Stress Strain, Principal Axes, and Mohr's Circle

General Stress State  in Principal Axes  System

This is a general state, we will rotate about the z-axis and compare the result to a general two-dimensional stress state.

σtensordiag = ({{σprinc_xx, 0, 0}, {0, σprinc_yy, 0}, {0, 0, σprinc_zz}}) ;

σtensordiag//MatrixForm

( σprinc xx 0 0 0 σprinc yy 0 0 0 σprinc zz )

Rotation about z-axis by angle θ

rotmat[θ_] := ({{Cos[θ], -Sin[θ], 0}, {Sin[θ], Cos[θ], 0}, {0, 0, 1}}) ;

rotmat[θ]//MatrixForm

( Cos [ θ ] - Sin [ θ ] 0 Sin [ θ ] Cos [ θ ] 0 0 0 1 )

Transformation to general two-dimensional stress state coordinate system by rotating the principal system by θ around z-axis

σrot = Simplify[Transpose[rotmat[θ]] . σtensordiag . rotmat[θ]] ;

σrot//MatrixForm

( Cos [ θ ] 2 σprinc xx + Sin [ θ ] 2 σprinc yy Cos [ θ ] Sin [ θ ] ( - σprinc xx + σprinc yy ) 0 Cos [ θ ] Sin [ θ ] ( - σprinc xx + σprinc yy ) Sin [ θ ] 2 σprinc xx + Cos [ θ ] 2 σprinc yy 0 0 0 σprinc zz )

( Cos [ θ ] 2 σprinc xx + Sin [ θ ] 2 σprinc yy Cos [ θ ] Sin [ θ ] ( - σprinc xx + σprinc yy ) 0 Cos [ θ ] Sin [ θ ] ( - σprinc xx + σprinc yy ) Sin [ θ ] 2 σprinc xx + Cos [ θ ] 2 σprinc yy 0 0 0 σprinc zz )

Writing the same equation in a slightly different way...

σrotalt = Collect[σrot//TrigReduce, {Cos[2θ], Sin[2 θ]}] ;

σrotalt//MatrixForm

( 1 2 Cos [ 2 θ ] ( σprinc xx - σprinc yy ) + 1 2 ( σprinc xx + σprinc yy ) 1 2 Sin [ 2 θ ] ( - σprinc xx + σprinc yy ) 0 1 2 Sin [ 2 θ ] ( - σprinc xx + σprinc yy ) 1 2 Cos [ 2 θ ] ( - σprinc xx + σprinc yy ) + 1 2 ( σprinc xx + σprinc yy ) 0 0 0 σprinc zz )

( 1 2 Cos [ 2 θ ] ( σprinc xx - σprinc yy ) + 1 2 ( σprinc xx + σprinc yy ) 1 2 Sin [ 2 θ ] ( - σprinc xx + σprinc yy ) 0 1 2 Sin [ 2 θ ] ( - σprinc xx + σprinc yy ) 1 2 Cos [ 2 θ ] ( - σprinc xx + σprinc yy ) + 1 2 ( σprinc xx + σprinc yy ) 0 0 0 σprinc zz )

Naming the coefficients of the two-dimensional state in the rotated principal system

( σlab xx σlab xy σlab xz σlab xy σlab yy σlab yz σlab xz σlab yz σlab zz ) = σrotalt ;

1. σlab xx in laboratory system rotated by θ from principal axis system

σlab xx

1 2 Cos [ 2 θ ] ( σprinc xx - σprinc yy ) + 1 2 ( σprinc xx + σprinc yy )

1 2 Cos [ 2 θ ] ( σprinc xx - σprinc yy ) + 1 2 ( σprinc xx + σprinc yy )

2. σlab yy in laboratory system rotated by θ from principal axis system

σlab yy

1 2 Cos [ 2 θ ] ( - σprinc xx + σprinc yy ) + 1 2 ( σprinc xx + σprinc yy )

1 2 Cos [ 2 θ ] ( - σprinc xx + σprinc yy ) + 1 2 ( σprinc xx + σprinc yy )

3. σlab xy in laboratory system rotated by θ from principal axis system

σlab xy

1 2 Sin [ 2 θ ] ( - σprinc xx + σprinc yy )

1 2 Sin [ 2 θ ] ( - σprinc xx + σprinc yy )

All z - components remain zero except the original diagonal term σlab zz :

σlab_yz

σlab_xz

σlab_zz

0

0

0

0

σprinc zz

σprinc zz

Equations 1, 2, and 3 are the equations that express a circle known as Mohr's circle of stress (see accompanying class notes). The equations show the way in which the stress tensor components in a two-dimensional state of stress (a "biaxial" stress state) vary with orientation of the coordinate system in which the stresses are described.

Notice that there are two invariants of the general stress :

The trace (or twice the offset of Mohr's Circle):

Simplify [ σlab xx + σlab yy ]

σprinc xx + σprinc yy

σprinc xx + σprinc yy

And the determinant

Simplify [ σlab xx σlab yy - ( σlab xy ) ^ 2 ]

σprinc xx σprinc yy

σprinc xx σprinc yy

These last two results are precisely the trace and determinant of the x and y terms in the original diagonal form of the stress state, thus illustrating the invariance of these quantities under rotation of coordinate rotations.

Example of Mohr's circle for two-dimensional body in uniaxial tension with σprinc xx = 10 MPa and all other stress components equal to zero

σprinc_xx = 10. ;

σprinc_yy = 0. ;

σprinc_xy = 0. ;

ParametricPlot [ { σlab xx , σlab xy } , { θ , 0 , π } , AxesLabel { normal stress , shear stress } , AspectRatio 1 , PlotLabel \t \t Mohr Circle for 10 MPa Uniaxial Tension , PlotStyle { Thickness [ 0.01 ] , Hue [ 1 ] } ]

[Graphics:HTMLFiles/Lecture-10_15.gif]

Graphics

Graphics

Comparing this plot with Figure 10-3 in the lecture notes, we see that the maximum and minimum tensile stresses are 10 and 0 MPa (from intercepts with x axis), as expected, and the maximum shear stress is ±5 MPa and it is experienced on a plane oriented at 2θ = 90° or θ = 45° to the tensile axis (remember that angles on Mohr's circle plots are twice the angle in the body).


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