1. Conservation Laws for Continua
Mass Conservation
Linear Momentum Conservation
Angular Momentum Conservation

2. Work-Energy Relations
Rate of mechanical work done on
a material volume
Conservation laws in terms of other stresses
Mechanical work in terms of other stresses

3. Principle of Virtual Work (alternative statement of BLM)If
for all
Then on

4. Thermodynamics Temperature Specific Internal Energy
Specific Helmholtz free energy
Heat flux vector
External heat flux
Specific entropy
First Law of Thermodynamics
Second Law of Thermodynamics

5. Transformations under observer changes
Transformation of space under a change of observer
All physically measurable vectors can be regarded as connecting two points in the inertial frame
These must therefore transform like vectors connecting two points under a change of observer
Note that time derivatives in the observer’s reference frame have to account for rotation of the reference frame

6. Some Transformations under observer changes

7. Some Transformations under observer changes
Objective (frame indifferent) tensors: map a vector from the observed (inertial) frame back onto the inertial frame
Invariant tensors: map a vector from the reference configuration back onto the reference configuration
Mixed tensors: map a vector from the reference configuration onto the inertial frame

8. Constitutive Laws
Equations relating internal force measures to deformation measures are known as Constitutive Relations
General Assumptions:
Local homogeneity of deformation
(a deformation gradient can always be calculated)
Principle of local action
(stress at a point depends on deformation in a vanishingly small material element surrounding the point)
Restrictions on constitutive relations:
1. Material Frame Indifference – stress-strain relations must transform consistently under a change of observer
2. Constitutive law must always satisfy the second law of
thermodynamics for any possible deformation/temperature history.

9. Fluids Conservation Laws Properties of fluids Kinematics
No natural reference configuration
Support no shear stress when at rest
Kinematics
Only need variables that don’t depend on ref. config
Conservation Laws

10. General Constitutive Models for Fluids
Objectivity and dissipation inequality show that constitutive relations must have form
Internal Energy
Entropy
Free Energy
Stress response function
Heat flux response function
In addition, the constitutive relations must satisfy
where

11. Constitutive Models for Fluids
Elastic Fluid
Ideal Gas
Newtonian Viscous
Non-Newtonian

12. Derived Field Equations for Newtonian Fluids
Unknowns:
Must always satisfy mass conservation
Combine BLM
With constitutive law. Also recall
Compressible Navier-Stokes
With density indep viscosity
For an incompressible Newtonian viscous fluid
Incompressibility reduces mass balance to
For an elastic fluid (Euler eq)

13. Derived Field Equations for Fluids
Recall vorticity vector
Vorticity transport equation (constant temperature, density independent viscosity)
For an elastic fluid
For an incompressible fluid
If flow of an ideal fluid is irrotational at t=0 and body forces are curl free, then flow remains irrotational for all time (Potential flow)

14. Derived field equations for fluids
For an elastic fluid
along streamline
Bernoulli
For irrotational flow
everywhere
For incompressible fluid

15. Normalizing the Navier-Stokes equation
Incompressible Navier-Stokes
Normalize as
Reynolds number
Euler number
Froude number
Strouhal number

16. Limiting cases most frequently used
Ideal flow
Stokes flow

17. Governing equations for a control volume (review)
Solving fluids problems: control volume approach
Governing equations for a control volume (review)

18. Example Steady 2D flow, ideal fluid
Calculate the force acting on the wall
Take surrounding pressure to be zero

19. Exact solutions: potential flow
If flow irrotational at t=0, remains irrotational; Bernoulli holds everywhere
Irrotational: curl(v)=0 so
Mass cons
Bernoulli

20. Exact solutions: Stokes Flow
Steady laminar viscous flow between plates
Assume constant pressure gradient in horizontal direction
Solve subject to boundary conditions

21. Exact Solutions: Acoustics
Assumptions:
Small amplitude pressure and density fluctuations
Irrotational flow
Negligible heat flow
Approximate N-S as:
For small perturbations:
Mass conservation:
Combine:
(Wave equation)

22. Wave speed in an ideal gas
Assume heat flow can be neglected
Entropy equation: so
Hence:

23. Application of continuum mechanics to elasticity
Material characterized by

24. General structure of constitutive relations
Frame indifference, dissipation inequality

25. Forms of constitutive relation used in literature
Strain energy potential

26. Specific forms for free energy function
Neo-Hookean material
Mooney-Rivlin
Generalized polynomial function
Ogden
Arruda-Boyce

27. Solving problems for elastic materials (spherical/axial symmetry)
Assume incompressiblility
Kinematics
Constitutive law
Equilibrium (or use PVW)
(gives ODE for p(r)
Boundary conditions

28. Linearized field equations for elastic materials
Approximations:
Linearized kinematics
All stress measures equal
Linearize stress-strain relation
Elastic constants related to strain energy/unit vol
Isotropic materials:

29. Elastic materials with isotropy

30. Solving linear elasticity problems spherical/axial symmetry
Kinematics
Constitutive law
Equilibrium
Boundary conditions

31. Some simple static linear elasticity solutions
Navier equation:
Potential Representation (statics):
Point force in an infinite solid:
Point force normal to a surface:

32. Simple linear elastic solutions
Spherical cavity in infinite solid under remote stress:

33. Dynamic elasticity solutions
Plane wave solution
Navier equation
Solutions: