Numerical verification of the existence of localization of the elastic energy for closely spaced rigid disks

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1 Nmerical verification of the existence of localization of the elastic energy for closely spaced rigid disks S. I. Rakin Siberian State University of transport Rssia, 6349, Novosibirsk, Dsy Kovalchk street, 9 www. rakinsi@ngs.r The problem of determining the thermomechanical characteristics of the system of closely spaced bodies considered by many athors, see [-9] and reviews on the problem in [, ]. For scalar problems, sch as problems of heat, electrostatic, etc., the localization effect was fond [-6] (so called Tamm shielding effect [6], consists in the fact that in a system of closely spaced highly condctive bodies, most energy is localized in the region between adjacent bodies. We can assme that the localization energy is enogh for a general property of closely spaced bodies. The aim is to stdy the effect of localization of energy in the closely spaced rigid bodies in an elastic medim.. Nmerical verification of localization of the elastic energy Keeping in mind the sstained interest in the problem of the behavior of systems of closely spaced elastic bodies (see [-9] and references in [, ], cold have been expected to have experimental data on the distribtion of local fields in sch systems. However, as in the case of the problem of thermal condctivity [], the athor of sch data cold not be fond. The reason seems to be the same as in the heat condction problem - the difficlty of measring fields in the narrow spaces between adjacent bodies. Therefore, as in [] as a research tool was chosen nmerical method. That is, a nmerical soltion of the problem of elasticity theory sing the program ANSYS. For the elasticity problem of nmerical calclations on ANSYS are highly accrate and can be regarded as the eqivalent of the experiment. Considered next plane problem of elasticity theory. There are two fairly hard disk placed in the relatively soft matrix linearly elastic and close to each other, see Fig.. Disks are moved as solids. The mtal displacement of the two drives on the plane can be represented as the sm of the following basic movements of disks relative to each other, "p-down", "left-right", "rotated on one side", "rotate in opposite directions."

2 a b c d Figre. Location and movement of the disc-inclsion in the matrix The calclations theoretically nlimited matrix replaced by a "sfficiently large" area. Namely, we consider disk of large radis R = 5, at the center of the disk were placed symmetrically arranged hard disks of the radis R = at a distance δ from each other. On the border of a large disk displacement were set eqal to zero. At the bondaries of disks - matrix assme that the conditions ideal contact. The characteristic size of the δ problem is the relative distance between the disks. For closely spaced disks is. The elastic properties of the matrix materials and disks as follows: Yong's modls of the matrix E =, the modls of disks E = (with sch a difference of Yong's modls the disks behave as almost absoltely rigid body, Poisson's ratio ν =. 3. mch less than. In the calclations, the range of variation is [ ] 4 δ ; For each of the basic movements of hard drives solved the problem of the plane theory of elasticity. Moving discs were taken of the same order with distance δ. The displacement along the Ox axis is denoted by. The displacement along the Oy axis is denoted by υ.. Basic movement "p-down" (Fig. a: For one disk =, υ = + δ. For another disc =, υ = δ.. Basic movement "left-right" (Fig. b: For one disk = δ, υ =. For another disc = +δ, υ =. 3. Basic movement "rotation in the same direction" (Fig. c: On the border of the disks the tangential movement ϕ = δ, the radial displacement r =. 4. Basic movement "rotation in different directions" (Fig.d: On the border of one disk the tangential displacement constant ϕ = δ, the radial displacement r =. For another disk the tangential displacement ϕ = δ, radial displacement r =. In the heat condction problem localization is most prononced for the energy and less prononced for the heat flow and gradients [, 3]. In or calclations, we observed a similar sitation: the localization is most prononced for energy, and less prononced for the strain and stress. Localization of elastic energy illstrated with Figs,3,4,5. The calclation was performed for the distance between the disks δ =.. For the nmerical characteristics of localization consider the elastic energy E mat of the whole matrix and elastic energy E in the neck between the discs (neck region was chosen in accordance with Figs,3,4,5. The ratio sh E E sh mat is the proportion of the total elastic energy of the R

3 matrix, concentrated in the neck between the disks (the size of the neck was taken / 3R. Ths, the total energy of elastic deformation of the matrix and the elastic energy of the disks in the neck, for the st, nd and 3rd base movements closely spaced hard disks are practically the same. That is, for these basic movements takes place the asymptotic (asδ localization of the elastic deformation energy of the matrix. Table. The share of total energy matrix entered between the discs in the neck E E sh mat Rotation, different directions left-right Rotation, the same direction p-down δ =.8 3% % 73% 3% δ =.8 3% 84% 8% 83% δ =.4 43% 89% 85% 87% δ =. 44% 9% 9% 9% δ =. 46 % 95% 93% 94% δ =. 5% 98% 98% 98% δ =. 5% 99.7% 99.6% 99.7% Fig. The energy of the elastic deformation nder disk moving p-down 3

4 Fig.3 The energy of the elastic deformation nder the disk moving left-right Fig.4 The energy of the elastic deformation nder the rotation disk in the same direction 4

5 Fig.5 The energy of the elastic deformation nder the rotation disk in different directions Table. Calclated vales for energy of basic movements Distance δ p-down left-right One direction Different directions Different directions / p-down δ =. 9.49е-3 3.е-3 3,9е-3,77е-3 8.7% δ =. 34,67е е е-5.85е-5 5.3% δ =. 4.93е е-7 5.е-7.9е-7.9% For the 4th basic movements - "rotation drives in different directions," a significant portion of the energy is collected in the neck between the discs. However, as can be seen from Table, in the neck it is collected abot half elastic energy, the second half of the energy is ot of the neck. That is, in this case, there is concentration of energy bt no localization (as defined in [3], the concentration of the local field in a region is to achieve a relatively high field in this region, and localization is collection of almost all the fields in this area. Calclated vales of the energy of deformation of the matrix (see Table show that for the 4th base movement, the energy is small compared with the energies corresponding to the st, nd and 3rd base movements (see the last colmn of Table. In this connection it can be neglected. 5

6 In the calclations presented in Table, the basic movements of movements in the "left-right", "p-down" assmed to be eqal δ. When rotating "one-way", "in different directions" is taken to be moving δ.. Formla for calclating the elastic energy of matrix by moving discs The linearity of the energy problem is a qadratic fnction of the variables,υ, ω. We need to calclate the coefficients of the qadratic form. Remark. In the problem of electrostatics and thermal condctivity [9], the energy is a qadratic fnction of the difference of potentials or temperatres of disks t : E = / ct where the coefficient c is the harmonic (electrostatic capacity. We need to calclate the capacity of analoges for the elasticity problem. By symmetry, the location of drives and symmetry bondary conditions for solving the problem for each of the basic movements have symmetry. It shold be noted that the types of symmetries for the basic movements of the "p-down" and "rotate in the same direction are the same." Used indexes: υ - movement of disks p down; - movement disks the left right; ω - rotating of disks in the same direction. To calclate the total energy is possible to consider only the first three basic movements, and the forth can be neglected. Strain in the neck between the discs between the two disks with arbitrary displacement drive is a linear combination of base strains,,, namely: = υ +. The corresponding energy E = ckl ( υ + ( υ kl kl + kl, where ckl - the elastic constants. Consider an isotropic elastic matrix, for which the non-zero elastic constants are []: c = c = λ +, c = c =, c = c = c = c =, µ λ, µ - Lame coefficients. λ µ The total energy E = E + + E + E3 E 4 6

7 ( Ei - energy region occpied i -th coordinate qarter can be calclated as the sm of the energies in the qarters of the Cartesian coordinate system. The soltions of problems for each type of basic movements have obvios symmetry (anti-symmetry with respect to the coordinate axes. With regard to the symmetry of soltions Ε = υ υ (( λ + µ (( υ + + ( υ + + λ ( + υ ( + σ + 4µ ( + υ Ε = υ υ (( λ + µ (( υ + ( υ ω ( υ + 4µ ( υ ω + λ ( υ + dxdy, dxdy and (also acconting that de to the symmetry of the problem E 3 = E, E 4 = E. Then Ε = ( Ε + Ε = λ(( υ + ( υ (( λ + µ (( υ + + ( υ + + ( υ + ( υ + + ( υ ( υ + + υ υ 4µ (( υ + + ( υ dxdy. When sqaring the sms in the integrands appear as sqares strain and strain pair prodcts. Given the symmetries of the integrals of certain pair prodcts deformations cancel each other and the total energy is expressed by the integral over W - the first qarter of the coordinate system E = ( λ + µ (( υ 4λ + ( υ + ( (( υ ( υ + 8µ (( υ + ( dxdy. + ( dxdy + dxdy + Expanding the brackets in ( and introdcing independent on 7 x, y mltipliers υ, υω, ω, the integral sign, we obtain the following expression for the total energy of deformation of the matrix:

8 Ε = Aυ + Bυω + Cω + D, The coefficients of the qadratic form (3 are given by the following integrals A = ( λ + µ (( + ( dxdy + 4λ dxdy + 8µ ( B = 4( λ + µ ( + dxdy + 4λ ( + dxdy + 6 C = ( λ + µ (( + ( dxdy + 4λ dxdy + 8µ, ( W D = ( λ + µ (( + ( dxdy + 4λ dxdy + 8µ ( dxdy, µ dxdy, dxdy. Note that in the qadratic form (3, which expresses the energy of deformation of the matrix in terms of the mtal movement of discs, there is the off-diagonal term B υω, which reflects the movement of the "p-down" and rotating in the same direction are copled. The presence of off-diagonal terms is a fndamental difference between vector problem of elasticity theory of scalar problems (heat condction, electrostatics, etc.. The analogy with the scalar problem (for example, the problem of the electrical capacitance takes place for the displacements "to the left to the right. To move "p and down" and "rotate in the same direction" there is no analog to scalar case, here the energy is the general qadratic form A υ B + C + υω ω. dxdy This is de to the fact that the movement of the "p-down" and "rotation in one direction" generate in neck the local deformations of similar form, namely shear deformation. The coefficients of the qadratic form (3 can be obtained by nmerically solving the problems of the theory of elasticity for the st, nd and 3rd movements of basic drives and the sbseqent nmerical evalation of integrals (4 (this can be done in the environment of ANSYS. As an example, the coefficients of the qadratic form (3 have been calclated for δ =, R =. For theseδ and R the following vales were obtained 6 6 A =.59, B =., C =.58, D = 8.5 It is seen, the coefficient B is not zero

9 3. Scaling formlas for energy The coefficients (5, as well as the harmonic (electrostatic capacity of bodies depend only on the geometry of the inclsions. In the case nder consideration inclde - discs. Therefore, the coefficients of (5 shold depend only on the vales δ and R. We show δ that in fact, the coefficients depend only on the ratio, that is, the problem is selfsimilar to the spatial variables. Consider the transformation of coordinates x xt, y = yt. = The Dirichlet problem for the theory of elasticity of a self-similar, which makes the soltion is not changed by the transformation (6. Consider integrals (5 giving the coefficients A, B, C, D of the qadratic form of energy (3. With the change of variables (6, the operators of differentiation are replaced in accordance with the formlas R x = t x, x = t y. ths, in the integrals in (4 will be a factor the Jacobean t, at the same time at this change of variables ( x, y = ( x, y t and by redcing these factors in the integrals (4 for change of variables (6 do not change, that is, they are invariants of the transformation ( 6. Then [], the integrals in (4 (i.e., δ the coefficients A,B,C,D depend only on the ratio. For example, coefficients (5 are the same for any δ and R whose ratio δ R = 5 We note that the considered problem describes the behavior of a pair of parallel fibers of circlar cross-section in a plane perpendiclar to its axis [, 3]. R. Notations R radis large disk; R radis small disks; δ the distance between small disks; E Yong's modls of the material matrix; E Yong's modls of the material disks; ν Poisson's ratio; movement by axes Ox ; υ movement by axes Oy ; r radial displacement; angle displacement; E energy. ϕ Indexes: sh neck; mat matrix. 9

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