Stress Concentrations in Three-Dimensional Electrostriction

1967 ◽  
Vol 34 (2) ◽  
pp. 431-436 ◽  
Author(s):  
T. E. Smith

Using the techniques employed in developing a Papkovich-Neuber representation for the displacement vector in classical elasticity, a particular integral of the kinematical equations of equilibrium for the uncoupled theory of electrostriction is developed. The particular integral is utilized in conjunction with the displacement potential function approach to problems of the theory of elasticity to obtain closed-form solutions of several stress concentration problems for elastic dielectrics. Under a prescribed uniform electric field at infinity, the problems of an infinite elastic dielectric having first a spherical cavity and then a rigid spherical inclusion are solved. The rigid spheroidal inclusion problem and the penny-shaped crack problem are also solved for the case where the prescribed field is parallel to their axes of revolution.

1995 ◽  
Vol 62 (2) ◽  
pp. 273-281 ◽  
Author(s):  
C. H. Kuo ◽  
L. M. Keer

The three-dimensional problem of a multilayered composite containing an arbitrarily oriented crack is considered in this paper. The crack problem is analyzed by the equivalent body force method, which reduces the problem to a set of singular integral equations. To compute the kernels of the integral equations, the stiffness matrix for the layered medium is formulated in the Hankel transformed domain. The transformed components of the Green’s functions and derivatives are determined by solving the stiffness matrix equations, and the kernels are evaluated by performing the inverse Hankel transform. The crack-opening displacements and the three modes of the stress intensity factor at the crack front are obtained by numerically solving the integral equations. Examples are given for a penny-shaped crack in a bimaterial and a three-material system, and for a semicircular crack in a single layer adhered to an elastic half-space.


2020 ◽  
Vol 22 (4) ◽  
pp. 1145-1156
Author(s):  
R. Selvamani ◽  
M. Mahaveer Sreejeyan ◽  
J. Rexy ◽  
B. Sriee Malvika

AbstractWave propagation in a thermo piezoelectric membrane immersed in an infinite fluid medium is discussed using three-dimensional linear theory of elasticity and thermos piezoelectricity. Three displacement potential functions are introduced to uncouple the equations of motion, heat and electric conduction equations. The frequency equations are obtained for longitudinal and flexural modes at the solid fluid interfacial boundary conditions. The numerical results are analyzed for PZT-4 material and the computed stress, strain, electric displacement and temperature distribution are presented in the form of dispersion curves and its characteristics are studied.


2015 ◽  
Vol 8 (1) ◽  
pp. 82-103
Author(s):  
Palaniyandi Ponnusamy

AbstractIn this paper, a mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of circular cross-sections immersed in inviscid fluid. The present study is based on the use of 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 modes of vibration and are studied based on Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity. The frequency equations of the coupled system consisting of cylinder and fluid are developed under the assumption of perfect-slip boundary conditions at the fluid-solid interfaces, which are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid. The computed non-dimensional frequencies are compared with Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity for longitudinal and flexural modes of vibrations. The dispersion curves are drawn for longitudinal and flexural modes of vibrations. Moreover, the dispersion of specific loss and damping factors are also analyzed for longitudinal and flexural modes of vibrations.


2000 ◽  
Vol 67 (4) ◽  
pp. 705-711 ◽  
Author(s):  
W. Q. Chen

This paper derives a general solution of the three-dimensional equations of transversely isotropic piezothermoelastic materials (crystal class, 6 mm). Two displacement functions are first introduced to simplify the basic equations and a general solution is then derived using the operator theory. For the static case, the proposed general solution is very simple in form and can be used easily in certain boundary value problems. An illustrative example is given in the paper by considering the symmetric crack problem of an arbitrary temperature applied over the faces of a flat crack in an infinite space. The governing integro-differential equations of the problem are derived. It is found that exact expressions for the piezothermoelastic field for a penny-shaped crack subject to a uniform temperature can be obtained in terms of elementary functions. [S0021-8936(00)01704-9]


2021 ◽  
Vol 16 (1) ◽  
Author(s):  
Annarita Perillo ◽  
Valeria Landoni ◽  
Alessia Farneti ◽  
Giuseppe Sanguineti

Abstract Purpose The purpose of this study is to evaluate inter- and intra-fraction organ motion as well as to quantify clinical target volume (CTV) to planning target volume (PTV) margins to be adopted in the stereotactic treatment of early stage glottic cancer. Methods and materials Stereotactic body radiotherapy (SBRT) to 36 Gy in 3 fractions was administered to 23 patients with early glottic cancer T1N0M0. Patients were irradiated with a volumetric intensity modulated arc technique delivered with 6 MV FFF energy. Each patient underwent a pre-treatment cone beam computed tomography (CBCT) to correct the setup based on the thyroid cartilage position. Imaging was repeated if displacement exceeded 2 mm in any direction. CBCT imaging was also performed after each treatment arc as well as at the end of the delivery. Swallowing was allowed only during the beam-off time between arcs. CBCT images were reviewed to evaluate inter- and intra-fraction organ motion. The relationships between selected treatment characteristics, both beam-on and delivery times as well as organ motion were investigated. Results For the population systematic (Ʃ) and random (σ) inter-fraction errors were 0.9, 1.3 and 0.6 mm and 1.1, 1.3 and 0.7 mm in the left-right (X), cranio-caudal (Y) and antero-posterior (Z) directions, respectively. From the analysis of CBCT images acquired after treatment, systematic (Ʃ) and random (σ) intra-fraction errors resulted 0.7, 1.6 and 0.7 mm and 1.0, 1.5 and 0.6 mm in the X, Y and Z directions, respectively. Margins calculated from the intra-fraction errors were 2.4, 5.1 and 2.2 mm in the X, Y and Z directions respectively. A statistically significant difference was found for the displacement in the Z direction between patients irradiated with > 2 arcs versus ≤ 2 arcs, (MW test, p = 0.038). When analyzing mean data from CBCT images for the whole treatment, a significant correlation was found between the time of delivery and the three dimensional displacement vector (r = 0.489, p = 0.055), the displacement in the Y direction (r = 0.553, p = 0.026) and the subsequent margins to be adopted (r = 0.626, p = 0.009). Finally, displacements and the subsequent margins to be adopted in Y direction were significantly greater for treatments with more than 2 arcs (MW test p = 0.037 and p = 0.019, respectively). Conclusions In the setting of controlled swallowing during treatment delivery, intra-fraction motion still needs to be taken into account when planning with estimated CTV to PTV margins of 3, 5 and 3 mm in the X, Y and Z directions, respectively. Selected treatments may require additional margins.


Author(s):  
J. R. Beisheim ◽  
G. B. Sinclair ◽  
P. J. Roache

Current computational capabilities facilitate the application of finite element analysis (FEA) to three-dimensional geometries to determine peak stresses. The three-dimensional stress concentrations so quantified are useful in practice provided the discretization error attending their determination with finite elements has been sufficiently controlled. Here, we provide some convergence checks and companion a posteriori error estimates that can be used to verify such three-dimensional FEA, and thus enable engineers to control discretization errors. These checks are designed to promote conservative error estimation. They are applied to twelve three-dimensional test problems that have exact solutions for their peak stresses. Error levels in the FEA of these peak stresses are classified in accordance with: 1–5%, satisfactory; 1/5–1%, good; and <1/5%, excellent. The present convergence checks result in 111 error assessments for the test problems. For these 111, errors are assessed as being at the same level as true exact errors on 99 occasions, one level worse for the other 12. Hence, stress error estimation that is largely reasonably accurate (89%), and otherwise modestly conservative (11%).


2019 ◽  
Vol 968 ◽  
pp. 496-510
Author(s):  
Anatoly Grigorievich Zelensky

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.


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