Analogues of the Kolosov–Muskhelishvili General Representation Formulas and Cauchy–Riemann Conditions in the Theory of Elastic Mixtures

1997 ◽  
Vol 4 (3) ◽  
pp. 223-242
Author(s):  
M. Basheleishvili
Keyword(s):  
Stress Tensor ◽  
Analytic Functions ◽  
Stress Vector ◽  
General Representation ◽  
Representation Formulas ◽  
Cauchy Riemann ◽  
Theory Of Elastic Mixtures

Abstract Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions. The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.

scholarly journals Unsteady Helical Flows of a Size-Dependent Couple-Stress Fluid

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Qammar Rubbab ◽  
Itrat Abbas Mirza ◽  
Imran Siddique ◽  
Saadia Irshad
Keyword(s):  
Stress Tensor ◽  
Couple Stress ◽  
Integral Transform ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Fluid Velocity ◽  
Axial Flow ◽  
Couple Stress Fluid ◽  
Stress Vector ◽  
Transform Method

The helical flows of couple-stress fluids in a straight circular cylinder are studied in the framework of the newly developed, fully determinate linear couple-stress theory. The fluid flow is generated by the helical motion of the cylinder with time-dependent velocity. Also, the couple-stress vector is given on the cylindrical surface and the nonslip condition is considered. Using the integral transform method, analytical solutions to the axial velocity, azimuthal velocity, nonsymmetric force-stress tensor, and couple-stress vector are obtained. The obtained solutions incorporate the characteristic material length scale, which is essential to understand the fluid behavior at microscales. If characteristic length of the couple-stress fluid is zero, the results to the classical fluid are recovered. The influence of the scale parameter on the fluid velocity, axial flow rate, force-stress tensor, and couple-stress vector is analyzed by numerical calculus and graphical illustrations. It is found that the small values of the scale parameter have a significant influence on the flow parameters.

Monogenic functions in commutative complex algebras of the second rank and the Lamé equilibrium system for some class of plane orthotropy

2019 ◽  
Vol 16 (3) ◽  
pp. 345-346
Author(s):  
Serhii Gryshchuk
Keyword(s):  
Stress Tensor ◽  
Analytic Functions ◽  
Deformation Tensor ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Monogenic Functions ◽  
Equilibrium System ◽  
Linear Combinations

We consider a class of plane orthotropic deformations of the form \(\varepsilon_{x} = \sigma_x + a_{12} \sigma_y\), \(\gamma_{xy} = 2 \left(p-a_{12}\right) \tau_{xy}\), \(\varepsilon_{y}= a_{12}\sigma_x+\sigma_y\), where \(\sigma_x\), \(\tau_{xy}\), \(\sigma_y\) and \(\varepsilon_{x}\), \(\frac{\gamma_{xy}}{2}\), \(\varepsilon_{y}\) are components of the stress tensor and the deformation tensor, respectively, real parameters \(p\) and \(a_{12}\) satisfy the inequalities: \(-1 \lt p \lt 1\), \(-1 \lt a_{12} \lt p\). A class of solutions of the Lamé equilibrium system for displacements is built in the form of linear combinations of components of ''analytic'' functions which take values in commutative and associative two-dimensional algebras with unity over the field of complex numbers.

scholarly journals On the complex q-Appell polynomials

2020 ◽  
Vol 74 (1) ◽  
pp. 31
Author(s):  
Thomas Ernst
Keyword(s):  
Imaginary Part ◽  
Generating Function ◽  
Complex Number ◽  
Analytic Functions ◽  
Complex Case ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
The Real ◽  
Cauchy Riemann

The purpose of this article is to generalize the ring of \(q\)-Appell polynomials to the complex case. The formulas for \(q\)-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex \(q\)-Appell polynomials are also \(q\)-complex analytic functions, we are able to give a first example of the \(q\)-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define \(q\)-complex Bernoulli and Euler polynomials. Previously, in order to obtain the \(q\)-Appell polynomial, we would make a \(q\)-addition of the corresponding \(q\)-Appell number with \(x\). This is now replaced by a \(q\)-addition of the corresponding \(q\)-Appell number with two infinite function sequences \(C_{\nu,q}(x,y)\) and \(S_{\nu,q}(x,y)\) for the real and imaginary part of a new so-called \(q\)-complex number appearing in the generating function. Finally, we can prove \(q\)-analogues of the Cauchy-Riemann equations.

scholarly journals Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line

2019 ◽  
Vol 16 (2) ◽  
pp. 200-214
Author(s):  
Sergiy Plaksa
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Real Axis ◽  
Analytic Functions ◽  
Boundary Value ◽  
Fredholm Integral Equations ◽  
Singular Line ◽  
Boundary Functions ◽  
Cauchy Riemann ◽  
The Given

We consider a generalized Cauchy-Riemann system with a rectilinear singular interval of the real axis. Schwarz boundary value problems for generalized analytic functions which satisfy the mentioned system are reduced to the Fredholm integral equations of the second kind under natural assumptions relating to the boundary of a domain and the given boundary functions.

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