Post on 25-Jan-2021
MECH 5312 – Solid Mechanics IIDr. Calvin M. Stewart
Department of Mechanical Engineering
The University of Texas at El Paso
Table of Contents
• Principal Strains
• Strain Invariants
• Volume Changes and Dilatation
• Strain Deviator
Principal Strains
• Principal stresses exists on an Arbitrary plane, n in a body where only normal stresses appear. Find two complementary planes that are orthogonal to n. We arrive at the principal stress tensor.
• What form does the corresponding principal strain tensor take?
1
2
3
0 0
0 0
0 0
σ
1
2
3
n
n
n
1 2 3n n n
Principal Strain Tensor
• Since, the principal stress tensor contains only normal stresses, the principal strain tensor will contain only normal strains.
• Our procedure for finding the principal strains follows
1
2
3
0 0
0 0
0 0
ε
1
2
3
n
n
n
1 2 3n n n
Principal Strain
• The principal directions coincide with Arbitrary plane, n through the body where the normal strain, εn is maximized and shear strain disappears, γ =0
0
0
0
xx n xy xz x
yx yy n yz y
zzx zy zz n
n
n
n
n ε I n 0
2 2 2 1x y zn n n
Strain Invariants
• The exact solution to the determinant
• resolves into a quadratic equation
• where J1, J2, and J3 are the strain Invariants.
det 0n ε I
3 2
1 2 3 0n n nJ J J
1
2 2 2
2
3 det
xx yy zz
xx yy yy zz zz xx xy xz yz
J tr
J
J
ε
ε
ExampleMathCAD
Volume Change and Dilatation
• A differential element is subject to normal strain in the x direction
• The length of the side increases by
• The new length is now
xx
xxdx
1 xx dx
Example, for dx
dx dx xxdx
1 xx dx
Volume Change and Dilatation
• If the differential element is subject to normal strains
• The lengths of the sides increase by
• The new lengths are now
, , xx yy zz
, , xx yy zzdx dy dz
1 , 1 , 1xx yy zzdx dy dz
dx
1 1 1def xx yy zzV dxdydz
The Deformed Volume, Vdef
V dxdydz
The Initial Volume, V
dy
dz
Volume Change and Dilatation
• Assuming finite displacement, the change in volume becomes
• The change in volume per unit volume is
• where is the dilatation and
0def xx yy zzV V V dxdydz
xx yy zz
V
V
1J
V dxdydz
Second order terms are neglected!
Strain Deviator
• Strain can be decomposed into volumetric and deviatoric parts
• The deviatoric strain can be found by rearranging
1
3d
ε I ε
1
3d
ε ε I
ExampleMathCAD
Calvin M. StewartAssistant ProfessorDepartment of Mechanical EngineeringThe University of Texas at El Paso500 W. University Blvd.Suite A126El Paso, Texas 79968-0717
Email: cmstewart@utep.eduURL: http://me.utep.edu/cmstewart/Phone: 915-747-6179Fax: 915-747-5019
Contact Information
mailto:cmstewart@utep.eduhttp://me.utep.edu/cmstewart/