About rational distribution of porosity at torsion of the bar of rectangular cross section

The article presents the issues of optimum design of structures by controlling material properties due to the level of porosity.
We consider the problem of pure torsion of a bar of rectangular cross section made of a material (steel) of a porous structure with variable cross-section porosity. We restrict ourselves to the case of elastic deformation, assuming that the maximum shear stress in the beam (${\tau}_{\mathrm{max}}$) does not exceed the fluidity limit of the material in shear (${\tau}_{T}$). It is known that the elastic characteristics of the material (module of shear G and the exertion of the fluidity start (${\sigma}_{T}$)) are functions of the porosity of the material. Smoothing the experimental data for the porous of steel by the method of least squares, we obtain the dependence $G(p), \sigma (p), \tau (p)$.
Since the variable in the cross section of the porosity leads to inhomogeneous material properties, we use the theory of inhomogeneous torsion of beams with the introduction of the functions of exertions $\Phi$. Boundary condition on the outer contour $L$ for the function of exertions $\Phi$ will be equality of $0$.
The solution of the boundary problem we replace by finding the minimum equivalent of functionality. For finding the minimum of the functional we use the method of finite elements. So we divide the cross section of the beam on rectangular triangles with sides, which are orientated to the coordinate $x$ and $y$. We approximate the function of the exertions $\Phi$ in each element of the linear spline. For other elements, the coefficients turn out to be the permutation of indices.
Substituting the obtained dependence in the functional, integrating and using the Ritz procedure, we obtain the system of equations, which is solved by the method of successive iterations.
The found value of function of the exertions $\Phi$ allows to calculate the components of the shear stress and consequently the shear stress in arbitrary point of the section.
Based on the obtained values of shear stresses, porosity is corrected in the cross section according to the scheme of successive approximations.
For instance, we consider a beam of rectangular cross section of $10 \times 5$ cm, which is made of porous steel. We divide it into $100$ pieces on the larger side and $50$ on the smaller one.
The numerical solution confirms that the minimum level of tangential stresses (near the corner points and in the center) porosity is higher, and the maximum level of tangential stresses (in the middle of sides) is below a certain average. In addition, the meaning of rotational moment for beams with rationally chosen porosity on the cross section ($М_{кр} = 2,19192 \, кНм$) $31.3 \%$ higher than for beams with average porosity ($М_{кр} = 1,6700 \, кНм$). That is the evidence on rationalization of the distribution of porosity in a cross section.