1. Continuum Mechanics (MTH487)
Lecture 30
Instructor
Dr. Junaid Anjum

2. Recap …
The principle of virtual work
Linear elasticity on

3. Aims and Objectives Linear elasticity (continued) Hooke’s law
Hooke’s law for isotropic media
Elastic constants (Young’s modulus, Poisson’s
ratio, bulk modulus, shear modulus)
Physical interpretation of elastic constants
problems

4. Linear Elasticity stress is a function of the strain
nonlinear
inelastic
linear
load
unload
Stress
Strain
load
unload
Stress
Strain
load
unload
Stress
Strain
stress is a function of the strain
complete recovery to the natural state upon
removal of the applied forces
: constitutive equation for elastic behavior
A body is linearly elastic if the stress depends linearly on the infinitesimal strain
where C, called the elasticity tensor
C: 81 components

5. Linear Elasticity
36 components

6. Linear Elasticity Strain as a function of stress
where the fourth-order tensor A is known as the compliance tensor; it is the inverse of C and therefore
not a tensor !!!

7. Linear Elasticity energy equation reduces to adiabatic, isothermal
: strain energy (purely mechanical)

8. Linear Elasticity : strain energy density
stress in terms of strain energy
linearized form …

9. Linear Elasticity
21 components

10. Linear Elasticity
Hooke’s law for isotropic media

11. Linear Elasticity
Problem 1. Show that for an isotropic linear elastic solid the principal axes of the stress and strain tensors coincide, and develop an expression for the relationship among their principal values.

12. Linear Elasticity
Hooke’s law for isotropic media: Elastic constants

13. Linear Elasticity Hooke’s law for isotropic media: Elastic constants
: Young’s modulus
: Poisson’s ratio

14. Linear Elasticity Physical interpretations …
Hooke’s law for isotropic media: Elastic constants
: shear modulus
: bulk modulus
Physical interpretations …
Uniaxial state of stress

15. Linear Elasticity Physical interpretations …
Hooke’s law for isotropic media: Elastic constants
Physical interpretations …
Simple Shear
Uniform triaxial tensions (or hydrostatic compression)
relates pressure to the change in volume

16. Linear Elasticity
Problem 3. For and calculate the normal strains for and

18. Things to remember : strain energy density : isotropic medium
: Young’s modulus
: Poisson’s ratio
: shear modulus
: bulk modulus