Constitutive Equations for Linear Elasticity and Newtonian Fluids
I.
The Generalized Hookes Law
The generalized Hookes law is a linear constitutive equation relating the Cauchy stress
to the infinitesimal strain.
C
(1)
Since this is a double inner produc
ME 5143
Homework #3
Spring 2014
Due 2/19
(1) The displacement vector has components given by:
u1 X 1 ,
Where
u2 X 2 ,
u3 X 3
X 1 , X 2 , X 3 are the reference coordinates and , and are parameters.
Determine the Euler-Almansi strain tensor, the Green-Lagra
ME 5143
Spring 14
Homework #1
Due 2/5
1 0 2
(1) Given S 0 1 2 evaluate ( a ) Skk ( b ) Sij Sij ( c ) S jk S jk ( d ) S pq Sqp
3 0 3
(2) Determine which of the following equations have the same meaning as ai Qij aj :
( a ) al Qlm am
(3)
( b )a p Qqpaq
( c
ME 5143
Homework #4
Spring 14 Due 2/26
1. Plane strain is define as a state of strain where zz , xz , yz 0 . Determine the form that
Hookes law must take for this case note that zz 0 . Show that xz , yz 0 and that
xx , yy , xy may be found given xx , yy
Solution of Boundary Value Problems in Linear Elasticity
I.
The Field Equations and Boundary Conditions for Boundary
Value Problems in Linear Elasticity
We summarize the previous developments to state the governing equations for the
displacement, strain a
Stress and Equilibrium
(I)
Surface Force and Body Force Distributions and Equilibrium
Consider the volume V shown in the figure. There are two types of forces such a
volume may experience, volumetric forces and surface forces. The usual volumetric force
t
Kinematics of Deformation
I.
Material and Spatial Coordinates
Consider the figure below depicting a chunk of material in the reference (material or
undeformed) configuration shown as Ro and the same chunk of material after loads have
been applied in a spa
ME 5143
Spring 14
Homework #1- Solution
1 0 2
(1) Given S 0 1 2 evaluate ( a ) Skk ( b ) Sij Sij ( c ) S jk S jk ( d ) S pq Sqp
3 0 3
Solution
(a) Skk S11 S22 S33 1 1 3 5
(b)
Sij S ij S1 j S1 j S 2 j S 2 j S 3 j S 3 j
S11 S11 S12 S12 S13 S13
S 21 S 21 S