Constitutive laws for strong geometric non-linearity

Constitutive Laws for Strong Geometric Non-Linearity

Luiz Bevilacqua

LNCC/MCT-Laboratório Nacional de Computação Científica. Av. Getúlio Vargas 333-Quitandinha-

25651-070. Petrópolis.RJ. Brazil

[e-mail]

This paper is divided into two different parts. The first one provides a brief introduction to the fractal geometry with some simple illustrations in fluid mechanics. We thought it would be helpful to introduce the reader into this relatively new approach to mechanics that has not been sufficiently explored by engineers yet. Although in fluid mechanics, mainly in problems of percolation and binary flows, the use of fractals has gained some attention, the same is not true for solid mechanics, from the best of our knowledge. The second part deals with the mechanical behavior of thin wires subjected to very large deformations. It is shown that starting to a plausible conjecture it is possible to find global constitutive equations correlating geometrical end energy variables with the fractal dimension of the solid subjected to large deformations. It is pointed out the need to complement the present proposal with experimental work.

Introduction.

The various approaches for constitutive laws involving large displacement and large strain, as a matter of fact, try to impose some reasonable upper bound to the quantity that is considered to be large. In some cases, however, it is not easy to establish a precise value to define how large is large. In this paper we consider the case of very strong geometric non-linearity. We are talking about configurations similar to packing a wire in a box in such way that it can be considered almost as a 3-D or 2-D solid (fig 1).

We believe that the classical approach used in solid mechanics to correlate strain and stress, even considering the well known corrections for the strain and stress tensors to take into account the lack of proportionality is not adequate in this case. The classical definition of the Euclidean metric to measure the kinematics of the deformation when this type of structure is under external loads is simply not practical. We mean that there must be easier ways to deal with this kind of geometry, from a global (non-local) point of view.

The use of fractal geometry has been applied successfully to investigate the structure of fracture propagation in two-dimensional solids see Borodich. Instead of making a time consuming analysis, following the fracture trajectory and possible bifurcation according to the random characteristics of the material microstructure, it is much simpler to investigate the fractal dimension that adequately represents the fracture pattern. The focus on the geometry implies in overlooking certain local variables. There is a trade off between local mechanical variables – stress and strain – and the global displacement or fracture paths. The use of a mixed approach is probably the more convenient. The geometry is better to figure out the fracture propagation pattern and once this pattern is determined the local variables can be calculated at the selected locations. The fractal geometry has been also successfully applied in geophysics to investigate fracture and surface characteristics of rocks and faults, see Turcotte.

As far as constitutive relations in solids are concerned, to the best of our knowledge, the subject has not been dealt yet in the current literature. We will try to essay an approach that could be more adequate relating global variables that are meaningful to the problem. The purpose here is to present a possible way to deal with this kind of problems starting from a plausible conjecture that needs to be tested and validated. In this sense this paper is provocative, intended to open the discussion about the theme and analyze the consistence of the fundamental hypothesis and the respective results.

To be concrete, we will concentrate our attention in finding what is the energy necessary to pack a wire in a flat box, in such a way that in the limit it could almost fulfill the plane confined between the two larger parallel sides of the box.

Before coming to this topic, it is worthwhile presenting an application of a non-Euclidean geometry to a fluid mechanics problem. The intention is to illustrate what a global constitutive law means.

Cluster Fractal Dimension in Fluid Mechanics.

The fractal geometry, see Feder, has been successfully used in several branches of mechanics. Particularly in fluid mechanics problems see Falconer and Feder, like percolation and fingering in viscous flow through porous media (Fig. 2) to give two examples. The fractal geometry is convenient to deal with phenomena where clustering is an essential characteristic.

Consider a random porous medium confined between two parallel plates (Fig. 2). A very viscous fluid initially at rest fills in the porous medium. Through a hole located at the bottom plate, air is injected displacing the viscous fluid in the medium.

It can be shown that instability occurs at the interface air-glycerol and the air front expands without preserving polar symmetry. Instead, the instability introduces a perturbation in the geometry of the interface inducing the fingering shown in the figure 2.

Instead of developing a local analysis of the problem, it is more convenient to deal with global characteristics if we are interested in global effects. For instance, it is possible to correlate the reduced radius R/Rg with the normalized mass or number of particles N/N0 encompassed in a circle of radius R. This correlation is only meaningful if a fractal geometry is introduced to measure the density of the set. A set of particles linked together along a line will have dimension 1 coinciding with the topological dimension of the R1. On the other hand if the set of particles fills in completely the plane, a two dimensional object is obtained coinciding with the topological dimension of the plane R2. Now the fingering shown in the figure 2 has an intermediate dimension etween 1 and 2. This fractal dimension indicates how densely the branches fill in the plane; it is called the cluster fractal dimension. Given the fractal dimension of this set, the volume is uniquely determined and for several purposes that is enough, the local characteristics, like velocity and pressure being irrelevant.

The introduction of a new metric allows the treatment of phenomena like fingering in binary flow trough a porous medium from a global point of view. The length of a particular branch, the conditions under which bifurcation occurs, local properties are not under consideration. The effort is oriented towards finding a proper cluster fractal dimension for the branching that appears when the injected fluid displaces the fluid initially at rest in the porous medium. It can be shown that the clustering of particles that follow a random walk trajectory and stop after touching a branch can simulate the spreading of the branches and the multiplication of the bifurcation points.

The accumulation of particles ( monomers) in one, two or three-dimensional space can be measured usually by:

(1)

Where r is a quantity depending on the geometry. Indeed, take a set of cubic particles, monomers, (R0 x R0 x R0). If the monomers are arranged along a line, with length R, the total number of particles covering this line is given by the expression (1) above with r = 1 and D = 1. If the monomers are arranged on a plane covering a square R x R, the number of monomers occupying the region is again given by (1) with r = 1 and D = 2. For a volume R x R x R it is found similarly r = 1 and D = 3.

Now natural phenomena not always follow this simple rule. The clustering of particles, simulated by random walk paths, that are typical in the case discussed here, is characterized by a geometry with a non-integer dimension D. The complete discussion of this problem is out of the scope of this paper. The reader can find a detailed exposition in Falconer for a mathematical approach or in Feder if the main interest is on applications. We will only show how this dimension appears in a global law.

If we plot Y= log (N(R)/ N0 ) vs. X = log(R/Rg) as depicted in the figure 3, where:

N(R): number of "particles" composing the branches or fingers within a region inside the circle of radius R

N0: total number of particles, that is, the number of particles composing all the branches inside the region limited by the circle of radius R0

R: an arbitrary radius with origin coinciding with the point of injection

Rg: the reduced radius of gyration corresponding to the portion of the branches within the circle of radius R, defined as:

Clearly in the interval -1.5 < X < 0 one finds:

(2.a)

or

(2.b)

where the exponent D is called the cluster fractal dimension of the fingering formation.

This correlation would have a minor importance if it would be valid just for a particular binary flow like air-glycerol. The interesting result however is that this relationship holds for a large spectrum of combinations of two fluids in a binary flow. This means that equation (2-a, b) is a constitutive equation for a binary flow in a porous medium displaying instability at the interface. It correlates the relative mass contents N/N0 within a region bounded by a circle of radius R and the reduced radius R/Rg.

This simple example illustrates the importance of a convenient metric to deal with a non conventional geometry that is more appropriated to correlate two variables characterizing some global aspect of a physical phenomenon.

The Fractal Representation of Nature: an Example

Before attempting to develop a theory to deal with very large displacements, as the case of a wire packed in a box, it will be helpful to explore in a simple example how the fractal geometry can be associated to the measurement of natural objects. For the readers interested in a detailed exposition of the fractal geometry we suggest the books by Falconer with a mathematical approach or Feder oriented more towards applications. A broader and very nice overview can be found in Schroeder.

One way of interpreting the fractal geometry is as a correction to the Euclidean geometry in order to make it better fitted to represent the real world. The set of real numbers is too dense to represent the real world. That is, real numbers in some sense are not real. On the other hand, the set of natural numbers while natural numbers is not dense enough to allow for a mathematical representation of nature. We could think of the fractal geometry as a kind of "desaccumulation" mechanisms over the Euclidean metric in the Rn, or a kind of accumulation mechanism over the set of the natural numbers.

We recognize that this is a kind of loose definition and intuitive interpretation of which is called fractal geometry, nevertheless it is quite motivating. It is worthwhile illustrating the ideas exposed above with a classical example. Suppose that we are interested in finding the length of a coastline or the boundary between two countries. Take the coastline of figure 5. Depending on the size of the rule used to measure the coast length between A and B, different values are found.

Plotting Y = log (L) vs. X = log (d), where L is the total length obtained using a rule of size equal d, the result is a straight line as shown in the figure 5. It is easily found that:

(3.a)

or

(3.b)

The constant b is a scaling factor. It is equal to the total length for the unit reference length, d = 1.

This is an expected result due to the irregular geometry of the contour and given that the straight rule of size d will not perfectly coincide with the piece of the coastline spanned between the two points coinciding with the extremities of the rule. It is also consistent to have the total length being approximated from bellow, that is, the smaller the rule the better the approximation.

If d is small enough the result obtained by adding up all the quantities obtained in each step following the coast line will eventually converge to the real length L0. This means that the straight line in figure 5 will change curvature for some d and become ultimately a segment parallel to the horizontal axis. Along this portion clearly the equations (3-a, b) are not valid anymore.

In general we can write:

LN = NdN (4)

That is, a rule of size dN would take N steps to cover the coastline. For dN sufficient small or N sufficient large, which is the same, dN @ L0/N and LN @ L0 .

The puzzling feature of this experiment is that for a smooth curve the horizontal portion of the curve representing log (L) vs. log (d) starts at a much larger value of d. So irregular curves are much more sensitive to the size of the rule then smooth curves. We may characterize the degree of regularity of a line by the exponent d appearing in the equation (3-b). It is however better to choose another parameter D = d+1 called the fractal dimension. So very large d and consequently D characterize large irregularities, d=0 or D=1, on the other hand are strictly valid for the straight line R1.

The question now arises, whether or not it is possible to represent the coastline by the straight line with angular coefficient d. That is, is it possible to characterize the geometry of a curve by this parameter D? A difficulty comes immediately from (3-b). Clearly the length L calculated according to this equation goes to infinity as d tends to zero. This limit value is not true for natural objects, and in this sense the representation fails.

Let us set temporally aside the coastline example and take a closer view on the geometry of irregular curves. Consider the sequence of curves constructed as shown in figure 6 (Feder, 1988) . The figure displays only the two first elements. The others are obtained using the law of formation easily deduced from the figure. The Koch curve is defined as the curve obtained when k® ¥. Now take the same technique used in the case of the coastline to calculate the length of the Koch curve. Successive sizes of the reference rule d will give out correspondent values for L (d). Plotting L against d a straight line with angular coefficient -d0 = D0 - 1 is obtained as shown in figure 5. In this case the representation is geometrically consistent since the length of the Koch curve measured in accordance to the Euclidean geometry from equation (4) reads:

(5.a)

Clearly Lk® ¥ as k® ¥.

So the Euclidean length of the Koch curve grows without limit and this result can found both with equation (3-b) or (4). We may therefore write:

(5.b)

From which follows as k® ¥:

This is the fractal dimension of the Koch curve of figure 6 and the associated angular coefficient -d₀ = 1- D₀ = 1 - log (4) / log (3) .

For a generalized Koch curve with l(1) = 1/q and N (1) = p we can write in general:

l(k) = (1 / q)k and N(k) = pk

and the fractal dimension reads:

(6)

It has been seen that Koch curves are consistent in the sense that the length given by (3-b) or (4) are equal. The length however grows without limit and we would like to have it finite. It is to say it is necessary to assign a proper metric to the curves of the Koch type in order to have a finite measure.

Let be a metric with measure in the mathematical sense, defined by M (Feder 1988; Falconer 1990):

(7)

Note that l is a fraction of the unit since the length assigned to the segment defining the initiator of the Koch curve was equal to 1. The exponents d' and d'' must make M finite but not zero. Let:

(8)

The only possibility to have Mk finite and different from zero as k® ¥ is that:

From which follows:

(9)

This expression for the fractal dimension D0 coincides with value given by (7), which was obtained starting from a different assumption. Choose d'' = 1, and get:

(10.a)

For d' = 1:

(10.b)

Returning to the coastline contour, we can say that the coastline of figure 5 has a fractal representation with dimension D = d + 1, where d is obtained form the figure 6.

The length of the corresponding fractal representation is finite provided that the metric introduced above ( Hausdorf - Besicovich ) is assigned to measure the length of the set.

Comparing with the Euclidean geometry this approach introduces a "correction" in the size of the segment l(k) or alternatively in the number of segments N(k) through the exponents d' or d''

Confined Wires : in Application

Let us turn now to the analysis of the fractal representation of relatively thin wires densely packed in a flat box. This configuration has a clear and natural correspondence with Koch type curves described in the previous section. The generalization of the theory presented in the sequel, to deal with, maybe, more important problems in mechanics, is not difficult.

Consider the following experiment. In a flat box with predominant dimensions BxB a flexible wire with diameter f is introduced trough a small hole. Assume that the wire is forced into the box to the point that there is no more room available inside and no further penetration is possible. The wire is then densely packed in the box.

Performing this hypothetical experiment for wires of the same material with different diameters f and different box sizes B, and plotting the total wire length forced into the box versus the wire diameter f, the curves shown in the figure 8 represent a possible expected behavior.

This conjecture is coherent with the basic geometric relationships assumed in the experiments. By inspection of the curves in the figure it is possible to arrive at the following conclusions:

i)- The curves are meaningful only for diameters f < B

ii)- In the limit as f ® B the corresponding wire length L that would possibly fit into the box is equal to B

iii)- For sufficiently small diameters relatively to the box size B, the length of the wire that could be forced in the box increases as the diameter decreases

iv)- For sufficiently small diameters relatively to the box size B, the rate of variation of the length L packed into the box with respect to the diameter is independent of B.

All these statements are fairly reasonable. There is no violation of any existent geometric relationship among the variables involved in the experiment. The last statement, iv) could be relaxed in order to obtain a more general theory. That is, the straight portion of the curves could fail to be parallel. For sake of simplicity, we will assume parallelism in what follows. The more general case doesn't present any serious analytical difficulty, but will not add any important contribution at this stage. Also for sufficiently small ratios f /B the hypothesis that the straight portion of the curves run parallel is more plausible.

We will introduce now a further simplification. It is expected that for f /B < < 1, the prolongation of the straight portion of the curves will be very close to OQ, and we put si = 0. Then for a given Bi it is possible to write:

(11.a)

or

(11.b)

That is introducing the normalized variables L/B and f /B the geometric behavior of the packing process can be represented by just one curve, for the same material as shown in the figure 9. Of course different materials will display different angular coefficients d. We will use dK for the angular coefficient to distinguish materials with different mechanical properties K.

Equation (11-b) can also be written under the form:

(12.a)

or using the fractal dimension DK as discussed before

(12.b)

For the geometry of the present case DK varies between 1 and 2. DK = 1 coincides with the topological dimension of a regular curve embedded in R1 and DK = 2 coincides with the topological dimension of the plane R2. DK close to 2 implies that the wire is densely folded and will cover the plane almost completely.

It is also plausible to expect that "soft" materials will be easier to fold than "hard" ones, displaying consequently a higher covering density of the plane. This means that DK is higher for "soft" materials than for "hard" materials. This translates into the energy density for different materials and geometries. The total strain energy retained in the wire after packing can be expressed as:

(13.a)

or

(13.b)

where is the average elastic component of the stress tensor along L, K and Kp are material constants and ^N_k the number of corners along the folded wire. For k sufficiently large this number will be approximately equal to pk, the number of elements in the kth pre-fractal term of the sequence representing the fractal curve. It is also possible to write, for a sufficient large k:

(14 .a)

where is the average size of the straight portion of the wire between two consecutive folds. It is reasonable to expect that is proportional to some power of B and of 1/f , as well. That is, big boxes will accommodate a greater number of folds. It is also possible to write:

(14.b)

The exponent "m" can be associated to a material property. For a same ratio f /B , the average size of depends on the material ductility. This interpretation will become apparent in the sequel. Combining expressions (14-a) and (14-b) we obtain:

(15)

Introducing expression (15) into (13-b) we obtain:

(16.a)

and recalling that when f ® B, ® 1 it is easily found that a =1/B and the expression above gives:

(16.b)

The first term within brackets in the right hand side of this equation represents the elastic energy and the second the energy due to the plastic deformation. It is now clear that the exponent "m" defines the portion of plastic energy stored in the wire. It is possible to distinguish three cases:

m>1: the plastic energy prevails and the elastic energy can be disregarded. Recall that (B/f )>>1.

m<1: the plastic energy stored in the wire is much smaller than the elastic energy.

m=1: the elastic and plastic energy stored in the wire are equivalent.

It is remarkable that the exponent "m" determines in a natural way the strain energy partition in its plastic and elastic portions.

In most of the cases we will be interested in ductile wires that store strain energy under plastic deformation, that is m>1. For this case

Disregarding then the first term within brackets in (16-a0 and using equation (12-b) we find:

(17)

The energy density per unit volume is now given by:

(18)

In a similar way, for sake of completeness, it can be found:

For m<1:

(19)

For m=1

(20)

The figure 10 shows E 3, for m>1,as function of f/B. Note that for a consistent correlation between L and ^b_k ^{= 4 - m}_k ^{- D}_k, it is required that b k >0. This means that the parameter Mk is limited. Indeed, mk <4-Dk.

This conclusion comes from the fact that L® ¥ as f/B® 0 or log (L)® ¥ as log (f/B)® -¥ in the equation (18) written in a log-log scale:

(21)

The specific energy has therefore a fractal dimension bk. this dimension depends on the material properties. The material with a characteristic parameter b1 is softer than the material b2. From the figure 10 we may conclude that for the same normalized diameter, softer materials store less energy than hard materials, which seems to be consistent.

For the cases m<1 and m=1, the results are similar.

Conclusion

We have tried to show in this paper that it is plausible to expect a fractal representation for thin wires packed into a flat box. The associated fractal dimension is probably a function of the material properties. It is expected "soft" wires to be more densely packed than "hard" wires. The specific energy - strain energy per unit volume of an ideal box BxBxB - varies with the normalized diameter f/B according to a power law, that is different for ductile materials and fragile materials. The specific energy for hard or fragile materials is characterized by the fractal dimension Dk while for ductile materials another parameter comes into play namely mk.

For both cases we come to the conclusion that the normalized specific energy presents also a fractal character. We believe that the theory is consistent and arrives to a global constitutive equation relating specific strain energy and the normalized wire diameter with respect to the characteristic length of the box.

Extensions of the theory for three-dimensional boxes and to other types of structures, like membranes, are not difficult. But we believe that it is maybe premature. It would be more useful to perform at least some simple experiments in order to test the theory. All the experiments referred to in this paper are hypothetical and are only good to give some hints for the preparation of real ones. They were also helpful to avoid serious mistakes or contradictions.

Finally as it was stated in the beginning this is provocative paper to open the discussion on how to derive global constitutive equations for cases where the classical Euclidean geometry fails to provide a convenient or practical metric, other kind of measure being preferable. Within this class of problems we would like to highlight the cases of biological configurations among others. The lung, for instance, has a geometry that is very much appropriate to be dealt within the context of the fractal geometry. To the best of our knowledge the dimension of associated fractal geometry representing the lung is not yet available. The fractal approach to problems in solid mechanics has not been extensively explored yet.

Literature

Manuscript received: May 1999. Technical Editor: Hans Ingo Weber.

• Feder J.; "Fractals", 1988, Plenum Press, New York and London, 283p.
• Falconer K., 1990, "Fractal Geometry: Mathematical Foundations and Applications", John Wile & Sons, Chichester, New York, 287p.
• Schroeder M., 1991, "Fractals, Chaos, Power Laws", W.H.Freeman, New York
• Panagiotopoulos P.D., 1990, "On the Fractal Nature of Mechanical Theories", Zeitschrift für Angewandte Mathematik und Mechanik, vol 70, pp. 258-260
• Panagiotopoulos P.D., 1991, "Fractal Approximation in the Theory of Elasticity", Zeitschrift für Angewandte Mathematik und Mechanik, vol.71, pp. 658-659
• Borodich F.M., 1997, "Some Fractal Models of Fracture", Journal of Mechanics and Physics of Solids, vol 45, pp 239-259
• Turcotte D.L., 1997, "Fractal and Chaos in Geology and Geophysics", Cambridge University Press, 390p.

Publication Dates

• Publication in this collection
  15 Dec 2000
• Date of issue
  2000

History

• Received
  May 1999