Approximate solution of three-dimensional problems in elasticity theory for a transversely isotropic medium

1969 ◽  
Vol 5 (8) ◽  
pp. 803-810 ◽  
Author(s):  
Yu. N. Nemish
Keyword(s):  
Approximate Solution ◽  
Elasticity Theory ◽  
Isotropic Medium ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium

scholarly journals On thermoelastic deformation of a transversally isotropic medium in 3D construction printing

2021 ◽  
Vol 2131 (3) ◽  
pp. 032024
Author(s):  
Yu Chirkunov ◽  
E Pikmullina ◽  
I Gasenko
Keyword(s):  
Differential Equations ◽  
Isotropic Medium ◽  
Group Analysis ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Thermoelastic Deformation ◽  
Transversally Isotropic ◽  
Materials Used ◽  
And Function

Abstract A three-dimensional dynamic model of a thermoelastic transversely isotropic medium is used to describe the thermoelastic deformation of materials with anisotropy of elastic properties with a selected direction of anisotropy. Such materials are layered and composite materials used in construction, mechanical engineering, aircraft and shipbuilding, soils in permafrost conditions, glaciers, as well as rocks (basalt, sandstone, marble, limestone, shale, and others). The study of this model, in particular, is relevant in connection with the use of 3D printers in construction. This is due to the fact that it is necessary to select the heating mode of the 3D printer head, in which cracks will not form during the cooling of the polystyrene concrete layers.We study this model using the group analysis methods, which is one of the most powerful and effective tools for obtaining exact solutions. The group stratification of the system of second-order differential equations defining this model is carried out. A system of first-order differential equations is obtained, which is equivalent to the equations of the original model. The solution describing a traveling wave for this system is obtained, that depends on arbitrary elements: parameters and function. For the specific sets of these elements, we study a deformation of a sphere and cube located inside a thermoelastic transversely isotropic medium with increasing time is found. The corresponding graphs are given.

Indentation of a Penny-Shaped Crack by an Oblate Spheroidal Rigid Inclusion in a Transversely Isotropic Medium

1984 ◽  
Vol 51 (4) ◽  
pp. 811-815 ◽  
Author(s):  
Y. M. Tsai
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Contact Stress ◽  
Rigid Inclusion ◽  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Penny Shaped Crack ◽  
Shaped Crack

The stress distribution produced by the identation of a penny-shaped crack by an oblate smooth spheroidal rigid inclusion in a transversely isotropic medium is investigated using the method of Hankel transforms. This three-part mixed boundary value problem is solved using the techniques of triple integral equations. The normal contact stress between the crack surface and the indenter is written as the product of the associated half-space contact stress and a nondimensional crack-effect correction function. An exact expression for the stress-intensity is obtained as the product of a dimensional quantity and a nondimensional function. The curves for these nondimensional functions are presented and used to determine the values of the normalized stress-intensity factor and the normalized maximum contact stress. The stress-intensity factor is shown to be dependent on the material constants and increasing with increasing indentation. The stress-intensity factor also increases if the radius of curvature of the indenter surface increases.

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