Linear elasticity of planar delaunay networks. Part II: Voigt ... - CiteSeerX

Aeta Meehaniea 84, 47--61 (1990)

ACTA MECHANICA | by Springer-Verlag 1990

Linear elasticity of planar Delaunay networks. Part Ih Voigt and Reuss bounds, and modification for centroids M. 0stoja-Starzewski and e. Wang, West Lafayette, I n d i a n a (Received July 26, 1989; revised January 29, 1990)

Summary. The linear elastic Delaunay network model developed in a previous paper is used to obtain further results on mechanical properties of graph-representable materials. First, we investigate the error involved in the uniform strain approximation -- a computationally inexpensive approach widely employed in the determination of effective moduli of granular and fibrous media. Although this approximation gives an upper bound on the macroscopic moduli, it results in very good estimates of their second order statistics. In order to derive a lower bound another window definition has to be introduced. Also, an energy-based derivation of both bounds is given. The final result relates to a modification of a Delaunay network so that its vertices correspond to the centroids of cells of the corresponding Voronoi tessellation; an increase of effective moduli and a decrease of their scatter are observed.

1 Introduction I n our recent paper [1] (henceforth referred to as P a r t I) we investigated continuum rand o m field approximations of effective moduli of linear elastic D e l a u n a y networks. The computational m e t h o d a d o p t e d was quite rigorous, but very costly. Only networks numbering not more t h a n 900 vertices eould be analyzed within reasonable c o m p u t a t i o n a l times of the Gould PN9080 and Cyber 205 computer systems available to us. On the other hand, there exists a possibility of using a uniform strain approximation, t h a t is, assuming a spatially homogeneous deformation field within a window 8 X 8, d is the ratio of scale L = 1/A-V of an approximating Representative Volume Element ( a V E ) A V to the average microsea!e d. This assumption results in a straightforward calculation of effective moduli e(x, co, 8). I t can handle networks with m u c h larger numbers of vertices. I t should be noted t h a t this uni/orm strain approximation is widely used in mechanics of granular materials (see e.g. [2, 3]), and in mechanics of polymer chain materials (see e.g. [4]). Thus, there arises a question concerning the correctness of this assumption and another one concerning the possibility of calculation of effective moduli using a uni/orm stress approximation. Our I)elaunay network model presented in [1] is naturally suited to answer these questions for a class of materials where the interactions between the neighboring vertices are in normal direction and linear elastic only. This investigation forms the m a i n goal of this paper. I t is well k n o w n t h a t the vertices of a D e l a u n a y network are not the eentroids of Diriehlet regions (cells) of its dual Voronoi tessellation. Hence, it is desirable to determine the difference in statistics of effective moduli of the original D e l a u n a y networks and the De-

48

M. Ostoja-Starzewski ~nd C. Wang

launay networks corrected for the centroids of cells. The latter networks are considered a better representation of interactions in certain granular media. This analysis is conducted using the structural mechanics method of P a r t I.

2 The uniiorm strain approximation The ansatz for the passage from a real discrete material microstructure to an equivalent continuum can be stated by postulating that, in the elastic response range, the strain energy of the microstructure of volume V 1 W = ~- ~ eFe eAuk

(1)

eEE

equals the strain energy of the continuum of the same volume

W =~

aije~dV.

(2)

V

Summation in (1) is taken over all edges of the graph G within V. Next, one introduces the constitutive laws in both the discrete and the continuum models. Thus, in the first case we have

~F~ = ~k~ eAui

(3)

where ~E~A ~ k i = ~L---T (~x~ -- Cxk) ("xi - - ~x~),

(4)

eE, eA and eL being the Young modulus, the cross sectional area and the length of the edge e -- (~fi) connecting the vertices located at ~x and ~x, respectively. ~or the continuum we have

~i~

=

Ciik~k~.

(5)

Upon introducing the above relations into (1) and (2), respectively, we obtain

W

1

~q) ~AuieAuk

(6)

and W=

1s I Ci~kue~,,e~1dV.

2

(7)

v

The problem of making a passage from the discrete to the continuum model -- t h a t is, of finding an effective stiffness tensor C of volume V -- necessitates a knowledge of the actual deformation field :u of all the vertices in the volume V. I n Section 3 of P a r t I, this problem was solved b y using structural mechanics method and an appropriate method of correlating this discrete deformation field with the strain field. However, all of our results obtained b y this method were limited to windows containing not more t h a n about 900 vertices. The basic reason for this limitation is that the exact method requires a solution of a large system of simultaneous linear algebraic equations (Eq. (7) of P a r t I), which poses very high demands on the computer time. As pointed out in the Introduction, a typical

Linear elasticity of planar Delaunay networks

49

approximation used in determining the effective moduli of discrete media is that of uniform strain resulting in the so-called Voigt bounds 13V. In the following we assess the error involved in using this approximation. The uniform strain approximation consists in assuming the displacement of every vertex of the network to conform with a uniform deformation field, that is (8)

Ui = a i j x j 2- b i.

The above relation is a small strain case of a more general formula pertaining to arbitrary strains and time-dependence describing the homogeneous (or affine) deformations [4]. According to (8), the deformation of a single edge of a Delaunay network is (see also [5]) (9)

A~u~ = ei~eLs

where 1

~; =: ~- (a~s + as~)

(10)

are the components of a spatially uniform homogeneous strain e, and ais = u i d . With the substitution of (9) into (6) the formula for strain energy of a discrete microstructure becomes =

1

%b 'L eL

(11)

e

On the other hand, by using the uniform strain approximation in (7) we find 1 W =- - ~ e~.,e 0 VC~skm

(12)

so that 1

w ~q~1,~eL ,, eL S" C~skm = "-~ ~,

(13)

The above is, strictly speaking, an expression for Cis~m(X, co, d), and is used for Monte Carlo-type calculations of the first and second order statistics of the random fields C(x, d). The plots of ensemble averages of thus calculated elastic moduli Cn and Caa (recall the matrix notation for Cis's defined by (4) in Part I) are shown in Fig. 1; the corresponding results for unmodified and modified Delaunay networks obtained in Part I are replotted here for comparison. Evidently, E{C~I} and E { C s , } where C ~ = E{CJ =- (C}

(14)

are independent of d -- a result to be expected in view of the statistical homogeneity of a Delaunay network's geometry and the uniform strain approximation. These calculations give an upper (Voigt) bound on the effective moduli, and in fact are close to the results of structural mechanics calculations for very small window sizes. The latter fact is explained by noting that the smaller the number of vertices in a window, the better is their displacement approximated by the uniform displacement field. It is interesting to make a comparison here with a study due to Bathurst and Rothenburg [2] concerning effective moduli of planar assemblies of discs with linear elastic normal and tangential interactions. Since the relation (9) would produce an overestimate of the

50

~ . Ostoja-Starzewski and C. Wang

actual average strain in an assembly of discs, a p r u d e n t modification was introduced b y these authors for normal (Aeu n) and tangential (Aeu t) relative displacements

deun(O) = ~(eiinini)' Aeut(O) = ~(e~jtini)

(15)

where the vectors n and t are coincident and orthogonal to the vector L joining the centroids of two contiguous discs, and ~ < 1 is the co-called stiffness reduction coefficient. The latter inequality was verified in [2] b y a computer simulation of an assembly of 1000 discs, and - - 0 . 9 5 and 0.93 were obtained for normal and tangential interactions, respectively. However, in view of our Fig. 1 it is inferred immediately t h a t ~ is not a constant h u t rather a decreasing function of ~ levelling off with (3 --~ c~. Formula (13) was used as a basis for the calculation of second order statistics of CH and C2~. Figure 2 shows the dependence of the coefficients of variation of these moduli on ~, while the variation of the root mean square value of the anisotropy index (defined b y (15) of P a r t I) with ~ is shown in Fig. 3. I n both these figures the results obtained in P a r t I are replotted for comparison. The autocorrelation functions Rc~, Re~ ~ and R e , for window size ~ ~ 2 are shown in Figs. 4a, 4b and 4c, respectively. The similarity of these graphs to those of Figs. 9a, 9b and 9c of P a r t I obtained b y the exact m e t h o d is striking. This similarity in autocorrelation function graphs was observed also for other window sizes. Thus we arrive at a very important result for planar D e l a u n a y networks, which m ~ y a p p l y to certain other r a n d o m microstructures as well: second order characteristics of r a n d o m fields of effective moduli can be obtained using the uniform strain approximation.

ECll and EC33 60-

o9

~

o

e

o9

o9

X

X

X

X

X

X

0

0

[]

0

N

50-

[]

[] ECll vs. 5, Delaunay network A EC33 vs. 8, Delaunay network

40-

0 E C l l vs. 5, Delaunay network,uniform strain V EC33 vs. 8, Delaunay network,uniform strain X ECll vs. 8, modified Delaunay network o EC33 vs. 5, modified Delaunay network * ECll vs. 8, modified Delaunay network,uniform strain + EC33 vs. 5, modified Delaunay network,uniform strain

30-

0

u0

70

u0

70

70

A

A

A

/,

A

A

I 8

I 12

I 16

I 20

I 24

10-

0-

I 0

2

I 4

| I 28 30

window edge length, 5 Fig. 1. Dependence of mean elastic moduli on window size

Linear elasticity of planar Delaunay networks

51

coefficients of variation of C l l and C33 .4

D coeff, vari. of C l l , Delaunay network coeff, vari. of C33, Delaunay network 0 coeff, vari. of C l l , Delaunay network, uniform strain V coeff, vari. of C33, Delaunay network, unifort strain X coeff, vari. of C l l , Modified Delaunay network o coeff, vari. of C33, Modified Delaunay network 9 coeff, vari. of C l l , Modified Delaunay network, uniform strain + coeff, vari. of C33, Modified Delaunay network, uniform strain

~Y .3 _

A

X7

.2

.1

o

P

P +

0

i

0

I 8

I

2 4

I

I

i

12 16 20 scale of window, 5

i

24

f

I

28 30

Fig. 2. Dependence of coefficients of variation of C11 and C33 on window size

rms value of anisotropy index of the effective moduli A .8--

A Delaunay network Delaunay network, uniform strain X the Modified Delaunay network + the Modified Delaunay network, uniform strain

.4--

X -t-

.2-

0 0

A []

I

l

2

4

Fig, 8. Dependence of the

I 8

rms

I I I 12 16 20 scale of window, 6

I 24

. I ~28 30

value of anisotropy index on window size

M. Ostoja-Starzewski and C. Wang

52 I.OQ

J

~

12

o o

~)

l .Oa,

.~lOg 9

0.0~1

b)

0

-?

1.0a '

12

""

9~

9

4t

o

a

e)

l~ig. 4. a, b and e (from top to bottom): Autocorrelation functions of Cll, C22 and C33 of the Delaunay network (window scale: ~ ~ 2; units in x and y axes are in terms of the nominal mean cell scale; uniform strain is assumed)

3 T h e u n i f o r m stress a p p r o x i m a t i o n 3.1 Foreword I n analogy to the statistical c o n t i n u u m ?~heories (see [6] a n d [7]) we w a n t to have the relation CR ~ C ~ Cv

(16)

Linear elasticity of planar Delaunay networks

53

in which the inequalities refer to the eigenvalues of Cn, C, and Cv. The i? tensor which refers in [6] to a single crystal would refer here to a single unit of the Delaunay network model. Such a unit is identified in the next subsection to be a single cell of the G' graph (i.e. G' tessellation). The l~euss and Voigt approximations provide the lower and upper bounds if the Hill condition

(~& (~s) = (~,~s)

(17)

holds. The necessary and sufficient conditions for (17) to hold are [6]: i) an ergodic hypothesis can be established, ii) the number of grains is infinite, iii) distributions of volume stress sources, if present, are not correlated to the local elastic moduli. I n our case the above are justified by the following: i) spatial homogeneity of Delaunay networks' statistics, ii) a very large number of vertices in any network G, i.e. d >~ l, iii) absence of forces applied to the internal vertices.

3.2 Determination o/the Reuss bounds At this stage it is natural to ask: can we perform and what do we obtain using a uniform stress approximation? In other words, can we find the so-called Reuss bound? The answer to this problem depends on the possibility of making stress a controllable quantity. We note at once, in analogy to the preceding developments on strain, t h a t the stress has to be controlled at the window boundary as well as in the window interior. The first requirement is analogous to the displacement-controlled boundary conditions used in all our foregoing analyses (Fig. 5 of P a r t I), while the second one is analogous to the uniform strain assumption. Figure 5 b of P a r t I is magnified here in Fig. 5 a, and we observe t h a t the boundary forces F c~) would now have to be applied along the axes of all two-force members of the graph cut b y the window edges. I t is clear t h a t some stress states - - simple shear for instance - - cannot be realized at all. The difficulty lies in our original definition of a wind o w 1. I n order to control the surface stress, a different choice of a window definition (i.e. laVE) has to be made. I n Fig. 5b we present such a choice. The idea is to separate a subgraph of original triangles from the graph G covered b y the square. The boundary forces are now applied at the outer vertices of this subgraph - - i.e. R V E - - or, alternatively, surface displacements are prescribed there. This definition is superior to the previous one in t h a t arbitrary surface forces, and hence surface tractions, as well as arbitrary surface displacements, and hence macro strains, m a y be prescribed. However, its drawback is t h a t the window shape is random, and makes it ambiguous which edges (in Fig. 5b) should be included in this subgraph. For the structural mechanics network thus defined we can definitely obtain the true effective moduli using the same structural mechanics approach as before. Also, the uniform strain approximation can be conducted in the same way as indicated in the previous sec1 l~elative merits and drawbacks of choosing various boundary conditions for circular domains of flowing granular media are discussed in [8].

54

M. Ostoja-Starzewski and C. Wang

(a)

(b)

Fig. 5 a. Window used for the uniform strain calculations

Fig. 5b. Window needed for the uniform stress formulation

tion, let us denote the corresponding Voigt b o u n d s b y @v. Finally, since the surface tractions can be prescribed arbitrarily we can h o p e to find the t~euss bounds e R too. At this stage we note t h a t there is no w a y to prescribe the stress at the scale of an edge e, since, in general, for two-force members

and hence a derivation analogous to the one for uniform strain, thanks to (9), is not possible. Therefore, the concept of a uni/orm stress approximation must be applied at the level of a single cell. Thus, we require t h a t all cells are under the same volume-average stress 1

~(x =

--

ffii

~

2r

(elieFi + ~ljeFi),

V~ E V.

(19)

eENa

B y a cell we mean a cell of the Voronoi tessellation (graph G') corresponding to a given vertex of G. I n (19) el is the vector from the vertex to the midpoint of egde e (recall the construction of G'), ~V is the cell's volume, and N , is the set of all edges incident at c~. I n addition, we require t h a t the force equilibrium conditions (the m o m e n t balance is satisfied trivially) be m e t Z ~

= 0,

i = 1, 2.

(20)

eE~Va

W e w a n t (19) and (20) to be satisfied at every given vertex independently of all other vertices for a n y given 6. Noting t h a t every vertex is under n = INJ forces due to n edges incident there, and t h a t these are two-force members, we see t h a t there is no w a y to satisfy these 5 equations for arbitrary 6 at vertices with n ~ 3 or 4, b u t there is no problem at vertices with n = 5. I n fact, for the vertices of this latter category we determine all the ~F's uniquely, while for vertices with 6 or more edges We cannot do so. Thus we could determine unambiguously the c o m p l e m e n t a r y energy U* at vertices with n = 5 only, b u t this would not work since D e l a u n a y networks have n = 6 on average. Fortunately, there is another and more general, albeit less precise, procedure. We recall from Section 2 t h a t the smaller the n u m b e r of vertices in a window, the better is their displacement approximated b y a homogeneous deformation. This result should be no less

Linear elasticity of planar Delaunay networks

55

t r u e / o r cells t h a n / o r square windows. Thus, we have a continuum approximation for a cell

a~j = Cijkm(~o) Skin

(21)

in which ~ signifies the realization of geometry and physical properties of cell ~, and s is the approximating uniform strain. Cijk,,(~o) in (21) is given by Cijkm(w) = ~

1 --

X~ e~ ki el ,~et j~,

(22)

e~N a

Hence, Ci~.k~((o) is anologous to Cr , ~o, ~) with (~ = 1. Now, if we prescribe the same for all the cells we obtain the Voigt bound (14). This is equivalent to an average Hooke's law

On the other hand, another way to interpret (21) is to consider 6 to be controllable. In that case we are faced with the following problem: find an effective equation for o -that is, find the lower (l~euss) bound R

6i1 = Cijkm(ekm}.

(24)

The solution is CR = (C-~(~)) -~.

(25)

Such a procedure of inversion, averaging and subsequent inversion yields the correct lower bound if the surface traction can be specified unambiguously, as is the case, for example, with polycrystalline materials. Since this is not the case with our Delaunay trusses, this procedure may not give a consistent lower bound. Indeed, this has been observed by us in limited computational experiments on regular Delaunay networks with random spring constants; in that case formula (25) yields moduli weakly dependent on ~, and close to (C(co, ~)} for 0 large. However, numerical results of C, CV, and Ca for the geometrically random Delaunay networks with the redefined window were not calculated due to the lack of resources.

3.3 The e n e r g y / o r m u l a t i o n

A generalization of the above results is obtained by studying the derivation of ttooke's laws from the potential and complementary energies. Thus, we observe that the problem of finding CR is equivalent to the problem of finding the ensemble average Of the relation

eij-

~U*(~)

~

(26)

where U* is a complementary energy of the given cell. It follows from (21) that U* is a random functional on stresses. Assuming sufficiency conditions for commutativity of differentiation and ensemble averaging, the solution is

~U*(o,) I

(27)

in which

(u*(~)): = (~.) ~j.

(2s)

M. Ostoja-Starzewski and C. Wang

56 In case U* is a quadratic form 1

U*(o~) = -~ ~jo~,~ij~(~)

(29)

relation (27) yields (25). We may also consider the problem of finding the l~euss bound from the potential energy under the condition of controllable stresses. Thus we have

~v(~)

(30)

in which U(w) is a random functional on strains. I t can be verified that an effective equation for a~j is

where

U ~I = (e& ~j

(32)

is an effective functional on average strains. Noting (28) we find u ~ = (u*(~o)).

(33)

Similar to the problem of finding the Reuss bounds, the problem of finding the moduli CV under controllable strains is equivalent either to the problem of ensemble average of the relation ~V(o~))

(34)

or to the problem of finding an effective equation for e~ in terms of complementary energy

U*(~)

~U*(~)

(35)

The solution to (34) is (we assume same type of sufficiency conditions as in (27))

in which (U(~o)} --~ e~j(a~i).

(37)

If U(co) is quadratic for every w C ~9, this results in (14). On the other hand, the solution to the second problem (relation (35)) is U * q -=

eii(aij}

(38)

= U *~/.

(39)

so that (U(~))

While the underlying graph in our study was assumed to be made of linear elastic edges, the foregoing development in terms of energies did not require this condition. Rather, the

Linear elasticity of planar Delaunay networks

57

whole derivation relied on the observation t h a t the true displactment field {"u, a ~ V} was very well approximated b y a homogeneous deformation of small windows. Now, using a thought experiment this should also hold true for nonlinear elastic graphs. However, since this paper is focused on linear elastic networks we leave it here as a conjecture. Finally, in the words of Strang [9] "since these nonlinear things are in front of us, why not take the last step?". I n our case it would be the setting up of the Legendre-Fenchel transform for a random situation. Thus we have for every co C/2

U*(a, ~)

= m a x [~: a ( ~ ) -

U(~, ~)]

(40)

g

and U(e, m) ~-- m a x [e(~o): a -- U*(a, w)].

(41)

r

Carrying out the ensemble averaging, we obtain from (40) for the case of control]able strains u*(