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 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.

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.

General theory of small elastic deformations superposed on finite elastic deformations

1952 ◽  
Vol 211 (1104) ◽  
pp. 128-154 ◽  
Keyword(s):  
General Theory ◽  
Elastic Material ◽  
Finite Deformation ◽  
Plane Surface ◽  
Thin Sheet ◽  
Potential Functions ◽  
Homogeneous Deformation ◽  
Infinitesimal Deformation ◽  
Elastic Deformations ◽  
Isotropic Elastic Material

Using tensor notations a general theory is developed for small elastic deformations, of either a compressible or incompressible isotropic elastic body, superposed on a known finite deformation, without assuming special forms for the strain-energy function. The theory is specialized to the case when the finite deformation is pure homogeneous. When two of the principal extension ratios are equal the changes in displacement and stress due to the small superposed deformation are expressed in terms of two potential functions in a manner which is analogous to that used in the infinitesimal deformation of hexagonally aeolotropic materials. The potential functions are used to solve the problem of the infinitesimally small indentation, by a spherical punch, of the plane surface of a semi-infinite body of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation symmetrical about the normal to the force-free plane surface. The general theory is also applied to the infinitesimal deformation of a thin sheet of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation by forces in its plane. A differential equation is obtained for the small deflexion of the sheet due to small forces acting normally to its face. This equation is solved completely in the case of a clamped circular sheet subjected to a pure homogeneous deformation having equal extension ratios in the plane of the sheet, the small bending force being uniformly distributed over a face of the sheet. Finally, equations are obtained for the homogeneously deformed sheet subjected to infinitesimal generalized plane stress, and a method of solution by complex variable technique is indicated.

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 ).

scholarly journals Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure, homogeneous deformation

1948 ◽  
Vol 240 (822) ◽  
pp. 491-508 ◽  
Keyword(s):  
Equilibrium State ◽  
Equilibrium States ◽  
Homogeneous Deformation ◽  
Actual State ◽  
Stress And Strain ◽  
Elastic Deformations ◽  
Isotropic Materials ◽  
Stress Components ◽  
Strain Components ◽  
Rigidity Modulus

The equilibrium of a cube of incompressible, neo-Hookean material, under the action of three pairs of equal and oppositely directed forces f 1 , f 2 , f 3 , applied normally to, and uniformly distributed over, pairs of parallel faces of the cube, is studied. It is assumed that the only possible equilibrium states are states of pure, homogeneous deformation. It is found that (1) when the stress components in the deformed cube are specified, the corresponding equilibrium state is uniquely determined (this is shown in § 6 of Part I). (2) When the three pairs of equal and oppositely directed forces f 1 , f 2 and f 3 are specified, (a) the corresponding equilibrium state is uniquely determined, provided that one or more of the forces f 1 , f 2 and f 3 is negative, i.e. is a compressional force, or, if they are all positive, provided that f 1 f 2 f 3 > (1/3E) 3 , where 1/3E is the constant of proportionality between the stress and strain components (analogous to the rigidity modulus of the classical theory of small elastic deformations of isotropic materials). (b) If f 1 , f 2 and f 3 are all positive and f 1 f 2 f 3 > (1/3E) 3 , then the equilibrium state is not necessarily uniquely determined. The number of equilibrium states which exist depends on the values of f 1 , f 2 , f 3 and 1/3E. The actual state of deformation which is obtained depends in general on the order in which the forces are applied.

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.

STUDI OF FRUIT LEATHER PROCESSING OF MANGGO

2013 ◽  
Vol 3 (2) ◽  
pp. 124-132
Author(s):  
Yessy Rosalina ◽  
Laili Susanti ◽  
Tatik Sulasmi
Keyword(s):  
Thin Sheet ◽  
Physical Appearance ◽  
The Other ◽  
Drying Temperature ◽  
Fruit Flesh ◽  
Leather Processing ◽  
High Yielding Varieties ◽  
Processing Techniques

Mango varieties Bengkulu is one of high yielding varieties from Bengkulu Province. The plants grow well and produce fruits almost year-around in all regions in the province of Bengkulu. Mango varieties Bengkulu have very large fruit, thick fruit flesh and slightly sour flavors. It makes Mango varieties Bengkulu is very suitable to be consumed in the form of a processed. Fruit leather is one form of processed fruit. Fruit leather is a thin sheet of fruit in dried form. The results showed that the best processing techniques of fruit leather for mango varieties Bengkulu is the process with addition of sugar by 20% and drying temperature at 60 ?C. The treatment produce fruit leather with the best physical appearance and flavor compared with the other treatments.

scholarly journals Stored Energy Welding Technology of Ultra-Thin Sheet Stainless Steel

2013 ◽  
Vol 749 ◽  
pp. 27-30
Author(s):  
Hao Bin Zhou ◽  
Ju Lian Ma ◽  
Xiang Qian Xu
Keyword(s):  
Stainless Steel ◽  
Spot Welding ◽  
Joint Strength ◽  
Thin Sheet ◽  
Welding Parameter ◽  
Welding Parameters ◽  
Dynamic Resistance ◽  
Stored Energy ◽  
Welding Technology ◽  
The Hard Problem

Considering the characteristics of the multilayered ultra-thin sheet stainless steel, by chosen the proper equipments and stored energy welding technology, the optimizing welding parameters are got from the condenser discharge spot welding experiments. The energy stored in the capacitor group is controlled by the energy required during welding and the discharge current is controlled by the size of dynamic resistance in resistance spot welding. So those control ways can better solve the hard problem of choosing welding parameter. The joint strength of ultra-thin sheet stainless steel is ensured by the reasonable quality assessment methods. The processed spot-welded joints are of good quality and appearance.

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