Rods, plates and shells

1968 ◽  
Vol 64 (3) ◽  
pp. 895-913 ◽  
Author(s):  
A. E. Green ◽  
N. Laws ◽  
P. M. Naghdi
Keyword(s):  
Continuum Mechanics ◽  
Three Dimensional ◽  
Elastic Rods ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Starting Point ◽  
Non Linear ◽  
Plates And Shells

AbstractWe discuss non-linear thermodynamical theories of rods and shells using the three-dimensional theory of classical continuum mechanics as a starting point. The three-dimensional theory is reduced to a two-dimensional theory for a shell, or plate, and a one-dimensional theory for a rod by employing an exact expansion for the displacement but an approximation for the temperature. For elastic rods and shells a method of approximation is suggested which brings the respective theories into correspondence with those of Green and Laws (1) and Green, Naghdi and Wain-wright(2).

Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates

1951 ◽  
Vol 18 (1) ◽  
pp. 31-38 ◽  
Author(s):  
R. D. Mindlin
Keyword(s):  
Uniqueness Theorem ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Dimensional Theory ◽  
Rotatory Inertia ◽  
One Dimensional ◽  
Edge Conditions

Abstract A two-dimensional theory of flexural motions of isotropic, elastic plates is deduced from the three-dimensional equations of elasticity. The theory includes the effects of rotatory inertia and shear in the same manner as Timoshenko’s one-dimensional theory of bars. Velocities of straight-crested waves are computed and found to agree with those obtained from the three-dimensional theory. A uniqueness theorem reveals that three edge conditions are required.

scholarly journals Derivation of the nonlinear bending–torsion model for a junction of elastic rods

2012 ◽  
Vol 142 (3) ◽  
pp. 633-664 ◽  
Author(s):  
Josip Tambača ◽  
Igor Velčić
Keyword(s):  
Cross Sections ◽  
Minimization Problem ◽  
Three Dimensional ◽  
Contact Forces ◽  
Elastic Rods ◽  
Nonlinear Bending ◽  
One Dimensional ◽  
Starting Point ◽  
Γ Convergence ◽  
The One

We derive the one-dimensional bending–torsion equilibrium model for the junction of straight rods. The starting point is a three-dimensional nonlinear elasticity equilibrium problem written as a minimization problem for a union of thin, rod-like bodies. By taking the limit as the thickness of the three-dimensional rods tends to zero, and by using ideas from the theory of Γ-convergence, we find that the resulting model consists of the union of the usual one-dimensional nonlinear bending–torsion rod models which satisfy the following transmission conditions at the junction point: continuity of displacement and rotation of the cross-sections; balance of contact forces and contact couples.

Membrane theory

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Three Dimensional ◽  
Thin Sheets ◽  
Two Dimensional ◽  
Leading Order ◽  
Membrane Theory ◽  
Dimensional Theory ◽  
Small Thickness

This chapter develops two-dimensional membrane theory as a leading order small-thickness approximation to the three-dimensional theory for thin sheets. Applications to axisymmetric equilibria are developed in detail, and applied to describe the phenomenon of bulge propagation in cylinders.

Transformations of two-dimensional layered zinc phosphates to three-dimensional and one-dimensional structures

2002 ◽  
Vol 12 (4) ◽  
pp. 1044-1052 ◽  
Author(s):  
Amitava Choudhury ◽  
S. Neeraj ◽  
Srinivasan Natarajan ◽  
C. N. R. Rao
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
One Dimensional ◽  
Zinc Phosphates

Mine Impact Burial Prediction From One to Three Dimensions

2008 ◽  
Vol 62 (1) ◽  
Author(s):  
Peter C. Chu
Keyword(s):  
Three Dimensional ◽  
Time History ◽  
Three Dimensions ◽  
Vertical Position ◽  
Burial Depth ◽  
Prediction Errors ◽  
Two Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Dimensional Models

The Navy’s mine impact burial prediction model creates a time history of a cylindrical or a noncylindrical mine as it falls through air, water, and sediment. The output of the model is the predicted mine trajectory in air and water columns, burial depth/orientation in sediment, as well as height, area, and volume protruding. Model inputs consist of parameters of environment, mine characteristics, and initial release. This paper reviews near three decades’ effort on model development from one to three dimensions: (1) one-dimensional models predict the vertical position of the mine’s center of mass (COM) with the assumption of constant falling angle, (2) two-dimensional models predict the COM position in the (x,z) plane and the rotation around the y-axis, and (3) three-dimensional models predict the COM position in the (x,y,z) space and the rotation around the x-, y-, and z-axes. These models are verified using the data collected from mine impact burial experiments. The one-dimensional model only solves one momentum equation (in the z-direction). It cannot predict the mine trajectory and burial depth well. The two-dimensional model restricts the mine motion in the (x,z) plane (which requires motionless for the environmental fluids) and uses incorrect drag coefficients and inaccurate sediment dynamics. The prediction errors are large in the mine trajectory and burial depth prediction (six to ten times larger than the observed depth in sand bottom of the Monterey Bay). The three-dimensional model predicts the trajectory and burial depth relatively well for cylindrical, near-cylindrical mines, and operational mines such as Manta and Rockan mines.

Theory of ultrasonic attenuation in one and two dimensional metals

1976 ◽  
Vol 54 (14) ◽  
pp. 1454-1460 ◽  
Author(s):  
T. Tiedje ◽  
R. R. Haering
Keyword(s):  
Ultrasonic Attenuation ◽  
Three Dimensional ◽  
Electronic Systems ◽  
Two Dimensional ◽  
Temperature Dependent ◽  
One Dimensional ◽  
The Difference

The theory of ultrasonic attenuation in metals is extended so that it applies to quasi one and two dimensional electronic systems. It is shown that the attenuation in such systems differs significantly from the well-known results for three dimensional systems. The difference is particularly marked for one dimensional systems, for which the attenuation is shown to be strongly temperature dependent.

Numerical exploration on snap buckling of a pre-stressed hemispherical gridshell

2021 ◽  
pp. 1-11
Author(s):  
Weicheng Huang ◽  
Longhui Qin ◽  
Qiang Chen
Keyword(s):  
Three Dimensional ◽  
Energy Functional ◽  
Elastic Rods ◽  
Geometrically Nonlinear ◽  
Local Response ◽  
Micro Electro Mechanical Systems ◽  
One Dimensional ◽  
Numerical Exploration ◽  
Planar Grid ◽  
Fundamental Understanding

Abstract Motivated by the observations of snap-through phenomena in pre-stressed strips and curved shells, we numerically investigate the snapping of a pre-buckled hemispherical gridshell under apex load indentation. Our experimentally validated numerical framework on elastic gridshell simulation combines two components: (i) Discrete Elastic Rods method, for the geometrically nonlinear description of one dimensional rods; and (ii) a naive penalty-based energy functional, to perform the non-deviation condition between two rods at joint. An initially planar grid of slender rods can be actuated into a three dimensional hemispherical shape by loading its extremities through a prescribed path, known as buckling induced assembly; next, this pre-buckled structure can suddenly change its bending direction at some threshold points when compressing its apex to the other side. We find that the hemispherical gridshell can undergo snap-through buckling through two different paths based on two different apex loading conditions. The first critical snap-through point slightly increases as the number of rods in gridshell structure becomes denser, which emphasizes the mechanically nonlocal property in hollow grids, in contrast to the local response of continuum shells. The findings may bridge the gap among rods, grids, knits, and shells, for a fundamental understanding of a group of thin elastic structures, and inspire the design of novel micro-electro-mechanical systems and functional metamaterials.

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