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.

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.

Extensional Waves in a Sandwich Plate With Interface Slip

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Peng Li ◽  
Feng Jin ◽  
Weiqiu Chen ◽  
Jiashi Yang
Keyword(s):  
Sandwich Plate ◽  
Collective Motion ◽  
Three Dimensional ◽  
Elastic Plates ◽  
Plate Equations ◽  
Elasticity Solutions ◽  
Interface Elasticity ◽  
Shear Motion ◽  
3D Elasticity ◽  
The Individual

The two-dimensional (2D) equations for thin elastic plates are used to study extensional motions of a sandwich plate with weak interfaces. The interfaces are governed by the shear-slip model that possesses interface elasticity and allows for a discontinuity of the tangential displacements at the interfaces. Equations for the individual layers of the sandwich plate are coupled by the interface conditions. Through a procedure initiated by Mindlin, the layer equations can be written into equations for the collective motion of the layers representing the extensional motion of the sandwich plate, and equations for the relative motions of the layers with respect to each other representing the symmetric thickness-shear motion of the sandwich plate. The use of plate equations results in relatively simpler models compared to the equations of three-dimensional (3D) elasticity. Solutions to a few useful problems are presented. These include the propagation of straight-crested waves in an unbounded plate with weak interfaces, the reflection of extensional waves at the joint between a perfectly bonded sandwich plate and a sandwich plate with weak interfaces, and the vibration of a finite sandwich plate with weak interfaces.

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 the general homogenization of von Kármán plate equations from three-dimensional nonlinear elasticity

2016 ◽  
Vol 15 (01) ◽  
pp. 1-49 ◽  
Author(s):  
Igor Velčić
Keyword(s):  
Nonlinear Elasticity ◽  
Three Dimensional ◽  
Von Karman ◽  
Elasticity Equations ◽  
Plate Equations ◽  
Von Kármán Plate ◽  
Von Kármán ◽  
Three Dimensional Elasticity

Starting from three-dimensional elasticity equations we derive the model of the homogenized von Kármán plate by means of [Formula: see text]-convergence. This generalizes the recent results, where the material oscillations were assumed to be periodic.

Effects of Three-Dimensional Pressure Gradients on High-Speed Turbulent Boundary Layers

2021 ◽  
Author(s):  
Scott J. Peltier ◽  
Brian E. Rice ◽  
Ethan Johnson ◽  
Venkateswaran Narayanaswamy ◽  
Marvin E. Sellers
Keyword(s):  
Boundary Layers ◽  
High Speed ◽  
Three Dimensional ◽  
Turbulent Boundary Layers ◽  
Pressure Gradients

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.

scholarly journals Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Energy Density ◽  
Constitutive Model ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
General Method

AbstractIn this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

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