U. Saravanan

Inflation, Extension, Torsion and Shearing of an Inhomogeneous Compressible Elastic Right Circular Annular Cylinder:

We study the inflation, extension, torsion and shearing of an isotropic inhomogeneous compressible annular right circular cylinder. Current approaches to homogenization that appeal to an equivalence in the stored energies could lead to serious errors in the estimate for stresses in a inhomogeneous body as stresses depend on the derivatives of the stored energy with respect to the deformation gradient. This is a serious drawback as many a time failures are determined by the stresses. The study demonstrates that, in particular, great caution should be exercised in homogenization, especially if an inhomogeneous body is to be approximated by a homogeneous body belonging to the same class. Comparison of local measures, such as stresses, reveal that their values in the case of the inhomogeneous body and its homogeneous counterpart can be both qualitatively and quantitatively far apart. Even the differences in global measures like the axial load, torque, etc., are found to be significant between the inhomogeneous body and its homogeneous counterpart. It is also shown that the material parameters characterizing the homogenous approximation gleaned from correlations from different experiments, performed on the same inhomogeneous body, can be quite different.

On the Role of Inhomogeneities in the Deformation of Elastic Bodies

It is quite common to approximate “mildly” inhomogeneous bodies as homogeneous bodies belonging to a certain constitutive class in view of the simplification that such an approximation accords. In this study, we investigate the consequences of such an assumption and we show that it is clearly inappropriate for many classes of inhomogeneous bodies. We choose specific boundary value problems to illustrate the fact that we could be grossly in error, both qualitatively and quantitatively, with regard to local measures such as stresses and strains. In the examples considered, we find that, for global quantities such as applied forces and moments, the error could be significant. Not only could the material parameters found from, say, an extension test and torsion test, which neglect the inhomogeneity of the body, be quite different from one that incorporates the inhomogeneity, but also the values for the material parameter in the homogenized approximation gleaned from these different experiments could be different. In the process of elucidating our thesis, we investigate an important class of deformations, which in view of the paucity of boundary value problems that have been solved for non-linear inhomogeneous solids is worth documenting in its own right.

A Comparison of the Response of Isotropic Inhomogeneous Elastic Cylindrical and Spherical Shells and Their Homogenized Counterparts

All real bodies are inhomogeneous, though in many such bodies the inhomogeneity is “mild” in that the response of the bodies can be “approximated” well by the response of a homogeneous approximation. In this study we explore the status of such approximations when one is concerned with bodies whose response is nonlinear. We find that significant departures in response can occur between that of a “mildly” inhomogeneous body and its homogeneous approximation (if the approximate model is restricted to a certain class), both quantitatively and qualitatively. We illustrate this fact within the context of a specific boundary value problem, the inflation of an inhomogeneous spherical shell. We also discuss the inappropriateness of homogenization procedures that lead to a homogenized stored energy for the body when in fact what is required is a homogenized model that predicts the appropriate stresses as they invariably determine the failure or integrity of the body.

Large amplitude oscillatory shear of unmodified and modified bitumen

The current binder testing protocols in the oscillatory domain use peak stress–strain data for material characterisation. The viscoelastic linearity limits are also based on such data. For a rigorous characterisation of the viscoelastic response of the binder, it is necessary that one records the complete waveform of the material response during oscillatory testing. This paper reports the waveform recorded for unmodified, crumb rubber modified and Styrelf modified bitumen during oscillatory loading. The waveform was collected for strain amplitudes of 1% and 5% at 30°C, 40°C and 50°C temperature. The linear and nonlinear behaviour of the material was studied using the geometrical symmetry of Lissajous plots. It was found that the material response was nonlinear. An appropriate frame invariant nonlinear constitutive model was used to predict the waveform response of all the binders tested.

Fidelity of the Estimation of the Deformation Gradient From Data Deduced From the Motion of Markers Placed on a Body That is Subject to an Inhomogeneous Deformation Field

Practically all experimental measurements related to the response of nonlinear bodies that are made within a purely mechanical context are concerned with inhomogeneous deformations, though, in many experiments, much effort is taken to engender homogeneous deformation fields. However, in experiments that are carried out in vivo, one cannot control the nature of the deformation. The quantity of interest is the deformation gradient and/or its invariants. The deformation gradient is estimated by tracking positions of a finite number of markers placed in the body. Any experimental data-reduction procedure based on tracking a finite number of markers will, for a general inhomogeneous deformation, introduce an error in the determination of the deformation gradient, even in the idealized case, when the positions of the markers are measured with no error. In our study, we are interested in a quantitative description of the difference between the true gradient and its estimate obtained by tracking the markers, that is, in the quantitative description of the induced error due to the data reduction. We derive a rigorous upper bound on the error, and we discuss what factors influence the error bound and the actual error itself. Finally, we illustrate the results by studying a practically interesting model problem. We show that different choices of the tracked markers can lead to substantially different estimates of the deformation gradient and its invariants. It is alarming that even qualitative features of the material under consideration, such as the incompressibility of the body, can be evaluated differently with different choices of the tracked markers. We also demonstrate that the derived error estimate can be used as a tool for choosing the appropriate marker set that leads to the deformation gradient estimate with the least guaranteed error.

Extension, inflation and circumferential shearing of an annular cylinder for a class of compressible elastic bodies:

This paper studies the classical problem of extension, inflation, and circumferential shearing of an annular cylinder for a new class of compressible elastic bodies wherein the left Cauchy–Green deformation tensor is given as a function of the Cauchy stress tensor. We use a semi-inverse method to study the problem by assuming forms for both the deformation field and the stress field. Focusing our attention on to three specific constitutive relations and two geometries corresponding to thick and thin annular cylinders, we study the qualitative features of the governing differential equations. The models are chosen so that they exhibit qualitatively different response features, one of them displaying a limiting stretch. The classical assumption that the hoop and axial stresses are nearly constant through the thickness of the thin annular cylinder subjected to inflation holds for this class of elastic bodies too. However, for thick-walled annular cylinders subjected to inflation at constant length and for a class of models that exhibits limiting stretch we find that stress “boundary layers” form and that the radial stretch is not monotonic.

Mechanical Properties of Human Saphenous Vein

Details about custom built experimental set up to perform inflation tests at constant length on blood vessels are presented. Using this displacement controlled set up we can apply and measure pressures up to 100 kPa and axial loads ranging up to 100 N. The surface deformation is determined from tracking twelve markers in 3D space using 2 CCD cameras. Discarded vein tissue from patients undergoing Coronary bypass surgery were collected and stored at 4oC in normal saline and experiments were completed within 24 hours from harvest. Inflation tests at different axial stretch ratios on saphenous vein were conducted. Results show that the deformation of the vein is not axially symmetric. These suggest that the vein is inhomogeneous and/or residually stressed not only in the radial direction, but also in the circumferential and/or axial direction. The loading and unloading path is not different, suggesting that the vein is being subjected to non-dissipative process. Checking for the incompressibility condition, results show that the vein is compressible. These results have implications in the development of constitutive models for the vein.

A study on the design and behavior of smart antenna

In this paper, a methodology to control the surface error on a Doppler antenna using the concept of a variable geometry truss structure is proposed. A genetic algorithm is used to find the optimal location and number of actuators with the objective to minimize the construction and runtime costs. The optimization also takes into account the limitations in actuation. A piezoceramic-based actuator is used to demonstrate the effectiveness of the methodology. A simple illustration of 2D trusses, which could form a part of a Doppler antenna structure, is used to show the efficacy of the method. An analysis of the effectiveness of such a design is presented. The influence of the variables in the problem is examined and observations made. This study concludes that the smart antenna concept is a viable, feasible and effective option in design.

Mechanical Properties of Abnormal Human Aortic and Mitral Valves

Details about custom built experimental set up to perform uniaxial and biaxial tests on planar soft tissues are presented. This displacement controlled set up can apply and measure loads ranging up to 100 N. The surface deformation is determined from tracking markers in 3D space using 2 CCD cameras. Discarded valve tissue from patients undergoing valve replacement surgery were collected and stored at 4oC in normal saline and experiments were completed within 24 hours from harvest. Uniaxial tests on aortic valve leaflets and biaxial tests on mitral valve leaflets were conducted. Results show that the deformation of these tissues is not homogeneous. The principal stretches in the plane orthogonal to the direction of stretching in the uniaxial stretch experiment are not the same but the principal directions do not change much with loading. Further, the results show that both the valve leaflets are compressible. The loading and unloading path is nearly the same and is not sensitive to the rate of displacement when it is varied between 200µm/s and 800µm/s. These results have implications in the development of constitutive models for these tissues.

Experiments to find constitutive relation for materials undergoing large deformation

In this article, details about state of art custom built experimental set up to perform biaxial tests on thin sheets made of polymers or soft biological tissues is presented. This displacement controlled set up can apply and measure loads ranging from 0.01 N to 100 N. Then, the uniform or non-uniform surface deformation is determined from tracking a set of markers in 3D space using 2 CCD cameras. Using this setup both elastic and viscoelastic properties of the material could be characterized and the assumption of incompressibility verified. Following Rivlin and Saunders,1 the stored energy corresponding to vulcanized rubber, for stretch ratios less than 1.5, is found by systematically varying one of the variables in the stored energy function. This is achieved by performing biaxial extension tests in which the stretch ratio along one direction is held constant but varied between protocols and the stretch ratio in the perpendicular direction is increased gradually from 1 to 1.5. Then, the predictive capability of the stored energy function is examined by comparing its prediction for the uniaxial extension test with the actual experimental results. However, unlike Rivlin and Saunders, the stored energy is allowed to be either a function of the invariants of left Cauchy-Green stretch tensor (Rivlin2) or the invariants of Hencky strain (Criscione et al3) or the principal stretch ratios (Ogden4). None of the stored energy functions evaluated in this study is able to consistently predict the available experimental data.