Thermal Stresses in Moderately Thick Elastic Plates

1959 ◽  
Vol 26 (3) ◽  
pp. 432-436
Author(s):  
B. E. Gatewood
Keyword(s):  
Temperature Gradient ◽  
Plane Stress ◽  
Thermal Stresses ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Elastic Plates ◽  
Plate Length ◽  
Moderately Thick Plates ◽  
Thick Elastic Plates ◽  
Thickness Variable

Abstract The three-dimensional stresses in the plate are investigated without using the plane-stress or plane-strain assumptions, the thickness of the plate being limited so that the normal stress in the thickness direction can be taken as a polynomial in the thickness variable. The temperature is taken as a polynomial in the thickness variable but with relatively large, though restricted, gradients with respect to the co-ordinates of the plane of the plate. For the case of the temperature constant in thickness variable, the stresses in the plane of the plate are presented as the plane-stress solution plus correcting terms due to the plate thickness, where the correcting terms involve the product of the temperature gradient and the ratio of the plate thickness to the plate length in the direction of the temperature gradient. In many cases the corrections are small even for moderately thick plates.

Three-Dimensional Elastic Stress Fields of Finite Thickness Plates with Elliptical Hole

2007 ◽  
Vol 353-358 ◽  
pp. 74-77
Author(s):  
Zheng Yang ◽  
Chong Du Cho ◽  
Ting Ya Su ◽  
Chang Boo Kim ◽  
Hyeon Gyu Beom
Keyword(s):  
Stress Concentration ◽  
Plane Strain ◽  
Plane Stress ◽  
Stress Concentration Factor ◽  
Concentration Factor ◽  
Maximum Stress ◽  
Elastic Stress ◽  
Finite Thickness ◽  
Plate Thickness ◽  
Three Dimensional

Based on detailed three-dimensional finite element analyses, elastic stress and strain field of ellipse major axis end in plates with different thickness and ellipse configurations subjected to uniaxial tension have been investigated. The plate thickness and ellipse configuration have obvious effects on the stress concentration factor, which is higher in finite thickness plates than in plane stress and plane strain cases. The out-of-plane stress constraint factor tends the maximum on the mid-plane and approaches to zero on the free plane. Stress concentration factors distribute ununiformly through the plate thickness, the value and location of maximum stress concentration factor depend on the plate thickness and the ellipse configurations. Both stress concentration factor in the middle plane and the maximum stress concentration factor are greater than that under plane stress or plane strain states, so it is unsafe to suppose a tensioned plate with finite thickness as one undergone plane stress or plane strain. For the sharper notch, the influence of three-dimensional stress state on the SCF must be considered.

Photoelastic Investigation of a Thick Plate With a Transverse Crack

1975 ◽  
Vol 42 (1) ◽  
pp. 9-14 ◽  
Author(s):  
G. Villarreal ◽  
G. C. Sih ◽  
R. J. Hartranft
Keyword(s):  
Plane Stress ◽  
Thick Plate ◽  
Three Dimensional ◽  
Stress Variation ◽  
Thick Plates ◽  
Stress Components ◽  
Photoelastic Investigation ◽  
Photoelastic Analysis ◽  
Moderately Thick Plates ◽  
Transverse Variation

The purpose of this investigation was to experimentally test one of the assumptions of a recent modified version of the theory of generalized plane stress. The form postulated by the theory for the stress variation through the thickness of a plate containing a crack will be compared with that obtained by three-dimensional photoelastic analysis. Specimens covering the range from thin to moderately thick plates were examined by the frozen stress technique. The experimentally measured transverse variation of the in-plane stress components σx and σy was in excellent agreement with that postulated by the theory.

scholarly journals A non-local asymptotic theory for thin elastic plates

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170249 ◽  
Author(s):  
R. Chebakov ◽  
J. Kaplunov ◽  
G. A. Rogerson
Keyword(s):  
Boundary Layers ◽  
Constitutive Relations ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Low Frequency ◽  
Elastic Plates ◽  
Local Behaviour ◽  
Elasticity Equations ◽  
Plate Equations ◽  
Non Local

The three-dimensional dynamic non-local elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size. The integral constitutive relations, incorporating the variation of an exponential non-local kernel across the thickness, are adopted. Long-wave low-frequency approximations are derived for both bending and extensional motions. Boundary layers specific for non-local behaviour are revealed near the plate faces. It is established that the effect of the boundary layers leads to the first-order corrections to the bending and extensional stiffness in the classical two-dimensional plate equations.

scholarly journals Edge wrinkling in elastically supported pre-stressed incompressible isotropic plates

2016 ◽  
Vol 472 (2193) ◽  
pp. 20160410 ◽  
Author(s):  
Michel Destrade ◽  
Yibin Fu ◽  
Andrea Nobili
Keyword(s):  
Plate Thickness ◽  
Three Dimensional ◽  
Organic Thin Films ◽  
Homogeneous State ◽  
Elastic Plates ◽  
Winkler Elastic Foundation ◽  
Soft Biological Materials ◽  
Elastically Supported ◽  
The Second Variation ◽  
Three Dimensional Elasticity

The equations governing the appearance of flexural static perturbations at the edge of a semi-infinite thin elastic isotropic plate, subjected to a state of homogeneous bi-axial pre-stress, are derived and solved. The plate is incompressible and supported by a Winkler elastic foundation with, possibly, wavenumber dependence. Small perturbations superposed onto the homogeneous state of pre-stress, within the three-dimensional elasticity theory, are considered. A series expansion of the plate kinematics in the plate thickness provides a consistent expression for the second variation of the potential energy, whose minimization gives the plate governing equations. Consistency considerations supplement a constraint on the scaling of the pre-stress so that the classical Kirchhoff–Love linear theory of pre-stretched elastic plates is retrieved. Moreover, a scaling constraint for the foundation stiffness is also introduced. Edge wrinkling is investigated and compared with body wrinkling. We find that the former always precedes the latter in a state of uni-axial pre-stretch, regardless of the foundation stiffness. By contrast, a general bi-axial pre-stretch state may favour body wrinkling for moderate foundation stiffness. Wavenumber dependence significantly alters the predicted behaviour. The results may be especially relevant to modelling soft biological materials, such as skin or tissues, or stretchable organic thin-films, embedded in a compliant elastic matrix.

ON THREE-DIMENSIONAL INITIAL-BOUNDARY PROBLEMS OF THICK ELASTIC PLATES’ DYNAMICS

2021 ◽  
Vol 3 ◽  
pp. 34-49
Author(s):  
Vladimir Stoyan ◽  
◽  
Sergey Voloshchuk ◽  
Keyword(s):  
Root Mean Square ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Initial Boundary ◽  
Accuracy Evaluation ◽  
Mean Square ◽  
Elastic Plates ◽  
Boundary Problems ◽  
Computational Mathematics ◽  
Thick Elastic Plates

Complex problems of three-dimensional elasticity theory for thick elastic plates with arbitrary geometry of their lateral surface are solved. Analytical dependencies of the components of the elastic dynamic displacements’ field of the plate’s inner points from the boundary-surface external-dynamic disturbing factors, defined by continuous functions or their values’ vectors, are constructed. It is assumed, that these disturbances have a classically defined powerful character, or are specified by a certain number of differential transformations of the field’s components of the plate’s dynamic displacement points. The absence of quantitative and qualitative restrictions on the determined transformations of the initial-boundary problems of the considered plates’ dynamics makes it incorrect and unsolvable by methods of classical and computational mathematics. The methodology of root-mean square mathematical modeling of discretely and continuously specified observations for the initial-boundary plate’s condition by the system of modeling functions and their values’ vectors is proposed in the paper. Constructed in this way field’s components of spatial-dynamic displacements of the plate’s points, precisely satisfying classical Lyame equation, with the available information on its initial-boundary condition, are agreed according to the root-mean square criterion. The problem of the obtained solutions’ ambiguity is investigated, their accuracy evaluation in accordance with the information on the external- dynamic condition of the investigated plate is conducted. The plate’s dynamics in the particular mode, for cases of information lack on external- dynamic influences on it and under the conditions of its geometric background according to spatial coordinates. The computer realization of the obtained mathematical results is engineeringly simple and can be easily implemented with the help of well-known methods of computational mathematics.

Thermal Stresses in Bodies Exhibiting Temperature-Dependent Elastic Properties

1952 ◽  
Vol 19 (3) ◽  
pp. 350-354
Author(s):  
H. H. Hilton
Keyword(s):  
Temperature Gradient ◽  
Elastic Properties ◽  
Plane Stress ◽  
Material Properties ◽  
Thermal Stresses ◽  
Temperature Gradients ◽  
Temperature Dependent ◽  
Steady State Temperature ◽  
Specific Temperature ◽  
Walled Cylinder

Abstract Expressions are derived for thermal stresses and strains due to a steady-state temperature gradient in a thick-walled cylinder and a circular thin plate, made of a material having temperature-dependent elastic properties. Two numerical examples are computed for specific temperature gradients and temperature-dependent elastic properties, which yield results showing that the maximum thermal stresses are appreciably lower and the maximum thermal strains are larger than the corresponding values obtained for temperature-independent properties. The validity of the thermal plane-stress assumptions is investigated and it is shown that such solutions, regardless of whether the material properties are temperature-dependent or constant, are only approximations. The smaller the temperature gradient the more closely are the plane-stress assumptions satisfied.

Theory of laminated elastic plates I. isotropic laminae

1988 ◽  
Vol 324 (1582) ◽  
pp. 565-594 ◽  
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Plane Stress ◽  
Three Dimensional ◽  
Laminated Plates ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Dimensional Solution ◽  
Stress Theory

In this paper (part I) we establish a theory for stretching and bending of laminated elastic plates in which the laminae are different isotropic linearly elastic materials. The theory gives exact solutions of the three-dimensional elasticity equations that satisfy all the interface traction and displacement continuity conditions, with no traction on the lateral surfaces; the only restriction is that edge boundary conditions can be satisfied only in an average manner, rather than point by point. The method, which is based on a generalization of Michell’s exact plane stress theory, yields exact solutions for each lamina. These solutions are generated in a very straightforward manner by solutions of the approximate two-dimensional classical equations of laminate theory and contain sufficient arbitrary constants to enable all the continuity and lateral surface boundary conditions to be satisfied. The values of the constants depend only on the lamina thicknesses and the elastic constants. Thus, for a given laminate and for any boundary-value problem , it is necessary only to solve the appropriate two-dimensional plane problem, and the corresponding exact three-dimensional laminate solution follows by straightforward substitutions. The two-dimensional solution may be derived by any of the available methods, including numerical methods. An important feature of the theory is that it determines the interfacial shearing tractions, as well as the in-plane stress components. The procedure is illustrated by applying the theory to three problems involving stretching and bending of laminated plates containing circular holes.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

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