1st BRAZILIAN CONGRESS OF TUNNELS AND UNDERGROUND STRUCTURES
ABOUT A METHOD FOR THE EXPERIMENTAL DETERMINATION OF THE LAW OF HOOKE FOR ANISOTROPIC ROCKY SOLIDS
Elysio R. F. Ruggeri1 Civil Engineer CREA 10.452/D - MG
SUMMARY In many situations, in dimensioning underground structures, the most realistic forecast of the mechanical behaviour of the fractured rocky solids should be done using a continuous, homogeneous, anisotropic, linear and elastic model (a CHALE model), as the isotropy could be utopian. However, one of the difficulties lies in the necessary establishment of the Law of Hooke, either for not having reliable information regarding axis and plains of elastic symmetry of the solid or for not being able to defend the adoption of determinable elastic constants by means of laboratory trials of body structures. The indirect determination of elastic constants of a solid by the measuring “in situ” of stress and strain tensors – core of the proposed method – may be a practical and economically advantageous mean in implanting and developing undertakings. This method supposes that the quoted measures are being done within a “representative volume element” of the solid, thus already taking into consideration its families of fractures. In this article, the method is briefly justified. Also, a discussion about some aspects related to its utility and the emergent improvement difficulties, starts. Practical fieldworks, in course, which will serve as base for new studies and communications, are being announced.
Key words: Hooke, rocky solid, anisotropy.
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FURNAS CENTRAIS ELÉTRICAS SA. – Caixa Postal 457 – CEP 74001-970 – Goiânia-GO- Brasil ruggeri@furnas.com.br
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FUNDAMENTAL HYPOTHESES (for the method development) H1 – The distribution of the fractures of a family in a representative volume element of a rocky solid, besides being homogeneous (independent of the element), presents well defined average and standard deviation. H2 – The homogeneous statistical distribution of the fractures and the triple state of stress to which the solid is submitted should allow it to be considered as a CHALE (Continuous, Homogeneous, Anisotropic, Linear and Elastic) body for which the Law of Hooke, that expresses the proportionality (weighted average) between stresses and strains is supposedly valid. H3 – The utilization of the small flat jack (SFJ) method is accepted to determine the stress and strain Cartesian tensors in a “point” of the wall in a circular section gallery properly opened in the solid. . POSITION OF THE PROBLEM Reference systems; notations We adopted an orthogonal (global) Cartesian reference system, O-xyz, with the origin in the center O of the right circular section of a gallery and base vectors: ˆi , ˆj e kˆ , respectively associated to Ox, Oy and Oz. The Oz axis is
tangent to the gallery axis, Ox points to the ground and Oy is taken so that the trihedral O-xyz is positive. About the circumference section (plan O-xy) we arbitrarily chose a point, in the neighborhood of which we will measure the stress and strain tensors (or dyadics). That neighborhood is defined by a fragment of right prism, which generators are parallel to the axis of the gallery, and which base, contained in the plain of the section, is a curvilinear quadrilateral presenting two opposed sides with straight lines (supports) converging in the center O, the other two being concentric circle arches in O. The generator is about 3.0 m long; the convergent sides of the section, about 0.5 m, and the circle arches about 1.0 m. The face of the prism that contains the point is, evidently, a fragment of the wall of the gallery which can be seen, without a significant mistake, as a 1.0 m wide by 3.0 m long rectangle. To the then defined prism fragment, we associate a (local) cylindrical system of reference in relation to which we can determine, with easiness, for the SFJ method, the dyadics of stress and strain related to the chosen point. Once these dyadics are calculated (each one in its reference system), we can refer them all to the global Cartesian system by rotations of the systems. The dyadics of the point, σ (stress) and ε (strain), with coordinates σij and εij (for i,j=x,y,z) in relation to Oxyz, can be expressed in the form of a 3x3 symmetrical matrix [1, sections 2.5 e 3.7]. For what interests us however, it is more convenient to represent them in a dyadic base (orthonormed) of the space of the symmetrical dyadics [2], µˆ 1 = ˆiˆi , µˆ 2 = ˆjˆj , µˆ 3 = kˆ kˆ ,
µˆ 4 =
1 ˆ ˆ ˆˆ 1 ˆˆ ˆ ˆ 1 ˆˆ ˆ ˆ ( jk + kj) , µˆ 5 = (ki + ik ) and µˆ 6 = ( i j + ji ) , 2 2 2
(01),
such base, as seen, generated from the vector base used to perform the measurements. In relation to that dyadic base, then, the dyadics can be represented by the column matrices 6x1 which transposes are: {σ}T = [σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] and {ε}T = [ε1 ε 2 ε 3 ε 4 ε 5 ε 6 ] ,
(02),
as long as we utilize the notation (of Voigt): σ1 = σ xx , σ 2 = σ yy , σ 3 = σ zz , σ 4 = 2 σ yz , σ 5 = 2 σ zx and σ 6 = 2 σ xy , and analogous representations for the strains. If desired, the dyadic of strains can be written in "engineering notation" and in relation to the orthonormed dyadic base (01), in the forms: ε1 = ε xx , ε 2 = ε yy , ε 3 = ε zz , ε 4 =
2 2 2 γ yz , ε 5 = γ zx and ε 6 = γ xy . 2 2 2
The Law of Hooke Independently of any reference system, the Law of Hooke can be poliadically expressed in the form ε = 4F : σ ,
(03),
where 4F represents the symmetric tetradic of flexibility (or compliance) of the solid. In relation to the orthonormed dyadic base (01), the Law of Hooke can be written in the matrix form
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ε1 F11 F12 F13 ε F22 F23 2 F33 ε 3 ε = 4 ε 5 sym. ε 6
F14 F24 F34 F44
F15 F25 F35 F45 F55
F16 σ1 F26 σ 2 F36 σ 3 . , F46 σ 4 F56 σ 5 F66 σ 6
(04),
or, still, in the corresponding compact matrix form where the matrices have evident meaning: {ε} =[ 4 F].{σ} ,
(05).
Matrix [4F] numerically represents the whole characteristic anisotropy of the solid, not having, in general, any particular form as the adopted reference systems are any. It is evident that if the direction of an axis of elastic symmetry or of an elastic symmetry plain is clear, it will be possible, without damage to the generality of the method, to choose a more convenient reference system (axis coinciding with symmetry axes, for instance), a case in which the matrix will present peculiar features (with many null elements, in the case of the example).
Geometrical interpretation and problem situation In the space of the symmetrical dyadics the Law of Hooke can be interpreted geometrically as a linear transformation of the space of the symmetrical dyadics in its own, which ruler is the flexibility tetradic [2]. From that point of view, the problem we intend to solve consists in determining the operator of that linear transformation knowing the corresponding pairs of stress/strain dyadics in different points of the solid, those dyadics which, as seen, can be determined by "in situ" measurements of normal stresses and elongations, by the SFJ method (H3 hypothesis).
PROBLEM SOLUTION For being more didactic, we will first present the solution of the problem as if the measurements (the data) were the true theoretical measures of the dyadics coordinates; afterwards we shall consider that those measures are not free from the uncertainties that accompany the measurements. In the first case we will still say that the measures are certain or not disturbed, unlike the second, in which they will be said uncertain or disturbed.
Solution with certain measures. Let N be the number (still incognito) of the solid’s points where we determine the corresponding stress/strain pairs of dyadics, σα e εα for α=1,2,...,N. For each value of α, that is, for each point of the solid, we will be able to write the Law of Hooke in form (03), or in form (05) if the dyadics and the tetradic are referred to the dyadic base (01). As, by hypothesis, the measures are correct, tetradic 4F is necessarily symmetric and for any corresponding pair (εε, σ), ε = 4 F : σ = σ : 4 F . For any set of six linearly independent stress dyadics (let those be the first six pairs among the N experimentally established) we can write, as we know [2]: 4
F = εuσu ,
(u=1,2,...,6)
(06),
the σu being the reciprocal (which necessarily exist) of the (independent) σu. (Reciprocal dyadics are dyadics which comply to the rule σ v : σ u = δ vu where the δ's are the deltas of Kronecker). Then,
being
σv : εu = σv : 4F : σu ,
or
σu : εv = σu : 4F : σv
(for
u,v=1,2,...,6),
results
in
σ v : ε u = σ u : ε v . The reciprocal is true, that is, if σ v : ε u = σ u : ε v then F= F . In fact, as from (06) we could 4
write, temporarily adopting the σu dyadics as base: that σ v : ε u = σ u : ε v :
4
4
F = (ε u : σ v )σ v σ u ; then,
4 T
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F T = (ε u : σ v )σ u σ v , or, considering
F T = (ε v : σ u )σ u σ v . Changing, in that expression, u for v and v for u (which does not alter
the sum), we find: 4 F T = (ε u : σ v )σ v σ u , that is, 4F = 4FT. In summary: 4
F = 4FT ⇔ σv : εu = σu : εv ,
(u,v=1,2,...,6)
(07).
We see, thus, that we only need to measure six corresponding pairs (σ σ,εε), with six independent stress dyadic, to determine the tetradic of flexibility.
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Regarding the dyadic base (01), the six equations (05) can then be written, simultaneously, in the following compact matrix form: [Ε] =[ 4 F].[Σ ] ,
(08),
where, remembering the form (04),
ε11 ε 21 [Ε] = εε 31 41 ε 51 ε 61
ε12 ε 22 ε 32 ε 42 ε 52 ε 62
... ε 1i ... ε 2i ... ε 3i ... ... ... ε 6i
... ε16 σ11 σ 21 ... ε 26 σ ε 36 , and, analogously, [Σ ] = 31 ... σ 41 ... ... σ 51 σ 61 ... ε 66
σ12 σ 22 σ 32 σ 42 σ 52 σ 62
... σ1i ... σ 2i ... σ 3i ... ... ... ... ... σ 6i
... σ16 ... σ 26 σ 36 , ... ... ... ... σ 66
(09),
the i-th column of [E] representing εj, the i-th column of [Σ Σ] representing σj. As the σ's are independent, the [Σ Σ] matrix is invertible. From (08) we obtain, then, the expression of [4F]: [ 4 F] = [Ε].[Σ ] −1 ,
(10).
If we had N > 6 pairs of measures (within which there were at least six pairs with independent stress dyadic), matrices (09) would have N columns each. In this case, we would post-multiply both members of (08) by [Σ Σ]T, we T Σ] is six) and we would simply write, would invert the [Σ Σ].[Σ Σ] product (which, in fact, is invertible because the rank of [Σ [ 4 F] = [Ε ].[Σ ] T .([Σ ].[Σ ] T ) −1 ,
(11).
Solution with uncertain measures
In that case, what really happens in practice, (07) not necessarily subsisting, the [4F] matrix given by (10), or by (11), may not be symmetrical (in general it is not). As it is not fit to discuss the applicability of the CHALE model to the solid (H2 hypothesis), nor the measurement method (H3 hypothesis), we should consider responsible the uncertainties, present in the measurements, for the asymmetry of [4F]. We have two ways to follow: 1) – admit the asymmetry of tetradic 4F calculated by (10) or (11). In that case, we must accept the calculation of the strain dyadic by the ε = 4 F : σ law when a σ dyadic (measured) is given, or the calculation of the stress dyadic by the inverted law σ = 4 R : ε when the ε dyadic is given, enabling to prove the existence of 4R (which will be the inverse of 4F whenever the state of stress is tridimensional). This second hypothesis could be the most frequent one once the determination of the strain dyadic of a point by the STT cell is relatively simple and cheap; 2) – Try to evaluate the most significant uncertainties of the measurements and to then calculate, by (10) or (11), the certain 4F tetradic, still not symmetric, which will exist with a certain variance, but “not very asymmetric”. We will next do some considerations about this problem.
Regarding measurement uncertainties.
The most complicated situation for analysis is the one in which the coordinates of a measured dyadic, denoted by εmeas, would be affected in various grades (or weights) by each one of the coordinates of the certain dyadic, ε. In that case, we must admit the existence of a “disturbance tetradic”, 4D, exhibiting a 6x6 full matrix, which transforms the certain dyadic, ε(0), into the uncertain: ε meas = 4 D : ε (0) . Although this tetradic may exist, its determination may be a problem of doubtful solution (if it exists) and, probably, of little reliability in practice. Let us symbolize by 4D(εε) and 4D(σσ) the disturbance tetradics (complete) for strains and stresses, respectively; and by X the uncertain tetradic of flexibility. It is true that tetradics 4D(εε) and 4D(σσ) are not scale tetradics (A4I type), for if they were, 4F and 4X would be directly proportional (parallel) and 4X would be, at least, always symmetric; which does not usually happen. If the measured dyadics, εu, σu, were directly proportional (parallel) to the corresponding certain dyadics ε (0)u , σ (0)u (each one with a proportionality factor) – let us say: 4
ε 1 = A 1 ε (0)1 , ε 2 = A 2 ε ( 0) 2 , ...
and
σ 1 = B1 σ ( 0)1 ,
σ 2 = B 2 σ ( 0) 2 , ... ,
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even so it will always be 4X = A1/B14G = A2/B24G = ..., that is, 4X would be at least symmetric (apart the relation A/B having to be constant); which, also, is not always true. Less complicated for analysis, and perhaps closer to reality, would be the situation where each coordinate of the measured dyadic was the homonymous coordinate of the certain dyadic affected in a certain degree. This means that any of the disturbance tetradics previously considered would be diagonal with respect to the work base (whichever base it were). Although that situation is substantially simpler than the one initially considered, the problem is still complex because that tetradic would have six coordinates and its determination (if possible) may not be obtained in an easy way. Let us discuss that approach a little more. Let us calculate (the non symmetric) 4X, with the uncertain dyadics, by the same expression (11) with which 4F is calculated with the certain dyadics. If 4D(εε) and 4D(σσ) would independ of the point where the measurements are made, the law of proportionality (Hooke) between the measured stress and strain dyadics, ε meas = 4 X : σ meas , could be written as follows: 4
D (ε ) : ε ( 0) = 4 X : 4 D ( σ ) : σ ( 0) ,
(12),
where the usage of parentheses in the second member is irrelevant. Replacing ε(0) in function of σ(0) in (12) (according to the Law of Hooke), we have: 4
D (ε ) :
4
F : σ ( 0) = 4 X : 4 D ( σ ) : σ (0) ,
or, being σ(0) any dyadic, 4 D (ε ) : 4
4
(13),
F = 4 X : 4 D ( σ ) . We finally obtain:
F = 4 D ( ε ) −1 : 4 X : 4 D ( σ ) ,
(14).
Expression (14) shows, then, how to obtain the certain flexibility tetradic knowing the disturbance tetradics of the certain stress and strain dyadics (tetradics independent of the point) and the uncertain flexibility tetradic 4X (which can be calculated by identical expression to (11)). The "disturbance factors" of the coordinates of the strain dyadic can be represented by 1+ai for i=1,2,...,6, where the ai are positive or negative random numbers (let us say, less than 20%); the factors of the stress dyadic coordinates are represented by 1+bi (the bi also being positive or negative random numbers, let us say less than 27%). As the disturbance tetradics are diagonals, they can be written as follows: 4 D (ε ) = 4 Ι + 4 α and 4 D (σ ) = 4 Ι + 4 β , where the diagonal tetradics 4α and 4β have evident representations. We can, now, write (14) in matrix expression: [ 4 F] = ( [ 4 Ι ] +[ 4 α ]) −1 . [ 4 X] . ( [ 4 Ι ] +[ 4 β]) ,
(15).
Being [4Ι]+[4α] a diagonal matrix and (1+a)-1=1-a+a2-a3+..., we can write (15) as follows: [ 4 F] = ( [ 4 Ι ] −[ 4 α ] +[ 4 α ] 2 −[ 4 α ] 3 + ...). [ 4 X] . ( [ 4 Ι ] +[ 4 β]) ,
(16).
As the ai are small numbers, their squares can be neglected with regards to themselves (0,22=0,04). In that case, we can give 4F an approximate expression in relation to 4α and 4β , as follows: [ 4 F] = ( [ 4 Ι ] −[ 4 α ]). [ 4 X] . ( [ 4 Ι ] +[ 4 β]) ,
(17),
or, also [ 4 F] = ( [ 4 Ι ] +[ 4 β] −[ 4 α ]). [ 4 X] −[ 4 α ]. [ 4 X].[ 4 β] ,
(18).
Expression (16), or its approximations (17) and (18), permit to calculate a value for 4F as long as 4α and 4β (which, by hypothesis, do not depend on the point) be given with some precision.
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FINAL CONSIDERATIONS. Way 1) – acceptance of the asymmetry - indicated in item “solution with uncertain measures”, can have its usefulness in works developed in a semi-empiric way. In these works, the Law of Hooke being known, the stress dyadic in a point of an excavation site, for instance, can be determined as long as the strain dyadic is measured in that point (let us say, using the STT cell method). The reliability of those results, in approximate terms, seems to be acceptable in the practical point of view.
Way 2) – evaluation of the most significant uncertainties - is necessary when a theoretical study of the behavior (global) of the solid using the Linear Theory of Elasticity is made, as the use of a tetradic, not symmetrical in the Law of Hooke, could bring some disturbance to the study. We know by the Theory of Elasticity that the strain energy density function (Green) existence in each point causes the symmetry of 4F. The counter-reciprocal of this proposition must be true: the asymmetry of 4F causes the non-existence of the strain energy density function, which, apparently, is an absurd, as we recognize the existence of energy stored in a solid. As (16) shows, to reach 4F from 4X, the evaluation of the disturbance tetradics is necessary, a not yet satisfactorily resolved problem. If we only have 4α and 4β estimates, we do not obtain more than one better estimate for 4 F, let us say 4F´, which, even so, is not symmetric. But 4F´ will not present a "strong asymmetry" as, by hypothesis, its main causes components of asymmetry have been eliminated. In that case, can the symmetric part of 4F´ be a reasonable estimate for 4F?. By the two pointed ways we will try to obtain compatible and coherent results with information already legalized by other means regarding the solid of the Serra da Mesa power plant. A great series of measurements of the stress and strain dyadics is already under way, since May/2003, in an instrumentation gallery especially dug in the solid of that plant during its construction. Those studies will be subject of future communications. ACKNOWLEDGEMENT All gratitude to the CENTRO TECNOLÓGICO DE ENGENHARIA CIVIL of FURNAS CENTRAIS ELÉTRICAS SA. BIBLIOGRAFICAL REFERENCES [1] – MASE, G. E., Theory and Problems of Continuum Mechanics, Mc Graw-Hill Book Company, New York, 1970 (ISBN 07-040663-4). [2] – RUGGERI, E. R. F., Fundamentals of Polyadic Calculus, under way.