Simple Stress and Strain 

Simple stresses and strain

Topics

• Stress

• Strain

• Stress strain curve

• Relation Between Elastic constants

Stress

Stress is defined as the force acting per unit area on an object, whenever a load is applied on an object it tends to deform, this causes an internal resistance to the deformation the resistive force created is opposite and equal in magnitude to the applied force, the value of this force acting per unit area is called as stress.

Stress is a tensor quantity, impact of stress on a material varies with respect to geometrical plane

Mathematically stress can be written as,

Stress = F/A

Unit of stress.

Unit of stress is N/m2 or pascal.

Types of stress

Based on impact on the object stress is classified it to various categories such as

1. Tensile Stress: This is the stress experienced by a material when it is subjected to a pulling force. It tends to elongate the material.

2. Compressive Stress: This occurs when a material is subjected to a pushing force, causing it to shorten or compress.

3. Shear Stress: This type of stress occurs when forces are applied parallel or tangential to a surface as a couple. It tends to cause layers of the material to slide past each other.

4. Bending Stress: This stress is experienced by a material when it is subjected to a bending force, causing it to bend or curve.

5. Torsional Stress: This occurs when a material is subjected to a twisting force, causing it to twist around its axis.

6. Volumetric Stress: Also known as hydrostatic stress, this type of stress occurs when a material is subjected to uniform pressure from all directions, leading to a change in volume without a change in shape.

7. Crushing stress: A localized compressive stress acting on the interface between two surfaces which are relatively at rest.

        Based on the reasons responsible for the stress it can be classified in to various categories such as

a.  Stress due to loading

This type of stress is caused due to the application of force on an object.

b. Thermal stresses

This type of stress is caused due the restrictions on expansion or contraction of a material due to the change in temperature.

c.  Residual stresses, etc…

These are stresses that remain on an material even after the removal of the original cause of stress, this is the result of inter-molecular forces.

Strain

         Strain is defined as the ratio of amount of deformation experienced by the body to the original dimension.

         Mathematically strain can be written as,

                     Strain (ɛ) = Δl / L

         Similar to stress strain is also classified in to various categories such as tensile, compressive, shear etc… based on the stress which causes it.

Stress-strain curve

Stress-strain curve explains the behavior of materials subjected to tensile loading stress strain diagram of materials varies widely due to the property variation such as brittleness etc..,

Stress-strain curve of ductile material is similar to the following;

Regions of a stress strain curve

When the load is increased gradually the strain on the body remains proportional to the stress till a certain value the range of the cure which satisfies the condition also called as Hooke’s law is called as line of proportionality the point at which the condition fails (A in the curve) is called as proportionality limit, this basically occurs till the inter-molecular bonds remain without breaking so that once the load is removed the molecules comes back to the original state.

After reaching the point A the value of stress don’t remain proportional to the strain, but the material Is capable of reaching back to its original state, once the load is removed till the point B, this point is called as Elastic limit after this point the material starts to deform plastically.

After the proportionality limit the material reaches the upper yield point “C'” at which atoms starts to dislocate and the material starts to yield plastically, after this point with out the application of stress, value of strain rises this is due to the movement of dislocation and the plot reaches the point ‘C’ Which is called as lower yield point, beyond this point application of stress is required to initiate deformation, at the lower yield point for some materials like low carbon steel due to breakage of bonds with impurities etc.., the graph goes wavy in nature indicating rise and fall of stress.

After reaching the lower yield point dislocations starts to pile up increasing the hardness of the material this occurs till the point ‘E’ called as ‘Ultimate tensile strength’ this region is called as strain hardening.

After reaching the point ‘E’ the material starts to fail by ‘necking’ the stress starts to fall with increase in strain and at the point ‘F’ the material fails which is called as fracture point.

Stress-Strain curve of a Brittle material

Unlike ductile material which undergo substantial amount of elastic and plastic deformation, brittle material fails with little deformation in sudden catastrophically.

         For brittle material yield point is nearly equal to the ultimate strength, thus Ultimate strength is considered for calculations instead of yield point.

         Youngs Modulus of brittle material’s are generally higher than ductile material and has capacity to withstand higher value of stress without yielding.

Relation between stress and strain

In practical application load applied on a material will be lesser than yield point thus ‘Hooks law’ is applicable.

Hook’s law states σ α ɛ

To remove the proportionality a constant is introduced called as ‘Proportionality constant’ or ‘Elastic constant’.

σ =K . ɛ

Based on the type of stress the proportionality constant varies

Tensile stress ; Young’s Modulus or Modulus of rigidity (E) is used

Shear stress ; Shears Modulus (G) is used

Volumetric stress ; Bulk Modulus (K) is used

Young’s Modulus and Shears Modulus remains constant for a material, but bulk modulus increases as pressure increases.

Relation Between Elastic constants

Whenever there is a tensile load applied on a direction there is a compressive strain induced perpendicular to the direction of load applied the relation between these strain’s is give by a ratio known as poisons ratio.

Poisons ratio = - Lateral strain / Longitudinal strain

Poisons ratio is always lesser than one for most of the cases there are some exceptions such as cork and skin which has poisons ratio as zero and negative respectively.

The relation between various constant’s of elasticity are given by

The relation between Young’s modulus and shear modulus is

E = 2G (1 + v) N/m2

The relation between Young’s modulus and Bulk modulus is:

E = 3K (1 - 2v) N/m2