Finite Element Simulations of Two Dimensional Peridynamic Models

Transcription

1 Finite Element Simulations of Two Dimensional Peridynamic Models Andrew T. Glaws Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Masters of Science in Mathematics Jeffrey T. Borggaard, Chair Lizette Zietsman Tao Lin May 7, 2014 Blacksburg, Virginia Copyright 2014, Andrew T. Glaws

2 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. Peridynamics has several key advantages over the classical theory of elasticity. The most notable of which is the ease with which fractures in the the material are handled. The goal here is to study the two theories and how they relate for problems in which the classical method is known to work well. While it is known that state-based peridynamic models agree with classical elasticity as the horizon radius vanishes, similar results for bond-based models have yet to be developed. In this study, we use numerical simulations to investigate the behavior of bond-based peridynamic models under this limit for a number of cases where analytic solutions of the classical elasticity problem are known. To carry out this study, the integralbased peridynamic model is solved using the finite element method in two dimensions and compared against solutions using the classical approach.

3 Contents 1 Introduction 1 2 Classical Elastic Theory The Equilibrium Equations The Stress-Strain Relations The Compatibility Equations Reduction to Two Dimensions and the Airy Stress Function Example Problem Peridynamic Theory Peridynamic States and the Equations of Motion Bond-based Peridynamics Microelastic Materials Boundary Conditions Fracture and Damage Finite Element Method The Weak Form Basis Functions Newton s Method and the Inner Integral iii

4 5 Numerical Results Example 1: Cantilever Beam Example 2: Plate with a Hole Example 3: Cracked Plate Conclusions 34 Bibliography 35 iv

5 List of Figures 2.1 The principle and shear stresses on each face are decomposed into their directional components An arbitrary body with directional components of the normal vector l 1, l 2, l The classical problem of the cantilevered beam with a shearing force on the free end. This problem will be returned to later The horizon H x of a point and a particular bond ξ = x x within the horizon The initial configuration of a point s horizon and its deformed image. There are no restrictions on how the deformation state vector can map bonds to their stressed state Various possible forms of the coefficient c( ξ ). The coefficient is defined to be zero outside the horizon and need not be continuous The imaginary boundary layers R 1 and R 2 are added to the true body R. Boundary conditions are applied within in these regions A plot of the pairwise force function against the streching of a bond. The force drops to zero once the bond has been stretched beyond the critical value s c The triangular reference element used to build the piecewise linear shape functions Example of the piecewise linear tent function satisfying φ i (x j ) = δ i,j An example horizon overlayed on the finite element mesh. Difficulties arise in dealing with those elements which intersect the boundary of the horizon Several examples of how partially contained elements are handled by constructing subelements v

6 5.1 The horizontal (1) and vertical (2) displacement maps for the cantilever beam problem based on the exact solution (a) and the classical elasticity solution (b) The peridynamic formulation of the cantilever beam requires that boundary layers with nonzero volume be added to the ends of the beam. Boundary conditions can then be applied in these regions The structure of the Jacobian matrices for various δ values The horizontal and vertical displacements based on the peridynamic method with decreasing horizons Maximum differences between the peridynamic and the classical solutions for (a) horizontal and (b) vertical displacements Boundary layers are added to three sides of the plate. Displacement conditions are applied in R D1 and R D2 and a tension force is applied in R N The von Mises stresses for the (a) classical elastic and (b) peridynamic models. The highest stresses appear just above the hole as expected The von Mises stress distribution found using the classical elasticity theory The cracked plate with imaginary boundary laryers on the top and bottom The stress distribution and errors for the cracked plate found using peridynamics. Three popular forms of the coefficient c( ξ ) are used vi

7 Chapter 1 Introduction The study of solid mechanics is a branch of continuum mechanics that seeks to understand the motion and deformation of bodies under load. An important area within this field is fracture mechanics, which deals with the formation and propogation of cracks. The imoprtance of the field is clear when one considers the structures, machinery, etc., that we rely upon not to fail. Traditionally, the theory of elasticity has been used as the means to mathematically explain solid mechanics. However, this approach cannot easily handle cracks in the material. Elasticity uses spatial derivatives to represent relationships between neighboring particles. These derivatives are undefined at discontinuities, or cracks, in the material. To deal with this issue, several techniques have been developed. For example, one may redefine the domain of the component so that the crack falls on the boundary or one may couple the system with additional equations to describe the motion of the crack through the material. However, these techniques require prior knowledge about where the crack will be located in the material and how it will grow. Thus, elasticity is not favorable when considering problems containing fractures. The theory of peridynamics overcomes this limitation by eliminating spatial derivatives in favor of integral equations to describe the relationships between particles. Since integrals are defined over discontinuities, the same set of equations are applied throughout the material regardless of the existence of any cracks. This non-local view of particle interactions allows for natural crack growth. Despite the obvious advantages of peridynamics in problems dealing with fracture, the focus of this study is not on crack formation or propogation. This topic is discussed briefly later in the document when we consider future research directions. However, the goal here is to compare the behavior of materials under the peridynamic theory and the classical elastic theory. 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

8 Andrew T. Glaws Chapter 1. Introduction 2 main components of the theory. Then, the peridynamic theory is discussed with some discussion concerning how one could incorporate crack formation and propogation. This is followed by an explanation of how the finite element method is applied to the peridynamic theory. Finally, some classical two dimensional problems are examined using both the classical elastic theory and peridynamics.

9 Chapter 2 Classical Elastic Theory The theory of elasticity is the classical method for studying bodies under load. This area has been studied extensively and only the major concepts are discussed here. Further details can be found in [8, 10, 14, 15]. A body can deform either elastically, where it returns to its original configuration once the load is removed, or plastically, where is does not return to its original configuration once the load is removed. As the name suggests, the theory of elasticity applies in the case of elastic deformations. In most continuum models, any body will deform elastically until some limiting 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 volume such as gravity or thermal forces. These body forces per unit volume are resolved into three components aligning with each coordinate axis, denoted X, Y, and Z. Alternatively, a force can act on the surface such as contact forces between two bodies. These forces per unit area are 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 divided into normal stresses, σ, and shear stresses, τ. For normal stresses, the subscript provides the direction of the force that is perpendicular to the plane on which the stress acts. For shear stresses, the first subscript is again the direction perpendicular to the plane on which the stress acts while the second subscript provides the direction of the shearing. It can be shown through the summation of moments that the shear stresses about each of the axes are equal. That is, τ xy = τ yx, τ yz = τ zy, τ xz = τ zx. (2.1) The stress threshold for plastic deformation mentioned earlier is important in the study of material failure. It has been shown that a material can fail despite none of the individual component stresses exceeding this threshold. Thus, a new value, called the von Mises stress 3

10 Andrew T. Glaws Chapter 2. Classical Elastic Theory 4 Figure 2.1: The principle and shear stresses on each face are decomposed into their directional components. equivalent or simply the von Mises stress, is defined as σ vm = (σ x σ y ) 2 + (σ y σ z ) 2 + (σ z σ x ) 2 + 6(τxy 2 + τyz 2 + τxz) 2. (2.2) 2 The von Mises stress equivalent is not truly a stress but a measure of the combined stresses and 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 different components of stress vary from point to point within a body. Consider the arbitrary body R shown in Figure 2.2. The total force applied to any point in R is decomposed into its directional body forces per unit volume and its direction surface forces per unit area as discussed earlier. The projection of the stresses onto the x axis is given by

11 Andrew T. Glaws Chapter 2. Classical Elastic Theory 5 Figure 2.2: An arbitrary body with directional components of the normal vector l 1, l 2, l 3. R (σ x l 1 + τ yx l 2 + τ zx l 3 ) ds, (2.3) where l 1, l 2, l 3 are the directional components of the outward normal. Next, the divergence theorem is applied to (2.3) resulting in R ( σx x + τ yx y + τ ) zx dv. (2.4) z 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 + τ yx y + τ ) zx z + X dv = 0. (2.5) Since the above holds for any generic body, the integrand can be assumed to be zero. A similar process in the y and z directions yields the three equilibrium equations σ x x + τ yx y + τ zx z + X = 0 τ xy x + σ y y + τ zy z + Y = 0 τ xz x + τ yz y + σ z z + Z = 0. (2.6)

12 Andrew T. Glaws Chapter 2. Classical Elastic Theory 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 along the x, y, and z axes, respectively. Strain is defined to relate the forces acting on a body to the actual deformation of the body. It is a measure of the intensity of the deformation in a particular direction. For example, the strain normal to the x axis can be thought of as the amount of elongation or contraction a particular element experiences given by u ε x = lim x 0 x = du dx. (2.7) Shear strain accounts for the intensity of the rotational deformation of any element and is given by γ xy = u y + v x. (2.8) By expanding this idea to the other directions, nine strain-displacement relations can be derived. However, any rotational deformation is not affected if the angle is measured from the x axis to the y axis or vice versa. Thus, the nine strain-displacement relations can be reduced to the six equations ε x = u x γ xy = γ yx = u y + v x ε y = v y γ xz = γ zx = u z + w x ε z = w z γ yz = γ zy = v z + w (2.9) y. It has been found through experiment that the relationship between stress and strain can be represented by σ x = 2Gε x + λe τ xy = Gγ xy σ y = 2Gε y + λe τ yz = Gγ yz σ z = 2Gε z + λe τ xz = Gγ xz, (2.10) where e = ε x + ε y + ε z (2.11) is called the volume strain. The constants G and λ are known as the Lamé constants and are related to the material s properties (bulk modulus E, and Poisson s ratio ν) through

13 Andrew T. Glaws Chapter 2. Classical Elastic Theory 7 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 deformations are geometrically allowable in the body. To illustrate this idea, consider a cube that will 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 should form between cubes). Mathematically, the need for the compatibility equations arises from the fact that the six strain components in (2.9) are expressed in terms of three displacement terms. Thus, it is to be expected that a relationship exists between the strain terms. The compatibility equations are not derived here but are given by 2 ε x y 2 2 ε y z 2 2 ε z x ε y x ε z y ε x z 2 = 2 γ xy x y = 2 γ yz x z = 2 γ xz x z 2 2 ε x y z = ( γ yz x x + γ xz y ( 2 2 ε y x z = y 2 2 ε z x y = z γ xz y ( γ xy z + γ xy z + γ ) xy z ) + γ yz x + γ yz x + γ xz y ). (2.14) To completely describe a loaded body using elasticity, three displacements, six stresses, and six strains must be found using three equilibrium equations, six stress-strain equations, and six strain-displacement relations. The problem is solved in the stress-strain realm while the compatibility equations ensure that the resultant deflections are smooth. The final component needed to solve the system of fifteen partial differential equations is the boundary conditions. These are generally described by surface traction equations having units force per area. These equations can be split into the various stress components along each axial direction. Alternatively, the boundary conditions can be provided in terms of displacements and depend on the strain-displacement equations to relate them to the rest of the system.

14 Andrew T. Glaws Chapter 2. Classical Elastic Theory Reduction to Two Dimensions and the Airy Stress Function A problem in elasticity may be reduced to two dimension in two ways depending on the geometry involved. Consider first a problem in which third dimension is much longer than the two cross-sectional dimensions, such as in an infinitely long beam. This is best handled using a technique called plane strain. Alternatively, if the third dimension is much smaller than 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 directions with no need to balance forces in the z direction assuming Z = 0. Additionally, the straindisplacement relations (2.9) reduce simply by allowing ε z = γ xz = γ yz = 0 and all other components to remain the same. The stress-strain equations become σ x = 2Gε x + λ(ε x + ε y ) τ xy = Gγ xy σ y = 2Gε y + λ(ε x + ε y ) τ yz = 0 σ z = λ(ε x + ε y ) τ xz = 0 (2.15) for plane strain, while for plane stress they are ε x = 1 E (σ x νσ y ) γ xy = τ xy G ε y = 1 E (σ y νσ x ) γ yz = 0 ε z = ν E (σ x + σ y ) γ xz = 0. (2.16) If the body forces F x and F y are negligible and the system is being represented in Cartesian coordinates, the equilibrium equations can be satisfied by a single scalar function known as the Airy stress function Φ(x, y). This function is related to the stresses in the two dimensional body by σ x = 2 Φ (x, y), y σ 2 y = 2 Φ (x, y), x τ 2 xy = 2 Φ (x, y). (2.17) x y Combining (2.17) with the equilbrium equations results in the two dimensional biharmonic equation 4 Φ = 4 Φ x Φ x 2 y + 4 Φ 2 y 4 = 0. (2.18)

15 Andrew T. Glaws Chapter 2. Classical Elastic Theory 9 Figure 2.3: The classical problem of the cantilevered beam with a shearing force on the free end. 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 long beam is attached to a wall at one end while a downward force is applied at the free end as shown in Figure 2.3. The stresses associated by such a system can be described by σ x = c 2 xy, σ y = 0, τ xy = c 1 c 2 2 y2. (2.19) Since no forces are being applied to the top and bottom boundaries of the beam, setting (τ xy ) y=±d = 0 yields c 2 = 2c 1 d 2. (2.20) A total shearing force per unit length F is applied over the end of the bar, resulting in d d τ xy dy = F. This is used to find the constant Thus, the forces may be rewritten as c 1 = 3F 4d. (2.21) σ x = 3F 2d xy, σ 3 y = 0, τ xy = 3F y2 (1 + 4d d ). (2.22) 2 Using the stress-strain and the strain-displacement equations to relate the displacements u and v to (2.22) results in

16 Andrew T. Glaws Chapter 2. Classical Elastic Theory 10 u x = 3F 2Ed xy, 3 v y = 3νF 2Ed xy, 3 u y + v x = 3F y2 (1 + 2Gd d ). 2 (2.23) Integrating (2.23) and enforcing conditions to prevent rigid body movement yields the final solution to the cantilever beam ( F u(x, y) = 4Gd νf ) y 3 3F 3 4Ed 3 4Ed 3 x2 y + v(x, y) = F 4Ed 3 x3 + 3νF 3F l2 4Ed 3 xy2 4Ed x + F l3 3 2Ed. 3 ( 3F l 2 4Ed 3F ) y, 3 4Gd (2.24)

17 Chapter 3 Peridynamic Theory The peridynamic theory was first introduced by Silling in [11]. Since then, peridynamics has been studied in both theoretical and computational frameworks, e.g. [7, 9]. Early methods were based on bond-based peridynamics which consider particle-particle interactions one at a time. Since then the theory has expanded to consider the infinitely many simultaneous interactions. Furthermore, it has been shown that the peridynamics theory converges to the classic elastic theory in the limiting case [13]. Computationally, peridynamics has been investigated 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 between adjacent particles in a body, the peridynamic theory allows for particles to interact over a finite distance through bonds. Given a component that occupies the region R and a particle in the body, x, peridynamics considers the relationship between x and all other particles, x, within the horizon of x, denoted by H x. In general, the horizon is defined by all particles within a ball of radius δ > 0 of the particle of interest. An example horizon is shown in Figure 3.1. The notation ξ = x x is used to identify a particular bond. H x = {x R : ξ = x x < δ}. (3.1) To understand how a particle influences other particles within its horizon, two state vectors are defined. At a given time t > 0, the deformation state vector is given by 11

18 Andrew T. Glaws Chapter 3. Peridynamic Theory 12 Figure 3.1: The horizon H x of a point and a particular bond ξ = x x within the horizon. (x 1 + u(x 1, t)) (x + u(x, t)) y(x 1, t) y(x, t) (x 2 + u(x 2, t)) (x + u(x, t)) Y [x, t] =. = y(x 2, t) y(x, t)., where H x = {x i } i=1. (x + u(x, t)) (x + u(x, t)) y(x, t) y(x, t) (3.2) The deformation state Y [x, t] : R 2 R 2 maps bonds from their reference state to their deformed 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 deformations that can occur around a single point. This is shown in Figure 3.2 where the deformed image can take potentially any shape. This generalization improves upon the classical theory in which spherical regions (such as a point s horizon) can only be mapped into ellipsoidal deformed states. Figure 3.2: The initial configuration of a point s horizon and its deformed image. There are no 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 vector is referred to as the force state vector and is denoted

19 Andrew T. Glaws Chapter 3. Peridynamic Theory 13 t(u(x 1, t) u(x, t), x 1 x) t(u(x 2, t) u(x, t), x 2 x) T [x, t] =., (3.4) t(u(x, t) u(x, t), x x) where t(u(x, t) u(x, t), x x), known as the force density vector, represents the force density applied to particle x by particle x. Using the notation η = u(x, t) u(x, t) for the relative deformation of two points, the force density vector becomes t(η, ξ). The force density 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 peridynamics 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 the deformation between the two particles but is defined by the cumulative deformation of all bonds in H x. This fact will not hold in future sections when a more specialized form of the peridynamic theory is examined. Additionally, since only particles within the horizon of x can influence the force state vector, it is assumed that x H x = T [x, t] x x = 0. (3.7) Once a constitutive model is set up, the basic peridynamic equation of motion is defined by the integral equation or ρü{x, t) = {T [x, t] x x T [x, t] x x }dv x + b(x, t) H x (3.8) ρü{x, t) = {t(u(x, t) u(x, t), x x) t(u(x, t) u(x, t), x x )}dv x +b(x, t), (3.9) H x where ρ is the mass density of the material and the term b, the loading force density, contains any external forces per unit reference volume acting at a point.

20 Andrew T. Glaws Chapter 3. Peridynamic Theory 14 In the case of static, but stressed materials, terms such as u and ü vanish. Thus, the equilibrium equation is 0 = {T [x, t] x x T [x, t] x x }dv x + b(x, t) (3.10) H x or 0 = {t(u(x, t) u(x, t), x x) t(u(x, t) u(x, t), x x )}dv x + b(x, t). (3.11) H x While the generality of state-based peridynamics offers obvious advantages, an alternative form of the peridynamic theory allows for simpler practical implementation but at the cost of added theoretical limitations [7]. 3.2 Bond-based Peridynamics A specialized version of the peridynamic theory exists in which the force density in a particular bond is determined only by the deformation of the individual bond and thus is independent of all other bonds within the horizon. Furthermore, the force density vectors can be assumed to be equal and opposite within a single bond. However, these simplifying assumptions result in several limitations in the theory. The most notable being that this model supports only one independent material constant. Thus, the Poisson s ratio is fixed at 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 the constitutive 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 ρü(x, t) = f(η, ξ) dv x + b(x, t) (3.14) H x

21 Andrew T. Glaws Chapter 3. Peridynamic Theory 15 and in the case of static materials 0 = f(η, ξ) dv x + b(x). H x (3.15) It should be noted that the pairwise force function need only be integrable, with no restrictions regarding smoothness or continuity with respect to either variable. Additionally from (3.7), it is assumed that f vanishes outside of H x. Thus, a given particle does not have any influence on any other particle that is outside of its horizon. Two important restrictions to the pairwise force function come from Newton s Third Law of Motion and the conservation of angular momentum. First, the model must respect equal and 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 particles to 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 where the scalar-valued function F satisfies f(η, ξ) = F (η, ξ) ξ + η, η, ξ, (3.18) ξ + η F (η, ξ) = F ( η, ξ) η, ξ. (3.19) From the above equations, it becomes clear that the interactions contained in a bond may be 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 contained 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.

22 Andrew T. Glaws Chapter 3. Peridynamic Theory 16 A material is called microelastic if there exists a scalar function, w(η, ξ), such that f(η, ξ) = w(η, ξ). (3.20) η This function, w, is referred to as the micropotential function and represents the energy per unit volume in a particular bond. The total energy density at a point can be found by adding up the energy in all the bonds in the horizon of the point, W = 1 2 H x w(η, ξ) dv x. (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 the body can be recovered. The microelastic potential depends on the intensity of the deformation of a bond. intensity is captured in This s(η, ξ) = ξ + η ξ, (3.22) ξ 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 bond to its reference state. A positive value of s indicates a lengthening of the bond while a negative value represents compression. The general form of the microelastic potential is Combining (3.20) and (3.23) yields w(η, ξ) = 1 2 c( ξ ) s2 (η, ξ) ξ. (3.23) f(η, ξ) = ( ) 1 η 2 c( ξ ) s2 (η, ξ) ξ = c( ξ ) s(η, ξ) ξ s η ξ + η = c( ξ ) s(η, ξ) ξ + η. (3.24) This equation provides the general framework for the peridynamic model of a microelastic material. It should be noted that while one must account for rigid body translations in the

23 Andrew T. Glaws Chapter 3. Peridynamic Theory 17 Figure 3.3: Various possible forms of the coefficient c( ξ ). The coefficient is defined to be zero outside the horizon and need not be continuous. system, rigid rotations are not an issue due to the presence of the unit directional vector of the 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 the constant 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 (0, ξ) η (3.25) η where f (0, ξ) is the Jacobian matrix of f evalutated at η = 0. While (3.25) can prove useful in simplifying numerical calculations, the loss of the stretching term results in significant η errors for problems with sufficiently large displacements. 3.4 Boundary Conditions In contrast to elasticity, no boundary conditions are required in order to solve the peridynamic equation of motion (due to the lack of any spatial derivatives). Thus, no separate equations are needed to solve the integral-based system. However, the body s interactions with its environment need to be captured in some form. External forces, which were presented either as principal or shear stresses or as body forces in the classical theory, are enforced via the loading force density, b(x). However, simply applying 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 volume is added to the material where the force is to be applied as shown in Figure 3.4. It has been suggested by numerical studies that this boundary layer be approximately equal to the

24 Andrew T. Glaws Chapter 3. Peridynamic Theory 18 Figure 3.4: The imaginary boundary layers R 1 and R 2 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 displacement condition on the fixed end in the cantilever beam problem from earlier. Similar to the case with surface forces, an imaginary boundary layer is added to the material along the segment where the condition is to be applied. The displacement is then enforced within this region while it is solved for in the rest of the body. The displacement cannot simply be enforced along the boundary of the body since the peridynamic equations are integral-based. Hence, the displacement boundary conditions must be enforced in a region with nonzero volume to ensure the conditions do not integrate to zero. As was the case with surface forces above, numerical experiments suggest that the imaginary 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 grow naturally. 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 a predefined value called the critical stretch. Thus, a new term is introduced into (3.24) where f(η, ξ) = c( ξ ) s(η, ξ) ξ + η µ(t, s), (3.26) ξ + η { 1 if s(t µ(t, s) = ) < s c for all t [0, t] 0 otherwise. (3.27) Thus, when a bond is stretched beyond the critical stretch, s c, the pairwise force function

25 Andrew T. Glaws Chapter 3. Peridynamic Theory 19 Figure 3.5: A plot of the pairwise force function against the streching of a bond. The force drops to zero once the bond has been stretched beyond the critical value s c. 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 stretch further and break resulting the growth of a crack in the material. The overall damage at a point is measured by φ(x, t) = 1 H x µ(t, s) dv x. (3.28) H x dv x 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 attached to the rest of the body.

26 Chapter 4 Finite Element Method The Finite Element Method (FEM), a technique commonly used to solve partial differential equations, may also be applied to solve integral equations such as (3.14) or (3.15). In general, the FEM begins with a reformulation of the problem into a weak form with the same solution as the original problem. Then a mesh is created by dividing the domain Ω into a finite number of elements. The approximate solution is found by solving the weak equation on each of the elements. 4.1 The Weak Form The generalized derivation of the weak form for problems with nonlocal operators can be found in [1]. Consider the equilibrium peridynamic equations given by (3.15) in two dimensions, and assume that the displacement function u belongs to the space S d = {v L 2 (R) L 2 (R) : v Rd = g d (x)} (4.1) or S n = L 2 0(R) L 2 0(R) = {v L 2 (R) L 2 (R) : v dx = 0} (4.2) with inner product u, v = u T v dx. (4.3) While the trial function set will vary depending on the boundary conditions of the problem, R Ω 20

27 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)) = f(u(x ) u(x), x x) dv x. (4.4) H x 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 Thus, the weak form of (3.15) is R H x [f(u(x ) u(x), x x)] T (v(x ) v(x)) dv x dv x. (4.6) 0 = 1 2 R [f(u(x ) u(x), x x)] T (v(x ) v(x)) dv x dv x H x R [b(x)] T v(x) dv x. (4.7) The function space S is then approximated by an N dimensional set S N. The goal is then to find a function in S N that approximates the unknown displacement function u(x) u N (x) = N u i φ i (x) S N, (4.8) i=1 where the set {φ 1 (x), φ 2 (x),..., φ N (x)} is a basis of S N. An approximate solution for the displacement of the body is calculated by finding the unknown coefficients {u 1, u 2,...u N }. Substituting (4.8) into (4.7) results in 0 = 1 [ ( N f u i φ 2 i (x ) R H x i=1 T N u i φ i (x), x x)] (v(x ) v(x)) dv x dv x i=1 [b(x)] T v(x) dv x. R (4.9) The test functions are then chosen to match the basis functions in the set S N and leads to the following system of equations

28 Andrew T. Glaws Chapter 4. Finite Element Method 22 Figure 4.1: The triangular reference element used to build the piecewise linear shape functions. 0 = 1 [ ( N f u i φ 2 i (x ) R for j = 1,..., N. H x i=1 T N u i φ i (x), x x)] (φ j (x ) φ j (x)) dv x dv x i=1 [b(x)] T φ j (x) dv x, R (4.10) 4.2 Basis Functions The basis functions φ i (x) are generally chosen as the Lagrange interpolating polynomials on each element. For simplicity, only piecewise linear polynomials are considered and each basis function is assumed to be nontrivial in only one spatial direction. A reference element is constructed with three finite element nodes as shown in Figure 4.1, and three nonzero linear polynomials are defined for each spatial component on this element. First, consider basis functions that are nontrivial only in the x 1 direction. The three basis functions are found by forcing ˆφ i (ˆx j ) = [δ ij 0] T where δ ij is the Kronecker delta function. Thus, if the finite element nodes of the reference element are indexed as shown, then the nonzero local basis functions in the x 1 direction are [ ] 1 x1 x ˆφ 1 (x) = 2, 0 [ ] x1 ˆφ 2 (x) =, 0 [ ] x2 ˆφ 3 (x) =. 0 (4.11) A linear mapping between the reference element and the true element is used to determine

29 Andrew T. Glaws Chapter 4. Finite Element Method 23 the basis functions on the appropriate elements within the body R. Additionally, any φ i where i does not correspond to one of the finite element nodes on the element is defined to be 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 {φ 1, φ 2,..., φ N }. Figure 4.2 shows the nontrivial component of one such tent function on a square body. Figure 4.2: Example of the piecewise linear tent function satisfying φ i (x j ) = δ i,j. 4.3 Newton s Method and the Inner Integral The solution to the weak form of the peridynamic equilibrium equation is determined by the coefficients {u 1, u 2,..., u N } from (4.8). Assuming a linear model and using the kernel (3.25) allows the coefficients to be solved directly from F u = b. However, to generalize f to the nonlinear case, the multidimensional Newton method must be employed to approximate the coefficients. This method requires an initial guess for the coefficients {ũ 1, ũ 2,..., ũ N } that is used to calculate the residual R j = 1 2 R H x [ ( N f ũ i φ i (x ) i=1 T N ũ i φ i (x), x x)] (φ j (x ) φ j (x)) dv x dv x i=1 [b(x)] T φ j (x) dv x, R (4.12) for j = 1,..., N. The Jacobian matrix, J, is constructed, for example using a forward difference method, and the update step s = J 1 R (4.13) is calculated and added to the previous guess of the coefficients. This process continues until the residual is sufficiently close to zero. It should be noted that the resultant Jacobian matrix will be sparse as the integration is restricted to the horizon. Furthermore, J will also

30 Andrew T. Glaws Chapter 4. Finite Element Method 24 be symmetric as a result of the linear admissibility condition. The structure of J outside of these two properties will depend on the indexing of the finite element nodes, which is unknown, and the radius of the horizon. In calculating (4.12), it should be noted that the traditional quadrature techniques used in finite elements must be modified slightly. Consider the double integral in (4.12). The inner integral over H x can be reduced to a function only of x I j (x) = 1 [ ( N f u i φ 2 i (x ) H x i=1 T N u i φ i (x), x x)] (φ j (x ) φ j (x)) dv x. (4.14) i=1 Thus, (4.12) simpifies to R j = R I j (x) dv x R [b(x)] T φ j (x) dv x. (4.15) This integral can be solved using traditional quadrature techniques. Let the collection of finite elements be K. For a given K K, define the quadrature nodes and weights as x K,q and w K,q, respectively. Then (4.15) can be approximated N R j w K,q I j (x K,q ) K K q=1 K K N w K,q [b(x K,q )] T φ j (x K,q ). (4.16) From (4.16) it becomes clear that the inner integral must be solved for each quadrature node I j (x K,q ). Consider a particular quadrature node x q on element K with horizon H xq shown in Figure 4.3. The inner integral I j is computed element-wise as is typical in the finite element method. However, multiple elements intersect the boundary of H xq and are therefore only partially contained in the horizon. The integral over these partial elements is approximated q=1 Figure 4.3: An example horizon overlayed on the finite element mesh. Difficulties arise in dealing with those elements which intersect the boundary of the horizon.

31 Andrew T. Glaws Chapter 4. Finite Element Method 25 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 shows a variety of scenarios that can arise and how subelements are constructed. Figure 4.4: Several examples of how partially contained elements are handled by constructing subelements.

32 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 the right end while a downward shearing force is applied to the left end. The exact solution to this problem is given by (2.24). Furthermore, a solution to this problem is shown using the finite element method to solve the classical equations of elasticity for this problem. The framework for this solution can be found in [5]. The horizontal and vertical displacement graphs of the exact and finite element elasticity solutions are presented in Figure 5.1 for the case l = 35, d = 2, F = 10, E = and ν = The finite element solution was computed with 200 nearly uniform linear elements corresponding to 260 displacement variables. Figure 5.1: The horizontal (1) and vertical (2) displacement maps for the cantilever beam problem based on the exact solution (a) and the classical elasticity solution (b). The numerical peridynamic solution begins with the construction of the cantilever beam with imaginary boundary layers on the left and right ends as shown in Figure 5.2. The boundary layer R D, which may be thought of as part of the wall to which the beam is attached, will experience zero displacement while a downward force will be placed in the region R N. These 26

33 Andrew T. Glaws Chapter 5. Numerical Results 27 Figure 5.2: The peridynamic formulation of the cantilever beam requires that boundary layers with nonzero volume be added to the ends of the beam. Boundary conditions can then be applied in these regions. conditions are applied in the boundary regions by u(x) = 0 [ ] 0 b(x) = F for x R D for x R N, (5.1) where the body force per unit area F < 0. To accelerate the convergence of the Newton steps, the linearized version of the peridynamic equation (3.25) is solved first. This solution is 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. As was mentioned earlier, these matrices are sparse. Furthermore, as δ increases the sparsity of the matrix decreases. This is caused by more elements falling into the radius of the horizon for any particular point.

34 Andrew T. Glaws Chapter 5. Numerical Results 28 Displacement maps based on the peridynamic theory are shown in Figure 5.4. These solutions are based on the cone-shaped coefficient c( ξ ) = 8E ( 1 ξ ). (5.2) πδ 3 δ Figure 5.4: The horizontal and vertical displacements based on the peridynamic method with decreasing horizons. As the horizon radius δ diminishes, the peridynamic solutions approach the finite element solution of the classical elasticity equations as well as their exact solution. Figure 5.5 displays this convergence again by comparing the infinity norm of the differences in the displacement solutions. Furthermore, the figure shows that increasing the density of the finite element mesh improved the agreement between the two methods. This supports the theorectical results given in [13]. Figure 5.5: Maximum differences between the peridynamic and the classical solutions for (a) horizontal and (b) vertical displacements.

35 Andrew T. Glaws Chapter 5. Numerical Results Example 2: Plate with a Hole The next example considers an infinite plate with a circular hole in the center. The plate is placed under uniaxial tension. This is another common problem in classical elasticity. The stresses are known to be highest on either side of the hole along the axis perpendicular to the applied force. The von Mises stress at these points are three times larger than the stresses far away from the hole. Computationally, this ratio can be approximated with a finite plate provided that the side length of domain of the solution is sufficiently larger than the radius of the hole. Symmetry in the problem can be exploited such that only one quarter of the plate must be considered. However, certain displacement boundary conditions must be applied. Consider the domain with imaginary boundary layers shown in Figure 5.6. The conditions in the boundary layers are [ ] u1 (x) u(x) = 0 [ ] 0 u(x) = u 2 (x) [ ] F b(x) = 0 for x R D1 for x R D2 for x R N (5.3) where u 1 (x) and u 2 (x) are unknown function. Thus, in R D1 the plate is fixed only in the x direction and in R D1 the plate is fixed in the y direction. Such behavior along these boundaries would be expected from the full plate due to symmetry. Figure 5.6: Boundary layers are added to three sides of the plate. Displacement conditions are applied in R D1 and R D2 and a tension force is applied in R N. The von Mises stress maps found using elasticity and peridynamics models are shown in Figure 5.7. In both cases, the highest stress appears in the just above the circular hole

36 Andrew T. Glaws Chapter 5. Numerical Results 30 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 ratio of 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 plate solved using the elasticity model. With a sufficiently large plate, the finite element elasticity solution approximates this ratio. Additionally, with a sufficiently small horizon radius, the peridynamic solution results in a stress ratio of approximately three. This further suggests that peridynamics is able to mimic the classical solution to problems in practice, provided that 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 the previous 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 Peridynamics Peridynamics Peridynamics

37 Andrew T. Glaws Chapter 5. Numerical Results 31 from the peridynamic formulation. Figure 5.8 shows the distribution of von Mises stresses found. This figure shows the high concentrations of stress at the ends of the crack, which one 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 and lower ends of the plate, shown in Figure 5.9. In these regions, the body force is [ ] 0 b(x) = F [ ] 0 b(x) = F for x R N1 for x R N2 (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. The equation contains the coefficient c( ξ ) which, as mentioned earlier, can take a wide range

38 Andrew T. Glaws Chapter 5. Numerical Results 32 of forms. In Figure 5.10, the cracked plate problem is solved using the peridynamic method for several common coefficients models: 1. Constant coefficient 2. Cone-shaped coefficient c( ξ ) = 8E πδ 3 c( ξ ) = 8E ( 1 ξ ) πδ 3 δ 3. Inverted cone-shaped coefficient c( ξ ) = 8E πδ 3 ( ξ All three coefficients proved relatively reliable as each manages to show the high concentrations 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 appears to perform the best. Conversely, the coefficient that performed the worst was the inverted cone. This agrees with the intuitive idea that in elastic media, the particles closer to the particle of interest should have the largest influence. δ ).

39 Andrew T. Glaws Chapter 5. Numerical Results 33 Figure 5.10: The stress distribution and errors for the cracked plate found using peridynamics. Three popular forms of the coefficient c( ξ ) are used.

40 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. Furthermore, as expected refining the mesh density improves the ability of the peridynamic to converge to the known elasticity solution. However, large increases in computational costs occur as the number of elements within each quadrature point s horizon increases leading to higher integration costs as well as higher nonlinear solver costs due to an increase in the overall size of the system. This spike in costs can be curbed somewhat through a precalculation of the horizon for each quadrature point as this is based on the unstressed reference configuration. Having justified the ability of the peridynamic solution to accurately mimic solutions for which classical elasticity is know to behave well, a logical next step would be to apply this technique to areas where elasticity has more difficulties, such as problems of crack propogation. Time dependence and fracture dynamics would have to integrated into the current finite element model to achieve this. Furthermore, the existing constitutive models for 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 this case, the technique of building subelements to handle the intersections of horizons with mesh elements would need to be improved upon. One suggestion is to incorporate isoparametric elements where needed. 34

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