Stress resolution

The element may be rotated about all three axes into a unique principal orientation in which all shear stresses vanish. The corresponding normal stresses in this principal orientation are termed the principal stresses. A stress state is characterised most succintly by the principal stresses, say ( σ₁, σ₂, σ₃ ), and failure theories - the next step in failure assessment - are expressed in terms of principal stresses. It is therefore necessary to examine how principal stresses are derived from Cartesian components.

Stress is a tensor entity, so complex tensor arithmetic must be applied in the three- dimensional case to evaluate stress components as the element rotates. We consider here only the simpler two dimensional case in which one direction is known to be principal, and resolution consists of element rotation only about that principal axis. Thus in the 2-dimensional sketch above, the z-axis is principal and x-y stress components vary as the element rotates about that axis. Fortunately the great majority of practical cases are two -dimensional since the relatively simple loading in conjunction with the natural choice of axes leads to one of the axes being principal automatically. For example there can be no stress on the free surface of a body, so the surface normal is an obvious choice for one of the three Cartesian axes - as there can be no shear on this surface, the axis is automatically a principal axis.
Although we consider only "two-dimensional stresses", it is important to remember always that stress states are essentially three dimensional.

In this course the positive convention adopted for the orientation of a plane (characterised by its normal) is shown at (a) below, and for normal and shear stress and strain at (b) - positive shear is counter-clockwise. The two sketches (c) illustrate the consistency of the positive senses for shear stresses and strains; the total shear strain (distortion) is γ = 2( γ/2). Using these conventions, the plane stress state at 'A' in the example of the previous section is :
                σ_x   =   -93,       σ_y   =   0,       τ_xy =   +34,       ( τ_yx =   -34)     MPa

We now examine the variation of normal and shear stress components as the inclination of the face on which they occur changes.
Consider the elemental unit cube ( size 1x1x1 ) under the known positive 2-dimensional stress components shown in ( i) below - the third dimension (z) is principal. Rotational equilibrium requires complementary shear, that is   τ_yx must equal   -τ_xy. This necessity has been incorporated into ( ii), from which it is apparent that the three components ( σ_x, σ_y, τ_xy ) together with the third principal are necessary to define the stress state. The three components are called the   Cartesian triad.

We wish to evaluate - in terms of the Cartesian triad - the stress components ( σ, τ) in the general direction θ, so we consider force equilibrium of the wedge element ( iii), one of whose faces is inclined at θ. The height of the wedge remains 1 unit, however the dimension in the x-sense becomes 1.tanθ, and the length of the hypotenuse is 1.secθ.
The force components on each face of the wedge are the stress components multiplied by the face area - these are shown in ( iv). For force equilibrium of the wedge :
in the σ-sense :-       σ.secθ   - ( σ_x + τ_xy.tanθ ) cosθ   - ( σ_y.tanθ + τ_xy ) sinθ   =   0
in the τ-sense :-       τ.secθ   + ( σ_x + τ_xy.tanθ ) sinθ   - ( σ_y.tanθ + τ_xy ) cosθ   =   0
        which on simplification give the required resolution equations :-
                σ   =   ¹/₂ ( σ_x + σ_y ) + ¹/₂ ( σ_x - σ_y ) cos 2θ + τ_xy sin 2θ
                τ   =                                 - ¹/₂ ( σ_x - σ_y ) sin 2θ + τ_xy cos 2θ

Similar equations but with different signs are encountered in the literature - sign differences arise from positive conventions other than the above.

The simultaneous occurence of sine and cosine terms in these equations makes it difficult to visualise how the resolved components ( σ, τ) vary as the direction θ changes. More easily interpreted equations result if the stress state is defined by the   basic triad ( σ_m, σ_a, θ_p ) rather than by the Cartesian triad. θ_p is the inclination of the plane of the   maximum principal stress with respect to the x-reference.
The basic triad is related formally to the Cartesian as follows :-
( 3a)     σ_m =  ¹/₂ ( σ_x + σ_y )
                σ_a cos 2θ_p = ¹/₂ ( σ_x - σ_y )       that is     σ_a = √[ {¹/₂ ( σ_x - σ_y ) }² + τ_xy² ]
                σ_a sin 2θ_p   =   τ_xy                                         θ_p = ¹/₂ arctan [ 2 τ_xy /( σ_x - σ_y ) ]
                              . . . . though these equations seldom need to be implemented.

Making these substitutions leads to resolution equations in the more meaningful form :-
( 4a)     σ   =   σ_m + σ_a cos 2 ( θ_p - θ )         { nomenclature explanation }
                τ   =               σ_a sin 2 ( θ_p - θ )

It is apparent that normal and shear stress components vary sinusoidally with direction θ (not unlike vector components) however the variation is second harmonic - that is stress components are the same along axes which lie at 180^o to one another. The two sinusoids are of the same amplitude σ_a and out of phase by 45^o.
σ_m is the constant component of the normal stress.

The principal and maximum shear stresses follow immediately as :-
( 5)       σ_max   =   σ_m + σ_a       ( at θ_p )
              σ_min   =   σ_m - σ_a       ( at θ_p - π/2 )
              τ_max   =             σ_a       ( at θ_p - π/4 )

These relations are often expressed graphically via Mohr's stress circle, in which σ_m and σ_a represent the circle's centre location and radius respectively. The   conventions require that angles on the circle, reckoned from the X-radius, are   double the corresponding angles on the element (which are measured from the x-reference), and in the   opposite sense. The reader should confirm that this construction satisfies equation ( 4a).

This script resolves the Cartesian components of a two-dimensional stress state into the principal components.

It is important to remember that   a stress-strain state is always essentially three- dimensional, involving three principals. We have addressed only resolution in two dimensions to obtain a single Mohr's circle involving the principals in these two dimensions - however two other circles must exist relating these two principals with the third.
As noted above the third (eg. automatic) principal is usually deduced from the nature of the problem. Two common situations arise when the state is either one of :

        plane stress :

If there is no stress orthogonal to the 1-2 resolution plane then   σ₃ = 0

        plane strain :

If there is no strain orthogonal to the 1-2 resolution plane then   ε₃ = 0 and it follows from ( 2) that   σ₃ = ν ( σ₁ + σ₂ )

Fig F completes the Mohr's circles for the example above (Fig D) assuming that the element is loaded in plane stress, that is the three principals are ( -600, -100, 0 ) MPa.
Fig G illustrates the three principals and three Mohr's circles for a completely unrelated stress state where two-dimensional resolution happens to relate to the largest (3-2) circle and σ₁ is the principal stress orthogonal to the resolution plane.

The outcome of stress resolution at an element must be a set of three principals - all three must be known before the element's safety can finally be assessed by application of an appropriate failure theory.

Strain resolution

Resolution of strain is generally unnecessary when assessing the safety of common engineering components, however the following description is given for completeness.

If a material behaves in a linear elastic manner then the directions of principal strains are identical to the directions of principal stresses, and all the preceding equations, and Mohr's circles, may be expressed in strain terms - provided that everywhere in the stress equations :

• normal stress, σ, is replaced by normal strain, ε, and
• shear stress, τ, is replaced by half the shear strain, ie. by γ/2.

The strain analogues of the foregoing are therefore :-
( 3b)     ε_m =  ¹/₂ ( ε_x + ε_y )     ;     ε_a = ¹/₂ [ ( ε_x - ε_y )² + γ_xy² ]^1/2     ;     θ_p = ¹/₂ arctan [ γ_xy /( ε_x - ε_y ) ]
( 4b)     ε     =   ε_m + ε_a cos 2 ( θ_p - θ )     ;         γ/2   =   ε_a sin 2 ( θ_p - θ )

Experimentally, surface stresses are found from strain gauges attached to accessible surfaces. It is useful therefore to be able to quickly interrelate stresses and strains for a plane stress state, for which, since the principal stress and strain directions coincide, it follows from ( 2) and ( 3) that :-
( 6)     E ε_m = ( 1 - ν ) σ_m     ;         E ε_a = ( 1 + ν ) σ_a
which relate the centres and radii of the two circles. If these circles are drawn to scale, and it is required that their circumferences coincide for ease of drawing, then it may be shown that :
( 7)     E $_ε = ( 1 + ν ) $_σ     ;         ( 1 + ν ) C_ε = ( 1 - ν ) C_σ         where

	$ε	is the scale of the strain circle ( strain units/mm )
	$σ	is the scale of the stress circle ( MPa/mm )
	Cε	is the distance (mm) of the strain circle centre from the shear axis
	Cσ	is the distance (mm) of the stress circle centre from the shear axis

Although the use of scaled Mohr's circles is not necessarily advocated, it is strongly recommended that the circles are at least sketched free-hand as an aid to interpretation of the equations. It is important that skill is developed in visualising the interplay between components and principals.
The following example demonstrates application of strain-to-stress transformation.