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Antiplane shear or antiplane strain [1] is a special state of strain in a body. This state of strain is achieved when the displacements in the body are zero in the plane of interest but nonzero in the direction perpendicular to the plane. For small strains, the strain tensor under antiplane shear can be written as
where the plane is the plane of interest and the direction is perpendicular to that plane.
The displacement field that leads to a state of antiplane shear is (in rectangular Cartesian coordinates)
where are the displacements in the directions.
For an isotropic, linear elastic material, the stress tensor that results from a state of antiplane shear can be expressed as
where is the shear modulus of the material.
The conservation of linear momentum in the absence of inertial forces takes the form of the equilibrium equation. For general states of stress there are three equilibrium equations. However, for antiplane shear, with the assumption that body forces in the 1 and 2 directions are 0, these reduce to one equilibrium equation which is expressed as
where is the body force in the direction and . Note that this equation is valid only for infinitesimal strains.
The antiplane shear assumption is used to determine the stresses and displacements due to a screw dislocation.
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.
Hooke's law is a law of physics that states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring, and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law since 1660.
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.
In elastodynamics, Love waves, named after Augustus Edward Hough Love, are horizontally polarized surface waves. The Love wave is a result of the interference of many shear waves (S-waves) guided by an elastic layer, which is welded to an elastic half space on one side while bordering a vacuum on the other side. In seismology, Love waves are surface seismic waves that cause horizontal shifting of the Earth during an earthquake. Augustus Edward Hough Love predicted the existence of Love waves mathematically in 1911. They form a distinct class, different from other types of seismic waves, such as P-waves and S-waves, or Rayleigh waves. Love waves travel with a lower velocity than P- or S- waves, but faster than Rayleigh waves. These waves are observed only when there is a low velocity layer overlying a high velocity layer/ sub–layers.
Euler–Bernoulli beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.
A neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, the stress-strain curve of a neo-Hookean material is not linear. Instead, the relationship between applied stress and strain is initially linear, but at a certain point the stress-strain curve will plateau. The neo-Hookean model does not account for the dissipative release of energy as heat while straining the material and perfect elasticity is assumed at all stages of deformation.
In continuum mechanics, a Mooney–Rivlin solid is a hyperelastic material model where the strain energy density function is a linear combination of two invariants of the left Cauchy–Green deformation tensor . The model was proposed by Melvin Mooney in 1940 and expressed in terms of invariants by Ronald Rivlin in 1948.
The Maxwell stress tensor is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.
A hyperelastic or Green elastic material is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material.
In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electromagnetic stress–energy tensor contains the negative of the classical Maxwell stress tensor that governs the electromagnetic interactions.
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.
Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.
The Flamant solution provides expressions for the stresses and displacements in a linear elastic wedge loaded by point forces at its sharp end. This solution was developed by A. Flamant in 1892 by modifying the three-dimensional solution of Boussinesq.
In solid mechanics, the Johnson–Holmquist damage model is used to model the mechanical behavior of damaged brittle materials, such as ceramics, rocks, and concrete, over a range of strain rates. Such materials usually have high compressive strength but low tensile strength and tend to exhibit progressive damage under load due to the growth of microfractures.
In continuum mechanics, a compatible deformation tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.
In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions. The typical thickness to width ratio of a plate structure is less than 0.1. A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem. The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads.
The Gent hyperelastic material model is a phenomenological model of rubber elasticity that is based on the concept of limiting chain extensibility. In this model, the strain energy density function is designed such that it has a singularity when the first invariant of the left Cauchy-Green deformation tensor reaches a limiting value .
The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love using assumptions proposed by Kirchhoff. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form.
The vibration of plates is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two. This suggests that a two-dimensional plate theory will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.
The Uflyand-Mindlin theory of vibrating plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1948 by Yakov Solomonovich Uflyand (1916-1991) and in 1951 by Raymond Mindlin with Mindlin making reference to Uflyand's work. Hence, this theory has to be referred to us Uflyand-Mindlin plate theory, as is done in the handbook by Elishakoff, and in papers by Andronov, Elishakoff, Hache and Challamel, Loktev, Rossikhin and Shitikova and Wojnar. In 1994, Elishakoff suggested to neglect the fourth-order time derivative in Uflyand-Mindlin equations. A similar, but not identical, theory in static setting, had been proposed earlier by Eric Reissner in 1945. Both theories are intended for thick plates in which the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. The Uflyand-Mindlin theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of one tenth the planar dimensions while the Kirchhoff–Love theory is applicable to thinner plates.
