ar

De

eridicrovanorree disededfew

ssumesand esaction.ar inteit the aena wch larhen tects coory descrepa

tinuum theory of elasticity, an early attempt was made by Voigt

across boundaries/interfaces or from one part of the body to anoth-er. Later Cosserat and Cosserat (1909) developed a mathematicalmodel based on couple stresses leading to a description of thestress elds via asymmetric tensors as opposed to the symmetricCauchy stress elds in the classical theory. From a kinematicalperspective, this theory enables non-local interactions via the

classically employed translational ones, for the material points

order gradients of the strain tensor are assumed to contribscale effecrent highe

generalized stresses conjugate to the gradients of strain.Apart from the limitation related to scale independenc

known that the mathematical setup in the classical contmechanics may not be quite appropriate in the context of severalother problems of fundamental interest in solid mechanics, viz.those including the spontaneous generation of cracks (Silling,2000). The inability to track and evolve a discontinuous eld maybe traced back to the kinematical requirement of a sufcientlysmooth, diffeomorphism-type deformation eld appearing in the Corresponding author.

E-mail address: royd@civil.iisc.ernet.in (D. Roy).

International Journal of Solids and Structures 59 (2015) 171182

Contents lists availab

International Journal of

.e ls(1887), who postulated the existence of a couple-traction alongwith the usual force-traction responsible for the force transfer

the internal work thereby bringing in the lengthconsequence, such formulations introduce diffehttp://dx.doi.org/10.1016/j.ijsolstr.2015.01.0180020-7683/ 2015 Elsevier Ltd. All rights reserved.ute tot. As ar order

e, it isinuuminstance, in problems involving high stress gradients at notchand crack tips, short wavelength dynamic excitations, the behaviorof granular solids, porous materials, modern-day engineeringnano-structures etc. (Eringen, 1976).

In order to circumvent such limitations of the classical con-

1971; Eringen and Kafadar, 1976; Eringen, 1999) and the refer-ences therein.

Gradient type non-local formulations (Mindlin, 1965; Mindlinand Eshel, 1968) are yet another approach to a generalized con-tinuum theory where, in lieu of micro-rotations, several higherClassical continuum mechanics ation of matter throughout the bodyof motion considering only the localof loads and those of inter-moleculthis theory. Such approximations limsical theory to macro-scale phenomlength scale of the loading is mumaterial length scales. However, wcomparable, the microstructural effand predictions of the classical theexperimental results. Substantial dia continuous distribu-tablishes the equationsBoth long range effectsractions are ignored inpplicability of the clas-here the characteristicger than the intrinsichese length scales areuld become signicantpart considerably fromncies are observed, for

and this allows an innitesimal volume element about a materialpoint to rotate independently of the translational motion. Thisidea, as formalized in the micropolar theory (Eringen, 1999),assumes the material micro-rotation to be independent of the con-tinuum macro-rotation (e.g. the curl of the displacement eld).Such a microstructure-motivated description of deformation pro-vided for the inclusion of length scale parameters in the constitu-tive equations which were otherwise absent. The development ofa structured generalized continuum theory only took place severaldecades later. Among numerous such contributions, we cite(Eringen and Suhubi, 1964; Nowacki, 1970; Kafadar and Eringen,1. Introduction incorporation of rotational degrees of freedom, along with theA micropolar peridynamic theory in line

Shubhankar Roy Chowdhury, Md. Masiur Rahaman,Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India

a r t i c l e i n f o

Article history:Received 1 October 2014Received in revised form 17 January 2015Available online 4 February 2015

Keywords:Micropolar peridynamicsLength scalesConstitutive correspondenceTimoshenko-type beamsPlane stress problems

a b s t r a c t

A state-based micropolar pto introduce additional mbring in the physically relemodeling via constitutive cAlong with a general thremodels for both the propoThe efcacy of the proposnumerical simulations of a

journal homepage: wwwelasticity

basish Roy , Narayan Sundaram

ynamic theory for linear elastic solids is proposed. The main motivation is-rotational degrees of freedom to each material point and thus naturallyt material length scale parameters into peridynamics. Non-ordinary typespondence is adopted here to dene the micropolar peridynamic material.mensional model, homogenized one dimensional Timoshenko type beammicropolar and the standard non-polar peridynamic variants are derived.models in analyzing continua with length scale effects is established viabeam and plane-stress problems.

2015 Elsevier Ltd. All rights reserved.

le at ScienceDirect

Solids and Structures

evier .com/locate / i jsols t r

The rest of the paper is organized as follows. Section 2 brieydescribes the state based PD theory and also gives a short accountof linear elastic micropolar theory. While Section 3 reports on asystematic derivation of a general 3D micropolar PD theory, theone dimensional adaptations of the theory are laid out in Sections4 and 5. This is followed by numerical illustrations and a fewconcluding remarks in Sections 6 and 7 respectively.

2. State based PD and micropolar elasticity

For completeness, a concise description of the state based PDtheory along with the constitutive correspondence is given in thissection. Equations of motion in the micropolar theory and the

al of Solids and Structures 59 (2015) 171182governing partial differential equations (PDEs). Therefore compu-tational methods for solving such problems using the classical the-ory either require a redenition of the object manifold so thatdiscontinuities lie on the boundary or some special treatment todene the spatial derivatives of the eld variables on a crackedsurface (Bittencourt et al., 1996; Belytschko and Black, 1999;Areias and Belytschko, 2005).

More recently Silling (2000) introduced a continuum theory, theperidynamics (PD), which is capable of addressing problemsinvolving discontinuities and/or long range forces. One of the mainfeatures of this theory is the representation of the equations ofmotion through integro-differential equations instead of PDEs. Thisrelaxes, to a signicant extent, the smoothness requirement of thedeformation eld and even allows for discontinuities as long as theRiemann integrability of the spatial integrals is ensured. Theseequations are based on a model of internal forces that the materialpoints exert on each other over nite distances. The initial model,the bond-based PD, treats the internal forces as a network of inter-acting pairs like springs. The maximum distance through which amaterial particle interacts with its neighbors via spring like inter-actions is denoted as the horizon. Such pair-wise forces howeverlead to an oversimplication of the model and in particular resultsin an effective Poissons ratio of 1/4 for linear isotropic elasticmaterials. This limitation has been overcome through a more gen-eral model, the state-based PD (Silling et al., 2007). According tothis theory, particles interact via bond forces that are no longergoverned by a central potential independent of the behavior ofother bonds; instead they are determined by the collective defor-mations of the bonds within the horizon of a material particle. Thisversion of the PD theory is applicable over the entire permissiblerange of Poissons ratio. Even though the PD has many attractivefeatures, the scarcity of strictly PD-based material constitutivemodels tends to limit its applicability. This difculty may howeverbe bypassed using a constitutive correspondence framework(Silling et al., 2007), which enables the use of classical materialmodels in a PD formulation.

In the present work, a novel proposal for a PD approach incor-porating micropolar elasticity is set forth. A set of state-basedequations of motion is derived for the micropolar continuum andthe constitutive correspondence utilized to dene the associatedmaterial model. Incorporation of additional physical informationvia the material length-scale parameters has been a primary moti-vation in the current development. Such an enhancement of themodel is expected to emulate more closely the physical behaviorof structures like nano-beams, nano-sheets, fracture characteristicsof thin lms, concrete structures etc. In this context, an earlierwork by Gerstle et al. (2007) on bond-based micropolar PD shouldbe mentioned, which, whilst eliminating the issue of xed Pois-sons ratio, does not offer a ready framework to incorporate therich repertoire of classical material models. The last work alsohas additional limitations in imposing the incompressibility con-straint, often employed in a wide range of models including thoseinvolving plastic deformation in metals (Silling et al., 2007). Alongwith a general three dimensional model, a one dimensional micro-polar PD model for a Timoshenko type beam is also derived in thiswork through an appropriate dimensional descent. For the purposeof comparison, a similar beam model based on the standard non-polar PD is derived. Effects of the length scale parameters on thestatic deformation characteristics of a beam under different bound-ary conditions are numerically assessed conrming the superiorityof the (new) micropolar model over the non-polar one. A couple oftwo dimensional planar problems, rst of a plate with a hole andthe other involving a plate with a central crack under tensile load-

172 S. Roy Chowdhury et al. / International Journing, also hold out similar observations. The theoretical develop-ment in this article is, however, limited to linear isotropic elasticdeformations only.linear elastic material model are also briey reviewed.

2.1. State based PD

Following the approach in Silling et al. (2007), a brief account ofthe state-based PD theory is presented below. PD is a non-localcontinuum theory that describes the dynamics of a body occupyinga region B0 R3 in its reference conguration and B0 R3 in thecurrent conguration. A schematic of the body is shown in Fig. 1.The bond vector n between a material point X 2 B0 and its neighborX0 2 B0, dened as n X0 X, gets deformed under the deforma-tion map v : B0 ! Bt . The deformed bond is given by the deforma-tion vector state Y (refer to Silling et al. (2007) for a precisedenition of states)

YXhni y0 y vX0 vX 1The family of bonds to be considered for a point X is given by itshorizon H dened as HX fn 2 B3jn X 2 B0; jnj < dg, whered > 0 is the radius of the horizon.

The state based PD equations of motion are of the followingintegro-differential form.

qXyX; t ZHX

fTX; thni TX n; thnigdVX0 bX; t 2

where q, T, b are the mass density, long range internal force vectorstate and externally applied body force density respectively.Superimposed dots indicate material derivatives with respect totime. This equation has been shown in Silling et al. (2007) to satisfythe linear momentum balance. In the standard non-polar PD theory,conservation of angular momentum is ensured by imposing thefollowing restriction on the constitutive relation.ZHX

TX; thni YX; thnidVX0 0; 8X 2 B0 3

Silling et al. (2007) have proposed a constitutive correspondence inorder to incorporate the classical material models within the PDFig. 1. Schematic PD body in the reference and current congurations.

below.

Averaged rotation state;HXhni 2h h 13

nal oFY ZHxjnjYhni ndVX0

K1 4

K ZHxjnjn ndVX0 5

Thni xjnjPK1n 6

Here F 2 R3 R3 is the non-local deformation gradient tensor,K 2 R3 R3 a non-local shape tensor, x a non-negative scalarinuence function (xjnj > 0; 8n 2 H) and P ~PF the rstPiolaKirchhoff stress obtained from the classical constitutive rela-tion for ~P written in terms of the non-local deformation gradient F.It has also been shown (Silling et al., 2007) that such correspon-dence is consistent with Eq. (3) and hence satises the conservationof angular momentum.

2.2. Micropolar elasticity

Linear elastic micropolar theory for an isotropic material isbriey reviewed here. For a more detailed exposition, the readeris referred to Eringen (1999). The micropolar kinematical relationsare given through

cji ui;j ekjihk; jji hi;j; i; j; k 1;2;3 7

where c is the innitesimal micropolar strain tensor and j the in-nitesimal wryness tensor. u and h respectively denote the displace-ment and micro-rotation vectors and e is the LeviCivita symbol orthe alternate tensor. ui,j denotes the partial derivative of ui withrespect to the spatial coordinate Xj. Einstein summation conventionholds for the expressions having repeated indices.

Micropolar equations of motions are obtained from the conser-vation of linear and angular momenta and they are written in thefollowing form.

rji;j bi qui; lji;j eijkrjk li Jhi 8

where r is asymmetric stress tensor and l the couple stress tensor.b, l, J respectively refer to the external body force density, bodycouple density and micro-inertia. In linear elastic micropolartheory, these equations of motion are supplemented with thefollowing constitutive relations.

rji kckkdji ~l gcji ~lcij; lji ajkkdji bjji cjij: 9

Here k and ~l are the Lam constants, and g, a, b, c are micropolarmaterial constants. More familiar constants like the shear modulusG, Youngs modulus E and Poissons ratio m are dened in terms ofthese parameters as follows.

G ~l g2; E G3k 2G

k G ; m k

2k 2G : 10

3. A micropolar PD theory

In developing a micropolar PD theory, we assume the existenceframework. This is obtained via a non-local denition of thedeformation gradient in terms of the deformation state and thenequating the PD-based internal energy to the classical one for thesame deformation. The correspondence relations are as shown

S. Roy Chowdhury et al. / International Jourof a non-local internal moment state l that is generated along withthe internal force state T as a response to the externally appliedbody force and body couple.Proof. Substituting Eqs. (2) and (14) in the left hand side of Eq.(15), one obtains the following equation: (with notational abuse,e.g., T is used in place of Thni and similarly for the other states).ZByqXyX;tdVX

ZBJXhX;tdVX

ZB

ZHyfTT0gdVX0dVX

ZBybdVX

ZB

ZHfll0gdVX0dVX

12

ZB

ZHy0 yfTT0gdVX0dVX

ZBldVX 16

Since no interaction exists beyond the horizon (H), all the inner3.2. Equations of motion

For the micropolar PD body, we postulate that the followingadditional equations, along with the equations of motion given inEq. (2), must be satised.

JXhX; t ZHX

flhni l0hnigdVX0

12

ZHX

Yhni fThni T0hnigdVX0 lX; t 14

where l0hni lX n; thni and a similar denition holds for T0. lis the externally applied body couple density.

In order to be physically admissible, it is necessary that thestates l and T satisfy the balance of linear and angular momentafor any bounded body B. While the former (i.e. the balance of linearmomentum) is known to be satised by the specic form of theequations of motion adopted in Eq. (2), the latter requires the sat-isfaction of the following identity.ZBy qXyX; tdVX

ZBJXhX; tdVX

ZBy bX; tdVX

ZBlX; tdVX 15

Proposition 3.1. Let the bounded body B be subjected to body forceand body couple density b and l respectively. Let the internal force andmoment vector states be T and l respectively. Then if Eq. (14) holds inB, balance of angular momentum will be satised, i.e., Eq. (15) willhold.3.1. Kinematics

Following the discussion on micropolar elasticity in Section 2.2,the micro-rotation h is introduced as additional degrees of freedomwithin the PD continuum in order to allow for rigid-type rotationof each material point. For a kinematical description of the PDbody, we need to dene a few new vector states as listed below.

Relative displacement state; UXhni u0 u vX0 X0 vX X 11

Relative rotation state; HXhni h0 h hX0 hX 12_ 1 0

f Solids and Structures 59 (2015) 171182 173integrals can be extended over the entire body (B) without affectingthe result. A change of variable X$ X0 and the subsequent applica-tion of Fubinis theorem lead to the following simplications:

al oZB

ZHfl l0gdVX0dVX

ZB

ZBfl l0gdVX0dVX

ZB

ZBfl0 lgdVX0dVX 0 17

12

ZB

ZHy0 yfTT0gdVX0dVX

12

ZB

ZBy0 yfTT0gdVX0dVX

12

ZB

ZBy0 fTT0gdVX0dVX

12

ZB

ZByfTT0gdVX0dVX

12

ZB

ZByfT0 TgdVX0dVX

12

ZB

ZByfTT0gdVX0dVX

ZB

ZByfTT0gdVX0dVX 18

Using Eqs. (17) and (18), Eq. (16) gets simplied to Eq. (15) and thusestablishes the balance of angular momentum. h

It is to be noted that for problems with small elastic deforma-tion, the second term of right hand side of Eq. (14) could beapproximated by 12

RHX n fThni T0hnigdVX0 .

3.3. Energy balance and constitutive relations

Wewill focus here only on the mechanical energy balance with-out considering any heat source or ux. Taking scalar products ofboth sides of Eq. (2) with the velocity _y and Eq. (14) with _h andtheir subsequent addition and integration over a nite sub-regionP B result in the following.ddt

ZP

q _y _y2

dVX ddtZP

J _h _h2

dVX ZP

ZBfT T0g _ydVX0dVX

12

ZP

ZBy0 y fT T0g _hdVX0dVX

ZP

ZBfl l0g _hdVX0dVX

ZPb _ydVX

ZPl _hdVX 19

Note that all the inner integrals in the expression above have beenextended to the whole body since there is no interaction beyond thehorizon, as indicated earlier.

The following identities (Eqs. (20)(22)) are needed for furthersimplication.

fThni T0hnig _y Thni _y0 T0hni _y Thni _y0 _y 20

12y0 yfThniT0hnig _h Thni

_h0

2yy0T0hni

_h2y0 y

!

Thni _h _h02

y0 y 21

lhni l0hnin o

_h lhni _h0 l0hni _h

lhni _h0 _h 22

Using the above identities, each of rst three terms on the righthand side of Eq. (19) could be split into two integrals as illustratedbelow.ZP

ZBfT T0g _ydVX0dVX

ZP

ZBT _y0 T0 _ydVX0dVX

ZP

ZBT _y0 _ydVX0dVX

ZP

ZBnP

T _y0 T0 _ydVX0dVX

174 S. Roy Chowdhury et al. / International JournZP

ZBT _y0 _ydVX0dVX 23where the antisymmetry of the integrand in rst integral is used toobtain the last step. Similarly,

12

ZP

ZBy0 y fT T0g _hdVX0dVX

ZP

ZBnP

T _h0

2 y y0 T0

_h2 y0 y

!dVX0dVX

ZP

ZBT

_h _h02

y0 ydVX0dVX 24

andZP

ZBfl l0g _hdVX0dVX

ZP

ZBnP

l _h0 l0 _hdVX0dVX

ZP

ZBl _h0 _hdVX0dVX 25

Using Eqs. (23)(25), Eq. (19) can be rewritten in the followingform, representing a power balance.

_KP WabsP WsupP 26

where KP RP q _y _y2 dVX RP J _h _h2 dVX is the kinetic energy in P. Theabsorbed power in P and the supplied power to P are respectivelygiven by

WabsP ZP

ZBT _y0 _y

_h _h02

y0 y( )

dVX0dVX

ZP

ZBl _h0 _hdVX0dVX 27

WsupPZP

ZBnP

T _y0 _h0

2yy0

!T0 _y

_h2y0 y

! !dVX0dVX

ZP

ZBnP

l _h0 l0 _hdVX0dVXZPb _ydVX

ZPl _hdVX 28

The above notions such as absorbed power, supplied power areborrowed from Silling and Lehoucq (2010). Since no other sourceof energy is considered, the absorbed power relates to the internalenergy EP as _EP WabsP. Also the form of integrals in (27)shows the additive character of the internal energy (i.e.,EP1 EP2 EP1 P2) thus ensuring the existence of an inter-nal energy density, e which therefore has the following rate form.

_eZBT _y0 _y

_h _h02

y0 y( )

dVX0 ZBl _h0 _hdVX0 29

For small deformation, we introduce an approximation to the aboveexpression leading to

_eZBT _y0 _y

_h _h02

X0 X( )

dVX0 ZBl _h0 _hdVX0 30

Using the denition of inner product of states given in Silling et al.(2007), the expression above is rewritten as follows.

_e T _Y _H_

X l _H T _U _H_

X l _H T _U

H_ l _H; 31

where UH_ UH

_

X is a composite state and its action on a bondn is given as U_hni u0 u h0h X0 X.

f Solids and Structures 59 (2015) 171182H 2

Assuming the energy density functional e to depend on YH_ and

H, the rate of e would be given by the following expression.

H xjnjrK1n _UH_hnidVX0

tr() is the trace operator, ()T the transpose operator and I thesecond order identity tensor.

The equations of motion Eqs. (2) and (14) along with theconstitutive model (38) complete the description of linear elasticmicropolar PD theory.

4. 1D micropolar PD beam

z

nal of Solids and Structures 59 (2015) 171182 175H

ZHxjnjlK1n _HhnidVX0 37

Note that the symmetry of shape tensor K is utilized in writing thelast expression. Comparing (37) with (31), the following constitu-tive relations are obtained.

Thni xjnjrK1n and lhni xjnjlK1n 38

Te with respect to UH_ and H respectively. Comparison of (31) with

(32) leads to the following constitutive relations (which are consis-tent with the second law of thermodynamics and also obtainable bythe ColemanNoll procedure (Tadmor et al., 2012)).

T eUH_ and l eH 33

3.3.1. Constitutive correspondenceIn order to make use of the micropolar material model

described in Eq. (9), the constitutive correspondence route isadopted and this calls for denitions of non-local strain and wry-ness tensors. Note that, for a continuously differentiable deforma-tion eld on B, the following holds.

UH_hni uX n uX hX n hX

2 X n X

ru e hn Ojn2j cnOjn2j 34

Hhni hX n hX rhn Ojn2j jnOjn2j 35where r is the gradient operator, e is the third order alternatetensor, and c and j are respectively the innitesimal micropolarstrain and wryness tensors. Analogous to Eq. (4), following are thenon-local approximations to c and j.

cUH_

ZHxjnjU

H_hni ndVX0

K1;

jH ZHxjnjHhni ndVX0

K1 36

Let wc; j be the internal energy density (per unit volume of themicropolar continuum) written in terms of the non-local kinematicquantities. Then equating eU

H_;H with wc; j, the PD model for

linear elastic micropolar materials is established. Following thecomputations below, explicit forms of the desired PD material mod-el could be obtained.

_eUH_;H _wc; j r : _c l : _j

r :ZHxjnj _U

H_hni ndVX0

K1

l :ZHxjnj _Hhni ndVX0

K1

ZHxjnjr : _U

H_hni nK1dVX0

ZHxjnjl : _H _hni nK1dVX0Z_eUH_;H eU

H_ _UH_ eH _H; 32

where eU_ and eH are the Frchet derivatives (Silling et al., 2007) of

S. Roy Chowdhury et al. / International Jourwhere r ktr c I ~l g c ~lc andl atr j I bj cjT 39x

ySince the discovery of carbon nano-tube in the early 1990s, anintensive research effort has been directed to the analysis of suchnano structures. Successful attempts have been made in develop-ing non-local continuum formulations to analyze static and vibra-tion characteristics of nano-rods, nano-beams, nano-lms etc.Being a non-local model incorporating material length scale para-meters, the proposed micropolar PD model provides a suitableframework for such problems. In the specic context of analyzingnano-rods and nano-beams, a one dimensional version of the pro-posed model would be of considerable interest, enabling one toobtain results with lesser computational effort as compared tothe full-blown 3D model.

A Timoshenko type beam model, set in the micropolar PDframework, is presented here.

4.1. Geometry and assumptions

Fig. 2 shows a schematic of the beam geometry. A Cartesiancoordinate frame is considered with its origin located at one endof the beam. The x axis is chosen along the length and lies on theplane containing neutral axes of the cross sections, y is along theneutral axis of the cross section containing the origin and z is alongthe thickness. The beam cross section is assumed to be symmetricabout the z axis. Our proposal for the PD micropolar beam is basedon a few assumptions; viz. the cross-sectional dimensions are sig-nicantly smaller than the axial dimension; material propertiesonly vary along the length; the body force and the body coupleare independent of the y axis; loading condition does not lead totwisting; displacement eld does not vary appreciably alongheight.

4.2. Approximations to displacement and micro-rotation elds

For a Timoshenko type beam, the three dimensional displace-ment and micro-rotation elds are approximated as follows.

uxx; y; z; t ux; t zwx; t; uyx; y; z; t 0;uzx; y; z; t wx; t 40

hxx; y; z; t 0; hyx; y; z; t hx; t; hzx; y; z; t 0 41where ux;uy;uz and hx; hy; hz are the Cartesian components of thedisplacement and micro-rotation elds. The approximations aboveallow expressing the three dimensional elds via several onedimensional descriptors, namely axial displacement u, macro-rotation w, transverse displacement w and micro-rotation h aboutthe y axis.Fig. 2. A schematic representation of the beam geometry.

Z xd

al o4.3. 1D kinematic states

The necessary one dimensional kinematic states for this beammodel are listed below.

Relative axial displacement state uxhx0 xi ux0 uxRelative transverse displacement state wxhx0 xi wx0wxRelative macro-rotation state wxhx0 xi wx0 wxRelative micro-rotation state hxhx0 xi hx0 hxAverage macro-rotation state w

_

xhx0 xi 12 wx0 wxAverage micro-rotation state h

_

xhx0 xi 12 hx0 hx

These states will be utilized to dene non-local one dimensionalstrain-like quantities and subsequently the constitutive relationsfor several one dimensional generalized force states (to be intro-duced in the next section) will also be given in terms of these kine-matic states.

4.4. Equations of motion

For the beam model at hand, the necessary equations of motionis obtainable by integrating the three dimensional equations ofmotions (Eqs. (2) and (14)) over the cross-section and introducingseveral one dimensional generalized force states. These equationshave the following form.

mux; t Z xdxd

Nxxhx0 xi Nxx0hx x0idx0 Bxx; t 42

mwx; t Z xdxd

Nzxhx0 xi Nzx0hx x0idx0 Bzx; t 43

~Jhx; t Z xdxdMyxhx0 xi Myx0hx x0i

dx0

12

Z xdxd

Nzxhx0 xi Nzx0hx x0i x x0dx0

12

Z xdxd

Msxhx0 xi Msx0hx x0i dx0 Lyx; t 44

The following equation is required in addition to Eqs. (42)(44). It isobtained by multiplying the rst of equation (2), i.e. the equationfor the motion in the x direction, with z and then integrating overthe cross-sectional area.

Iqwx; t Z xdxd

Mxhx0 xi Mx0hx x0i dx0

12

Z xdxd

Msxhx0 xi Msx0hx x0i dx0 Lzx; t 45

Denitions of the force states and moment states introduced inEqs. (42)(45) are listed below.

Nxxhx0 xi ZA

ZATXhx0 xixdA0dA;

Nzxhx0 xi ZA

ZATXhx0 xizdA0dA;

Myxhx0 xi ZA

ZAlXhx0 xi

ydA0dA;

Msxhx0 xi ZA

ZATXhx0 xixz0 zdA0dA

176 S. Roy Chowdhury et al. / International JournMxhx0 xi 12

ZA

ZATXhx0 xixz0 zdA0dA_e1 xd

Nxhx0 xi _uhx0 xidA0

Z xdxd

Nzhx0 xi _whx0 xi _h_

hx0 xix0 x

dA0

Z xdxdMyhx0 xi _hhx0 xidA0

Z xdxd

Mhx0 xi _whx0 xidA0

Z xdxd

Mshx0 xi_w_

hx0 xi _h_

hx0 xi

dA0 46

Using a similar denition for inner product of states in one dimen-sion and introducing the required composite states, the equationabove can be written in the following compact form.

_e1 Nx _u Nz _wh_

xMy _hM _wMs _w

h_ 47

where wh_

xhx0 xi hx0 xi h

_

hx0 xix0 x and wh_hx0 xi

w_

hx0 xi h_

hx0 xi.Assuming the energy density functional e1 to depend on the

states u, wh_

x, h, w and w

h_, similar constitutive relations as in

Eq. (33) may be obtained as

Nx e1u; Nz e1wh_x; My e1h; M e1w and Ms e1w

h_

48

4.5.1. Constitutive correspondenceA similar strategy as described in Section 3.3.1 is adopted here

to establish the constitutive correspondence in one dimension. Inthe classical micropolar description of a beam (Ramezani et al.,2009), an expression for strain energy rate per unit length in termsof one dimensional forces and moments (or bending moment) andtheir conjugate strain-like quantities can be written as follows.

_w1 ANxx _cu ANxz _cwh ANzx _cwh AMxy _jh M _jw 49

where cu @u@x, cwh @w@x h, cwh w h, jh @h@x and jw @w@x.Denitions of the one dimensional forces and moments used in

(49) along with the constitutive relations (see Eq. (9)) are givenbelow..

Nxx 1AZArxxdA Ecu; Nxz

1A

ZArxzdA ~l gcwh ~lcwh

Nzx 1AZArzxdA ~l gcwh ~lcwh;Also the integrated body forces and couples that appear in the beamequations of motion are derived as

Bxx; t ZAbxdA; Bzx; t

ZAbzdA; Lzx; t

ZAzbxdA;

Lyx; t ZAlydA

Further, m RA qdA, ~J RA JdA. The area element dA = dydz anddA0 dy0dz0. d is the maximum (radial) distance over which the longrange interaction is considered.

4.5. Constitutive equations

To nd the constitutive relations for these one dimensionalnon-local force and moment states, we dene the internal energyper unit length (e1) of the beam. Eq. (30), using approximate defor-mation elds (see Section 4.2), is integrated over the cross-section-al area leading to the following expression.

f Solids and Structures 59 (2015) 171182Mxy 1AZAlxydA bjh and M

1A

ZAzrxxdA EIjw 50

We introduce the following one dimensional, non-local, strain-likequantities that are useful in establishing the constitutivecorrespondence.

cuu Z xdxd

x1jx0 xjuhx0 xix0 xdx0

K11

cwhwh_

x

Z xdxd

x1jx0 xjwh_

xhx0 xix0 xdx0

K11

cwhwh_

Z xdxd

x1jx0 xjwh_hx0 xix0 x2dx0

K11

jhh Z xdxd

x1jx0 xjhhx0 xix0 xdx0

K11

jww Z xdxd

x1jx0 xjwhx0 xix0 xdx0

K11 51

Iqwx;tZ xdxd

Mxhx0 xiMx0hxx0i dx0

12

Z xdxd

Nzxhx0 xiNzx0 hxx0i xx0dx0 Lzx;t 55

The above equations are derived following the same steps asoutlined in Section 4. For the derivation of Eq. (55), constitutiverestrictions for the non-polar material given by Eq. (3) are imposedin addition. The force states, body force and couple have the samedenitions given in the previous section.

The required constitutive equations are

Nxhx0 xi x1jx0 xjAEcuK11 x0 x;Nzhx0 xi x1jx0 xjAGcwwK11 x0 x;Mhx0 xi x1jx0 xjEIjwK11 x0 x; 56

S. Roy Chowdhury et al. / International Journal of Solids and Structures 59 (2015) 171182 177Here x1jx0 xj > 0 for jx0 xj 6 d and x1jx0 xj 0 otherwise.The one dimensional shape tensor is given by K1

R xdxd x1jx0 xj

x0 x2dx0.cu, cwh, cwh, jh and jw are respectively the non-local approxima-

tions to cu, cwh, cwh, jh and jw. Replacing the local strain terms inEq. (49) by their non-local counterparts and using the fact that_e1 _w1, the following relations are obtained using Eq. (46).Nxhx0 xi x1jx0 xjAEcuK11 x0 xNzhx0 xi x1jx0 xjA~l gcwh ~lcwhK11 x0 xMyhx0 xi x1jx0 xjAbjhK11 x0 xMhx0 xi x1jx0 xjEIjwK11 x0 xMshx0 xi x1jx0 xjA~l gcwh ~lcwhK11 x0 x2 52Eqs. (42)(45) and the constitutive relations in (52) completelydescribe the micropolar Timoshenko type PD beam.

5. Non-polar PD beam

For the state based non-polar PD as described in Section 2.1, asimilar beam model is derived. The beam geometry, different rele-vant assumptions, the approximation of the displacement eld andthe necessary states are similar to those noted in Section 4.Dynamics of this non-polar 1D model is captured through thefollowing equations of motion.

mux; t Z xdxd

Nxxhx0 xi Nxx0hx x0i dx0 Bxx; t 53

mwx; t Z xdxd

Nzxhx0 xi Nzx0hx x0i dx0 Bzx; t 54Fig. 3. Transverse displacement of one dimensional beam: (a) canwhere the non-local strains, except cww, are dened in Eq. (51).Denition of cww is as follows.

cwwww_ x Z xdxd

x1jx0 xjww_ xhx0 xix0 xdx0

K11

with ww_xhx0 xi whx0 xi w

_

hx0 xix0 x.In the context of non-polar PD beam modeling, the reader is

referred to OGrady and Foster (2014) where an EulerBernoullitype PD beam model is derived.

6. Representative numerical examples

In this section, three illustrative examples, viz. the bending ofa thin beam, the in-plane tensile response of a plate with a cir-cular hole and response of a centrally cracked plate under uni-axial tensile load (mode 1), are considered to demonstrate theeffects of micropolarity as incorporated in the state based PDtheory. While the proposed one-dimensional model (seeSections 4 and 5) is employed for the bending analysis of abeam, a somewhat straightforward 2D plane-stress type reduc-tion of the 3D theory, the details of which are given in Section 3,is adopted for the analysis of a plate with a hole and a platewith a central crack.

For the purpose of numerical implementation, body (1D or 2D)is discretized into a nite number of nodes in its reference con-guration. Each node is associated with nite length in case of1D body and nite area in the 2D case. These nodes are typicallytermed as PD particles. All the governing equations are then writ-ten for these PD particles along with a Riemann-sum typeapproximation (summed over these PD particles) for the integralquantities in these equations (Eqs. (2), (14), (42)(45), (53)(55)).tilever beam (b) xed beam and (c) simply supported beam.

al o178 S. Roy Chowdhury et al. / International JournThis discretization results in a set of algebraic equations with dis-placement and microrotation at the different PD particles asunknowns, which are solved for after proper imposition of bound-ary conditions. Details of numerical implementation of a typical PDalgorithm may be found elsewhere, e.g. Breitenfeld et al. (2014).

6.1. One-dimensional beam

Deformation characteristics of nano- or micro-beams arestrongly affected by the internal length scale parameters of the

Fig. 4. Macro rotation of one dimensional beam: (a) cantilev

Fig. 5. Micro rotation of one dimensional beam: (a) cantilev

Fig. 6. Tip deection of cantilever beam (a) thickness vs. tip deection for constant lengf Solids and Structures 59 (2015) 171182constituent materials. Such beams generally show stiffer responsethan what is predicted by classical (local) theories. The proposedone dimensional micropolar PD beammodel is expected to capturesuch stiffer response correctly owing to the extra information it car-ries on the material length scale parameters. One may contrast thiswith the non-polar PD, which should fail to predict such responsedespite carrying some length scale information that, whilst beinginherent in a PD formulation, may still be physically irrelevant.

For the numerical implementation, the geometric and materialproperties considered for the beam model are:

er beam (b) xed beam and (c) simply supported beam.

er beam (b) xed beam and (c) simply supported beam.

th scale (0.01 m) (b) length scale vs. tip deection for constant thickness (0.05 m).

L 1 m; h 0:05 m; b 0:05 m; E 20 GPa; m 0:3;Fig. 7. Plate with a hole subjected to uniform tension.

S. Roy Chowdhury et al. / International Journal og G=20; b G=5000:

where L; h and b are the length, thickness and width of beam respec-tively. The beam is subject to the following loading conditions.

Bx 0; Bz 103 N=m and Ly 0 N m=m:Please note that while these parameters may not be physically rep-resentative, they are adequate for the purpose of demonstration ofour technique.

The beam is analyzed for three different boundary conditions,namely xed-free (a cantilever beam), xedxed (a xed beam)and pinnedpinned (a simply supported beam), using both non-polar and micropolar PD models. As a standard practice in PD,the boundary conditions are applied, in each case, over a patch ofnodes considered beyond the actual length of the beam. Withl b=2Gp as the denition for the length scale parameter, itmay be observed that the beam thickness considered is of the sameorder as l = 0.01 and hence it is anticipated that the length scaleparameter would considerably inuence the deformation charac-teristic of the beam. As reported in Figs. 3 and 4, the non-polarFig. 8. Stress (ryy) distribution: (a) non-polar PD (b) micropolar PD.stress concentration near the hole is signicantly lesser than whatis predicted by the classical theory. Therefore, in the context of 2Delastostatic problems, this example is chosen here to verify theeffectiveness of the proposed micropolar PD theory over the stan-dard non-polar one.

To reduce the computational overhead, a plate of very smalldimensions (length 0.02 m, width 0.005 m and thicknesst = 0.001 m with a central circular hole of radius a = 0.00125 m) isconsidered here for illustrative purposes. The plate is subjectedto a uniaxial uniform tension p0 106 N=m as shown in Fig. 7.Plane-stress constitutive relations are used to solve this problemusing both non-polar PD and proposed micropolar PD models.The plate material is assumed to have the following properties:

E 100 GPa; m 0:3; g G=1:5; b G=160000

In order to make the length scale effect conspicuous, the hole radiusis chosen less than the length scale parameter value (the latterbeing 0.0018 m) as dened in Section 6.1.

Although the force and moment states are obtained by solvingthe micropolar PD model, we have chosen to report the compo-nents of the equivalent nonlocal Cauchy-type stress in order todisplay the results in more familiar forms. Displacement andmicro-rotation vectors, obtained by solving the PD model, are usedto compute the non-local strain and wryness using the denitionsgiven in (36). Constitutive relations given in (9) are appropriatelyreduced for the plane stress case and evaluated using the non-localstrain and wryness. Non-local stress and couple stress componentsthus computed are reported in some of the following gures. APD solutions are seen to be insensitive to the material length scaleparameter. Exact solutions considering the EulerBernoulli modelare also reported in Figs. 3 and 4. Low thickness to length ratioof the beams allows the EulerBernoulli type model to be applica-ble in this case. Both EulerBernoulli and non-polar Timoshenko-type PD models yield comparable results and fail to reect theanticipated stiffening of the beams at that length scale. The pro-posed micropolar PD model, on the other hand, successfully bringsout the expected length scale effects in the transverse deectionand rotation proles of the beam (see Figs. 3 and 4). Fig. 5 showsthe computed micro-rotations for each of the three beams.

An additional numerical parametric study is done for the can-tilever beam to demonstrate the effect of the length scale on itstransverse tip deection. Keeping the length scale parameter xedat 0.01 m, as the thickness is reduced, the deection tends to blowup when the non-polar PD model is employed. However the pro-posed micropolar PD model remedies the situation by obtaininga deformation prole whose magnitude is smaller by many orders(Fig. 6(a)) and thus remains consistent with the physically expect-ed scenario. Fig. 6(b) shows the result of another study where thecantilever beam thickness is kept xed at 0.05 m and the tipdeection is reported for different length scale values. While thenon-polar PD model again shows up length scale insensitive tipdeection, the micropolar PD deection is clearly affected by thevarying length scale. As the length scale value is reduced from0.05 m to zero, the length scale effect diminishes and both themicropolar PD and non-polar PD schemes lead to the samesolution.

6.1.1. Elastic plate with a holeThe static in-plane tensile response of a plate with a central cir-

cular hole is known to display a strong inuence of the materiallength scale parameter as the hole radius becomes equal or lessthan the length scale value (Mindlin, 1963). In such a case, the

f Solids and Structures 59 (2015) 171182 179similar treatment is adopted to represent the results through thenon-polar PD model too. Fig. 8 shows the normal stress (ryy)distributions via both the PD models.

Fig. 9. Shear stresses: (a) rxy ryx (non-polar PD) (

Fig. 10. Couple stresses: (a) lzx (micropolar PD) (b) lzy (micropolar PD).

Fig. 11. Normalized stress (ryy/(p0t)) distribution along h = 0.

180 S. Roy Chowdhury et al. / International Journal oIt may be observed that the incorporation of the length scaleparameter in the micropolar PD results in lower normal stressesnear the hole than those through the non-polar PD model by a fac-tor of nearly 1.3. While Fig. 9(a) shows the shear stress compo-nents obtained in the non-polar PD case to be symmetric, theasymmetric nature of the shear stress components (rxyryx) basedon the micropolar PD theory is evident in Fig. 9(b) and (c). Distri-butions of non-zero couple stresses, especially near the hole, inthe micropolar PD model are shown in Fig. 10. Such stresses areignored in the non-polar PD variant. Essentially the considerationof couple stresses in the micropolar PD theory entails an inherentasymmetry of the stress tensor and an incorporation of the intrin-sic material length parameter is responsible for reducing the nor-mal stresses in the vicinity of the hole.

In order to provide an explicit comparison of the normal stress-es (ryy) obtained from both the models, the normalized stress dis-tribution (ryy=p0) along the radial line h = 00 is reported in Fig. 11.This gure clearly shows that the stress concentration factor(dened as the normalized stress at the hole edge i.e. at r = a and

b) rxy (micropolar PD) (c) ryx (micropolar PD).f Solids and Structures 59 (2015) 171182h = 0) is predicted higher in non-polar PD case, with a much lesserconcentration being obtained when the micropolar PD model issolved. Effect of the length scale in reducing the stress concentra-tion is thus numerically validated through the proposed micropo-lar PD model.

6.1.2. Elastic plate with a central crackIn continuing our studies to assess the effect of micropolarity

within the PD framework, we take up, as the nal example, another2D plane stress problem of a plate with a central crack undermode one loading condition. Solution to a similar problem hasbeen discussed, using a non-polar PD scheme, in (Breitenfeldet al., 2014). A plate of length 0.005 m, width 0.005 m andthickness t = 0.001 m with a central crack of size 2a = 0.00125 mis considered here. The plate is subjected to a uniaxial uniformtension p0 106 N=m such that it creates a mode 1 type loading.Plane-stress constitutive relations are used to solve this problemusing both non-polar and micropolar PD models. The plate materi-al is assumed to have the following properties:

E 100 GPa; m 0:3; g 1:8G; b G=160000:

The crack size here is considered to be less than the length scale ofthe material and therefore micropolarity should reduce the stressconcentration at the crack tip. Fig. 12 shows the ryy distribution

nal oS. Roy Chowdhury et al. / International Jourfor both non-polar and micropolar PD cases and exhibits certainqualitative differences in the stress distributions. These results cor-respond to the same discretization level and the horizon radius.Fig. 13 shows the normalized stress (ryy/(p0t)) distribution alongthe crack. It reveals the effect of micropolarity on the solution inthe form of a reduction of the stress concentration at the cracktip. Note that it is the smearing effect over the horizon, inducedby the PD model, that renders the crack-tip stress nite, whose val-ue however gets inuenced by micropolarity. A more detailed studyto understand the effect of length scales introduced via the horizonand micropolarity on the crack tip stress eld and convergence ofthe numerical solutions will be pursued elsewhere.

7. Conclusions

A three dimensional, state-based micropolar PD model has beendeveloped, incorporating physically consistent nonlocal particleinteractions through the use of an additional length scale para-meter. This length scale parameter is in addition to the horizonradii present in the standard PD theory. The material model in thiswork has been restricted to the micropolar isotropic elastic case.The incorporation of micropolar effects in the PD model equips itto better predict the deformation of continua where the lengthscale effect is signicant, e.g. very thin beams. It is also expected

Fig. 12. Stress (ryy) distribution: (a)

Fig. 13. Normalized stress (ryy/(p0t)) distribution along the direction of crack.to be useful to model other, similar continua like nano-beamsand nano-sheets. In addition, homogenized one dimensionalmicropolar as well as non-polar PD beam models have beenderived. In lieu of the full-blown 3D continuum, the use of dimen-sionally reduced models such as the micropolar PD beam involvessignicantly lower computational overhead. The non-polar variantof the one dimensional PD model is shown to be material lengthscale insensitive and therefore predicts unphysically large defor-mation. This is remedied by developing a micropolar PDmodel thataccounts for length scale effects, thus restoring the physicality ofthe response. A similar observation holds even for planar elasto-static problems. An application of the proposed micropolar PDmodel for dynamical systems and its extensions incorporating

non-polar PD (b) micropolar PD.f Solids and Structures 59 (2015) 171182 181plasticity and damage will be considered elsewhere. Indeed, it iswith such problems as damage propagation, which violate materialcontinuity and whose evolution necessarily demands length scales,that the full spectrum of advantages of the proposed micropolar PDscheme will be adequately exploited.

Acknowledgment

DR thanks the Defense Research and Development Organiza-tion, Government of India, for partially funding this researchthrough Grant #DRDO/0642.

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182 S. Roy Chowdhury et al. / International Journal of Solids and Structures 59 (2015) 171182

A micropolar peridynamic theory in linear elasticity1 Introduction2 State based PD and micropolar elasticity2.1 State based PD2.2 Micropolar elasticity

3 A micropolar PD theory3.1 Kinematics3.2 Equations of motion3.3 Energy balance and constitutive relations3.3.1 Constitutive correspondence

4 1D micropolar PD beam4.1 Geometry and assumptions4.2 Approximations to displacement and micro-rotation fields4.3 1D kinematic states4.4 Equations of motion4.5 Constitutive equations4.5.1 Constitutive correspondence

5 Non-polar PD beam6 Representative numerical examples6.1 One-dimensional beam6.1.1 Elastic plate with a hole6.1.2 Elastic plate with a central crack

7 ConclusionsAcknowledgmentReferences