Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors

<p style='text-indent:20px;'><inline-formula><tex-math id="M1">\begin{document}$ M $\end{document}</tex-math></inline-formula>-eigenvalues of partially symmetric nonnegative tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we establish two upper bounds for the maximum <inline-formula><tex-math id="M2">\begin{document}$ M $\end{document}</tex-math></inline-formula>-eigenvalue of partially symmetric nonnegative tensors, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.</p>

DURABILITY ASSESSMENT OF BENDING STRUCTURES MADE OF NONLINEAR ELASTIC MATERIAL

Experimental studies and field tests indicate that the effect of corrosive media leads to significant changes in the physical and mechanical characteristics of structural materials. The article proposes a mathematical model that allows predicting the negative impact of aggressive media and assessing the durability of bent structures.

Sharp Bounds on the Minimum M-Eigenvalue of Elasticity M-Tensors

The M-eigenvalue of elasticity M-tensors play important roles in nonlinear elastic material analysis. In this paper, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors without irreducible conditions, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.

M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors

M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.

Use of generalized finite difference method for calculation of beams from nonlinear elastic material

The introduction into practice of construction of structures made of high-strength steels and other materials having a nonlinear deformation diagram caused the active development of the nonlinear theory of calculation of structures. Replacing Hooke’s law with nonlinear dependencies between stresses and strains leads to so-called physical nonlinearity. For the calculation of such structures, the experimentally obtained dependences between stresses and strains are described using analytical expressions. A number of variants of such approximations have been proposed by various researchers. In this paper, we consider the calculation of beams of symmetrical cross-section made of nonlinear elastic material, for which the dependence between stresses and strains is described by a cubic parabola. This approximation ensures the symmetry of the diagram σ with respect to the tension σ – ɛ compression, and also gives a good match with the experimental curve. The use of the generalized finite difference method for solving the problem allows to reduce the system of nonlinear differential equations to the system of algebraic equations, for the solution of which the method of successive approximations is used. Studies have shown that the proposed method of calculation makes it possible to obtain a fairly accurate solution for a small number of elements. In addition, the presented calculation algorithm is convenient for programming and numerical calculation. As an example of systems that allow calculation using the considered algorithm, beam elements of building structures can act. The calculation of a beam from a nonlinear elastic material on the action of a concentrated force is given.

A new method for determination of Poisson’s ratio of woven fabric at higher stresses

Determination of Poisson’s ratio and shear module of complex, nonlinear elastic material, such as woven fabric, is a challenge for researchers in the field of textile mechanics. In the standard method of determining the Poisson’s ratio, the transverse fabric strain is measured by a 1% tensile extension. In this way, there is no information about changing the Poisson’s ratio at higher tensile extensions, and the methodology itself is unsuitable for larger extensions because of woven fabric buckling. In this research, a device has been designed, which can be built on a dynamometer and which has the ability to measure transverse forces in fabric during tensile test. A mechanical model is developed from which it is possible to calculate Poisson’s ratio throughout the fabric stress–strain curve.

Modal analysis of masonry structures

This paper presents a new numerical tool for evaluating the vibration frequencies and mode shapes of masonry buildings in the presence of cracks. The algorithm has been implemented within the NOSA-ITACA code, which models masonry as a nonlinear elastic material with zero tensile strength. Some case studies are reported, and the differences between linear and nonlinear behaviour are highlighted.