Contribution to the Mathematical Description of Stress-Strain Behavior of Elastomers

1980 ◽  
Vol 53 (4) ◽  
pp. 836-841 ◽  
Author(s):  
K. Tobisch
Keyword(s):  
Energy Density ◽  
Density Function ◽  
Mathematical Description ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Pure Shear ◽  
Simple Extension ◽  
Stress Strain ◽  
Strain Behavior ◽  
Strain Energy Density Function

Abstract Based on the hypothesis of Valanis and Landel that the strain energy density function W(λx,λy,λz)could be represented as the sum of three identical functions ω(λi) of the principal extension ratios λi(i=x,y,z), an expression for ω(λi) is suggested which is distinguished by its relative simplicity. The stress-strain relations developed from this expression are tested successfully by applying them to experimental results of other authors. The types of strain which were examined were simple extension, biaxial extension and pure shear; the elongations were to about 700%.

On the Small Strain Behavior of Peroxide Crosslinked Natural Rubber

1986 ◽  
Vol 59 (1) ◽  
pp. 130-137 ◽  
Author(s):  
Gregory B. McKenna ◽  
Louis J. Zapas
Keyword(s):  
Energy Density ◽  
Density Function ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Scaling Law ◽  
Small Strain ◽  
Constant Length ◽  
Strain Behavior ◽  
Small Strains ◽  
Strain Energy Density Function

Abstract Torque and normal force measurements on a cylinder subjected to torsion at constant length were used to study the behavior of NR crosslinked with 5 phr dicumyl peroxide. The derivatives of the strain-energy density function ∂W/∂I1 and ∂W/∂I2 were calculated from the data using the scaling law of Penn and Kearsley. The new results extend the limit of small strains at which the strain-energy density function derivatives have been measured to γ<0.005 and further confirm our previous results that for peroxide-crosslinked NR, ∂W/∂I2 does not become negative at small strain, contrary to several reports in the literature. Reduced stress was determined for the rubber by using the approach of Kearsley and Zapas to calculate the derivative w′(λ) of the Valanis-Landel form of the strain energy function. The results were compared with the measured values for reduced stress in tension and compression at small strains. While the deviation between the predictions and the experimental behavior do not exceed 6%, the characters of the calculated and measured reduced stress plots are different. The measurements in torsion were not obtained at small enough strains to enable direct comparison with the extension/compression behavior at |ε|<0.002. Extrapolation of the results did not produce the anomalous cusp observed in the reduced stress for 0.998<1/λ<1.002 which was reported in our previous study. The fact that torsional data do not show the cusp offers support to the Kearsley suggestion that at these extremely small deformations, rubber compressibility may play an important role in the stress-strain behavior. This could also explain the apparent discrepancy between the predicted Valanis-Landel behavior and the observed behavior. Future work involving higher precision experiments is required to resolve the matter.

The strain energy density function

1986 ◽  
pp. 237-253
Author(s):  
G. C. Sih ◽  
J. G. Michopoulos ◽  
S. C. Chou
Keyword(s):  
Energy Density ◽  
Density Function ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Strain Energy Density Function

An adaptive meshing automatic scheme based on the strain energy density function

1997 ◽  
Vol 14 (6) ◽  
pp. 604-629 ◽  
Author(s):  
A. Hernández ◽  
J. Albizuri ◽  
M.B.G. Ajuria ◽  
M.V. Hormaza
Keyword(s):  
Energy Density ◽  
Density Function ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Adaptive Meshing ◽  
Strain Energy Density Function

Large Deformation Analysis of the Arterial Cross Section

1971 ◽  
Vol 93 (2) ◽  
pp. 138-145 ◽  
Author(s):  
B. R. Simon ◽  
A. S. Kobayashi ◽  
D. E. Strandness ◽  
C. A. Wiederhielm
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Numerical Integration ◽  
Density Function ◽  
Cross Section ◽  
Finite Deformation ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Exponential Form ◽  
Strain Energy Density Function

Possible relations between arterial wall stresses and deformations and mechanisms contributing to atherosclerosis are discussed. Necessary material properties are determined experimentally and from available data in the literature by assuming the arterial response to be a static finite deformation of a thick-walled cylinder constrained in a state of plane strain and composed of an incompressible, nonlinear elastic, transversely isotropic material. Experimental justification from the literature and supporting theoretical considerations are presented for each assumption. The partial derivative of the strain energy density function δW1/δI , necessary for in-plane stress calculation, is determined to be of exponential form using in situ biaxial test results from the canine abdominal aorta. An axisymmetric numerical integration solution is developed and used as a check for finite element results. The large deformation finite element theory of Oden is modified to include aortic material nonlinearity and directional properties and is used for a structural analysis of the aortic cross section. Results of this investigation are: (a) Fung’s exponential form for the strain energy density function of soft tissues is found to be valid for the aorta in the biaxial states considered; (b) finite deformation analyses by the finite element method and numerical integration solution reveal that significant tangential stress gradients are present in arteries commonly assumed to be “thin-walled” tubes using linear theory.

Sign in / Sign up

Export Citation Format

Share Document