Copyright c 2003 Tech Science Press CMES, vol.4, no.1, pp.177-196, 2003 Analysis of Materials with Strain-Gradient Effects: A Meshless Local Petrov-Galerkin(MLPG) Approach, with Nodal Displacements only Z.Tang, S. Shen and S.N. Atluri 1 Abstract: A meshless numerical implementation is reported of the 2-D Fleck-Hutchinson phenomenologi- cal strain-gradient theory, which fits within the frame- work of the Toupin-Mindlin theories and deals with first- order strain gradients and the associated work-conjugate higher-order stresses. From a mathematical point of view, the two-dimensional Toupin-Mindlin strain gradi- ent theory is a generalization of the Poisson-Kirchhoff plate theories, involving, in addition to the fourth-order derivatives of the displacements, also a second-order derivative. In the conventional displacement-based ap- proaches in FEM, the interpolation of displacement re- quires C 1 –continuity (in order to ensure convergence of the finite element procedure for 4 th order theories), which inevitably involves very complicated shape func- tions. These shape functions involve large numbers of degrees of freedom in every element, including nodal dis- placements, nodal rotations (i.e. first order gradients of displacement), and even higher order derivatives. C 1 - continuous methods are mostly feasible only for one- dimensional problems. The standard approach for solv- ing Bernoulli-Euler beam problems is by employing C 1 - continuous Hermite cubic shape functions, interpolating both displacements and rotations (i.e., slopes). For two- dimensional problems, such as involving plate and shell analysis, C 1 -continuous methods are very complicated, and formulations for three-dimensional problems as they arise from strain gradient theories become more or less intractable. The high computational cost and large num- ber of degrees of freedom soon place such formulations beyond the realm of practicality. Recently some mixed and hybrid formulations, requiring only C 0 -continuity, have also been developed and applied to strain gradient plasticity or elasticity problems. But they are even more problematic. While some of the developed elements 1 Center for Aerospace Research and Education, University of California, Irvine 5251 California Ave, Suite 140 Irvine, CA 92612, USA have been subjected to the Patch test and other bench- mark problems, a rigorous numerical analysis is missing: mathematical proofs of consistency and stability have not been demonstrated, and the rate of convergence has not been established. The large number of nodal degrees of freedom is still inevitable in such mixed methods. It is evident that currently, no efficient finite element methods are available for strain gradient theory formulations. In this paper, a truly meshless approach, the Meshless Local Petrov-Galerkin Method(MLPG), is introduced for higher-order gradient theories that trace their roots to the work of Toupin. The degrees of freedom consist of only nodal displacements, i.e. nodal rotational degrees of free- dom are not used. A numerical analysis of the method is presented, covering the usual ground of consistency, sta- bility and hence, convergence in several examples. All the results show that, when solving fourth-order elliptic problems such as those arising in gradient theories of ma- terial behavior, the MLPG method is superior to primal or mixed finite element procedures. keyword: MLPG, Gradient theory, MLS 1 Introduction The most widely used theories of continuum solid me- chanics involve non-polar materials that are also sim- ple in the sense of Noll (1965). Material point rota- tions are neglected, and the Cauchy stress is assumed to be a functional of the deformation, only through its first gradient. This local assumption has, for long, proved to be adequate when the wavelength of a defor- mation field is much larger than the dominant micro- structural length scale of the material. However, when the two length scales are comparable, this assumption is questionable. An interest in polar materials has ex- isted at least since the pioneering work of the Cosser- ats(1907,1909). The rotational degrees of freedom intro- duce couple stresses and body couples. Rich theories of
Analysis of Materials with Strain-Gradient Effects: A Meshless LocalPetrov-Galerkin(MLPG) Approach, with Nodal Displacements only
Z.Tang, S. Shen and S.N. Atluri1
Abstract: A meshless numerical implementation isreported of the 2-D Fleck-Hutchinson phenomenologi-cal strain-gradient theory, which fits within the frame-work of the Toupin-Mindlin theories and deals with first-order strain gradients and the associated work-conjugatehigher-order stresses. From a mathematical point ofview, the two-dimensional Toupin-Mindlin strain gradi-ent theory is a generalization of the Poisson-Kirchhoffplate theories, involving, in addition to the fourth-orderderivatives of the displacements, also a second-orderderivative. In the conventional displacement-based ap-proaches in FEM, the interpolation of displacement re-quires C1 –continuity (in order to ensure convergenceof the finite element procedure for 4th order theories),which inevitably involves very complicated shape func-tions. These shape functions involve large numbers ofdegrees of freedom in every element, including nodal dis-placements, nodal rotations (i.e. first order gradients ofdisplacement), and even higher order derivatives. C1-continuous methods are mostly feasible only for one-dimensional problems. The standard approach for solv-ing Bernoulli-Euler beam problems is by employing C 1-continuous Hermite cubic shape functions, interpolatingboth displacements and rotations (i.e., slopes). For two-dimensional problems, such as involving plate and shellanalysis, C1-continuous methods are very complicated,and formulations for three-dimensional problems as theyarise from strain gradient theories become more or lessintractable. The high computational cost and large num-ber of degrees of freedom soon place such formulationsbeyond the realm of practicality. Recently some mixedand hybrid formulations, requiring only C 0-continuity,have also been developed and applied to strain gradientplasticity or elasticity problems. But they are even moreproblematic. While some of the developed elements
1 Center for Aerospace Research and Education,University of California, Irvine5251 California Ave, Suite 140Irvine, CA 92612, USA
have been subjected to the Patch test and other bench-mark problems, a rigorous numerical analysis is missing:mathematical proofs of consistency and stability have notbeen demonstrated, and the rate of convergence has notbeen established. The large number of nodal degrees offreedom is still inevitable in such mixed methods. It isevident that currently, no efficient finite element methodsare available for strain gradient theory formulations.In this paper, a truly meshless approach, the MeshlessLocal Petrov-Galerkin Method(MLPG), is introduced forhigher-order gradient theories that trace their roots to thework of Toupin. The degrees of freedom consist of onlynodal displacements, i.e. nodal rotational degrees of free-dom are not used. A numerical analysis of the method ispresented, covering the usual ground of consistency, sta-bility and hence, convergence in several examples. Allthe results show that, when solving fourth-order ellipticproblems such as those arising in gradient theories of ma-terial behavior, the MLPG method is superior to primalor mixed finite element procedures.
keyword: MLPG, Gradient theory, MLS
1 Introduction
The most widely used theories of continuum solid me-chanics involve non-polar materials that are also sim-ple in the sense of Noll (1965). Material point rota-tions are neglected, and the Cauchy stress is assumedto be a functional of the deformation, only throughits first gradient. This local assumption has, for long,proved to be adequate when the wavelength of a defor-mation field is much larger than the dominant micro-structural length scale of the material. However, whenthe two length scales are comparable, this assumptionis questionable. An interest in polar materials has ex-isted at least since the pioneering work of the Cosser-ats(1907,1909). The rotational degrees of freedom intro-duce couple stresses and body couples. Rich theories of
such continua were worked out in the 1960s(Truesdell& Toupin 1960, Toupin 1962, Mindlin 1964) The terms”micropolar” and ”micromorphic” continua were coinedto refer to solids in which the directors attached to ma-terial points are respectively rigid and deformable (Erin-gen, 1976). Such theories have applications in electri-cally polarized media, granular materials and in biolog-ical tissues, among other areas. Since they introduced alength scale, ”Cosserat continua” have been proposed asregularized descriptions of softening materials (Borst &Sluys, 1991), which otherwise demonstrate deformationslocalized onto bands of vanishing width.
The earliest work involving higher-order gradients of de-formation appears to be that of Cauchy (1851). Suchcontinua were termed ”materials of grade n” by Trues-dell and Noll(1965), referring to the order of the high-est derivative of the deformation. Linear and nonlinearelasticity were considered, in light of the role playedby the second gradient of displacement, i.e., the straingradient, by Truesdell and Toupin (1960), and later byToupin (1962). The balance laws included a higher-orderstress—“couple stress” in this case, and hence requiredhigher-order boundary conditions on the “couple stress”traction and strain. Assuming a stored energy function,the couple stress was related, via formal procedures, tothe strain gradient. Shortly after that, Mindlin (1964)proposed a linear theory that encompassed the theoriesof the Cosserats and of Toupin as special cases. Theformalism of Toupin and Mindlin was recently extendedto the deformation and flow theories of plasticity (Fleck& Hutchinson, 1997), motivated by some experimentalobservations of the apparent length scale dependence ofplasticity in metals, when deformations vary at scales ofthe order of a micron (Stelmashenko et al, 1993; Flecket al. 1994; Stolken et al, 1998). Variants of this straingradient plasticity theory have also appeared (Gao et al,1999; Huang et al, 2000; Chen et al, 2002). These the-ories have been widely applied to studying length scale-dependent deformation phenomena in metals. Polar andhigher-order continuum theories have been applied tolayered materials, composites and granular media, in ad-dition to polycrystalline metals.
The solution of the initial and boundary value problemsposed in terms of the higher-order theories is not straight-forward: the governing differential equation and bound-ary conditions are complicated (Toupin, 1962) and ana-lytic solutions are restricted to the simplest cases. Com-
putational difficulties also arise. While boundary condi-tions are easier to treat in the variational setting, require-ments of regularity dictate that, in FEM the displacementmust be a C1 function over the domain. The degrees offreedom include nodal displacements and displacementgradients. The situation is partially analogous to clas-sical Bernoulli-Euler beam and Poisson-Kirchhoff platetheories in one and two spatial dimensions respectively.
Finite element formulations incorporating C1 displace-ment fields are therefore a natural first choice for straingradient theories. For example, the use of Specht’s tri-angular element (Specht, 1988) for the special case ofcouple stress theory was examined (Xia & Hutchinson,1996). The element contains displacement derivatives asextra nodal degrees of freedom(denoted as DOF subse-quently), and C1-continuity is satisfied only in a weakaveraged sense along each side of the element; therefore,the element is not a strict C1 element. Furthermore, theelement fails to deliver an accurate pressure distributionfor an incompressible, non-linear solid (Xia & Hutchin-son, 1996). There is a reliable rectangular C1 element(Zienkiewicz and Taylor, 1994), but its shape and num-bers of DOF are obviously strong limitations.
The lack of robust C1-continuous elements then drovethe development of various C0-continuous elements forcouple stress theory in last decades (Herrmann, 1983;Xia & Hutchinson, 1996; Shu & Fleck, 1998; Shu etal. 1999, Amanatidou & Aravas, 2002; Engel et al.2002; Tenek and Aifantis 2002; etc.). Finite element for-mulations for the Fleck-Hutchinson strain gradient plas-ticity theory have been developed with plate elementsas a basis, but were generally found to perform poorly(Xia & Hutchinson, 1996). Mixed and hybrid formu-lations have also been developed in the same work andelsewhere: Shu et al(1999) introduced some C0 elementtypes, where nodal degrees of freedom include nodaldisplacements and corresponding gradients, and kine-matic constraints between displacement gradients are en-forced via the Lagrange multiplier method. Their lowest-order triangular element requires 28 unknowns per ele-ment, and their lowest-order quadrilateral element 38 un-knowns; Amanatidou and Aravas(2002) proposed mixedC0-contiunity finite element formulations, where everyelement includes around 70 nodal degrees of freedom in2-D problems. From these numbers it is evident that cur-rently, no efficient finite element methods are availablefor strain gradient theory formulations.
Analysis of Materials with Strain-Gradient Effects: A MLPG Approach, with Nodal Displacements only 179
The importance of the role of a rotation-free approxima-tion has been realized for a long time, especially whensolving problems involving high order PDE. For exam-ple, in the thin-plate bending theories, if there is no needfor independent rotation fields, it is consequently pos-sible to develop methods that do not involve rotationaldegrees of freedom. Basically this spirit was well es-tablished in finite difference formulations (Timoshenko,1959), but finite element counterparts to this idea seemedimpossible due to the necessity of employing slope de-grees of freedom in order to satisfy continuity require-ments. In the last decade, a number of procedures hasbeen proposed(Babuka and Zlamal 1973; Baker 1977;Phaal and Calladine 1992a, 1992b; Onate and Cervera1993; Onate and Zarate, 2000), based on a combinationof finite difference, finite element, finite volume and adhoc concepts, leading to elements with only displace-ment degrees of freedom. Typically, these elements arevery simple, i.e. the three-node triangle has dominated,and they all involve non-locality. By non-locality wemean that the curvature in an element depends on the dis-placement field in that element and its neighbors. Conse-quently, these elements fall outside the classical frame-work. Furthermore, C1-continuity requirements are sim-ply ignored (for the most part, elements of this typeare viewed pejoratively: they are frequently describedas non-conforming, incompatible, and variational crimes(Strang et al, 1973)), but these elements seem to performfairly well and have the great advantage of eliminatingaltogether the rotational degrees of freedom. In the lin-ear case, this leads to a saving of computer solution timeand in the non-linear case it additionally amounts to asubstantial reduction in complexity. The reason for thisis that, in the linear case, rotations have a vector spacestructure (i.e., R3), whereas in the non-linear case theyhave a non-linear group structure. Algorithms to preservethe group structure are very complicated, especially indynamics. Thus, eliminating the rotation field ab initio isa very attractive proposition.
During the last two decades, the technique of meshlessinterpolation of trial & test functions has been attracting agreat attention. Meshless approximations, such as Mov-ing Least Square(MLS), Reproducing Kernel (RK) andPartition of Unity Method (PU), possess intrinsic non-local properties. Unlike a typical finite element method,these non-local properties of meshfree approximationsconfer an arbitrary degree of smoothness on solutions
and have been applied to various problems (Belytschko etal. 1994, Duarte et al. 1996, Babuska et al. 1997, Wend-land 1999, Atluri et al. 1998, Atluri et al.1999, Atluri etal. 2000, Atluri et al. 2001, Gu and Liu 2001, Atluri &Shen 2002a,b). While an analysis of a class of meshlessmethods has appeared recently and provided a possibilityto cope with high order elliptic problem conveniently, thereal connection between them has not been made.
The purpose of the current paper is to report the de-tails of a numerical approach, the Meshless Local Petrov-Galerkin (MLPG) method, to the Toupin-Mindlin formu-lation of strain gradient theories. The Meshless LocalPetrov-Galerkin (MLPG) method was first introduced byAtluri et. al. in 1998, wherein the weak form of govern-ing equations are based on local sub-domains, instead ofwhole domain in question. Subsequently the test functiondomains are also intentionally localized to sub-domainsand, the space for test function may be completely dif-ferent from that of trial function. MLPG is well-knownas a truly meshfree method, in which two characteristicsare guarantied: one is a non-element interpolation tech-nique, and the other is a non-element approach for inte-grating the weak form. Most of the element-free methodsare based on the non-element interpolation techniques,such as the Shepard interpolation technique (Shepard,1968) and other approximations mentioned above, whichdo not need any elements for constructing the interpola-tion functions for the unknown variables. However, mostof the meshless methods such as EFG, PKPM, and hp-clouds method, still require a global background meshfor numerical integration of the global weak form. Atluriand Shen in their pioneering work (2002a,b) have de-picted the framework and application of MLPG methodsystematically, and successfully developed some fast androbust approaches. It should be noted that the MLPGconcept is independent of a meshless interpolation tech-nique, and it can be combined with any meshless interpo-lation technique, such as PUM, or PKPM. Furthermore,it can be shown that almost every other meshless methodproposed in literature could be considered as a specialcase of MLPG.
In this paper, the main features of the MLPG are re-viewed. The application of the MLPG to gradient the-ory is described in detail. To study the accuracy of thepresent method, convergence tests are carried out, andseveral 2-D problems of gradient theory have been ana-lyzed. From these tests, the MLPG method is found to
give quite accurate results. The remarkable accuracy inthese numerical simulations shows promising character-istics for solving general problems of materials whoseconstitutive laws involve strain-gradients.
2 Review of a linear elastic strain gradient theory
Toupin (1962) and Mindlin (1963, 1964) developed atheory of linear elasticity whereby the strain energy den-sity w depends upon both the symmetric strain tensorεi j ≡ 1
2 (ui, j + u j,i)and the second gradient of displace-ment, ηi jk ≡ uk,i j. The corresponding compatibility equa-tions are (Mindlin,1964):
∇∇∇ ×εεε× ∇∇∇ = 0 (1a)
∇∇∇ ×ηηη = 0 (1b)
The strain energy function w is assumed to be a convexfunction, with respect to its arguments (ε, η) for eachpoint x of a solid of volume V . The total energy Wstoredin the solid is determined by the displacement field u(x)within V
W(u) ≡∫
Vw(ε(u),η(u);x)dx (1c)
with ε, η being derived from u, as discussed above.
The energy variation of the solid due to an arbitrary vari-ation of the displacement u is:
δW =∫
V(σi jδεi j + τi jkδηi jk)dx (2)
σi j ≡ ∂w∂εi j
; τi jk ≡ ∂w∂ηi jk
. (3)
w =12
λεiiε j j +µεi jεi j +a1ηi j jηikk +a2ηiikηk j j
+a3ηiikη j jk +a4ηi jkηi jk +a5ηi jkη jki (4a)
σi j =∂w∂εi j
= 2µεi j +λεkkδi j (4b)
τi jk =∂w
∂ηi jk= a1(ηippδjk +η jppδik)
+a2(ηkppδi j +12
ηppiδjk +12
ηkk jδik)
+a3(2ηppkδi j)+a4(2ηi jk)+a5(η jki +ηik j) (4c)
Use of the divergence theorem transforms (3) into
δW = −∫
V[(σ jk, j − τi jk,i j)δuk]dx
+∫
S[(σ jk − τi jk,i)n jδuk + τi jkniδuk, j]ds (5)
where S is the surface bounding the volume V . Stationar-ity of the energy integral, with respect to the variations ofthe displacement field, provides the equilibrium relation
σ jk, j − τi jk,i j = 0 (6)
To identify the required boundary conditions, we notethat is not independent of on the surface S, because, ifis known on S, so are the tangential gradients of at S.Therefore, six independent displacement boundary con-ditions are required for a correct formulation of the prob-lem, e.g. prescribed values for ui, i = 1,2,3 and theirnormal derivatives, at S.
In order to identify the independent traction boundaryconditions, We separate the last integral of (5):
by decomposing the gradient into a tangential gradientD jδuk and a normal gradient n jDδuk, viz.,
δuk, j = D jδuk +n jDδuk, (8a)
where
D j ≡ (δjp −n jnp)∂p; D ≡ np∂p (8b)
and denotes the partial derivative with respect to x p. Theterms in (7) may be resolved, further, in more than oneway. For the first term on the right hand side of (7),which contains the non-independent variation , we write(Mindlin, 1964)
where is the alternating tensor. By Stokes’s theorem, theintegral, over a smooth surface, of the last term in (10)vanishes. If the surface has an edge C, formed by theintersection of two portions, S1 and S2, of S, Stokes’stheorem gives
∫S
nqeqpm∂p(eml jnlniτi jkδuk)dS =∮
c[nim jτi jk]δukds
(11)
where m j = eqpmsmnl and the sm are the components ofthe unit vector tangent to C. The bold face brackets [] in (11) indicate the enclosed quantity is the differencebetween the values on S1 and S2.
Now substitute (7), (9), (10) (11) into (5) and obtain thefollowing final form of the principle of virtual work
To summarize, the displacement field u(x) must satisfythree equilibrium equations given by the relation (6), andeither the six traction boundary conditions given by (13)and (14), or the six displacement boundary conditions
ui(x) = u0i (x), Dui(x) = v0
i (x), i = 1,2,3, (15)
(or a mixture of them). A corollary of the above principleof virtual work is the stationarity principle
∫V
(σi jδεi j + τi jkδηi jk)dx = 0 (16)
for any (ε, η) derived kinematically from a displacementfield u(x) and for any (σ,τ) satisfying the equilibriumcondition(5), and the zero traction conditions t k = Rk = 0.It should be noted that, for convenience, we assume thatthe body double force (and also the body force) vanishesin this paper.
3 Review of Meshless Local Petrov-Galerkin(MLPG) method
The initial idea of meshless methods may date back to thesmooth particle hydrodynamics (SPH) method for mod-eling astrophysical phenomena (Gingold and Monaghan,1977). After Nayroles et al. published their work of Dif-fuse Element Method in 1992, the research about mesh-less methods turned to be very active. Since then, severalmeshless methods, such as Element Free Galerkin (EFG)by Belytschko et al. (1994), Reproducing Kernel ParticleMethod (RKPM) by Liu et al. (1996), the Partition ofUnity Finite Element Method (PUFEM) by Babuska andMelenk (1997), hp-cloud method by Duarte and Oden1996, Natural Element Method (NEM) by Sukumar etal. (1998) and Meshless Galerkin methods using RadialBasis Functions (RBF) by Wendland (1999), have alsobeen reported in literature. The major differences in thesemeshless approaches focus on the techniques for interpo-lating the trial function. Although no mesh is requiredin these approximations for the trial and test functionsfor the unknowns, shadow elements(background meshes)are constructed for the integration of the weak-form, orof the ‘energy’.
The meshless local Petrov-Galerkin (MLPG) methodwas first introduced by Atluri & Zhu (1998a, b) andAtluri et al. (1999) for solving linear and non-linearboundary value problems. The weak forms of governingPDE are based on local sub-domains, which may over-lap each other in order to cover whole question domain.The spaces for trial and test functions may be differentfrom each other(Atluri and Shen 2002a,b): the nodal trialfunction may correspond to any one of Moving LeastSquare(MLS), PU, Shepard function, or RBF types of
interpolations; and the test function may be totally dif-ferent, which may correspond to any one of the above oreven a Dirac delta function, the heaviside step function,the Gaussian weight function of MLS, a special form ofthe fundamental solution to the differential equation, orany other convenient function in the support domain, Ω te
(Fig.1), of the test function.
The Schematics of the MLPG method are shown in Fig.1, where the support domain of weight function at anynode I is usually taken as a circle in 2-D cases with itsradius rI . The interpolation of trial function at any pointx should involve all the nodes whose support domainscover this point. Also the node based test domain (alsotaken as a circle in 2-D for convenience) with radius rJ
could have a different size from that of shape function. Itshould be noted that, in Fig.1 the domain of support ofthe test function ΩI
te is synonymous with the node-basedsub-domain.
I te Ω
J tr Ω
x x
Ite
Ω
Γ Ω ∂ =Γ Ι I te s
ss I te L Γ= Ω ∂ Υ
J
I
Γ
sL
The nodes which influence the interpolation at x
Ite
Ω
for an internal node sIte L=Ω∂
Figure 3.1: Schematics of the MLPG method
There are no more background meshes needed for pur-pose of the integration of the weak form in the MLPGmethod, i.e., all pertinent integrals can be easily evalu-ated over well-shaped sub-domains, and on their bound-aries. For instance, in 2-D cases, we may define allthe sub-domains that are based on inner nodes (i.e. in-side the domain) as circular domains without intersec-tion with global boundaries. Every sub-domain based onthe boundary node is a part of circle. Then integrationof sub-domain based weak form may be simply evalu-ated on the circles (or part of a circle). Fig 3.2 showsthe intersections between supports of nodal shape func-tions and some specific inner sub-domain(i.e. test do-main). Also one may define inner sub-domains that in-
trJnode I Ω
x2
x1 ΩIte
Ω ,k I tr
Figure 3.2: The nodal shape function has a differentform in each small region ΩI,k
tr , which is the intersectionof ΩI
te and ΩJtr.
tersect global boundaries in more general. There are sev-eral approaches introduced by Atluri et al.(2002a, b) tohandle those more complicate cases.
4 MLGP Method for Materials with Strain-Gradient Effects
4.1 Weak forms based on sub-domains
Now we proceed to apply MLPG method to solids in-volving gradient theories. Re-write the governing equa-tion (6) in the domain Ω bounded by the boundary Γ asfollows,
σ jk, j − τi jk,i j = 0 , (17)
with boundary conditions :
ui(x) = u0i (x),Dui(x) = v0
i (x), i = 1,2,3, on ΓΓΓu (18a)
or
ti(x) = t0i (x),Ri(x) = R0
i (x), i = 1,2,3 on ΓΓΓt (18b)
or a mixture of them, where , are the prescribed displace-ments and their normal derivatives on boundary, and ,are the surface traction and the double stress traction.uk denotes a displacement field satisfying (17) with cor-responding constitutive relationships. The constitutiveequations and strain-displacement relationship are de-fined in preceding sections.
Analysis of Materials with Strain-Gradient Effects: A MLPG Approach, with Nodal Displacements only 183
Meshless methods based on global Galerkin formula-tion usually begin with the global weak form of govern-ing equation over the entire domain Ω. However in thepresent local Petrov-Galerkin formulation, we start froma weak form over a local sub-domain Ωs, and use theMLS approximation to develop a truly meshless method,where the local sub-domain Ωs is located entirely insidethe global domain Ω. The local sub-domian Ω s is conve-niently taken to be a sphere (in 3-D, or a circle in 2-D)centered at a point x in domain. A generalized local weakform of the differential equation (17) and the boundaryconditions (4.2a,b), over a local sub-domain Ω s, can bewritten as:
∫Ωs
(σ jk, j − τi jk,i j)ρkdx−α∫
Γsu
(uk −u0k)ρkds
−β∫
ΓsDu(vk −v0
k)ρkds = 0 (19)
with local boundary conditions
uk(x) = u0k(x),Duk(x)≡ vk = v0
k(x),k = 1,2,3, on ΩΩΩs
(20a)
or
tk(x) = t0k (x),Rk(x) = R0
k(x),k = 1,2,3 on ΩΩΩs (20b)
or mixture of them, where , ρk are trial and test func-tion, respectively, and vk = Duk = uk,ini,. Γs is a part of∂Ωs, over which the boundary conditions are specified.In general, ∂Ωs = Γs ∪Ls, with Γs being a part of the lo-cal boundary located on the global boundary, and let L s
being the other part of the local boundary over which noboundary condition is specified, i.e., Γ s = ∂Ωs ∩Γ, (seeFig. 1). For a sub-domain located entirely inside theglobal domain, there is no intersection between ∂Ω s andΓ, Ls= ∂Ωs. Then Γsu and ΓsDu denote boundary condi-tions of displacement and normal gradient of displace-ment, respectively. α , β are two constants in penaltyfunctions for handling essential boundary conditions. Aslong as the sub-domains cover whole question domains,the weak forms based on local sub-domains are equiva-lent with weak form defined on whole domain(Atluri andShen, 2002a, b).
From equations in the preceding sections, it is straight-
forward to obtain following weak form,
∫∂Ωs
tkρkds+∫
∂Ωs
RkDρkds
+∫
∂Ωs
nqeqpm∂p(eml jnlniτi jkρk)dS
−∫
Ωs
(σ jkρk, j + τi jkρk,i j)dx−α∫
Γsu
(uk −u0k)ρkds
−β∫
ΓsDu
(vk −v0k)ρkds = 0 (21)
According to (11), the third term in left hand side van-ishes if the sub-domain does not intersect with globalboundary Γ, i.e. Ωs ∪Γs =Ø. In two-dimensional casesthe integral over global boundary Γ may be replaced by
∑γ
[nim jτi jk]γρk (22)
where the sum over γ refers to any corners that may ex-ist on the global bounding curve of the two-dimensionalbody. It should be noted that, in all the coming exam-ples, (22) will vanish identically on boundaries, so thatfor convenience we will not repeat this term in the finalweak form and consequent procedures in calculations.Then we may re-arrange (21) into
α∫
Γsu
ukρkds +β∫
ΓsDu
vkρkds
+∫
Ωs
(σ jkρk, j + τi jkρk,i j)dx−∫
Γsu+Lstkρkds
−∫
ΓsDu+LsRkDρkds = α
∫Γsu
u0kρkds +β
∫ΓsDu
v0kρkds
+∫
Γst
t0k ρkds+
∫ΓsR
R0kDρkds (23)
wherein Γst and ΓsR denote boundary conditions of sur-face traction and the double stress traction, respectively.All the natural boundary conditions have been imposedin (23). In MLPG method (Atluri and Shen 2002a,b),test functions ρk are treated as functions that are differ-ent from the trial functions. For convenience, we may
localize test functions onto sub-domain, such that ρ k andDρk vanish on the boundary Ls. Then (23) could be sim-plified further to be
α∫
Γsu
ukρkds+β∫
ΓsDu
vkρkds+∫
Ωs
(σ jkρk, j + τi jkρk,i j)dx
−∫
Γsu
tkρkds−∫
ΓsDu
RkDρkds = α∫
Γsu
u0kρkds
+β∫
ΓsDu
v0kρkds+
∫Γst
t0k ρkds+
∫ΓsR
R0kDρkds (24)
Recalling the definitions of τ i jk, ηi jk, and in precedingsections, due to the presence of derivative of higher orderstress in the surface traction (see equation 2.12), we mustintroduce at least C2-continuity interpolation scheme forthe trial function, in order to guarantee the convergenceof numerical solution.
4.2 Meshfree interpolation for trial function and testfunction
In general, meshless methods use a local interpolation oran approximation to represent the trial function , usingthe values (or the fictitious values) of the unknown vari-able at some randomly located nodes in the local vicinity.A variety of local interpolation schemes that interpolatethe data at randomly scattered points(without the needfor a mesh) in two or more independent variables are cur-rently available(Atluri and Shen, 2002a,b):
i. Shepard Functions (Shpard, 1968), which have aconsistency of zeroth order only;
ii. Moving Least Squares interpolation (MLS) (Lan-caster & Salkauskas, 1981; Nayroles, Touzot & Vil-lon, 1992), which generalizes Shepard’s approach,implicitly, to the case of shape functions of high or-der consistency;
iii. Partition of Unity Methods ( Bauska & Melenk,1997), which generalizes Shepard’s approach, ex-plicitly, to higher orders of consistency;
iv. Reproducing Kernel Particle Methods (Liu, Chen,Uras & Chang, 1996), which are identical to theMLS approach, if the kernel is identical to theweight functin of a MLS approximation, and is ren-dered to be higher order consistent by same basis;
v. Radial Basis Functions (Wendland, 1995), whichuse random points in the domain, and generate therequired trial & test functions.
The moving least-square method, which is adopted inthis paper, is generally considered to be one of the bestschemes to interpolate data with a reasonable accuracy.Basically the MLS interpolation does not pass throughthe nodal data, as shown in Fig. 4.1. Here we give a briefsummary of the MLS approximation. For details of theMLS approximation, see Belytschko et al. (1994) andAtluri et al. (1999).
Consider a domain in question with control points forboundaries (i.e. nodes on boundaries) and some scat-tered nodes inside, where every node has its nodal value(fictitious) and influence radius(radius for local weightfunction). Now for the distribution of trial function atany point x and its neighborhood located in the problemdomain Ω, uh(x) may be defined by
uh (x) = pT (x)a(x)∀x ∈ Ωx (25a)
where pT (x)=[p1(x), p2(x), . . . ,pm(x)] is a completemonomial basis of order m, and a(x) is a vector con-taining coefficients a j(x), j=1, 2, . . . ,m which are func-tions of the space co-ordinates x=[x1,x2,x3]T . The com-monly used bases in 2-D or 3-D are the linear basis, dueto their simplicity. In Zhu et al. (1998) and Atluri etal. (1998), both linear and quadratic basis are used, andthe results show that both bases possess high accuracy.In the present high order elliptic problem, we will beginwith the quadratic basis
pT (x) = [1,x1,x2,x21,x1x2,x2
2] (25b)
for 2-D, wherein m=6.
The coefficient vector a(x) is determined by minimizinga weighted discrete L2 norm, which can be defined as
J (x) =N
∑I=1
wI (x)[pT (xI)a(x)− uI]2 (26)
where wI(x), is a weight function associated with thenode I, with wI(x)> 0 for all x in the support of wI (x),xI denotes the value of x at node I, N is the number ofnodes in Ωx for which the weight functions wI (x)> 0.
Analysis of Materials with Strain-Gradient Effects: A MLPG Approach, with Nodal Displacements only 185
uh(x) uI^
xI x
Figure 4.1: The distinction between uI and uI.
Here it should be noted that u I , I=1, 2,. . . , N, in equation(26), are the fictitious nodal values, and not the actualnodal values of the unknown trial function u h(x), in gen-eral (see Fig. 4.1 for a simple one-dimensional case forthe distinction between uI and uI).
Solving for a(x) by minimizing J in equation (26), andsubstituting it into equation (4.9), give a relation whichmay be written in the form of an interpolation functionsimilar to that used in the FEM, as
The MLS approximation is well defined, only when thematrix in Eq. (29) is non-singular. φI(x) is usually calledthe shape function of the MLS approximation, corre-sponding to the nodal point x I . FromEqs. (28) and (30),it may be seen that φI(x)=0 when wI(x)=0. The fact thatφI(x) vanishes, as defined, for x not in the support ofnodal point, xI preserves the local character of the mov-ing least squares approximation. The nodal shape func-tion is complete up to the highest order of the basis. Thesmoothness of the nodal shape function is determined bythat of the basis and weight function.
The choice of the weight function is more or less arbi-trary, as long as the weight function is positive and con-tinuous and satisfies the continuity requirement in weakform. Since C2 continuity is needed in current computa-tion, (see similar approach by Long and Atluri, 2002) theweight function corresponding to node I may be written
as
wI (x) =
(1− (dI/
rI)2)3 , 0 ≤ dI ≤ rI
0, dI ≥ rI(41a)
where dI = |x−xI | is the distance from node xI to pointx and rI is the size of the support for the weight functionwI (and thus determines the influence domain of nodexI). The size of support, rI of the weight function wI as-sociated with node I should be chosen such that r I shouldbe large enough to have sufficient number of nodes cov-ered in the domain of definition of every sample point(n ≥ m), in order to ensure the regularity of A . Usually,we have to guarantee all the nodes covered in the domainof definition of every sample point can not be on a singleline if linear interpolation space is applied; all of thesenodes can not be on two lines if quadratic space is ap-plied. A very small rI may result in a relatively largenumerical error while using the Gauss numerical quadra-ture to calculate the entires in the system stiffness ma-trix. On the other hand, rI should also be small enough tomaintain the local character of the MLS approximation.It can be easily seen that the weight function (4.25) pos-sesses C2 continuity. Thus as discussed in detail in Atluriand Shen (2002a,b), the MLS, and the corresponding trialfunctions are C2 continuous over the entire domain.
The consistency condition, namely the ability of theMLS interpolation (27) to exactly represent the jth[j=1,2,. . . .m] component of monomials in (4.9b), impliesthat in general
N
∑I=1
φI(x)xlIi = xl
i ∀x ∈ Ω, ∀l ≤ k (41b)
where kis the highest order in the base. In fact, the linearbasis assures that the MLS approximation has the linearcompleteness. Thus, it can reproduce any smooth func-tion and its first derivative with arbitrary accuracy, as theapproximation is refined. Quadratic or higher order basehas similar characteristics.
A generalization of the MLS interpolation scheme usingthe data for the derivative of a function, in addition to thevalue of the function itself, at a finite number of nodes,can be found in Atluri et al (1999).
The choice of test function is somewhat arbitrary in theMLPG method. The various features of the MLPG havebeen illustrated (Atluri and Shen 2002a,b), by solvinglinear Poisson’s equation with use of different kinds
Analysis of Materials with Strain-Gradient Effects: A MLPG Approach, with Nodal Displacements only 187
of the test functions, including MLS weight func-tion (MLPG1), the collocation Dirac’s Delta function(MLPG2), discrete least squares (i.e. error func-tion, MLPG3), modified fundamental solution to thePDEs(MLPG4), Heaviside step function(MLPG5) andalso nodal shape functions (i.e local Galerkin method,MLPG6). In MLPG6, the integrand is far more com-plex than in the five other MLPGs, due to the fact thatthe test function comes from the same space of the trialfunction. Furthermore, the partition method must be usedto obtain a convergent result (Atluri,et al 1999); sim-ilar situation exists in MLPG3, where, additionally, atrial function with higher order continuity requirementis needed; MLPG4 would be very attractive as long asthe fundamental solution of PDE is available, althoughthe integration with the singularity has to be handled;the concept of MLPG2 is very similar with that of theCollocation Method, where the numerical accuracy isfound to be sensitive to the distribution and density of thenodes; the main purpose of MLPG1 is to avoid evaluatingthe integration on the local boundary, because the MLSweight function always vanishes over the local boundary(in equation 4.7, we have to look for a test function suchthat its normal gradient over the local boundary also van-ishes); by taking constant as the test function over everysub-domain, domain integral is altogether avoided in theMLPG5, which is widely considered as a robust, fast andalso accurate approach, when solving 2nd order BVPs.
However, in current problem (from the local weak form(23)), the feature of MLPG5 can not show its advantages( usually taken to eliminate the need of domain inte-gration, but can not avoid all the integration over localboundary): if we consider a constant (i.e. Heaviside stepfunction) as the test function over every sub-domain, thelast term of right hand side of (23) vanishes (also does thelast term of left hand side ), so that the double stress trac-tion can not be imposed. In other hand, selecting othersimple function, such as linear function etc., to be a testfunction in each local test domain, may eliminate one ofthe two domain integrals in (23), but incurs all the localboundary integration inevitably.
Then we resort to the concept of MLPG1. It is seenthat the local weak form over Ωs involves both the firstderivative as well as the second derivative of ρk in do-main integrand. Thus the test function for every sub-domain at least needs C1 continuity over the local sub-domain. In order to eliminate the integral over the local
boundary, we consider a localized spline weight functionto be the test function, which is defined as:
ρI (x)=
1−6
(dIRI
)2+8(
dIRI
)3−3(
dIRI
)4, 0 ≤ dI ≤ RI
0, dI ≥ RI
(42)
such that ρk and Dρk vanish on the boundary Ls re-quired to obtain (24), where RI is the size of the supportfor the weight function, which is usually much smallerthan size of support domain of shape function. It shouldbe noted that the test functions satisfying these require-ments are not unique and could have other forms, such
as(
1− (dI/RI)2)2
, or even the current MLS weight
function(
1− (dI/RI)2)3
, or the proper Gaussian weight
functions, etc. All of these choices come from the frame-work of MLPG1 and it is found that there is no obviousdifference among the rates of convergence when apply-ing any of weight functions above as test function (Atluriand Shen 2002a, b). And that is confirmed again in cur-rent calculations.
4.3 Discretization of the weak form with plane strainassumption
The plane strain assumption is taken in all of examplesin this paper. In general , the displacement fields are
where ρJ denotes test function (localized spline weightfunction) on node J based sub-domain. ‘i’ denotes nodenumber and N is the total number of nodes, and
It can be easily seen that the system stiffness matrix inthe present method is banded but unsymmetrical. Thelocations of the non-zero entries in the system ‘stiffness’matrix depend upon the nodes located inside the domainof influence of the node.
5 Numerical examples
In this section, patch tests and two numerical exam-ples, including: 1. a bimaterial system under uniformshear and 2. an infinite plane subjected to a remote uni-form tension, are presented. The strain-gradient materialmodel is as described earlier. Compared with analyticalresults, the MLPG method has shown its high accuracywhen coping with gradient theory.
5.1 Patch test
The patch test was conducted with a known simple dis-placement field (Fig. 5.1)
u(x1,x2) =
x1
x2
(48)
which satisfies the compatibility equations (2.0a, 2.0b)and governing equation (6). Also a uni-axial tractionboundary problem was used in the patch test (Fig.5.2)
The formulation passed the tests for both kinds of bound-ary conditions, where the range of the radius of testdomain varies from 1.0 to 3.5 times of the minimumnodal distancedmin and that of radius of support of nodalshape function varies from 3.0 to 5.0 times of dmin andat least one of ai (i=1,5) in equation (4a) does not van-ish. “Pass” here means that the linear displacement field(patch test 1) and the simplest (constant) conventional
Figure 5.1: Patch test 1: essential boundary conditionsare prescribed in this patch test, where the dimension ofthe domain in question is 1 x 1.
U = 0
U=
0
2
1
u2=t1=Du1=R2=0
u 1=
t 2=
Du 2
=R
1=0
t1=R1=R2=0, t2=σ
t 1=
t 2=
R1=
R2=
0
Figure 5.2: Patch test2: mixed boundary conditions areenforced, approximating the problem of an infinite planewith uni-axial traction, for the patch test.
stress field(patch test 2) were reproduced, respectively,within quadrature and roundoff errors: using double pre-cision the maximum error in the displacement or stressfield was of order 10−7 or 10−6 respectively. The mainreason why a higher accuracy could not be obtained isthat the shape function is very complicated and the squareof the derivatives could not be integrated accurately (seeAtluri and Shen, 2002a,b). So we can not expect the sameaccuracy in the patch test as for finite elements.
5.2 Numerical examples
Due to the complexity and difficulty of a gradient theory,the obtainable analytical solutions are only restricted tosome simple problems. In this section, we will focus ontwo problems : 1. boundary layer analysis; and 2. thestress field analysis in an infinite plate, with a hole, sub-jected to bi-axial tension p at infinity, under a plane strainassumption.
5.2.1 Boundary layer analysis
Higher-order gradient theories predict the existence ofboundary layers adjacent to in-homogeneities such asinterfaces. Consider, for example, a bimaterial systemcomposed of two perfectly bonded half planes of elasticstrain gradient solids, subjected to a remote
Material #1
Material #2
o x1
x2
σ21∞
σ21∞
Figure 5.3: Notation and geometry of a bimaterial underuniform shear
shear stress σ∞21 as shown in Fig. 5.3. For the strain gra-
dient solid specified by equations (2.0-2.15) , the shearstrain ε12 has a continuous but non-uniform distribution
Analysis of Materials with Strain-Gradient Effects: A MLPG Approach, with Nodal Displacements only 191
within a boundary layer adjacent to the interface. In aspecific quantitative example, we shall make the follow-ing arbitrary choice of constitutive parameters. The shearmodulus µ of material 1 is taken to be twice that of ma-terial 2, i.e. µ1 = 2µ2 . For each material i, the constantsa3 and a4 (as defined in equation (4a)) are equal to 1
2 µl2i ,
i = 1,2, while a1,a2,a5 vanish. Here is usually called theinternal length scale for materials with strain gradient ef-fects.
An analytical solution is presented here briefly (Shu etal. 1999). For this bimaterial system, the conventionalelasticity theory dictates that the shear stress is uniform;and the shear stain jumps in magnitude at the interface,from ε12 = σ∞
12/2µ1 in material 1, to ε12 = σ∞12/2µ2 in
material 2. By including strain gradient effects, a con-tinuously distributed shear strain can be obtained. In thisproblem, the only non-zero displacement, strain, stressand higher-order stress are u1,ε12,σ12 and τ221, respec-tively, and they are functions of the co-ordinate x 2 only.From the constitutive equation (4c), it follows that
σ12 = 2µiε12 and τ221 = 2µl2i η221 = 4µl2
i∂ε12
∂x2(49)
in material i. Substitution of the above relations into thegoverning equation (6) leads to
∂ε12
∂x2− l2
i∂3ε12
∂x32
= 0 (50)
where li =√
2li. The general solution to the above ordi-nary differential equation is
ε12 = d1 +d2ex2/l1 (51)
and
ε12 = d4 +d5ex2/l1 (52)
Here d1 to d6 are six constants that need to be determined.The general solution is subject to the following boundaryconditions:
i. ε12 → σ∞12/2µ1 as x2 →−∞
andε12 → σ∞
12/2µ2 as x2 → +∞and at the interface,
ii. continuity of traction:(σ−
12τ221,2)|x2→0− = (σ−12τ221,2)|x2→0+
iii. continuity of double stress traciton:τ221|x2→0− = τ221|x2→0+
iv. continuity of strain: ε12|x2→0− = ε12|x2→0+
then the particular solution, satisfying all these condi-tions, is
x1
x2
o
50l
50l
t2=R1=R2=0, t1=σ∞21
t2=R1=R2=0, t1=-σ∞21
t 1=
Du 1
=D
u 2=
0, t 2
=σ∞
21
t 1=
Du 1
=D
u 2=
0, t 2
=-σ
∞21
Figure 5.4: A typical nodal pattern for boundary layeranalysis, where m = 1.0 and the uniform nodal distancein x1 direction is 2.5l.x2 < 0 for material #1 and x2 > 0for material #2. Besides the boundary conditions shownabove, we set u1 = u2 = 0 at left-bottom corner and u2 =0 at right-bottom corner to avoid the rigid movement.
ε12 =σ∞
12
2µ1
1+
µ1 −µ2
µ2
µ2 l2
µ1l1 +µ2
l 2
ex2/l1
for x2 < 0
(53a)
and
ε12 =σ∞
12
2µ2
1+
µ2 −µ1
µ1
µ1 l1
µ1l1 +µ2
l 2
e−x2/l2
for x2 > 0
(53b)
Defining ε12 = σ∞12(µ1 + µ2)/(µ1µ2) as an average shear
strain, solutions may be written in a dimensionless form
where µ1 = 2µ2 and l1 = l2 = l have been used withoutloss of generosity.
A numerical model with a typical nodal distribution isdescribed in Fig. 5.4, where the domain with dimen-sion of 50l x 50l is taken to model the bimaterial sys-tem. The minimum nodal distance in the x2 direction ismtimesl,where the value of m varies from 0.5-3.5. Nodaldistance in x1 direction is uniform, which is 2.5 times ofthe minimum nodal distance in the x2 direction.
The results are presented in Fig.5.5. The numerical so-lution converges quickly to the exact solution, with anincreasing refinement of nodal distances, i.e. decreasingm. It is interesting to note that the strain calculated at theinterface is accurate even for a very coarse nodal pattern,when m is 3.5.
5.2.2 An infinite plate with a hole
The problem in section 5.2.1 could be regarded as one-dimensional problem, although it is solved in the frame-work of a general 2-D plane strain problem. In order toexamine the convergence property of the approach in areal 2-D problem, we consider an infinite solid contain-ing a circular cylindrical hole. The solid is subjected toa remote bi-axial uniform tension, as shown in Figure5.6. An analytical solution was obtained (Amanatidouetc. 2002) for this problem, for the special case of a cou-ple stress solid, i.e. the strain energy density per unit vol-ume w depends upon strain and upon that part of straingradients which can be expressed as
w =12
λεiiεkk +µεi jεi j
+12
l2[λη i j jηikk +µ(ηi jkηi jk +ηi jkηk ji)] (55)
i.e. a2,a3 vanish in (4a) and a1 = 12 l2λ, a4 = a5 = 1
2 l2µ.
The problem is axially symmetric and the displacementfield is of the form
-10 -5 0 5 10
0.35
0.4
0.45
0.5
0.55
0.6
0.65m=3.5
m=0.5
x2 / l ε 1
2/ε
12
Figure 5.5: An accuracy study for various nodal pat-terns, where solid line denotes the analytical solution. Itis noticed that a relatively coarse nodal pattern, i.e.m =3.5, gives fairly accurate results. In all of these calcula-tions, the radius of the node-based test domain (not con-stant) is twice that of the shortest distance from that nodeto other ones, i.e. 2dshortest . The radius of support of thenodal shape function is 4.5dshortest .
a x1
x2
σ∞22= p
σ∞22= p
σ∞11
= p
σ∞11
= p
Figure 5.6: Notation and geometry of an infinite planesubjected to bi-axial remote tension.
Analysis of Materials with Strain-Gradient Effects: A MLPG Approach, with Nodal Displacements only 193
t1=R1=R2=0, t2= p
t 1=
p,
t 2=
R1=
R2=
0
u2=t1=Du1=R2=0
u1=
t 2=
Du
2=
R1=
0
t1=t2=R1=R2=0
Figure 5.7: The quarter of hole in the domain is divideduniformly into 10 partitions and total number of nodes indomain is 165, where the radius of circle is a.
u(r) =p
2µ
(1−2ν)r +
a2
r
+lc
[ar
K1
(al
)− (1−2v)K1
(rl
)](56)
where
c =1−2v
2K0
(al
)+
1−v2
(4la
+al
)K1
(al
)(57)
and Kn(x) are the well-known ‘modified Bessel functionsof the second kind’.
The problem is solved numerically using the MLPGmethod. One-quarter of the plate is analyzed; the nodaldistributions used in the calculations are shown in Fig.5.7and Fig. 5.8, where the total number of nodes are 165 and2091, respectively. The numerical simulations are carriedout for ν = 0.3 and a = 3l. The domain with dimensionsof 10a x 10a is taken to approximate the quarter of theinfinite plate.
The following boundary conditions are prescribed (seeFig.5.7)
Figure 5.8: The quarter of hole in the calculated domainis divided uniformly into 40 partitions and total numberof nodes in domain is 2091. The boundary conditions areas same as those in Fig.5.7.
• u2 = t1 = Du1 = R2 = 0 on bottom boundary• u1 = t2 = Du2 = R1 = 0 on left boundary• t2 = R1 = R2 = 0, t1 = p on right boundary• t1 = R1 = R2 = 0, t2 = p on right boundary• t1 = t2 = R1 = R2 = 0 on surface of the hole
The numerical solution agrees very well with the exactsolution. The displacement and stress fields are includedin Fig. 5.9. and Fig. 5.10, respectively. When the totalnumber of nodes is 2091 (shown in Fig.5.8), a remark-able accuracy is obtained. It should be noted that theclassical elasticity solution with l = 0 predicts a stressconcentration σθθ|r=a = 2.0p and a value σrr|r=a = 0,whereas the present gradient elasticity solution with a =3l predicts σθθ|r=a = 1.94p and σrr|r=a = 0.16p.
In all of these calculations, radius of node-based test-function domain (not constant) is the shortest distancefrom that node to other ones, i.e. dshortest . The radius ofsupport of the nodal shape function is around 3.0d shortest .We have noticed that increasing the node density woulddefinitely improve the accuracy in general, but we can notblindly increase radius of test domain R I or/and supportsize of nodal shape function rI. Otherwise, numerical in-stability may occur.
Figure 5.9: Variation of ur for the plate with a hole, withdifferent total number of nodes, where the solid curvedenotes the exact solution.
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
/r a
σθθ/p
σrr/p
Numerical solution with 165 nodes Numerical solution with 2091 nodes
Figure 5.10: Variations of σθθand σrr for the plate witha hole, with different total number of nodes, where thesolid curves denote the exact solution.
6 Concluding Remards
The Meshless local Petrov-Galerkin (MLPG) method hasbeen developed for materials within the Toupin-Mindlinframework of strain gradient type constitutive theory. In-trinsic non-local properties of the meshless interpolationleads to real rotation-free approaches, where displace-ments are the only nodal degrees of freedom. The MLPGmethod defines the weak form of governing equations onsub-domains, which consequently helps to evaluate inte-gration on some well-shaped region, e.g. circle in 2-D,and no mesh is needed. All of these advantages havebeen combined in calculations presented in this paperand quite accurate results have been obtained. The re-markable accuracy in these numerical simulations showspromising characteristics of MLPG for solving generalproblems of material in elasticity, where strain-gradienteffects may be important.
Acknowledgement: This work is supported by NASA,and ONR, with Dr. I.S. Raja and Dr. Y.D.S. Rajapakseas the cognizant program officials. Authors also thankDr. Z. Han for his helpful discussion and constructivesuggestions.
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