scholarly journals Large elastic deformations of isotropic materials. V. The problem of flexure

1949 ◽  
Vol 195 (1043) ◽  
pp. 463-473 ◽  
Keyword(s):  
Energy Function ◽  
Incompressible Material ◽  
Curved Surface ◽  
Curved Surfaces ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Axis Ii ◽  
Strain Invariants ◽  
Deformed State ◽  
Derivatives Of

A cuboid of highly elastic incompressible material, whose stored-energy function W is a function of the strain invariants, has its edges parallel to the axes x, y and z of a rectangular Cartesian co-ordinate system. It can be bent so that: (i) every plane, initially normal to the x -axis, becomes part of the curved surface of a cylinder whose axis is the z -axis; (ii) every plane, initially normal to the y -axis, becomes a plane containing the z -axis; (iii) there is no displacement parallel to the z -axis. It is found that such a state of flexure can be maintained by the application of surface tractions only, and these are calculated explicitly in terms of the derivatives of W with respect to the strain invariants. The surface tractions are normal to the surfaces on which they act, in their deformed state. Those acting on the surfaces initially normal to the x -axis are uniform over each of these surfaces. The assumption is then made that the stored-energy function W has the form, originally suggested by Mooney (1940), for rubber, W = C 1 ( λ 2 1 + λ 2 2 + λ 2 3 -3) + C 2 ( λ 2 2 λ 2 3 + λ 2 3 λ 2 1 + λ 2 1 λ 2 2 -3), where C 1 and C 2 are physical constants for the material and λ 1 , λ 2 , λ 3 are the principal extension ratios. For this case—and therefore for the incompressible neo-Hookean material (Rivlin 1948 a, b, c ), which is obtained from this by putting C 2 = 0—it is found that the flexure can be maintained without the application of surface tractions to the curved surface, provided that 2( a 1 - a 2 ) ( r 1 r 2 ) ½ = r 2 1 - r 2 2 , where ( a 1 - a 2 ) is the initial dimension of the cuboid, parallel to the x -axis, and r 1 and r 2 are the radii of the curved surfaces. When this condition is satisfied, the system of surface tractions applied to a boundary initially normal to the y -axis is equivalent to a couple M , proportional to ( C 1 + C 2 ). It is also found that the surface tractions applied to a boundary normal to the z -axis has a resultant F 2 proportional to ( C 1 - C 2 ).

Large elastic deformations of isotropic materials VI. Further results in the theory of torsion, shear and flexure

1949 ◽  
Vol 242 (845) ◽  
pp. 173-195 ◽  
Keyword(s):  
Elastic Material ◽  
Simple Form ◽  
Energy Function ◽  
Simple Extension ◽  
Thick Sheet ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Special Cases ◽  
Strain Invariants ◽  
Uniform Extension

The forces necessary to produce certain simple types of deformation in a tube of incompressible, highly elastic material, isotropic in its undeformed state, are discussed. The first type of deformation may be considered to be produced by the following three successive simpler deformations: (i) a uniform simple extension, (ii) a uniform inflation of the tube, in which its length remains constant, and (iii) a uniform simple torsion, in which planes perpendicular to the axis of the tube are rotated in their own plane through an angle proportional to their distance from one end of the tube. Certain special cases of this deformation are considered in greater detail employing a simple stored-energy function of the form lf=C'1(/1-3)+C2(/2-3), where Cx and C2 are physical constants for the material and Ix and /2 are the strain invariants. The second type of deformation considered is that in which the simpler deformations (i) and (ii) mentioned above are followed successively by simple shears about the axis of the tube and parallel to it. The forces which must be applied are calculated for the simple form of stored-energy function given above. Finally, the simultaneous simple flexure and uniform extension normal to the plane of flexure of a thick sheet is discussed, and a number of the results obtained in a previous paper (Rivlin 19496) are generalized.

scholarly journals Large elastic deformations of isotropic materials IV. further developments of the general theory

1948 ◽  
Vol 241 (835) ◽  
pp. 379-397 ◽  
Keyword(s):  
Energy Function ◽  
Equations Of Motion ◽  
Incompressible Material ◽  
Surface Forces ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Incompressible Materials ◽  
Scalar Invariants ◽  
Compressible Materials ◽  
Motion Boundary

The equations of motion, boundary conditions and stress-strain relations for a highly elastic material can be expressed in terms of the stored-energy function. This has been done in part I of this series (Rivlin 1948 a ), for both the cases of compressible and incompressible materials, following the methods given by E. & F. Cosserat for compressible materials. The stored-energy function may be defined for a particular material in terms of the invariants of strain. The form in which the equations of motion, etc., are deduced, in the previous paper, does not permit the evaluation of the forces necessary to produce a specified deformation unless the actual expression for the stored-energy function in terms of the scalar invariants of the strain is introduced. In the present paper, the equations are transformed into forms more suitable for carrying out such an explicit evaluation. As examples, the surface forces necessary to produce simple shear in a cuboid of either compressible or incompressible material and those required to produce simple torsion in a right-circular cylinder of incompressible material are derived.

Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber

1951 ◽  
Vol 243 (865) ◽  
pp. 251-288 ◽  
Keyword(s):  
Energy Function ◽  
Pure Shear ◽  
Thin Sheet ◽  
The Other ◽  
Homogeneous Deformation ◽  
Simple Extension ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Isotropic Materials ◽  
Deformation Curves

It is shown in this part how the theory of large elastic deformations of incompressible isotropic materials, developed in previous parts, can be used to interpret the load-deformation curves obtained for certain simple types of deformation of vulcanized rubber test-pieces in terms of a single stored-energy function. The types of experiment described are: (i) the pure homogeneous deformation of a thin sheet of rubber in which the deformation is varied in such a manner that one of the invariants of the strain, I 1 or I 2 , is maintained constant; (ii) pure shear of a thin sheet of rubber (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained at unity, while the other is varied); (iii) simultaneous simple extension and pure shear of a thin sheet (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained constant at a value less than unity, while the other is varied); (iv) simple extension of a strip of rubber; (v) simple compression (i.e. simple extension in which the extension ratio is less than unity); (vi) simple torsion of a right-circular cylinder; (vii) superposed axial extension and torsion of a right-circular cylindrical rod. It is shown that the load-deformation curves in all these cases can be interpreted on the basis of the theory in terms of a stored-energy function W which is such that δ W /δ I 1 is independent of I 1 and I 2 and the ratio (δ W /δ I 2 ) (δ W /δ I 1 ) is independent of I 1 and falls, as I 2 increases, from about 0*25 at I 2 = 3.

Large elastic deformations of isotropic materials IX. The deformation of thin shells

1952 ◽  
Vol 244 (888) ◽  
pp. 505-531 ◽  
Keyword(s):  
Numerical Integration ◽  
Elastic Deformation ◽  
Energy Function ◽  
Thin Shells ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Isotropic Materials ◽  
Large Elastic Deformation ◽  
Circular Diaphragm

The theory of the large elastic deformation of incompressible isotropic materials is applied to problems involving thin shells. The inflation of a circular diaphragm of such a material is studied in detail. It is found that the manner in which the extension ratios and curvatures vary in the immediate neighbourhood of the pole of the inflated diaphragm can be determined analytically. However, in order to determine their variation throughout the inflated diaphragm a method of numerical integration has to be employed. Although this is, in principle, valid for any form of the stored-energy function, the calculations are carried through only for the Mooney form. Finally, the problem of the inflation of a spherical balloon, which has already been dealt with by Green & Shield (1950), is discussed in further detail.

Wrinkling of a Fiber-Reinforced Membrane

2008 ◽  
Vol 76 (1) ◽  
Author(s):  
E. Shmoylova ◽  
A. Dorfmann
Keyword(s):  
Boundary Conditions ◽  
Energy Function ◽  
Mechanical Response ◽  
Transverse Isotropy ◽  
Base Material ◽  
Strain Energy Function ◽  
Transversely Isotropic ◽  
Incompressible Material ◽  
Fiber Reinforced ◽  
Relaxed Energy

In this paper we investigate the response of fiber-reinforced cylindrical membranes subject to axisymmetric deformations. The membrane is considered as an incompressible material, and the phenomenon of wrinkling is taken into account by means of the relaxed energy function. Two cases are considered: transversely isotropic membranes, characterized by one family of fibers oriented in one direction, and orthotropic membranes, characterized by two family of fibers oriented in orthogonal directions. The strain-energy function is considered as the sum of two terms: The first term is associated with the isotropic properties of the base material, and the second term is used to introduce transverse isotropy or orthotropy in the mechanical response. We determine the mechanical response of the membrane as a function of fiber orientations for given boundary conditions. The objective is to find possible fiber orientations that make the membrane as stiff as possible for the given boundary conditions. Specifically, it is shown that for transversely isotropic membranes a unique fiber orientation exists, which does not affect the mechanical response, i.e., the overall behavior is identical to a nonreinforced membrane.

Motion Analysis and Simulation of Complex Curved Surface WEDM System

2008 ◽  
Vol 392-394 ◽  
pp. 151-155
Author(s):  
Tong Wang ◽  
K. Jiang ◽  
Shu Qiang Xie ◽  
Shuang Shuang Hao
Keyword(s):  
Electrical Discharge Machining ◽  
Three Dimensional ◽  
Simulation Method ◽  
Curved Surface ◽  
Polar Coordinates ◽  
Curved Surfaces ◽  
Machining Quality ◽  
Motion Parameters ◽  
Dimensional Graph ◽  
Analysis Of Motion

In this paper, the characteristics and general laws of cutting complex curved surface by wire electrical discharge machining (WEDM) system are studied. Based on analysis of motion parameters the universal mathematical model of polar coordinates is derived. Moreover, the simulation of WEDM system is introduced, which is carried out by using language Visual C++ and the three dimensional graph software OpenGL.This simulation method is helpful in improving machining quality and productivity of complex curved surfaces, and is fundation for establishing CAD/CAPP/CAM technology in WEDM.

Digital Planimetry With a New Adaptive Calibration Procedure Results in Accurate and Precise Wound Area Measurement at Curved Surfaces

2020 ◽  
pp. 193229682095934
Author(s):  
Piotr Foltynski ◽  
Piotr Ladyzynski
Keyword(s):  
Systematic Error ◽  
Calibration Procedure ◽  
Curved Surface ◽  
Area Measurement ◽  
Curved Surfaces ◽  
Wound Area ◽  
Optical Scanner ◽  
Adaptive Calibration ◽  
Relative Errors ◽  
Commercial Device

Background: The purpose of this study was to determine the accuracy of wound area measurement at a curved surface using a digital planimetry (DP) with the newly proposed adaptive calibration. Methods: Forty wound shapes were printed and placed at the side surfaces of cylinders with diameters of 9.4 and 6.2 cm. Area measurements were carried out using a commercial device SilhouetteMobile (Aranz, New Zealand) and the planimetric app Planimator. Planimetric area measurements were carried out using 2 one-dimensional calibration markers placed above and below the wound shape. The method of adaptive calibration for DP was described. Reference area values of wound shapes were obtained by pixel counting on digital scans made with an optical scanner. Relative errors (REs) and relative differences (RDs) for area measurements were analyzed. Results: The median of REs for the DP with adaptive calibration (DPwAC) was equal to 0.60% and was significantly smaller than the median for the SilhouetteMobile device (SMD) (2.65%), and significantly smaller than the median for the DP (2.23%). The SD of RDs for the DPwAC of 0.87% was considerably lower than for the SMD (6.45%), and for the DP without adaptive calibration (2.51%). The mean of RDs for the DPwAC (0.082%) was not significantly different from zero, which means that the systematic error was not present for the DPwAC. Conclusions: The use of the adaptive calibration in DP to measure the areas at curved surface resulted in a significant increase of accuracy and precision, and removal of systematic error. The DPwAC revealed 4.4 times lower error and 7.4 times higher precision of area measurement at curved surfaces than the SMD.

The three kinds of degree distributions and nash equilibrium on the limiting random network

2019 ◽  
Vol 20 (05) ◽  
pp. 2050033
Author(s):  
Dong Han ◽  
Min Xia
Keyword(s):  
Nash Equilibrium ◽  
Random Network ◽  
Sufficient Conditions ◽  
Curved Surface ◽  
High Dimensional ◽  
Game Model ◽  
Curved Surfaces

A generalized dynamically evolving random network and a game model taking place on the evolving network are presented. We show that there exists a high-dimensional critical curved surface of the parameters related the probabilities of adding or removing vertices or edges such that the evolving network may exhibit three kinds of degree distributions as the time goes to infinity when the parameters belong to the super-critical, critical and sub-critical curved surfaces, respectively. Some sufficient conditions are given for the existence of a regular Nash equilibrium which depends on the three kinds of degree distributions in the game model on the limiting random network.

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