Formulation of the Stress Distribution Due to a Concentrated Force Acting on the Boundary of Viscoelastic Half-Space

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1 Formulaion of he Sress Disribuion Due o a Concenraed Force Acing on he Boundar of Viscoelasic Half-Space Yun eng and Debao Zhou Deparmen of Mechanical and Indusrial Engineering Universi of Minnesoa, Duluh MN 8 USA peng04@d.umn.edu and dhou@d.umn.edu. Absrac - This paper presens he derivaive of he sress disribuion in he condiion ha a poin (or concenraed) force acs on he surface of a viscoelasic half-space. This soluion is obained hrough he combinaion of he soluion o he sress disribuion from he forces in boh normal and angenial direcions. The angenial-force viscoelasic problem is defined as he applicaion of a angenial poin force acing on he surface of a viscoelasic half-space and he normal-force viscoelasic problem is relaed o a normal force in he same siuaion. The elasic-viscoelasic correspondence principle is used in he derivaion of he sress disribuion. Two ime-dependen funcions, also called deermining funcions, are uilied o epress he muliplicaion of he eernal force and he ime-dependen viscoelasic maerial properies, i.e. he volumeric and deviaoric relaaion funcions and oisson s raio. Using he boundar condiions and he equilibrium condiions of a half-space, he mahemaical epressions of he deermining funcions can be eplicil obained. The sress disribuion in a viscoelasic maerial under general poin force (wih boh normal and angenial componens) is hen obained. These soluions can be used o formulae he sress disribuion in he crushing or cuing of linear viscoelasic maerials. Ke words: Sress disribuion linear viscoelasic angenial poin force half-space. I INTODUCTION Cuing operaions are involved in he process of man viscoelasic maerials. Modeling he relaionship beween he cuing force and sress disribuion can help predic he fracure due o cuing and idenif differen maerials along he cuing pah. Cuing problem can be modeled as a bel-shaped area force acing on he surface of a half-space. The sud of he sress disribuion due o a poin force will be he firs sep in modeling he cuing sress disribuion. A cuing force can be considered as he resulan of a normal and a angenial componen acing on he conac surface beween he ool and he maerial. The sress disribuion in an elasic half-space due o a normal poin force and ha due o a angenial poin force has been modeled b Boussinesq and Cerrui, respecivel. Correspondingl, he are called Boussinesq s problem and Cerrui s problem. For viscoelasic maerials, Talbl (00) raised he mehod b subsiuing he muliplicaion of eernal force and viscoelasic maerial funcions wih ime-dependen funcions. This paper will concenrae on he formulaion of sress disribuion in a viscoelasic under a angenial poin force. The simplified model is shown in Fig.. Due o he asmmer resuling from he angenial force, onl Caresian coordinaes could be used. This makes our formulaion much more complicaed. Fig. : A angenial force ( ) acs on he surface of a half-space Furhermore, using he soluion formulaed in his paper, a soluion o he problem in viscoelasic maerials under boh normal and angenial poin forces can be obained. This generalied soluion can be used in man furher problems, for eample, he food slicing cu problems formulaed b Zhou and McMurra (0). The remainder of his paper is as follows: angenial-force viscoelasic problem is eplained in Secion II and he soluion o his problem is derived in Secion III. The general soluion is shown in Secion IV. The conclusions are drawn in Secion V. II STATEMENT OF A VISCOELASTIC HALF-SACE UNDE TANENTIAL-FOCE A. Tangenial oin Force In his problem, a reference frame O- is defined on he half-space as shown in Fig., where he boundar of he half-space is a = 0, and a angenial poin force, which could be a funcion of ime, is applied a he origin along he -ais, and he posiive -ais poins owards he inerior of he half-space. The sress soluion o his problem for elasic maerials was given b Cerrui in 88 hrough he use of singulariies from poenial heor. The resuls were also presened b Love (97). The displacemen disribuions a poin (,, ) inside he half-space bod are:

2 e ( ) ( ) u e v e 4 ( ) ( ) v w ( ) ( ) where u e, v e and w e denoe he displacemen in he posiive -, - and -ais direcions in elasic case,, is he eernal angenial force, is he elasic shear modulus and is he oisson s raio. The sress disribuions can hen be obained using he kinemaic equaions and consiuive equaions for elasic maerials. The sresses a poin (,, ) saisf he following boundar condiions for his problem: dd 0 ( ) dd 0 dd 0 ( ) dd 0 dd 0 ( ) dd 0, where he hree equaions in lef column represen he force balance, and he hree equaions in righ column represen he orque balance. B. Viscoelasic Half-Space under Tangenial oin Force In our derivaion, he viscoelasic effec is aken ino consideraion based on he original angenial-force elasic problem. The difference beween elasic and viscoelasic problems lies in he consiuive equaions. In an elasic maerial, he relaionship beween sress and srain can be described b Hooke s Law. However, in a viscoelasic problem, he maerial will show elasic behaviors like solids and also show viscous behaviors like fluids. This changes he relaionship beween sress and srain. During he calculaion of he sress, insead of mulipling srain wih maerial properies, such as Young s Modulus and oisson s raio, he convoluions of srain wih relaaion funcions are used. These relaionships are shown as: d d d d d d d d d, where and are he deviaoric and volumeric relaaion funcions, and / is he mean srain. Therefore, we sae he angenial-force viscoelasic problem as follows: finding ou he soluions o he sress disribuion of a half-space bod under a poin angenial force applied o he surface, wih boundar condiions and sress-srain relaionships saisfied. III SOLUTIONS TO TANENTIAL-FOCE VISCOELASTIC OBLEM A. Displacemens For he linear viscoelasic case, b appling he elasic-viscoelasic correspondence principle and replacing and wih and, The following epressions can be obained: u u u v v v 4 w w w where u, v and w denoe he displacemens in he posiive direcions of he -, - and -ais wih: u u ( ) ( ) v v ( ) ( ) w w. ( ) ( ) In following secions, all derivaions will be conduced wih he erms of and carried ou individuall. B. Normal Srains The normal srains are obained via he derivaive of displacemens wih coordinaes. The obained normal srains are: u u u v v v = 4 w w w where: u ( ) ( ) ( ) u ( ) ( ) ( ) v ( ) ( ) ( ) v ( ) ( ) ( ) w ( ) ( ) w ( ) ( ) ( ) ( ) Thus he mean srain is: 4 and he mean sress is d d d 4

3 C. Shear Srains The shear srains, and can be obained as: v w v w v w = u w u w u w 4 u v u v u v where v w v w 0. u w u w 0. u v ( ) ( ) ( ) u v. ( ) ( ) ( ) Noed is ha he coefficiens of in and are ero, hen he srains can be wrien as 0 = 0 4 u v u v D. Sresses Wih srains known, he sresses can be obained from viscoelasic consiuive equaions (Zhang 994, p. 6) as: d d 4 d 4 ( ) ( ) ( ) d ( ) ( ) ( ) d d 4 d 4 ( ) ( ) ( ) d ( ) ( ) ( ) d d 4 d 4 d d d 4 d d 4 d d 4 ( ) ( ) ( ) d. ( ) ( ) ( ) E. Boundar Condiions Now consider he boundar condiions. Inegraing he sress componen and noicing ha convoluion will no ake par in spaial inegrals, he firs boundar condiion becomes: d dd d 0 4 The above equaion means ha he resulan force in direcion on he O-- plane cancels he eernal force. I is worh o menion ha he inegrals of he oher sress componens are all eros because all of hem are odd funcions of wih he inegraions over,. The value of he above inegral should be ero as required b he balance of force. Then he following epression can be obained: d F. Equilibrium Equaion Bringing all he sresses ino he firs equilibrium equaion E 0 will ield: E d 4 u u v u w d 4 u u v 0, where he coefficiens of d and are: u u v u w u u v u w u u v u u v. ()

4 Thus here is: E d d B eliminaing he same coefficiens conaining he coordinaes, here is: d d 0 () The second equilibrium equaion will ield he same equaion relaionship as given in (). The hird equilibrium equaion 0 has been saisfied auomaicall.. Soluion o Tangenial-Force Viscoelasic roblem and are linearl independen in Equaions () and (). Thus, and can be uniquel solved b solving he Volerra inegral equaions of he second kind (Zhang 994, p.0-6). iven he iniial condiion (0) = 0, all sress and srain componens are ero a = 0 and 0 0, 0 0, he soluion o can be obained. Noed is ha he deailed derivaions are no included in his par for he purpose of concision since he deailed derivaions and he proof of he uniqueness can be found in he paper b Talbl (00). Manipulaing equaions () and (), here is: d 6 Viewing in whole and solving his Volerra inegral equaion, can be obained wih previousl solved. Taking advanage of () and (), he erm of can be eliminaed and he sress disribuions can be wrien as follows: ( ) ( ) ( ) ( ) ( ) ( ) d. ( ) ( ) ( ) d d () Comparing wih classic Cerrui s soluions (Johnson, 98 p.69-70), we found ha he erm ()- *dψ in viscoelasic soluions plas he same role as he erm (-μ) in elasic soluions. IV STESS DISTIBUTION IN A VISCOELASTIC HALF- SACE UNDE ENEAL OINT FOCES We now consider a general force wih boh normal and angenial componens, in which wo angenial forces () and (), along - and - ais respecivel and one normal force () along -ais are applied a he origin. The displacemen and sress disribuions are obained using he superposiion of he displacemen and sress disribuion b (), () and (). The sress and displacemen disribuions for he problem when onl one angenial force ()= () along -ais has been discussed in previous discussions. For disambiguaion, we rewrie he soluion wih proper superscrip (i) (i =, or ) o denoe he applied forces. Therefore we have:,,,,,. where he epressions of he righ hand side are he same as in () onl wih () replaced b (). Fig. : Model for coordinae s ransformaion When here is a single angenial force () applied on he O-- plane a poin O along -ais, he sress disribuion is obained hrough he calculaion of he frame roaion. In his mehod, as shown in Fig., we firs roae he frames around -ais b 90 and denoe he new coordinae ssem as O-. The corresponding Jacob mari is: cos(90 ) sin(90 ) sin(90 ) cos(90 ) The coordinaes are ransformed as ' ' ' ' ' and ' ' ' ' The corresponding sress ensor is ransformed as ' ' ' ' ' ' ' ' ' ' T ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' Correspondingl, he deermining funcions can be rewrien as (Talbl, 00):

5 i i i d, i,,, 0 0 i i i U i d, i,, In coordinaes O-, () is applied a he origin along -ais. Based on he soluion we obained in (), he sress in O- ssem can be obained as: ' ' ' ' ' ' d '( ' ') ' ( ' ') ' ( ' ') ' ' ' ' ' ' ' ' ' ' ' d '( ' ') ' ( ' ') ' ( ' ') ' ' ' ' ' ' (4) ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' d. '( ' ') ' ( ' ') ' ( ' ') ' where ' ' ' ' Afer ransformaion o O- coordinaes, sresses in (4) can be wrien as: d ( ) ( ) ( ) () d. ( ) ( ) ( ) When a single normal force () is applied a poin O and along he posiive -ais, he sress disribuion is shown in (6). Noed is ha his epression has been given b Talbl (00). d ( ) ( ) ( ) d 4 d 0 0. Based on (4), () and (6), we provide he soluion of sress disribuions o he general problem as follows: i i,, i i i i,, i i i i,. i i (6) Similarl, he displacemen disribuions o viscoelasic problem u, v and w as follows: i u u, i i v v, i i w w, i where he componens are: u ( ) ( ) v 4 ( ) ( ) w ( ) ( ) ( ) ( ) u v 4 w ( ) ( ) ( ) ( ) u v 4 ( ) ( ) w The resuls will be used in our fuure research abou he formulaion of he cuings for linear viscoelasic maerials, in which a disribuive force is used insead of a ij r, poin force. We could obain he sress response o a cerain disribuive force b considering he poin force as a funcion of and, replacing b, b, and hen inegraing he corresponding sress or displacemen componens for and in - plane. and b V CONCLUSION The soluions o he sress and displacemen disribuions in a viscoelasic half-space under a poin force are presened in his paper. The mehod is based on he elasic-viscoelasic corresponding principle. The soluion o he displacemen of an elasic half-space under a angenial poin force was used as he displacemen soluion o he viscoelasic problem. Based on he equilibrium equaions and boundar condiions of a viscoelasic half-space, we obained he soluion o he wo ime-dependen deermining funcions via he Volerra inegral equaions of he second kind. Then, he sress disribuion due o a angenial force in -ais direcion is solved based on a frame roaion mehod.

6 Finall, b combining he soluions when here is onl a normal force componen, we ge he resuls for he generalied case, where an arbirar force wih hree nonero componens in, and direcions, is applied. These resuls could be furher used o solve he sress disribuion in a viscoelasic half-space under a disribuive force, b aking he inegral over he area where he disribuive force is applied. VI EFEENCES Boussinesq, J., Applicaion des oeniels a I Eude de l Euqilibre e du Mouvemen des Solides Elasiques, auhier-villars, aris, (88). Love, A. E. H., A Treaise on he Mahemaical Theor of Elasici, 4h Ediion. Cambridge Universi ress (97). Talbl, L. K., Boussinesq's viscoelasic problem on normal concenraed force on a half-space surface, Mechanics of Time- Dependen Maerials. :-9, (00). Xu, Z., Elasici Mechanics (Tan Xing Li Xue) h Ediion, Higher-level Educaion ublicaion, in Chinese, (996). Zhang, C. Y., Viscoelasic Fracure Mechanics (Nian Tan Xing Duan Lie Li Xue), Huahong Universi of Science and Technolog ress, in Chinese, (994). Zhou, D. and McMurra,., Slicing Cus on Food Maerials Using oboic Conrolled aor Blade, Modelling and Simulaion in Engineering, Acceped and o be published in Nov 0, Hindawi ublishing Corporaion, (0). Johnson, K. L., Conac Mechanics, Cambridge Universi ress, (98).