There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Electrodynamics Of Continuous Media

THE appearance of a treatise in English upon the mathematical theory of elasticity is an event the potential importance of which may be judged by the that the author, in his frequent suggestions for collateral reading, refers to only three such, those of Southwell, Timoshenko, and Love. In spirit and content Sokolnikoff}s book differs greatly from each and all of these. It may be described by a possible sub-title: “A pure mathematician surveys topics related to certain problems in the mathematical theory of elasticity”. It is symptomatic of the change in outlook of American mathematics over the past few decades. Mathematical Theory Of Elasticity Prof. I. S. Sokolnikoff with the collaboration of Asst. Prof. R. D. Speche. Pp. xi + 373. (New York and London: McGraw-Hill Book Co., Inc., 1946.) 22s. 6d.

Mathematical Theory Of Elasticity

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Some Basic Problems Of The Mathematical Theory Of Elasticity

Finite elements for functionally graded Reissner–Mindlin plates

Here, we put the equivalent shear modulus of metamaterials under scrutiny. A micromorphic model for the equivalent shear modulus of metamaterials is developed. This model represents a metamaterial as a multiscale material that exhibits a microstructural strain (micro-strain) field, which is independent of the macroscopic strain (macro-strain) field of the material. The mechanisms by which the shear modulus of a metamaterial would change and depend on the material size are discussed. It is revealed that the shear modulus of a metamaterial is size and microstructure topology-dependent as long as the micro-strain field is significant and different from the macro-strain field. The conditions at which the shear modulus would be zero, negative, or positive are also defined. A unique material parameter ( β ) that depends on the microstructure topology is introduced to define the conditions of the material stability. It is revealed that a metamaterial with a negative shear modulus is stable as long as β 2 0 . The proposed micromorphic model is also employed to determine the equivalent shear modulus of composite metamaterials of the form of coated inclusions embedded in a matrix material. It is demonstrated that a composite metamaterial with a giant shear modulus can be produced by employing either one phase of a negative modulus or an interfacial layer between the inclusion and the matrix. In addition, the implementation of an interface between the inclusion and the matrix would give a composite metamaterial with a negative shear modulus.

On the equivalent shear modulus of composite metamaterials

The equations of generalized piezo-thermoelasticity within the frame of dual-phase-lag model are used to study the effect of gravitational force on the behavior of a half-space. Analytic expressions for the displacement components, temperature, stress and strain tensors components are obtained using normal mode analysis. Numerical results for the field quantities of practical interest are given in the physical domain and illustrated graphically. Comparison is made between the results predicted by Lord–Shulman theory and dual-phase-lag (DPL) model. The results obtained by applying both of the L–S theory and DPL model are shown to be very close to each other except in determining one of the components of the electric displacement where the results differ and in general the effect of the presence of gravity is to weaken the absolute values of the physical quantities except in the case of the same component of the electric displacement.

Effect of gravity on piezo-thermoelasticity within the dual-phase-lag model

A model of nonlinear thermo-electroelasticity in extended thermodynamics

In this article, we study the flexoelectricity induced in a prismatic anisotropic bar due to torsion. The simplified strain gradient elasticity theory is considered in this study. The bar is unifor...

Flexoelectric effect induced in an anisotropic bar with cubic symmetry under torsion

A plane problem of uncoupled thermomagnetoelasticity for an infinite, elliptical cylinder carrying a steady axial current by a boundary integral method

A. R. El Dhaba

Papers

On the equivalent shear modulus of composite metamaterials

Effect of gravity on piezo-thermoelasticity within the dual-phase-lag model

A model of nonlinear thermo-electroelasticity in extended thermodynamics

Flexoelectric effect induced in an anisotropic bar with cubic symmetry under torsion

A plane problem of uncoupled thermomagnetoelasticity for an infinite, elliptical cylinder carrying a steady axial current by a boundary integral method