Romanian Academy

About: Romanian Academy is a archive organization based out in Bucharest, Romania. It is known for research contribution in the topics: Population & Nonlinear system. The organization has 3662 authors who have published 10491 publications receiving 146447 citations. The organization is also known as: Academia Română & Societatea Literară Română.

The exponentiated Hencky-Logarithmic strain energy: Part II: Coercivity, planar polyconvexity and existence of minimizers

Abstract: We consider a family of isotropic volumetric–isochoric decoupled strain energies $$F \mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll}\frac{\mu}{k}\,e^{k\,\|{\rm dev}_n{\rm log} {U}\|^2}+\frac{\kappa}{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}({\rm log} U)]^2}&\text{if}& \det\, F > 0,\\+\infty &\text{if} &\det F\leq 0,\end{array}\right.$$ based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, \({\kappa=\frac{2\mu+3\lambda}{3} > 0}\) is the infinitesimal bulk modulus with \({\lambda}\) the first Lame constant, \({k,\hat{k}}\) are dimensionless parameters, \({F= abla \varphi}\) is the gradient of deformation, \({U=\sqrt{F^T F}}\) is the right stretch tensor and \({{\rm dev}_n{\rm log} {U} ={\rm log} {U}-\frac{1}{n}{\rm tr}({\rm log} {U})\cdot{1\!\!1}}\) is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy $$\begin{array}{ll}F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U)&:={\mu}\,\|{\rm dev}_n{\rm log} U\|^2+\frac{\kappa}{2}\,[{\rm tr}({\rm log} U)]^2,\end{array}$$ which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family \({W_{_{\rm eH}}}\) are polyconvex for \({k\geq \frac{1}{3},\,\widehat{k}\geq \frac{1}{8}}\) , extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann’s polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations.

Inter-comparison of the g-, f- and p-modes calculated using different oscillation codes for a given stellar model

Abstract: In order to make asteroseismology a powerful tool to explore stellar interiors, different numerical codes should give the same oscillation frequencies for the same input physics. Any differences found when comparing the numerical values of the eigenfrequencies will be an important piece of information regarding the numerical structure of the code. The ESTA group was created to analyze the non-physical sources of these differences. The work presented in this report is a part of Task 2 of the ESTA group. Basically the work is devoted to test, compare and, if needed, optimize the seismic codes used to calculate the eigenfrequencies to be finally compared with observations. The first step in this comparison is presented here. The oscillation codes of nine research groups in the field have been used in this study. The same physics has been imposed for all the codes in order to isolate the non-physical dependence of any possible difference. Two equilibrium models with different grids, 2172 and 4042 mesh points, have been used, and the latter model includes an explicit modelling of semiconvection just outside the convective core. Comparing the results for these two models illustrates the effect of the number of mesh points and their distribution in particularly critical parts of the model, such as the steep composition gradient outside the convective core. A comprehensive study of the frequency differences found for the different codes is given as well. These differences are mainly due to the use of different numerical integration schemes. The number of mesh points and their distribution are crucial for interpreting the results. The use of a second-order integration scheme plus a Richardson extrapolation provides similar results to a fourth-order integration scheme. The proper numerical description of the Brunt-Vaisala frequency in the equilibrium model is also critical for some modes. This influence depends on the set of the eigenfunctions used for the solution of the differential equations. An unexpected result of this study is the high sensitivity of the frequency differences to the inconsistent use of values of the gravitational constant (G) in the oscillation codes, within the range of the experimentally determined ones, which differ from the value used to compute the equilibrium model. This effect can provide differences for a given equilibrium model substantially larger than those resulting from the use of different codes or numerical techniques; the actual differences between the values of G used by the different codes account for much of the frequency differences found here.