Laws of elasticity

The study of flow and elasticity dates to antiquity. Practical rheology existed for centuries before Hooke and Newton proposed the basic laws of elastic response and simple viscous flow, respectively, in the seventeenth century. Further advances in understanding came in the mid-nineteenth century with models for viscous flow in round tubes. The introduction of the first practical rotational viscometer by Couette in 1890 (1,2) was another milestone. 

The mechanical properties of a material play an important role in powder flow and compaction by influencing particle-particle interaction and cohesion, that is to say, by influencing the true area of contact between particles. For example, Hertz [26] demonstrated that both the size and shape of the zone of contact followed simply from the elastic properties of a material. Clearly then, the true area of contact is affected by elastic properties. From the laws of elasticity, one can predict the area of contact between two elastic bodies. More recent work has demonstrated, however, that additional factors must be taken... 

To better understand the nature and features of these vibrations, bonds can be considered as springs. Given this analogy, the behaviour of these molecular springs approximately follows Hooke s law of elasticity. In physics, Hooke s law relates the strain on a body (spring) to the force (load or mass) causing the strain . In essence, molecular bonds follow this linear relationship, where the... 

Three equations are basic to viscoelasticity (1) Newton s law of viscosity, a = ijy, (2) Hooke s law of elasticity. Equation 1.15, and (3) Newton s second law of motion, F = ma, where m is the mass and a is the acceleration. One can combine the three equations to obtain a basic differential equation. In linear viscoelasticity, the conditions are such that the contributions of the viscous, elastic, and the inertial elements are additive. The Maxwell model is ... 

A general relation containing the law of elasticity and the law of viscosity as extreme cases can be introduced as... 

Since the stress has units of force/area and the strain is dimensionless, the modulus has units of force/area. Equation (7.98) is Hooke s law of elasticity and it is valid for all solids at sufficiently small strains. 

For the measurement of stress by x-rays we have developed three working equations, namely, (16-14), (16-17), and (16-26). Each contains an appropriate stress constant K, by which diffraction line shift is converted to stress. Furthermore each was derived on the assumption that the material under stress was an isotropic body obeying the usual laws of elasticity. This assumption has to be examined rather carefully if a calculated value of K is to be used for stress measurement. 

The most straightforward type of lattice minimisation is performed at constant volume, where the dimensions of the basic imit cell do not change. A more advanced type of calculation is one performed at constant pressure, in which case there are forces on both the atoms and the unit cell as a whole. The lattice vectors are considered as additional variables along with the atomic coordinates. The laws of elasticity describe the behaviour of a material when... 

Newton came across the Robert Hooke s famous book Micrographia that was published around 1964. Robert Hooke is famous for providing the law of elasticity in 1660. It states that for relatively small deformations of an object, the displacement or size of the deformation is directly proportional to the deforming force or load. It is said that Hooke got this idea while working with Robert Boyle (1627-1691) on whose name is a law that states that for a fixed amount of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportional. In 1678, Hooke described the inverse square law to describe planetary motion. Later on Newton provided a universal law of gravitation that is stated as follows ... 

Combining equation (1.28) with the Hooke s law of elasticity and Newton s law of viscosity, one can obtain ... 

Hooke, Robert (1635-1703) English physicist, who worked at Oxford University, where he assisted Robert Boyle. Among his many achievements were the law of elasticity see Hooke s law), the watch balance wheel, and the compound microscope. In 1665, using his microscope to study vegetable tissues, he saw little boxes , which he named cells . 

The JKR theory was developed in the early 1970s to account for the adhesion between spherical bodies, especially fine particles (see Powder Adhesion), making elastic contact. Previously, most studies of elastic contacts had presumed that bodies like steel railway wheels behaved as though adhesion was zero. In other words, no adhesion interaction seemed to be acting in most engineering contact situations. Negligible adhesion force is generally detected as a wheel is lifted from a rail, and the size of the contact spot between wheel and rail can usually be predicted accurately from the laws of elastic deformation, without any molecular attractive forces. 

The stress field of a screw dislocation is pure shear. As indicated earlier, high strains exist in the core region and, therefore, Hooke s Law of elasticity does not apply and so will not be considered. The dislocation line is parallel to the z axis there are no displacements in the x and y directions and the other stress components are zero ... 

Molecular dynamics computations for a system of N(= 108) hard spheres were first performed by Alder and co-workers. In the case of hard spheres the motion of each particle is determined by the laws of elastic collisions. When a force on a particle / can be represented by the negative gradient of a given potential function, then the classical equations of motion may be written in the following form... 

Here, represents the Cauchy stress tensor, p is the mass density, and ft and m, are the body forces and displacements in the i direction within a bounded domain Q. The two dots over the displacements indicate second derivative in time. The indices i and j in the subscripts represent the Cartesian coordinates x, y, and z. When a subscript follows a comma, this indicates a partial derivative in space with respect to the corresponding index. For the special case of elastic isotropic solids, the stress tensor can be expressed in terms of strains following Hooke s law of elasticity, and the strains, in turn, can be expressed in terms of displacements. The resulting expression for the stress tensor is... 

Fibers are subject to forces or loads when they are processed or in use. It is important to understand how fibers deform (elongate, compress, twist) or break as a function of apphed force. When the force is relatively small and does not exceed the elastic limit, fibers obey the Hooke s law, i.e., the law of elasticity. Hooke s law was discovered by the English physicist Robert Hooke in 1660, and it states the deformation of an object is directly proportional to the applied force. In addition, the object returns to its original shape and size upon the removal of the force. 

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