Finite Element Simulations of Two Dimensional Peridynamic ...Finite Element Simulations of Two Dimensional Peridynamic Models Andrew T. Glaws (ABSTRACT) This thesis explores the science

Finite Element Simulations of Two Dimensional PeridynamicModels

Andrew T. Glaws

Thesis submitted to the Faculty of theVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Masters of Sciencein

Mathematics

Jeffrey T. Borggaard, ChairLizette Zietsman

Tao Lin

May 7, 2014Blacksburg, Virginia

Copyright 2014, Andrew T. Glaws

Finite Element Simulations of Two Dimensional Peridynamic Models

Andrew T. Glaws

(ABSTRACT)

This thesis explores the science of solid mechanics via the theory of peridynamics. Peridy-namics has several key advantages over the classical theory of elasticity. The most notableof which is the ease with which fractures in the the material are handled. The goal here isto study the two theories and how they relate for problems in which the classical methodis known to work well. While it is known that state-based peridynamic models agree withclassical elasticity as the horizon radius vanishes, similar results for bond-based models haveyet to be developed. In this study, we use numerical simulations to investigate the behaviorof bond-based peridynamic models under this limit for a number of cases where analyticsolutions of the classical elasticity problem are known. To carry out this study, the integral-based peridynamic model is solved using the finite element method in two dimensions andcompared against solutions using the classical approach.

Contents

1 Introduction 1

2 Classical Elastic Theory 3

2.1 The Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 The Stress-Strain Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 The Compatibility Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Reduction to Two Dimensions and the Airy Stress Function . . . . . . . . . 8

2.5 Example Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Peridynamic Theory 11

3.1 Peridynamic States and the Equations of Motion . . . . . . . . . . . . . . . 11

3.2 Bond-based Peridynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Microelastic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5 Fracture and Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Finite Element Method 20

4.1 The Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Newton’s Method and the Inner Integral . . . . . . . . . . . . . . . . . . . . 23

iii

5 Numerical Results 26

5.1 Example 1: Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.2 Example 2: Plate with a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.3 Example 3: “Cracked” Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Conclusions 34

Bibliography 35

iv

List of Figures

2.1 The principle and shear stresses on each face are decomposed into their direc-tional components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 An arbitrary body with directional components of the normal vector `1, `2, `3. 5

2.3 The classical problem of the cantilevered beam with a shearing force on thefree end. This problem will be returned to later. . . . . . . . . . . . . . . . . 9

3.1 The horizon Hx of a point and a particular bond ξ = x′−x within the horizon. 12

3.2 The initial configuration of a point’s horizon and its deformed image. Thereare no restrictions on how the deformation state vector can map bonds totheir stressed state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Various possible forms of the coefficient c(||ξ||). The coefficient is defined tobe zero outside the horizon and need not be continuous. . . . . . . . . . . . . 17

3.4 The imaginary boundary layers R1 and R2 are added to the true body R.Boundary conditions are applied within in these regions. . . . . . . . . . . . 18

3.5 A plot of the pairwise force function against the streching of a bond. Theforce drops to zero once the bond has been stretched beyond the critical valuesc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 The triangular reference element used to build the piecewise linear shapefunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Example of the piecewise linear tent function satisfying φi(xj) = δi,j. . . . . 23

4.3 An example horizon overlayed on the finite element mesh. Difficulties arise indealing with those elements which intersect the boundary of the horizon. . . 24

4.4 Several examples of how partially contained elements are handled by con-structing subelements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

v

5.1 The horizontal (1) and vertical (2) displacement maps for the cantilever beamproblem based on the exact solution (a) and the classical elasticity solution (b). 26

5.2 The peridynamic formulation of the cantilever beam requires that boundarylayers with nonzero volume be added to the ends of the beam. Boundaryconditions can then be applied in these regions. . . . . . . . . . . . . . . . . 27

5.3 The structure of the Jacobian matrices for various δ values. . . . . . . . . . . 27

5.4 The horizontal and vertical displacements based on the peridynamic methodwith decreasing horizons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.5 Maximum differences between the peridynamic and the classical solutions for(a) horizontal and (b) vertical displacements. . . . . . . . . . . . . . . . . . . 28

5.6 Boundary layers are added to three sides of the plate. Displacement conditionsare applied in RD1 and RD2 and a tension force is applied in RN . . . . . . . 29

5.7 The von Mises stresses for the (a) classical elastic and (b) peridynamic models.The highest stresses appear just above the hole as expected. . . . . . . . . . 30

5.8 The von Mises stress distribution found using the classical elasticity theory. . 31

5.9 The cracked plate with imaginary boundary laryers on the top and bottom. . 31

5.10 The stress distribution and errors for the cracked plate found using peridy-namics. Three popular forms of the coefficient c(||ξ||) are used. . . . . . . . . 33

vi

Chapter 1

Introduction

The study of solid mechanics is a branch of continuum mechanics that seeks to understand themotion and deformation of bodies under load. An important area within this field is fracturemechanics, which deals with the formation and propogation of cracks. The imoprtance ofthe field is clear when one considers the structures, machinery, etc., that we rely upon notto fail.

Traditionally, the theory of elasticity has been used as the means to mathematically explainsolid mechanics. However, this approach cannot easily handle cracks in the material. Elastic-ity uses spatial derivatives to represent relationships between neighboring particles. Thesederivatives are undefined at discontinuities, or cracks, in the material. To deal with thisissue, several techniques have been developed. For example, one may redefine the domainof the component so that the crack falls on the boundary or one may couple the systemwith additional equations to describe the motion of the crack through the material. How-ever, these techniques require prior knowledge about where the crack will be located in thematerial and how it will grow. Thus, elasticity is not favorable when considering problemscontaining fractures.

The theory of peridynamics overcomes this limitation by eliminating spatial derivatives infavor of integral equations to describe the relationships between particles. Since integrals aredefined over discontinuities, the same set of equations are applied throughout the materialregardless of the existence of any cracks. This non-local view of particle interactions allowsfor natural crack growth.

Despite the obvious advantages of peridynamics in problems dealing with fracture, the focusof this study is not on crack formation or propogation. This topic is discussed briefly laterin the document when we consider future research directions. However, the goal here is tocompare the behavior of materials under the peridynamic theory and the classical elastictheory. Thus, the problems examined are ones for which elasticity is known to perform well.

The thesis begins with an overview of classical elasticity, including brief derivations of the

1

Andrew T. Glaws Chapter 1. Introduction 2

main components of the theory. Then, the peridynamic theory is discussed with some discus-sion concerning how one could incorporate crack formation and propogation. This is followedby an explanation of how the finite element method is applied to the peridynamic theory. Fi-nally, some classical two dimensional problems are examined using both the classical elastictheory and peridynamics.

Chapter 2

Classical Elastic Theory

The theory of elasticity is the classical method for studying bodies under load. This areahas been studied extensively and only the major concepts are discussed here. Further detailscan be found in [8, 10,14,15].

A body can deform either elastically, where it returns to its original configuration once theload is removed, or plastically, where is does not return to its original configuration oncethe load is removed. As the name suggests, the theory of elasticity applies in the case ofelastic deformations. In most continuum models, any body will deform elastically until somelimiting stress threshold is reached. This threshold is dependent on the materials properties.

Forces can be applied to the body in two different ways. Some forces act on the entire volumesuch as gravity or thermal forces. These body forces per unit volume are resolved into threecomponents aligning with each coordinate axis, denoted X, Y , and Z. Alternatively, a forcecan act on the surface such as contact forces between two bodies. These forces per unit areaare called stresses, and as above, are decomposed into their directional components

Consider the differential element depicted in Figure 2.1. The stresses on the body are dividedinto normal stresses, σ, and shear stresses, τ . For normal stresses, the subscript providesthe direction of the force that is perpendicular to the plane on which the stress acts. Forshear stresses, the first subscript is again the direction perpendicular to the plane on whichthe stress acts while the second subscript provides the direction of the shearing. It can beshown through the summation of moments that the shear stresses about each of the axesare equal. That is,

τxy = τyx, τyz = τzy, τxz = τzx. (2.1)

The stress threshold for plastic deformation mentioned earlier is important in the study ofmaterial failure. It has been shown that a material can fail despite none of the individualcomponent stresses exceeding this threshold. Thus, a new value, called the von Mises stress

3

Andrew T. Glaws Chapter 2. Classical Elastic Theory 4

Figure 2.1: The principle and shear stresses on each face are decomposed into their directionalcomponents.

equivalent or simply the von Mises stress, is defined as

σvm =

√(σx − σy)2 + (σy − σz)2 + (σz − σx)2 + 6(τ 2

xy + τ 2yz + τ 2

xz)

2. (2.2)

The von Mises stress equivalent is not truly a stress but a measure of the combined stressesand allows one to determine the proximity of the current stress configuration to failure.

Analysis of using the elastic theory depends on three key components:

1. the equilibrium equations,

2. the stress-strain relations, and

3. the compatibility conditions.

2.1 The Equilibrium Equations

The derivation of the equilibrium equations depends upon the realization that the differentcomponents of stress vary from point to point within a body. Consider the arbitrary bodyR shown in Figure 2.2. The total force applied to any point in R is decomposed into itsdirectional body forces per unit volume and its direction surface forces per unit area asdiscussed earlier.

The projection of the stresses onto the x axis is given by


Figure 2.2: An arbitrary body with directional components of the normal vector `1, `2, `3.

∫∫∂R

(σx`1 + τyx`2 + τzx`3) dS, (2.3)

where `1, `2, `3 are the directional components of the outward normal. Next, the divergencetheorem is applied to (2.3) resulting in

∫∫∫R

(∂σx∂x

+∂τyx∂y

+∂τzx∂z

)dV. (2.4)

Similarly, the total body force in the x direction is found by integrating X over the volume.Thus, balancing all forces along this axis requires

∫∫∫R

(∂σx∂x

+∂τyx∂y

+∂τzx∂z

+X

)dV = 0. (2.5)

Since the above holds for any generic body, the integrand can be assumed to be zero. Asimilar process in the y and z directions yields the three equilibrium equations

∂σx∂x

+∂τyx∂y

+∂τzx∂z

+X = 0

∂τxy∂x

+∂σy∂y

+∂τzy∂z

+ Y = 0

∂τxz∂x

+∂τyz∂y

+∂σz∂z

+ Z = 0.

(2.6)


2.2 The Stress-Strain Relations

Let u(x, y, z), v(x, y, z), and w(x, y, z) denote the displacements of a particle in a body alongthe x, y, and z axes, respectively. Strain is defined to relate the forces acting on a body tothe actual deformation of the body. It is a measure of the intensity of the deformation in aparticular direction. For example, the strain normal to the x axis can be thought of as theamount of elongation or contraction a particular element experiences given by

εx = lim4x→0

4u4x

=du

dx. (2.7)

Shear strain accounts for the intensity of the rotational deformation of any element and isgiven by

γxy =∂u

∂y+∂v

∂x. (2.8)

By expanding this idea to the other directions, nine strain-displacement relations can bederived. However, any rotational deformation is not affected if the angle is measured fromthe x axis to the y axis or vice versa. Thus, the nine strain-displacement relations can bereduced to the six equations

εx =∂u

∂xεy =

∂v

∂yεz =

∂w

∂z

γxy = γyx =∂u

∂y+∂v

∂xγxz = γzx =

∂u

∂z+∂w

∂xγyz = γzy =

∂v

∂z+∂w

∂y.

(2.9)

It has been found through experiment that the relationship between stress and strain canbe represented by

σx = 2Gεx + λe τxy = Gγxy

σy = 2Gεy + λe τyz = Gγyz

σz = 2Gεz + λe τxz = Gγxz,

(2.10)

where

e = εx + εy + εz (2.11)

is called the volume strain. The constants G and λ are known as the Lame constants andare related to the material’s properties (bulk modulus E, and Poisson’s ratio ν) through


G =E

2(1 + ν)(2.12)

and

λ =νE

(1 + ν)(1− 2ν). (2.13)

2.3 The Compatibility Equations

Maintaining compatibility within the system ensures that the strains and resulting deforma-tions are geometrically allowable in the body. To illustrate this idea, consider a cube thatwill be placed under a load. Prior to loading, the body is divided into many smaller cubes.When the load is applied, the cubes should still fit together smoothly (i.e. no gaps shouldform between cubes). Mathematically, the need for the compatibility equations arises fromthe fact that the six strain components in (2.9) are expressed in terms of three displacementterms. Thus, it is to be expected that a relationship exists between the strain terms. Thecompatibility equations are not derived here but are given by

∂2εx∂y2

+∂2εy∂x2

=∂2γxy∂x∂y

2∂2εx∂y∂z

=∂

∂x

(−∂γyz∂x

+∂γxz∂y

+∂γxy∂z

)∂2εy∂z2

+∂2εz∂y2

=∂2γyz∂x∂z

2∂2εy∂x∂z

=∂

∂y

(−∂γxz

∂y+∂γxy∂z

+∂γyz∂x

)∂2εz∂x2

+∂2εx∂z2

=∂2γxz∂x∂z

2∂2εz∂x∂y

=∂

∂z

(−∂γxy

∂z+∂γyz∂x

+∂γxz∂y

).

(2.14)

To completely describe a loaded body using elasticity, three displacements, six stresses, andsix strains must be found using three equilibrium equations, six stress-strain equations, andsix strain-displacement relations. The problem is solved in the stress-strain realm while thecompatibility equations ensure that the resultant deflections are smooth.

The final component needed to solve the system of fifteen partial differential equations isthe boundary conditions. These are generally described by surface traction equations havingunits force per area. These equations can be split into the various stress components alongeach axial direction. Alternatively, the boundary conditions can be provided in terms ofdisplacements and depend on the strain-displacement equations to relate them to the rest ofthe system.


2.4 Reduction to Two Dimensions and the Airy Stress

Function

A problem in elasticity may be reduced to two dimension in two ways depending on thegeometry involved. Consider first a problem in which third dimension is much longer thanthe two cross-sectional dimensions, such as in an infinitely long beam. This is best handledusing a technique called plane strain. Alternatively, if the third dimension is much smallerthan the others, such as for a thin plate, plane stress is used.

In both cases, the equilibrium equations (2.6) will remain the same in the x and y directionswith no need to balance forces in the z direction assuming Z = 0. Additionally, the strain-displacement relations (2.9) reduce simply by allowing εz = γxz = γyz = 0 and all othercomponents to remain the same. The stress-strain equations become

σx = 2Gεx + λ(εx + εy) τxy = Gγxy

σy = 2Gεy + λ(εx + εy) τyz = 0

σz = λ(εx + εy) τxz = 0

(2.15)

for plane strain, while for plane stress they are

εx =1

E(σx − νσy) γxy =

τxyG

εy =1

E(σy − νσx) γyz = 0

εz =−νE

(σx + σy) γxz = 0.

(2.16)

If the body forces Fx and Fy are negligible and the system is being represented in Cartesiancoordinates, the equilibrium equations can be satisfied by a single scalar function known asthe Airy stress function Φ(x, y). This function is related to the stresses in the two dimensionalbody by

σx =∂2Φ

∂y2(x, y), σy =

∂2Φ

∂x2(x, y), τxy = − ∂2Φ

∂x∂y(x, y). (2.17)

Combining (2.17) with the equilbrium equations results in the two dimensional biharmonicequation

∇4Φ =∂4Φ

∂x4+

∂4Φ

∂x2∂y2+∂4Φ

∂y4= 0. (2.18)


Figure 2.3: The classical problem of the cantilevered beam with a shearing force on the freeend. This problem will be returned to later.

2.5 Example Problem

A classic problem in elasticity is that of the cantilever beam [14]. For this problem, a longbeam is attached to a wall at one end while a downward force is applied at the free end asshown in Figure 2.3. The stresses associated by such a system can be described by

σx = c2xy, σy = 0, τxy = −c1 −c2

2y2. (2.19)

Since no forces are being applied to the top and bottom boundaries of the beam, setting(τxy)y=±d = 0 yields

c2 = −2c1

d2. (2.20)

A total shearing force per unit length F is applied over the end of the bar, resulting in−∫ d

−d τxy dy = F . This is used to find the constant

c1 =3F

4d. (2.21)

Thus, the forces may be rewritten as

σx = − 3F

2d3xy, σy = 0, τxy = −3F

4d(1 +

y2

d2). (2.22)

Using the stress-strain and the strain-displacement equations to relate the displacements uand v to (2.22) results in


∂u

∂x= − 3F

2Ed3xy,

∂v

∂y=

3νF

2Ed3xy,

∂u

∂y+∂v

∂x= − 3F

2Gd(1 +

y2

d2).

(2.23)

Integrating (2.23) and enforcing conditions to prevent rigid body movement yields the finalsolution to the cantilever beam

u(x, y) =

(F

4Gd3− νF

4Ed3

)y3 − 3F

4Ed3x2y +

(3Fl2

4Ed3− 3F

4Gd

)y,

v(x, y) =F

4Ed3x3 +

3νF

4Ed3xy2 − 3Fl2

4Ed3x+

Fl3

2Ed3.

(2.24)

Chapter 3

Peridynamic Theory

The peridynamic theory was first introduced by Silling in [11]. Since then, peridynamics hasbeen studied in both theoretical and computational frameworks, e.g. [7, 9]. Early methodswere based on bond-based peridynamics which consider particle-particle interactions one ata time. Since then the theory has expanded to consider the infinitely many simultaneousinteractions. Furthermore, it has been shown that the peridynamics theory converges tothe classic elastic theory in the limiting case [13]. Computationally, peridynamics has beeninvestigated using both direct and finite element methods [2–4,6, 12].

3.1 Peridynamic States and the Equations of Motion

Unlike the traditional theory of elasticity which focuses only on contact forces betweenadjacent particles in a body, the peridynamic theory allows for particles to interact over afinite distance through bonds. Given a component that occupies the region R and a particlein the body, x, peridynamics considers the relationship between x and all other particles,x′, within the horizon of x, denoted by Hx. In general, the horizon is defined by all particleswithin a ball of radius δ > 0 of the particle of interest. An example horizon is shown inFigure 3.1. The notation ξ = x′ − x is used to identify a particular bond.

Hx = x′ ∈ R : ||ξ|| = ||x′ − x|| < δ. (3.1)

To understand how a particle influences other particles within its horizon, two state vectorsare defined. At a given time t > 0, the deformation state vector is given by

11

Andrew T. Glaws Chapter 3. Peridynamic Theory 12

Figure 3.1: The horizon Hx of a point and a particular bond ξ = x′−x within the horizon.

Y [x, t] =

(x1 + u(x1, t))− (x+ u(x, t))(x2 + u(x2, t))− (x+ u(x, t))

...(x∞ + u(x∞, t))− (x+ u(x, t))

=

y(x1, t)− y(x, t)y(x2, t)− y(x, t)

...y(x∞, t)− y(x, t)

,where Hx = xi∞i=1.

(3.2)

The deformation state Y [x, t] : R2 → R2 maps bonds from their reference state to theirdeformed configuration at time t by

Y [x, t]〈ξ〉 = y(x′, t)− y(x, t). (3.3)

One advantage of the peridynamic deformation state vector is the generality of the deforma-tions that can occur around a single point. This is shown in Figure 3.2 where the deformedimage can take potentially any shape. This generalization improves upon the classical theoryin which spherical regions (such as a point’s horizon) can only be mapped into ellipsoidaldeformed states.

Figure 3.2: The initial configuration of a point’s horizon and its deformed image. There areno restrictions on how the deformation state vector can map bonds to their stressed state.

The second state vector describes the interactions between particles in a body. This vectoris referred to as the force state vector and is denoted


T [x, t] =

t(u(x1, t)− u(x, t),x1 − x)t(u(x2, t)− u(x, t),x2 − x)

...t(u(x∞, t)− u(x, t),x∞ − x)

, (3.4)

where t(u(x′, t) − u(x, t),x′ − x), known as the force density vector, represents the forcedensity applied to particle x by particle x′. Using the notation η = u(x′, t) − u(x, t) forthe relative deformation of two points, the force density vector becomes t(η, ξ). The forcedensity vector builds on (3.2) by mapping bonds to their resultant forces. Thus,

T [x, t]〈ξ〉 = t(η, ξ) (3.5)

yields the force that x′ exerts on x in it’s deformed state. Any constitutive model of peri-dynamics is given by the how the deformation state influences the force state. That is,

T [x, t] = T (Y [x, t]). (3.6)

It should be noted that the force in a particle bond, x′ − x, is not determined solely by thedeformation between the two particles but is defined by the cumulative deformation of allbonds in Hx. This fact will not hold in future sections when a more specialized form of theperidynamic theory is examined. Additionally, since only particles within the horizon of xcan influence the force state vector, it is assumed that

x′∈ Hx =⇒ T [x, t]〈x′ − x〉 = 0. (3.7)

Once a constitutive model is set up, the basic peridynamic equation of motion is defined bythe integral equation

ρux, t) =

∫Hx

T [x, t]〈x′ − x〉 − T [x′, t]〈x− x′〉dVx′ + b(x, t) (3.8)

or

ρux, t) =

∫Hx

t(u(x′, t)−u(x, t),x′−x)−t(u(x, t)−u(x′, t),x−x′)dVx′ +b(x, t), (3.9)

where ρ is the mass density of the material and the term b, the loading force density, containsany external forces per unit reference volume acting at a point.


In the case of static, but stressed materials, terms such as u and u vanish. Thus, theequilibrium equation is

0 =

∫Hx

T [x, t]〈x′ − x〉 − T [x′, t]〈x− x′〉dVx′ + b(x, t) (3.10)

or

0 =

∫Hx

t(u(x′, t)− u(x, t),x′ − x)− t(u(x, t)− u(x′, t),x− x′)dVx′ + b(x, t). (3.11)

While the generality of state-based peridynamics offers obvious advantages, an alternativeform of the peridynamic theory allows for simpler practical implementation but at the costof added theoretical limitations [7].

3.2 Bond-based Peridynamics

A specialized version of the peridynamic theory exists in which the force density in a par-ticular bond is determined only by the deformation of the individual bond and thus isindependent of all other bonds within the horizon. Furthermore, the force density vectorscan be assumed to be equal and opposite within a single bond. However, these simplifyingassumptions result in several limitations in the theory. The most notable being that thismodel supports only one independent material constant. Thus, the Poisson’s ratio is fixedat ν = 0.25 while the bulk modulus is allowed to remain free.

In the case of bond-based peridynamics, the pairwise force function, which contains theconstitutive information for the bond, is introduced. It is defined as

f(u(x′, t)−u(x, t),x′−x) = t(u(x′, t)−u(x, t),x′−x)−t(u(x, t)−u(x′, t),x−x′), (3.12)

or more concisely as

f(η, ξ) = t(η, ξ)− t(−η,−ξ). (3.13)

The peridynamic equation of motion is then updated by inserting (3.12) into (3.9) to obtain

ρu(x, t) =

∫Hx

f(η, ξ) dVx′ + b(x, t) (3.14)


and in the case of static materials

0 =

∫Hx

f(η, ξ) dVx′ + b(x). (3.15)

It should be noted that the pairwise force function need only be integrable, with no restric-tions regarding smoothness or continuity with respect to either variable. Additionally from(3.7), it is assumed that f vanishes outside of Hx. Thus, a given particle does not have anyinfluence on any other particle that is outside of its horizon.

Two important restrictions to the pairwise force function come from Newton’s Third Lawof Motion and the conservation of angular momentum. First, the model must respect equaland opposite forces acting within the body. Thus, from (3.12)

f(η, ξ) = t(η, ξ)− t(−η,−ξ) = −[t(−η,−ξ)− t(η, ξ)] = −f(−η,−ξ). (3.16)

The above relation is known as the linear admissibility condition. The second constraint,referred to as the angular admissibility condition, requires the force between two particlesto act only along the vector between them

(ξ + η)× f(η, ξ) = 0, ∀η, ξ. (3.17)

Combining (3.16) and (3.17), it is possible to write f in its most general form as

f(η, ξ) = F (η, ξ)ξ + η

|ξ + η|, ∀η, ξ, (3.18)

where the scalar-valued function F satisfies

F (η, ξ) = F (−η,−ξ) ∀η, ξ. (3.19)

From the above equations, it becomes clear that the interactions contained in a bond maybe viewed as a possibly nonlinear “spring” between the two particles. The nature of this“spring” is determined by the properties of the material under stress.

3.3 Microelastic Materials

All of the fundamental information concerning the material properties of the body is con-tained in the pairwise force function (3.13). While (3.18) and (3.19) will always hold true,the form of f will depend on the properties of the material being modeled.


A material is called microelastic if there exists a scalar function, w(η, ξ), such that

f(η, ξ) =∂w(η, ξ)

∂η. (3.20)

This function, w, is referred to as the micropotential function and represents the energyper unit volume in a particular bond. The total energy density at a point can be found byadding up the energy in all the bonds in the horizon of the point,

W =1

2

∫Hx

w(η, ξ) dVx′ . (3.21)

A microelastic body behaves similar to a body experiencing classical elastic deformation.That is, the deformations on the body are not permanent and the energy exerted on thebody can be recovered.

The microelastic potential depends on the intensity of the deformation of a bond. Thisintensity is captured in

s(η, ξ) =||ξ + η|| − ||ξ||

||ξ||, (3.22)

referred to as the bond-strain or “stretching” term. Notice that ξ + η represents the final,deformed bond between x and x′. In this way, s compares the length of the deformed bondto its reference state. A positive value of s indicates a lengthening of the bond while anegative value represents compression.

The general form of the microelastic potential is

w(η, ξ) =1

2c(||ξ||) s2(η, ξ) ||ξ||. (3.23)

Combining (3.20) and (3.23) yields

f(η, ξ) =∂

∂η

(1

2c(||ξ||) s2(η, ξ) ||ξ||

)= c(||ξ||) s(η, ξ) ||ξ|| ∂s

∂η

= c(||ξ||) s(η, ξ)ξ + η

||ξ + η||.

(3.24)

This equation provides the general framework for the peridynamic model of a microelasticmaterial. It should be noted that while one must account for rigid body translations in the


Figure 3.3: Various possible forms of the coefficient c(||ξ||). The coefficient is defined to bezero outside the horizon and need not be continuous.

system, rigid rotations are not an issue due to the presence of the unit directional vector ofthe deformed bond.

The only constraint on the coefficient c(||ξ||) is that it maintain the integrability of f(η, ξ).Thus, the form of this coefficient can vary greatly. Common coefficient functions include theconstant function, the regular cone, and the inverted cone, shown in Figure 3.3.

The microelastic pairwise force function can be linearized using the Taylor series expansion.The linearized form of (3.24) is

f(η, ξ) =∂f

∂η(0, ξ)η (3.25)

where∂f

∂η(0, ξ) is the Jacobian matrix of f evalutated at η = 0. While (3.25) can prove use-

ful in simplifying numerical calculations, the loss of the stretching term results in significanterrors for problems with sufficiently large displacements.

3.4 Boundary Conditions

In contrast to elasticity, no boundary conditions are required in order to solve the peridy-namic equation of motion (due to the lack of any spatial derivatives). Thus, no separateequations are needed to solve the integral-based system. However, the body’s interactionswith its environment need to be captured in some form.

External forces, which were presented either as principal or shear stresses or as body forcesin the classical theory, are enforced via the loading force density, b(x). However, simplyapplying the forces on the boundary is not sufficient as they will integrate to zero. Thus,to effectively simulate a surface force, an imaginary boundary layer with nonzero volumeis added to the material where the force is to be applied as shown in Figure 3.4. It hasbeen suggested by numerical studies that this boundary layer be approximately equal to the


Figure 3.4: The imaginary boundary layers R1 and R2 are added to the true body R.Boundary conditions are applied within in these regions.

radius of the horizon, δ.

Some problems may call for displacement boundary conditions such as the zero displacementcondition on the fixed end in the cantilever beam problem from earlier. Similar to the casewith surface forces, an imaginary boundary layer is added to the material along the segmentwhere the condition is to be applied. The displacement is then enforced within this regionwhile it is solved for in the rest of the body.

The displacement cannot simply be enforced along the boundary of the body since theperidynamic equations are integral-based. Hence, the displacement boundary conditionsmust be enforced in a region with nonzero volume to ensure the conditions do not integrateto zero. As was the case with surface forces above, numerical experiments suggest that theimaginary boundary region have thickness δ.

3.5 Fracture and Damage

One of the significant benefits of peridynamics is the ability for cracks to appear and grownaturally. Cracks form in the peridynamic model when bonds between particles are broken,that is when particles no longer interact. A bond breaks when the bond stretches beyond apredefined value called the critical stretch. Thus, a new term is introduced into (3.24)

f(η, ξ) = c(||ξ||) s(η, ξ)ξ + η

||ξ + η||µ(t, s), (3.26)

where

µ(t, s) =

1 if s(t′) < sc for all t′ ∈ [0, t]0 otherwise

. (3.27)

Thus, when a bond is stretched beyond the critical stretch, sc, the pairwise force function


Figure 3.5: A plot of the pairwise force function against the streching of a bond. The forcedrops to zero once the bond has been stretched beyond the critical value sc.

between x and x′ vanishes permanently as shown by Figure 3.5. When a bond is broken,the overall force density at the point x is reduced. This can cause nearby bonds to stretchfurther and break resulting the growth of a crack in the material.

The overall damage at a point is measured by

φ(x, t) = 1−∫Hxµ(t, s) dVx′∫HxdVx′

. (3.28)

Notice that 0 ≤ φ(x, t) ≤ 1 and as the body is deformed the damage increases montonically.When φ(x, t) = 1, the particle x has broken every bond in its horizon is no longer attachedto the rest of the body.

Chapter 4

Finite Element Method

The Finite Element Method (FEM), a technique commonly used to solve partial differentialequations, may also be applied to solve integral equations such as (3.14) or (3.15). Ingeneral, the FEM begins with a reformulation of the problem into a weak form with thesame solution as the original problem. Then a mesh is created by dividing the domain Ωinto a finite number of elements. The approximate solution is found by solving the weakequation on each of the elements.

4.1 The Weak Form

The generalized derivation of the weak form for problems with nonlocal operators can befound in [1]. Consider the equilibrium peridynamic equations given by (3.15) in two dimen-sions, and assume that the displacement function u belongs to the space

Sd = v ∈ L2(R)× L2(R) : v|Rd= gd(x) (4.1)

or

Sn = L20(R)× L2

0(R) = v ∈ L2(R)× L2(R) :

∫Ω

v dx = 0 (4.2)

with inner product

〈u,v〉 =

∫RuTv dx. (4.3)

While the trial function set will vary depending on the boundary conditions of the problem,

20

Andrew T. Glaws Chapter 4. Finite Element Method 21

in general it will be referred to as S. Define the operator Lu : S → S by

L(u(x)) =

∫Hx

f(u(x′)− u(x),x′ − x) dVx′ . (4.4)

Then the weak form of the peridynamic equilibrium equation is

0 = 〈Lu,v〉+ 〈b,v〉, (4.5)

where v ∈ S is called the test function. Given the anti-symmetry of the kernel provided by(3.13) and assuming that 〈Lu,v〉 <∞, the inner product can be rewritten as

〈Lu,v〉 = −1

2

∫R

∫Hx

[f(u(x′)− u(x),x′ − x)]T (v(x′)− v(x)) dVx′ dVx. (4.6)

Thus, the weak form of (3.15) is

0 =1

2

∫R

∫Hx

[f(u(x′)−u(x),x′−x)]T (v(x′)−v(x)) dVx′ dVx−∫R

[b(x)]Tv(x) dVx. (4.7)

The function space S is then approximated by an N dimensional set SN . The goal is thento find a function in SN that approximates the unknown displacement function

u(x) ≈ uN(x) =N∑i=1

uiφi(x) ∈ SN , (4.8)

where the set φ1(x),φ2(x), ...,φN(x) is a basis of SN . An approximate solution for thedisplacement of the body is calculated by finding the unknown coefficients u1, u2, ...uN.Substituting (4.8) into (4.7) results in

0 =1

2

∫R

∫Hx

[f

(N∑i=1

uiφi(x′)−

N∑i=1

uiφi(x),x′ − x

)]T(v(x′)− v(x)) dVx′ dVx

−∫R

[b(x)]Tv(x) dVx.

(4.9)

The test functions are then chosen to match the basis functions in the set SN and leads tothe following system of equations


Figure 4.1: The triangular reference element used to build the piecewise linear shape func-tions.

0 =1

2

∫R

∫Hx

[f

(N∑i=1

uiφi(x′)−

N∑i=1

uiφi(x),x′ − x

)]T(φj(x

′)− φj(x)) dVx′ dVx

−∫R

[b(x)]Tφj(x) dVx,

(4.10)

for j = 1, ..., N .

4.2 Basis Functions

The basis functions φi(x) are generally chosen as the Lagrange interpolating polynomialson each element. For simplicity, only piecewise linear polynomials are considered and eachbasis function is assumed to be nontrivial in only one spatial direction. A reference elementis constructed with three finite element nodes as shown in Figure 4.1, and three nonzerolinear polynomials are defined for each spatial component on this element. First, considerbasis functions that are nontrivial only in the x1 direction. The three basis functions arefound by forcing φi(xj) = [δij 0]T where δij is the Kronecker delta function. Thus, if thefinite element nodes of the reference element are indexed as shown, then the nonzero localbasis functions in the x1 direction are

φ1(x) =

[1− x1 − x2

0

],

φ2(x) =

[x1

0

],

φ3(x) =

[x2

0

].

(4.11)

A linear mapping between the reference element and the true element is used to determine


the basis functions on the appropriate elements within the body R. Additionally, any φi

where i does not correspond to one of the finite element nodes on the element is defined tobe the zero function on this element. In this manner, a collection of piecewise linear “tent”functions for all spatial direction is defined on R by φ1,φ2, ...,φN. Figure 4.2 shows thenontrivial component of one such tent function on a square body.

Figure 4.2: Example of the piecewise linear tent function satisfying φi(xj) = δi,j.

4.3 Newton’s Method and the Inner Integral

The solution to the weak form of the peridynamic equilibrium equation is determined by thecoefficients u1, u2, ..., uN from (4.8). Assuming a linear model and using the kernel (3.25)allows the coefficients to be solved directly from Fu = b. However, to generalize f to thenonlinear case, the multidimensional Newton method must be employed to approximate thecoefficients. This method requires an initial guess for the coefficients u1, u2, ..., uN that isused to calculate the residual

Rj =1

2

∫R

∫Hx

[f

(N∑i=1

uiφi(x′)−

N∑i=1

uiφi(x),x′ − x

)]T(φj(x

′)− φj(x)) dVx′ dVx

−∫R

[b(x)]Tφj(x) dVx,

(4.12)

for j = 1, ..., N . The Jacobian matrix, J , is constructed, for example using a forwarddifference method, and the update step

s = −J−1R (4.13)

is calculated and added to the previous guess of the coefficients. This process continuesuntil the residual is sufficiently close to zero. It should be noted that the resultant Jacobianmatrix will be sparse as the integration is restricted to the horizon. Furthermore, J will also


be symmetric as a result of the linear admissibility condition. The structure of J outsideof these two properties will depend on the indexing of the finite element nodes, which isunknown, and the radius of the horizon.

In calculating (4.12), it should be noted that the traditional quadrature techniques used infinite elements must be modified slightly. Consider the double integral in (4.12). The innerintegral over Hx can be reduced to a function only of x

Ij(x) =1

2

∫Hx

[f

(N∑i=1

uiφi(x′)−

N∑i=1

uiφi(x),x′ − x

)]T(φj(x

′)− φj(x)) dVx′ . (4.14)

Thus, (4.12) simpifies to

Rj =

∫RIj(x) dVx −

∫R

[b(x)]Tφj(x) dVx. (4.15)

This integral can be solved using traditional quadrature techniques. Let the collection offinite elements be K. For a given K ∈ K, define the quadrature nodes and weights as xK,q

and wK,q, respectively. Then (4.15) can be approximated

Rj ≈∑K∈K

N∑q=1

wK,qIj(xK,q)−∑K∈K

N∑q=1

wK,q[b(xK,q)]Tφj(xK,q). (4.16)

From (4.16) it becomes clear that the inner integral must be solved for each quadrature nodeIj(xK,q). Consider a particular quadrature node xq on element K with horizon Hxq shown inFigure 4.3. The inner integral Ij is computed element-wise as is typical in the finite elementmethod. However, multiple elements intersect the boundary of Hxq and are therefore onlypartially contained in the horizon. The integral over these partial elements is approximated

Figure 4.3: An example horizon overlayed on the finite element mesh. Difficulties arise indealing with those elements which intersect the boundary of the horizon.


by constructing subelements. Once the area of the intersection is sufficiently approximated,new quadrature points and weights must be found on these subelements. Figure 4.4 showsa variety of scenarios that can arise and how subelements are constructed.

Figure 4.4: Several examples of how partially contained elements are handled by constructingsubelements.

Chapter 5

Numerical Results

5.1 Example 1: Cantilever Beam

The cantilever beam from earlier is revisited here. Recall, that a long beam is fixed at theright end while a downward shearing force is applied to the left end. The exact solutionto this problem is given by (2.24). Furthermore, a solution to this problem is shown usingthe finite element method to solve the classical equations of elasticity for this problem. Theframework for this solution can be found in [5]. The horizontal and vertical displacementgraphs of the exact and finite element elasticity solutions are presented in Figure 5.1 forthe case ` = 35, d = 2, F = −10, E = 3 × 106 and ν = 0.25. The finite element solutionwas computed with 200 nearly uniform linear elements corresponding to 260 displacementvariables.

Figure 5.1: The horizontal (1) and vertical (2) displacement maps for the cantilever beamproblem based on the exact solution (a) and the classical elasticity solution (b).

The numerical peridynamic solution begins with the construction of the cantilever beam withimaginary boundary layers on the left and right ends as shown in Figure 5.2. The boundarylayer RD, which may be thought of as part of the wall to which the beam is attached, willexperience zero displacement while a downward force will be placed in the region RN . These

26

Andrew T. Glaws Chapter 5. Numerical Results 27

Figure 5.2: The peridynamic formulation of the cantilever beam requires that boundarylayers with nonzero volume be added to the ends of the beam. Boundary conditions canthen be applied in these regions.

conditions are applied in the boundary regions by

u(x) = 0 for x ∈ RD

b(x) =

[0F

]for x ∈ RN ,

(5.1)

where the body force per unit area F < 0. To accelerate the convergence of the Newtonsteps, the linearized version of the peridynamic equation (3.25) is solved first. This solutionis then used as the initial guess to the general nonlinear form of the peridynamic equation(3.24).

Figure 5.3: The structure of the Jacobian matrices for various δ values.

The structure of the Jacobian matrices for various horizon radii is shown in Figure 5.3. Aswas mentioned earlier, these matrices are sparse. Furthermore, as δ increases the sparsity ofthe matrix decreases. This is caused by more elements falling into the radius of the horizonfor any particular point.


Displacement maps based on the peridynamic theory are shown in Figure 5.4. These solutionsare based on the cone-shaped coefficient

c(||ξ||) =8E

πδ3

(1− ||ξ||

δ

). (5.2)

Figure 5.4: The horizontal and vertical displacements based on the peridynamic methodwith decreasing horizons.

As the horizon radius δ diminishes, the peridynamic solutions approach the finite elementsolution of the classical elasticity equations as well as their exact solution. Figure 5.5 displaysthis convergence again by comparing the infinity norm of the differences in the displacementsolutions. Furthermore, the figure shows that increasing the density of the finite elementmesh improved the agreement between the two methods. This supports the theorecticalresults given in [13].

Figure 5.5: Maximum differences between the peridynamic and the classical solutions for (a)horizontal and (b) vertical displacements.


5.2 Example 2: Plate with a Hole

The next example considers an infinite plate with a circular hole in the center. The plate isplaced under uniaxial tension. This is another common problem in classical elasticity. Thestresses are known to be highest on either side of the hole along the axis perpendicular to theapplied force. The von Mises stress at these points are three times larger than the stressesfar away from the hole. Computationally, this ratio can be approximated with a finite plateprovided that the side length of domain of the solution is sufficiently larger than the radiusof the hole.

Symmetry in the problem can be exploited such that only one quarter of the plate must beconsidered. However, certain displacement boundary conditions must be applied. Considerthe domain with imaginary boundary layers shown in Figure 5.6. The conditions in theboundary layers are

u(x) =

[u1(x)

0

]for x ∈ RD1

u(x) =

[0

u2(x)

]for x ∈ RD2

b(x) =

[F0

]for x ∈ RN

(5.3)

where u1(x) and u2(x) are unknown function. Thus, in RD1 the plate is fixed only in thex direction and in RD1 the plate is fixed in the y direction. Such behavior along theseboundaries would be expected from the full plate due to symmetry.

Figure 5.6: Boundary layers are added to three sides of the plate. Displacement conditionsare applied in RD1 and RD2 and a tension force is applied in RN .

The von Mises stress maps found using elasticity and peridynamics models are shown inFigure 5.7. In both cases, the highest stress appears in the just above the circular hole


Figure 5.7: The von Mises stresses for the (a) classical elastic and (b) peridynamic models.The highest stresses appear just above the hole as expected.

and diminishes quickly further away from the hole. Furthermore, Table 5.2 gives the ratioof the maximum von Mises stress to the average stress on the boundary for several cases.As mentioned earlier, this ratio is known to be three in the case of a truly infinite platesolved using the elasticity model. With a sufficiently large plate, the finite element elasticitysolution approximates this ratio. Additionally, with a sufficiently small horizon radius, theperidynamic solution results in a stress ratio of approximately three. This further suggeststhat peridynamics is able to mimic the classical solution to problems in practice, providedthat δ is small.

5.3 Example 3: “Cracked” Plate

The final example examines a rectangular plate with a thin “crack” in the middle. As in theprevious example, a tension force is applied at the ends of the plate parallel to the crack.Again, the solution is found using the classical elasticity method and compared with results

Table 5.1: Stress ratios for elasticity and peridynamic models with varying horizon radii.

Model δ Stress Ratio

Elasticity N/A 3.04717

Peridynamics 0.100 3.23288




from the peridynamic formulation. Figure 5.8 shows the distribution of von Mises stressesfound. This figure shows the high concentrations of stress at the ends of the crack, whichone might expect to lead to further crack growth.

Figure 5.8: The von Mises stress distribution found using the classical elasticity theory.

To build the peridynamic problem, imaginary boundary layers are added to the upper andlower ends of the plate, shown in Figure 5.9. In these regions, the body force is

b(x) =

[0F

]for x ∈ RN1

b(x) =

[0−F

]for x ∈ RN2

(5.4)

with F > 0 is applied to simulate the pulling force from the elastic case.

Figure 5.9: The cracked plate with imaginary boundary laryers on the top and bottom.

In Chapter 3, we introduced the peridynamic equation for the microelastic materials. Theequation contains the coefficient c(||ξ||) which, as mentioned earlier, can take a wide range


of forms. In Figure 5.10, the cracked plate problem is solved using the peridynamic methodfor several common coefficients models:

1. Constant coefficient

c(||ξ||) =8E

πδ3

2. Cone-shaped coefficient

c(||ξ||) =8E

πδ3

(1− ||ξ||

δ

)3. Inverted cone-shaped coefficient

c(||ξ||) =8E

πδ3

(||ξ||δ

).

All three coefficients proved relatively reliable as each manages to show the high concen-trations of stress near the ends of the crack with low stresses appearing along each side.However when compared to the solution from elasticity, the cone-shaped coefficient appearsto perform the best. Conversely, the coefficient that performed the worst was the invertedcone. This agrees with the intuitive idea that in elastic media, the particles closer to theparticle of interest should have the largest influence.


Figure 5.10: The stress distribution and errors for the cracked plate found using peridynam-ics. Three popular forms of the coefficient c(||ξ||) are used.

Chapter 6

Conclusions

Several typical benchmark problems in classical elasticity have been examined numerically.The peridynamic method is shown to converge well to elasticity as the horizon diminishes.This solidifies the notion of peridynamics as the nonlocal analog to the elastic model. Fur-thermore, as expected refining the mesh density improves the ability of the peridynamic toconverge to the known elasticity solution. However, large increases in computational costsoccur as the number of elements within each quadrature point’s horizon increases leadingto higher integration costs as well as higher nonlinear solver costs due to an increase in theoverall size of the system. This spike in costs can be curbed somewhat through a precalcu-lation of the horizon for each quadrature point as this is based on the unstressed referenceconfiguration.

Having justified the ability of the peridynamic solution to accurately mimic solutions forwhich classical elasticity is know to behave well, a logical next step would be to applythis technique to areas where elasticity has more difficulties, such as problems of crackpropogation. Time dependence and fracture dynamics would have to integrated into thecurrent finite element model to achieve this. Furthermore, the existing constitutive modelsfor fracture described above are highly simplified and is an important area for further study.A computational challenge would be to extend this study to three dimensional regions. In thiscase, the technique of building subelements to handle the intersections of horizons with meshelements would need to be improved upon. One suggestion is to incorporate isoparametricelements where needed.

34

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[2] F. Bobaru and Y. D. Ha. Adaptive refinement and multiscale modeling in 2d peridy-namics. Journal for Multiscale Computational Engineering, 9:635–659, 2011.

[3] X. Chen and M. Gunzburger. Continuous and discontinuous finite element methodsfor a peridynamics model of mechanics. Computer Methods in Applied Mechanics andEngineering, 200:1237–1250, 2010.

[4] L. F. Alves S. A. Silling E. Askari F. Bobaru, M. Yang and J. Xu. Convergence, adap-tive refinement, and scaling in 1d peridynamics. International Journal for NumericalMethods in Engineering, 77:852–877, 2009.

[5] J. Koko. Vectorized Matlab codes for linear two-dimensional elasticity. Scientific Pro-gramming, 15:157–172, 2007.

[6] R. W. Macek and S. A. Silling. Peridynamics via finite element analysis. Finite Elementsin Analysis and Design, 43:1169–1178, 2007.

[7] E. Madenci and E. Oterkus. Peridynamic Theory and Its Applications. Springer, NewYork, NY, 2014.

[8] W. A. Nash. Strength of Materials. McGraw-Hill, New York, NY, 1972.

[9] M. Zimmermann S. A. Silling and R. Abeyaratne. Deformation of a peridynamic bar.Journal of Elasticity, 73:173–190, 2003.

[10] A. S. Saada. Elasticity Theory and Applications. Krieger, Malabar, FL, 1993.

[11] S. A. Silling. Reformulation of elasticity theory for discontinuities and long-range forces.Journal of the Mechanics and Physics of Solids, 48:175–209, 1999.

[12] S. A. Silling and E. Askari. A meshfree method based on the peridynamic model ofsolid mechanics. Computers & Structures, 83:1526–1535, 2005.

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Andrew T. Glaws Chapter 6. Conclusions 36

[13] S. A. Silling and R. B. Lehoucq. Convergence of peridynamics to classical elasticitytheory. Journal of Elasticity, 93:13–37, 2008.

[14] S. Timoshenko and J. N. Goodier. Theory of Elasticity. McGraw-Hill, New York, NY,1951.

[15] A. C. Ugural and S. K. Fenster. Advanced Strength and Applied Elasticity. PrenticeHall, Upper Saddle River, NJ, 2003.

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