How Is Normal Strain Derived in Continuum Mechanics?

Click For Summary
SUMMARY

The derivation of normal strain in continuum mechanics involves the Taylor expansion of the square root function, specifically sqrt(1+e), where "e" represents the small strain. The equation length(ab) ≈ dx + ∂u/∂x*dx is reached by assuming that the squares of the gradients are negligible and applying the approximation sqrt(1+e) ≈ 1 + e/2. This simplification is crucial for understanding how small deformations affect material behavior under stress.

PREREQUISITES
  • Understanding of continuum mechanics principles
  • Familiarity with Taylor series expansions
  • Knowledge of strain and deformation concepts
  • Basic calculus, particularly differentiation
NEXT STEPS
  • Study the derivation of strain measures in continuum mechanics
  • Learn about the implications of small strain assumptions in material science
  • Explore advanced topics in elasticity theory
  • Investigate the application of Taylor series in engineering problems
USEFUL FOR

Students and professionals in mechanical engineering, civil engineering, and materials science who are looking to deepen their understanding of strain analysis and continuum mechanics principles.

nissan4l0
Messages
11
Reaction score
0
http://imgur.com/SnHyP

What are the mathematical steps and assumptions to reach the conclusion that length(ab) ≈ dx + ∂u/∂x*dx ?

If you consider the the squares of the gradients to be negligible, you still have a square root and multiplication by the constant "2". What other assumptions do we make to derive the final equation?
 
Physics news on Phys.org
You make a Taylor expansion of sqrt(1+e), where "e" is regarded to be small.
Then, you have, approximately, sqrt(1+e)=1+e/2.
 

Similar threads

How Is Normal Strain Derived in Calculus?
  • · Replies 3 ·
Replies
3
Views
3K
High School Having trouble deriving the volume of an elliptical ring toroid
  • · Replies 4 ·
Replies
4
Views
4K
Graduate Continuum Mechanics - Deformation gradient
  • · Replies 4 ·
Replies
4
Views
3K
MHB Normalization of a gaussian wavefunction
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad "Undo" Second Derivative With Square Root?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad How do you normalize this wave function?
  • · Replies 31 ·
2
Replies
31
Views
6K
Graduate Understanding Mechanics of Materials: Derivation of Strain Energy
  • · Replies 8 ·
Replies
8
Views
6K
Quantum Mechanics: The First Step in Proving the Constancy of a Normalization
  • · Replies 5 ·
Replies
5
Views
2K
Graduate How Is the Heisenberg Uncertainty Principle Derived?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate How do you derive relativistic tensors in an orthonormal basis?
  • · Replies 2 ·
Replies
2
Views
3K