Three-dimensional colour functions for stress state visualisation Martin Bedna ´ rik n , Igor Kohu ´t Geophysical Institute, Slovak Academy of Sciences, Du ´bravska ´ cesta 9, 845 28 Bratislava, Slovak Republic article info Article history: Received 25 February 2011 Received in revised form 3 May 2012 Accepted 6 May 2012 Available online 19 May 2012 Keywords: Mohr’s circle Failure criterion Brazilian test HSV colour specification abstract Three-dimensional colour functions were designed for visually agreeable and computationally efficient stress state visualisation in failure analysis. The colours they generate depend on three aspects of the stress state: the distance to failure, the stress character (compressive or tensile), and the relative magnitude of the intermediate principal stress. Thus, the colour of a single pixel carries information about all three scalar components of the stress tensor at a point. By discarding the intermediate principal stress information, a three-dimensional colour function degenerates to a family of two-dimensional colour functions with free palette choice parameter. We demonstrate the performance of our colouring technique in the application example – visualisation of the stress field in the Brazilian test. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction The number of independent ‘‘scalar components’’ (the term used, e.g., in Zehner, 2006) of a second-rank tensor in 3D space, such as stress tensor, is three (three eigenvalues; three invariants). Thus, the three channels of RGB colours are ideally suited for coding all three of the scalar components. Their RGB coding leaves, however, no free channels for principal directions, which require another three. The fact that stress tensors cannot be represented by colours only should not be used as an excuse for idle colouring. We are not satisfied with the present situation where stress visualisation approaches fall into two extreme categories – those attempting to show the complete stress tensor (e.g. Crossno et al., 2005, using Mohr’s circles as glyphs; Hotz et al., 2004, transforming the tensor field into a metric, visualised by a texture aligned to the eigenvector field; Hashash et al., 2003, assessing the properties of different types of glyphs) and those showing just one chosen component of the stress tensor, (common in many commercial finite-element programs). Our aim is to fill the gap between these two and enrich the stress visualisation toolbox by a new, yet very natural approach to show as much as can be expressed by an RGB colour. 2. Stress colouring concept The purpose of our stress colouring method is to indicate regions of interest in a stress field. We are interested in seeing in colour – how close to failure the stressed solid material is – where the stress states are compressive and where tensile – what the relative magnitude of the intermediate principal stress is The RGB colour space is continuous and bounded. The physical quantities we wish to map into the RGB space should have the same qualities. Let us see whether they satisfy the requirements. The first quantity is the distance to failure. When we talk about stress, we always imply there is also a stressed solid material. This material is at the same time the most natural (and very often, the only available) device for measuring the size of the deviatoric part of the stress. In the real world, the gauge of the device offers only two possible readings: failed or not. In our computations, however, we can make the scale continuous. We simply evaluate the relevant failure criterion and pick up its output value without applying to it any failed-unfailed threshold. This way we obtain the most natural continuous failure distance measure. It can be further normalised to the / 1, 1S range. The second quantity, the compressive-tensile indicator of the stress state character, is in fact the sign of the (mean) stress with output values { 1, 0, 1}. We propose some natural ways to make this quantity too continuous with values within the range /1, 1S. Let us call this continuous quantity the (mean) stress sign measure. We shall see in the application example that there are situations where the sign of the mean stress is not the most interesting and representative indicator of the stress state character. The third quantity, the intermediate principal stress s 2 , is by its nature continuous and bounded to the range /s 3 , s 1 S. The only remaining simple task is to normalise it to the /1, 1S range. We are confident that these three quantities represent the triplet of stress tensor scalar characteristics most relevant to failure analysis (cf. shape descriptors t, c, R corresponding to Coulomb criterion in Kratz et al., 2010). For this triplet, our colour Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/cageo Computers & Geosciences 0098-3004/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cageo.2012.05.010 n Corresponding author. Tel.: þ421 259410613; fax: þ421 259410607. E-mail addresses: [email protected], [email protected] (M. Bedna ´ rik). Computers & Geosciences 48 (2012) 117–125
Transcript
Computers & Geosciences 48 (2012) 117–125
Contents lists available at SciVerse ScienceDirect
Computers & Geosciences
0098-30
http://d
n Corr
E-m
Martin.B
journal homepage: www.elsevier.com/locate/cageo
Three-dimensional colour functions for stress state visualisation
Martin Bednarik n, Igor Kohut
Geophysical Institute, Slovak Academy of Sciences, Dubravska cesta 9, 845 28 Bratislava, Slovak Republic
Three-dimensional colour functions were designed for visually agreeable and computationally efficient
stress state visualisation in failure analysis. The colours they generate depend on three aspects of
the stress state: the distance to failure, the stress character (compressive or tensile), and the relative
magnitude of the intermediate principal stress. Thus, the colour of a single pixel carries information about
all three scalar components of the stress tensor at a point. By discarding the intermediate principal stress
information, a three-dimensional colour function degenerates to a family of two-dimensional colour
functions with free palette choice parameter. We demonstrate the performance of our colouring
technique in the application example – visualisation of the stress field in the Brazilian test.
& 2012 Elsevier Ltd. All rights reserved.
1. Introduction
The number of independent ‘‘scalar components’’ (the term used,e.g., in Zehner, 2006) of a second-rank tensor in 3D space, such asstress tensor, is three (three eigenvalues; three invariants). Thus, thethree channels of RGB colours are ideally suited for coding all threeof the scalar components. Their RGB coding leaves, however, no freechannels for principal directions, which require another three. Thefact that stress tensors cannot be represented by colours only shouldnot be used as an excuse for idle colouring. We are not satisfied withthe present situation where stress visualisation approaches fall intotwo extreme categories – those attempting to show the completestress tensor (e.g. Crossno et al., 2005, using Mohr’s circles as glyphs;Hotz et al., 2004, transforming the tensor field into a metric,visualised by a texture aligned to the eigenvector field; Hashashet al., 2003, assessing the properties of different types of glyphs) andthose showing just one chosen component of the stress tensor,(common in many commercial finite-element programs). Our aim isto fill the gap between these two and enrich the stress visualisationtoolbox by a new, yet very natural approach to show as much as canbe expressed by an RGB colour.
2. Stress colouring concept
The purpose of our stress colouring method is to indicate regionsof interest in a stress field. We are interested in seeing in colour
–
how close to failure the stressed solid material is – where the stress states are compressive and where tensile
ll rights reserved.
x: þ421 259410607.
–
what the relative magnitude of the intermediate principalstress is
The RGB colour space is continuous and bounded. The physicalquantities we wish to map into the RGB space should have thesame qualities. Let us see whether they satisfy the requirements.
The first quantity is the distance to failure. When we talkabout stress, we always imply there is also a stressed solidmaterial. This material is at the same time the most natural(and very often, the only available) device for measuring the sizeof the deviatoric part of the stress. In the real world, the gauge ofthe device offers only two possible readings: failed or not. In ourcomputations, however, we can make the scale continuous. Wesimply evaluate the relevant failure criterion and pick up itsoutput value without applying to it any failed-unfailed threshold.This way we obtain the most natural continuous failure distancemeasure. It can be further normalised to the /�1, 1S range.
The second quantity, the compressive-tensile indicator of thestress state character, is in fact the sign of the (mean) stresswith output values {�1, 0, 1}. We propose some natural ways tomake this quantity too continuous with values within the range/�1, 1S. Let us call this continuous quantity the (mean) stresssign measure. We shall see in the application example that thereare situations where the sign of the mean stress is not the mostinteresting and representative indicator of the stress statecharacter.
The third quantity, the intermediate principal stress s2, is by itsnature continuous and bounded to the range /s3, s1S. The onlyremaining simple task is to normalise it to the /�1, 1S range.
We are confident that these three quantities represent thetriplet of stress tensor scalar characteristics most relevant tofailure analysis (cf. shape descriptors t, c, R corresponding toCoulomb criterion in Kratz et al., 2010). For this triplet, our colour
M. Bednarik, I. Kohut / Computers & Geosciences 48 (2012) 117–125118
function generates its colour, which can be understood intuitivelyin the following way:
–
the closer to failure, the more intensive the colours – the more compressive the stress, the more reddish the hues;
the more tensile, the more greenish
–
Fig. 1. Angle of sight (grey) of the Mohr’s circle perpendicular diameter seen from
as the intermediate principal stress moves from one outerprincipal stress to another, more blue is mixed in and thecolours turn from warm to cold ones. For instance, redbecomes magenta, green becomes cyan.
Reading of the colour maps generated by our colour functionsrequires nothing more (and nothing less) than a naked healthy eyeand a healthy brain with a basic feeling for colour. Everyone thusequipped is able to tell which one of the colour spots placed next toeach other is shinier, which is greener and less red than the others,and which is warmer or colder – in other words, to evaluate threedistinct colour qualities at once. Physiological colour reading iscomparative – unlike colorimeters, we are not reading isolatedcolour spots, but always compare them with their environment. Ifour colour maps are read physiologically, we are sure they arereadable, with a great deal of benefit to the reader.
plane origin.
3. From principal stresses to colourable quantities
In this section, we expect some basic experience with Mohr’scircles and failure criteria, on the level presented in Jaeger and Cook(1979). From the stress tensor given at a point, let us compute theprincipal stresses s1, s2, s3, s1Zs2Zs3 (in this section is adoptedthe convention of positive compression). We want to design thesimplest colouring scheme that makes sense. At this moment, weare tempted to see the candidates for arguments of the simplestpossible colour function directly in the principal stresses. A colour-ing scheme that takes the principal stresses for separate quantitiesand associates them with individual colour channels would, how-ever, fail to deliver physically reasonable and intuitively readablestress colour maps. The principal stresses at a point are not threeseparate quantities – only in their mutual context can they beunderstood as a stress state. This fact should be reflected in theconstruction of stress colour functions. This is also our mainobjection against separate colour or contour maps of individualprincipal stresses. The colourable quantities (or colourables) wepropose have one trait in common: they are calculated from at leasttwo principal stresses (exception is admissible only in one colour-able, as we shall see). Our colourables recode the common contextof the three principal stresses, which would be lost if they werecoloured as separate quantities.
Let us start with the simplest colourable m involving the leastnumber of principal stresses – only the outermost stresses s1, s3.It is based on the mean (hence m) stress sm¼(s1þs3)/2. Accord-ing to sign(sm), we talk about tensile or compressive stress states.In some situations, this can be an unjustified oversimplification –let us think about a stress state, where sm is positive and the onlytensile (i.e. negative) principal stress is s3. Is this stress statecompressive? In some sense yes, but our common sense says no.In fact, it is rather the sign of s3 which decides about thecharacter of the stress state. There are several ways to deal withit. Let us introduce the more sophisticated one first. Let us draw aMohr’s circle with diameter (s3,0)�(s1,0), and through the centreof the circle (sm,0), draw the perpendicular diameter abscissa(Fig. 1). Now we calculate the angular length of this abscissa asseen from Mohr plane origin (0,0). This angle of sight varies from0 for very distant or zero diameter Mohr’s circles, to p for nonzerodiameter circles centred at (0,0), with the singularity for zerodiameter circles at (0,0).
That is the main idea behind the expressions
m¼0 for s2
mþt2m ¼ 0
sm smj j
s2mþt2
mfor s2
mþt2ma0 ,
8<: ð1Þ
and
m¼0 for 9sm9þ9tm9¼ 0sm
9sm9þ 9tm9 for 9sm9þ9tm9a0 ,
8<: ð2Þ
where 9tm9¼(s1�s3)/2 is the maximum shear stress – the radiusof the Mohr’s circle. The output range of both (1) and (2) is/�1,1S. The quantity m in (1) adheres strictly to the geometricmeaning of the angle of sight, unlike in (2). Nevertheless, bothexpressions are capable of distinguishing stress states which,according to the simple sign(sm) criterion, are the same. Forinstance, mA(�1/2,1/2) represents the ‘‘mixed’’ stress states, inwhich one of the principal stresses has a different sign than theother two, m¼1/2 means biaxial or uniaxial compression, m¼1means hydrostatic compression with s1¼s2¼s3¼sm. In the realworld, the difference between biaxial and hydrostatic compres-sion is huge, although both are compressions with thesame sign(sm). Thus, in (1) and (2), we generalised sign(sm)to a much more useful quantity: stress sign measure. Instead oftm and sm can be used octahedral shear and normal stresses
The stress sign measure m can be defined differently, depend-ing on the user’s needs. For instance, the linear interpolationformula
m¼signðMÞ for Mmax ¼Mmin
2M�Mmax�MminMmax�Mmin
for MmaxaMmin
(ð3Þ
can be used, where M is the absolute stress sign measure andMmax, Mmin are the maximum and the minimum of M we want tocolour. Stresses sm, soct or s3 can be used as M. For MA/Mmin,MmaxS and MmaxaMmin, the normalised stress sign measuremA/�1,1S. There can be stress fields where m according to(1) or (2) covers only a small part of /�1,1S. In these cases, wecan also use (3) i.e. renormalize m by setting Mmax¼mmax, Mminammin, M¼m to assure optimum usage of the colour palette.
M. Bednarik, I. Kohut / Computers & Geosciences 48 (2012) 117–125 119
The next colourable variable is the distance to failure d
measured by the failure criterion chosen. All three principalstresses enter the three-dimensional failure criteria, whereas onlyoutermost stresses s1, s3 are used in the two-dimensional ones.The Coulomb criterion belongs to the latter group and we will useit as an example in explaining d. The usual way to write it is
9t9¼ S0þs tan f, ð4Þ
where S0 is cohesive strength and f is the internal friction angle,s is the normal stress and t is the shear stress at failure. The otherCoulomb criterion form
9tm9¼ S0cos fþsmsin f ð5Þ
(cf. Jaeger and Cook, 1979) is more convenient for computationpurposes. It has a simple geometric meaning: (sm,0) is the centreand 9tm9 the radius of the Mohr’s circle that touches both lines (4).If for the fixed (sm,0) the 9tm9 is smaller than in (5), the materialdoes not fail. If it is equal, the failure occurs only at (s,7t) givenby Eq. (4). If it is greater, the failure can occur at any point of theMohr’s disc where it exceeds the wedge between the two lines(4). The relationship between (4) and (5) is shown in Mohr planein Fig. 2.
DC ¼ 9tm9�ðS0cos fþsmsin fÞ ð6Þ
can be understood as a failure distance measure according to theCoulomb criterion or Coulomb failure measure.
Let us denote the failure distance measure for a general failurecriterion as D. Within the investigated region, let us set themaximum Dmax and minimum Dmin values of D we want to showby colour. These Dmax and Dmin do not necessarily need to be theactual maximum and minimum values of D attained within theregion. Let us now map the values D, DA/Dmin,DmaxS to d, dA/�1,1S by linear interpolation
d¼0 for Dmax ¼Dmin
2D�Dmax�DminDmax�Dmin
for DmaxaDmin:
(ð7Þ
For De/Dmin,DmaxS, no colour will be generated. The quantityd is then the normalised or relative failure distance measure. Itsadjectives will be often omitted and the reader will have to infer
Fig. 2. Geometric meaning of Coulomb failure criterion formulae (4) (gr
either from the notation or context whether we are talking aboutD or d.
The last colourable a (as aspect ratio) is the relative magnitudeof the intermediate principal stress s2, defined by the linearinterpolation formula
a¼0 for s1 ¼ s3
2s2�s1�s3s1�s3
for s1as3:
(ð8Þ
As s1Zs2Zs3, a does not exceed the range /�1,1S. Con-ceivable are situations where a according to (8) covers only asmall part of the interval /�1,1S. In these cases, we can also usea formula analogous to (3) i.e. renormalize a to assure optimumusage of the colour palette. The user may prefer to switch thepolarity of a.
In structural geology, the more naturally defined stress shape
ratio
F¼s2�s3
s1�s3ð9Þ
(after Ramsay and Lisle, 2000, p.789, but adapted to our signconvention) is used to characterize the stress tensor shape. Theoutput range of F, however, is /0,1S. It would not fit ourintention to have (d, m, a) within a cube region /�1,1S�/�1,1S�/�1,1S, aimed at deriving the simplest possiblecolour function formulae. Nevertheless, the conversion betweena (8) and F (9) (for s1as3) is very simple
F¼ ð1þaÞ=2: ð10Þ
Now the stress state is characterised by a new triplet ofcolourable quantities (d, m, a). They will be used as three inde-pendent parameters in the stress colour function construction.
4. From colourable quantities to colours
The HSV parametrization of the RGB colour space can beadapted easily to the purposes of stress imaging: the stress signmeasure m can be represented by changes of hue h from red(dish)to green(ish) hues. The representation of the distance to failure d
depends on the choice of background: for colour function to be
ey) and (5) (black). See Section 3 and Table of symbols for details.
Fig. 3. Warm and cold colour fans for use on black (left, s¼1, v varies) and white (right, v¼1, s varies) backgrounds. The colour angle c¼2pu determines the hue (Eqs. (11)
and (12)).
M. Bednarik, I. Kohut / Computers & Geosciences 48 (2012) 117–125120
used on white background, d will be represented by saturation s.For use on black background, d will be coupled with brightness v.To represent the relative magnitude of the intermediate principalstress, let us first go through all values of h to generate therainbow cockades for white (s increasing with the radius (dþ1)/2and v¼1), and black (v increasing with the radius (dþ1)/2 ands¼1) backgrounds. Next, let us cut out of each cockade two pies –warm colour sector and cold colour sector (Fig. 3). The colourpalette of one of the pies will be attributed to the minimum valueof the intermediate principal stress magnitude parameter a, theother one to the maximum value of a. The palettes for the a valuesin between will be obtained by interpolation of these two outer-most palettes.
Having outlined the principles, let us now go into details of thewhite background colour function construction. In HSV colourspecification, the hue is equal to the angle parameter. As the firststep to the colour function construction, let us dissociate them.From now on, the angle parameter u, u¼c/2p, will be anindependent variable and the hue will be its function h(u). Byshifting and inversion of the original angle scale of the lower, coldcolour fan, we produce a new cold-hue function hC, which yieldsmagenta for u¼0 and cyan for u¼1/3 (Fig. 3)
hC ¼�uþ5=6 ð11Þ
The hue function for the fan of warm hues remains simply
hW ¼ u: ð12Þ
Let us choose a convention of red(dish) hues for compressiveand green(ish) for tensile stress states. Then the pair of huescorresponding to stress sign parameter m, mA/�1,1S, will beobtained by substituting
u¼ ð1�mÞ=6 ð13Þ
into (11) and (12).The parametrization of the failure measure is straightforward
– the higher d, the higher the saturation s, sA/0,1S, of the colour
s¼ ð1þdÞ=2: ð14Þ
Colour mixture formula is not available for HSV colours. Beforewe can enjoy simple RGB blending (19), we have to perform(h, s, v) to (r, g, b) conversion (Yang and Yang, 2008). Let us denoteH¼6h, its fractional part DH¼ frac(H), where f racðHÞ ¼H�bHc andthe floor function bHc is for HZ0 identical with the integer part of
H, further p¼1�s, q¼1� DH s, t¼1�(1�DH)s. Then
ðr, g, bÞ ¼ vU
ð1, t, pÞ for bHc ¼ 0
ðq, 1, pÞ for bHc ¼ 1
ðp, 1, tÞ for bHc ¼ 2
ðp, q, 1Þ for bHc ¼ 3
ðt, p, 1Þ for bHc ¼ 4
ð1, p, qÞ for bHc ¼ 5
:
8>>>>>>>>><>>>>>>>>>:
ð15Þ
Using (15) with v¼1, we obtain the warm colour cW and itscold counterpart cC as (r, g, b) triplets
cW ¼ð1, 1�ð1�6hW Þs, 1�sÞ for uA/0, 1=6Þ
ð1þð1�6hW Þs, 1, 1�sÞ for uA/1=6, 1=3S
(ð16Þ
cC ¼ð1�ð5�6hCÞs, 1�s, 1Þ for uA/0, 1=6Þ
ð1�s, 1�ð6hC�3Þs, 1Þ for uA/1=6, 1=3S
(ð17Þ
The relative magnitude of the intermediate principal stress a
determines the warmth – the proportion
w¼ ð1þaÞ=2, ð18Þ
(cf. (10)) in which cW and cC are to be mixed to give the finalcolour c representing distance to failure, stress sign and inter-mediate stress magnitude
c¼wcWþð1�wÞcC : ð19Þ
5. Collection of colour functions
Let us start with the white background colour function. Themost general form can be written as
cðs, u, wÞ
¼ð1�sð5�6hCÞð1�wÞ, 1�sð1�6whW Þ, 1�swÞ for uA/0, 1=6Þ
1�sð1�wð2�6hW ÞÞ, 1þ3sð1�2hCÞð1�wÞ, 1�swð Þ for uA/1=6, 1=3S:
(
ð20Þ
If hW ðuÞ, hCðuÞ, s, w are given by (11), (12), (14) and (18), then(20) simplifies to
cðd, m, aÞ ¼1
4
3þa�dþmþadþdm�a9m9ð1þdÞ
3þa�d�mþad�dm�a9m9ð1þdÞ
3�a�d�ad
0B@
1CA
T
: ð21Þ
M. Bednarik, I. Kohut / Computers & Geosciences 48 (2012) 117–125 121
The black background colour function reads
cBlkðs, u, wÞ ¼ sð�4þ5wþ6hCð1�wÞ, 6whW , 1�wÞ for uA/0, 1=6Þ
wð2�6hW Þ, 4�3w�6hCð1�wÞ, 1�wð Þ for uA/1=6, 1=3S
(
ð22Þ
in full, and
cBlkðd, m, aÞ ¼1
4
1þaþdþmþadþdm�a9m9ð1þdÞ
1þaþd�mþad�dm�a9m9ð1þdÞ
1�aþd�ad
0B@
1CA
T
ð23Þ
simplified. Note that in (22) s does not have the meaning ofsaturation, but of brightness value. Let us find a purely trilinearapproximation clin( d,m,a) of c( d,m,a) (21) as a linear interpolationof the outer values of m
clinðd, m, aÞ ¼1
2ðð1�mÞ cðd, �1, aÞþð1þmÞ cðd, 1, aÞÞ: ð24Þ
Fig. 4. Constant a sections of the d, m, að ÞA �1, 1h i � �1, 1h i � �1, 1h i cube coloured b
trilinear white background colour function (25), black background colour function (23
Analogous formula is valid for cBlklin ðd, m, aÞ, cBlkðd, m, aÞ. The
approximations have the form
clinðd, m, aÞ ¼1
4
3�dþmþdm
3�d�m�dm
3�a�d�ad
0B@
1CA
T
, ð25Þ
cBlklin ðd, m, aÞ ¼
1
4
1þdþmþdm
1þd�m�dm
1�aþd�ad
0B@
1CA
T
: ð26Þ
In failure analysis, the least important information is themagnitude of the intermediate principal stress. If we discard itand restrict the colour imaging only to showing the distance tofailure and stress sign, 2D stress colour functions will suffice forthis purpose. We propose the 2D stress colour functions which aredegenerate versions of the 3D colour functions given in previoussection: of the fans ranging from warmest to coldest (Fig. 3), wehave to choose the most pleasant one. We recommend the choice
y 3D colour functions, from top to bottom: white background colour function (21),
), trilinear white background colour function (26).
M. Bednarik, I. Kohut / Computers & Geosciences 48 (2012) 117–125122
of the warmest ‘‘rasta’’ fan. In that case, substituting w¼1 to (21)and (23) yields the desired two-dimensional colour functions. Theconstant-a-sections of ( d,m,a) cubes coloured by various colourfunctions are shown in Fig. 4.
6. An example: colouring the Brazilian test
In plane stress and strain 2D problems, out-of-plane principalstress s2 depends linearly on s1, s3
s2 ¼ kðs1þs3Þ: ð27Þ
where k¼0 for plane stress and k¼n for plane strain states, nbeing Poisson’s ratio (here, the principal stresses are no longersorted and indexed according to their size).
Thus, 2D colour functions are perfectly appropriate for show-ing the stress state in these 2D problems, and the use of 3D colourfunctions is really unnecessary, though still conceivable. The use
Fig. 5. Brazilian test setup. The cylindrical specimen (white) is diametrically
compressed by two jaws (grey).
Fig. 6. Coulomb criterion with tensile cut-off in (sm,tm) and (s3,s1) s
of our 2D colour functions gives us the opportunity to treat a as afree parameter and see its effect on the stress colour map.
The purpose of the Brazilian or diametrical compression test isto find the tensile strength st of a material. If an unconfinedisotropic cylinder is compressed from its upper and lower flanks(Fig. 5), a horizontally tensile and vertically compressive stressstate develops in the vicinity of the cylinder axis.
In the case of compression distributed uniformly along twodiametrically opposite load application strips with finite angularwidths 2a, the stress field in the cylinder cross-section solving thestatic equilibrium boundary value problem formulated (after Maand Hung, 2008) in polar coordinates (r,y), 0rrrR, �pryrp
sr,rþðtry,yþsr�syÞ=r¼ 0, ð28Þ
try,rþðsy,yþ2tryÞ=r¼ 0, ð29Þ
srðR, yÞ ¼�p for 9y9ra or 9y9Zp�a0 for ao9y9op�a ,
(ð30Þ
tryðR, yÞ ¼ 0 for �p=2ryrp=2 ð31Þ
is according to Hondros (1959)
sr ¼�2p
p aþX1n ¼ 1
1� 1�1
n
� �r2
� �r2n�2sin 2na cos 2ny
!, ð32Þ
sy ¼�2p
p a�X1n ¼ 1
1� 1þ1
n
� �r2
� �r2n�2sin 2na cos 2ny
!, ð33Þ
try ¼2p
pX1n ¼ 1
ð1�r2Þr2n�2sin 2na sin 2ny, ð34Þ
where p is the magnitude of the applied compression, R is theouter diameter of the cross-section and r¼ r=R. For the coordi-nate setup and sign convention, see Fig. 1 in Ma and Hung (2008).As for the stresses, the convention obeys elasticity theory stan-dards, with tension positive and compression negative.
Ma and Hung (2008) found the closed form to (32)–(34). Theirexpressions are, however, complicated by the use of inverse
paces (full thick line). The unit of all axes is cohesive strength S0.
Fig. 8. Stress field in the cross-section of cylinder under Brazilian test coloured by
two-dimensional black background colour function (23) for a¼1 – zoomed view of
the load application region (cf. Fig. 7). Contours of principal stress s3 are white;
contours of failure distance according to Coulomb criterion with tensile cut-off D9C
(43) are cyan. Note that the colour map delineates very nicely both the boundary
between the pure compression zone (s3 compressive) and zone of s3 tensile, and
the valley between two peaks of failure distance function D9C . The contour values
of both quantities are given as multiples of cohesive strength S0.
M. Bednarik, I. Kohut / Computers & Geosciences 48 (2012) 117–125 123
function (arctan) and hence not very suitable for generating thesolution in all the subregions of the cross-section. Interestingly,the software Mathematicas 6 by Wolfram Research was able tofind the closed form of the sums:
sr ¼�2p
p ðaþLþQ Þ, ð35Þ
sy ¼�2p
p ðaþL�Q Þ, ð36Þ
try ¼2p
pð1�r2Þ
2ð1þr2Þsin 2a sin 2y
ð1þr4Þð1þr4�4r2cos 2a cos 2yÞþ2r4ðcos 4aþcos 4yÞð37Þ
These formulae are valid for the whole region at once –however, at the expense of numerical computations with complexnumbers. Fortunately, Mathematica can perform this task.
Let us have the principal stresses indexed as
s3 ¼ smþ9tm9, s1 ¼ sm�9tm9, s2 ¼ 2ksm, ð40Þ
where
sm ¼ ðsrþsyÞ=2, 9tm9¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2
ry�ðsr�syÞ2=4
q: ð41Þ
We can check that the mean stress sm ¼ ðs1þs3Þ=2 is, in thecase of diametrical compression, purely compressive over thewhole region. Thus, none of the stress sign measures m based onsm introduced previously would be a stress sign indicator suitablefor the Brazilian test. The only principal stress that varies fromcompressive to tensile over the region is s3, the others arecompressive (or zero). Therefore, we will base the stress charactermeasure on s3 and use the formula
m¼2s3�s3max�s3min
s3max�s3min, ð42Þ
where s3max is the actual maximum s3max ¼ s39r ¼ 0 and the mostvivid colours were obtained by setting s3min ¼�s3max. This givesm¼0 at the free boundary (e.g. r¼1, y¼p/2) where s3¼0.
To get a realistic picture of the failure danger, let us use theCoulomb criterion with tensile cut-off (formulated, as typical forfailure criteria, with the inverted stress sign convention assumed:tension negative, compression positive) and express the formula
Fig. 7. Stress field in the cross-section of cylinder under Brazilian test coloured by c
parameter a. Left disc – white background functions, right disc – black background fun
half-discs with the contours of D9C (43). The contour values of both quantities are given
of Paul (1961) in (sm,tm) space (Fig. 6)
D¼D9C ¼ 9tm9�f
9CðsmÞ, ð43Þ
where
f9CðsmÞ ¼
sm�st for smosmt
S0cos fþsmsin f for smZsmt,
(ð44Þ
smt ¼ ðS0cos fþstÞ=ð1�sin fÞ, ð45Þ
and sto0 is assumed.In the plane strain case k¼n, and for realistic values of
Poisson’s ratio, we obtain over the whole region D9C 4D
9C12 and
D9C 4D
9C23, where D
9C12 ¼ 9tm129�f
9Cðsm12Þ, D
9C23 ¼ 9tm239�f
9Cðsm23Þ,
9tm129¼ 9s1�s29=2, sm12 ¼ ðs1þs2Þ=2, 9tm239¼ 9s2�s39=2,sm23 ¼ ðs2þs3Þ=2. Thus, in a plane strain case with the failuredistance measure based on the Coulomb criterion, it is sufficientto consider just s1, s3.
To get d, we use (7) with the setting Dmax ¼D9r ¼ 0 and Dmin ¼
�Dmaxþ2D9 r¼ 1
y¼ p=2
.
olour functions (21) and (23), shown in quarterdisks for varying palette choice
ction. Western half-discs shown with the contours of principal stress s3, eastern
as multiples of cohesive strength S0. See Section 6 and Table of symbols for details.
Fig. 9. Performance of white and black background colour functions (21) and (23) within the zoomed regions (cf. Fig. 7) for palette choice parameter values a¼�1, a¼0,
a¼1 (from left to right).
Table A1Table of symbols.
Symbol Quantity
a relative magnitude of the intermediate principal stress s2 in 3D colour functions or palette choice parameter in 2D colour functions
b blue component in RGB colour specification
c colour vector (r,g,b)
c() white background colour function
cBlkðÞ black background colour function
clin() trilinear white background colour function
cBlklin ðÞ
trilinear black background colour function
cC cold colour
cW warm colour
D absolute failure distance measure
DC absolute failure distance measure based on Coulomb criterion
D9C
absolute failure distance measure based on Coulomb criterion with tensile cut-off
d relative (normalised) failure distance measure
f9C ðsmÞ
Coulomb criterion with tensile cut-off as function of mean stress
g green component in RGB colour specification
h hue component in HSV colour specification
hC or hC(u) cold hue function
hW or hW(u) warm hue function
k material constant equal to 0 in plane stress and to Poisson’s ratio in plane strain
L part of the solution of the diametrically compressed disc
M absolute stress sign measure
m relative (normalised) stress sign measure
p magnitude of the diametrical compression of the disc
Q part of the solution of the diametrically compressed disc
R radius of the disc
r red component in RGB colour specification or
polar coordinate in radial direction, depending on the context
s saturation component in HSV colour specification
S0 cohesive strength of the material
u colour angle (normalised to /0,1S) in HSV colour specification
v value (brightness) component in HSV colour specification
w warmth i.e. the proportion in which the warm and the cold colours are mixed
a half of the arc, on which the load is applied in the Brazilian test
y polar coordinate in angular direction
n Poisson’s ratio of the material
r polar coordinate in radial direction, normalised to /0,1Ss normal stress at failure
s1 the most compressive principal stress
s2 intermediate principal stress or
principal stress in the z- direction in the diametrical compression
s3 the least compressive principal stress
sc uniaxial compressive strength
sm the mean stress i.e. the mean of the two outermost principal stresses, sm¼(s1þs3)/2
soct octahedral normal stress,soct ¼ ðs1þs2þs3Þ=3
sr trr component of the stress tensor in polar coordinates
st uniaxial tensile strength
sy tyy component of the stress tensor in polar coordinates
try component of the stress tensor in polar coordinates
f angle of internal friction of the material
c colour angle (in radians) in HSV colour specification
M. Bednarik, I. Kohut / Computers & Geosciences 48 (2012) 117–125124
The material parameters used in the numerical test of stresscolouring are n¼0.3, f¼p/6, uniaxial compressive strengthsc ¼ 2S0cos f=ð1�sin fÞ, uniaxial tensile strength st¼�2sc/25.
The setting of the st/sc ratio is inspired by the experimental dataof Cadorin et al. (2001). With pressure p¼6S0 and a¼p/40(i.e.4.51, as in Ma and Hung, 2008), the 3 maxima of the failure
M. Bednarik, I. Kohut / Computers & Geosciences 48 (2012) 117–125 125
criterion D9C are slightly above the critical zero level and approxi-
mately equally high. The minima of D9C are reached in the vicinity
of the load application (Fig. 7). The regions bounded by purplelines are zoomed in Figs. 8 and 9.
7. Discussion
The stress colour maps clearly show why the cylinder sub-jected to Brazilian test splits along the plane running from oneload application strip to one diametrically opposite: beneath thestrips, the material is so compressed that it cannot fail. It actuallyforms two indenters (cf. black/white half-moons in the colouredcross-section details in Fig. 9). At their fronts develop zones ofalmost uniaxial pressure (s3 ¼ 0, shown by yellow and bluestreaks in Fig. 9 right and left, respectively), where the failurestarts. It further spreads to the favourably-stressed central zone.Through our stress colour map, we can cover the whole cross-section in sufficient detail. If instead we use any kind of glyphs,we lose the stress imaging resolution due to the very large areathe glyphs require on the screen in order to be readable. Ofcourse, we should not compare incomparables – the glyphs showthe orientation of principal stress axes, whereas the colours donot. We should, moreover, keep in mind that the 3D glyphs haveto be presented on a 2D screen and themselves need to becoloured to differentiate between compressive and tensile princi-pal stresses. Therefore, the effective information value theydeliver may be lower than expected, especially for user withweak 3D imagination. As there can be users who do not intui-tively understand colour maps, there can well be others who donot intuitively understand glyphs, hyperstreamlines and hyper-streamsurfaces. The advantages of various stress imaging techni-ques can be emphasised and disadvantages suppressed by theircombination. For instance, our colour map can provide an over-view of interesting spots in the stress field. Then the user canexplore in detail points of interest: by repeated mouse clicking ona chosen point, the glyph or tags with (s1,s2,s3) or (D, M, a)values will appear. By dragging and dropping, the glyph can berotated, enlarged, etc. The other way round, the readability ofglyphs may be enhanced by colouring them according to ourmethod.
8. Summary
The process of attributing a certain colour to a given stressstate according to proposed colour functions has three main steps
1.
Computing the principal stress values (s1,s2,s3), 2. Converting principal stress values (s1,s2,s3) into colourables
(d, m,a)
3. Substituting (d,m,a) into the colour function that meets the
user’s preferences (background, colourfulness).
The proposed colour functions are simple and the colour mapsthey produce colourful and meaningful – intuitively interpretablein terms of the quantities characterizing the stress state.
Acknowledgements
The authors are grateful to the Slovak grant agency VEGA(grants no. 2/0107/09, 1/0461/09, 1/0747/11 and 2/0067/12) andthe Slovak Research and Development Agency (grant APVT-51–002804) for the partial support of this work. The Mathematicas
8 software by Wolfram Research was used for the derivation ofsome formulae and for plotting colour figures. For plotting theblack and white graphics or combining raster and vector graphics,OCADs 6 software by OCAD AG was used.
Appendix. Table of symbols.
See Table A1.
References
Cadorin, J.F., Jongmans, D., Plumier, A., Camelbeeck, T., Delaby, S., Quinif, Y., 2001.Modelling of speleothems failure in the Hotton cave (Belgium). Is thefailure earthquake induced? Netherlands Journal of Geosciences/Geologie enMijnbouw 80 (3–4), 315–321.
Crossno, P., Rogers, D.H., Brannon, R.M., Coblentz, D., Fredrich, J.T., 2005. Visualiza-tion of geologic stress perturbations using Mohr diagrams, IEEE. Transactionson Visualization and Computer Graphics 11 (5), 508–518. September–October2005.
Hashash, Y.M.A., Yao, J.I.-C., Wotring, D.C., 2003. Glyph and hyperstreamlinerepresentation of stress and strain tensors and material constitutive response.International Journal for Numerical and Analytical Methods in Geomechanics27 (7), 603–626. 2003.
Hondros, G., 1959. The evaluation of Poisson’s ratio and the modulus of materialsof a low tensile resistance by the Brazilian (indirect tensile) test withparticular reference to concrete. Australian Journal of Applied Science 10,243–268.
Hotz, I., Feng, L., Hagen, H., Hamann, B., Joy, K., Jeremic, B., 2004. Physically basedmethods for tensor field visualization. Proceedings of the 15th IEEE Visualiza-tion Conference (Vis 04), pp. 123-130.
Jaeger, J.C., Cook, N.G.W., 1979. Fundamentals of Rock Mechanics, third ed.Chapman and Hall, London 593 pp.
Kratz, A., Meyer, B., Hotz, I., 2010. A Visual Approach to Analysis of Stress Tensor Fields.ZIB-Report 10–26 (August 2010), 15pp. /http://opus4.kobv.de/opus4-zib/front-door/deliver/index/docId/1191/file/ZR-10-26.pdfS, [accessed 10 November 2011].
Ma, C.-C., Hung, K.-M., 2008. Exact full-field analysis of strain and displacement forcircular disks subjected to partially distributed compressions. InternationalJournal of Mechanical Sciences 50 (2), 275–292.
Paul, B., 1961. A modification of the Coulomb–Mohr theory of fracture. Journal ofApplied Mechanics 28, 259–268.
Ramsay, J.G., Lisle, R.J., 2000. The techniques of modern structural geology.Applications of Continuum Mechanics in Structural Geology, 3. AcademicPress, London 360 pp.
Yang, C.-K., Yang, H.-L., 2008. Realization of Seurat’s pointillism via non-photo-realistic rendering. The Visual Computer 24 (5), 303–322.
Zehner, B., 2006. Interactive exploration of tensor fields in geosciences usingvolume rendering. Computers and Geosciences 32 (1), 73–84.
