How to Calculate and Solve for Thermal Stress in Relation to Poisson’s Ratio | Polymer Deformation

Last Updated on December 11, 2021

The image above represents thermal stress in relation to poisson’s ratio.

To compute for thermal stress in relation to poisson’s ratio, four essential parameter are needed and these parameters are Elastic Modulus (E), Linear Coefficient of Thermal Expansion (α), Change in Temperature (ΔT) and Poisson’s Ratio (μ).

The formula for calculating thermal stress in relation to poisson’s ratio:

σ = ^EαΔT / _{1 – μ}

Where:

σ = Thermal Stress in Relation to Poisson’s Ratio
E = Elastic Modulus
α = Linear Coefficient of Thermal Expansion
ΔT = Change in Temperature
μ = Poisson’s Ratio

Let’s solve an example;
Find the thermal stress in relation to poisson’s ratio when the elastic modulus is 2, the linear coefficient of thermal expansion is 7, the change in temperature is 9 and the poisson’s ratio is 11.

This implies that;

E = Elastic Modulus = 2
α = Linear Coefficient of Thermal Expansion = 7
ΔT = Change in Temperature = 9
μ = Poisson’s Ratio = 11

σ = ^EαΔT / _{1 – μ}
σ = ^(2)(7)(9) / _{1 – 11}
σ = ¹²⁶ / _-10
σ = -12.6

Therefore, the thermal stress in relation to poisson’s ratio is -12.6 Pa.

Calculating the Elastic Modulus when the Thermal Stress in Relation to Poisson’s Ratio, the Linear Coefficient of Thermal Expansion, the Change in Temperature and the Poisson’s Ratio is Given.

E = ^{σ(1 – μ)} / _αΔT

Where:

E = Elastic Modulus
σ = Thermal Stress in Relation to Poisson’s Ratio
α = Linear Coefficient of Thermal Expansion
ΔT = Change in Temperature
μ = Poisson’s Ratio

Let’s solve an example;
Find the elastic modulus when the thermal stress in relation to poisson’s ratio is 10, linear coefficient of thermal expansion is 2, the change in temperature is 3 and the poisson’s ratio is 4.

This implies that;

σ = Thermal Stress in Relation to Poisson’s Ratio = 10
α = Linear Coefficient of Thermal Expansion = 2
ΔT = Change in Temperature = 3
μ = Poisson’s Ratio = 3

E = ^{σ(1 – μ)} / _αΔT
E = ^{10(1 – 3)} / ₍₂₎₍₃₎
E = ^10(-2) / ₆
E = ^-20 / ₆
E = -3.33

Therefore, the elastic modulus is -3.33.

Calculating the Linear Coefficient of Thermal Expansion when the Thermal Stress in Relation to Poisson’s Ratio, the Elastic Modulus, the Change in Temperature and the Poisson’s Ratio is Given.

α = ^{σ(1 – μ)} / _EΔT

Where:

α = Linear Coefficient of Thermal Expansion
σ = Thermal Stress in Relation to Poisson’s Ratio
E = Elastic Modulus
ΔT = Change in Temperature
μ = Poisson’s Ratio

Let’s solve an example;
Find the linear coefficient of thermal expansion when the thermal stress in relation to poisson’s ratio is 2, the elastic modulus is 1, the change in temperature is 3 and the poisson’s ratio is 5.

This implies that;

σ = Thermal Stress in Relation to Poisson’s Ratio = 2
E = Elastic Modulus = 1
ΔT = Change in Temperature = 3
μ = Poisson’s Ratio = 5

α = ^{σ(1 – μ)} / _EΔT
α = ^{2(1 – 5)} / ₁₍₃₎
α = ^-8 / ₃
α = -2.67

Therefore, the linear coefficient of thermal expansion is -2.67.

Calculating the Change in Temperature when the Thermal Stress in Relation to Poisson’s Ratio, the Elastic Modulus, the Linear Coefficient of Thermal Expansion and the Poisson’s Ratio is Given.

ΔT = ^{σ(1 – μ)} / _Eα

Where:

ΔT = Change in Temperature
σ = Thermal Stress in Relation to Poisson’s Ratio
E = Elastic Modulus
α = Linear Coefficient of Thermal Expansion
μ = Poisson’s Ratio

Let’s solve an example;
Find the change in temperature when the thermal stress in relation to poisson’s ratio is 8, the elastic modulus is 4, the linear coefficient of thermal expansion is 2 and poisson’s ratio is 7.

This implies that;

σ = Thermal Stress in Relation to Poisson’s Ratio = 8
E = Elastic Modulus = 4
α = Linear Coefficient of Thermal Expansion = 2
μ = Poisson’s Ratio = 7

ΔT = ^{σ(1 – μ)} / _Eα
ΔT = ^{8(1 – 7)} / ₄₍₂₎
ΔT = ^8(-6) / ₈
ΔT = ^-48 / ₈
ΔT = -6

Therefore, the change in temperature is -6.

Calculating the Poisson’s Ratio when the Thermal Stress in Relation to Poisson’s Ratio, the Elastic Modulus, the Linear Coefficient of Thermal Expansion and the Change in Temperature is Given.

μ = 1 – (^EαΔT / _σ)

Where:

μ = Poisson’s Ratio
σ = Thermal Stress in Relation to Poisson’s Ratio
E = Elastic Modulus
α = Linear Coefficient of Thermal Expansion
ΔT = Change in Temperature

Let’s solve an example;
Find the poisson’s ratio when the thermal stress in relation to poisson’s ratio is 45, the elastic modulus is 5, the linear coefficient of thermal expansion is 2 and the change in temperature is 3.

σ = Thermal Stress in Relation to Poisson’s Ratio = 45
E = Elastic Modulus  = 5
α = Linear Coefficient of Thermal Expansion = 2
ΔT = Change in Temperature = 3

μ = 1 – (^EαΔT / _σ)
μ = 1 – (^5(2)(3) / ₄₅)
μ = 1 – (³⁰ / ₄₅)
μ = 1 – 0.66
μ = 0.34

Therefore, the poisson’s ratio is 0.34.

Nickzom Calculator – The Calculator Encyclopedia is capable of calculating the thermal stress in relation to poisson’s ratio.

Now, Click on Polymer Deformation under Materials and Metallurgical

Now, Click on Thermal Stress in Relation to Poisson’s Ratio under Polymer Deformation

The screenshot below displays the page or activity to enter your values, to get the answer for the thermal stress in relation to poisson’s ratio according to the respective parameter which is the Elastic Modulus (E), Linear Coefficient of Thermal Expansion (α), Change in Temperature (ΔT) and Poisson’s Ratio (μ).

Now, enter the values appropriately and accordingly for the parameters as required by the Elastic Modulus (E) is 2, Linear Coefficient of Thermal Expansion (α) is 7, Change in Temperature (ΔT) is 9 and Poisson’s Ratio (μ) is 11.

Finally, Click on Calculate

As you can see from the screenshot above, Nickzom Calculator– The Calculator Encyclopedia solves for the thermal stress in relation to poisson’s ratio and presents the formula, workings and steps too.