Module 1: Elasticity 1. Thus the elastic behavior implies the absence of any permanent deformation. Elasticity has been developed following the great achievement of Newton in stating the laws of motion, although it has earlier roots. The need to understand and control the fracture of solids seems to have been a first motivation. Leonardo da Vinci sketched in his notebooks a possible test of the tensile strength of a wire.

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Module 1: Elasticity 1. Thus the elastic behavior implies the absence of any permanent deformation. Elasticity has been developed following the great achievement of Newton in stating the laws of motion, although it has earlier roots. The need to understand and control the fracture of solids seems to have been a first motivation. Leonardo da Vinci sketched in his notebooks a possible test of the tensile strength of a wire.

Galileo had investigated the breaking loads of rods under tension and concluded that the load was independent of length and proportional to the cross section area, this being the first step toward a concept of stress. Every engineering material possesses a certain extent of elasticity. The common materials of construction would remain elastic only for very small strains before exhibiting either plastic straining or brittle failure.

However, natural polymeric materials show elasticity over a wider range usually with time or rate effects thus they would more accurately be characterized as viscoelastic , and the widespread use of natural rubber and similar materials motivated the development of finite elasticity.

While many roots of the subject were laid in the classical theory, especially in the work of Green, Gabrio Piola, and Kirchhoff in the mid's, the development of a viable theory with forms of stress-strain relations for specific rubbery elastic materials, as well as an understanding of the physical effects of the nonlinearity in simple problems such as torsion and bending, was mainly the achievement of the British-born engineer and applied mathematician Ronald S.

Rivlin in the's and 's. Simultaneously, Navier had developed an elasticity theory based on a simple particle model, in which particles interacted with their neighbours by a central force of attraction between neighboring particles.

Later it was gradually realized, following work by Navier, Cauchy, and Poisson in the 's and 's, that the particle model is too simple. Most of the subsequent developments of this subject were in terms of the continuum theory. George Green highlighted the maximum possible number of independent elastic moduli in the most general anisotropic solid in Green pointed out that the existence of elastic strain energy required that of the 36 elastic constants relating the 6 stress components to the 6 strains, at most 21 could be independent.

In , Lord Kelvin showed that a strain energy function must exist for reversible isothermal or adiabatic response and showed that temperature changes are associated with adiabatic elastic deformation. The middle and late 's were a period in which many basic elastic solutions were derived and applied to technology and to the explanation of natural phenomena.

Adhmar-Jean-Claude Barr de Saint-Venant derived in the 's solutions for the torsion of noncircular cylinders, which explained the necessity of warping displacement of the cross section in the direction parallel to the axis of twisting, and for the flexure of beams due to transverse loading; the latter allowed understanding of approximations inherent in the simple beam theory of Jakob Bernoulli, Euler, and Coulomb. Heinrich Rudolf Hertz developed solutions for the deformation of elastic solids as they are brought into contact and applied these to model details of impact collisions.

Solutions for stress and displacement due to concentrated forces acting at an interior point of a full space were derived by Kelvin and those on the surface of a half space by Boussinesq and Cerruti. In Kelvin had derived the basic form of the solution of the static elasticity equations for a spherical solid, and this was applied in following years for calculating the deformation of the earth due to rotation and tidal force and measuring 1 Applied Elasticity for Engineers.

During the process of deriving such equations, one can consider all the influential factors, the results obtained will be so complicated and hence practically no solutions can be found. Therefore, some basic assumptions have to be made about the properties of the body considered to arrive at possible solutions. Under such assumptions, we can neglect some of the influential factors of minor importance.

The following are the assumptions in classical elasticity. The Body is Continuous Here the whole volume of the body is considered to be filled with continuous matter, without any void.

Only under this assumption, can the physical quantities in the body, such as stresses, strains and displacements, be continuously distributed and thereby expressed by continuous functions of coordinates in space.

However, these assumptions will not lead to significant errors so long as the dimensions of the body are very large in comparison with those of the particles and with the distances between neighbouring particles.

The Body is Perfectly Elastic The body is considered to wholly obey Hooke's law of elasticity, which shows the linear relations between the stress components and strain components. Under this assumption, the elastic constants will be independent of the magnitudes of stress and strain components. The Body is Homogenous In this case, the elastic properties are the same throughout the body.

Thus, the elastic constants will be independent of the location in the body. Under this assumption, one can analyse an elementary volume isolated from the body and then apply the results of analysis to the entire body.

The Body is Isotropic Here, the elastic properties in a body are the same in all directions. Hence, the elastic constants will be independent of the orientation of coordinate axes.

The Displacements and Strains are Small The displacement components of all points of the body during deformation are very small in comparison with its original dimensions and the strain components and the rotations of all line elements are much smaller than unity.

Hence, when formulating the equilibrium equations relevant to the deformed state, the lengths and angles of the body before deformation are used. In addition, when geometrical equations involving strains and displacements are formulated, the squares and products of the small quantities are neglected. Therefore, these two measures are necessary to linearize the algebraic and differential equations in elasticity for their easier solution.

Although, elasticity, mechanics of materials and structural mechanics are the three branches of solid mechanics, they differ from one another both in objectives and methods of analysis. Mechanics of materials deals essentially with the stresses and displacements of a structural or machine element in the shape of a bar, straight or curved, which is subjected to tension, compression, shear, bending or torsion. Structural mechanics, on the basis of mechanics of materials, deals with the stresses and displacements of a structure in the form of a bar system, such as a truss or a rigid frame.

The structural elements that are not in form of a bar, such as blocks, plates, shells, dams and foundations, they are analysed only using theory of elasticity. Moreover, in order to analyse a bar element thoroughly and very precisely, it is necessary to apply theory of elasticity.

Although bar shaped elements are studied both in mechanics of materials and in theory of elasticity, the methods of analysis used here are not entirely the same. When the element is studied in mechanics of materials, some assumptions are usually made on the strain condition or the stress distribution. These assumptions simplify the mathematical derivation to a certain extent, but many a times inevitably reduce the degree of accuracy of the results obtained.

However, in elasticity, the study of bar-shaped element usually does not need those assumptions. Thus the results obtained by the application of elasticity theory are more accurate and may be used to check the appropriate results obtained in mechanics of materials. While analysing the problems of bending of straight beam under transverse loads by the mechanics of materials, it is usual to assume that a plane section before bending of the beam remains plane even after the bending.

This assumption leads to the linear distribution of bending stresses. In the theory of elasticity, however one can solve the problem without this assumption and prove that the stress distribution will be far from linear variation as shown in the next sections. Further, while analysing for the distribution of stresses in a tension member with a hole, it is assumed in mechanics of materials that the tensile stresses are uniformly distributed across the net section of the member, whereas the exact analysis in the theory of elasticity shows that the stresses are by no means uniform, but are concentrated near the hole; the maximum stress at the edge of the hole is far greater than the average stress across the net section.

The theory of elasticity contains equilibrium equations relating to stresses; kinematic equations relating the strains and displacements; constitutive equations relating the stresses and strains; boundary conditions relating to the physical domain; and uniqueness constraints relating to the applicability of the solution.

Module 2: Analysis of Stress 2. To examine these internal forces at a point O in Figure 2. If the plane is divided into a number of small areas, as in the Figure 2. From this the concept of stress as the internal force per unit area can be understood.

Assuming that the material is continuous, the term "stress" at any point across a small area DA can be defined by the limiting equation as below. Stress is sometimes referred to as force intensity. In the limit as DA0, the above corresponding axes. Taking the ratios. These values of the three intensities are defined as the "Stress components" associated with the X-face at point O. The stress components parallel to the surface are called "Shear stress components" denoted by t.

The shear stress component acting on the X-face in the y-direction is identified as txy. The stress component perpendicular to the face is called "Normal Stress" or "Direct stress" component and is denoted by s. This is identified as sx along X-direction. From the above discussions, the stress components on the X -face at point O are defined as follows in terms of force intensity ratios. Passing through that point, infinitely many planes may be drawn.

As the resultant forces acting on these planes is the same, the stresses on these planes are different because the areas and the inclinations of these planes are different. Therefore, for a complete description of stress, we have to specify not only its magnitude, direction and sense but also the surface on which it acts. For this reason, the stress is called a "Tensor". Figure 2. Of these three are direct stresses and six are shear stresses.

In matrix notation, it is often written as. Let us now define a new stress term sm as the mean stress, so that. Imagine a hydrostatic type of stress having all the normal stresses equal to sm, and all the shear stresses are zero.

We can divide the stress tensor into two parts, one having only the "hydrostatic stress" and the other, "deviatorial stress". The hydrostatic type of stress is given by.

Here the hydrostatic type of stress is known as "spherical stress tensor" and the other is known as the "deviatorial stress tensor".

It will be seen later that the deviatorial part produces changes in shape of the body and finally causes failure. The spherical part is rather harmless, produces only uniform volume changes without any change of shape, and does not necessarily cause failure. In indicial notation, the co-ordinate axes x, y and z are replaced by numbered axes x1, x2 and x3 respectively.

The components of the force DF of Figure 2. The definitions of the components of stress acting on the face x1can be written in indicial form as follows:. Here, the symbol s is used for both normal and shear stresses. In general, all components of stress can now be defined by a single equation as below. Thus stresses could be one-dimensional, two-dimensional or three-dimensional as shown in the Figure 2. Such a state is described by stresses s x , s y and txy and the X and Y body forces Here z is taken as the independent co-ordinate axis.

We shall now determine the equations for transformation of the stress components s x , s y and txy at any point of a body represented by infinitesimal element as shown in the Figure 2. Consider an infinitesimal wedge as shown in Fig. It is required to determine the stresses s x and t xy , that refer to axes x , y making an angle q with axes X, Y as shown in the Figure.

Let side MN be normal to the x axis. Considering s x and t xy as positive and area of side MN as unity, the sides MP and PN have areas cosq and sinq, respectively. Equilibrium of the forces in the x and y directions requires that.

The normal and shear stresses on the x' plane MN plane are obtained by projecting Tx and Ty in the x and y directions. These are the principal directions, along which the principal or maximum and minimum normal stresses act.

When Equation 2.

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## Applied Elasticity_T.G. Sitharam n L.govinda Raju

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## Applied Elasticity

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## Applied Elasticity for Engineers

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