LECTURE 7

ANALYSIS OF STRAINS

CONCEPT OF STRAIN

Concept of strain : if a bar is subjected to a direct load, and hence a stress the bar will change in length. If the bar has an original length L and changes by an amount dL, the strain produce is defined as follows:

Strain is thus, a measure of the deformation of the material and is a nondimensional Quantity i.e. it has no units. It is simply a ratio of two quantities with the same unit.

Since in practice, the extensions of materials under load are very very small, it is often convenient to measure the strain in the form of strain x 10-6 i.e. micro strain, when the symbol used becomes m Î.

Sign convention for strain:

Tensile strains are positive whereas compressive strains are negative. The strain defined earlier was known as linear strain or normal strain or the longitudinal strain now let us define the shear strain.

Definition: An element which is subjected to a shear stress experiences a deformation as shown in the figure below. The tangent of the angle through which two adjacent sides rotate relative to their initial position is termed shear strain. In many cases the angle is very small and the angle it self is used, ( in radians ), instead of tangent, so that g = Ð AOB - Ð A'OB' = f

Shear strain: As we know that the shear stresses acts along the surface. The action of the stresses is to produce or being about the deformation in the body consider the distortion produced b shear sheer stress on an element or rectangular block

This shear strain or slide is f and can be defined as the change in right angle. or The angle of deformation g is then termed as the shear strain. Shear strain is measured in radians & hence is non dimensional i.e. it has no unit.So we have two types of strain i.e. normal stress & shear stresses.

Hook's Law :

A material is said to be elastic if it returns to its original, unloaded dimensions when load is removed.

Hook's law therefore states that

Stress ( s ) a strain( Î )

Modulus of elasticity : Within the elastic limits of materials i.e. within the limits in which Hook's law applies, it has been shown that

Stress / strain = constant

This constant is given by the symbol E and is termed as the modulus of elasticity or Young's modulus of elasticity

Thus

The value of Young's modulus E is generally assumed to be the same in tension or compression and for most engineering material has high, numerical value of the order of 200 GPa

Poisson's ratio: If a bar is subjected to a longitudinal stress there will be a strain in this direction equal to s / E . There will also be a strain in all directions at right angles to s . The final shape being shown by the dotted lines.

It has been observed that for an elastic materials, the lateral strain is proportional to the longitudinal strain. The ratio of the lateral strain to longitudinal strain is known as the poison's ratio .

Poison's ratio ( m ) = - lateral strain / longitudinal strain

For most engineering materials the value of m his between 0.25 and 0.33.

Three dimensional state of strain : Consider an element subjected to three mutually perpendicular tensile stresses sx , syand sz as shown in the figure below.

If sy and sz were not present the strain in the x direction from the basic definition of Young's modulus of Elasticity E would be equal to

Îx= sx/ E

The effects of sy and sz in x direction are given by the definition of Poisson's ratio m ' to be equal as -m sy/ E and -m sz/ E

The negative sign indicating that if syand sz are positive i.e. tensile, these they tend to reduce the strain in x direction thus the total linear strain is x direction is given by

Principal strains in terms of stress:

In the absence of shear stresses on the faces of the elements let us say that sx , sy , sz are in fact the principal stress. The resulting strain in the three directions would be the principal strains.

i.e. We will have the following relation.

For Two dimensional strain: system, the stress in the third direction becomes zero i.e sz = 0 or s3 = 0

Although we will have a strain in this direction owing to stresses s1& s2 .

Hence the set of equation as described earlier reduces to

Hence a strain can exist without a stress in that direction

Hydrostatic stress : The term Hydrostatic stress is used to describe a state of tensile or compressive stress equal in all directions within or external to a body. Hydrostatic stress causes a change in volume of a material, which if expressed per unit of original volume gives a volumetric strain denoted by Îv. So let us determine the expression for the volumetric strain.

Volumetric Strain:

Consider a rectangle solid of sides x, y and z under the action of principal stresses s1 , s2 , s3 respectively.

Then Î1 , Î2 , and Î3 are the corresponding linear strains, than the dimensions of the rectangle becomes

( x + Î1 . x ); ( y + Î2 . y ); ( z + Î3 . z )

hence the

ALITER : Let a cuboid of material having initial sides of Length x, y and z. If under some load system, the sides changes in length by dx, dy, and dz then the new volume ( x + dx ) ( y + dy ) ( z +dz )

New volume = xyz + yzdx + xzdy + xydz

Original volume = xyz

Change in volume = yzdx +xzdy + xydz

Volumetric strain = ( yzdx +xzdy + xydz ) / xyz = Îx+ Îy+ Îz

Neglecting the products of epsilon's since the strains are sufficiently small.

Volumetric strains in terms of principal stresses:

As we know that

Strains on an oblique plane

(a) Linear strain

            

Consider a rectangular block of material OLMN as shown in the xy plane. The strains along ox and oy are Îx and Îy , and gxy is the shearing strain.

Then it is required to find an expression for Îq, i.e the linear strain in a direction inclined at q to OX, in terms of Îx ,Îy , gxy and q.

Let the diagonal OM be of length 'a' then ON = a cos q and OL = a sin q , and the increase in length of those under strains are Îxacos q and Îya sin q ( i.e. strain x original length ) respectively.

If M moves to M', then the movement of M parallel to x axis is Îxacos q + gxy sin q and the movement parallel to the y axis is Îyasin q

Thus the movement of M parallel to OM , which since the strains are small is practically coincident with MM'. and this would be the summation of portions (1) and (2) respectively and is equal to

This expression is identical in form with the equation defining the direct stress on any inclined plane q with Îx and Îy replacing sx and sy and ½ gxy replacing txy i.e. the shear stress is replaced by half the shear strain

Shear strain: To determine the shear stain in the direction OM consider the displacement of point P at the foot of the perpendicular from N to OM and the following expression can be derived as

In the above expression ½ is there so as to keep the consistency with the stress relations.

Futher -ve sign in the expression occurs so as to keep the consistency of sign convention, because OM' moves clockwise with respect to OM it is considered to be negative strain.

The other relevant expressions are the following :

Let us now define the plane strain condition

Plane Strain :

In xy plane three strain components may exist as can be seen from the following figures:

Therefore, a strain at any point in body can be characterized by two axial strains i.e Îx in x direction, Îy in y - direction and gxy the shear strain.

In the case of normal strains subscripts have been used to indicate the direction of the strain, and Îx , Îy are defined as the relative changes in length in the co-ordinate directions.

With shear strains, the single subscript notation is not practical, because such strains involves displacements and length which are not in same direction.The symbol and subscript gxy used for the shear strain referred to the x and y planes. The order of the subscript is unimportant. gxy and gyx refer to the same physical quantity. However, the sign convention is important.The shear strain gxy is considered to be positive if it represents a decrease the angle between the sides of an element of material lying parallel the positive x and y axes. Alternatively we can think of positive shear strains produced by the positive shear stresses and viceversa.

Plane strain :

An element of material subjected only to the strains as shown in Fig. 1, 2, and 3 respectively is termed as the plane strain state.

Thus, the plane strain condition is defined only by the components Îx , Îy , gxy : Îz = 0; gxz= 0; gyz= 0

It should be noted that the plane stress is not the stress system associated with plane strain. The plane strain condition is associated with three dimensional stress system and plane stress is associated with three dimensional strain system.

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