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Computational Materials
Published in Theodore H.H. Pian, Chang-Chun Wu, Hybrid and Incompatible Finite Element Methods, 2005
Theodore H.H. Pian, Chang-Chun Wu
In the equations above, Tijkl is a fourth-order unit tensor Tijkl=1/2(δikδjl+δilδjk), where ij is the Kronecker symbol. Npqkl is also a fourth-order unit tensor, and Mpk and Hpk are second-order unit tensors.
Applied Mathematic Technologies in Nonlinear Mechanics of Thin-Walled Constructions
Published in Mangey Ram, S. B. Singh, Mathematics Applied to Engineering and Management, 2020
and where θ=ε11+ε22+ε33 is volume deformation (relative volume change), T1=‖δij‖ is unit tensor, δij is Kronecker symbol. Deformation intensity is defined by the formula () ei=23(ε11−ε22)2+(ε22−ε33)2+(ε33−ε11)2+6(ε122+ε132+ε232).
Numerical analysis for free flow through side rectangular orifice in an open channel
Published in ISH Journal of Hydraulic Engineering, 2021
where Ui is the time-averaged velocity component in xi; t is the time; xi (i = 1,2,3) is the perpendicular axis; is the density of fluid; P is the time-averaged pressure of fluid; is kinetic viscosity of fluid; ; is Kronecker symbol; Fi is the volumetric force acting on per unit weight; k = is turbulent kinetic energy per unit mass; is turbulent viscosity coefficient; G = is shear stress production term; is dissipating rate of turbulent kinetic energy. , are general constants with the values of 0.09, 1.44, 1.92, 1.0 and 1.3.
Thermoelastic processes analyzer for piecewise homogeneous conductive structures subjected to pulsed electromagnetic actions
Published in Journal of Thermal Stresses, 2018
Roman Musii, Nataliya Melnyk, Veronika Dmytruk
Here is the deformation tensor; is the deformer; is the del operator; is the unit tensor; is the Kronecker symbol; , are the dyadic product of the vectors and ; , are the coefficient of linear thermal expansion and the Poisson's ratio, respectively; is the shear modulus; is Young's modulus; is the density of the -th solid material. All thermophysical and physicomechanical characteristics of the materials of the constituent solids are considered to be constant.
Numerical study of the Richtmyer–Meshkov instability induced by non-planar shock wave in non-uniform flows
Published in Journal of Turbulence, 2019
Zhen Wang, Tao Wang, Jingsong Bai, Jiaxin Xiao
The simulations are run with a large-eddy simulation code MVFT (multi-viscous flow and turbulence). The governing equations of MVFT are the Favre-filtered compressible multi-viscous-flow Navier-Stokes equations which can be written as where i and j are the three directions of x and y. Here , , , and stand for the resolved-scale fluid density, the velocity, the pressure, and the total energy per unit mass, respectively; N refers to the types of fluids; is the volume fraction of the sth fluids which satisfies ; is the diffusion coefficient, with υ the kinematic viscosity and Sc the Schmidt number; is the Newtonian fluid viscous stress tensor, with the Kronecker symbol; is the subgrid scale stress tensor; and is the energy flux per unit time and space in resolved scales, is the resolved heat conduction coefficient with the dynamical viscosity, the specific heat of fluid, and the Prandtl number, is the temperature of fluid; is the energy flux per unit time and space in subgrid scales, is the subgrid heat conduction coefficient with the turbulent dynamical viscosity, and the turbulent Prandtl number. The equation of state adopts the ideal gas state form.