**STRAIN IN PHOTOELASTIC PARTICLE **

**ASSEMBLIES **

**H.G.B. ALLERSMA **

**TR diss) **

**1560 **

**STRAIN IN PHOTOELASTIC PARTICLE **

**ASSEMBLIES **

**STRAIN IN PHOTOELASTIC PARTICLE **

**ASSEMBLIES **

### by

### Henderikus G.B. Allersma

### Delft University of Technology

### The Netherlands

### Delft, 1987

**TR dia **

**1560 **

This research was carried out at the Geotechnical Laboratory of the Department of Civil Engineering of the Delft University of Technology, the Netherlands. I am grateful to the technical staff of the laboratory, A. Mensinga, J. van Leeuwen and J.J. de Visser, who modified ideas in real devices and took care for the preparation of several test models.

1 INTRODUCTION 1 2 THEORY OF MEASUREMENT 7

2.1 STRESS TENSOR 7 2.1.1 Contact forces 7 2.1.2 Stress tensor in granular material 8

2.1.3 Stress distribution in a particle 10 2.1.4 Relation of interparticle stress and bulk stress 12

2.2 STRAIN TENSOR 15 2.2.1 Relative displacement of particles 15

2.2.2 Determination of the strain tensor 18 2.3 OPTICAL STRESS MEASUREMENT AT A MATERIAL POINT 19

2.3.1 Light 19 2.3.2 Polarisation 20

2.3.3 Double refraction 22 2.3.4 Optical filter system 23 2.3.5 Particle as sensors 31 2.3.6 Optical averaging inhomogeneous stress 33

2.4 DISPLACEMENT MEASUREMENT 36 2.4.1 Detection of marked particles 36

2.4.2 Determination of centre mark 38

2.4.3 Elimination of noise 40

2.5 DATA PROCESSING 41 2.5.1 Collected data 41 2.5.2 Principal stress trajectories 42

2.5.3 Stress distribution 45

2.5.4 Strain tensor 53

3 TEST SETUP 58 3.1 OPTICAL MEASURING DEVICE 58

3.1.1 Mechanical part 58 3.1.2 Optical system 60 3.1.3 Electronic circuit 63 3.1.4 Software and measuring procedure 66

3.2 MODELLING 73 3.2.1 Production of crushed glass 73

3.2.4 Preparation of a model 81 3.2.5 Loading systems and sensors 83

4 APPLICATIONS 85 4.1 SHEAR 85 4.1.1 Introduction 85

4.1.2 Theory of shear 87 4.1.3 Shear devices used 95 4.1.4 Measuring results 98

4.1.5 Discussion 115 4.2 EXPERIMENTS WITH LABORATORY-SCALE PENETROMETERS 121

4.2.1 Introduction 121 4.2.2 Models used 121 4.2.3 Measuring results 124

4.2.4 Discussion 141 4.3 EXPERIMENTS WITH A LABORATORY-SCALE HOPPER 148

4.3.1 Test setup 149 4.3.2 Measured results 150

4.3.3 Discussion 150 5 DISCUSSION 155 5.1 POSSIBILITIES AND LIMITATIONS 155

5.2 ACCURACY 157 NOTATION 160 REFERENCES 162 SAMENVATTING 168

The development of material models and calculation procedures, to predict the mechanical behaviour of non-cohesive packed particle assemblies is still in full swing. In particular, numerical calculations based on the finite element method have proved very powerful. Contrary to the calculation techniques, however, little progress has been made in the development of new measuring principles. Sensoring and data collection have indeed been improved and more accurate devices are developed to investigate the behaviour of a sample of granular material. The measuring principle, however, is in general still based on the same concept as many years ago, namely: measurement of boundary loads or stress and displacement of boundary segments. The stresses and strains in the interior of a sample have to be estimated from the boundary conditions in this case, which is possible, only, if the stress distribution and deformation field are uniform. The information obtained by convential measuring methods is therefore very restricted if a granular material is subjected to complex boundary conditions. The boundary conditions in model tests relating to practical problems are in general too complex to yield useful information about the conditions in the interior of the granular material, so that it is not possible to analyse such problems in detail.

Because the actual mechanisms cannot be visualised in most tests, it is not properly possible to verify the results of calculations. Although the numerically calculation techniques that have been developed are highly advanced, the quality of the results greatly depends on the stress-strain relations used. The stress-strain relations are mainly based on the behaviour of samples of granular material which are subjected to a known uniform stress condition. A typical device for material testing is the triaxial apparatus, in which a sample can be subjected to very well defined boundary stresses. This good control over the boundary stresses, however, restricts the scope to simulate conditions which are more realistic in practice, such as stress

rotation and continuous deformation.

It is well known that the behaviour of a packed particle assembly depends greatly on the stress history and stress path in the material. It is therefore not sufficient to investigate a sample under simple loading conditions only; but it is also necessary to know the mechanical behaviour of a sample when it is subjected to a stress path similar to the one occuring in reality. In the first instance, this requires knowledge about the stress path to which a granular material is subjected in a specific practical situation and, secondly, a device is required, which can simulate this stress condition. Several devices have been developed in which more realistic stress conditions can be simulated. Examples are the simple-shear apparatus (e.g. Roscoe, 1970) torsional-shear apparatus (e.g. Symes et al., 1982), directional-shear apparatus (Arthur et al. 1981) and multiaxial cubic test cell (Mould et al. 1982). For a particular problem a choice has to be made as to which material testing device gives the most useful results. Because clear information about the expected stress path is. often not available, many assumptions have to be made before the carefully measured material properties can be used in calculations.

To obtain more information about the stress-strain behaviour in practical problems and to verify the results of calculation procedures, a more advanced measuring method is required which would enable stress and strain components in the interior of the granular material to be visualised.

Up to now it was possible only to obtain detailed information about the strain in the interior of the granular material. Two methods are available for the strain measurements. The first method, developed by Roscoe et al.(1963), uses marked particles which are distributed in the granular material, such as lead shot. Because the density of lead is much more than, for example, that of sand the position of the lead shot in successive stages of a test can be located on a photograph by means of X-rays. This method has been used by e.g. Bransby et al. (1975) to investigate the flow of granular material in model tests of

hoppers. The second method, which was developed by Butterfield et al., (1970), uses stereo-photogrammetry to determine the relative displacements in a granular material. In this method two successive photographs of the surface of a sand sample are used. The texture of the particle surface in combination with a non-uniform deformation show a mountainous area in a stereo viewer. The difference in altitude can be converted into displacements by means of a stereoscopic plotting machine. The

strain tensor at a material point can be derived from the relative displacements of three points in a representative region.

The measurement of the stress distribution in the granular material, however, proved to be more complicated. As contrasted with elastic materials, there is no unique relation between

stress and strain which can be used to determine the stress increment in a region with a known deformation. A more direct method is therefore required to obtain information about the

stress distribution. It is not possible to realise this by means of electrical strain gauges because the large number of sensors, which would have to be distributed within the granular material

to perform systematic measurements, would unacceptably disturb the behaviour.

At present only the optical method of stress analysis known as photoelasticity is available for investigating in detail the state of stress in a granular material. The fundamentals of this technique were established by Brewster, who in 1811 discovered the law which describes the phenomenon of polarisation by reflection. Another important step was achieved in 1852, when Herapath discovered that crystals of a complex salt containing quinine, hydriodic acid and sulphuric acid, possess the ability to absorb light which oscillates in a specific plane. The needle-shaped crystals, with a girth diameter which is considerably smaller than the wave-length of visible light, could be used to produce synthetic polarisation sheets. This technique was developed in the first decades of 1900. The production of

visualise double refraction in models of transparent materials. In 1930 Colter and Filon, both of the University of London, demonstrated that the doubly refractive property could be utilized to visualise stresses in elastic materials. Since then, photoelasticity has increasingly become a useful tool for solving problems in structural engineering practice. The theory of photoelasticity and many applications thereof are for example described in Frocht's book (1946). In the last ten years the use of the photoelastic method has declined dramatically because many complex boundary value problems in structural engineering can now be solved with powerful computers and numerical calculation procedures.

In 1957 it was demonstrated by Dantu as well as by Wakabayashi, that photoelasticity could be used to visualise the transmission of force in a packed particle assembly. The optical phenomenon in a granular material, however, had to be interpreted

in a quite different way than was usual in elastic materials. It was not possible to create sharp isoclinics and isochromatics for analysing the stress distribution. Instead of isochromatics a pattern of clear stripes could be observed, which were assumed to represent major principal stress trajectories.

The transmission of force in a granular material was analysed in detail by De Josselin de Jong and Verruijt (1969). They performed tests with cylindrical disks of photoelastic material to simulate a two dimensional assembly of a granular material. The isochromatics were used to determine the magnitude and direction of the contact forces between the disks. This technique was later used by Drescher and De Josselin de Jong

(1972) to investigate aspects of a mathematical model for the flow of granular material. They described a method for transforming the distribution of the discrete contact forces and displacements in a region into a second-rank tensor, so that the test results could be expressed in terms as usual be employed in soil mechanics. They found experimentally that a tensor describes the distribution of the contact forces in a region fairly satisfactorily. Much research on the distribution of the

contact forces and the fabric structure in two-dimensional analogues of disks was performed by Konishi, Oda and Nemat-Nasser

(1982). They used, for example, elliptical disks of photoelastic material to investigate the anisotropic behaviour of a granular material. The test technique with disks is not suitable for investigating the stress distribution in model tests of practical problems because an assembly of disks has a very discrete character and is restricted to two-dimensional assemblies only.

In 1976 , Drescher published an experimental study on flow rules for granular material in which he again used crushed glass as the optically sensitive material. Because attempts undertaken by Wakabayashi (1959) and De Josselin de Jong (1960) to determine the magnitude of stress in crushed glass, by means of compensators, did not yield satisfactory results only information about the stress direction could be obtained in samples of crushed glass. It was shown in a qualitative way by Wakabayashi (1957,1959), Drescher and De Josselin de Jong (1972) and Oda and Konishi (1974) that the average direction of the light stripes visible in circularly polarised light approximately coincide with the lines of action of the major principal stress. Manual measurement of the directions from photographs, however, was rather subjective and very time consuming, and the direction of the stripes established in photographs appeared to be dependent on deviations in the optical filters, on the wave length distribution of the light source and on the sensitivity of the film to a particular wave-length. Because very realistic scale models can be prepared with crushed glass, it would be very 'useful if more information about stresses and strains could be

obtained by using the optical properties of glass.

The main purpose of this study was to develop optical methods, for determining stresses and strains in assemblies of photoelastic granular material and to design a device for performing systematic measurements in scale models of practical problems and material testing devices. In the first instance a connection was established between the distribution of the

contact forces at a material point and the optical phenomenon in the granular material. This resulted in D a method for determining two stress components in a three-dimensional plane strain sample, 2) a calculation procedure for determining absolute stresses, and 3) a prototype of a device for performing the measurements (Allersma, 1982a). At a later stage a computer-controlled optical device was developed which was equipped also with a digital camera. Stresses and displacements can now be measured simultaneously. The measuring method has been applied to analysing the mechanical behaviour of granular materials in several tests, which simulate practical problems and material testing devices.

The theoretical background of the photoelastic measuring method, the determination of the deformation and the data processing are described in Chapter 2. In Chapter 3 the layout, control and operation of the automated optical measuring device are described, and the production process of the crushed glass and the preparation, loading and sensoring of a model are explained. Some applications of the optical measuring method are presented in Chapter 4. The stress-strain behaviour under large shear deformations has been investigated, and several measurements of penetration tests are presented and discussed. Finally it is shown how the stress distribution changes in a hopper if some material flow has been taken place. In Chapter 5

the possibilities, accuracy and limitations of the measuring method are discussed, and suggestions for further development of

2 THEORY OF MEASUREMENT

2.1 STRESS TENSOR

2.1.1 Contact forces

If a large number of particles are brought together there will be contact points between the particles. At each contact point there acts a force due to gravity and/or external loads. The direction of the forces and the friction and cohesion between the particles determine whether particles move with respect to each other or not. At micro scale the distribution of the contact forces is very inhomogeneous. Besides body forces and external load there are several other factors which influence the magnitude and distribution of the contact forces, such as;

- the density of the particle assembly, - the strength of the particles,

- the shape of the particles,

- the elasticity of the individual particles, - the particle size.

The first four items influence the number of contact points between the particles. If the assembly is packed more densely, more contact points between the particles are to be expected. If the averaged force transmission is the same, the magnitude of the individual contact forces will be smaller. The strength of the particles is responsible for the amount of crushing at the contact points, which results in a redistribution of the contact forces. The number of contact points between the particles is directly influenced by the shape and the elasticity of the particles. The magnitude of the contact forces is more or less inversely proportional to the particle size. If the particles are very small, the magnitude of the contact forces comes closer to that of the atomic forces. This explains, for example, why an initially non-cohesive granular material shows a cohesive behaviour when it is moulded to a fine powder.

the contact forces with a second-rank tensor, because it is then possible to apply the fundamentals of continuum mechanics. It has to be realised, however, that a second-rank tensor gives a simplified image of reality. The discrete character of granular material and information on, for example, the number of contact forces and the magnitudes of the individual contact forces is not included in a tensor. At present, however there is no better way to symbolise the force distribution. Therefore the optical measuring method has been so designed, that parameters are obtained which can be used to derive the components of a tensor, so that the distribution of the contact forces is described in a similar way, as is usual in soil mechanics.

2.1.2 Stress tensor in granular material

In continuum mechanics, stress is defined as the resultant force acting on a unit area. This definition is based on homogeneous materials, which means that the elementary particles, i.e. atoms, are very small in comparison with the stress gradients. If continuum mechanics is applied to granular material, areas have to be considered which contain sufficient particles. On the other hand, it is not permitted to consider too large area units, because in that case information on the stress gradient is lost in samples with a non-uniform stress distribution. A unit area can be assumed to be representative if a small deviation in surface does not significantly change the average stress. However, it is not always suitably possible to define a representative area in a granular material, because large gradients in stress and strain may occur. The thickness of a shear band, for example, is about ten particle diameters, so that the contact forces of only a few particles are representative of the stress state. This is of course very little in comparison with homogeneous materials and it is not quite clear how such phenomena have to be interpreted.

If the macro stress in a region is not very inhomogeneous, the averaging procedure, which is described by Drescher and

De Josselin de Jong (1972) can be used to transform discrete contact forces into stress compcAaents. If a region with volume V

is considered with a non-uniform stress state* o. ., which is in *

equilibrium, then the average stress o. . is defined by

### «ij = è f

v### °ij

d v ( 2 : l )which, by using Gauss's divergence theorem, can be written as

### *ij = hi

x### i

( m ) T### j

( m ) ( 2 : 2 )m=l in which

u = number of discrete forces intersected by the boundary of V

x. = i-co-ordinate of the intersection point of

the m-th contact force (■ i=l,2,3)

T (m)= j _c o m p o n e nt of the m-th force ( j=l,2,3)

Eq. 2:2 was used by Drescher and De Josselin de Jong to determine the stress tensor in a region of a two dimensional assembly of photoelastic disks. The contact forces were determined from the isoclinics in the disks, using a procedure developed by De Josselin de Jong and Verruijt (1969). Contrary to usual photoelastic measuring techniques, it was possible in this way to determine the complete stress tensor in a representative region. Since the distribution of the contact forces was known, it was possible to plot a Maxwell diagram of the forces which intersect a hypothetical circular boundary. An example of such a diagram is presented in Fig.2:1. The more closely the diagram approaches an ellipse, the better the force distribution can be described by a tensor. Taken into account that a relatively small number of particles were considered, fairly good agreement with an ellipse is observed.

The analysis of Drescher and De Josselin de Jong has demonstrated that the distribution of the contact forces can be described fairly well with a second-rank tensor, and agreement can be expected to be better if the region under consideration

**o 3 10 kg **

Fig.2:1 Maxwell diagram of a circular region in a two-dimensional assembly of 150 disks (from Drescher and De Josselin de Jong, 1972).

contains more particles.

Because the contact forces create stresses in the disks, the stress tensor which describes the distribution of the contact forces in a region can also be derived by averaging all the internal stresses of all the disks in a region considered, as is shown by eq.2:l. It was found possible to determine two components of the average stress tensor in a body of photoelastic material by optical means. Because the whole volume of the body is considered in this method, the body may be of any shape. Since the optical averaging procedure is also valid over the thickness of a sample, the average stress components can also be derived in three-dimensional assemblies of arbitrarily shaped particles of photoelastic material. Because the optical averaging can be performed automatically, systematic measurements can be performed in scale models with more realistic particles.

2.1.3 Stress distribution in a particle

The particles play an important role in the optical measuring technique. In fact, they are used as gauges which translate the discrete contact forces into tensor components and they make these components measurable. In the optical measuring method

employed the stress components are accumulated vectorially over the surface of a region. The stress distribution in the region considered therefore need not to be homogeneous. However, because the magnitude of the principal stress difference (a -a ) , which can be measured is limited, it is necessary to know how the stresses vary at micro scale. Information on the stress distribution at micro scale can be obtained by calculating the stress distribution in a body which is loaded by discrete forces. If a particle is schematised as a circular disk the stress distribution can be calculated with an analytical solution based on elastic theory (Timoshenko and Goodier, 1951). In the case of a diametrical load the distribution of the principal stress difference and the principal stress direction, ty is given by

4P (a2-xz-y2)

*(a -a ) = (2:3) *

1 2 ira Cx2+(a-y)z]Cx2+(a+y)23

2xy

***-.-r-r-7**

**(2:4)**

x -y +a in which

P = the applied diametrical load a = the radius of the disk

x,y = the co-ordinates of a point in the disk (-a<x<a and -a<y<a)
The distribution of (a -a ) and ip in a diametrically loaded disk
is represented in Fig. 2:2. As may be expected the distribution
of* (a -a ) and x|> is not uniform. It appears, however, that the *

gradients are smooth and that the variation of the principal stress difference is not very large in the most significant region. Only in a small region close to the contact points are too large principal stress differences likely to occur. However, this region is so small that no significant influence on the total result is be expected.

It is not necessary to consider disks which are subjected to more point loads, because the peak values of (a -a ) decrease in

**M **

Fig.2:2 The distribution of the principal stress difference and the major principal stress direction in a diametrically load ed disk.

this case. Furthermore, it is not to be expected that the shape of the boundary has much influence on the variation in the principal stress difference, so that it is not necessary to investigate a great variety of particle shapes.

2.1.4 Relation of interparticle stress and bulk stress

A material point in a particle assembly is a volume which contains sufficient particles for obtaining representative averaged values of the stress components. In the case of a three-dimensional plane strain sample, of which the surface is large in comparison with the thickness, a material point is, for example, a cylinder with a length equal to the thickness of the

Fig.2:3 Distribution of the contact forces at the boundary of a cylindrical material point.

sample. Such a cylindrical material point contains a number of particles over the thickness as well as over the surface. Each particle is loaded by a number of contact forces. In Fig.2:3 it is demonstrated schematically how the contact forces are distributed over the hypothetical cylindrical boundary. As will be the case in practice, the individual contact forces are not all plotted perpendicularly to the axis of the cylinder. Furthermore, a cohesionless granular sample has to be supported at the entire boundary, so that in reality the stress condition is three-dimensional. However, if the thickness of the sample is sufficiently large, the force components perpendicular to the plane of the sample are assumed to have no significant influence on the plane deformation.

The averaged value of the components of the tensor, which describe the averaged stress in the cylindrical material point,

Fig.2:4 Representation of the contribution of one contact force to the averaged stress in a cylindrical region.

can be calculated with eq.2:2. In Fig.2:4 it is shown, how a m-th contact force T has to be resolved into different components.

It is assumed that a sample can be loaded in such a way that

one of the principal stresses, e.g.* a , is perpendicular to the *

boundary of the sample and therefore* a and a act in the plane *

of the sample. The state of stress at a point of a plane sample is represented in a Mohr diagram in Fig.2:5. It is to be noted that compressive stress has a negative sign. The averaging procedure is based on the whole volume of the material point. In reality, however, only a part of the volume is occupied by the granular material, while the pores are filled with air or a

liquid. The average stress in the particle bodies is therefore larger than calculated with the averaging procedure. Although this point is debatable it is neglected, because a discussion leads to new details such as the phenomenon that some particles

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An optical method can be applied to determine averaged stress components which refer to a material point in a plane strain sample. It can be demonstrated with the theory, which describes two dimensional photoelasticity, that the optical averaging over the thickness and surface of a material point is almost the same as the averaging procedure with eq.2:l.

2.2 STRAIN TENSOR

2.2.1 Relative displacement of particles

If a granular material is loaded, the particles will move in relation to each other. The displacements of the individual particles cause some macro deformation of a sample. To investigate the strain behaviour of a sample in detail, it is necessary to know the displacements in a representative region. Because it is not convenient to use the displacements of all the individual particles in formulas, a method is required to characterise the displacements at a material point. In the same way as the stresses it is usual to describe the relative

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Ux.x Uy.y

Fig.2:6 Representation of the strain components in a Mohr di agram (relative displacement diagram).

displacements between the particles at a material point by means of a second-rank tensor. The components of the strain tensor at a material point are represented in a Mohr diagram in Fig.2:6. The

strain tensor is in general not symmetrical, due to the rotation, u of the material.

In Fig.2:7 the distribution of the directions of the displacements of a tensorial deformation field is plotted. It is

assumed that UJ=0 and that the volume remains constant: hence the

principal strains E and E have the same absolute value. To demonstrate that the displacements in a granular material can be described reasonably well with a tensor, a similar pattern has been created by means of two negatives on which the particle structure of two successive stages of a plane sample are fixed

(Fig.2:8). If the field is subjected to a homogeneous deformation the major directions of strain become visible. This technique was used by De Josselin de Jong, 1959, to analyse the deformation in a two dimensional assembly of rods. Although good averaging is obtained over a large amount of particles, this technique is not always suitable for obtaining detailed information on the deformation, because a large region with a homogeneous deformation is required.

*.• .' * l t t \ \ \ \ *

Fig.2:7 Calculated distribution of the directions of the dis placements of a tensorial deformation field.

Fig.2:8 Distribution of the directions of displacements, visualised in a plane strain granular material.

2.2.2 Determination of the strain tensor

Contrary to the optical stress measurement, it is not practicable to obtain strain components which are based on continuous averaging of the displacements over the volume of a material point. Therefore the determination of the strain tensor has to be based on the displacements of some discrete points. The displacements of only three points are required to calculate the strain components at a material point. At present there are two measuring principles available for determining the displacement of a discrete point. In the most advanced techniques two successive exposures of the particle structure at the surface of a sample are used to determine the displacement of a point by means of the stereo-photogrammetric method (Butterfield et al.,1970). The advantage of this method is that the displacement at the surface can be measured more or less continuously, so that large strain gradients can be observed. A disadvantage is, however, that the digitisation of the displacements requires expensive precision devices and is time consuming. Because the measuring procedure cannot be automated in a simple way, it is not practicable to apply this method to the interpretation of a large number of photographs. Further, it is not simply possible to follow the pattern of a single point, which is for example useful in tests with large deformations.

A simpler technique for determining the displacements of discrete points is to measure the co-ordinates of labelled particles in successive stages of a test. If a plane sample is not transparent, particles with a higher density e.g. lead shot, can be distributed in a plane in the sample and X-rays can be used to fix the actual position of lead balls on a photograph. This method was used, for example, by Bransby et al.,1973 to investigate the deformation in plane hoppers. This technique is very attractive in tests with crushed glass. Because the assembly of glass particles is made transparent by saturating the pores with a liquid which has the same refraction index as glass, black marks can be made visible simply by normal light.

Furthermore particles, of the same density can be used such as black coloured glass. Because, due to small deviations of the refraction index of glass, the particle structure is visible in an assembly too, a stereo observation of successive photographs can be used to support the interpretation of the displacements of marked particles.

Several types of digitisers are available to determine the co-ordinates of the marks from photographs. In general, however, the digitisation of the co-ordinates is rather time-consuming, and if cross hairs are used it is not always easy to define a particular point on similar marks of successive tests. Furthermore, the photographic step is not convenient. To prevent errors due to strength of the film material, it is necessary to use glass plates with a light-sensitive layer. However, a disadvantage is that a camera which handles glass plates is not

suitable for automation.

To eliminate the photographic step, a digital camera is applied in this research to determine the position of marks directly in a test model. The advantage of this method is that the procedure can be completely automated and the results are directly available, so that the progress of a test can be observed in real time.

2.3 OPTICAL STRESS MEASUREMENT AT A MATERIAL POINT

2.3.1 Light

Light is used as the medium for determining the stress distribution and deformation in a transparent assembly of optically sensitive glass particles. The major advantage of light is that it has no effect on the mechanical behaviour of the particles. Only secondary effects, such as heat production of a light source, can influence the measurement, because temperature gradients causes additional internal stresses in glass. For this reason a laser source appeared to be very suitable, because an intensive light beam is produced with minimum production of heat

radiation. Furthermore, light with a very small range in wave length is obtained, so that the mathematical description of photoelasticity is in very close agreement with the reality. For this purpose a He-Ne laser is used, which produces light with a wave-length of X=633nm. Light is a complex phenomenon and several theoretical models have been developed in order to describe this phenomenon completely. For this purpose, however, the classical ether-wave theory of Huygens is quite adequate. According to this theory, space is filled with a hypothetical perfect elastic medium, and light is a wave phenomenon caused by a disturbance in this medium. The disturbance consists of vibrations of the elementary particles. It is assumed that the vibrations take place in a direction perpendicular to the direction of propagation of the wave. In ordinary light the elementary particles are subjected to many harmonic waves, which vibrate in random directions and which differ in amplitude and phase. The motion of a particle is therefore entirely random as is schematised in Fig.2:9. It is not possible to describe ordinary light with simple mathematical formulas. To perform photoelastic measurements ordinary light has to be modified into a well-defined phenomenon which can be realised by means of a polarisation filter.

2.3.2 Polarisation

The random behaviour of ordinary light can be modified into a well-defined wave motion by means of polarisation. Polarisation can be achieved by dichroic crystals, such as herapatite

(discovered by Herapath in 1852). The physical model assumes that ordinary light is converted into two mutually perpendicular plane-polarised beams when it enters the crystal. Because the absorptive capacity of dichroic crystals is much greater for one of these beams, only one plane-polarised beam is transmitted

(Fig.2:10). In plane-polarised light the light vector vibrates harmoniously in parallel planes, as is shown schematically for

Fig.2:9 Diagram of the motion of ordinary light, projected on a plane perpendicular to the light beam.

**Unpolaiüad Inddint U(ht **

**CourUty M. Orabau and Polanid Corporation **

Fig.2:10 Schematic representation of the production of polar ised light by a dichroic crystal (from Frocht, 1946).

Fig.2:11 Graphic representation of the wave motion of polar ised light.

one plane in Fig.2:11. According to the wave theory polarised light'can be described by

Asin(ü)t) w = 2ir- rad/s (2:5)

were a = the actual amplitude at time t A = the maximum amplitude

id = the angular velocity

c = the propagation speed of light x = the wave-length of the light used

The intensity I of a plane-polarised light beam is defined by

I = 2A* (2:6)

2.3.3 Double refraction

Isotropic transparent materials, such as glass, become temporarily double-refractive when subjected to shear stresses. The physical background of this phenomenon is that the velocity of propagation of light which vibrates in the principal stress directions is different. The velocity of propagation decreases if the stress, acting on the plane of vibration, increases. To describe the double-refractive property the following physical model is assumed (Fig.2:12)

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**POLARIZER **
**1/4-**** \ PLHTE ****'. CIRCULRR **
**"LIGHT **
**ELLIPTICAL LIGHT **
**ROTATING FILTER **
**LIGHT SENSOR **

**(2 **

**(2**

Fig.2:13 Lay-out of the optical filter system.

In optics permanent double-refractive filters are used to produce light with particular properties. A typical example of such a filter is a quarter-wave plate. In this filter the retardation between the waves which vibrate in the direction of the principal axes is exactly a quarter of the wave-length of the light used.

2.3.4 Optical filter system

The application of stress optics to particle assemblies requires some properties of the optical filter system which are usually not of importance for stress analysis in bodies consisting of a homogeneous material.

Although the assembly of glass particles is made transparent with a liquid, the absorption of light due to refraction and reflection is not homogeneous over the surface of the plane sample and depends on the deformation. It is therefore necessary that the intensity of the incident light, including losses due to absorption by the optical filters, can be measured when the model is stressed.

the micro stresses in the particles of a small region. The accumulation of the optical effects over the surface and thickness of the light beam should therefore be in agreement with the vectorial addition of the micro stresses.

The layout of the optical system which best meets the requirements is shown schematically in Fig.2:13. Monochromatic, linearly polarised light, produced by a He-Ne laser, is modified into circularly polarised light by means of a so called quarter-wave plate. If the circular-polarised light beam has been

transmitted through a model of double-refractive material, an additional phase difference S causes an elliptical motion of the light vector. A rotating polarisation filter (analyser) and a light sensor are used to analyse the elliptically polarised light.

The initial polarised light beam, which can be described with eq.2:5, enters the quarter-wave plate in such a way that the plane of polarisation is incline at 45 with respect to the optical axes of the plate. The light wave is resolved in the directions of the optical axes q and q and can be described by

qx = iAJ~2cos(wt) (2:7a)

q2 = |Af2cos(ut) (2:7b)

When the light leaves the filter, the retardation has caused a phase difference of TT/2 between the perpendicular waves. The equations describing the vibration in the direction of the optical axes now become

q = |Af2cos(ujt) (2:8a)

q2 = |A4"2C os (ut-^r = |Af2sin(üJt) (2:8b)

the light is obtained which can be written in two dimensions with

r = |AJ~2 (2:9a)

**()> = u i t ■ (2:9b) **

were r and* $ are the length and the direction of the light vector *

at time t, respectively. The intensity of this circular polarised light, known as the initial light intensity, I , is

I = 2r2= A2 (2:10)

o

When the circular-polarised light enters the temporary double-refractive material of the test model, the circular motion is resolved into two perpendicular waves again, which vibrate in the directions of the principal stresses, as represented by

a = iAJTsin(wt) (2:lla)

**2 2 **

ai = |AJ~2cos(wt) (2:llb)

were a and a are the amplitudes of the light wave in the* a *

and* a direction, respectively. The propagation velocity of the *

two waves are not equal, and the velocity difference e, which is proportional to the difference of the principal stresses, is given by

E = K(a -a ) (2:12) were K is an optical material constant.

The velocity difference causes a phase retardation S between the waves which is proportional to the thickness s of the model, the wave-lenght X of the light used and the average velocity c' of the light in the double-refractive material. The expression

for S becomes

S = ^ ^ r ( a -a ) (2:13)

XC 1 2

In in a test X, c', K and s are constants; this equation can therefore be written as

(a* -a ) = MS (2:14) *

were M is a factor taking account of all parameters influencing the retardation S.

The phase difference causes an elliptical motion of the light when it leaves the model, as is represented schematically in Fig.2:14. This motion can be described with

a2 = iAJ~2sin(0Jt+S) (2:15a)

*8LI* = ^AJTcosdjUt) (2:15b)

If the light enters the second polarisation filter (analyser) the light vector r (Fig.2:14) will be resolved into two perpendicular components, but only one component, which coincides with the polarisation plane will be transmitted. The amplitude of the

transmitted wave depends on the angle* y between the polarisation *

plane and the* a -axis and can be described by *

a = ^A.T2Csin(ut+S) sinY+cos(wt) cosyD (2:16)

The value of wt at which a is a maximum, A , can be found by

**y y **

differentiating eq.2:16. After some rearrangement this gives

wt = a r c t g f — .c o^S.* A = $ (2:17) *

^ (.cotgy+sinSj T

and thus

A = ?AJTCsin(0+6)sin>f+cos(t)cosY3 (2:18)

analyzer

Fig.2:14 Graphical representation of the light motion when it has been transmitted through the test model.

max 0 Pmax. Pmin. rotation analyzer F i g . 2 : 1 5 L i g h t i n t e n s i t y I a s a a n g l e p of t h e a n a l y s e r . 180

The light intensity as a function of* y is now *

I = 2A2 (2:19)

Y Y

Because the maximum amplitudes in eq.2:15 are equal, it can be concluded that the position of the ellipse is always symmetrical with respect to the principal stresses; so in the case of

Fig.2:14, I is a maximum if* y= 45° and a minimum if -y= 135°. *

Substituting these extreme values into eq.2:17 gives respectively

<(,= 45°-| and <p = 135°-|

Using eq.2:18, the maximum and minimum amplitudes are found to be

A = iAf2(cos4+sin|) (2:20a)

is 2 2 2

A _ = iAJ~2(cosf-sin4) (2:20b)

135 &* A £» *

Remembering that I =A , the maximum and the minimum light intensity can be written as

I = 2A2 = I (1+sinS) (2:21a)

max is o

I . = 2A2 = 1 (1-sinS) (2:21b)

m m 135 o

If the directions of the principal stresses are not known, the maximum and the minimum light intensity can be found numerically by rotating the analyser through 180 and measuring, for example, at every degree the light intensity (Fig.2:15). The intensity of the incident light can be calculated from the average light intensity Ï as follows

I = Ï = ^(1 +1 . ) (2:22) o 2 max min

Fig.2:16 Difference of the maximum and the minimum light in tensity as a function of the phase retardation.

retardation, for example, with
,1 -I _{max r }
E +1 ._{max m m ' }* , *
,1 -l . .
o . _ f max min^
S = arcsinlj —= 1
(2:23)

If the initial position of the analyser is known, the direction of the principal stresses can be deduced from the rotation angle B of the analyser at maximum light intensity. In the case of Fig.2:15 the principal stress directions are

t|) = B -45

T l "max (2:24

a)
*\\> = B +45 *

2 max (2:24b)

A simplified equation can be derived to calculate I , because o

I is apparently a harmonic function of* y with a period of 180 . *

An equivalent equation is then

(I -I)sin2Y+I _{max } (2:25)

Substituting the expressions for the intensities gives

1 = 1 (sinSsin2Y+l)

were* y may also be written as y = P-ip ■ *

The range of S is limited because I -I . decreases if S

3 max m m

becomes larger than —IT radians (Fig.2:16). However, a favourable circumstance is that the principal stress difference in particle assemblies is usually small in comparison with that found in tests on homogeneous cohesive materials. If a sample of glass particles with a thickness of 40 mm is used in a model test, a maximum principal stress difference of about 700 kN/m can be measured. In the tests described in this paper the maximum principal stress was in general not more than 250 kN/m (8<—ir).

2.3.5 Particles as sensors

The mathematical description of the optical determination

of (a -a ) and the principal stress directions \|> and i|) is based

on a plane stress model of photoelastic material, with a homogeneous stress distribution at the material point considered.

In the case of a plane model of crushed glass the individual particles are used as an optical sensor which translate the distribution of the contact forces into optically measurable

stress components. Apart from optical imperfections of the granular material, several other phenomena are not in agreement with the assumptions on which the mathematical description of the behaviour of light in double-refractive material is based. The following deviations can be noted;

- the model is not homogeneous;

- the stress distribution in the particles is not plane;

- the stress distribution is not homogeneous over the surface and thickness of a material point.

It is not to be expected that the inhomogeneous character of a granular sample has a significant influence on the validity of the mathematical description of the optical measurement. Indeed, the stresses in the optically sensitive material of a particle assembly are larger than if the same space were filled with a homogeneous optically sensitive material. However, because the thickness of the model includes also the pore volume, the

Fig.2:17 Vectorial addition of two Mohr circles.

relation between the retardation S and* (a -a ) in eq.2:13 is in *

1 2 ^

agreement with the definition of stress in a granular material. Moreover, the linear relation between the parameters in eq.2:13 demonstrates that the optical constant for an assembly of crushed material is theoretically the same as for a homogeneous material. In tests it was attempted to approach the plane stress condition as much as possible. However, the micro stresses in the particles cannot be prevented from deviating from the plane state of stress. This is not in agreement with the two-dimensional stress optics. Although there is no mathematical or experimental evidence, it is presumed that the optically determined stress components describe a two-dimensional stress state in the plane of the sample, which is in agreement with the averaged stress.

Contrary to the assumptions made for the theoretical analysis of the optical filter system, the stress distribution is not homogeneous over the surface and thickness of a material point considered. Since the volumetric averaged stress is obtained by vectorial addition of the components of the micro stresses, the optical measurement is required to produce a similar result. Due to this requirement several combinations of optical filters are not suitable for this application. A circular polariscope (Frocht, 1946), for example, adds the principal stress

differences over the surface as scalars, while in the case of polarisers on both sides of the model rotating synchronously

(e.g. Redner, 1976),. equal stresses inclined at 45° are suppressed. It is demonstrated below that the filter system used in this research adds the stress components almost vectorially. A n additional advantage of this filter system over other systems

is that a separation is obtained between the directions of* a *

and* a . In particular, this is convenient in samples with a *

complex stress distribution.

2.3.6 Optical averaging inhomogeneous stress

The optically measured components of the stress tensor are the length S of a vector and its orientation 2tJ). In a Mohr diagram this vector is the line segment between the pole and the centre of the Mohr circle. The addition of stress tensors can be performed by vectorial addition of all the line segments of the Mohr diagrams. An example of a two-dimensional state of stress is shown in Fig. 2:17. The optically measurable parameters S, ijj and S', ij)' represent two regions with a different state of stress, and S", i))" is the result of the vectorial addition. The mathematical relation is

6" = JtS2+S'Z+2SS'cos2(ijj -U>' ) D (2:27a)

\|>" = i-arcos[ij-rrcos2ij) -»-|7rcos2tJ)'] (2:27b)

The optical addition of the tensor components can be separated into an addition over the thickness of a test model and an addition over the cross-section of the light beam.

The optical addition over the surface can be investigated with eq.2:26. If the surface of a material point is divided into two equal parts with different stresses eq.2:26 becomes

The maximum and the minimum value of I and the values of 6 to match can be found numerically. The values for S" and t|j" can now be calculated from eq.2:23 and eq.2:24 and verified with the

results of eq.2:27. Some numerical examples of the vectorial and optical addition are given in Table.2:1.

two regions with different stress * i 0 0 0 0 0 0

### s

10 20 2 5 20 20 6' 10 40 12 5 20 20*** ;**0 10 45 45 45 30 optical addition surface S" 20 58.1 12.1 7.1 28.0 34.5

*** ;**0 6.7 40 23 23 15 thickness S" 20 59.1 12.3 7.1 28.0 34.5

**<)>;•**0 6.0 40 23 21 14 vectorial addition S" 20 59.2 12.2 7.1 28.3 34.6

*** ;**0 6.7 40.3 22.5 22.5 15

Table 2:1. Comparison of optical and vectorial addition of optically measurable stress components.

It appears that the optical addition is in reasonable agreement with the vectorial addition. The error is a maximum if l^-ijri = 45° and increases if S+S' or |S-S'| increases.

Also, it can be deduced from eq.2:28 that the average light intensity during the rotation of the analyser is independent of the stress distribution in the measuring region. The sum of the light intensity, over a rotation of 180° of the analyser, can be derived with

**ïïr i ■> **

E IQ* = <±I s i n S s i n 2 ( 8 - i | ) )+±I s i n S ' s i n 2 ( B - i | ) ' ) + I )dB ( 2 : 2 9 ) *

P 0 J 2 o r ^ l 2 o ^ Ti o ^

Since 2P ranges from 0 to 2 IT, the first two terms do not contribute to the summation. The contribution of the last term corresponds with 1801 . The averaged light intensity during rotation is Ï = I , so that I can be derived from the averaged light intensity also if the stress distribution is inhomogeneous

**2 \ d2 HNHLYSER **

Fig.2:18 Behaviour of light transmitted through two layers of double-refractive material with different stresses.

over the cross section of the light beam.

The optical averaging of the stress components over the thickness is independent of the filter system used. To investigate the agreement between the vectorial and optical addition, it is assumed that the light described by eq.2:15 is transmitted through a second layer of double-refractive material with a different retardation and principal stress direction. The two perpendicular waves of eq.2:15 are transformed into

a' = a cosot-a sina

2 2 1

*a.' = a cosa+a sina *

1 1 2

were a' and a' are the vibrations in the* a' and a' direction of *

1 2 1 2

the second layer, respectively, and* a. is the inclination *

between* o and a' (Fig.2:18). When the light leaves the second *

layer an additional retardation S' has taken place. Substituting the functions of a and a , eq.2:30 becomes .

1 2 ^

a^ = 2A^2Ccosasin(wt+S+S')-sinacosCut+S')D (2:31a)

(2:30 )

a' = ~A4~2Ccosotcos((jüt )+sin<xsin(wt+S) H

Assuming that* a coincides with the x-axis the vibration in the *

direction of the polarisation plane can be described with

a = a'cos( v-oO+a' sin(Y-a) (2:32)

*y l ' z ' *

The intensity of the transmitted light is defined by

I = 2AZ (2:33)

Y* y *

were A is the maximum amplitude of the light wave at the actual position of the polariser. I and the maximum and minimum light intensity during the rotation of the analyser are determined numerically and the values of 5' and i|)" are calculated with eq.2:23 and eq.2:24. If the results are compared with the vectorial addition (Table.2:1) it is shown that a similar deviation is found as with the addition over the surface. Also, it can be shown numerically that I = Ï if the stress

o

distribution is not homogeneous over the thickness of the sample. It is not necessary to investigate the addition over more than two regions or layers with different stresses, since the motion of the light will always be elliptical (harmonic waves with the same frequency are added). Because the distribution of the micro stresses is more or less random, positive and negative deviations can take place. It is therefore not to be expected that the optical summation will result in a large error.

2.4 DISPLACEMENT MEASUREMENT

2.4.1 Detection of marked particles

The deformation in a plane strain sample is determined from the co-ordinates of marked particles in successive stages of a test. Because the assembly of crushed glass is transparent, normal light can be used to observe the position of black

LIGHT SOURCE LENS IMRGE

**1^1 **

**>****MRRK**

**■■-¥■■****CRUSHED**

**GLHSS**

**\**Fig.2:19 Diagram of the optical system to detect a black mark in an assembly of crushed glass.

particles which are situated in a plane halfway through the thickness of a sample. To eliminate the photographic step a digital camera is used to determine the co-ordinates of the marks directly in the plane sample. Since the area considered and the number of marks are rather large (e.g. 900 cm with 40 to 100 marks) it is not possible to analyse the total area accurately enough all at once. Therefore the position of the digital camera is controlled by an accurate x-y scanner, so that the area of the sample can be divided into small square elements which can be analysed consecutively.

The most important part of the camera is an integrated circuit with a window of about 6 mm square which contains a matrix of 128x128 light-sensitive pixels (diodes). Electronics in the camera converts the light intensity, received by each pixel, into a voltage which is connected to an output line for a short time. The voltage of the pixels can be read by a computer. A synchronisation signal, which is also produced by the camera is used to separate the signals of the pixels. If the image of a region with a black mark, which has a diameter of about 2 mm, is focused one-to-one on the window of the sensor (Fig.2:19), a number of pixels will receive a low light level. The output voltage of a pixel is converted into black (1) or white (0), depending on whether the voltage is respectively lower or higher than a reference value. The status of each pixel is read line by line and is stored sequentially in the computer memory. The

**COLUMN **
**127 **
**ROM **
**127 **
o
0
o
128
o
1
**0 **
129
o
2
130
o
127
255
**o **
**16383 **

Fig.2:20 Relation between the memory locations and the row and column number of a picture element.

relation between the row and the column number and the actual memory location is represented in Fig.2:20.

2.4.2 Determination of centre mark

The position of the centre of the image of a black particle, with respect to the upper left corner of the pixel matrix can be derived from the row and the column number of the dark pixels. If the number of the memory location of the first pixel is assumed to be zero, the co-ordinates of a pixel ( x

corresponds with the memory location n can be calculated from

**y ) ** w h i c h

**xP**

** =**

** i n t**

**S)**

**s **

y = n s - k y „
' p JP
( 2 : 3 4a)
( 2 : 3 4b)
were k is the number of pixels in a row and s is the distance between the pixels. The centre of the image of a black mark

co-ordinates of all the black pixels and dividing the results by the total amount t as formulated by

x =* ^Zx (2:35*a)

m t p

y = ir£y (2:35b)

-'m t* Jp *

The absolute position of a mark can be calculated from the known co-ordinates of the upper left-hand corner of the pixel matrix with respect to the origin of the scanner.

To prevent errors in the interpretation of the image of a mark several control parameters are used, such as

- the number of dark pixels of the image of a mark;

- the number of dark pixels at the boundary of the pixel matrix; - the measured displacement of a mark.

The number of dark pixels which is covered by a mark can be used to determine the surface of the black image, so that the computer can distinguish a mark from a small accidental dirty spot. The number of black pixels at the boundary of the pixel matrix indicates if a mark is close to a boundary segment of the model. If the measured displacement of a mark after a deformation step is unreasonably large, e.g. in comparison with measured displacements of boundary segments, it can be concluded that there is something wrong.

Because the whole image of a mark is used to determine the centre,, the accuracy is not significantly dependent on the shape of the mark. However, the assembly of crushed glass is not perfectly transparent due to small deviations in the refraction index and dirt, so that the image of a mark is often accompanied by more or less randomly distributed black spots (noise). To increase the accuracy a procedure has been developed to eliminate the noise before the co-ordinates of a mark are determined definitively.

b

**O **

*m~ *

**O **

Fig.2:21 Example of an image of a mark and noise on the pixel matrix of the image sensor.

2.4.3 Elimination of noise

A machine language routine has been developed to eliminate noise in an image of a black mark. An example of an image is represented in Fig.2:21. The mark is located by a circle, while some irregular indicated regions are supposed to be black fields due to dirt in the assembly. Because the surface of the mark dominates and the noise is distributed more or less at random, the main point of the black pixels is always situated within the boundary of the image of the mark. In Fig.2:21 the provisional centre of the mark is indicated by the intersection point of the lines aa' and bb'. To clean the image the program starts with a pixel in the first row which is located on line aa'. The pixel number is reduced until the left-hand part of the boundary of the matrix is reached. The contents of the corresponding memory locations are checked for black or white. If a white pixel has been observed once, all the following pixels in that segment of the row are also set to white. Next, the right-hand segment of the row is processed on the same way. This procedure is repeated

for all the rows, and the columns are similarly processed. The result is that all the black spots which are not connected with the black region of the mark are eliminated, so that the centre of the mark can now be determined definitively.

2.5 DATA PROCESSING

2.5.1 Collected data

During the measurement of the parameters which describe the condition in the granular material of a test model a considerable amount of data is produced. Since the digitised conditions of several tests are required to analyse the stress strain behaviour of the granular material, it is necessary that the data of each stage can be read easily by a computer. Therefore the measured data of the different stages is stored in files on a disk. The file name includes the name of the test and a number which represents the sequence. The following data are stored in a file - name, reference number of the measurement, date, time, number

of strain gauges, thickness of the sample;

- co-ordinates of the angular points of the boundary, the loca tion of strain gauges;.

- number of rows and columns of the nodal points at which the stress components are measured, the distance between the nodal points, and the co-ordinates of the first nodal point of the mesh;

- number of rows and columns of black marks in the field; - number of marks at the boundary;

- a matrix containing the relative principal stress differences; - a matrix containing the principal stress directions;

- a matrix containing the co-ordinates of the black marks;, - the output of the strain gauges.

The data is collected by a single-board microcomputer which controls the measuring device (Section 3.4.1). The digits are transmitted to a more powerful computer, which stores the data and graphically displays the elementary parameters of the

Fig.2:22 a) Graphical representation of the distribution of the principal stress directions, based on measurements at 100 discrete points, b) Visible pattern of light stripes, viewed with a circular polariscope.

stresses, displacements and strain gauges graphically. The course of a test can be conveniently monitored in this way. In a later stage the data of one or more tests are subjected to a more detailed analyses.

2.5.2 Principal stress trajectories

Since the digital,point information of the principal stress directions is not very convenient for visual inspection, a computer program has been developed which converts the discrete measurements into a continuous regular pattern. The pattern is formed by curved lines, principal stress trajectories, of which the tangent at any particular point represents the principal stress direction at that point of the sample. A computer plot of such a pattern is shown in Fig.2:22a. The plotting procedure is