Constitutive Relationship Between Stress and Strain
For One-Dimensional Stress System
For a uniaxial or one-directional stress condition (i.e., normal stress acting only in one direction), the relationship between stress and strain follows Hooke’s Law. According to this law, when a material is loaded within its elastic range, the normal stress developed in the body is directly proportional to the strain it produces. This implies that the ratio of normal stress to corresponding strain remains a constant as long as the material stays within the elastic limit. This constant is known as the Modulus of Elasticity or Young’s Modulus.
Normal stress / Corresponding strain = Constant
or
σ / e = E
Where:
σ = Normal stress
e = Strain
E = Young’s Modulus
e = σ / E ...(1.7 A)
The above expression provides the stress–strain relation for normal stress acting in a single direction.
For Two-Dimensional Stress System
Before establishing the stress–strain relationship for a 2-dimensional stress condition, it is necessary to define longitudinal strain, lateral strain, and Poisson’s ratio.
1. Longitudinal Strain
When a body is subjected to an axial tensile force, its length increases. At the same time, its dimensions perpendicular to the direction of load decrease. Thus, the body undergoes deformation both along the direction of load and at right angles to it.
The ratio of the change in length to the original length is called longitudinal strain (or linear strain). It also represents deformation per unit length in the direction of the applied force.
Let:
L = Original length
P = Axial tensile force
δL = Increase in length due to P
Longitudinal strain = δL / L ...(1.7 B)
2. Lateral Strain
The strain produced in a direction perpendicular to the applied load is known as lateral strain. If a rectangular bar of length L, breadth b, and depth d is pulled axially, its length will increase while both breadth and depth will reduce.
Let:
δL = Increase in length
δb = Reduction in breadth
δd = Reduction in depth
Longitudinal strain = δL / L
and
Lateral strain = δb / b or δd / d ...(1.7 C)
3. Poisson’s Ratio
Poisson’s ratio is the ratio of lateral strain to longitudinal strain. It remains constant for a given material when it is loaded within the elastic limit. It is represented by the symbol μ (mu).
Poisson’s ratio, μ = Lateral strain / Longitudinal strain ...(1.7 D)
Since lateral strain occurs opposite in sign to longitudinal strain, algebraically:
Lateral strain = − μ × Longitudinal strain ...(1.7 E)
4. Relationship Between Stress and Strain (Two-Dimensional System)
Consider a rectangular element ABCD subjected to normal stresses σx and σy acting mutually perpendicular to each other (refer to Fig. 1.5a).
Let:
σx = Stress in x-direction
σy = Stress in y-direction
Strain due to σy:
Stress σy produces strain in both x and y directions.
Strain in y direction = σy/E
Lateral strain in x direction = − μ × (σy/E)
Strain due to σx:
Stress σx produces strain in x and y directions.
Strain in x direction = σx/E
Lateral strain in y direction = − μ × (σx/E)
Total strain in x direction is the combined effect of σx and σy:
ex = (σx/E) − μ(σy/E) ...(1.7 F)
Total strain in y direction:
ey = (σy/E) − μ(σx/E) ...(1.7 G)
The above equations describe the stress–strain relationship for a two-dimensional stress condition. In these formulas, tensile stresses are considered positive and compressive stresses negative.
Three-Dimensional Stress System
Figure 1.5(b) illustrates a three-dimensional element subjected to three mutually perpendicular normal stresses — σ₁, σ₂, and σ₃ — acting along the x, y, and z axes respectively.
To understand the total strain in each direction, consider the strain produced by each stress independently.
Effect of stress σ₁:
σ₁ produces strain along the x-axis equal to σ₁/E.
It also produces lateral strains in the y and z directions, each equal to − μ × (σ₁/E).
Effect of stress σ₂:
σ₂ produces strain along the y-axis equal to σ₂/E, and lateral strains in the x and z
directions equal to − μ × (σ₂/E).
Effect of stress σ₃:
σ₃ produces strain along the z-axis equal to σ₃/E, and lateral strains in the x and y
directions equal to − μ × (σ₃/E).
Total Strains in the Three Directions
1. Total strain in x-direction due to stresses σ₁, σ₂ and σ₃:
e₁ = (σ₁/E) − μ(σ₂/E) − μ(σ₃/E) ...(1.7 H)
2. Total strain in y-direction due to stresses σ₁, σ₂ and σ₃:
e₂ = (σ₂/E) − μ(σ₃/E) − μ(σ₁/E) ...(1.7 I)
3. Total strain in z-direction due to stresses σ₁, σ₂ and σ₃:
e₃ = (σ₃/E) − μ(σ₁/E) − μ(σ₂/E) ...(1.7 J)
These three equations collectively form the stress–strain relationship for a body subjected to three orthogonal normal stresses in a three-dimensional stress system.