INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING. VOL. 2. 159-174(1970)

ON THE NUMERICAL SOLUTION OF A CLASS OF PROBLEMSIN A LINEAR FIRST STRAIN-GRADIENT

THEORY OF ELASTICITY

J. T. ODEN,· D. M. RIGSlly.t AND O. CORNET'ri

• Professor of ElIgi1le('ri1lg M(·c!lIl11ic.\'. t S('lIior Research Assistant a1ld t Gradl/ole SlIIdentDeparlll/(·nt of f:.'1I.fJineeri1lg.Ullil'<'rsily of A/abama i1l fI,lI/tst,ille. HI/llls/'ille. A/abama

SUMMARY

This paper is concerned with the development of general discrete models for the analysis of boundary-value problems in the first strain-gradient theory of elasticity, Extensions of the finite element methodare constructed for this purpose. and general equations of motion arc derived for finite elements of aclass of micro-polar materials which are characterized by strain energy functions involving strains andsecond gradients of strains or displacements. The notion of generalized nodal doublets is introduced.The problem of a composite consisting of a strain-gradient sensitive microlayer embedded betweensemi-infinite bodies is examined as an example problem, Some of the results arc compared with avail-able exact solutions.

INTRODUCTION

Although the idea of couple stresses was introduced by Voigt in his 1887 study of elastic crystals, 1

the first attempts at developing general notions of micro-structure in elastic materials is usuallyattributed to E. and F. Cosserat.2.3 Until recent years. the work of these early investigatorswas virtually unnoticed; but a paper of Toupin4 and a series of papers by Mindlin and hiscollaborators5-7 have apparently revived interest in the subject. Of particular interest are theso-called 'strain-gradient' theories of elasticity in which the potential energy of deformation isassumed to be a function of not only the strain. as in the classical theories. but also first- andhigher-order gradients of the strain. A general. non-linear strain-gradient theory of elasticity,including general constitutive equations and boundary conditions. was first presented by Toupin.4Subsequently. Mindlin5 investigated linear versions of the theory in which three forms of thestrain energy functions were considered; and more recently Mindlin and Eshel8 presented detailedcomparisons of these forms. M indlin6 regards the strain-gradient theory as a low frequency. longwavelength micro-deformation approximation of the more general micro-structure theories.Applications of linear strain-gradient theories to various boundary-value problems have beenconsidered by a number of authors. including. for example, Sternberg and Muki.9 Day andWeitsman.1O Cook and Weitsmanll and Hazen and Weitsman.12 These authors have shown. inparticular. that the consideration of the strain-gradient theories has a significant efTect on thestress concentrations around discontinuities.

The basic equations of the linear first strain-gradient theories of elasticity arc considerablymore involved than those of the classical theory. and solutions to specific boundary-valueproblems may be correspondingly more difficult if attacked by classical methods. The presentpaper is concerned with the development of general discrete models for the analysis of boundary-value problems in the linear first strain-gradient theory of elasticity. Extensions of the finiteelement concept are constructed for this purpose, and general equations of motion are deriverl for

Reeeh'ed 20 August /968Rel'ised 20 Jalll/ary /969

@ 1970 by John Wiley & Sons. LId.

159

160 J. T. ODEN. D, M. RIGSBY ANO D. CORNETT

tinite e1cments of a class of micro-polar materials (i,e. materials which arc strain-gradient sensi-tive) described by strain energy functions which involve quadratic forms in the strains and thesecond gradients of displacements. I t is shown that this approach leads to the concept of generalizednodal double forces called doublets. corresponding to double stresses and the usual generalizednodal forces which. in the discrete model, correspond to strcsscs. An alternate formulation leadsto the notion of genernlized nodal couples. The development of consistent linite element modelsof various fields is necessarily different than in the classical case. for nodal values of the displace-ment gradients must now be specified in addition to the displaccments themselves.

Following this introduction. basic equations of thc first strain-gradient thcory of elasticity arcreviewed. Finite clement models of thc displacemcnt field appropriate to thc problcm arc thendiscussed and several specitic forms of these approximations arc presented. The approximateficlds arc then introduced into appropriate energy balances for a finite element. and generalequations of motion are obtained. Numcrical results of applications of the finite element modelto problems with known solutions are then examined.

FIRST STRAIN-GRADIENT THEORY OF ELASTICITY

Following the work of Mindlin and Eshel.8 we review briefly in this section several basic ideas ofthe first strain-gradient theory of elasticity. For an arbitrary continuous media. the basic kinematicvariables needed are defined by the following linear relations:

(I)

1Yij = 1(u1j + Uj.,J = strain

Wij = !(Uj.i-Ui.j) = rotation

Wi = !l'ijl.: ul.:,j = rotation vector

Kjjl.: = ul.:,ij = second gradient of displacement

Kij/; = !(UI.:,ji +ui.l,) = gradient of strain

Kij = ~ejll.: Uk.li = gradient of rotation

Rijk = A(uk,ij+Ui,jk+Uj.ki) = symmetric part of Kiik

In these equations. ui = u;(xj• t) arc the Cartesian components of the displacement field of acontinuum. Xj being a system of rectangular Cartesian co-ordinates. and the commas dcnotepartial differentiation with respect to xj(i,e. lIi.j == cu,.jaXj: UI.:.ij == cZUJd.\'i ih).

We now consider an elastic body constructed of a material which is characterized by a strainenergy function IV which represents the strain energy per unit of undeformed volume, At thispoint. we depart from the classical theory and assume that 1-1' is a function of not only the strainsbut also gradients of the strain. Since we intend to confine our attention to linear theories. wetake IV to be a quadratic form in Yij and Kiik' Kijk or Ki{ Mindlin6 and Mindlin and Eshel8considered thrce such forms for the strnin energy density for isotropic bodies:

IV = If'(Yij, Kijd = 'f'(Yij' Kijl.:) = W(Yij· Kij' Kiik) (2)where

It' = PYii Yjj +µ'Yij Yij+ (I. Kiik Kkjj +az Kijj {<ikk +a:l Kiik KjjA' +al Kijl.: Kijk +af) Kijk Kkji (3)

Ii, 1\ .' A A • A A • A A • A A • A A (4)T = 'l/lY,i Yjj + µ'Yij Yij + lIll'iik Kkjj +1I2"iji Kikk + lIa Kiih" Kjjk + 1I4Kijk Kij/ ..+ (/5 Kijl.: Kkji

I-V = !/\Yii Yjj +µ'Yij Yij + 2dl i<ij Kij + 2dz Kij Kji +~01 Riij Kkkj + O2 Kijk Kijk +.lEijk Kij Kkl/llll (5)Here A and µ are the usual Lame constants and al• az •.... al. a2. .1 are micro-moduli associatedwith strain-gradient sensitive materials. These three forms arc equivalent to one another andsimple relations betwecn the micro-moduli of each form can be obtained.

A CLASS OF 1'lWnUMS IN A LINEAR FIRST STRAIN-GRADIENT THEORY Of EI.ASTICITY 161

Components l1ij of the strcss tensor are derived from the strain energy functions as follows:elF elf' clf'

aij = ~ = -;c-- = ~ = aji (6)<Yij ('Yij <Yij

Likewise

(7)

(8)

(9)

and('IT'

il1..=--.llI ..=O (10)I] CKij II

whcre {lijk' Pi}k' and i1ijk are components of double stress and 11Iij is the deviator of the couplestress tensor. In Refcrence 8 it is shown that

Pijk = (.Lijk + {lkij - P'jki ( II)Il1ij = 1£jpq{liPIf (12)

fl.ijk = HP'ijk +P'jki + {lki) (13)Thus. the doublcts Pi}k and P'ijk and couple stresses 11Iij can be calculated once thc doublets (Lijt;

are known.If the strain-gradient equations arc viewed as long wavelength limit-of-dill'ercnce equations

for a simple crystal lattice in the micro-structure of the material. velocity gradient terms do notappear in the kinctic energy density T. and T is given by the usual formula

T= !Pliili; (14)where p is the mass density and Ii; arc the components of velocity. Then the equations of motionfor the continuum can be shown to be:8

ajk.j - ii.;jk.ij + Fk = piik (15)wherc Fk are the components of body force per unit volume. Other forms of equation (15) interms of PUk' 111U and fl.;jk can be obtained by using the relation given in cquations (II )-( 13).Various forms of the associated traction boundary conditions can be found in Rcfercnce 8.

Now consider a linite volume r and surface arca A of such a material which is subjected tobody forces Fl." body couples C I.' and body doublets <l>ij throughout. as well as surface tractions Sj.surface couples //Ij and surface double stresses Ji-ij/:' Then. according to Reference 8. a principleof conservation of energy can be postulated of the form

d .• .eft 1.(T+W)dl'= 1.(Fklik+CkWk+ID(jklYik)dr+ I (Sk1ik+//IkWk+lI;flijkYjk)dA (16)

• I • I • A

in which lll(jk') is the symmetric part of lfljk and /Ii are the components of a unit vector normal to A.The right side of cquation (16) rcpresents n. the total mechanical power developed hy the externalforces. couplcs and doublets. Other forms of equation (16) can. of course. be derived usingequations (II). (12) and (13).

FINITE ELEMENT APPROXIMATIONS

The tinite clement conccpt involves the representation of a continuous body by a collection of alinite number or component parts called finite elements.13 Ordinarily these flnite clements arc taken

162 J. T. ODEN. D. M. RIGSBY ANO O. CORNETT

to be of vcry simple gcometric shapes. and various ficlds are approximated locally over euchelement in terms of their values at prescribed nodal points in the element and on its boundaries.On connecting the clements appropriately together. a discrete model ofthc original field is obtainedin which lhe field is described by a finite number of its values at discrete nodal points in the modeland by piecewisc upproximations over each finite clement. The appealing aspect of this procedureis that the clements can be considered to be completely disconnected for the purpose of defining thelocal field approximations. Thus. the 'behaviour' of a typical element can be described locally. inan approximatc manner. without consideration of its ultimate location in the modcl. the behaviourof other elcmcnts or its mode of connection with adjacent elements. This makes the procedureparticularly well suited for bodies with irregular geometry and boundary conditions. Severalgeneralizations of these ideas have recently been presented.14,15

In the application of the finite element concept to classical elasticity problcms. the mostprimitive type of finite element approximation is the simplex model in which functions arerepresented by piecewise linear approximations over each finite element. Tetruhedral. triangularor lineal finite elcments are cmployed in three-. two- and one-dimensional simplcx approxima-tions. rcspectively. and the local fields over each element are uniquely determined in terms of thcirvalues at the vertices of thcse elements. Moreover. since the process of asscmbling elementsamounts to matching nodal values of adjacent finite elements. simplex clements lead to anapproximation of the given function which is continuous throughout the model. thereby satisfyingone of the basie criteria for the convergence of the model to the actual field.

For the prcsent type of problem. however. the often used simplex approximations cannot beused-at least without a great deal of difficulty-because they lead to displacement gradientswhich arc uniform over each finite element. Consequently. the second gradients of displacementsvanish over each element and the basic ingredients of the strain-gradient theories are lost.Therefore. for the problem at hand it is necessary to develop higher-order linite clement approxima-tions which arc capable of depicting second gradients of displacement over a finite clement.

Consider. then. u discrete model of a continuous elastic solid that consists of a finite number Eof finite elements connccted together at various nodal points in such a way that no gaps or dis-continuities occur in thc model that are not present in the actual body it rcpresents. Consider inparticular a typical finite element e which. for the moment. is isolated from the rest. and IctUj(x, r) denote the local displacement over the element. Thc local field is approximated over theelement by functions of the form

u· = ./.,,(x) liy' +m ,,(xl li ,... (I7)• 'P. • 1 T. J . '.I

where i. j = 1.2.3. and the repeated nodal indices N are summed from I to Nr• Nr being the totalnumber of nodal points of element e. In this equation. USi are the components of displacement ofnode N of the element and USij are the displacement gradients at node N of the clement. That is.if x;, (i = 1.2.3; N = 1.2 .. , .. Nfl denote the co-ordinates of node N.

.j - d cul\'~)_/1,.(.\ ,.l = USi an (Ix) = USi.} (18)

(19)}The interpolating functions i/Js(x) and 'P.\ix) are defined so as to have the properties

ci/Js(x.l1),/1v(X1I) = S"11' ,,' = 0. .. . . ('x'

C'sPS,.(xJ1)sPSi(X.lt) = 0, ex} = Sij S,l/.\'

wherc Sij and tiM'" arc Kronccker deltas. Thus. thc functions i/J",(x) and 'P.,)x) may he regardedas generalized Hcrmite interpolation functions. since the approximate field in equation (17)

A CLASS OF I'ROIlI.FMS IN A LINEAR FIRST STRAIN-GRADIENT THEORY or ELASTICITY 163

coincides with the displacement field and its first partial derivatives at cach nodc point of theelement. Examples of thesc functions arc to be citcd subscqucntly.

To obtain the Ilnite clement approximation of the entire displacement field over thc assembledsystem of elements. clement identification labels (e) are affixed to each local field (e.g. lIj(rl) andglobal values of displacement and displacement gradient at a node u of the assemhly of elementsare denoted UIl; and U:1iJ respectively. Then. a mathematically formal statement of the relationbetween local and global values at incident nodal points is given by References 16-18:

(20)

where 11.\';1,) and lISiJfrl denote local values of IIi and lIiJ at node N of c1cmcnt e. U;li and U;liJdenote corresponding global values at node u of the assembly of elements. and n~{iis definedas being unity if node N of element e is identical to node u of the assembled system and zero ifotherwise. The repeatcd global node index ..l in equation (20) is to be summed from I to G. Gbeing the total number of nodes in the assembled system of elements. The flnal finite clementrepresentation of the displacement field lii(X) over the entire collection of E elements is given by

Jo: Jo:Ui(x) = ~ lIi(x) = ~ n~i(ifJ~) U.li+ ?({j U.li)

e=1 Ce) c=l(21 )

where .p~{1= .p~) (x) and <P~vJ= 9~J (x) are the interpolating functions of equation (17) correspond-ing to finite clement e,

In general. the functions .pc:,:' and S"~:) arc required to be such that the field Ui(x) and its firstpartial derivatives are continuous throughout the model. Such conditions in turn impose restric-tions on the shape of the elcments. and are introduced to fulfil one of the basic requirements formonotonic convergence of the finite elemcnt model to that of the actual field as the networkof elements is relined. Below. several examples of admissible local field approximations arcconsidered.

Three-dimensional elel/l('I//

The tetrahedron with live nodes. onc at each vertex and a flfth located at an arbitrary point inthe element. is an example of a three-dimensional finite element on which a function and its firstpartial derivatives can be approximated (Figure I). Each component of displacement Iii can be

o 0 Figurc I, Higher-order finitc clcments

164 J, T. ODEN. D. M. RIGSBY AND D. CORNETT

approximated by the gcneral thrce-dimensional cubic

IIi = Ai + Bijx} + Ciik xi xk + Diikm xi x~·XIII

where(22)

(23a)

(23b)

with i.j. k. m = 1.2.3. For each IIi' there are twenty independent coefficients Ai' Bi}' Cijk andDiiklll' There arc determined in terms of the nodal values IIXi and uYi.i by the sixty independentconditions

IIlx1,·.) = Ii .. C'II;(X\.)• .\'1' ---'---('xi - IIXiJ (24)

whcre N = I.2. 3. 4.5 and X\' are the co-ordinates of node N. On solving cquation (24). the resultsare introduced into equation (22) and the local displacement fields arc put in the form ofequation (17).

TIl'o-dimel/siol/al elemelll

For plane problems, a triangular element with four nodes. one at each vertex and one at itscentroid. can be used. Actually. thc location of the fourth node is arbitrary. but the centroid ofthe element is often a convenicnt choice. In this ease the values of the local field and of its firstpartial derivatives are prescribed at nodes I. 2 and 3 but only the function itself is prescribed atnode 4, Similar approximations have been llsed for plane stress problems by Felippa.18 A cubicapproximation is again uscd:

II" = A" + B""pxP + Co:h xfJ :'(Y + Dal(Xl )3+ D"2(Xl)2 X2+ Dn3 Xl(X2)2+ D"i.\·2)3 (25)

where C"fJY = Co:yfJand a.. f3. y = 1.2. For each component of displacement III and 112 there are tenunspecified coefficients. These are determined from the conditions

N = I.2. 3.4 }

N = 1. 2. 3(26)

Then equation (17) acq uires a slightly modified form. since '?4"(X) == O.

One-dimensiol/al element

A one-dimensional element is considered as a third example. In this case. one-dimensionalHermite interpolation polynomials arc used and the displacement III = II(X) is given by the cubic

Ii = A + Bx+ Cx2+ D.\.3 (27)

with A. B. C and D determined from the conditions

- DII(,",' .1

) 1liLr - oX(:1I(X2)

112 ..r = ex(28)

where Xl and X2 are the co-ordinates of nodes I and 2 of the element. Solving equation (28) andintroducing the results into equation (27). it is found that equation (17) is given by

(29)

A CI.ASS OF I'ROBLEMS IN A LINEAR FIRST STRAIN-GRAOIENT THEORY OF ELASTICITY 165

Here Y's(.\") = Y's.r(.r) (N = 1.2)

.pI = 1-3g2+2fl.Y'I = L(g - 2f+ fl)·

where g = (x-xl)/L and I. = .\'2-.\']'

.p2 = 3f- 2f192 = L( - g2+ f1)

(30)

Other elements

It is emphasized that the above elements are cited only as examples. Several other types of finiteelements can be developed which are capable of being used in first strain-gradient problems. Intwo-dimensional space. for example. rectangular elements over which the local field is given bytwo-dimensional Hermite interpolation polynomials. each given by sixteen-term bicubic poly-nomials. can bc used. Similar functions were used by Bogner and othersJ9 to develop flat plateelements. The higher-order plane stress elements presented by Tocher and Hartz20 could also beused. As a final comment in this regard. wc cite the one-dimcnsional clemen I in which thc secondderivatives of displacemcllt /I.u arc specificd at each nodc. as well as the dcrivatives II,,F' In Ihiscase. a fifth-degree polynomial uniquely detcrmined the local field in terms of the values of /I. /I.,r

and 1I'.r•r at each of the two nodes.2l

KINEMATICS AND EQUATIONS OF MOTION OF A FINITE ELEMENT

We now consider an clastic body of matcrial charactcrized by strain energy functiolls of thc formgivcn in equation (2). The body is of arbitrary shape. with arbitrary boundary conditions. and isin motion due to the action of a general system of external forces. couples and doublets. Followingthe usual procedure. we set out to construct a discrete model of this systcm by representing thebody by a collection of a finite number E of finite elements connected togethcr at their nodes. Forthe moment. the particular type of finite elements is unimportant. We now temporarily coniineour attention to a typical tinite clement e. Assuming that the displacement field throughout theelement is given approximately by equation (17), we fino from equation (I) that for the finiteclement

Yij: H.pS.ill.\'j+.p.\'JII.\·i+?\·k.ill.\·j,k+?st.:.jll.\·i.0 '\

_Wi - ~f·ij/,N}s.j liSt.:+ rp.\·rj 1I.\'I:.r )

K ... , = 11.\ ... /.l \' •. +~\' ../I \", rI)~ f .. 1) . ~ T. r.l} .~.

(31)

(32)

(33)

Here N = I.2..... Nr; i.j. k.r = I.2. 3: and the element identification label e has been omitted forconvenience. Other kinematical variables can be evaluatcd by simply introducing equation (17)into the appropriate equation in expression (I). However. the kinematics of the element is satis-factorily specified by equations (17) and (31 ) if an energy function of the form given by equation (3)is used.

Let WITI denote the strain energy density for finite element e. Then.according to equations (2).(3). (6). (7) and (31). the strcsses and doublct stresses in thc finite element arc givcn in terms of thenodal values IISi and IISi.j by

I (C IY(rI (I IYI..) Iaij = -2 ~+-;-- = Gij(x; IIsr; IIsr,J

.. CYij CYji

_ I (nT'(,.) il/T'ld) _Itjjk = -2 ~+~ = fjjk(x; /lsr: /lSr.J

Cl<.ijk ( Kjik

12

166 J. T. OOEN. D. M. RIGSBY ANO U, CORNEn

wherc

G;J(x; IIsr: IISr.J = >'D;j(.pS r IISr + 'PSk.r IIsr,J + µ(YJ.\·J 11.\';+ .ps.; liS} + 'P.\'k.; IIS},k + 'PSkJ IISi,k)

(34)F;j(;(x; "Sr; lI.\'r,) = !(dl SjkD;II/+(1\ S;k Djm+403 Dij Dkll/) (.pN.rr"SIII +'P.\'jJ,rrIlSm./l)

+ (til 8ij 8 kil, + U2 Sjk 8illl + O2 S;(; 8jm) (.pS.mr ".\·r + 'Ps I,.mr "Sr,p) + 2(1,.(.pS.ij "Sk + 'P.\'r.ij IIS(;.r)

+05( 8jm Da + S;m D}I) (.pS.k/llIl.\'1 + 'PSr.(;/lIII.\'I.r) (35)

Let TCd denote the kinetic energy density for the element. Noting that

Ji'l,) = (.pS,i Gij + .p.\·,;k Fu:j) lisj + ('P.Vr,; Gij + 'PSr,;k Fik}) li.\'j,r (36)

and introducing equation (\7) into the time derivative of equation (14). we find that the totaltime-rate-of-change of energy (kinetic plus internal) for the finite element is

r (t'd+ JV,el) dr = fIl.\'.l1 iiSj li.l1J+kS.lIr iisJ ,i.lIj,r + k.\'.lIr ii.l1j,r liSj + i.\'.I/ir ii").i li.l/j,r.. J ••

+ li.\·j J' (.p.\.,; Gij + YJS.iIJik}) d,,+ liN},r r ('PSr,; Gij + 'P.\'r,;k Fit.:}) dlJ (37)I', .IJ',

where I', is the total volume of the elemcnt and

fIlS,lI = /' p.p.\,.p.ude (38).. J"!"

kN.l1r = r p.pS'P.ltrdv (39)... '"f'

is.llir = /' P'Ps; 'PJ/r de (40).. ,. r

Here fIlS.lJ is the so-called consistent mass matrix for the element. k,\i;Jlr represents a couplinginertia doublet at node /If due to a unit acceleration at node N. and the array (\,Jtir representsthe inertial effect of nodal doublets acting 011 the element. The terms containing k.\'.Itr and i,\·.llir

in cquation (37) do not appear in thc traditional finite elcmcnt formulation. They may be inter-prcted as being duc to the 'micro-structure' cndowed in the element by assigning it to the higher-order displacement field defined in equation (17).

It is also interesting to note that any finite element equation represcnts a global description ofa relation that. in the limit. pertains to a point in a continuum. We see that even though the micro-effects of velocity gradients were ignored in dcfining the kinetic energy density in equation (14).terms equivalent to velocity-gradient effects (as represented by the terms involving liwJ inequation (37» appcar in the finite element equations. This suggests that higher-order finiteelement formulations are related naturally to the ideas of micro-structure in a continuum. Indeed.the basic equations describing micro-effects in clasticity follow after highcr-order kinematicalterms are introduced in describing the 'micro-deformation' of a differential element.6 This isprccisely the procedure that is followcd in constructing finite element models of continuous media.Thus. a continuum with micro-structures is thc limiting case of a finite element model constructedof higher-order finitc elements. These observations also suggest that in the case in which strain-gradient equations are viewed as a moderately long wavelength limit of difference equations ofsimplc crystal latticcs. thc tcrms involving nodal velocity gradicnts in equation (37) can beomitted, In fact. thesc terms should vanish in the limit if TId is to approach T of equation (14)as the finite elcment network is refined.

A CLASS or "ROllLBIS IN A LINEAR FIRST STRAIN-GRADIINT THEORY or EI.ASTICITY 167

Returning to equation (37). we see that the mechanical power 12(" developed by the externalforces. couples and doublets acting on the elcment is given by

where

and

P.n = r (Fkifls+lflllllkCnifis.m+II>likllf.\'J)d".. J.,

+ r (SkifJ.\'+~f'"l1ktllllifis.III+IIi1i.iiklf.\·J)dA, A •

d.\·jk = r (Fk /fiS} + ~E"'"k Cn /fiSj.1II + 11>(IIIkl ?\j.lIl) dv, r,

(41 )

(42)

(43)

(44)

Here PSk is the generalized force acting at node N of element e and d.\'jk is a gencralized 110daldouble-force acting at node N of element e. Both the nodal forces and the nodal doublets are'consistent' in the sense that the mechanical power in equation (41) is the Same as that developedby the distributed external forces Fk' Sk' couples Cn• til". and doublets <l>(jkl and µijk due to theassumed veloci ty field calculated using equation (17).

According to cquation (16). energy is conserved in the finite clement if

r (T1rl+I¥lr)dr = n(d, 1',

Thus. substituting equations (37) and (41) into equation (44) and simplifying. we arrive at theenergy balance for a finite element:

[mv.l/ iiMi + k S.\I, ii.l/j.,+ r (ifJS.i Gij + ifiS.ikFiki) dr - P.Yi] li.\'j + [kllS, ii.l/j + i.llSI, ii.llj.i• J'f

+ {, (?\,.i G(j+'Ps,.ikFikj)dr-dsi,l li.Vi., = 0 (45)• 1'.

Now the nodal values IISi and USiJ arc regarded as linearly independent functions of time. andequation (45) must hold for arbitrary values of the nodal vclocities lisj and nodal velocitygradients. It follows that the terms in brackcts in equation (45) must vanish. Thus. we arrive atthe pair of equations

tIIS.1I ii,l/j+k.\'M,ii.llj.,+ /' (ifIS,iGij(X; 11.11,; 11.11,.,)+ ifiS.ik Fikj(X; 1I;It,: IIs,..J)dr = psN) (46)I'

kS.l/, iillj+ i.ltSl, iiltj.i+ /' (?S,.i Gij(x; ".1/,; 11.11,...)+?\·,.ik Fik/X; 11.11,;11.11,»or = d.\·j,U),I'

(47)

Equations (46) and (47) are the general eqlla/ions olmo/ioll of a finite element of an clastic bodyconstructed of a material characterized by a strain-energy function of the form in equation (3).The functions Gij( ) and Fiki ) arc defined in equations (34) and (35) and we are reminded onceagain that the repeated indices are to be summed throughout their admissible range. In this casei.j. k. r. s = 1.2.3; M. N = 1.2 ..... N,. where N,. is the total number of nodcs of elemcnt e,

168 J. T. llI>EN. O. ~1. RIGSBY AND D. CORJ\ETT

Equations (46) and (47) pertain to a typical finite elclllcnt. It is now a simple Illatter to conncctall of the linite elements togethcr and to obtain the corresponding global equations of motionfor the entirc discrete model of thc body. From the invariance of mechanical power. we find

f;P:,j = ~ n.~ip~}.

"

f;

D:'jr = I O\'i d\'~ir

(48)

wherc P.:.J and D ':'jr arc the global generalized forces and double forces at node 2i of the con-nected systcm. n~iis the array givcn in equation (20) and L' is the totalnumhcr of finite elements.Introducing cquations (46) and (47) into equation (48) and incorporating equation (20) into theresult. we arrive at the global equations of motion

,.; .\1 U.. + v () + '\~qlcl I [.J. G ( .. nIl') " . qlr)" )! .:.r· rj '":,rr ('i.I' .:.....~- ':'.\ 'f'.\',i ij X. ~"rJJ u.lJr· ~-rJt v'l·r."

~ .. J"I'

(49)

le'L' 0 + f ['J' + '\~(lIe) j [ , C' (x' net') U . (lId U )"'':'1'1' rj :'l'ir l'i.1 "-' ~".:..\'. ~.\'r.i Ij.'· ~"r.1t .1tr· --I'.lt J·r..'

e .. J r

in which

M.:.1• = 2: n~~JII/~~!IJiW.t,

'.:.rir = L nKU.{fyiri:WlIr

(50)

(51 )

(52\

(53)

Equations (49) and (50) represcnt the final system of lincar differential equations describing thcmotion of thc finite clement modcl of the body, Upon applying appropriate boundary conditions.these are solved for the nodal displacements U':'i and displacement gradients U':'J r' Local valucsI/Si and U'·I.J arc calculated using equation (20) and the clement strains. strain gradients. stressesand douhle stresscs are evaluated using equations (31 )-(35).

SAMPLE PROBLE1vlS

To demonstrate the application of the theory developed in this paper. we now cxamine somcsimple special cases of the general finite clement equations, In particular. we consider the staticproblem of the elastic half-space. suhjected to uniform stress at infinity. in which two strain-gradient-sensitive materials arc welded together in such a way that a microlaycr of width 211results, This geometry might bc used to reprcscnl a simplilied model of a compositc materialformed by microplates or whiskers embedded in a soft matrix.lU as indicated in Figure 2, Thisproblem was investigated by Day and Weitsnuln.lU who obtained exact solutions using the firststrain-gradient theory of elasticity,

In the prescnt case the displaccment field of each element is of the form given in equation (27)with thc nodal displacements liS and displacement gradients ".\'.r (N = 1.2) being the unknowns.The strain energy function reduces to

(55)

A CLASS OF PROIlLBIS IN A LINEAR fiRST STRAIN-GRADIENT THEORY OF EI.ASTICITY 169

where 111 = I. 2 and A(J)'µ(II.IW are material constants for the exterior material and AW' µ'(2t./12

are those for the microlayer. Here A(,,,). µ'(",. are the Lame constants and.., I

12 =~.., (t1'1+li'2+lia+(j'I+ri'S) = ~2 (361+262)1\+ _µ. 1\+ µ.

where al ..... as arc the microllloduli for equation (3) and al and at are those of equation (5). Wemay also have use for the ratio

(56)

which is a measure of discontinuity of the classical strain at the intcrface of the microlayer andthe exterior material.

Figure 2, Composite body

According to equation (32), thc stress al11 in the microlayer and the stress an) outside themicro]ayer arc given by

(57)and from equation (33).

(58)

For purposes of comparing our results with those of Reference ]0. wc arc also interested in thedouble stress µ122 which can be shown to be given by

Ji.122 = dlll·xx

Following Day and WeitsmanY' we also introduce a parameter c.: such that

al= 1-32O:/2(A+2µ.), 62=c.:J2(A+2µ)

Then. if p = It/I. equation (59) can be writtcn in the form

1 =(ud _ . _ .. 1(111) (Arµ'122 - (I 2.:'«1111) 3 (ml+2µ11ll1)µ'.,r.rI P(m)

(59)

(60)

(61 )

170 J. T. aDEN. D. M. RIGSUY ANO O. CORNETT

Also. in thc present cxample thc equations of motion (46) and (47) for an element reduce to thclinear static relations

(2RL 8s1 (LR 4S) (R 12sItil =)5"+£ ul •.r+ - 30+£ u2 •.r+ 10+ U, (111-112)

(2RL 8S) (·LR 4S) (.R 12S')

tl2 =)5"+£ 112•.r+ -30+£ IIl·.c+TO+ U, (lIl-U2)

where R = A + 2µ and s = R12/2.

DISCUSSION

(62)

(63)

(64)

(65)

In the examples considered. we examine two cases characterized by the following values of thematerial parameters f3. Pl' P2' ctl and L"t2-Case I: f3 = PI= 10. P2 = 20. Il:l = Cl:2 = 0,1; Case II:f3 = P2 = 10, PI = 2. Q;1 = ():2 = 0,1. for each case, finite element solutions were obtained for3. 5, 10 and 15 elements inside the microlaycr O!!;, x/h ~ I. For the results shown the length of thefinite element modcl was 411. This rather short model gave surprisingly good results as comparedwith the exact solution. for all solutions cited here, the externally applied force per unit area isunity.

Figures 3 and 4 show the dimensionless displacement R(2) ulh plotted as a function of dimension-less distance x/h. Notice that the discontinuity in strain at the interface which is predicted by theclassical (non-polar) theory of elasticity does not appear in the strain-gradient solution. This morerealistic result is. of course. due to the fact that second gradients of displacement are now used tocharacterize the deformation so that it is possible to maintain continuity in the first derivatives ofthe displacement. It is also noted that for ."\/h> I thc displacements obtaincd from thc strain-gradient solution do not approach those of thc classical solution. The finite element solution isscen to be in good agreement with the exact solution for all cases considered.

In Figures 5 and 6 the Cauchy stress au = au is shown as a function of the dimensionlessdistance x/h. We observc that with only three finite elcments in the microlayer. good results areobtained. Unlike the classical case. we see that the stress experiences a finite discontinuity at theinterface x/h = I. According to Reference 10. the value of the dimensionless stress always liesbetween the classical value of unity and a value equal to the parameter f3. With PI finite, the stressapproaches f3 as pz becomes infinitely larger and approaches unity as PI becomes infinite with P2finite. If f3 > 1·0 (that is. if the material of the microlayer is stifTer than that outside the microlayer).then the stress at the interface is always greater than the classical stress. We note that these strain-gradient effects are of a local character and are significant only in a neighbourhood of the inter-face. As xl" becomes larger, the stress approaches values predicted by classical theory.

Figure 7 shows the dimensionless double stress (1/")µ.rYlI = (I/h)µ122 plotted versus x/" forfifteen finite elements. As expected. finite element solutions for the double stress are less accuratethan the corresponding solutions for displacements and stresses since they arc functions of thesecond derivatives of an approximate displacement field. Averaged values of the doublc stress overeach element are in qualitative agreement with the exact solutions.

"r:>'"'"

z>rZ"'",:>;;c

~i$r-".,.....::;;

:>

:n-I;;c:>ZI

Cl;;c:>S2mZ-I

-I:l:".,

~-<

=;;ctr.-I

::;;

::-:n-I

:5-I-<

DoyOnd Weltsmon,f', '2,pz'10-- JI "Firite element, 0 I

I IDay CTod Wettsmon,p, :10.Pz: 20-- (/ PFinite element 0 / I

/

'i'/>' I

7

9

8

10

oo 05 10 15 20

xlnFigure 4. Displaccmcnts. Fifteen c1cmcnts in microlaycr

-:::," 61 Classical SolutionI~N

(\J:NI~ 5

c4Q)

EQ)u015.oJ> 3£5

2

201,5

Classical S:lIuhon

Day and Wellsman. P, : 2, Pl : 10 -, -l',iFlrute element 0 .. I

I ; IDoyondWeIIsmCln,P, :IO,Pz'20-- , ...i,

Finite element, 0 9 :, i

oo 1-0

xlnFigure 3. Displacemcnts. Three clements in micralaycr

lOrI

9~I

I8

7

-:::," 6

]< 5

cE 11Q)u0a.oJ>£5 3

2"

-.J

-..IN

4 .. ¢ 4

Day ond Weitsman.p' :2,P2 :10--- Day and Weltsrnan,p,:2,p2: 10--I Finite element, 0 I Flnlfe element, 0r Doyor]d Weitsmon. P, :lO.Pz :20--- I Doycnd Weltsmon, P, :10pz: 20--II Finite element, 0 I Fmlte element, 0

3 l- I I '-

I 3 I :-'I I 0I ::;II '"'zI I,I ::;I

I{ " 1/ ~.. 2, " 2" " b d' ;<lb

I,I ;:;

til

11VJVI III:::lVI ~~ -<I I Cf)

Cf)

/ I ?r :>Classical Solution z

cClassical Solution /Jl I / ~- 1

/. -0--0 (),,- ....a---a 0/if ' fl..(J ;<l.....zU/ ?;:J ~

(

01 I t , I 00 05 10 15 20 0 05 10 1·5 20

x/h xJhFigurc 5. Cauchy stress UU• Three elemenls in microlayer Figure 6. Cauchy stress UU. Fifteen c1cmcnts in microlaycr

A CLASS OF !'IWIII.EMS II" A LINEAR FIRST STRAIN-GRAOIENT THEORY OF t:LAS1IC'lTY 173

008 r 0 Day and Weitsman, P, :2, P2' 10- -

Finite element, 0

007 ~ 9 Dayond Weltsmon, ~ '10,~=20--

I \ F',n1te element, 0

0{)6 ~ IqI qI

~ 005 I \\i: I \

-1<: ? \.;, 004

~'"~ Iin \

'" 0,03 J \:0g I q0,02 Q \

I q\ 0

001 r p' q/ J Q0.-0

0 05 10 t'5 20

x/hFigure 7. Double stress fir •• III. Fifteen elements in microlayer

ACKNOWLEDGEMENT

Support of this work by the National Aeronautics and Space Administration through GeneralResearch Grant. NGL 01-002-001. is greatly appreciated.

REFERENCES

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109-124 (1968).9. E. Sternberg and R. Muki. 'The effect of couple-stresses on the stress concemrations around a crack'. 111/.

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Soc, cir·. EIIKr,r. 94. EM3. 773-795 (1968),13. M. J. Turncr. R. W. Clough. H. C. Martin and L. J. Topp. 'Stiffness and deflcction analysis of complex

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nonlincar viscoclaslicily·. 0"1'. tlreor. & a(l(ll. Atedl. Proc. SOIlfI/('aslem COlli TIL.,o. 1/(11'1. M,'c/, .. IVSECT AM, Pergamon Press. Oxford (1968).

15. J. T. aden, 'A gcncral theory of finite elemenls·. 1111, j, III/III. lHelh. ElIglIg, I. 205-221: 247--259 (1969).

174 J, T, aDEN. D. M. RIGSny ANI) D. CORNEtT

16. J. W, Wissmann. 'Nonlinear slructural analysis: tensor formulation', Pmc. COlli Maldx ,'vieth. Stl'/l/'I.Afech .. Wrigill-Patterson Air Force Base/A.F. Flight Dynamics Lab .. TR·66·110. Dayton. Ohio (1966),

17, J. T. Oden. 'Numerical formulalion of nonlinear elaSlicity problems'. J. ElIglIg Struct. DiL', Alii, Soc. cil'.ElIgrs. 93. No. ST.l, 235-255 (1967).

Ill. C. A. Felippa. 'Refined finite element analysis of linear and nonlinear lwo·dimensional structures'. Struct.E/lglIg Slrllcl. ;\tech. Rep. SESI. 66 ,22. University of California (1966). Also University Microfilms Inc,.Ann Arbor. Michigan (1966).

19, F. K. Bogner. R. L. Fox and L. A. Schmit. 'The generation of interelement. compatible stiffness and massmatrices by the use of interpolation formulas'. Proc. COlli i'vlalrix Melh, Strucl. Mech .. Wright-PattersonAir Force Base/AF. Flight Dynamics Lab .. TR·66-80. Dayton. Ohio (1966).

20, J. L. Tocher and B. J. Hartz .. Higher-order finite elements for plane stress', J. ElIgllg Mech, Div.. Alii. Soc.cil', ElIgrs. 93. No. EM4. 149-172 (1967),

21, H. L. Langhaar and S. C. Chu. 'Piecewise polynomials and the panition method for ordinary differentialequations'. Del'. Iheo/'. & appl. Al('ch. Pmc. SOl/theaSlem COlli Theo. IIppl. Mech .. IV SECTAM. PergamonPress. Oxford (1968).