Characteristic Entities in PhotoStress Method

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The creation of isoclines with regards to angle parameter is a periodical phenomenon with period . With synchronous rotation of the polariser and the analyser, isoclinic fringes continuously change from the isoclines of angle parameter up to angle parameter in line with rotation angle of the polariser and the analyser. They are distributed along whole analysed surface of the object under examination simultaneously according to the changes in directions of principal normal stresses. The isoclinic fringe runs through every point of the surface under consideration with a specific angle parameter as there is only one direction of principal stresses, i.e. direction or . However, there are points, fringes and surfaces which remain dark even during simultaneous rotation of the polariser and the analyser. Both principal normal stresses in such points, fringes and surfaces have identical magnitude in every direction. Through these points, fringes or surfaces are running isoclines of all angle parameters. These are called singular points, fringes or surfaces ^[3].

Angle parameter changes from point to point along any curve S which runs through the point P (Figure 2).

Figure 2. General example of an isocline and stress trajectory (isostatic line)

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However, there is the S_i curve which runs through the point P. Along this curve the angle parameter remains constant, while

(1)

Provided that stress is known, and , the equation of such isocline can be expressed mathematically as a relation

(2)

where represent normal stresses, is shear stress and represents isoclinic angle parameter. One specific isocline is related to every relevant constant.

• the isocline with angle parameter connects points in which shear stress and normal stress . On one side, the shear stress >0 and on the other side the shear stress <0.

• angle parameter α changes on the examined surface continuously and, as a result, other isoclines following each other do not intersect. Only one isocline can run through the analysed point, except for singular points ^{[1, 5]}.

• the edge of the plane object subject to examination, which is not shear-loaded in tangential direction, has only principal normal stress other than zero ^{[1, 5]}.

3. Singular Points

Points, in some cases lines and surfaces, which remain dark permanently even when rotating intersected polarisation filters, are called singular points, lines and surfaces. Shear stresses and the difference of principal normal stresses in singular points equal zero. In singular points the object is loaded with hydrostatic tension (> 0) or pressure (< 0), or is in a stressless state ( = 0). Any direction in these points can be considered the direction of principal normal stresses. Isoclines of all angle parameters intersect in singular points. As a result, isostatic lines in the area of singular point are largely curved what makes the drawing more difficult. The position of singular points can be identified on the basis of the picture of isochromatics, which appear as dark points, fringes or surfaces ^{[3, 4]}.

According to rotation direction, singular points can be divided into two groups:

• positive singular points

• negative singular points.

When rotating the analyser clockwise (to the right) and isoclinic fringes are moving equally around the singular point, the singular point is positive (Figure 3).

Figure 3. Positive singular point

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Isoclinic fringes are moving in the opposite direction around negative singular point, i.e. anti-clockwise (Figure 4). Singular points which occur on the object's circumference are always negative.

Figure 4. Negative singular point

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Singular points can as well be classified as points of first order, points of second order and points of a higher order. Such classification respects the number of dark lines which run through given singular point. Singular points of second and a higher order are unstable with respect to the nature of mathematical expressions of stress components. It is manifested through decomposition to singular points of lower orders (e.g. point of second order breaks down to two points of first order) even after a small change in the shape of the object subject to examination. Figure 5 depicts singular points of different orders which occur in a network of isostatic curves ^{[3, 4]}.

Figure 5. Singular points

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4. Isostatic Lines

Isostatic lines are lines which are characteristic for inner stress states. They provide informative overview of the behaviour of the object subject to examination. There are zero shear stresses along isostatic lines. Their tangents give the direction of principal normal stresses in every point of the analysed object ^[1].

Isoclinic lines do not offer direct representation of the direction of principal stresses. On the other hand, they provide us with data which are necessary to gain a set of isostatic lines (stress trajectories) of I and II type. Stress trajectories can be obtained via graphical construction, while the base for such construction of isostatic lines of I and II type is the set of isoclinic lines. Graphical construction of isostatic lines of I and II type is described in detail in given literature ^{[4, 5]}.

Isostatic lines can be expressed mathematically with equation

(3)

where angle is defined by the expression which derives from the Mohr's circle (Figure 6) ^{[3, 4]}.

Figure 6. Mohr's circle of stresses

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Considering the Mohr's circle (Figure 6):

(4)

Adjusting the relation (2) and after substitution for

(5)

a differential equation of isostatic lines of I type emerges in the following form:

(6)

Considering isostatic lines of II type which in every point are perpendicular to isostatic lines of I type:

(7)

Given differential equations (6) and (7) define both types of isostatic lines.

Figure 7 shows the set of isostatic lines of I and II type in a half plane loaded by a single force ^[3].

Figure 7. Set of isostatic lines in a half plane loaded by a single force

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5. Isochromatic Fringes

Isochromatic fringes are known as connecting lines of points along which the difference of principal normal stresses is constant. They appear on an illuminated photoelastic coating as lines of single (iso) colour (chromos). They can be observed under circular polarised light ^{[3, 6]}. Figure 8 shows isochromatic fringes during gradual diametrical tensional loading of the photoelastically coated sample ^[2].

Figure 8. Isochromatic fringes

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Linear dependency between temporary birefringence and the difference of principal strains or principal normal stresses enables us to determine the plane stress state on the surface of a photoelastically coated object. There is a constant phase shift of light waves along individual isochromatic fringes, i.e. isochromatics are geometrical locations of constant birefringence points ^{[7, 8]}.

When white light is used, photoelastic pattern appears as a series of gradual isochromatic areas of different colours, while every area represents a different birefringence level related to that particular deformation in the object subject to examination. The colour of every area (isochromatic fringe) uniquely identifies the birefringence or the fringe order (deformation level) throughout the whole area. The characteristics of colourful isochromatic fringes are listed in Table 1 ^{[1, 5]}.

The orders of isochromatic fringes in the photoelastic coating are proportional to the difference between principal strains in the coating and on the surface of the object subject to examination. This linear relation can be expressed in terms of the following relation:

(8)

where represent principal strains in the given photoelastic coating,

- fringe constant of optical sensitivity of the photoelastic coating,

- fringe order identified in the measurement point during zero loading of the object under analysis (initial compensator value),

- fringe order identified in the measurement point during loading of the object under analysis (final compensator value),

- maximum shear deformation.

Table 1. Characteristics of colourful isochromatic fringes

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