Effect of couple-stresses on elastic stress distribution has been investigated in a semi-infinite medium under the action of a dynamic pressure on the boundary. As a particular example this pressure has been assumed to be a pulse of pressure moving uniformly along the boundary. It is found that the effect of couple-stress on shear stresses is predominant on the boundary surface

Tiwari, G. R.

English

Defence Science Journal

COUPLE-STRESS IN A, SEM -INFINITE MEDIUM UNDER %HE ACTION OF A DYNAMIC PRESSURE ON THE BOUNDARY G. R. TIWARI Shri G.  S. Tech. Institute, Indore (Received 26 April 1969 ; revised 7 November 19699 Effect of couple-stresses on elastic stress distribution has been investigated in a semi-iffinite medium under the aotion of a dynamic pressure on the boundary. As a parMcular example this pressure has been assumed to be apulse of pressure moving uniformly along the boundary. It is found that the effect of couple-stress on shear stresses is predominant on the boundary surface. Classical theory of elasticity assumes the stress tensor to be symmetric, though i t  was pointed out much earlier by Cosserate brothers1 that there may be physical phenomenon where the stress tensor is not symmetric. For example, when the couple-stresses are present, the shear stress on the surface x = constant in the direction of y is not equal to the shmr stress2 on the surface y = constant in the direction of x. In  the present paper the effects of couple-stresses in a semi-infinite elastic medium under the action of a variable pressure on the boundary has been investigated. As a particular case a pulse of pressure moving uni- formly along the boundary has been considered. The effect of couple-stress on shear stresses along the boundary, is proportional to the parameter of couple-stress for certain intemaI of time. This effect is predominant on the boundary surfay. C O S S E R A T E ' B  E Q U A T I O N S  O F  X O T I O N  . . The equations of motion in cartesian coordinates that hold in a stressed body when the couple-stresses are taken into account, besides the usual normal and shearing strwesl*3 is written. The consideration of forces alohg x and y axes gives the equations of motions.4. if the body forces are negleoted. The considerstion of the moments yieldsa-* if the body couples are omitted. It is to be noted that the equation (2) docs not contain inertia terms. is8 ~ ) E P .  801. d. , VOL. 20, f ULY 1950 ,- F U N D A M E N T A L  E Q U A T I O N 8  In a state of plane strain, the non-va-uishing components (u ,  v) of the displacement veotor are funotions of x and y. The strain components are associated with the stress components, while the local rotation component is associated with the couple -st~esses PX and py . The couple-stress px produces cur- vature Kz parallel to the x-axis and py produces the curvature K g  parallel to y-axis. It has $een shown by Mindlin2 that z? so that bhich is the compatibility equation between curvatures. Compatibility conditions between cmatures and the strain components are found by eliminating w from (4)  and (5)  as The cmatwes have been assumed proportional to the couple stressesa. 1 K , =  - 1 4B ' x  , Kg =,'= PY (8) where B is a modulus of curvature of the material. When B = 0, the classical results are obtained. On substituting the values of Kx and Kg from (7)  in (8) a id  using (3), we get , In the case of plane strain the normal strains (ezz , eg y )  are related to the normal stresses (0% ; Q) as ux = X (eta + e y l )  + 1 ag = X (em + egg) + ~ P Y Y  J whereX= . Bv and p = -- ' are Lame's elastic coostan&, B is the (1 + v) (1-24 2 (I+.) Youig's modulus and v is the Pbisson ratio. Symmetric Part 2 1 Ta = - (%y + 7y2) 2 I of the shear stresses (rZ1 , ryZ) produces the shear atrain erg and hence of the shear stresses.produces the rotation W ,  and the relation between T.4 and rn8i be obtained with the help of equations (2), (5) and (8), Also, the equations (2), (7) and (8) yield From (11) and (13), we get on substituting from (5)? 1 (14) 2B [ v s  ($) - ~2 (%)I = T~~ - 5y J 1 These equations yield - 8" r u n  = (lava + 1) - - (12va-1) - P 3x es) 1 - ray = (E2v2 + 1) - - P .  * where - B B 12=2(l+v) - - - . E - C1 Substituting the values af oy) , ctY from (10) and the values of Tx9, , ryz from (46) ia (I), we get aav 3% sf. (16) (A+P) (% + -) + p p  ( v g -  2mg a2u + c ~ v + = p  - ata L Let where + and #I are functions of x , y and t , then equation (16) yield where = c?/c28 , c12 = (A + 2p)/p , ~2~ = and T = clt. From (18) can be determined + and $ if the boundary conditions are prescribed. S T A T E M E N T  O F  T H E  P R O B L E M  A N D  B O U N D A R Y  C O N D I T k O ' N S  The distribution of stresses in the semi-infinite elastic medium y 2 0 when a variable pressure P ( x ,7 ) is applied to the boundary y = 0 is considered. The x-axis is taken along the boundary and the y-axis pointing into the medium. The boundary conditions are - - P (z, T), r d a  = O, p y  = . O o l ~ y = O  (19) where P ( x , T ) is piecewise continuous and absolutely integrable in ( - a .<_ a: < oo ). &reover, all the components of stress, displaceme* and couple-stress tend to zero as y tends to infinity. Performing over the equations (18), the Laplace tramform, assuming that V a a ~(x,y,O)'=-- #(a:,y,O)=$(x,y,O)= - $ ( L , y , O )  = o  a? - a~ (20) we get p+l = v2+* and = V2& - 12V4$l -F J $7 d ~ ,  +,=I  + e  d ~ .  where +I = 0 0 NOW let W ~ , Y , P ) =  j --a (22) and 155 be the Fourier transforms of #I and respeotively. Mdtiplping equation .ei& and integrating with respect to x between the limits (-* , * ), we get The solutions of these equations are $2 = A exp i--y (t.2 + p2P-)  + BI exp 4 Y (5% + pz)t 3, , = A, exp iy (t2 + P>)$ I + C exP -P-Y ( P 2  + PI2)$ 1 + Cl exp {y (t2 + ~ 2 ~ ) f  I + D exp {-Y (6' + P ~ ~ F  } (23) where A , Bl , Al , C , Cl, and D are constants, and '. 1 * pr" - { 1 + (1--4Fv12)' 1 2P In order to satisfy the condition that the coqponents of stress, displacement and couple- stress tend to zero as y tends to  idnity,  we take & = A l  - C l = O .  The inversion theorem for the Fourier transform~ieldk - X exp ( - i t @  ) at Performing LapIaoe transform on (17) and substituting for 41 and from &bove we get 192 BE*. $a, 5.. VOL. 20, JULY 1970 C where ti and v are the Laplace transforms of u and v r e s ~ ~ 1 y ; -  - ,.a - .  Taking Laplace transform of (lo), (15) and (19) and substituti6g for' ii and ;it from (25) , we get /' - -i& c exp L* { - y (P + P P ~  ]+ (2t2 tp;2+12pl)- D exp ( -- y (t2 -I- p22f )] e d t ,  , J m -. - B r p, = - [ [ itp12C exp - y (F + p12)* 1 + exp * .  --a0 -it% ( - Y ( s ~ + P ~ ' ~ ) ] B  dt ,' B t 4 t = - s [ [ p i  ( t2+  pi2) cexp{ --YIP + P ~ ~ ) ) + I I ~ ( I ~ + K )  D ~ V  -00 { - Y (ta + p2%f } ] b d6 . (26) < i - - - . - -  - where a, , up , r y z ,  7t.y , and r , ~ ,  are Laplaoe tranefom ,of corrixponding, stress J and couple-strcm components. . - - - The boundary conditions (19) yield 1 4 4 - ( + p2p22; A - Ci t  (P 3- pIP) - Di6 (Sea + pz2) = - where and P, (z , p) is the Laplace transform of P (3, 7).  \ Solving equations (27) we get- 91 p2 (5 * P) A = - -  1 where .-. = p; (k2  + pZ2)l/2 ( 2 f 2  + p: - 12p14) / - p12 ([2 + p12)lIJ (2t2 + pZ2 - 1 ~ 2 1 ,  1 * '  g2 = p22 (t2 + pz2)1L9 ( ( P + 3 Bay ) Substituting *he values of A,  C and D in (25) and (-26), the components of displacement stress and couple stress on integrating and taking the inverse Laplace transform can b oalculated. For showing the, effect of couple stresses we calculate T~~ - ry r .  From (26) we iget . is4 Dm. $01. J., VOL. 20, J~ULY, 1940 As a particular case a pulse of pressure moving uniformly with velocity vo along I he ' 8  boundary is considered, that is P ( x ,  7 )  = P B ( x - f l 1 ~ )  where & = V O / ~ I  &oh gives P2 (i! , p) = Pa ( [PI + i p  ) (31 j Substituting for C and D from (28) and for P2 (I , p) from (31) in (30) and using the relation" If ( 5 ) S ( f  - a ) d S = f ( a )  - -00 - - 4% , ~ j ~ & ~ ( - )  Pe - *PIPI we get rXy - ryz  = - - - 4 =P 2 , p + Y  on neglecting terms containing h;gher powers of 1, and writing On taking inverse Laplace transforma we get from (32) 2 0 < 7 < x/&. From this result it is obvious that the shear stress along the bounding surface (y = 0 )  i s>  J- proportional to 1 esp 7 - y ( T  %%ffl ) provided r is greater than x/Bl. In ' 1 '  other words it can be said that the shear stress at any point on the bounding surfme is proportional to 1, the parameter of couple-stress. At any point inside the semi-infinite - a i u m  TX,, r yX is proportional to I exp ( -- yl451.).  That is, the etrect of couple- , stress is predominhnt on tlie bounding surface because the parameter, I, is generalty small. When, 0 < T < x/Sl (or 0 < t < X / T ) ~ ) ,  T ,  = T ~ X .  Which is the classical result. It means that there is no effect of couple-stress on shear stresses when 0 < T < x/&. A C K N O W L E D G E M E N T  Author expresses his sincere thanks to Dr. G. Paria for the help and guidance in pre paring this paper. R E F E R E N C E S  1. COSSERAT, E., & COYSERAT, I?., ''Theories dee Corps Deformables" (A. Herman et fils.), Paris, 1909. 2. MINDLIN, R. D., Ezper. dlech., 8 (1963), 1. t 3. M ~ D L I N ,  R. D. & TIERSTEIN, H. F., Arch. Rat. Mech. Anal.,ll (1962),415. 4 .  MISICU, M., Rev. Mech. Appel., 8 (1963), 921. .W 5.  ~NEDDEN, I. N., "Fourier Transforms", Soc. 50, (McGraw-Hill Book Company), New York, 1951. I I I ERDELYI, "Tables of Integral Transforms", Vol. I Bateman project (McCraw-Hill Book Company), New York, 1954. 

Couple Stress in a Semi Infinite Medium under the Action of a Dynamic Pressure on the Boundary

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Couple Stress in a Semi Infinite Medium under the Action of a Dynamic Pressure on the Boundary

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