Journal of Fractional Calculus and Nonlinear Systems, 2021
In this paper, we determine some expansion formulae of the incomplete I-functions in affiliation ... more In this paper, we determine some expansion formulae of the incomplete I-functions in affiliation with the Leibniz rule for the Riemann-Liouville type derivatives. Further, expansion formulae of the incomplete $\overline{I}$-function, incomplete $\overline{H}$-function, and incomplete H-function are conferred as extraordinary instances of our primary outcomes.
Moroccan Journal of Pure and Applied Analysis, 2020
In this article, we have derived some integral transforms of the polynomial weighted incomplete H... more In this article, we have derived some integral transforms of the polynomial weighted incomplete H-functions and incomplete ̄H-functions. The obtained image formulas are of general nature and may, as special cases, give rise to integral transforms involved with the H-functions and ̄H-functions.
In this paper, we derive certain Chebyshev type integral inequalities connected with a fractional... more In this paper, we derive certain Chebyshev type integral inequalities connected with a fractional integral operator in terms of the generalized Mittag-Leffler multi-index function as a kernel. Our key findings are general in nature and, as a special case, can give rise to integral inequalities of the Chebyshev form involving fractional integral operators present in the literature.
The desire for present article is to derive from the application of fractional calculus operators... more The desire for present article is to derive from the application of fractional calculus operators a transformation that expresses a potentially useful incomplete hypergeometric function in various forms of a countable sum of lesser-order functions. Often listed are numerous (known or new) specific cases and implications of the findings described herein
In this paper, we assess an integral containing incomplete H-functions and utilize it to build up... more In this paper, we assess an integral containing incomplete H-functions and utilize it to build up an expansion formula for the incomplete H-functions including the Bessel function. Next, we evaluate an integral containing incomplete H̅-functions and use it to develop an expansion formula for the incomplete H̅-functions including the Bessel function. The outcomes introduced in this paper are general in nature, and several particular cases can be acquired by giving specific values to the parameters engaged with the principle results. As particular cases, we derive expansions for the incomplete Meijer ${}^{(\Gamma )}G$ G ( Γ ) -function, Fox–Wright ${}_{p}\Psi _{q}^{(\Gamma )}$ Ψ q ( Γ ) p -function, and generalized hypergeometric ${}_{p}\Gamma _{q}$ Γ q p function.
Mathematical structures are used in the areas of biology, physics, engineering and social science... more Mathematical structures are used in the areas of biology, physics, engineering and social sciences. A model will help us interpret the system, analyze the impacts of various components, and make behavioral predictions. In this paper, we developed an internal blood pressure model that involves incomplete I-functions. Next, while taking a course in the constraints of incomplete Ifunctions, we give a few special cases of our model, and also mention some known results.
The principal aim of this article is to establish certain generalized fractional integral inequal... more The principal aim of this article is to establish certain generalized fractional integral inequalities by utilizing the Marichev-Saigo-Maeda (MSM) fractional integral operator. Some new classes of generalized fractional integral inequalities for a class of n (n ∈ N) positive continuous and decreasing functions on [a, b] by using the MSM fractional integral operator also derived.
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2013
ABSTRACT In this paper, a mode-I crack problem is address for a piezoelectric plate cut along two... more ABSTRACT In this paper, a mode-I crack problem is address for a piezoelectric plate cut along two-equal-collinear semi-permeable cracks. The plate being subjected to combined in-plane mechanical and electric displacement loads, respectively. Problem is formulated using Stroh formalism and solved using complex variable technique. Closed form expressions are derived for crack opening displacement, crack opening potential drop, field intensity factors, mechanical and total energy release rates. Theoretical derivations are validated by exact solutions existing in literature. Numerical examples considered for poled ceramics PZT-4, PZT-5H, and PZT-7A show the effect of applied mechanical and electrical displacement loadings on field intensity factors, mechanical and total energy release rates. Moreover, the effect of inter-crack distance as well as different types of crack-face electric boundary conditions on field intensity factors, mechanical and total energy release rates are presented graphically, discussed, and concluded.
ABSTRACT The problem of two unequal collinear straight cracks weakening a poled transversely isot... more ABSTRACT The problem of two unequal collinear straight cracks weakening a poled transversely isotropic piezoelectric ceramic is addressed under semi-permeable electric boundary conditions on the crack faces. The plate has been subjected to combined in-plane normal(to the faces of the cracks) mechanical and electric loads. Problem is formulated employing Stroh formalism and solved using complex variable technique. The elastic field, electric field and energy release rate are obtained in closed analytic form. A case study is presented for poled PZT-5H cracked plate to study the effect of prescribed mechanical load, electric load, inter-crack distance and crack lengths on crack arrest parameters stress intensity factor (SIF), electric displacement intensity factor (EDIF) and mechanical and total energy release rates (ERR). Moreover a comparative study is done of impermeable and semi-permeable crack face boundary conditions on SIF, EDIF and ERR, and results obtained is presented graphically. It is observed that the effect of dielectric medium in the crack gap cannot be ignored.
A strip-saturation model is proposed for a transversely isotropic piezoelectric plane weakened by... more A strip-saturation model is proposed for a transversely isotropic piezoelectric plane weakened by two collinear equal cracks, when developed saturation zones at the interior tips of the cracks get coalesced. The plane is subjected to unidirectional, normal (to the crack length) in-plane tension and electric displacement. The developed saturation zones are arrested by distributing over their rims the normal, cohesive, unidirectional saturationlimit electrical displacement. The solution is obtained using Stroh formulation and complex variable technique. Closed form expressions are derived for crack opening displacement (COD), crack potential drop (COP), field intensity factors, length of saturation zone, energy release rate. Case study carried out for PZT-4 to show the effects of inter-crack distance on the stress intensity factor. The variations of energy release rates are plotted for PZT-4, PZT-5H and BaTiO 3 to study the effects of the geometry of the two cracks.
In this paper, a mode-I crack problem is address for a piezoelectric plate cut along two-equal-co... more In this paper, a mode-I crack problem is address for a piezoelectric plate cut along two-equal-collinear semi-permeable cracks. The plate being subjected to combined in-plane mechanical and electric displacement loads, respectively. Problem is formulated using Stroh formalism and solved using complex variable technique. Closed form expressions are derived for crack opening displacement, crack opening potential drop, field intensity factors, mechanical and total energy release rates. Theoretical derivations are validated by exact solutions existing in literature. Numerical examples considered for poled ceramics PZT-4, PZT-5H, and PZT-7A show the effect of applied mechanical and electrical displacement loadings on field intensity factors, mechanical and total energy release rates. Moreover, the effect of inter-crack distance as well as different types of crack-face electric boundary conditions on field intensity factors, mechanical and total energy release rates are presented graphica...
A plane problem for a poled transversely isotropic piezoelectric plane cut along two equal collin... more A plane problem for a poled transversely isotropic piezoelectric plane cut along two equal collinear straight cracks is considered. It is assumed that the electrical yielding occur at the continuations of the cracks due to the applied mechanical and electrical loadings. We model these crack continuations as the zones with constant cohesive saturation limit electrical displacement. Stroh formalism and complex variable technique is adopted to obtain the analytic solution of the problem. Closed form expressions are derived for developed saturation zone length, crack opening displacement; crack opening potential drop, stress intensity factors, energy release rate. A qualitative numerical case study is presented for PZT-4, PZT-5H, and BaTiO3 ceramics to study the effects of various parameters viz. developed saturation zone length and prescribed load, stress intensity factor, energy release rate, and crack opening displacement on crack growth resistance. Energy release rate and stress int...
The multiple-crack problems for piezoelectric ceramics till now have not yet address the crack op... more The multiple-crack problems for piezoelectric ceramics till now have not yet address the crack opening arrest problem. The present work addresses this paucity. A 2-D strip-electro-mechanical yielding model is proposed for a transversely isotropic piezoelectric media weakened by two internal equal collinear straight cracks. The infinite boundary is prescribed with combined uniform constant in-plane mechanical and electrical loads. Developed mechanical and electric strip zones are arrested by prescribing over their rims uniform, normal, cohesive yield point stress and saturation limit electric displacement. Two cases are considered when saturation zone is bigger than developed yield zone and vice-versa. Stroh formulation together with complex variable technique is employed to obtain the solution. Closed form expressions are derived for saturation zone length, yield zone length, crack opening displacement (COD), crack opening potential jump (COP) and energy release rate (ERR). An illus...
Journal of Fractional Calculus and Nonlinear Systems, 2021
In this paper, we determine some expansion formulae of the incomplete I-functions in affiliation ... more In this paper, we determine some expansion formulae of the incomplete I-functions in affiliation with the Leibniz rule for the Riemann-Liouville type derivatives. Further, expansion formulae of the incomplete $\overline{I}$-function, incomplete $\overline{H}$-function, and incomplete H-function are conferred as extraordinary instances of our primary outcomes.
Moroccan Journal of Pure and Applied Analysis, 2020
In this article, we have derived some integral transforms of the polynomial weighted incomplete H... more In this article, we have derived some integral transforms of the polynomial weighted incomplete H-functions and incomplete ̄H-functions. The obtained image formulas are of general nature and may, as special cases, give rise to integral transforms involved with the H-functions and ̄H-functions.
In this paper, we derive certain Chebyshev type integral inequalities connected with a fractional... more In this paper, we derive certain Chebyshev type integral inequalities connected with a fractional integral operator in terms of the generalized Mittag-Leffler multi-index function as a kernel. Our key findings are general in nature and, as a special case, can give rise to integral inequalities of the Chebyshev form involving fractional integral operators present in the literature.
The desire for present article is to derive from the application of fractional calculus operators... more The desire for present article is to derive from the application of fractional calculus operators a transformation that expresses a potentially useful incomplete hypergeometric function in various forms of a countable sum of lesser-order functions. Often listed are numerous (known or new) specific cases and implications of the findings described herein
In this paper, we assess an integral containing incomplete H-functions and utilize it to build up... more In this paper, we assess an integral containing incomplete H-functions and utilize it to build up an expansion formula for the incomplete H-functions including the Bessel function. Next, we evaluate an integral containing incomplete H̅-functions and use it to develop an expansion formula for the incomplete H̅-functions including the Bessel function. The outcomes introduced in this paper are general in nature, and several particular cases can be acquired by giving specific values to the parameters engaged with the principle results. As particular cases, we derive expansions for the incomplete Meijer ${}^{(\Gamma )}G$ G ( Γ ) -function, Fox–Wright ${}_{p}\Psi _{q}^{(\Gamma )}$ Ψ q ( Γ ) p -function, and generalized hypergeometric ${}_{p}\Gamma _{q}$ Γ q p function.
Mathematical structures are used in the areas of biology, physics, engineering and social science... more Mathematical structures are used in the areas of biology, physics, engineering and social sciences. A model will help us interpret the system, analyze the impacts of various components, and make behavioral predictions. In this paper, we developed an internal blood pressure model that involves incomplete I-functions. Next, while taking a course in the constraints of incomplete Ifunctions, we give a few special cases of our model, and also mention some known results.
The principal aim of this article is to establish certain generalized fractional integral inequal... more The principal aim of this article is to establish certain generalized fractional integral inequalities by utilizing the Marichev-Saigo-Maeda (MSM) fractional integral operator. Some new classes of generalized fractional integral inequalities for a class of n (n ∈ N) positive continuous and decreasing functions on [a, b] by using the MSM fractional integral operator also derived.
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2013
ABSTRACT In this paper, a mode-I crack problem is address for a piezoelectric plate cut along two... more ABSTRACT In this paper, a mode-I crack problem is address for a piezoelectric plate cut along two-equal-collinear semi-permeable cracks. The plate being subjected to combined in-plane mechanical and electric displacement loads, respectively. Problem is formulated using Stroh formalism and solved using complex variable technique. Closed form expressions are derived for crack opening displacement, crack opening potential drop, field intensity factors, mechanical and total energy release rates. Theoretical derivations are validated by exact solutions existing in literature. Numerical examples considered for poled ceramics PZT-4, PZT-5H, and PZT-7A show the effect of applied mechanical and electrical displacement loadings on field intensity factors, mechanical and total energy release rates. Moreover, the effect of inter-crack distance as well as different types of crack-face electric boundary conditions on field intensity factors, mechanical and total energy release rates are presented graphically, discussed, and concluded.
ABSTRACT The problem of two unequal collinear straight cracks weakening a poled transversely isot... more ABSTRACT The problem of two unequal collinear straight cracks weakening a poled transversely isotropic piezoelectric ceramic is addressed under semi-permeable electric boundary conditions on the crack faces. The plate has been subjected to combined in-plane normal(to the faces of the cracks) mechanical and electric loads. Problem is formulated employing Stroh formalism and solved using complex variable technique. The elastic field, electric field and energy release rate are obtained in closed analytic form. A case study is presented for poled PZT-5H cracked plate to study the effect of prescribed mechanical load, electric load, inter-crack distance and crack lengths on crack arrest parameters stress intensity factor (SIF), electric displacement intensity factor (EDIF) and mechanical and total energy release rates (ERR). Moreover a comparative study is done of impermeable and semi-permeable crack face boundary conditions on SIF, EDIF and ERR, and results obtained is presented graphically. It is observed that the effect of dielectric medium in the crack gap cannot be ignored.
A strip-saturation model is proposed for a transversely isotropic piezoelectric plane weakened by... more A strip-saturation model is proposed for a transversely isotropic piezoelectric plane weakened by two collinear equal cracks, when developed saturation zones at the interior tips of the cracks get coalesced. The plane is subjected to unidirectional, normal (to the crack length) in-plane tension and electric displacement. The developed saturation zones are arrested by distributing over their rims the normal, cohesive, unidirectional saturationlimit electrical displacement. The solution is obtained using Stroh formulation and complex variable technique. Closed form expressions are derived for crack opening displacement (COD), crack potential drop (COP), field intensity factors, length of saturation zone, energy release rate. Case study carried out for PZT-4 to show the effects of inter-crack distance on the stress intensity factor. The variations of energy release rates are plotted for PZT-4, PZT-5H and BaTiO 3 to study the effects of the geometry of the two cracks.
In this paper, a mode-I crack problem is address for a piezoelectric plate cut along two-equal-co... more In this paper, a mode-I crack problem is address for a piezoelectric plate cut along two-equal-collinear semi-permeable cracks. The plate being subjected to combined in-plane mechanical and electric displacement loads, respectively. Problem is formulated using Stroh formalism and solved using complex variable technique. Closed form expressions are derived for crack opening displacement, crack opening potential drop, field intensity factors, mechanical and total energy release rates. Theoretical derivations are validated by exact solutions existing in literature. Numerical examples considered for poled ceramics PZT-4, PZT-5H, and PZT-7A show the effect of applied mechanical and electrical displacement loadings on field intensity factors, mechanical and total energy release rates. Moreover, the effect of inter-crack distance as well as different types of crack-face electric boundary conditions on field intensity factors, mechanical and total energy release rates are presented graphica...
A plane problem for a poled transversely isotropic piezoelectric plane cut along two equal collin... more A plane problem for a poled transversely isotropic piezoelectric plane cut along two equal collinear straight cracks is considered. It is assumed that the electrical yielding occur at the continuations of the cracks due to the applied mechanical and electrical loadings. We model these crack continuations as the zones with constant cohesive saturation limit electrical displacement. Stroh formalism and complex variable technique is adopted to obtain the analytic solution of the problem. Closed form expressions are derived for developed saturation zone length, crack opening displacement; crack opening potential drop, stress intensity factors, energy release rate. A qualitative numerical case study is presented for PZT-4, PZT-5H, and BaTiO3 ceramics to study the effects of various parameters viz. developed saturation zone length and prescribed load, stress intensity factor, energy release rate, and crack opening displacement on crack growth resistance. Energy release rate and stress int...
The multiple-crack problems for piezoelectric ceramics till now have not yet address the crack op... more The multiple-crack problems for piezoelectric ceramics till now have not yet address the crack opening arrest problem. The present work addresses this paucity. A 2-D strip-electro-mechanical yielding model is proposed for a transversely isotropic piezoelectric media weakened by two internal equal collinear straight cracks. The infinite boundary is prescribed with combined uniform constant in-plane mechanical and electrical loads. Developed mechanical and electric strip zones are arrested by prescribing over their rims uniform, normal, cohesive yield point stress and saturation limit electric displacement. Two cases are considered when saturation zone is bigger than developed yield zone and vice-versa. Stroh formulation together with complex variable technique is employed to obtain the solution. Closed form expressions are derived for saturation zone length, yield zone length, crack opening displacement (COD), crack opening potential jump (COP) and energy release rate (ERR). An illus...
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