Vol 25, No 3 (2019)

We establish solvability and correctness criteria for two Fredholm type linear integro-differential operators B2, B4 encompassing up to second and fourth powers, respectively, of a differential operator A� with aknown inverse I = A�−1. We also derive explicit solution formulae to corresponding initial and boundary value problems by using the inverse of the differential operator. The approach is based on the theory of the extensions of linear operators in Banach spaces. Three example problems for ordinary and partial integro-differential operators are solved.

In the article the Cauchy problem for the one system of the differential equations of the fourth order is received in the plane of two independent variables. This system of the hyperbolic differential equations of the fourth order does not contain derivatives less than the fourth order. The regular solution of the Cauchy problem for the system of the hyperbolic differential equations of the fourth order is explicitly built. The solution of the Cauchy problem for the system of the hyperbolic differential equations of the fourth order is found by the Riman method. In the paper the matrix of Riman for the system of the hyperbolic differential equations of the fourth order is constructed also. The matrix of Riman is expressed through hypergeometrical functions of matrix argument.

In the present work, photoelastic and finite element methods have been employed to study the near crack tip fields in isotropic linear elastic cracked bodies under mixed mode loading. The investigated fracture results have been obtained for a series of cracked specimens by testing plates with two parallel cracks, two inclined parallel cracks, three-point bend specimens, four-point bend specimens, inclined edge crack triangular shape specimens subjected to symmetric three point bend loading. The multi-parameter Williams series expansion is used for the crack tip field characterization. Digital photoelasticity method is utilized for determination of the Williams series expansion’s coefficients. The unknown coefficients in the multi-parameter equation are determined using a linear least squares method in an over-deterministic manner. Together with the experimental determination of the fracture mechanics parameters finite element method is invoked to describe the crack tip stress field. Coefficients of higher order terms are either found numerically by finite element method. A good agreement is found between the numerical and experimental results. The significant advantages using multi-parameter equations in the analysis of the stress field are shown and the errors that a study with a limited number of terms produce is demonstrated. The comparison with finite element analysis highlighted the importance and precision of the photoelastic observation for the evaluation of the fracture mechanics parameters. The experimental SIF, T-stress values and coefficients of higher-order terms estimated using the digital photoelasticity method for the extensive series of cracked specimens are compared with finite element analysis (FEA) results, and are found to be in good agreement.