Research Article A Finite Points Method Approach for Strain Localization … · 2019. 7. 31. ·...

Research ArticleA Finite Points Method Approach for Strain Localization Usingthe Gradient Plasticity Formulation

Luis Pérez Pozo,1 Andy Campos,2 Sheila Lascano,2

Sergio Oller,3 and Antonio Rodríguez-Ferran4

1Department of Mechanical Engineering, Advanced Center for Electrical and Electronic Engineering (Basal Project FB0008),Universidad Técnica Federico Santa Maŕıa, Avenida España 1680, Valparaı́so, Chile2Department of Mechanical Engineering, Universidad Técnica Federico Santa Maŕıa, Avenida España 1680, Valparaı́so, Chile3International Center for Numerical Methods in Engineering (CIMNE), Universitat Politècnica de Catalunya (UPC),Campus Norte UPC, 08034 Barcelona, Spain4Laboratori de Càlcul Numèric (LaCàN), Universitat Politècnica de Catalunya (UPC), Campus Norte UPC, 08034 Barcelona, Spain

Correspondence should be addressed to Luis Pérez Pozo; [email protected]

Received 10 July 2014; Accepted 9 October 2014; Published 11 December 2014

Academic Editor: Jose Merodio

Copyright © 2014 Luis Pérez Pozo et al.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The softening elastoplastic models present an unsuitable behavior after reaching the yield strength: unbounded strain localization.Because of the material instability, which is reflected in the loss of ellipticity of the governing partial differential equations, thesolution depends on the discretization. The present work proposes to solve this dependency using the meshless Finite PointsMethod. This meshfree spatial discretization technique allows enriching the governing equations using gradient’s plasticity andintroducing an internal length scale parameter at the material model in order to objectify the solution.

1. Introduction

The strain localization phenomenon is usually a precursorof material failure. In geomaterials such as soils, concrete,and rocks, strain localization has been observed in manysituations such as triaxial tests in the lab and earth excavationprocesses in the field [1].

If the classical continuum theory is directly applied toelastoplastic softeningmaterials, a loss of ellipticity of the gov-erning partial differential equations occurs, and the boundaryvalue cannot correctly describe the physical problem [2].Thestrains are localized in zones that have a width related tothe characteristic length of the subdomains that are used inthe discretization. When these subdomains become smaller,the width of the strain band also decreases without a lowerbound. If the discretization length tends to zero, there is noenergy dissipation, which does not make physical sense [3].To overcome the loss of ellipticity and the related problems,an implicit or explicit length scale must be incorporated into

the material description or the formulation of the boundaryvalue problem [4]. These kinds of regularization techniques,called localization limiters, are usually based on various formsof enriched continuum theories; see [5–9].

Several proposed strategies consider an internal materiallength scale. One strategy introduces nonlocal theories andgradient methods into the model [10]. These concepts can berelated by a Taylor series expansion, as shown by Peerlings etal. [11].

The nonlocal constitutive theories hold that the localstate of the material at a given point may not be sufficientto evaluate the stress at that point. This statement can bephysically justified by the fact that no real material is an idealcontinuous medium; on a sufficiently small scale, the effectsof heterogeneity and discontinuous microstructure becomenonnegligible [7].

The gradient plasticity formulation [12–15] introduceshigher-order gradient terms of a nonlocal variable into the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 782079, 12 pageshttp://dx.doi.org/10.1155/2014/782079

2 Mathematical Problems in Engineering

differential equation that governs its evolution. The originsof these gradient terms are discussed in Groma and Vörös[16] for different cases that are related to atomic distance,dislocations, and finite grain size. Here, the internal materiallength scale is implicitly introduced through the regulariza-tion parameter. The choice of the variable to be representednonlocally depends on the described material behavior andthe chosen approach [17–21].

Furthermore, meshless or meshfree methods possessintrinsic nonlocal properties. The approximation functionsare not locally constructed because the support size of theweight function is greater than the nodal spacing; therefore,the approximation is inherently nonlocal. These nonlocalproperties are used to incorporate an intrinsic length scalethat regularizes the problems of material instabilities [4].

Meshless methods are a family of numerical techniquesthat do not require a mesh. In these methods, the body orthe domain is discretized using a collection of points. It isdivided into the local interpolation subdomains, which arealso called clouds.The clouds consist of one central point (thestar node) and several neighboring points. Generally, thesemethods are computationally efficient and easy to implement,and they have been successfully used in several applications.The general characteristics, classifications, advantages, anddisadvantages of these methods have been described in [22–25].

This work is focused on a meshless Finite Points Method(FPM) for the strain localization analysis through a gra-dient plasticity formulation. This method uses an implicitnonlocal approach to represent the global material behavior.The present contribution solves the solution-discretizationdependency using the meshless FPM. This meshfree spa-tial discretization technique allows enriching the governingequations using the gradients’ plasticity and introducingan internal length scale parameter at the material model.The same shape functions that conform to the meshlessapproximation are used in the gradient additional terms,which reduces the computational cost.

The paper is organized as follows. Section 2 introducesthe basic formulation of the meshless FPM and providesa brief overview of its main features. Section 3 focuses onthe basic formulation of strain gradient plasticity. Section 4introduces some aspects related to the numerical implemen-tation. In Section 5, we give several numerical examples.Conclusions are drawn in Section 6.

2. The Finite Point Method (FPM)

In this section, we review the basic formulation of the mesh-less FPM and provide a brief overview of its main features.

This method was proposed by Oñate et al. [26, 27] tosolve convective-transport and fluid-flow problems. Its appli-cation has been extended to advection-diffusion transport[28], incompressible-flow problems [29], elasticity [30, 31],solid dynamics [32], solidification modeling [33], nonlinearmaterial behavior problems [34–39], adaptive refinement[40, 41], and the large deflection analysis of flexible plates[42]. The lack of dependence on a mesh or an integration

procedure is an important feature that makes the FPM a trulymeshless method.

To obtain the final system of discrete equations, theFPM approximates the local solution of a partial differentialequation at each point of the discretized domain usinga weighted-least-square technique and a point-collocationprocedure. Because the approximation procedure used by thismethod is local, it is necessary to define a subdomain Ω𝑘 foreach node that contains the neighboring nodes, which areselected using a suitable criterion [43, 44]. This collection ofpoints is called a cloud, and its referential central point isthe star node. For example, a relevant aspect in the definitionof clouds is that their superposition must produce the entiredomainΩ,

𝑁𝑝

⋃

𝑘=1

Ω𝑘 = Ω, (1)

where 𝑁𝑝 is the total number of nodes. Note that thedefinition of clouds is the basic, initial step in implementingthe FPM approximation using fixed weighted least squares.

2.1. Fixed Weighted-Least-Square Approximation. With thedefined discretized domain, let us define a function 𝑢(x),which is approximated by �̂�(x); �̂�(x) is only valid in the cloudΩ𝑘 associated with the star node xk. The function 𝑢(x) is alinear combination of known functions p(x),

𝑢 (x) ≅ �̂� (x) = p𝑇 (x)𝛼k, ∀x ∈ Ω𝑘, (2)

where p(x) is the vector that represents the basis of 𝑚linearly independent functions and 𝛼k is a vector of constantparameters that are only valid in Ω𝑘. The elements of theinterpolation basis may belong to any function family. Never-theless, for simplicity, the first𝑚monomials polynomials areused. Examples of this basis in 2D are p𝑇(x) = [1, 𝑥, 𝑦] for𝑚 = 3 and p𝑇(x) = [1, 𝑥, 𝑦, 𝑥2, 𝑥𝑦, 𝑦2] for𝑚 = 6.

Because (2) is valid for all𝑁𝑐 points of the 𝑘th subdomain,the approximations �̂�(X) conform to a Vandermonde systemthat is given by the relation

u (Xk) ≅ û (Xk) = P (Xk) ⋅ 𝛼k, (3)

where

Xk = [xk,1 ⋅ ⋅ ⋅ xk,Nc]𝑇

,

u (Xk) = [𝑢 (xk,1) ⋅ ⋅ ⋅ 𝑢 (xk,Nc)]𝑇

,

û (Xk) = [�̂� (xk,1) ⋅ ⋅ ⋅ �̂� (xk,Nc)]𝑇

,

𝛼k = [𝛼𝑘,1 ⋅ ⋅ ⋅ 𝛼𝑘,𝑁𝑐]𝑇

,

P (Xk) = [[[

p (xk,1)...

p (xk,Nc)

]]

]

.

(4)

Mathematical Problems in Engineering 3

In general, the number of points 𝑁𝑐 that conform to thecloud is greater than the number of functions 𝑚 that definethe basis; hence, thematrixP(Xk) is usually rectangular.Thus,the interpolation property is lost, and the problem must beaddressed with a numerical approximation. The coefficientsof the vector 𝛼k must be determined so that the weightedsums of the squared differences between the exact values𝑢(x) and the approximated values �̂�(x) of each point areminimized according to

min{

{

{

𝑁𝑐

∑

𝑗=1

𝑤(xj) ⋅ (𝑢(xj) − �̂�(xj))2}

}

}

, (5)

where𝑤(xj) is a fixed weighting function that is defined inΩ𝑘and evaluated for the node xj; see [26, 27]. The minimizationprocess described by (6) leads to the expression for the vector𝛼k:

𝛼k = A−1(Xk)B (Xk)𝜆 (Xk) , (6)

where 𝜆(Xk) represents the unknown parameters that aresought on the cloud. Ω𝑘 is defined as

𝜆 (Xk) = [𝜆 (xk,1) ⋅ ⋅ ⋅ 𝜆 (xk,Nc)]𝑇

. (7)

Additionally, the matrices A(Xk), B(Xk), and W(Xk) aregiven as

A (Xk) = P (Xk)W (Xk)P𝑇 (Xk) ,

B (Xk) = P𝑇 (Xk)W (Xk) ,(8)

andW(Xk) is an𝑁𝑐 × 𝑁𝑐 diagonal matrix defined by

W (Xk) = [diag (𝑤 (xk,1) ⋅ ⋅ ⋅ 𝑤 (xk,Nc))] , (9)

where the weighting functions 𝑤(xj) are derived to have unitvalues near the star node and zero values outside the Ω𝑘subdomains. Under the FPM, the common selection is thenormalized Gaussian,

𝑤(xj) ={{

{{

{

exp (− (ℎ𝑗/𝜁)) − exp (− (𝑟/𝜁))1 − exp (− (𝑟/𝜁))

, if ℎ𝑗 ≤ 𝑟,

0, if ℎ𝑗 > 𝑟,(10)

where ℎ𝑗 is the distance between the star node and the pointxj, 𝑟 = 𝑞 ⋅ ℎmax (max. of ℎ𝑗) is a reference distance, and𝜁 = 𝛽 ⋅ 𝑟. A detailed description of the effects of the constantparameters 𝑞 and 𝛽 on the numerical approximation and theguidelines to set their values are presented in [45]. Other

x

u, w

𝜆(xk,j)

w(xk,j)

xk−2 xk−1 xk xk+1 xk+2

u(xk,j)

û(xk,j)

Ωk

Figure 1: Weighted least squares procedure.

considerations in selecting the function 𝑤(xj) can be foundin [26, 27, 46]. Finally, replacing (7) in (2) leads to

�̂� (x) = N𝑇 (x)𝜆 (Xk) , (11)

where N(x) is the shape functionmatrix, which is defined as

N (x) = p𝑇 (x)C (Xk) , (12)

where C(Xk) = A−1(Xk)B(Xk). Note that according to theleast-square nature of the approximation, 𝑢(x) ≅ �̂�(x) ̸=𝜆(x); see Figure 1. In other words, the local values of theapproximating function do not fit the nodal unknown values.Indeed, �̂�(x) is the true approximation, which we use tosatisfy the differential equation and the boundary conditions.In this context, 𝜆(x) are simply the unknown parameters thatwe aim to determine. According to these concepts and (9), itis possible to obtain

�̂�𝑥 (x) = N𝑇

𝑥(x)𝜆 (Xk) , �̂�𝑥𝑥 (x) = N

𝑇

𝑥𝑥(x)𝜆 (Xk) ,

(13)

where (⋅)𝑥 and (⋅)𝑥𝑥 denote the first and second space deriva-tives, respectively. Note that these derivatives are computedby taking the derivative of the basis functions p(x) in (2).

2.2. Discrete-Equation System and the Point CollocationScheme. Let us assume a boundary value problem governedby the set of differential equations with boundary conditions,

𝐴 (𝑢) = 𝑏 in Ω,

𝐵 (𝑢) = 𝑡 on Γ𝑡,

𝑢 − 𝑢𝑝 = 0 on Γ𝑢,

(14)

where𝐴 and𝐵 are differential operators and𝑢 is the unknownfunction of the problem. In solid mechanics problems, 𝐴and 𝐵 correspond to the equilibrium equations, 𝑢 is thedisplacement field, 𝑏 denotes the body forces that act on thedomain Ω, 𝑡 denotes the external tractions on Γ𝑡, and 𝑢𝑝


ΩI

I

dymin = hy

dxmin = hx

(a)

ΩI

Idymin = hy

dxmin = hx

(b)

Figure 2: Definition of stabilization length for (a) the interior and (b) the boundary nodes.

represents the prescribed values of the unknown function onΓ𝑢.

Using the point collocation scheme [47], the differentialequations (14), and the final FPM approximation, which isdefined in (11), one obtains the system of discrete equations

[(�̂�)]𝑖 − 𝑏𝑖 = 0 in Ω, 𝑖 = 1,𝑁𝑝,

[𝐵(�̂�)]𝑗 − 𝑡𝑗 = 0 on Γ𝑡, 𝑗 = 1,𝑁𝑡,

(�̂� − 𝑢𝑝)𝑘= 0 on Γ𝑢, 𝑘 = 1,𝑁𝑢,

(15)

where𝑁𝑝 is the number of points inΩ and𝑁𝑡 and𝑁𝑢 are thenumbers of points on the boundaries Γ𝑡 and Γ𝑢, respectively.

2.3. Stabilization Procedure. The collocation procedure usu-ally leads to an ill-conditioned system of equations withunstable and inaccurate results mainly because the pointcollocation method cannot satisfy the equilibrium over acloud of finite-sized points by simply sampling the equationat the star node in the cloud. These deficiencies are morepronounced in clouds near the boundary Γ𝑡 because theclouds in these regions are not symmetric [30].

A stabilized form of the governing equation is used inthis work.This formulation is derived from the finite calculus(FIC) procedure that was described in [28, 48]. The FICmethod is based on imposing the typical balance laws ofmechanics over a finite-sized domain. Then, the unknownfields are approximated within the finite domain using aTaylor series expansion and retaining the higher-order termsover those used in the standard infinitesimal approach [26–28, 30]. This method naturally introduces new terms inthe governing differential equation, and these terms have

stabilization features. The stabilized form of (15), which usesthe FIC method, is

[𝐴(�̂�) −1

2ℎ𝑘

𝜕

𝜕𝑥𝑘

𝐴(�̂�)]

𝑘

− 𝑏𝑘 = 0 in Ω,

[𝐵(�̂�) −1

2ℎ𝑘𝑛𝑘

𝜕

𝜕𝑥𝑘

𝐴(�̂�)]

𝑘

− 𝑡𝑘 = 0 on Γ𝑡,

(�̂�𝑘 − 𝑢𝑝) = 0 on Γ𝑢,

(16)

where 𝑛𝑘 denotes the components of the unit normal tothe boundary Γ𝑡 and ℎ𝑘 denotes the stabilization-lengthparameters that are obtained from

h𝑇 = [𝑑𝑥min 𝑑𝑦min] ; (17)

see Figure 2.The efficiency of the FIC stabilization procedure in the

context of FPM and solid mechanics has been presented in[30–32, 35, 41].

3. Strain Gradient Plasticity Formulation

In this section, we present the basic formulation of straingradient plasticity. We start from the definition of the non-local approach and obtain the implicit gradient form of theaccumulated plastic strain.

The nonlocal variable 𝑔(x) is defined from the internalvariable 𝑔(x) as the weighted average over a domainV in thevicinity of point x [49]:

𝑔 (x) = ∫V

𝜗 (x, 𝜉) 𝑔 (𝜉) 𝑑𝜉, (18)

where the weight function 𝜗 depends only on the distance |x−𝜉|; hence, 𝜗 ≡ 𝜗(x, 𝜉). A possible choice for 𝜗 is the Gaussianweight function, as proposed in [50].


The weight function 𝜗 introduces the internal lengthscale 𝑐, which is considered a material parameter with thedimension of length. This parameter defines the dimensionof the neighborhood that affects the nonlocal function. Notethat the average is extended to the whole domain V, butbecause of the shape of theweight function the internal lengthscale 𝑐 defines the region of the body that surrounds point x,which significantly influences the behavior at that point [51].

In this work, the local accumulated plastic strain 𝜅(x)is considered the evolutionary internal variable. Then, theexplicit nonlocal gradient formulation 𝜅(x) of this plasticstrain becomes [11]

𝜅 (x) = 𝜅 (x) − 𝑐2∇2𝜅 (x) . (19)

To obtain the implicit gradient formulation of the accu-mulated plastic strain 𝜅, (19) is rewritten as

𝜅 (x) = 𝜅 (x) + 𝑐2∇2𝜅 (x) . (20)

Several advantages of using an implicit formulation instead ofan explicit formulation have been reported in the literature.For example, the continuity requirements for the shapefunctions in numerical simulations are lower, the dynamicalbehavior is more realistic, a better approximation of theunderlying nonlocal integral model is provided, and thesingularities in the strain field are more adequately damped(see [11, 52–54]).

The implicit formulation of (20) introduces the gradienteffects and is accompanied by the boundary condition

∇𝜅 (x) ⋅ n = 0, (21)

where n is the outward unit normal to the boundary. In theliterature, there is no agreement respect to the physical senseof this boundary condition. Nevertheless, some thermody-namical aspects related to the nonlocal residual energy havebeen interpreted by Polizzotto [55].

In the nonlocal gradient plasticity formulation, the scalaryield function is defined by

𝑓 (𝜎, 𝜅, ∇2𝜅) = 𝜎 (𝜅) − 𝜎

th(𝜅, ∇2𝜅) , (22)

where 𝜎 is the stress tensor and the threshold stress insoftening is given by

𝜎th(𝜅, ∇2𝜅) = 𝜎

th0+ ℎ ⋅ (𝜅 − 𝑐

2∇2𝜅) , (23)

and parameter ℎ is the linear softening modulus.More details about strain gradient plasticity and some

thermodynamics aspects can be found in [1–4, 21, 56].

4. Numerical Implementation

Some aspects related to the numerical implementation of thisproposal are shown in this section.

The elastoplastic constitutive model is expressed by thestress-strain relationship

𝜎 = D : (𝜀 − 𝜀𝑝) , (24)

where 𝜀 and 𝜀𝑝 are the total and the plastic strain tensors,respectively, andD is the elastic moduli tensor.

The rate-type stress-strain relationship obtained from(24) is given by

�̇� = D : (�̇� − �̇�𝑝) , (25)

which can be rewritten as

�̇� = D : (�̇� − ̇𝛾m) , (26)

where ̇𝛾 is the plastic multiplier, which is given by

̇𝛾 = �̇�, (27)

andm is the gradient to the yield surface, which is defined in(22).

However, the yield function 𝑓(𝜎, 𝜅, ∇2𝜅) (Equation (22))and the plastic multiplier ̇𝛾 (Equation (27)) evolve accordingto the loading-unloading or the Kuhn-Tucker conditions andthe consequent consistency (or persistency) condition, whichestablishes that the stress state should persist on the yieldsurface for the plastic flow to occur:

𝑓 (𝜎, 𝜅, ∇2𝜅) ⩽ 0,

̇𝛾 ⩾ 0,

𝑓 (𝜎, 𝜅, ∇2𝜅) ⋅ ̇𝛾 = 0.

(28)

During the plastic flow, the stress point must remain onthe yield surface according to the equation

̇𝑓 (𝜎, 𝜅, ∇2𝜅) =

dd𝑡(𝜎, 𝜅, ∇

2𝜅) = 0, (29)

which is known as the Prager’s consistency condition.Considering the above dependencies, (29) can be rewrit-

ten as

𝜕𝑓

𝜕𝜎: �̇� +

𝜕𝑓

𝜕𝜅�̇� +

𝜕𝑓

𝜕∇2𝜅∇2�̇� = 0, (30)

which corresponds to the consistency condition equationunder plastic deformation gradients.

The plastic consistency equation for a time step Δ𝑡 can beapproximated using a Taylor series to yield

d𝑓d𝑡(𝜎, 𝜅, ∇

2𝜅)

𝑡+Δ𝑡

≈ 𝑓 (𝜎, 𝜅, ∇2𝜅)𝑡+𝜕𝑓

𝜕𝜎

𝑡

: Δ𝜎 +𝜕𝑓

𝜕𝜅

𝑡

Δ𝜅 +𝜕𝑓

𝜕∇2𝜅

𝑡

Δ∇2𝜅,

(31)

where the increments correspond to

Δ𝜎 =𝑡+Δ𝑡𝜎 − 𝑡𝜎, (32a)

Δ𝜅 =𝑡+Δ𝑡𝜅 −𝑡𝜅, (32b)

Δ∇2𝜅 =𝑡+Δ𝑡∇2𝜅 −𝑡∇2𝜅. (32c)


xlw = 10nm

L0 = 100nm

up

Figure 3: Uniaxial tensile test statement.

0 0.004 0.008 0.012 0.016 0.020

0.5

1

1.5

2

Displacement

Forc

e

21 nodes41 nodes81 nodes


(a)

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

x

Equi

vale

nt p

lasti

c str

ain



×10−3

(b)

Figure 4: Uniaxial tensile test: stress-displacement curves for several discretizations.

If after a time step Δ𝑡, the plastic flow condition has beenreached at any point; that is, (d𝑓/d𝑡)(𝜎, 𝜅, ∇2𝜅)|𝑡+Δ𝑡 = 0; (31)can be reworked to give

𝜕𝑓

𝜕𝜎

𝑡

: Δ𝜎 +𝜕𝑓

𝜕𝜅

𝑡

Δ𝜅 +𝜕𝑓

𝜕∇2𝜅

𝑡

Δ∇2𝜅 = − 𝑓 (𝜎, 𝜅, ∇

2𝜅)𝑡.

(33)

It should be noted that, in this work, all gradient terms areobtained using the same shape functions that conform to theFPMmeshless approximation (see (13)):

∇2𝜅 ≈ N𝑇

𝑥𝑥(x) 𝜅 (Xk) , (34)

which reduces the computational cost.More details about the numerical implementation issues

can be found in [2, 10, 57–59]. On the other hand, numericalimplementation aspect in the meshfree context can be foundin [1, 4, 60].

5. Numerical Examples

5.1. Uniaxial Tensile Test. This numerical problem is basedon the reference solution that was presented by de Borst andMuehlhaus [2].The central tenth of the bar is weakened (10%reduction in the Young’s modulus) to induce localization,as shown in Figure 3. The numerical tests are displacementcontrolled. The nondimensional geometrical and materialparameters are summarized in Table 1.

Table 1: Uniaxial tensile test. Geometrical and material parameters(nondimensional).

Meaning Symbol ValueElasticity modulus 𝐸 20000Initial yield stress 𝜎𝑦 2Softening modulus ℎ −2000Length of bar 𝐿0 100Length of weakened part 𝑙𝑤 10Internal length scale 𝑐 5Prescribed displacement 𝑢𝑝 0.02

For an internal length scale of 𝑐/𝐿0 = 0.05, differentequidistant discretizations have been studied (see Figure 4).As shown in that figure, dependence on the discretizationbecomes negligible for nodal densities of 161 and beyond.

5.2. Biaxial Compression Test. Next, the classical biaxialcompression test [61] is analyzed using the nonlocal plasticmodel with the proposed linear softening. The weakenedimperfection at the left bottom corner of the geometry isused to illustrate the discretization insensitivity, as shownin Figure 5. The numerical results are compared with thoseshown by Rodŕıguez-Ferran et al. [58]. The initial yield stressof the weakened area is reduced by 10%, and the study isperformed for three different discretizations.The nondimen-sional material parameters are summarized in Table 2.


10

10

60

120

y

x

�p

Figure 5: Biaxial compression test: problem statement.

0 0.005 0.01 0.0150

0.2

0.4

0.6

0.8

1

1.2

RB/(b 0𝜎y)

�/L0

37 × 19 nodes61 × 31 nodes

85 × 43 nodesRodŕıguez-Ferran et al. (2011)

Figure 6: Biaxial compression test: results for different discretizations.

Table 2: Biaxial compression test: material parameters (nondimen-sional).

Meaning Symbol ValueElasticity modulus 𝐸 11920Poisson’s ratio ] 0.3Initial yield stress 𝜎

𝑦100

Softening modulus ℎ −400Internal length scale 𝑐 0.5Prescribed displacement V𝑝 −1.5

The results are clearly discretization-insensitive and con-sistent with Rodŕıguez-Ferran et al. [58], as shown in theforce-displacement curves in Figure 6 and the deformationpatterns and the equivalent plastic strain in Figure 7.

5.3. Sensitivity Test for Biaxial Compression. This exampleis a variation of the biaxial compression test, which waspresented in the previous section. The goal is to illustrate theimperfection-size insensitivity (Figure 10). The initial yield

Table 3: Sensitivity test for biaxial compression: material parame-ters (nondimensional).

Meaning Symbol ValueElasticity modulus 𝐸 11920Poisson’s ratio ] 0.3Initial yield stress 𝜎

𝑦100

Softening modulus ℎ −400Internal length scale 𝑐 0.5Prescribed displacement V𝑝 −2.4

stress of the weakened areas was reduced by 10%, and thestudy was performed for two imperfection sizes, as shownin Figure 8. The numerical results are compared with thoseshown by Rodŕıguez-Ferran et al. [58]. The nondimensionalmaterial parameters are summarized in Table 3.

The results are clearly insensitive to the size of theimperfection and consistent with Rodŕıguez-Ferran et al.[58], as shown in the force-displacement curves in Figure 6


0.13166

0.11697

0.10228

0.087588

0.072897

0.058206

0.043515

0.028824

0.014133

−0.00055735

(a)

0.13096

0.11635

0.10175

0.087141

0.072535

0.057929

0.043322

0.028716

0.01411

−0.00049631

(b)

0.15054

0.13374

0.11694

0.10014

0.083336

0.066535

0.049733

0.032932

0.016131

−0.00066976

(c)

Figure 7: Biaxial compression test: deformation pattern and equivalent plastic strain for (a) 37 × 19 nodes, (b) 61 × 31 nodes, and (c) 85 × 54nodes.

in addition to the deformation patterns and the equivalentplastic strain in Figure 9.

5.4. DiagonallyWeakened Biaxial Tensile Test. In this numer-ical example, a localization strain band is induced in aweakened area on the diagonal, as shown in Figure 11. Thenumerical results are compared with those shown by Liebeet al. [62]. The initial yield stress of the weakened area wasreduced by 20%, and the study is performed for two differentdiscretizations. The nondimensional material parameters aresummarized in Table 4.

The results are discretization-insensitive and consistentwith Liebe et al. [62], as shown in the reaction-displacementcurves in Figure 12, in addition to the deformation patternsand the equivalent plastic strain in Figure 13.

6. ConclusionsA meshless Finite Point Method for the strain localizationanalysis using a gradient plasticity formulation has beenpresented.

Table 4: Diagonally weakened biaxial tensile test: material parame-ters (nondimensional).

Meaning Symbol ValueElasticity modulus 𝐸 50000Poisson’s ratio ] 0.3Initial yield stress 𝜎𝑦 25Softening modulus ℎ −15Internal length scale 𝑐 0.5Prescribed displacement V𝑝 0.2

This meshfree spatial discretization technique allowsenriching the governing equations using the gradients’ plas-ticity and introducing an internal length scale parameter atthe material model.

The differentiability order of the polynomial base ofapproximation allows us to use the same shape function toapproximate the displacements field and the internal plasticdeformation variable, which reduces the computational cost.


60

120

25

5

5

55

y�p

x

(a)

60

120

15

25

25

45

y�p

x

(b)

Figure 8: Sensitivity test for biaxial compression: (a) a small and (b) a large imperfection.

0.055101

0.048952

0.042802

0.036652

0.030502

0.024353

0.018203

0.012053

0.0059035

−0.00024624

(a)

0.062572

0.05551

0.048449

0.041387

0.034325

0.027264

0.020202

0.01314

0.0060783

−0.00098337

(b)

Figure 9: Sensitivity test for biaxial compression: the deformation patterns and the equivalent plastic strain for (a) small and (b) largeimperfections.

0 0.004 0.008 0.012 0.016 0.020

0.2

0.4

0.6

0.8

1

1.2

RB/(b 0𝜎y)

�/L0

5 × 5

25 × 25

5 × 5

25 × 25 (Rodŕıguez-Ferran et al. (2011))(Rodŕıguez-Ferran et al. (2011))

Figure 10: Sensitivity test for biaxial compression: normalized force-displacement curves for the two imperfection sizes.


100

100

y�p

x

12.5/√2

Figure 11: Diagonally weakened biaxial tensile test: problem statement.

0 0.05 0.1 0.15 0.2 0.25 0.30

50

100

150

200

250

Reac

tion

17 × 17 nodes33 × 33 nodesLiebe et al. (2003)

Edge displacement �p

Figure 12: Diagonally weakened biaxial tensile test: reaction-displacement curves for two discretizations.

0.049571

0.043829

0.038087

0.032344

0.026602

0.02086

0.015118

0.0093752

0.0036329

−0.0021094

Figure 13: Diagonally weakened biaxial tensile test: examples of the deformation pattern and the equivalent plastic strain for a discretizationof 33 × 33 nodes.


The numerical results, which were obtained using theproposed technique, demonstrate that the solution and theresponse are independent of the discretization density.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

Luis Pérez Pozo acknowledges the financial support from theChilean Agency CONICYT (FONDECYT project 1140583).

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