Seminar khoa học của GS.TS Julius Kaplunov

Vào 9h00, ngày 23/12/2024 Viện IAST tổ chức buổi trao đổi học thuật tại Phòng họp A104 với nội dung chi tiết như sau: 

GS.TS Julius Kaplunov trình bày về "A dynamic theory for a thin functionally graded plate"

Tóm tắt:

3D dynamic equations in linear elasticity for a transversely inhomogeneous isotropic layer are analysed. The thickness of the layer is assumed to be small in comparison with a typical wavelength, while Young’s modulus, Poisson’s ratio and mass density are defined as arbitrary functions in the transverse variable.  The same asymptotic scaling, as in the homogeneous setup, is adapted for 3D displacements and stresses. In this case, characteristic time and length scales are related to each other as in the classical Kirchhoff theory for plate bending. As usual, all the sought for quantities are expanded in series in terms of a small geometric parameter corresponding to the relative thickness. At leading order, we arrive at a 2D fourth order equation for bending motion taking the same form as that in the Kirchhoff theory to within the expressions for constant coefficients, which are now given by certain integrals across the thickness involving variable problem parameters. The asymmetry of the 3D original problem with respect to the transverse variable results in presence of quasi-static extension governed at leading order by 2D equations similar to those for the generalised plane stress. It is remarkable that, in spite of asymmetry, the aforementioned problem can be decoupled, i.e. the bending sub problem is solved independently from the extension one, whereas the latter is treated with terms determined from the solution of the bending sub problem in its right-hand side. It is also worth noting that the adhoc engineering formulations for functionally graded plates do not take into consideration the possibility of such decoupling leading to differential equations of a doubled order. The next order asymptotic approximation for plate bending is also derived. The equation of motion is still of the fourth order as at the leading order approximation, i.e. it does not support spurious solutions similar to the homogeneous setup. Higher-order corrections come through a mixed fourth order time space derivative. However, the constant coefficients in the refined 2D bending equation are expressed through rather sophisticated multiple integrals across the thickness. Nevertheless, decoupling of bending motions and extension de formations occur at higher order as well. Any comparison with related adhoc refined plate bending does not seem to be fruitful. The point is that the asymptotic cross section thickness variation of the displacement field at higher order nontrivially depends on variable problem parameters and cannot be approximated through simple polynomials typical of many engineering assumptions. In addition, adhoc models usually deal with stress resultants and stress couples and therefore neglect the peculiarities of the cross-thickness variation of the stress field, which is of particular importance namely for functionally graded structures. The developed asymptotic framework is also valid for layered plates with piecewise uniform problem parameters. It also allows various extensions, including analysis of functionally graded shells, coatings and interfacial layers.