LECTURE 7
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
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.
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.
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
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
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 .
For most engineering materials the value of m his between 0.25 and 0.33.
If s
The effects of s The negative sign indicating that if s
In the absence of shear stresses on the faces of the elements let us say that s i.e. We will have the following relation.
Although we will have a strain in this direction owing to stresses s Hence the set of equation as described earlier reduces to
Hence a strain can exist without a stress in that direction
_{v}. So let us determine the expression for the volumetric strain.
Consider a rectangle solid of sides x, y and z under the action of principal stresses s Then Î
hence the
New volume = xyz + yzdx + xzdy + xydz Original volume = xyz Change in volume = yzdx +xzdy + xydz Volumetric strain = ( yzdx +xzdy + xydz ) / xyz = Î Neglecting the products of epsilon's since the strains are sufficiently small.
As we know that
Consider a rectangular block of material OLMN as shown in the xy plane. The strains along ox and oy are Î Then it is required to find an expression for Î 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 Î If M moves to M', then the movement of M parallel to x axis is Î 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 Î
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
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 Î In the case of normal strains subscripts have been used to indicate the direction of the strain, and Î 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 g
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 Î 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. |