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

Example2-6 (Solecki)

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

Example

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

Deviatoric Stress

• Deviatoric stress is the stress that drives shape change

D m S S I

Example

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

Symbolic ManipulationOpen MathCAD

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