**Branch :**First Year-Engineering Syllabus

**Subject :**Elements of Mechanical Engineering

## Elastic Stress – strain Relations

**Elastic Stress – strain Relations:**** **In this section we shall generalize the elastic behavior, so as to arrive at the relations which connect all the six components of stress with the six components of elastic stress. Futher, we would restrict overselves to linearly elastic material.

Before writing down the relations let us introduce a term ISOTROPY

**ISOTROPIC: ** If the response of the material is independent of the orientation of the load axis of the sample, then we say that the material is isotropic or in other words we can say that isotropy of a material in a characteristics, which gives us the information that the properties are the same in the three orthogonal directions x y z, on the other hand if the response is dependent on orientation it is known as anisotropic.

Examples of anisotropic materials, whose properties are different in different directions are

(i) Wood

(ii) Fibre reinforced plastic

(iii) Reinforced concrete

**HOMOGENIUS: **A material is homogenous if it has the same composition through our body. Hence the elastic properties are the same at every point in the body. However, the properties need not to be the same in all the direction for the material to be homogenous. Isotropic materials have the same elastic properties in all the directions. Therefore, the material must be both homogenous and isotropic in order to have the lateral strains to be same at every point in a particular component.

**Generalized Hook's Law: ** We know that for stresses not greater than the proportional limit.

These equation expresses the relationship between stress and strain (Hook's law) for uniaxial state of stress only when the stress is not greater than the proportional limit. In order to analyze the deformational effects produced by all the stresses, we shall consider the effects of one axial stress at a time. Since we presumably are dealing with strains of the order of one percent or less. These effects can be superimposed arbitrarily. The figure below shows the general triaxial state of stress.

Let us consider a case when s_{x} alone is acting. It will cause an increase in dimension in X-direction whereas the dimensions in y and z direction will be decreased.

Therefore the resulting strains in three directions are