1. Continuum Mechanics (MTH487)
Lecture 31
Instructor
Dr. Junaid Anjum

2. Recap : strain energy density : isotropic medium : Young’s modulus
: Poisson’s ratio
: shear modulus
: bulk modulus

3. Linear Elasticity
Problem 1. If and calculate the resulting strain tensor

4. Linear Elasticity
Problem 2. If the strain energy density is generalized in the sense that it is assumed to be a function of the deformation gradient components instead of the small strain components, that is, if , make use of the energy equation and the continuity equation to show that in this case

6. Linear Elasticity
Problem 2. For an isotropic elastic medium, express the strain energy density in terms of
the components of
the invariants of

8. Linear Elasticity
Problem 3. Show that

9. Linear Elasticity
Problem 4. Establish the following relation between the rate of working and the rate of change of energy

11. Fundamental Laws and Equations
Material Derivatives of Volume integrals
Let any scalar, vector or tensor property of the collection of particles occupying the current volume V be represented by the integral
: distribution of the property per unit volume
since is a fixed volume in the referential configuration,

12. Fundamental Laws and Equations
Material Derivatives of Volume integrals
: Transport Theorem
amount created in volume
amount entering through the bounding surface

13. Fundamental Laws and Equations
Material Derivatives of Surface integrals
: distribution of the property over the surface
Material Derivatives of Line integrals
for properties of particles lying on the spatial curve C

14. Fundamental Laws and Equations
Conservation of mass, Continuity equation:
The measure of mass may be a function of the space variables and time.
: mass of the volume in current configuration
defines the mass density of the body.
gives the total mass of the body in current configuration.
: mass of the body in the reference configuration.
The law of conservation of mass asserts that the mass of a body, or of any portion of the body, is invariant under any motion, that is, remains constant in every configuration.

15. Fundamental Laws and Equations
Conservation of mass, Continuity equation:
Continuity equation in Eulerian form
: for incompressible media (density of the individual
particles remain constant)

16. Fundamental Laws and Equations
Conservation of mass, Continuity equation:
Since the law of conservation of mass requires the mass to be the same in all configurations
hence in material description
which is known as the Langrangian or material form of the continuity equation

17. Fundamental Laws and Equations
Linear momentum principle, Equations of motion
Let a material continuum body having a current volume V and bounding surface S be subjected to surface traction and distributed body forces as shown in the figure. In addition, let the body be in motion the velocity field
: linear momentum of the body
the principle of linear momentum states that the time rate of change of the linear momentum is equal to the resultant forces acting on the body.

18. Fundamental Laws and Equations
Linear momentum principle, Equations of motion
: equilibrium equations

19. Fundamental Laws and Equations
Linear momentum principle, Equations of motion
A fluid at rest (or in a state of rigid body motion) is incapable of sustaining any shear stress whatsoever. Hence stress vector on an arbitrary element of surface at any point in a fluid at rest is
where the proportionality constant is the thermostatic pressure or as it is frequently called, the hydrostatic pressure.
i.e. the hydrostatic pressure is equal to the mean normal stress.
For a fluid in motion, the shear stresses are not usually zero, and in this case we write
where is the viscous stress tensor which is zero for fluid at rest. The pressure is the thermodynamic pressure.

20. Fundamental Laws and Equations
Linear momentum principle, Equations of motion
showing that for a fluid in motion is not equal to the mean normal stress. Also the above relation reduces to for a fluid at rest
non linear : Stokesian fluid
linear : Newtonian fluid
: viscous properties of the fluid
for homogeneous isotropic Newtonian fluid,
where and are viscosity coefficients

21. Fundamental Laws and Equations
Linear momentum principle, Equations of motion
hence the mean normal stress for a Newtonian fluid is
The Stokes condition assures that for a fluid at rest the mean normal stress equals the negative pressure p.

22. Fundamental Laws and Equations
Linear momentum principle, Equations of motion
invoking Stokes condition,
for incompressible fluids

23. Equations of motion Steady flow
If the velocity components of a fluid are independent of time, the motion is called a steady flow.
Furthermore, if the velocity field is constant and equal to zero everywhere, the fluid is at rest, and the theory for this condition is called hydrostatic. For this the Navier-Stokes equations are simply
assuming barotropic condition between and
additionally, considering body forces to be conservative

24. Equations of motion
governing equation for steady flow of a barotropic fluid with conservative body forces.

25. Equations of motion The Bernoulli Equation
Equation of motion for barotropic fluid with conservative body forces
Streamline: A space curve, tangent to which at each point gives the direction of the velocity vector.
integrating above equation along the streamline
: Bernoulli Equation
: differential tangent vector along the streamline
G constant for steady flows (may vary from streamline to streamline)
unique constant G, if the flow is irrotational