Mohr circles for strain, simplified

GEOLOGICAL JOURNAL, VOL. 22, 11S132 (1987)

Mohr circles for strain, simplified

Susan H. Treagus Department of Geology, The University, Manchester M13 9PL, U.K.

The Mohr circle is a well known representation of two-dimensional strain. It is commonly used to illustrate strain algebra, or derive a strain ellipse from particular strain data. The established sign convention is positive (anticlockwise) shear strain on the upward ordinate, after Brace. However, in practice, confusion may easily arise from inconsistent conventions in standard text books.

A Mohr-circle convention is proposed here which represents clockwise shear strain on the upward -y’ ordinate (i.e. upside-down). It simplifies the Mohr circle by representing both single and double angles in their natural sense. Single angles may be traced from rock to Mohr circle directly which is not the case for the Brace circle. Deformed brachiopods and stretched belemnites are used to illustrate this simplified Mohr circle.

The pole to the Mohr circle is a useful addition which allows strain data to be represented in their true orientation. The pole method only works on the y’ clockwise-up Mohr circle.

The Mohr circle is useful to demonstrate simple-shear geometry and algebra. Successive Mohr circles with poles may be used to map strain across a heterogeneous simple-shear zone.

KEY WORDS Mohr circle Strain ellipse Simple shear

1. The Mohr circle for strain

1 a. Introduction The Mohr circle for strain is widely used in structural geology, both as a

representation of two-dimensional strain algebra, and as a construction for strain determination. A Mohr diagram sensu strict0 is a representation of three-dimensional stress (Mohr 1905). However, Nadai (1950) demonstrated that a similar graph could be used for strain tensors, either A vs y or A‘ vs y’. Geological applications of these two types of Mohr diagram for strain were pioneered by Brace (1961), but subsequently the reciprocal diagram (A‘ vs y’) has received more attention and is popularly called the Mohr diagram for strain.

I shall reserve the term Mohr diagram for strain for a representation of three- dimensional strain (i.e. a strain ellipsoid). Such a diagram consists of three (Mohr) circles and the area bounded by them: for a review and new applications, see Treagus (1986). Confusion may arise from literature which uses the term ‘Mohr diagram’ to describe a single Mohr circle. Throughout this paper I use the term Mohr circle for strain for the representation of two-dimensional strain (i.e. any strain ellipse).

0072-1050/87/020119-14$07.00 0 1987 by John Wiley & Sons, Ltd.

Accepted November 1986

120 S . H. TREAGUS

Figure 1. (a) The usual Mohr strain circle convention, with +y’ up. (b) The Mohr circle with clockwise (negative) y’ up. Both Mohr circles show the single (+) and double (2+) angle conventions. (c) The strain ellipse for both Mohr circles showing point L and its angle of shear +,-. Angle 9 has the same sense as in (b), but not as in (a). A is quadratic stretch, X’ reciprocal quadratic stretch (=l/A): y’ is the reciprocal shear value = $A, where y = tan+. Principal axes are denoted by subscripts 1 and 2.

Brace (1961) and Ramsay (1967) showed that the Mohr circle (A’ vs y’) provides a useful method of constructing the strain ellipse for particular sets of strain data. More recently, Ramsay and Huber (1983) and Ragan (1985) present many examples of the use of the Mohr circle for strain in structural geology. However, a comparison between the descriptions of Mohr circles in the geological literature from 1961 to 1986 reveals many inconsistencies in the sign convention used for

MOHR CIRCLES FOR STRAIN 121

y'. Brace (1961) presented the graph with anticlockwise shear positive (Figure la) which is in line with accepted stress convention, and the convention of anticlockwise angles positive. But a comparison of Mohr-circle constructions for simple strained brachiopod problems reveals considerable licence and inconsistency of convention (compare Ramsay 1967, fig. 5.57, with Ramsay and Huber 1983, fig. 8.30, Ragan 1968, fig. 5.10 with 1985 fig. 10.14, and Allison 1984 fig. 2).

In the following account, Mohr-circle conventions will be compared and assessed and a simple Mohr circle presented, illustrated by examples. The pole to the Mohr circle, described by Cutler and Elliott (1983) and Allison (1984), will be further clarified and applied to Mohr circle constructions for simple shear.

1 b. Mohr-circle conventions The most widely used Mohr-circle convention is shown in Figure l a . This may

be called Brace's convention (cf. Brace 1961; Means 1976; Ramsay and Huber 1983; Ragan 1985). Anticlockwise shear strain is represented on the upper semicircle, clockwise on the lower one. The problem with this convention is highlighted by Means (1976): angles are measured on the diagram in a sense opposite to the sense on the real object. This is apparent from a comparison of the sense of 4 in Figures l a and lc: clockwise on the strain ellipse, anticlockwise on the Mohr circle.

Most users of the Mohr circle in this convention use the double angle 24, measured from the centre of the circle (Figure la). Thus, the equations

(A' + X i ) (A; - A;) cos 24) A' = 1 __ - 2 2

(1;- A;) . sin 24 y' = 2--

+ ") with radius 2 can be simply verified on a Mohr circle centred at -

@-3. The double-angle convention is particularly convenient in Mohr-circle 2 construction, once it is remembered that each angle is not only doubled but its sign changed.

In some cases it may be considered appropriate to disregard the sign of y' and present all Mohr-circle data on an upper Mohr semicircle (cf. examples in Ramsay 1967). Such a A' vs /y'/ graph is valid since A' vs y' Mohr circles are symmetrical about their A ' axes. However, the half Mohr circle no longer fully represents the angular relationships of the strain ellipse: all data are mirrored into a single unidentified ellipse quadrant.

The convention proposed in this paper is shown in Figure lb . It is an upside- down version of Figure l a . Clockwise shear strain (-y') is represented on the upper semicircle, anticlockwise lower. Means (1976) suggested that such a reverse of the normal convention would simplify the Mohr circle. I shall call this convention the y r clockwise-up Mohr circle.

Ic. The y' clockwise-up Mohr circle The convention shown in Figure l b has two advantages over the accepted

(Brace's) convention. (1) Angles are plotted in the same sense as in nature (4 in Figures l b and c are both clockwise). This is true for both single and double angles. (2) Single angles may be traced directly from nature to the Mohr

122 S. H. TREAGUS

circle (e.g. +). These two points overcome the previous inconveniences of the Mohr circle-the procedure to convert angles to double angles and change their signs.

The accepted sign convention for y’ is not changed. The simple Mohr circle merely chooses to represent - y‘ (clockwise) on the upper graph and + y’ (anticlockwise) on the lower.

Examples in section 2 will demonstrate the simplicity of this Mohr-circle convention for construction. In section 3 it will be shown how the pole is added to a circle of this convention (Cutler and Elliott 1983) to introduce orientational data.

2. Examples of Mohr-circle construction for strain data

2a. Example 1: brachiopods This example considers two idealized deformed brachiopods (Figure 2a) used

by Ramsay (1967, pp. 236-7) and Ramsay and Huber (1983, pp. 13 and 145), and similar to Ragan’s example (1985, p. 196). The angular strain data provided by the two brachiopods are shown in Figure 2b. Both hinge lines have suffered clockwise angular shear (+A, JIB) and are separated by angle a.

1. Construct the angular shear ‘gradients’ for A and B (Figure 2c). 2. Draw an arbitrary circle on tracing paper with two radii (A and B) at angle

2a (Figure 2d). 3. Fit the circle (Figure 2d) on to the Mohr graph (Figure 2c) until the A , B

points fall on their correct shear gradients (Figure 2e). This is the Mohr circle for the brachiopods on a relative scale (nA’, n-y’).

4. Compute the principal strain ratio nA;/nA’, = A;/X;. This is found to be 9, which is a strain ellipse of axial ratio R = 3 (Figure 2f). Note that the principal axes (A,, A,) are not necessarily strain ellipsoid axes, and that absolute strains cannot be determined from angular strain data alone.

5. Draw chords A-A; and B-A; on Figure 2e. These now show the single-angle relationships of the specimen. Angle + is the orientation of A, to hinge line A.

6. Rotation of the sample (Figure 2a) into the position in Figure 2f shows the direct relationship of the simple Mohr circle to the strain data. The hinge lines of brachiopods A and B are now parallel to the A-A2 and B-A, chords, and the principal elongation (A,) is parallel to the A’ axis of the graph.

2b. Example 2: belemnites Figure 3a shows two stretched belemnites and an extension lineation, after

Ramsay and Huber (1983, fig 6.b). The example is similar to one given in Ragan (1985, p. 190). The data are reciprocal quadratic stretches for the two belemnites (A;, A,) and their angles (+*, +BN) to the extension lineation, taken as A,.

A; and A; are determined in Figure 3a in the manner described by Ramsay (1967, p 248), from the original (I) and final ( L ) lengths:

I = sum of belemnite fragment lengths (a, + a2 + a3 . . .) L = total stretched length (fragments + gaps) A‘ = (l/L)2.

This method is suitable for belemnites which have an even distribution of belemnite fragments and gaps, as in this example. However, many natural stretched belemnites have more variable fragment:gap ratios indicative of a sequential fragmen- tation history. Ferguson’s progressive unstraining method (Ferguson 1981;


- n

Strain ellipse

Figure 2 . Example 1. (a) Schematic brachiopods A and B and their angular data (b). The two angular shears ($A and $B) are represented by gradients on a Mohr diagram (c). An arbitrar circle d) is drawn on tracing film with A and B as radii and A”B as 2a. Circle (d) is fitted onto Jagram ((c) in (e) to find the ositions of A and B (triangles). The positions of hinge lines A and B relative to A , are given to boPd chords which are parallel by the brachiopod hinges as redrawn in (f) together with the derived strain ellipse. See Figure 5 for a true orientational representation.

124 S. H. TREAGUS

_ _ - - - - _ _ - _ _ - - -_- - - - -

(b)

a;= 0.54

&= 0.73

'pn= 32'

I&= 60°

A

9o-(p, 7 E

0 .2 .4 .6

I 7' ( c )

'2

3 XSCALE .8 1 .o

Figure 3. Example 2. (a) Two belemnites and a stretching lineation (A,) and Ramsay and Huber (1983, fig. 6.6). Values A, and A; are derived as described in the text, and inset with +,, and +=, in (b). Construction of the simple Mohr circle follows the order of chord numbers in (b) and (c), as described fully in the text.


Ferguson and Lloyd 1984) would be more appropriate for such data, although calculations are less simple.

The data are inset in Figure 3b, as measured before final drafting in Figure 3a: values are not the same as those given in Ramsay and Huber (1983, p. 96). There are several ways to construct the Mohr circle to fit these data. The one presented here is a modified single-angle form of Ragan’s (1985, pp. 190-1) construction, drawn in the simple y’ clockwise-up convention.

1. Draw three ordinate lines (Figure 3b) to represent the y’ axis, A; and A;, on an appropriate scale. The abscissa (i.e. A‘ axis) is not drawn: the construction depends on its determination.

2. Choose a suitable position on the A; ordinate for belemnite A: this will be expected to fall on the upper hemisphere of the unknown Mohr circle because it is clockwise of X I .

chord 1 in Figure 3b. Now draw another chord (chord 2) at angle (90-4,) to chord 1. The intersection of chord 2 with A, line determines the position of belemnite B (dot).

4. Draw a chord from B with inclination & (chord 3). The intersection of chords 1 and 3 determines A; (circle).

5. The A’ axis is drawn through A, (Figure 3c). Chord 4 is drawn from A perpendicular to chord 1, and chord 5 at B perpendicular to chord 3: they intersect at A;.

6. The Mohr circle is drawn on A;, A;. In Figure 3c, A; = 0.43 and A; = 0-83, giving a strain ellipse with principal axes 1-52 and 1.09. Because the data in Figure 3a are represented with the stretchi.1; lineation across the page (laterally), the belemnites are parallel to their Mohr-circle chords. This would not be the case with a Brace-convention Mohr circle which would be an upside-down version of Figure 3c, which could not be traced directly onto the field data (Figure 3a).

3. Draw a chord from point A (triangle) inclined at

3. The pole to the Mohr circle

The pole to the Mohr circle for strain is described in Cutler and Elliott (1983) as a method of orienting a series of compatible strain ellipses. Although it is not clear in Cutler and Elliott’s description, the Mohr circle must be drawn in the y’ clockwise-up convention. The method of orienting data in terms of a Mohr pole does not work with the more accepted Brace convention: angles are not represented in the correct sense. There may also be some confusion from the convention given in Allison’s (1984) review of the pole to the Mohr circle. The anticlockwise shear in Allison’s figure 2 is equivalent to the usual definition of clockwise shear, followed in this paper (personal communication).

The pole is a special point on the y’ clockwise-up Mohr circle which allows true orientations to be represented. It is the unique point of intersection of all lines drawn in their actual orientation from their Mohr-circle points. Figure 4 shows the position of the pole (P) for four orientations of the same strain ellipse. A rotation of this strain ellipse in space (a to d) is shown by a change in P (starred) on otherwise identical Mohr circles. The pole is most simply constructed in these examples by drawing the true orientation of XI as a chord (solid line) and marking its intersection with the circle, P, the pole. Any other direction, such as L , can be drawn in its true orientation and will also intersect the circle at this pole. All

126 S. H. TREAGUS

Figure 4. The pole to the Mohr circle, illustrated for four different orientations of the same strain ellipse: (a)-(d). For each, line L (at + to A , ) is represented by a broken line, the A , axis by a solid line and the pole (P) starred.

MOHR CIRCLES FOR STRAIN

A@

127

Figure 5. Example 1 (Figure 2) re-illustrated using the Mohr circle pole, P. (a) is the true orientation of the data; (b) the pole (star) allows the true hingeline orientation (bold), and the principal axes (bold broken) to be represented on the constructed Mohr circle. Compare with Figure 2f.

angles (such as +) are represented truly, in value and sense, at the pole, because of the geometry of the circle.

The addition of the pole to the simple y' clockwise-up Mohr diagram enables orientation as well as values of two-dimensional strain to be recorded. It will be noted that Figure 4a, where A1 is oriented parallel to the A' axis and P = A;, is the Mohr circle representation given in the previous examples (Figures 2 and 3). This would seem to be an appropriate choice of orientation for unoriented two- dimensional strain data. Figure 5 shows Example 1 represented in its actual (Figure 5a) orientation on the constructed Mohr circle. The hinge lines of brachiopods A and B are represented by bold lines and the strain axes by broken lines. A similar brachiopod problem was illustrated by Allison (1984, figure 2).

4. Mohr circles for simple shear

Simple-shear strain is well suited to Mohr-circle representation. True simple shear is two-dimensional and can be completely represented by a Mohr circle. The special geometry of simple shear, and algebraic expressions arising from this, can be illustrated very simply. A state of variable simple shear such as a progressive increase of shear strain to model a heterogeneous shear zone, can be represented with the aid of Mohr-circle poles as Cutler and Elliott (1983) demonstrated for other types of compatible strain fields.

4a. A Mohr simple-shear construction A simple-shear strain ellipse may be constructed very simply. Its essential

characteristics are the value of shear strain, y = tan +, in a direction of zero strain (Figure 6a). This direction, the shear direction (S) is represented in Figure 6b: at S, -y' = y because X' = 1. There will only be one plane-strain @;.A; = 1) ellipse on

128 S. H. TREAGUS

Y'

Figure 6. Simple shear on the Mohr circle. (a) Shear direction, S, angle of shear, JI, and shear strain y; (b) is their Mohr representation. Note that A' = 1 at S; (c) Construction to determine the centre, C, of the Mohr circle, and trigonometric values. Triangle S1C is enlarged in (d); (e) is the Mohr circle for this simple shear, A , is at angle 0 to the shear direction, S; (f) is the simple-shear strain ellipse.

which S falls. This ellipse could be found by using a set of pre-drawn plane-strain ellipses such as those used in Treagus (1983, fig. 4). Alternatively, a simple construction procedure may be followed.

1. Drop a perpendicular from the shear gradient, OS, to the A' axis (Figure 6c;

2. Bisect (D-1) at C: its coordinates are (1 + y2/2, 0). This is the centre of the line SD). From trigonometry, distance D-1 is yz.

Mohr circle.


Figure 7. Simple shear and the pole to the Mohr circle. (a) The same simple shear as in Figure 6: (b) Its Mohr circle (with abbreviated y‘ axis) showing true orientations with respect to the pole (star). The shear direction (S) is the horizontal chord and A , and h2 (broken chords) are parallel in (a) and (b).

3 . Draw a circle centred at C, through S, the Mohr circle for this simple-shear

4. Construct the S-A; chord: its inclination, 0, defines the angle of A , to the

Figures 6c-6e illustrate the essential algebra of true simple shear. It can be seen

strain (Figure 6e).

shear direction, in the correct sense (Figure 6f).

that tan 26 = 2ly

A; + A; At C, A’ = 1 + y2/2, and also A ’ = ~ 2 . Solution of these two equations, together with the plane-strain relationship &.A; = 1 reveals that

y 1 (y2 + 4)t A’j A’f = (Treagus 1981).

2 1 7 2

From the two similar +s right-angled triangles in Figure 6c, it can be shown that = tan 0, A 3 = cot 6 (Treagus 1981).

4b. Simple shear using Mohr-circle poles A system of right-lateral simple shear (S) is represented in Figure 7a. Its Mohr

circle is shown in Figure 7b. The shear direction, S, is represented by the bold lateral line from point S: it re-intersects the circle at point P (starred) which is the pole for this example. The directions of A , and X2 can now be drawn from P to their Mohr circle points, and these directions (broken lines) represent the true principal-axis orientations. Had the direction of simple shear not been represented laterally, the pole would have had a different position, as illustrated in Figure 4.

Cutler and Elliott (1983) first introduced the Mohr-circle pole as a method of ‘mapping’ compatible strains on a Mohr diagram. A system of heterogeneous

S. H. TREAGUS 130

Y (b) al= 1

0'

/-- LfC 0-

Y .t:

S

(C )

Figure 8. Progressive simple shear represented by Mohr circles and oles (a) Three values of simple shear strain (yl, y2, y3) increasing from S , to S,. (b) Their three MoRr circles of increasing size, poles P,, P,, P3 and orientations and elongations 6 , , e2. 03. The pole curve for the three simple shears, treated a s r g r e s s i v e , is,given on diagram 3 by the dotted curve; (c) The three simple-shear strain ellipses an their initial circle. Note the parallelism of the principal axes to their pole representations on the Mohr circles above.

simple shear is a special example of strain compatibility (Cutler and Cobbold 1985), well suited to a Mohr graphical representation. Figure 8a shows three simple shears (1, 2, 3) which could model a simple-shear gradient in a heterogeneous shear zone. The three Mohr circles are constructed in Figure 8b, each pole located, and orientations of hl (broken lines) drawn. The three simple-shear strain ellipses and their principal axes are drawn in Figure 8c.


The ‘pole curve’ (Cutler and Elliott 1983) is shown by dots in Figure 8b, diagram 3. It represents the pole trajectory for an infinite number of Mohr circles representing a continuous y gradient from 0 through 0.5 and 1.0 to 1-5. Any gradual strain gradient can be represented and oriented by a pole curve as Cutler and Elliott (1983) showed. However, the sharp refraction of compatible strain expected at clear competence contrasts, illustrated on Mohr circles by Treagus (1983), would be represented by ‘pole jumps’ not continuous pole curves.

5. Conclusions

The Mohr circle for strain is simple to use and teach if it is drawn with clockwise shear upwards. Angles on the circle are then measured in the same sense as in nature. The use of single angles, in combination with more well-used double angles, allows data to be transferred or traced directly from specimen to Mohr circle.

The pole to the Mohr circle requires a y’ clockwise-up diagram although this had not previously been made clear. The pole adds a new dimension to the Mohr strain graph: it allows all lines to be constructed in their true orientation. All angular relationships are correctly represented from a particular pole because of circle geometry.

Simple shear can be represented easily on Mohr circles, and the relevant algebra derived. Progressive simple shear in shear zones may be mapped on a Mohr graph by a series of expanding Mohr circles and their unique pole curve.

Acknowledgements. I should like to thank colleagues in the Department of Geology at Manchester University for having unwittingly initiated this attempt to simplify the Mohr circle for strain. Jack Treagus and Iain Allison are thanked for their comments on the manuscript, Catherine Hardy for typing, and Richard Hartley for drawing the figures.

References

Allison, I. 1984. The pole of the Mohr diagram. Journal of Structural Geology, 6, 331-333. Brace, W. F. 1961. Mohr construction in the analysis of large geologic strain. Builetin of the Geological

Cutler, J. M. and Cobbold, P. R. 1985. A geometric approach to two-dimensional finite strain

- and Elliott, D. 1983. The compatibility equations and the pole to the Mohr circle. Journal of

Ferguson, C. C. 1981. A strain reversal method for estimating extension from fragmented rigid

- and Lloyd, G. E. 1984. Extension analysis of stretched belemnites: a comparison of methods.

Society of America, 72, 1059-1080.

compatibility: interpretation and review. Journal of Structural Geology, 7, 727-735.

Structural Geology, 5 , 287-297.

inclusions. Tectonophysics, 79, T43-52.

Tectonophysics, 101, 196-206. Means, E. D. 1976. Stress and Strain. Springer-Verlag, New York. 339 pp. Mohr, 0. 1905. AbhandlunRen aus dem Gebiete der Technischen Mechanik. Ernst & Sohn. Berlin. Nadai, A. 1950. Theory of plow and Fracture of Solids. McGraw-Hill, New York. 572 pp. Ragan, D. M. 1968. Structural Geology. An Introduction to Geometrical Techniques. John Wiley &

- 1985. Structural Geology. an Introduction to Geometrical Techniques. Third Edition. John Wiley

Ramsay, J. G. 1967. Folding an1Fracturing of Rocks. McGraw-Hill, New York. 568 pp. - and Huber, M. I. 1983. The Techniques of Modern Structural Geology. Volume 1 : Strain Analysis.

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- 1983. A theory of finite strain variation through contrasting layers, and its bearing on cleavage

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Mohr circles for strain, simplified