mech 5312 solid mechanics ii - mechanical engineeringme.utep.edu/cmstewart/documents/me5312/lecture...
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MECH 5312 – Solid Mechanics IIDr. Calvin M. Stewart
Department of Mechanical Engineering
The University of Texas at El Paso
Table of Contents
• Principal Stresses
• Stress Invariants
• Principal Directions
• Octahedral Stress
• Hydrostatic and Deviatoric Stresses
• 3D Mohr’s Circle
• Stress Analysis
• Symbolic Manipulation
Principal Stress
• The principal directions coincide with Arbitrary plane, n through the body where the normal stress, σn is maximized and shear stress disappears, τ =0
0
0
0
xx n xy xz x
yx yy n yz y
zzx zy zz n
n
n
n
n S I n 0
2 2 2 1x y zn n n
Stress Invariants
• To solve for the principal stresses, σ1 , σ2 , and σ3 take the determinant of the inner portion and solve
• or use the Stress Invariants
det 0n S I
3 2
1 2 3 0n n nI I I
1 tr xx yy zzI S
2 2 2
2 xy yz xz xx yy yy zz zz xxI
3 detI S
1 2 3
Principal Directions
• The principal directions, n1, n2, and n3 can then be obtained by plugging each principal stress into
• and solving for n using the 4 equations.
n S I n 0
2 2 2 1x y zn n n
Octahedral Stresses
• Consider a stress element with only principal stresses.
• If we section this element into a system of planes that are equally inclined with respect to each other.
• Theses are the octahedral planes with the stress element.
Octahedral Stresses
• The Octahedral normal stress and shear stress can be calculated as follows
• In terms of principal stress
• In term of general stress
Hydrostatic Stress
• Cauchy stress can be decomposed into volumetric (hydrostatic/spherical) and shear (deviatoric) parts.
• Hydostatic stress is the stress that drives volume change and is often called mean stress
1tr
3 3
xx yy zz
H m oct
S
d H S S I
Mohr’s Circle
• A graphical approach to see how normal and shear stress change with the angle of the arbitrary plane, n
1 2
max,12
2 3
max,23
1 3
,max max,13
2
2
2abs
Calvin M. StewartAssistant ProfessorDepartment of Mechanical EngineeringThe University of Texas at El Paso500 W. University Blvd.Suite A126El Paso, Texas 79968-0717
Email: [email protected]: http://me.utep.edu/cmstewart/Phone: 915-747-6179Fax: 915-747-5019
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