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Separation

The term separationrefers to any technique for determining one of the principal

stresses uniquely. It must be remembered that when we use the basic relation

1 2

Nf

h (1)

we determine only the difference between the principal stresses not these stresses

separately.

Boundary condition

Perhaps the most important separation method consists of applying the

boundary condition at a free edge of the specimen (Fig. 1). If t and n denote the

tangential and normal directions at a free boundary, then = 0 because there is no

shear traction applied at the boundary, and therefore the normal stresses nand tare

principal stresses.

Fig.1. Stress state at a free boundary.

Furthermore, n= 0 because there is no normal traction applied to the surface,

either. The principal stresses are now separated because one of them, n, is equal to

zero and the other one, t, must account for the difference 1 - 2:

n

1 2 t

0,

Nf

h

(1)

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Note that the signof the tangential normal stress is not resolved; it may be positive,

negative, or zero.

Shear-difference method

To separate the principal stresses in the interior of a specimen, more work is

required. A method that is particularly useful because it does not require any additional

experimental apparatus is the shear-difference method, which is based on stress

equilibrium, as will be shown subsequently. Before we examine this method, let us first

observe what can be determined easily at any point in the specimen using a

combination of isochromatics and isoclinics. The isochromatics provide the principal-stress difference

1 2

Nf

h

and therefore the radius Rof Mohrs circle

1 2Nf

R2 2h

(2)

as shown in Fig. 2.

Fig.2. Calculation of from isochromatics and isoclinics at an arbitrary point.

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Meanwhile, a complete family of isoclinics will provide values of the principal-

stress orientation a throughout the specimen. Thus, at any point in the specimen, the

angle 2 in Mohrs circle can be found. Observe from Fig. 2 that the shear stress

has the value

xy

NfRsin2 sin2

2h (3)

and that sufficient information is available from the isochromatics and isoclinics to

determine this value anywhere in the specimen.

We now consider how to use values ofalong certain lines in the specimen

to separate the principal stresses. Recall from stress equilibrium (Fig. 3) that, in the

absence of body forces,

Fig.3. Differential stress element.

xyx 0

x y

everywhere in the body. Along a line y= const (Fig. 4), dy= 0, and therefore

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x x

x

xy x

xy

d dx dyx y

dx 0y y

dxy

(4)

Fig.4. Integration path for the shear-difference method.

In the shear-difference method, we write the central-difference approximation

for

xy

y

as

xy xy xy

y y

where

xy(x) and

xy(x) denote the values of shear stress along the lines y= y+ and y

= y- , respectively, and then integrate Eqn. (4) from x= x0 to an arbitrary point xalong

the line y= const:

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