en31 intro lect

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 observers
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 knownas 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

• Properties of fluids
• No natural reference configuration
• Support no shear stress when at rest
• Kinematics
• Only need variables that dont dependon 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 Newtonianviscous 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 t0
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
Strouhal number
Froude 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
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 t0, 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

(gives ODE for p(r)

• Equilibrium (or use PVW)

• 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