Both the displacement and stress are given on a part S of the boundary of the domain. This problem is densely solvable while data of compact support in the interior of S fail to belong to the range of the problem. Hence the problem is ill-posed which makes the standard calculi of Fourier integral operators inapplicable. If S is real analytic the Cauchy-Kovalevskaya theorem applies to guarantee the existence of a local solution. We invoke the special structure of the oscillation equation to derive explicit conditions of global solvability and an approximation solution.
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I, Makhmudov O., and Niyozov I. E. "Regularization of the Cauchy Problem for the System of Elasticity Theory in R up (m)." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/2998/.

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Jung, Michael, and Todor D. Todorov. "On the Convergence Factor in Multilevel Methods for Solving 3D Elasticity Problems." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601510.

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The constant gamma in the strengthened Cauchy-Bunyakowskii-Schwarz inequality is a basic tool for constructing of two-level and multilevel preconditioning matrices. Therefore many authors consider estimates or computations of this quantity. In this paper the bilinear form arising from 3D linear elasticity problems is considered on a polyhedron. The cosine of the abstract angle between multilevel finite element subspaces is computed by a spectral analysis of a general eigenvalue problem. Octasection and bisection approaches are used for refining the triangulations. Tetrahedron, pentahedron and hexahedron meshes are considered. The dependence of the constant $\gamma$ on the Poisson ratio is presented graphically.
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Reddy, Annem Narayana. "Two Inverse Problems In Linear Elasticity With Applications To Force-Sensing And Mechanical Characterization." Thesis, 2010. https://etd.iisc.ac.in/handle/2005/2075.

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Two inverse problems in elasticity are addressed with motivation from cellular biomechanics. The first application is computation of holding forces on a cell during its manipulation and the second application is estimation of a cell’s interior elastic mapping (i.e., estimation of inhomogeneous distribution of stiffness) using only boundary forces and displacements. It is clear from recent works that mechanical forces can play an important role in developmental biology. In this regard, we have developed a vision-based force-sensing technique to estimate forces that are acting on a cell while it is manipulated. This problem is connected to one inverse problem in elasticity known as Cauchy’s problem in elasticity. Geometric nonlinearity under noisy displacement data is accounted while developing the solution procedures for Cauchy’s problem. We have presented solution procedures to the Cauchy’s problem under noisy displacement data. Geometric nonlinearity is also considered in order to account large deformations that the mechanisms (grippers) undergo during the manipulation. The second inverse problem is connected to elastic mapping of the cell. We note that recent works in biomechanics have shown that the disease state can alter the gross stiffness of a cell. Therefore, the pertinent question that one can ask is that which portion (for example Nucleus, cortex, ER) of the elastic property of the cell is majorly altered by the disease state. Mathematically, this question (estimation of inhomogeneous properties of cell) can be answered by solving an inverse elastic boundary value problem using sets of force-displacements boundary measurements. We address the theoretical question of number of boundary data sets required to solve the inverse boundary value problem.
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Reddy, Annem Narayana. "Two Inverse Problems In Linear Elasticity With Applications To Force-Sensing And Mechanical Characterization." Thesis, 2010. http://etd.iisc.ernet.in/handle/2005/2075.

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Two inverse problems in elasticity are addressed with motivation from cellular biomechanics. The first application is computation of holding forces on a cell during its manipulation and the second application is estimation of a cell’s interior elastic mapping (i.e., estimation of inhomogeneous distribution of stiffness) using only boundary forces and displacements. It is clear from recent works that mechanical forces can play an important role in developmental biology. In this regard, we have developed a vision-based force-sensing technique to estimate forces that are acting on a cell while it is manipulated. This problem is connected to one inverse problem in elasticity known as Cauchy’s problem in elasticity. Geometric nonlinearity under noisy displacement data is accounted while developing the solution procedures for Cauchy’s problem. We have presented solution procedures to the Cauchy’s problem under noisy displacement data. Geometric nonlinearity is also considered in order to account large deformations that the mechanisms (grippers) undergo during the manipulation. The second inverse problem is connected to elastic mapping of the cell. We note that recent works in biomechanics have shown that the disease state can alter the gross stiffness of a cell. Therefore, the pertinent question that one can ask is that which portion (for example Nucleus, cortex, ER) of the elastic property of the cell is majorly altered by the disease state. Mathematically, this question (estimation of inhomogeneous properties of cell) can be answered by solving an inverse elastic boundary value problem using sets of force-displacements boundary measurements. We address the theoretical question of number of boundary data sets required to solve the inverse boundary value problem.
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Lee, Cheuk Yu. "Fundamental solution based numerical methods for three dimensional problems: efficient treatments of inhomogeneous terms and hypersingular integrals." Phd thesis, 2016. http://hdl.handle.net/1885/117204.

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In recent years, fundamental solution based numerical methods including the meshless method of fundamental solutions (MFS), the boundary element method (BEM) and the hybrid fundamental solution based finite element method (HFS-FEM) have become popular for solving complex engineering problems. The application of such fundamental solutions is capable of reducing computation requirements by simplifying the domain integral to the boundary integral for the homogeneous partial differential equations. The resulting weak formulations, which are of lower dimensions, are often more computationally competitive than conventional domain-type numerical methods such as the finite element method (FEM) and the finite difference method (FDM). In the case of inhomogeneous partial differential equations arising from transient problems or problems involving body forces, the domain integral related to the inhomogeneous solutions term will need to be integrated over the interior domain, which risks losing the competitive edge over the FEM or FDM. To overcome this, a particular treatment to the inhomogeneous term is needed in the solution procedure so that the integral equation can be defined for the boundary. In practice, particular solutions in approximated form are usually applied rather than the closed form solutions, due to their robustness and readiness. Moreover, special numerical treatment may be required when evaluating stress directly on the domain surface which may give rise to hypersingular integral formulation. This thesis will discuss how the MFS and the BEM can be applied to the three-dimensional elastic problems subjected to body forces by introducing the compactly supported radial basis functions in addition to the efficient treatment of hypersingular surface integrals. The present meshless approach with the MFS and the compactly supported radial basis functions is later extended to solve transient and coupled problems for three-dimensional porous media simulation.
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Book chapters on the topic "Elasticity - Cauchy's Problem"

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Diaz, J. B. "Solution of the Singular Cauchy Problem for a Singular System of Partial Differential Equations in the Mathematical Theory of Dynamical Elasticity." In Teoria delle distribuzioni, 179–89. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-10967-6_4.

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Anand, Lallit, Ken Kamrin, and Sanjay Govindjee. "Isotropic linear elasticity." In Introduction to Mechanics of Solid Materials, 69–97. Oxford University PressOxford, 2022. http://dx.doi.org/10.1093/oso/9780192866073.003.0006.

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Abstract This chapter introduces the constitutive equation for a linear isotropic elastic material. The notion of a quadratic strain-energy function is used to derive the general form of the linear elastic stress-strain relation for fully anisotropic materials. The restriction of this relation for isotropic materials is then introduced in various forms, providing definitions for the common elastic constants—namely, shear modulus, bulk modulus, Young’s modulus and Poisson’s ratio; the interrelations between these moduli are also discussed. Thermoelastic stress-strain relations are presented for the isotropic case. Tables of values of elastic moduli for common materials are also included. The linear isotropic stress-strain relations are then used in the equations of equilibrium to generate the Navier–Cauchy displacement equations of equilibrium. Common boundary conditions are described, and the boundary-value problem for isotropic linear elasticity is stated.
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Sutton, Adrian P. "Hooke’s law and elastic constants." In Physics of Elasticity and Crystal Defects, 29–54. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198860785.003.0003.

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Hooke’s law and elastic constants are introduced. The symmetry of the elastic constant tensor follows from the symmetry of stress and strain tensors and the elastic energy density. The maximum number of independent elastic constants is 21 before crystal symmetry is considered, and this leads to the introduction of matrix notation. Neumann’s principle reduces the number of independent elastic constants in different crystal systems. It is proved that in isotropic elasticity there are only two independent elastic constants. The directional dependences of the three independent elastic constants in cubic crystalsare derived. The distinction between isothermal and adiabatic elastic constants is defined thermodynamically and shown to arise from anharmonicity of atomic interactions. Problems set 3involves the derivation of elastic constants atomistically, the numbers of independent elastic constants in non-cubic crystal symmetries, Cauchy relations, Cauchy pressure, invariants of the elastic constant tensorand compatibility stresses.
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Sutton, Adrian P. "Stress." In Physics of Elasticity and Crystal Defects, 9–28. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198860785.003.0002.

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The concept of stress is introduced in terms of interatomic forces acting through a plane, and in the Cauchy sense of a force per unit area on a plane in a continuum. Normal stresses and shear stresses are defined. Invariants of the stress tensor are derived and the von Mises shear stress is expressed in terms of them. The conditions for mechanical equilibrium in a continuum are derived, one of which leads to the stress tensor being symmetric. Stress is also shown to be the functional derivative of the elastic energy with respect to strain,which enables the stress tensor to be derived in models of interatomic forces. Adiabatic and isothermal stresses are distinguished thermodynamically and anharmonicity of atomic interactions is identified as the reason for their differences. Problems set 2 containsfour problems, one of which is based on Noll’s insightful analysis of stress and mechanical equilibrium.
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Crispo, Francesca. "A new proof of existence in the 𝐿³-setting of solutions to the Navier–Stokes Cauchy problem." In Interactions between Elasticity and Fluid Mechanics, 115–34. EMS Press, 2022. http://dx.doi.org/10.4171/esiam/3/5.

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Eller, M., V. Isakov, G. Nakamura, and D. Tataru. "Uniqueness and Stability in the Cauchy Problem for Maxwell and Elasticity Systems." In Nonlinear Partial Differential Equations and their Applications - Collège de France Seminar Volume XIV, 329–49. Elsevier, 2002. http://dx.doi.org/10.1016/s0168-2024(02)80016-9.

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Conference papers on the topic "Elasticity - Cauchy's Problem"

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Bolikulov, Furkat, and Ниёзов Икбол. "The Cauchy problem for the system of elasticity." In International scientific conference "Ufa autumn mathematical school - 2021". Baskir State University, 2021. http://dx.doi.org/10.33184/mnkuomsh1t-2021-10-06.70.

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MAKHMUDOV, O. I., and I. E. NIYOZOV. "REGULARIZATION OF THE CAUCHY PROBLEM FOR THE SYSTEM OF ELASTICITY THEORY IN Rm." In Proceedings of the 5th International ISAAC Congress. WORLD SCIENTIFIC, 2009. http://dx.doi.org/10.1142/9789812835635_0006.

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LIN, JUAN. "STABILITY OF CAUCHY TYPE INTEGRAL APPLIED TO THE FUNDAMENTAL PROBLEMS IN PLANE ELASTICITY." In Proceedings of the Third International Conference. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814327862_0032.

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Dyakonov, Radimir G., and Yuri M. Grigor'ev. "An analytical method of regularized solving of the ill-posed Cauchy problem in the elasticity theory." In 9TH INTERNATIONAL CONFERENCE ON MATHEMATICAL MODELING: Dedicated to the 75th Anniversary of Professor V.N. Vragov. AIP Publishing, 2021. http://dx.doi.org/10.1063/5.0043788.

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Momeni, S. Alireza, and Mohsen Asghari. "A Study on a Grade-One Type of Hypo-Elastic Models." In ASME 2014 12th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/esda2014-20123.

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In Hypo-elastic constitutive models an objective rate of the Cauchy stress tensor is expressed in terms of the current state of the stress and the deformation rate tensor D in a way that the dependency on the latter is a homogeneously linear one. In this work, a type of grade-one hypo-elastic models (i.e. models with linear dependency of the hypo-elasticity tensor on the stress) is considered for isotropic materials based on the objective corotational rates of stress. A positive real parameter denoted by n is involved in the considered type. Different values can be selected for this parameter, each selection leads to a specific model within the class of grade-one hypo-elasticity. The spin of the associated corotational rate is also dependent on the parameter n. In the special case of n=0, the corresponding hypo-elastic model reduces to a grade-zero one with the logarithmic rate of stress; noting that this rate is a corotational rate associated with the logarithmic spin tensor. Moreover, by choosing n=2, the model reduces to a grade-one hypo-elastic model with the Jaumann rate, i.e. the corotational rate associated with the vorticity spin tensor. As case studies, the simple shear problem is investigated with utilizing the considered type of hypo-elastic models with various values for parameter n, and the curves for the stress-shear response are depicted.
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Delale, F., and X. Long. "The General Fracture Problem of a Functionally Graded Thermal Barrier Coating (TBC) Bonded to a Substrate." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-60711.

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In this paper we consider the general fracture problem of a functionally graded thermal barrier coating (TBC) bonded to a substrate. Functionally Graded Materials (FGMs) used in TBCs are usually made from ceramics and metals. Ceramics provide thermal and corrosion resistance while metals provide the necessary fracture toughness and heat conductivity. The volume fractions of the constituents will usually vary from 100% ceramic at the surface to 0% at the interface continuously providing seamless bonding with the metal substrate. To study the general fracture problem in the TBC we consider an arbitrarily oriented crack in an FGM layer bonded to a half plane. The elastic properties of the FGM layer are assumed to vary exponentially, while those of the half plane are homogeneous. The elastic properties are continuous at the interface. As shown in [1], then the governing elasticity equations become partial differential equations with constant coefficients. Using the transform technique, and defining the crack surface displacement derivatives as the unknown auxiliary functions, the mixed-mode crack problem is reduced to a system of Cauchy type singular integral equations. It is shown that at the crack tips the stresses still possess the regular square-root singularity, making it possible to use the classical definition of stress intensity factors. The singular integral equations are solved numerically using a Gaussian type quadrature and the mode I and mode II stress intensity factors are calculated for various crack lengths and crack orientations. Also the crack surface displacements are computed for different crack inclinations. It is observed that the crack orientation, crack length and the nonhomogeneity parameter affect the stress intensity factors significantly.
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Mehraban, Arash, Jed Brown, Henry Tufo, Jeremy Thompson, Rezgar Shakeri, and Richard Regueiro. "Efficient Parallel Scalable Matrix-Free 3D High-Order Finite Element Simulation of Neo-Hookean Compressible Hyperelasticity at Finite Strain." In ASME 2021 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/imece2021-70768.

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Abstract The paper investigates matrix-free high-order implementation of finite element discretization with p-multigrid preconditioning for the compressible Neo-Hookean hyperelasticity problem at finite strain on unstructured 3D meshes in parallel. We consider two formulations for the matrix-free action of the Jacobian in Neo-Hookean hyperelasticity: (i) working in the reference configuration to define the second Piola-Kirchhoff tensor as a function of the Green-Lagrange strain S(E) (or equivalently, the right Cauchy-Green tensor C = I+2E), and (ii) working in the current configuration to define the Kirchhoff stress in terms of the left Cauchy-Green tensor τ(b). The proposed efficient algorithm utilizes the Portable, Extensible Toolkit for Scientific Computation (PETSc), along with the libCEED library for efficient compiler optimized tensor-product-basis computation to demonstrate an efficient nonlinear solution algorithm. We utilize p-multigrid preconditioning on the high-order problem with algebraic multigrid (AMG) on the assembled linear Q1 coarse grid operator. In contrast to classical geometric multigrid, also known as h-multigrid, each level in p-multigrid is related to a different approximation polynomial order p, instead of the element size h. A Chebyshev polynomial smoother is used on each multigrid level. AMG is then applied to the assembled Q1 (trilinear hexahedral elements), which allows low storage that can be efficiently used to accelerate convergence to a solution. For the compressible Neo-Hookean hyperelastic constitutive model we exploit the stored energy density function to compute the stored elastic energy density of the Neo-Hookean material as it relates to the deformation gradient. Based on our formulation, we consider four different algorithms each with different storage strategies. Algorithms 1 and 3 are implemented in the reference and current configurations respectively and store ∇Xξ, det(∇ξX), and ∇Xu. Algorithm 2 in the reference configuration stores, ∇Xξ, det(∇ξX), ∇Xu, C−1, and λ log (J). Algorithm 4, in the current configuration, stores det(∇ξX), ∇xξ, τ, and μ – λ log(J). x refers to the current coordinates, X to the reference coordinates, and ξ to the natural coordinates. We perform 3D bending simulations of a tube composed of aluminum (modulus of elasticity E = 69 GPa, Poisson’s ratio v = 0.3) using unstructured meshes and polynomials of order p = 1 through p = 4 under mesh refinement. We explore accuracy-time-cost tradeoffs for the prediction of strain energy across the range of polynomial degrees and Jacobian representations. In all cases, Algorithm 4 using the current configuration formulation outperforms the other three algorithms and requires less storage. Similar simulations for large deformation compressible Neo-Hookean hyperelasticity as applied to the same aluminum material are conducted with ABAQUS, a commercial finite element software package which is a state-of-the-art engineering software package for finite element simulations involving large deformation. The best results from the proposed implementations and ABAQUS are compared in the case of p = 2 on an Intel system with @2.4 GHz and 128 GB RAM. Algorithm 4 outperforms ABAQUS for polynomial degree p = 2.
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