Showing papers by "Kumbakonam R. Rajagopal published in 2020"

A Review of Implicit Constitutive Theories to Describe the Response of Elastic Bodies

Abstract: Implicit constitutive relations are discussed for elastic, electro- and magneto-elastic and thermo-elastic bodies. Approximations of implicit constitutive equations for electro-elastic bodies are developed when the displacement gradient is small, while in the case of the constitutive equations for thermo-elastic bodies approximations are developed when both the displacement gradient and the heat flux are small. Boundary-value problems are studied confining attention to some specific forms of implicit constitutive relations.

Gibbs free energy based representation formula within the context of implicit constitutive relations for elastic solids

Abstract: We derive a representation formula for a class of solids described by implicit constitutive relations between the Cauchy stress tensor and the Hencky strain tensor. Using a thermodynamic framework, we show that the Hencky strain tensor can be obtained as the derivative of the specific Gibbs free energy with respect to a stress tensor related to the Cauchy stress tensor. Unlike previous studies that have considered implicit relations between the Cauchy stress tensor and the Hencky strain we work with quantities that allow us to split the deformation into two parts. One part is connected to deformations that change the volume and the other to deformations where volume is preserved. Such a decomposition allows us to clearly characterise the interplay between the corresponding parts of the stress tensor, and to identify additional restrictions regarding the admissible formulae for the Gibbs free energy. We also show that if the constitutive relations of this type are linearised under the small strain assumption, then one can transparently obtain linearised models with density/pressure/stress dependent elastic moduli in a natural manner.

Modeling of the Aorta: Complexities and Inadequacies.

Abstract: The aorta is a very complex organ comprising three layers, consisting of four kinds of tissues. It is an anisotropic, inhomogeneous, multiconstituent, and living organ that presents both a formidable challenge and a tremendous opportunity to a modeler to mathematically characterize its structure. Unfortunately, even the most sophisticated models in vogue do not faithfully take into consideration its various complexities, falling very short of putting into place a reasonable model, as they ignore many of the quintessential features that need to be taken into account. In this article, we address the various features that need to be taken into account to develop a meaningful model of the aorta.

Density-driven damage mechanics (D3-M) model for concrete I: mechanical damage

Abstract: Damage in concrete has been modelled using various approaches such as fracture mechanics, continuum damage mechanics and failure envelope theories. This study proposes a new approach to model the i...

Segmental Variations in the Peel Characteristics of the Porcine Thoracic Aorta

Abstract: Aortic dissection occurs predominantly in the thoracic aorta and the mechanisms for the initiation and propagation of the tear in aortic dissection are not well understood. We study the tearing characteristics of the porcine thoracic aorta using a peeling test and we estimate the peeling energy per unit area in the ascending and the descending segments. The stretch and the peel force per unit width undergone by the peeled halves of a rectangular specimen are measured. We find that there can be significant variation in the stretch within the specimen and the stretch between the markers in the specimen varies with the dynamics of peeling. We found that in our experiment the stretch achieved in the peeled halves was such that it was in the range of the stretch at which the stress-stretch curve for the uniaxial experiment starts deviating from linearity. Higher peeling energy per unit area is required in the ascending aorta compared to the descending aorta. Longitudinal specimens required higher peeling energy per unit area when compared to the circumferential specimens.

Density driven damage mechanics (D3-M) model for concrete II: fully coupled chemo-mechanical damage

Abstract: In Part I, a density driven damage mechanics (D3-M) approach and its application to model mechanical damage in concrete are presented. In this study, chemical and chemo-mechanical damage in concret...

Chemo-mechanical coupling and material evolution in finitely deforming solids with advancing fronts of reactive fluids

Abstract: A new stabilized method is presented for coupled chemo-mechanical problems involving chemically reacting fluids flowing through deformable elastic solids. A mixture theory model is employed wherein kinematics is represented via an independent set of balance laws for each of the interacting constituents. A significant feature of the mixture model is the interactive force field in the momentum balance equations that couples the constituents implicitly at the level of the governing system of equations. The constitutive relations for the constituents in the mixture model are based on maximization of the rate of entropy production. Since each constituent is not discretely modeled and the interactive effects are mathematically coupled at the local continuum level, the resulting system serves as a physics-based reduced-order model for the complex microstructure of the material system. When constitutive equations are substituted into the balance laws, they give rise to a system of coupled nonlinear PDEs. Evolving nonlinearity and coupled chemo-mechanical effects give rise to spatially localized phenomena, namely boundary layers, shear bands, and steep gradients that appear at the reaction fronts. For large reaction rates, the balance of mass of the fluid becomes a singularly perturbed equation (reaction-dominated), which may exhibit boundary and/or internal layers. Likewise, for large reaction rates and/or low diffusivity, the balance of linear momentum for the fluid constituent also becomes a singularly perturbed PDE. Presence of these features in the solution requires stable numerical methods, and we present a variational multiscale (VMS)-based stabilized finite element method for the initial-boundary value problem. Mathematical attributes of the method are investigated via a range of numerical test cases that involve diffusion of chemically reacting fluids through nonlinear elastic solids. Enhanced stabilization features and higher spatial accuracy of the models and the methods are highlighted.

Viscoelastic transitions exhibited by modified and unmodified bitumen

Abstract: The working temperature of a bituminous pavement can typically range from 75∘C to −20∘C. Bitumen shows a wide spectrum of mechanical behaviour in this temperature range and these include those of a...

Modeling Approaches and Some Physical Considerations Concerning Thermodynamics and the Theory of Mixtures Applied to Time-Dependent Behaviors in Heterogeneous Materials

Abstract: Following a brief review of the theory of continuum mixtures, recent developments and applications emphasizing time-dependent behaviors of heterogeneous materials are described. Common approximations of mixture theory and related continuum homogenization schemes such as assigning material properties, boundary conditions and body forces are considered. Approaches to imposing restrictions due to the second law are discussed; traditional employment of the Clausius-Duhem inequality enforced for arbitrary processes is contrasted with maximization of the rate of entropy production applied to a functional space suitably restricted to conform with constitutive postulates. Remarks are made concerning related homogenization and thermodynamic developments, including poroelasticity, volume-averaging, local accompanying state, and linear irreversible thermodynamics. Applications to fibrous composite materials, and to additional classes of heterogeneous materials, are briefly discussed.

Modeling deformation induced anisotropy of light-activated shape memory polymers

Abstract: Light-activated shape memory polymers (LASMPs) are a new generation of active materials that undergo phase transformations upon light irradiation at different wavelength and intensities. This type of material shows much promise in its use in adaptive and shape reconfiguration structures for applications in biomedical and aerospace engineering. LASMPs exhibit nonlinear viscoelastic behavior due to the rearrangements in the macromolecular networks of the polymers when they are mechanically deformed. The deformed configuration can have a different symmetry group than the original undeformed configuration. In this study we assume that the body has multiple natural configurations in order to incorporate the microstructural changes in the viscoelastic polymers due to mechanical loading and light irradiation. We also account for the directional preference of the new network that is formed due to stretching the polymers. In this paper, the constitutive models developed for elastic and viscoelastic LASMPs in our previous work (Yuan and Zhi, 2017; Zhi and Yuan, 2017) are modified to include the changes in the symmetry group of the new network due to mechanically stretching the polymer. The models are implemented within ABAQUS finite element (FE) through the use of user-material subroutines (UMAT).

Implicit constitutive relations for describing the response of visco-elastic bodies

Abstract: A constitutive relation is proposed for viscoelastic bodies that is a generalization of the classic Kelvin–Voigt model, wherein the left Cauchy–Green tensor, the symmetric part of the velocity gradient, and the Cauchy stress tensor are implicitly related. The model developed includes several models that are being used in the literature to describe the elastic and viscoelastic response of bodies. In this paper, we study special homogeneous deformations of a slab within the context of the implicit viscoelastic model.

Analysis of reclaimed asphalt blended binders using linear and nonlinear viscoelasticity frameworks

Abstract: This paper presents a comprehensive analysis of blended asphalt binders extracted from mixtures that contain various amounts of reclaimed asphalt pavements (RAP). The analysis is performed within a thermodynamically consistent non-linear viscoelastic (NVE) modeling framework, which has the advantage of accounting for the contributions of each of the constituents (i.e. virgin binder and RAP binder) to the response of the blended binder. In order to calibrate and validate the NVE model, the RAP blended binders were subjected to different testing protocols: frequency sweep, multiple stress creep and recovery, repeated creep and recovery with multiple stress levels, random creep and recovery, and stress relaxation. For the binders used in this study, linear viscoelasticity (LVE) was suitable to model the frequency sweep and multiple stress creep and recovery data, but it did not capture the repeated creep and recovery with multiple stress levels results that involved higher stress and strain levels. The NVE model, on the other hand, was successful in describing the repeated creep and recovery with multiple stress levels results. The validation was achieved by comparing the NVE model predictions with the random creep and recovery and stress relaxation test results. Finally, the paper recommends a new method, based on the parameters of the NVE model, to evaluate the rutting resistance of blended asphalt binders.

Implicit constitutive relations for visco-elastic solids: Part II. Non-homogeneous deformations

Abstract: A constitutive relation was developed in Part I for describing the response of a class of visco-elastic bodies, wherein the left Cauchy–Green tensor, the symmetric part of the velocity gradient, and the Cauchy stress tensor are related through an implicit constitutive relation. Here, we study a boundary value problem within the context of the model namely the inhomogeneous deformation of the body, corresponding to the response of an infinitely long slab due to the influence of gravity.

A thermomechanical description of the mold filling process in roll-to-roll nanoimprinting lithography

Abstract: We derive the constitutive equations for a viscoelastic fluid that is undergoing a continuous mold filling process at the nanoscale within the context of a thermomechanical framework The governing equations are obtained by substituting the constitutive equations into the appropriate balance laws From the general characteristics inherent to the roll-to-roll nanoimprinting lithography processes, we derive suitable initial, boundary and simplifications of the governing equations Since the general problem is nonlinear and appears to be intractable both analytically and numerically with the analytical and computational tools available currently, we simplify the problem by assuming the displacement gradient and its time rate to be small, resulting in an integro-differential problem with dynamic boundary conditions To numerically obtain the evolution of the free surface of the fluid within the mold during filling, we propose a numerical scheme based on the Marker and Cell method for the simplified problem We present and discuss a representative sample of the results from the numerical simulations of the nondimensional equation by considering changes to key process and transport parameters that affect mold filling

High Amplitude Torsional Shear of Porcine Thoracic Aorta

Abstract: Studying the mechanics of aortic tissue is a crucial component in understanding its behavior under healthy as well as diseased conditions. Wall shear stress and circumferential stress have been largely accepted as significant factors in arterial growth and remodeling as a response to changes in flow and pressure. But, experimental studies on aortic tissues have largely focused on uniaxial and biaxial tests which are more suited for investigating circumferential stress. This may be explained by the inherent convenience of gripping tissue and then applying deformations in the uniaxial and biaxial modes. As such, the behavior of aortic tissue under shear has been left relatively unstudied. We propose to study the response of porcine aortic wall tissue under cyclic constant torsional shear strain rate for high amplitude (50%) and at different shear strain rates (4%/s and 40%/s). Three to four 12.5 mm diameter samples are obtained from the descending porcine aorta. Initial results clearly indicate a non linear response for the moment as a function of the angle of twist while many popular models predict a linear response for the arterial wall even under large shear strain.

Mathematical Modeling of Rock Glacier Flow with Temperature Effects

Abstract: This paper stems from the interest in the numerical study of the evolution of Boulder Clay Glacier in Antarctica, whose morphological characteristics have required the revision of the basis for most of the recent mathematical models for glacier dynamics. Bearing in mind the need to minimize the complexity of the mathematical model, we have selected the constitutive equation of rock glacier ice recently presented by two of the authors. Here, this model is extended in order to include temperature effects. In addition to the effects of climate change, it is also necessary to take into consideration the non-negligible level of melting due to temperature changes induced by normal stresses arising from the interactions of ice and the rock fragments that are within the rock glacier. In fact, local phase transition that occurs leading to the release of water implies significant modifications of ice viscosity, the main intrinsic factor driving the flow. In this paper we derive the model that describes the flow of rock glaciers that takes into consideration the effects of temperature and the normal stresses generated by the ice and rock fragments interactions.

Corrigendum to "Reversal of flow of a non-Newtonian fluid in an expanding channel" [Int. J. Non-Linear Mech. 101 (2018) 44-45]

Abstract: In their study “Reversal of flow of non-Newtonian fluid in an expanding channel” (Harley et al., 2018), the authors assume a similarity transformation to study the problem of the flow between two intersecting planes. After deriving their equations, the authors purport to obtain a numerical solution to the problem. This solution is incorrect as a similarity solution of the form sought is not possible. Here, we show that the similarity solution of the form sought leads to an overdetermined system of equations for which there is no solution possible that satisfies the appropriate boundary conditions.

Some Remarks on the Equilibrium of Granular Materials Described by Constitutive Relations That Depend on the Gradients of the Density or Volume Fraction

Abstract: One of the ways of determining the properties of granular materials is to subject the material to a tri-axial state of stress with three distinct eigenvalues, in a cubical tri-axial testing equipment, the material tested being in a static state. A constitutive relation for granular materials that does not allow the body to be in tri-axial static equilibrium would not be a reasonable model as experiments clearly show that granular materials can exist in such tri-axial states. In this short note, we observe that a large class of models that have been developed and used to describe the response of granular materials will not allow them to be in a static tri-axial state of stress with three different principal stresses.

A Model to Describe the Response of Arctic Sea Ice

Abstract: In this paper we develop a model for the flow of Arctic sea ice within the context of the theory of interacting continua that takes into account the change of phase between the two constituents, ice and water. After documenting the general balance laws for mass, linear and angular momentum, energy, the second law of thermodynamics and the volume additivity constraint, we discuss the specific constitutive relations that are to be used for the various quantities that appear in the balance laws. Ice is modeled as a non-Newtonian fluid that is a generalization of the usual model due to Glen to take into account the ability of ice to develop normal stress differences in simple shear flow, while water is modeled as a Navier–Stokes fluid. Constitutive relations are discussed for the change of phase, the interaction forces such as the drag, etc. In order to make the problem amenable to analysis, we simplify the governing equations, keeping the quintessential features of the problem of interest in mind.

Numerical sensitivity analysis of a rock glacier flow model versus detection of an internal sliding occurrence

Abstract: This work concerns the rock glacier flow model introduced, in its basic form, by Kannan and Rajagopal in [1] and extended with inclusion of temperature effects by Kannan, Rajagopal, Mansutti and Urbini in [2]. This one is based on the general conservation laws (momentum, mass and energy) and takes into account the effect of shear rate, pressure and rocks and sand grains volume fraction onto viscosity, also by implementing the effects of local pressure melting point variation. Here we present the results of a sensitivity analysis of the parameters developed by shooting the location of the internal sliding occurence, induced by the presence of rocks and sand grains trapped within the interstices of the glacier, and the value of the shear velocity. The case of the Murtel-Corvatsch glacier in Switzerland is considered for the availability of the detailed description based on measured data published by Arenson, Hoelzle and Springman in [3]. The numerical results obtained improve those ones presented in [1] and show clearly the contribution of each numerical and functional parameter of the model. They also exhibit a very good agreement with observations which makes this modelling approach very promising for general application. [1] Kannan, K., Rajagopal, K.R.: A model for the flow of rock glaciers. Int. J. Non-lin. Mech., 48, pp. 59– 64 (2013) [2] Kannan, K., Mansutti, D., Rajagopal, K.R. and Urbini, S.: Mathematical modeling of rock glacier flow with temperature effects, in Mathematical Approach to Climate Change and its Impacts (P. Cannarsa, D. Mansutti and A. Provenzale, eds.), pp. 137-148, Springer-INDAM series, vol.38 (2020) [3] Arenson, L., Hoeltzle, M. and Springman, S.: Borehole Deformation Measurements and Internal Structure of Some Rock Glaciers in Switzerland. Permafrost and Periglacial Processes, 13, pp. 117-135 (2002).