Cavitation in nonlinear elasticity and associated problems

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The repulsion property in nonlinear elasticity and a numerical scheme to circumvent it:The repulsion property in nonlinear elasticity

For problems in the Calculus of Variations that exhibit the Lavrentiev phenomenon, it is known that a \textit{repulsion property} may hold, that is, if one approximates the global minimizer in these problems by smooth functions, then the approximate energies will blow up. Thus, standard numerical schemes, like the finite element method, may fail when applied directly to these types of problems. In this paper we prove that a generalised repulsion property holds for variational problems in three dimensional elasticity that exhibit cavitation.We propose a numerical scheme that circumvents the repulsion property, which utilizes an adaptation of the Modica and Mortola functional for phase transitions in liquids. In our scheme, the phase function is coupled, via the determinant of the deformation gradient, to the stored energy functional. We show that the corresponding approximations by this method satisfy the lower bound $\Gamma$--convergence property in the multi-dimensional, non--radial, case. The convergence to the actual cavitating minimizer is proved for a spherical body, in the case of radial deformations

Study of the Pharmacognostic Characterization and Antimicrobial Activity of the Medicinal Plant \u3cem\u3eCassia obtusa\u3c/em\u3e L.

The species, Cassia obtusa L., consists of small herbs found in tropical and subtropical regions and have wide application in herbal formulations. Leaf, stem, and fruit are used to cure various ailments in human beings. In fact, plants produce a diverse range of bioactive molecules making them a rich source of different types of medicines researches in bioactive substances might lead to the discovery of new compounds that could be used to formulate new and most potent antimicrobial drugs to over come the problem of resistant to the currently available antibiotics. The main objective of the present investigations is to analyze the fluorescence characters and evaluate the antimicrobial activity of crude extract of leaf, stem and fruit against selected gram positive and gram negative bacteria. The leaf, stem and fruit powder of the plants showed varying degree of antibacterial activity against all the tested bacteria

Cavitation of a spherical body under mechanical and self-gravitational forces

In this paper, we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider two types of problems: the displacement problem in which the outer boundary of the body is subjected to a Dirichlet-type boundary condition, and the one with zero traction on the boundary but with an internal pressure function. For a spherically symmetric body occupying the unit ball, the minimization is done within the class of radially symmetric deformations. We give conditions for the existence of such minimizers, for satisfaction of the Euler-Lagrange equations, and show that for large displacements or large internal pressures, the minimizer must develop a cavity at the centre. We discuss a numerical scheme for approximating the minimizers for the displacement problem, together with some simulations that show the dependence of the cavity radius and minimum energy on the displacement and mass density of the body.</p

On the convergence of a regularization scheme for approximating cavitation solutions with prescribed cavity volume size

Let $\Omega\in\mathbb{R}^n$, $n=2,3$, be the region occupied by a hyperelastic body in its reference configuration. Let $E(\cdot)$ be the stored energy functional, and let $x_0$ be a flaw point in $\Omega$ (i.e., a point of possible discontinuity for admissible deformations of the body). For V>0 fixed, let $u_V$ be a minimizer of $E(\cdot)$ among the set of discontinuous deformations $u$ constrained to form a hole of prescribed volume $V$ at $x_0$ and satisfying the homogeneous boundary data $u(x)=Ax$ for $x\in\partial \Omega$. In this paper we describe a regularization scheme for the computation of both $u_V$ and $E(u_V)$ and study its convergence properties. In particular, we show that as the regularization parameter goes to zero, (a subsequence) of the regularized constrained minimizers converge weakly in $W^{1,p}(\Omega\setminus{{\mathcal{B}}_{\delta}(x_0)})$ to a minimizer $u_{V}$ for any \delta>0. We obtain various sensitivity results for the dependence of the energies and Lagrange multipliers of the regularized constrained minimizers on the boundary data $A$ and on the volume parameter $V$. We show that both the regularized constrained minimizers and $u_V$ satisfy suitable weak versions of the corresponding Euler--Lagrange equations. In addition we describe the main features of a numerical scheme for approximating $u_V$ and $E(u_V)$ and give numerical examples for the case of a stored energy function of an elastic fluid and in the case of the incompressible limit

On the Structure of Linear Dislocation Field Theory

Uniqueness of solutions in the linear theory of non-singular dislocations, studied as a special case of plasticity theory, is examined. The status of the classical, singular Volterra dislocation problem as a limit of plasticity problems is illustrated by a specific example that clarifies the use of the plasticity formulation in the study of classical dislocation theory. Stationary, quasi-static, and dynamical problems for continuous dislocation distributions are investigated subject not only to standard boundary and initial conditions, but also to prescribed dislocation density. In particular, the dislocation density field can represent a single dislocation line. It is only in the static and quasi-static traction boundary value problems that such data are sufficient for the unique determination of stress. In other quasi-static boundary value problems and problems involving moving dislocations, the plastic and elastic distortion tensors, total displacement, and stress are in general non-unique for specified dislocation density. The conclusions are confirmed by the example of a single screw dislocation.Comment: This is the published versio

On the convergence of a regularization scheme for approximating cavitation solutions with prescribed cavity volume size

Let $\Omega\in\mathbb{R}^n$, $n=2,3$, be the region occupied by a hyperelastic body in its reference configuration. Let $E(\cdot)$ be the stored energy functional, and let $x_0$ be a flaw point in $\Omega$ (i.e., a point of possible discontinuity for admissible deformations of the body). For V>0 fixed, let $u_V$ be a minimizer of $E(\cdot)$ among the set of discontinuous deformations $u$ constrained to form a hole of prescribed volume $V$ at $x_0$ and satisfying the homogeneous boundary data $u(x)=Ax$ for $x\in\partial \Omega$. In this paper we describe a regularization scheme for the computation of both $u_V$ and $E(u_V)$ and study its convergence properties. In particular, we show that as the regularization parameter goes to zero, (a subsequence) of the regularized constrained minimizers converge weakly in $W^{1,p}(\Omega\setminus{{\mathcal{B}}_{\delta}(x_0)})$ to a minimizer $u_{V}$ for any \delta>0. We obtain various sensitivity results for the dependence of the energies and Lagrange multipliers of the regularized constrained minimizers on the boundary data $A$ and on the volume parameter $V$. We show that both the regularized constrained minimizers and $u_V$ satisfy suitable weak versions of the corresponding Euler--Lagrange equations. In addition we describe the main features of a numerical scheme for approximating $u_V$ and $E(u_V)$ and give numerical examples for the case of a stored energy function of an elastic fluid and in the case of the incompressible limit

Sorption of Arsenic(III) from wastewater using Prosopis spicigera L. wood (PsLw) carbon-polyaniline composite

Water contamination by toxic heavy metal ions causes a serious public health problem for humans. The present work reports the development of a new adsorbent of PsLw carbon-polyaniline composite by direct oxidation polymerisation of aniline with PsLw carbon for the removal of arsenic (As).  The structure and morphologies of the adsorbent were characterised by Fourier transform infrared spectroscopy (FTIR) and Scanning electron microscopy (SEM). The ability of the adsorbent for the removal of As(III) was estimated by batch and kinetic studies. The optimum adsorption behaviour of the adsorbent was measured at pH=6.0. The equilibrium process was found to be in good agreement with Langmuir adsorption isotherm and the maximum adsorption capacity was 98.8 mg/g for an initial concentration of 60 mg/L at 30 °C. The kinetic study followed pseudo-second-order kinetics. Thermodynamic parameters predict the spontaneous, feasible and exothermic nature of adsorption. Column operation was carried out to remove As(III) bulk and column data obeys the Thomas model. The results indicated that PsLw carbon-polyaniline composite can be employed as an efficient adsorbent than polyaniline for removal of As(III) from wastewater