On large elastic deformation of prestressed right circular annular cylinders
We formulate and study inflation, extension and twisting of prestressed cylindrical shells that are isotropic in the stress free configuration. We establish that if the prestresses vary only radially in the annular cylinder then a deformation field of the form r=r(R), θ=ΘΩZ, z=λZ is possible in annular cylinders made of any incompressible material and find sufficient conditions for the deformation to be possible when made of compressible materials. When the material is capable of undergoing large elastic deformations and has a non-linear constitutive relation, for the cases studied here, there is up to 26 percent variation in the boundary loads required to engender a given boundary displacement between the prestressed and stress free annular cylinders. On the other hand, the difference in the realized deformation field is only marginal (less than 2 percent). These are unlike the case wherein the material obeys Hooke's law and undergoes small deformations. This study has some relevance to the deformation of blood vessels. © 2010 Elsevier Ltd. All rights reserved.
Modeling the non-dissipative response of vulcanized unfilled rubber
Here the possibility of developing a better three dimensional model for rubber when Hencky strain is written as an explicit function of the Cauchy stress is examined. Based on the experimental evidence, vulcanized unfilled rubber is assumed to be isotropic, incompressible, and deforming from a stress-free reference configuration. Vulcanized unfilled rubber undergoes strain-induced crystallization when uniaxial stretch exceeds 2. Therefore, in this study, a non-dissipative model is sought for rubber only till the von Mises stress is below a critical value. Using the biaxial response reported in Kawabata et al. (1981), the material functions are arrived adopting Rivlin and Saunders methodology. The material constants in the determined material functions are found using biaxial tests for four other experimental studies on rubber reported in the literature. The ability of the arrived constitutive model to describe the uniaxial and equal biaxial response of the same rubber is evaluated. The proposed model is benchmarked with established models in the classical framework. Among the models studied the proposed model performed as well as the other models for Kawabata et al. data set and marginally better than the other models for the remaining four experimental data sets. However, while the proposed model performance does not critically depend on the experiments used to deduce the material parameters, for the other models the material parameters need to be found from fixed biaxial experiments for better performance.
A new set of biaxial and uniaxial experiments on vulcanized rubber and attempts at modeling it using classical hyperelastic models
Using a custom built experimental setup, biaxial and uniaxial extension tests are performed on the same specimen. In biaxial tests, the stretch ratio along y direction is held constant and the stretch ratio in the perpendicular x direction is increased gradually from 1 to 1.4. Stored energy function is obtained from seven different biaxial tests performed by varying the stretch ratio along the y direction at which it is held fixed. Stored energy functions are proposed for the tested rubber sheets by allowing the stored energy to be a function of the principal invariants of right Cauchy-Green deformation tensor or the invariants of the Hencky strain or the principal stretch ratios. To evaluate the performance of the developed stored energy functions, they are compared with 10 existing constitutive relations available in literature in terms of their ability to capture their biaxial response and predicting capability for the uniaxial experiments on the rubber specimens studied. This study reveals that the stored energy function developed in this study as a function of the principal invariants of the right Cauchy-Green deformation tensor predicts the uniaxial behavior the best and capture the biaxial response second best among the models studied for the tested material. However, even this model on an average has 25 percent error in the predicted stresses.