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1
Content available Applications of implicit constitutive theory for describing the elastic response of rocks and concrete
Bustamante R., Rajagopal K. R.
Archives of Mechanics
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2022
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Vol. 74, nr 6
513--547
EN
An implicit constitutive relation is proposed for elastic bodies, when the gradient of the displacement is assumed to be very small, and as a result the strains are small. The resulting constitutive relation is a non-linear relationship between the linearized strain and the stress. The model is used to fit data for rock and concrete. Some boundary value problems are studied within the context of homogeneous deformations, and also a problem with inhomogeneous deformations is analyzed, namely the inflation of a circular annulus. The predictions of this new implicit constitutive relation are compared with the predictions of the constitutive equations for linearized elastic bodies.
2
Content available remote Modeling Gum Metal and other newly developed titanium alloys within a new class of constitutive relations for elastic bodies
Kulvait V., Málek J., Rajagopal K. R.
Archives of Mechanics
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2017
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Vol. 69, nr 3
223--241
EN
Many titanium alloys and even materials such as concrete exhibit a nonlinear relationship between strain and stress, when the strain is small enough that the square of the norm of the displacement gradient can be ignored in comparison to the norm of the displacement gradient. Such response cannot be described within the classical theory of Cauchy elasticity wherein a linearization of the nonlinear strain leads to the classical linearized elastic response. A new framework for elasticity has been put into place in which one can justify rigorously a nonlinear relationship between the linearized strain and stress. Here, we consider one such model based on a power-law relationship. Previous attempts at describing such response have been either limited to the response of one particular material, e.g. Gum Metal, or involved a model with more material moduli, than the model considered in this work. For the uniaxial response of several metallic alloys, the model that is being considered fits experimental data exceedingly well.
3
Content available remote Euler–Bernoulli type beam theory for elastic bodies with nonlinear response in the small strain range
Janečka A., Průša V., Rajagopal K. R.
Archives of Mechanics
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2016
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Vol. 68, nr 1
3--25
EN
The response of many new metallic alloys as well as ordinary materials such as concrete is elastic and nonlinear even in the small strain range. Thus, using the classical linearized theory to determine the response of bodies could lead to a miscalculation of the stresses corresponding to the given strains, even in the small strain regime. As stresses can determine the failure of structural members, such miscalculation could be critical. We investigate the quantitative impact of the material nonlinearity in the Euler–Bernoulli type beam theory. The governing equations for the deflection are found to be nonlinear integro-differential equations, and the equations are solved numerically using a variant of the spectral collocation method. The deflection and the spatial stress distribution in the beam have been computed for two sets of models, namely the standard linearized model and some recent nonlinear models used in the literature to fit experimental data. The predictions concerning the deflection and the spatial stress distribution based on the standard linearized model and the nonlinear models are considerably different.
4
Content available remote Stress concentration due to an elliptic hole in a degrading linearized elastic solid
Kulvait V., Malek J., Rajagopal K. R.
International Journal of Applied Mechanics and Engineering
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2010
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Vol. 15, no 1
35-49
EN
A computational analysis is carried out of boundary-value problems associated with a planar generalized linearized elastic solid body with an elliptic hole in it, with a fluid diffusing through the solid. This diffusion of fluid can either enhance or degrade the load carrying capacity of the body, based on how the material moduli of the solid depend on the concentration of the fluid, that is whether the presence of the fluid degrades or strengthens the material. We investigate the nature of the solution when the aspect ratio tends to zero, a problem relevant to the stress singularity at crack tips.
5
Content available remote On some issues concerning the modeling of the motion of fluids
Rajagopal K. R., Tao L., Chen G. Q.
International Journal of Applied Mechanics and Engineering
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2005
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Vol. 10, no 3
475-487
EN
We address some issues regarding the use of the Lagrangian description and convected frames in describing fluid motions. We also discuss the implications of Brownian motion on modeling the macroscopic motion of fluids and the schemes of filtered simulations. The relevance of these issues to the modeling of turbulence is discussed in detail.
6
Content available remote A note on the flows of inhomogeneous fluids with shear-dependent viscosities
Anand M., Rajagopal K. R.
Archives of Mechanics
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2005
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Vol. 57, nr 5
417--428
EN
Inhomogenous fluids have not been studied with the intensity that they deserve. In fact, many studies that are supposedly concerned with the response of inhomogeneous fluids are not directed at inhomogeneous fluids, and this stems from not recognizing the fact that the properties of a fluid varying in its current configuration does not mean that the fluid is inhomogeneous. Here, we show that mild variations in the properties of the fluid which might warrant it being approximated as a homogeneous fluid with average properties could lead to significant errors in the computation of both global and local quantities, associated with the flow.
7
Content available remote Couette flows of fluids with pressure dependent viscosity
Rajagopal K. R.
International Journal of Applied Mechanics and Engineering
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2004
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Vol. 9, no 3
573-585
EN
For liquids such as water, for a range of pressures for which the density remains nearly constant, the viscosity could change by several orders of magnitude. Thus, such liquids could be approximated as incompressible liquids whose viscosity depends on the pressure. Stokes (1945) recognized this possibility and the fact that the viscosity could be considered to be independent of the pressure in only special flows. That this is indeed so has been verified in the case of numerous liquids (Bridgman, 1931). In this short study, we allow the viscosity of the fluid to be dependent on the pressure and investigate the consequence of the effect of gravity in simple flows such as Couette flows between parallel plates. We find that gravity can have a profound effect on the structure of the flow. Its presence leads to the concentration of vorticity adjacent to one of the plates.
8
Flows of bubbly liquids
Tao L., Rajagopal K. R.
Applied Mechanics and Engineering
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1999
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Vol. 4, no 3
407-442
EN
This paper deals with the modeling of flows of a Newtonian liquid dispersed with bubbles. We provide an averaging procedure that combines and gainfully exploits area, volume and ensemble averaging methods, and arrives at the governing equations for bubbly liquids. Expressions are suggested for the interaction mechanisms and other constitutive quantities that appear as a consequence of the averaging. Restrictions on the constitutive quantities due to the second law of thermodynamics are delineated and a specific boundary value problem solved within the context of the theory. It is found that the prediction of the theory compares favorably with experimental observations for the specific boundary value problem under consideration.
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