Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 380

and subsequent use of the double angle trigonometric formulas sin 2 y ¼ 2 sin y

cos 2 sin 2 cos y and cos 2 y ¼ y y yield the following:

s x 0 ¼ð 1 = 2 Þðs x þ s y Þþð 1 = 2 Þðs x s y Þ cos 2 y þ t xy sin 2 y

s y 0 ¼ð 1 = 2 Þðs x þ s y Þð 1 = 2 Þðs x s y Þ cos 2 y t xy sin 2 y;

t x 0 y 0 ¼ð = Þðs x s y Þ y þ t xy cos 2 y:

1 2 sin 2 (A.143)

These are formulas for the stresses s x 0 , s y 0 , and t x 0 y 0 as functions of the stresses

s x , . Note that the sum of the first two equations in

(A.143) yields the following expression, which is defined as 2 C ,

s y and t xy , and the angle 2 y

2 C s x 0 þ s y 0 ¼ s x þ s y : (A.144)

s x 0 þ s y 0 ¼ s x þ s y is a repetition of the result (A.90) concerning
the invariance of the trace of a tensor, the first invariant of a tensor, under change of
basis. Next consider the following set of equations in which the first is the first of
(A.143) incorporating the definition (A.144) and transposing the term involving C
to the other side of the equal sign, and the second equation is the third of (A.143):

The fact that

s x 0 C ¼ð 1 = 2 Þðs x s y Þ cos 2 y þ t xy sin 2 y;

t x 0 y 0 ¼ð 1 = 2 Þðs x s y Þ sin 2 y þ t xy cos 2 y:

If these equations are now squared and added we find that

2 2

R 2 ðs x 0 C Þ þðt x 0 y 0 Þ ¼ (A.145)

where,

2 2

R 2 ð 1 = 4 Þðs x s y Þ þðt xy Þ : (A.146)

Equation (A.145) is the equation for a circle of radius R centered at the point
s x 0 ¼ 0. The circle is illustrated in Fig. A.3 .

The points on the circle represent all possible values of

C , t x 0 y 0 ¼

t x 0 y 0 ; they are
determined by the values of C and R , which are, in turn, determined by

s x 0 , s y 0 and

s x , s y , and

t xy . The eigenvalues of the matrix are the values of the normal stress s x 0 when the

s

circle crosses the R , as may

be seen from Fig. A.3 . Thus Mohr's circle is a graphical analog calculator for the
eigenvalues of the two-dimensional second order tensor

s x 0 axis. These are given by the numbers C þ R and C

, as well as a graphical

s

Q T representing the transformation of
components. The maximum shear stress is simply the radius of the circle R ,an
important graphical result that is readable from Fig. A.3 .
As a graphical calculation device, Mohr's circles may be extended to three
dimensions, but the graphical calculation is much more difficult than doing the

s 0 ¼

analog calculator for the equation Q s