Stress-Strain Theory

1
Stress-Strain Theory
Under action of applied forces, solid bodies
undergo deformation, i.e., they change shape and
volume. The static mechanics of this deformations
forms the theory of elasticity, and dynamic
mechanics forms elastodynamic theory.
2
Strain Tensor
After deformation
Displacement vector u(x) x- x
3
Strain Tensor
After deformation
(1)
into equation (1)
4
Strain Tensor
After deformation
(2)
5
Problem
V gt C
6
Problem
V gt C
V lt C
Elastic Strain Theory
Elastodynamics
7
Acoustics
8
Acoustics
9
Acoustics
No Shear Resistance No Shear Strength
10
Acoustics
dw, du ltlt dx, dz
dx
11
Acoustics
really small
big small
big small
Infinitrsimal strain assumption elt.00001
Dilitation
dx
12
1D Hookes Law
pressure
strain
-k
du
Infinitrsimal strain assumption elt.00001
F/A
Pressure is F/A of outside media acting on face
of box
13
Hookes Law
Infinitrsimal strain assumption elt.00001
F/A
Pressure is F/A of outside media acting on face
of box
14
Hookes Law
Dilation
e
k
U
Infinitrsimal strain assumption elt.00001
e
k
-
(
)
P
S

xx
Bulk Modulus
15
Newtons Law
ma F
-dxdz
16
Newtons Law
17
Newtons Law
(Newtons Law)
(Hookes Law)
Divide (1) by density and take Divergence
Take double time deriv. of (2) substitute (2)
into (3)
18
Newtons Law
Constant density assumption
19
Constant density assumption
Body Force Term
F
20
(No Transcript)
21
Divergence
U(xdx,z)dz
0
gtgt 0
No sources/sinks inside box. What goes in must
come out
Sources/sinks inside box. What goes in might not
come out
U(xdx,z)
U(x,z)
(x,z)