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Magnetostrictive materials that couple mechanical and magnetic domains have been widely explored for use in sensors and actuators. These materials often exhibit a nonlinear material response under applied magnetic fields, which limits the use of linear constitutive models. Furthermore, the nonlinear constitutive relations tend to be implicit in nature. Hence, a finite element scheme that can handle the implicit relationship between the mechanical (stresses and strains) and magnetic (magnetic flux density and magnetic field) quantities is proposed in order to arrive at solutions to boundary value problems. In the proposed scheme, while the physical requirements of equilibrium and strain–displacement relation are satisfied point-wise, the constitutive relations hold in a weak integral sense. A fully coupled magnetostrictive plane stress rectangular element is developed based on the proposed scheme and its efficacy in arriving at solutions to coupled field boundary value problems is illustrated by subjecting the element to standard loading conditions.

A thermodynamic framework is proposed to capture the dissipative response of metals. In contrast to the conventional practice, a stressed reference configuration is assumed instead of a stress-free configuration. The second law of thermodynamics is converted into equality by prescribing a non-negative rate of dissipation function. Stress in the reference configuration evolves with time to satisfy the second law of thermodynamics. Appropriate constitutive prescriptions are made to model the experimentally observed cyclic response of metals such as mild steel, aluminum alloy, and titanium, to uniaxial loading. An appropriate choice of material parameters captures well the experimentally observed response for different loading protocols.

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.

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.

Time-dependent strains in reinforced concrete (RC) members are usually estimated using approximate algebraic methods. This paper presents an exact method for estimating the time-dependent s...

Knowing the current state of a bridge is of interest for a variety of reasons. Some parameters that determine the current state of a bridge are the material properties and boundary conditions. Using strain measurements obtained from a slow-moving vehicle on a bridge, the boundary condition and material properties are determined through a mechanistic-based approach. Observing that the sign of the curvature would change at locations near the support when a load passes over a bridge with end rotational restraints, a methodology for determining the boundary conditions is proposed and validated. The linear elastic properties of the material that the bridge is made up of is determined from the strain measured at locations where the stress is independent of the material property. In this procedure, the structure is analyzed assuming some material properties and the stress at the measured point is determined. Then, the material parameters in the isotropic Hooke’s law are determined so that the stress estimated from the experimentally determined strains agrees with that obtained from the analysis with arbitrarily assumed material parameters. A prestressed high-performance concrete pi-shaped girder tested under a three-axle slow-moving load with strains measured at different locations is used to bring out the efficacy and appropriateness of the proposed methodologies. The mean value of Young’s modulus of the prestressed concrete bridge agrees well with the experimentally determined Young’s modulus.

Here mixture theory is used to capture the changes in cement concrete exposed to sodium sulfate till cracks develop. Toward this, the mixture is assumed to be made of eleven constituents of which the sodium sulfate and water move relative to themselves and the remaining nine solid constituents. The nine solid constituents constrained to move together are the eight relevant chemical constituents in concrete that react with sodium sulfate and all the other remaining chemical constituents of concrete that do not react with sulfates. Constitutive assumptions needed to be made within this mixture theory framework are the same as those reported by Gouder and Saravanan (Acta Mech 227(11):3123–3146, 2016). Within this framework of mixture theory, the radial ingress and reaction of sodium sulfate solution with the concrete cylinder sealed at top and bottom, exposed to a constant concentration of sodium sulfate at its outer surface, are formulated. The resulting nonlinear governing differential equations are converted into a system of nonlinear algebraic equations using a forward finite difference scheme in space and a backward difference in time. The nonlinear algebraic equations are solved simultaneously using constrained minimization technique till the water reaches the center of the cylinder. The results obtained for ingress without chemical reactions agree with those predicted by Fick’s equation. The axial expansion of the cylinder and the increase in the value of Young’s modulus of the part of concrete which reacted with sulfates agree qualitatively with the experiments.

The test method of ASTM C512 (ASTM. 2015. Standard test method for creep of concrete in compression. ASTM C512/C512M. West Conshohocken, PA: ASTM) dictates the use of a spring-loaded creep ...

Recently, a thermodynamically consistent non-linear constitutive equation has been developed to describe the large deformation cyclic response of viscoelastic polyamides (see [ 17 ]). In this paper, two boundary value problems within the context of the above model, namely the stress relaxation of a right circular annular cylinder subject to twisting, and the inflation of a sphere are studied. In addition to solving the above problems numerically, investigation of the merits and pitfalls of studying the same boundary value problem for a special class of inhomogeneous body and its homogenized counterpart is undertaken. This study finds that for moderate strains the differences in relaxation time between the actual inhomogeneous body and its homogenized counterparts may be significant.

We develop a methodology for solving truss problems involving viscoelastic materials where, of all the member forces that satisfy the nodal force equilibrium equation and nodal displacements that satisfy the displacement boundary conditions, those member forces and nodal displacements that satisfy the constitutive relation are sought. Since a rate type viscoelastic constitutive relation involves the rate of the stress or strain, this study explores the use of member forces, nodal displacements, and support reactions or their rates as independent variables. Assuming small deformations, the nodal force equilibrium and the displacement boundary condition results in a linear equality constraint between the independent variables. Then we find the unknown independent variables such that the root mean squared error in the constitutive relation of the members of the truss is minimized subject to the satisfaction of the linear constraint at selected times. The objective function is evaluated at selected times or integrated over subintervals of time. We explore six possible solution methods and benchmark them for their accuracy and efficiency. We study statically determinate and indeterminate truss whose members are modeled using rate and integral type viscoelastic constitutive relations for creep and oscillatory loading. For the standard linear solid model, we find that the proposed methods are more accurate than ABAQUS and, at times, require lesser computational wall time. We also demonstrate the applicability of the proposed methodology to fractional order and nonlinear viscoelastic constitutive relations.

It is customary to use a displacement-based formulation to seek solutions to boundary value problems for its computational efficacy. In displacement-based formulations, it is convenient to prescribe the constitutive relation for stress as an explicit function of displacement gradient. However, from a general theoretical point of view, the stress and displacement gradient could be related by an implicit function. This study develops techniques to solve boundary value problems when linearized strain and stress are related by an implicit function. Here both the stresses and the displacement are taken as unknowns. The stress field is constructed such that it satisfies the equilibrium equations identically within the element and the traction continuity requirements between the elements. A continuously differentiable displacement field is constructed, and the linearized strain is computed from this displacement field. Then, the unknown parameters in the stress and displacement field are estimated such that the constitutive relation holds in weak integral sense. Though in this procedure the number of unknowns has increased in comparison with the displacement formulation, both the strength and serviceability condition can be checked directly without any post-processing. Also, in this procedure, both the equilibrium equations and continuity of displacement are met exactly. The equation that is not satisfied exactly is the constitutive relation, which is an approximation anyway. The efficacy and accuracy of this method are benchmarked by studying some standard problems. Planar and three-dimensional truss elements have been developed and benchmarked. Then, a rectangular plane element is implemented and its performance recorded.

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.

Concrete exposed to sulfates deteriorates. Here, an attempt is made to see whether the framework of mixture theory can be used to model the changes that occur in concrete exposed to sodium sulfate. Toward this, diffusion and reaction of sulfates with concrete is modeled within the framework of mixture theory. Appropriate choices are made for the Helmholtz free energy and interaction momentum so that the process of diffusion of sulfates in concrete can be captured. As expected in mixture theory, diffusion causes deformation of the solid. The parameters in the mixture theory model are determined by comparing the steady-state concentration profile of the diffusing sodium sulfate solution and inlet velocity of the fluid with that of Fick’s solution for the same boundary conditions. An assumption is made for the mass production term to capture the reaction of sulfates with concrete. The rate constant and order of the reaction are estimated using the concentration of gypsum, calcium hydroxide and ettringite reported in the literature, for various duration of exposure of cement pastes to known concentration of sodium sulfate solution. Finally, the governing equations for the combined problem of steady-state diffusion and reaction of sulfates with concrete are presented. A numerical scheme to solve the governing equations is outlined. Long-term concentration profiles of sodium sulfate predicted by the framework agree qualitatively with the experimentally observed profile reported in the literature.

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.

A methodology for obtaining implicit constitutive representations involving the Cauchy stress and the Hencky strain for isotropic materials undergoing a non-dissipative process is developed. Using this methodology, a general constitutive representation for a subclass of implicit models relating the Cauchy stress and the Hencky strain is obtained for an isotropic material with no internal constraints. It is shown that even for this subclass, unlike classical Green elasticity, one has to specify three potentials to relate the Cauchy stress and the Hencky strain. Then, a procedure to obtain implicit constitutive representations for isotropic materials with internal constraints is presented. As an illustration, it is shown that for incompressible materials the Cauchy stress and the Hencky strain could be related through a single potential. Finally, constitutive approximations are obtained when the displacement gradient is small.

All bodies are inhomogeneous at some scale but experience has shown that some of these bodies can be idealized as a homogeneous body. Here we examine which bodies can be idealized as a homogeneous body when they are subjected to a non-dissipative mechanical process. This is done by studying circumstances in which an inhomogeneous body admits pure stretch homogeneous deformations. Then, we devise experiments wherein these circumstances are prevented. If homogeneous deformation is observed in these devised experiments, the body could be modeled as a homogeneous body. We limit our analysis to a class of isotropic elastic bodies deforming from a stress free reference configuration whose Cauchy stress is explicitly related to left Cauchy-Green deformation tensor. It is further assumed that the constitutive relation is differentiable function of the position vector of material particles in the stress free reference configuration. Then, we find that a cuboid made of compressible and isotropic material could be modeled as a homogeneous body if it deforms homogeneously due to the application of the normal stresses on all of its six faces and the magnitude of the normal stresses on three orthogonal faces are different. A cuboid made of incompressible and isotropic material could be modeled as a homogeneous body, if it deforms homogeneously in two different biaxial experiments, such that the plane in which the forces are applied in the two biaxial experiments is mutually orthogonal. � 2014 Elsevier Ltd. All rights reserved.

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. � 2013 Copyright Taylor and Francis Group, LLC.

Major challenge in heart valve tissue engineering for paediatric patients is the development of an autologous valve with regenerative capacity. Hybrid tissue engineering approach is recently gaining popularity to design scaffolds with desired biological and mechanical properties that can remodel post implantation. In this study, we fabricated aligned nanofibrous Bio-Hybrid scaffold made of decellularized bovine pericardium: polycaprolactone-chitosan with optimized polymer thickness to yield the desired biological and mechanical properties. CD44+, αSMA+, Vimentin+ and CD105- human valve interstitial cells were isolated and seeded on these Bio-Hybrid scaffolds. Subsequent biological evaluation revealed interstitial cell proliferation with dense extra cellular matrix deposition that indicated the viability for growth and proliferation of seeded cells on the scaffolds. Uniaxial mechanical tests along axial direction showed that the Bio-Hybrid scaffolds has at least 20 times the strength of the native valves and its stiffness is nearly 3 times more than that of native valves. Biaxial and uniaxial mechanical studies on valve interstitial cells cultured Bio-Hybrid scaffolds revealed that the response along the axial and circumferential direction was different, similar to native valves. Overall, our findings suggest that Bio-Hybrid scaffold is a promising material for future development of regenerative heart valve constructs in children.

In this article, the appropriateness of modelling asphalt using linear viscoelastic constitutive relations is explored. Towards this, a rate-type nonlinear viscoelastic model recently proposed by Santoshreddy et al. (2011) is used and it is shown that the stress relaxation and creep response of asphalt are influenced by the type of experiment, initial rate of displacement (or loading) and the stress and strain measure used. Furthermore, it is shown that the creep and stress relaxation functions are not dependent functions as required by the linear viscoelastic theory, even in the range of values of engineering strain and stress in which the linearity property holds approximately. These observations lead one to conclude that linear viscoelastic models maybe inappropriate to model asphalt. � 2012 Copyright Taylor and Francis Group, LLC.

Plastic zone method of advanced analysis, which uses shell elements to model the entire structure, is the most accurate method available to predict the ultimate strength and behavior of steel frames. The disadvantage of such full shell plastic zone models is that it is computationally expensive and hence its use is limited to small structures. Beam elements in commercial finite element packages can model residual stress and capture spread of plasticity, but cannot model local buckling of plates that the member is made up of, which leads to unloading and failure in steel frames. A hybrid model using shell elements only in the regions vulnerable to elastic or inelastic local buckling and beam elements in other locations could overcome this limitation of full beam element model. The issues in using this hybrid model are, knowing a priori the location and length of the shell element region and connecting the beam and shell regions without any artificial stress concentrations or incompatible displacements. In this study, in addition to addressing these issues, the hybrid model is systematically evaluated by studying its performance in structural elements. It is seen that the hybrid model strength predictions has an average error of only 0.91% but requires on an average 83% less computational time when compared to the full shell plastic zone models. � 2011 Elsevier Ltd.

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.

Using a special model that belongs to a new class of elastic bodies wherein the Cauchy–Green stretch is given in terms of the Cauchy stress and its invariants, within the context of the spherical inflation of a spherical annulus, we show that interesting phenomena like the development of “stress boundary layers” manifest themselves. We consider two cases of boundary value problems, one in which there is a cavity in a sphere and the other in which there is a rigid spherical inclusion in a sphere. We show that in the case of a rigid inclusion, it is possible for a pronounced “stress boundary” layer to develop, in that the values of the stresses within this boundary layer that is adjacent to a spherical inclusion are much larger than external to it. We also show that in the case of both the cavity and a rigid inclusion, the stress concentration is an order of magnitude higher than the increase in the deformation gradient, that is, the stress and the stretch do not scale in a similar manner. While the stress adjacent to a rigid inclusion can be 2500 times the applied radial stress, the maximum stretch, which occurs at the rigid inclusion is about 10. While the variation in the stresses are linear in thin walled annular regions, we find that in thick walled annular regions, the variation of the stresses is non-linear.

We study several simple boundary value problems for a special class of inhomogeneous, isotropie, compressible elastic solids: inflation of a spherical shell; bending, stretching and shearing of a rectangular block; and straightening, stretching and shearing of a sector of a hollow cylinder. The purpose of the study is twofold: to establish new exact solutions to boundary value problems concerning inhomogeneous, isotropie, compressible elastic solids that have technological relevance (to the deformation of layered composites); and to reaffirm a thesis concerning an inherent difficulty with regard to the homogenization techniques that are in vogue that appeal to the stored energy in the homogenized body being the same as that for the inhomogeneous body. In addition to establishing new exact solutions for a class of inhomogeneous elastic solids, the study reinforces the thesis of Saravanan and Rajagopal that great care ought to be exercised in approximating even a very mildly inhomogeneous body by an equivalent homogeneous body when large deformations are involved.

Here we investigate whether the deformation observed in an experiment in which the porcine circumflex coronary artery is subjected to inflation at constant length included in the class, $$r$$ $$=$$ $$r(R)$$ , $$\theta$$ $$=$$ $$\Theta$$ , $$z$$ $$=$$ $$\Lambda Z$$ . We find that this is not the case and discuss its implications in the study of the mechanics of this artery. Moreover, we identify and quantify the uncertainty in the value of the invariants of the left Cauchy–Green tensor inferred from the 2D motion of markers affixed to the surface of the test specimen, and suggest that 3D tracking of markers is needed due to inherent bending and twisting induced by pressurization in vitro.

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.

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.

We show that even when the constitutive relations for the elastic response of compressible bodies satisfy the empirical, Baker–Ericksen and ordered forces inequalities and ellipticity restrictions, they could result in physically unrealistic response. We pick a constitutive relation that satisfies the above inequalities and is also globally elliptic but its response to both uniaxial extension and inflation at constant length is physically unrealistic, based on the available experimental observations. However, the Coleman–Noll, pressure–compression, tension–extension and invertibility of force-stretch conditions and that proposed in Carroll and McCarthy (1995) are violated by this constitutive relation.

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.

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. Copyright � 2013 by ASME.

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 = Θ + Ω Z , z = λ Z 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.

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.

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 (Rivlin 2 ) or the invariants of Hencky strain (Criscione et al 3 ) or the principal stretch ratios (Ogden 4 ). None of the stored energy functions evaluated in this study is able to consistently predict the available experimental data.

There are several problems wherein it would be convenient to have representations for the stress from a stressed configuration as reference. We assume that the body undergoes finite elastic deformations (i.e., non-dissipative processes) from a stressed reference configuration but it is possible that a dissipative (inelastic) process could be the cause for the origin of the stresses in the reference configuration. We restrict ourselves to determining the representation for the stress in a body, from a stressed reference configuration, whose symmetry group in the stress free configuration (that is accessible from the stressed reference configuration) coincides with the full orthogonal group. We also find restrictions on the constitutive relation so that we have a model that exhibits physically acceptable response characteristics. � 2008 Elsevier Ltd. All rights reserved.

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.

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.