Strain Energy due to Change in Volume given Principal Stresses Solution

STEP 0: Pre-Calculation Summary
Formula Used
Strain Energy for Volume Change = ((1-2*Poisson's Ratio))/(6*Young's Modulus of Specimen)*(First Principal Stress+Second Principal Stress+Third Principal Stress)^2
Uv = ((1-2*𝛎))/(6*E)*(σ1+σ2+σ3)^2
This formula uses 6 Variables
Variables Used
Strain Energy for Volume Change - (Measured in Joule per Cubic Meter) - Strain Energy for Volume Change with no distortion is defined as the energy stored in the body per unit volume due to deformation.
Poisson's Ratio - Poisson's Ratio is defined as the ratio of the lateral and axial strain. For many metals and alloys, values of Poisson’s ratio range between 0.1 and 0.5.
Young's Modulus of Specimen - (Measured in Pascal) - Young's Modulus of Specimen is a mechanical property of linear elastic solid substances. It describes the relationship between longitudinal stress and longitudinal strain.
First Principal Stress - (Measured in Pascal) - First Principal Stress is the first one among the two or three principal stresses acting on a biaxial or triaxial stressed component.
Second Principal Stress - (Measured in Pascal) - Second Principal Stress is the second one among the two or three principal stresses acting on a biaxial or triaxial stressed component.
Third Principal Stress - (Measured in Pascal) - Third Principal Stress is the third one among the two or three principal stresses acting on a biaxial or triaxial stressed component.
STEP 1: Convert Input(s) to Base Unit
Poisson's Ratio: 0.3 --> No Conversion Required
Young's Modulus of Specimen: 190 Gigapascal --> 190000000000 Pascal (Check conversion here)
First Principal Stress: 35 Newton per Square Millimeter --> 35000000 Pascal (Check conversion here)
Second Principal Stress: 47 Newton per Square Millimeter --> 47000000 Pascal (Check conversion here)
Third Principal Stress: 65 Newton per Square Millimeter --> 65000000 Pascal (Check conversion here)
STEP 2: Evaluate Formula
Substituting Input Values in Formula
Uv = ((1-2*𝛎))/(6*E)*(σ123)^2 --> ((1-2*0.3))/(6*190000000000)*(35000000+47000000+65000000)^2
Evaluating ... ...
Uv = 7582.1052631579
STEP 3: Convert Result to Output's Unit
7582.1052631579 Joule per Cubic Meter -->7.58210526315789 Kilojoule per Cubic Meter (Check conversion here)
FINAL ANSWER
7.58210526315789 7.582105 Kilojoule per Cubic Meter <-- Strain Energy for Volume Change
(Calculation completed in 00.004 seconds)

Credits

Created by Vaibhav Malani
National Institute of Technology (NIT), Tiruchirapalli
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Verified by Sagar S Kulkarni
Dayananda Sagar College of Engineering (DSCE), Bengaluru
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13 Distortion Energy Theory Calculators

Distortion Strain Energy
Go Strain Energy for Distortion = ((1+Poisson's Ratio))/(6*Young's Modulus of Specimen)*((First Principal Stress-Second Principal Stress)^2+(Second Principal Stress-Third Principal Stress)^2+(Third Principal Stress-First Principal Stress)^2)
Tensile Yield Strength by Distortion Energy Theorem Considering Factor of Safety
Go Tensile Yield Strength = Factor of Safety*sqrt(1/2*((First Principal Stress-Second Principal Stress)^2+(Second Principal Stress-Third Principal Stress)^2+(Third Principal Stress-First Principal Stress)^2))
Tensile Yield Strength by Distortion Energy Theorem
Go Tensile Yield Strength = sqrt(1/2*((First Principal Stress-Second Principal Stress)^2+(Second Principal Stress-Third Principal Stress)^2+(Third Principal Stress-First Principal Stress)^2))
Tensile Yield Strength for Biaxial Stress by Distortion Energy Theorem Considering Factor of Safety
Go Tensile Yield Strength = Factor of Safety*sqrt(First Principal Stress^2+Second Principal Stress^2-First Principal Stress*Second Principal Stress)
Strain Energy due to Change in Volume given Principal Stresses
Go Strain Energy for Volume Change = ((1-2*Poisson's Ratio))/(6*Young's Modulus of Specimen)*(First Principal Stress+Second Principal Stress+Third Principal Stress)^2
Strain Energy due to Change in Volume with No Distortion
Go Strain Energy for Volume Change = 3/2*((1-2*Poisson's Ratio)*Stress for Volume Change^2)/Young's Modulus of Specimen
Distortion Strain Energy for Yielding
Go Strain Energy for Distortion = ((1+Poisson's Ratio))/(3*Young's Modulus of Specimen)*Tensile Yield Strength^2
Volumetric Strain with No Distortion
Go Strain for Volume Change = ((1-2*Poisson's Ratio)*Stress for Volume Change)/Young's Modulus of Specimen
Stress due to Change in Volume with No Distortion
Go Stress for Volume Change = (First Principal Stress+Second Principal Stress+Third Principal Stress)/3
Total Strain Energy per Unit Volume
Go Total Strain Energy per Unit Volume = Strain Energy for Distortion+Strain Energy for Volume Change
Strain Energy due to Change in Volume given Volumetric Stress
Go Strain Energy for Volume Change = 3/2*Stress for Volume Change*Strain for Volume Change
Shear Yield Strength by Maximum Distortion Energy Theorem
Go Shear Yield Strength = 0.577*Tensile Yield Strength
Shear Yield Strength by Maximum Distortion Energy Theory
Go Shear Yield Strength = 0.577*Tensile Yield Strength

Strain Energy due to Change in Volume given Principal Stresses Formula

Strain Energy for Volume Change = ((1-2*Poisson's Ratio))/(6*Young's Modulus of Specimen)*(First Principal Stress+Second Principal Stress+Third Principal Stress)^2
Uv = ((1-2*𝛎))/(6*E)*(σ1+σ2+σ3)^2

What is strain energy?

Strain energy is defined as the energy stored in a body due to deformation. The strain energy per unit volume is known as strain energy density and the area under the stress-strain curve towards the point of deformation. When the applied force is released, the whole system returns to its original shape. It is usually denoted by U.

How to Calculate Strain Energy due to Change in Volume given Principal Stresses?

Strain Energy due to Change in Volume given Principal Stresses calculator uses Strain Energy for Volume Change = ((1-2*Poisson's Ratio))/(6*Young's Modulus of Specimen)*(First Principal Stress+Second Principal Stress+Third Principal Stress)^2 to calculate the Strain Energy for Volume Change, Strain Energy due to Change in Volume given Principal Stresses formula is defined as the energy stored in a body due to deformation. This energy is the energy stored when volume changes with zero distortion. Strain Energy for Volume Change is denoted by Uv symbol.

How to calculate Strain Energy due to Change in Volume given Principal Stresses using this online calculator? To use this online calculator for Strain Energy due to Change in Volume given Principal Stresses, enter Poisson's Ratio (𝛎), Young's Modulus of Specimen (E), First Principal Stress 1), Second Principal Stress 2) & Third Principal Stress 3) and hit the calculate button. Here is how the Strain Energy due to Change in Volume given Principal Stresses calculation can be explained with given input values -> 7.6E-9 = ((1-2*0.3))/(6*190000000000)*(35000000+47000000+65000000)^2.

FAQ

What is Strain Energy due to Change in Volume given Principal Stresses?
Strain Energy due to Change in Volume given Principal Stresses formula is defined as the energy stored in a body due to deformation. This energy is the energy stored when volume changes with zero distortion and is represented as Uv = ((1-2*𝛎))/(6*E)*(σ123)^2 or Strain Energy for Volume Change = ((1-2*Poisson's Ratio))/(6*Young's Modulus of Specimen)*(First Principal Stress+Second Principal Stress+Third Principal Stress)^2. Poisson's Ratio is defined as the ratio of the lateral and axial strain. For many metals and alloys, values of Poisson’s ratio range between 0.1 and 0.5, Young's Modulus of Specimen is a mechanical property of linear elastic solid substances. It describes the relationship between longitudinal stress and longitudinal strain, First Principal Stress is the first one among the two or three principal stresses acting on a biaxial or triaxial stressed component, Second Principal Stress is the second one among the two or three principal stresses acting on a biaxial or triaxial stressed component & Third Principal Stress is the third one among the two or three principal stresses acting on a biaxial or triaxial stressed component.
How to calculate Strain Energy due to Change in Volume given Principal Stresses?
Strain Energy due to Change in Volume given Principal Stresses formula is defined as the energy stored in a body due to deformation. This energy is the energy stored when volume changes with zero distortion is calculated using Strain Energy for Volume Change = ((1-2*Poisson's Ratio))/(6*Young's Modulus of Specimen)*(First Principal Stress+Second Principal Stress+Third Principal Stress)^2. To calculate Strain Energy due to Change in Volume given Principal Stresses, you need Poisson's Ratio (𝛎), Young's Modulus of Specimen (E), First Principal Stress 1), Second Principal Stress 2) & Third Principal Stress 3). With our tool, you need to enter the respective value for Poisson's Ratio, Young's Modulus of Specimen, First Principal Stress, Second Principal Stress & Third Principal Stress and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Strain Energy for Volume Change?
In this formula, Strain Energy for Volume Change uses Poisson's Ratio, Young's Modulus of Specimen, First Principal Stress, Second Principal Stress & Third Principal Stress. We can use 2 other way(s) to calculate the same, which is/are as follows -
  • Strain Energy for Volume Change = 3/2*Stress for Volume Change*Strain for Volume Change
  • Strain Energy for Volume Change = 3/2*((1-2*Poisson's Ratio)*Stress for Volume Change^2)/Young's Modulus of Specimen
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