Saravanan Umakanthan

The validity of some of the assumptions made during systematic development of constitutive relation – homogeneity, isotropy, and non-dissipative response – is examined experimentally by testing plain concrete cylinders in the circumferential displacement controlled uniaxial compression tests. The tested cylinders indicate that the surface strains are not uniform and that the principal direction of the strain varies with the magnitude of the applied load. Hence, these cylinders are either deforming from a stressed state or anisotropic or both. The percentage dissipation computed as the area between the loading and unloading curves normalized using the area under the unloading curve is less than 10 percent up to 90 percent of the peak load. Therefore the mechanical response of concrete can be considered as non-dissipative. Assuming that the magnitude of the residual stresses present in the concrete panels would be small compared to the stresses arising due to the applied load, these residual stresses are ignored. Consequently, using the implicit constitutive theory framework for isotropic and compressible materials undergoing a non-dissipative process from a stress-free reference configuration, a three-dimensional constitutive relation for plain concrete is proposed. The material parameters in the proposed model are estimated from the uniaxial compression test and equal biaxial compression test. These parameters are expressed using initial Young's modulus, initial Poisson's ratio, maximum uniaxial compressive stress, the axial and transverse strain corresponding to the maximum uniaxial compressive stress, maximum equal biaxial compressive stress and the equal biaxial strain corresponding to maximum equal biaxial compressive stress. The ability of the proposed model to capture the response of various grade and type of concrete in ten biaxial experiments reported in the literature is examined. It is found that despite the limiting assumptions, the compression-compression, compression-tension and tension-tension stress strain response of the concrete in these ten experiments are adequately captured by the proposed model with mean R 2 value of 0.94.

Many bodies of biological and engineering interest have a fibrous and layered structure. Hence, it is believed that these bodies are inhomogeneous and made up of anisotropic material. Classical mechanical experiments used to find the required material symmetry in the constitutive relation cannot distinguish inhomogeneous bodies made of isotropic material and homogeneous bodies made of anisotropic material. Therefore, it is of interest to find an alternative hypothesis so that inhomogeneity and anisotropy can be determined independent of the other. This study finds that the principal (or eigen) direction of the left Cauchy-Green deformation tensor, B does not vary with the magnitude of the applied uniaxial load at a given location whenever the body - homogeneous or inhomogeneous - is made of isotropic and hyperelastic material and the deformations are measured from a stress free reference configuration. In general, the principal direction of the left Cauchy-Green deformation tensor varies with the magnitude of the uniaxial load when the body is made up of anisotropic material. Thus, it is concluded that if the variation in the principal direction of B with the magnitude of the applied uniaxial load is experimentally investigated then one could ascertain whether the body is made up of isotropic or anisotropic material.

This work documents the formulation of an axial element that could handle implicit constitutive relations. This formulation uses nodal displacements, member forces, and reaction forces as primary unknowns. For some assumed values of these primary unknowns satisfying the nodal equilibrium equations, the computed stretch and stress in the members need not be in agreement with the constitutive relation. Hence, these primary unknowns are found such that the error in the constitutive relation for the members are minimized subjected to the constraint that the nodal equilibrium equations hold. Axial element developed using this concept is used to analyze a few trusses. The results of this formulation are in agreement with the results obtained from ABAQUS ® when stress is an explicit function of the stretch ratio. Also, some statically determinate as well as indeterminate trusses are solved for an implicit constitutive relation and the reasonableness of the obtained results established.

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.