Peridynamics

Abstract

In this chapter, we present an integral representation of continuum mechanics which serves as the underlying theory for the peridynamics model. Practical implementation of this model into simulations illustrate the power of this approach. In particular, mechanical deformation from impact and external strain display that the peridynamic model captures failure processes that are difficult, if not impossible, to obtain from other continuum models.

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5.1 Introduction

The applied materials science is closely linked with engineering design through the common interest in properties of materials. The mechanical engineers are concerned with functionality and reliability of engineered products while materials scientists explore the link between the macro-scale properties and atomistic/molecular level structure of the constituent materials. Considerations surrounding damage and failure of materials are important parts of the design process. While an engineered part must be practical and functional, it must also serve reliably for a predictable amount of time. Analyses that simulate the response under real physical conditions are commonly utilized during the design of engineering systems, which include, among others, stress analysis with the purpose of identifying concentrations of stress. The interest in finding locations of high stress concentrations is almost exclusively due to concerns of fracture and failure, which is directly related to reliability of an engineering system.

Traditionally, the prediction of deformations and stresses have been performed using the finite element method (FEM), which is a well-established and robust method. While providing accurate predictions for stress components, traditional FEM’s capability of capturing realistic material response in modeling and predicting failure is questionable. This is due to the underlying assumption that the material remains continuous during deformation; there is no inherent mathematical formulation accounting for possibility of discontinuities arising, which is the case when a crack initiates. One possibility is to include the crack geometry in the analysis. However, since FEM is formulated based on partial differential equations, computations at/near the geometric discontinuities defined by crack faces and crack tips break down due to spatial derivatives being undefined. While a number of remedies are proposed to address this shortcoming, they involve cumbersome special techniques that treat fracture as a special case rather than including it as an inherent material behavior.

In addition to the difficulties of handling discontinuities mathematically, one must be aware that fracture is almost always a dynamic phenomenon which involves propagation of cracks. In engineered systems with complex three-dimensional geometries, cracks rarely propagate along a flat plane; they often turn, branch, oscillate and stop. A predictive computational approach must be able to address this variable nature of fracture in three-dimensions and in a dynamic sense. This task has proven to be most challenging in traditional FEM approaches. In recent past, a number of enhancements have been proposed; two such methods that gained general acceptance in the fracture prediction community are (i) XFEM [1] and (ii) cohesive zone methods [2]. Both of these methods made significant improvements to FEM’s handling of problems involving fracture , they require externally supplied kinetic relations for crack growth without providing a damage model as part of the constitutive relations.

Silling [3, 4] introduced a new theory, peridynamic (PD) theory , that represents the continuum mechanics in integral form, which removes the mathematical artifacts present in classical formulations involving spatial derivatives. In the PD theory, internal forces are expressed through nonlocal interactions between the material points within a continuous body, and damage is part of the constitutive model. Additionally, interfaces between dissimilar materials have their own properties and damage can propagate when and where it is energetically favorable for it to do so. The integral equation representation allows crack initiation and growth simultaneously at multiple sites with arbitrary paths.

In this chapter, first, peridynamic theory formulation and its numerical implementation are presented briefly. A number of applications are considered in order to demonstrate the capabilities of the theory.

5.2 Integral Representation of Continuum Mechanics

In PD theory, a point in a solid body interacts with all points within its range as illustrated in Fig. 5.1. Even though the discretization considers material points (as opposed to elements), the body is assumed continuous. The PD theory reformulates the deformation of a continuous body in terms of integral equations without the spatial derivative terms of the classical continuum theory. This feature allows PD theory formulation to apply everywhere in the solid body even when geometric and material discontinuities are present.

Fig. 5.1

Kinematic description of a pair of PD material points

In peridynamic theory , motion of a material point x is evaluated through analysis of its interactions with other material points ${\mathbf{x^{\prime}}}$ in the body. There exists a distance beyond which the influence of ${\mathbf{x^{\prime}}}$ on motion of x become negligible. This distance is termed horizon and the region defined by the horizon is called the neighborhood of x. For example, the circular region in the initial configuration shown in Fig. 5.1 is the neighborhood of x. The equations describing the motion of x at the deformed configuration at time t is written as

$$ \rho \frac{{\partial^{2} {\mathbf{u}}}}{{\partial t^{2} }} = \int\limits_{H} {\left\{ {{\mathbf{T}}\left( {{\mathbf{x}},t;{\mathbf{x}} - {\mathbf{x^{\prime}}}} \right) - {\mathbf{T}}\left( {{\mathbf{x^{\prime}}},t;{\mathbf{x^{\prime}}} - {\mathbf{x}}} \right)} \right\}dH + {\mathbf{b}}({\mathbf{x}},t)} $$

(5.1)

in which ρ is the mass density, u is the displacement vector of material point x, and ${\mathbf{b}}({\mathbf{x}},t)$ is the body load vector. The integral is the vector summation of internal forces between the material points x and ${\mathbf{x^{\prime}}}$, and is taken over the volume surrounding the material point, H, which defines the neighborhood of x. The initial distance between x and ${\mathbf{x^{\prime}}}$ is denoted by the vector ξ shown in Fig. 5.1, initial configuration. At time t, the body is deformed with x and ${\mathbf{x^{\prime}}}$ moving to new positions with corresponding displacement vectors u and ${\mathbf{u^{\prime}}}$, respectively. The initially spherical (circular in 2-D) neighborhood of x becomes, in general, an irregularly shaped domain. The force vectors between x and ${\mathbf{x^{\prime}}}$ are described formally by force state vectors ${\mathbf{T}}({\mathbf{x}},t;{\varvec{\upxi}})$ and ${\mathbf{T}}({\mathbf{x^{\prime}}},t;{\varvec{\upxi}})$, former belonging to x and latter to ${\mathbf{x^{\prime}}}$, then the corresponding deformed configuration is described by the deformation state vector Y. It should be noted that the force state vectors are functions of time t, initial positions x and ${\mathbf{x^{\prime}}}$, and the relative position ξ, rendering the PD formulation Lagrangian . Detailed derivations related to force and deformation state vectors can be found in Silling et al. [1] and Madenci and Oterkus [2].

In this chapter, the force state vectors, ${\mathbf{T}}({\mathbf{x}},t;{\varvec{\upxi}})$ and ${\mathbf{T}}({\mathbf{x^{\prime}}},t;{\varvec{\upxi}})$, describing the interaction between the material points x and ${\mathbf{x^{\prime}}}$ will be assumed to have same magnitude and opposite directions. This assumption leads to the so-called “bond-based” peridynamic formulation with the following PD equation of motion

$$ \rho \frac{{\partial^{2} {\mathbf{u}}}}{{\partial t^{2} }} = \int\limits_{H} {{\mathbf{f}}({\mathbf{x}},{\mathbf{x^{\prime}}},t;{\varvec{\upxi}}) dH + {\mathbf{b}}({\mathbf{x}},t)} $$

(5.2)

where the interaction between x and ${\mathbf{x^{\prime}}}$ is described by the response function f. It is explicitly written for an isotropic material as

$$ {\mathbf{f}}({\varvec{\upeta}},{\varvec{\upxi}}) = \frac{{{\varvec{\upxi}} + {\varvec{\upeta}}}}{{\left| {{\varvec{\upxi}} + {\varvec{\upeta}}} \right|}}\mu cs $$

(5.3)

in which ξ is the relative position $({\varvec{\upxi}} = {\mathbf{x}} - {\mathbf{x^{\prime}}})$ and η is the relative displacement $({\varvec{\upeta}} = {\mathbf{u^{\prime}}} - {\mathbf{u}})$. Based on the relative position and displacements, the stretch between x and ${\mathbf{x^{\prime}}}$ can be defined as

$$ s = \frac{{\left| {{\varvec{\upxi}} + {\varvec{\upeta}}} \right| - \left| {\varvec{\upxi}} \right|}}{{\left| {\varvec{\upxi}} \right|}} $$

(5.4)

The stretch in Peridynamic theory is analogous to the strain in classical continuum theory. The relationship between the force and stretch, (5.3), involves a material parameter, c, which is commonly referred to as the bond constant. Its value is evaluated by calculating the strain energy of a PD domain and equating it to the corresponding strain energy that is found under classical continuum mechanics formulation; Silling and Askari [3] used isotropic extension as the loading for this operation to find c as

$$ c = \frac{18\kappa }{{(\pi \delta^{4} )}} $$

(5.5)

Finally, the binary function μ is utilized for introducing failure between two material points as follows

$$ \mu (t,{\varvec{\upxi}}) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{if}}\;s(t^{\prime},{\varvec{\upxi}}) \,<\, s_{0} \;{\text{for}}\;{\text{all}}\;0\, \le\, t^{\prime} \,\le\, t} \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. $$

(5.6)

where $s_{0}$ is the critical stretch. When the stretch between a material point pair exceeds the critical stretch, the interaction between the material points is permanently terminated. During the simulation, each interaction is checked against this criterion $(s < s_{0} )$ at each time step and number of broken interactions emanating from each material point is recorded. Thus, damage at a material point is defined as the ratio of broken interactions to the original number of interactions; damage is a value between zero (no damage) and one (completely detached from other material points). Therefore, damage at a material point along a well-defined crack face in a brittle fracture problem is expected to be around 0.5.

5.3 Practical Implementation

Solution of peridynamic equations of motion (5.1) or (5.2) is achieved through numerical means. This involves discretization of the solution domain into small and simple sub-volumes. These sub-volumes should be simple 3-D shapes as they are used in numerical volume integration of PD equations; the simplest sub-volume that enables the easiest numerical handling of PD analysis is a cubic sub-volume. In this case, a peridynamic material point is placed at the center of the cubic sub-volume. After discretization, the volume integral of (5.2) can be written for material point i in terms of summations as

$$ \rho {\ddot{\mathbf{u}}}_{i}^{n} = \mathop \sum \limits_{j} {\mathbf{f}}\left( {{\mathbf{u}}_{i}^{n} ,{\mathbf{u}}_{j}^{n} ,{\mathbf{x}}_{i}^{n} ,{\mathbf{x}}_{j}^{n} } \right)V_{j} + {\mathbf{b}}\left({{\mathbf{x}}_{i}^{n}} \right) $$

(5.7)

in which $V_{j}$ represents the volume of the sub-volume represented by material point j, and the superscript n denotes the time step. This equation is written at each material point i to calculate the acceleration vector at point i by equating its product with the mass density to the forces acting on point i within its horizon.

The numerical implementation of the PD theory involves a number of practical issues surrounding the volume integration, time marching and convergence. Each of these issues have been addressed in varying levels of detail in literature; a comprehensive discussion of these topics and further considerations can be found in [2]. A brief summary is provided herein.

The summation of (5.7) approximately represents the volume integral of (5.2) where the volume is the neighborhood of point i defined by a sphere with a certain radius (horizon). In the summation representation, each sub-volume is lumped in a material point located at the mass center of the sub-volume. Thus, if the material point j who belongs to the neighborhood of material point i is within this radius, its whole volume is included in the summation. As it can be foreseen, this situation may pose inaccuracies for material points j that are near the boundary of the sphere where part of the volume may fall outside of the sphere. A similar situation arises when the material point j narrowly falls outside the horizon while a non-trivial portion of its volume is still inside the horizon. These two scenarios should be addressed in order to increase the accuracy of the computations. The remedy may involve a simple algorithm that estimates the amount of volume that falls within the horizon and adjust the contribution accordingly; this is implemented by [4].

Another common issue related to numerical implementation is the effect of free surfaces that exist within the horizon of material the material point i. Evaluation of the material parameters of PD model (5.5) involves comparison of energy quantities between classical continuum mechanics and PD theory; this is performed based on the assumption that the sphere defining the neighborhood of a material point is completely populated (whole sphere). However, as with every finite geometry, existence of physical boundaries will contradict this assumption and a correction must be made. Silling [4] describes this operation as the surface correction. The computed material parameters of PD must be corrected according to the local geometry of the material point i. Detailed discussion on surface corrections is given in [2].

The peridynamic equation of motion, while not having spatial derivatives, contains time derivative term (acceleration of material point i). In the numerical implementation of PD, a time marching scheme must be employed. The PD simulation starts with complete list of known positions, velocities and accelerations (initial conditions). The acceleration term on the left hand side of (5.7) is expressed in terms of a combination of displacements and velocities belonging to current (known) and next (unknown) time steps such that a complete list of displacements, velocities and accelerations in the next time step is calculated. This is a rather routine operation common to many transient problems of continuum mechanics; any traditional method can be employed (e.g., finite difference, Runge-Kutta, Adams-Bashforth). The well-known error considerations apply; a stability criterion for each of these methods can be constructed to find a suitable time step size.

The peridynamic theory works with the equations of motion, and thus a dynamic formulation by nature. However, in many problems of mechanics, a static (or steady-state) solution is sought after. Kilic [5] addressed this issue by implementing an adaptive dynamic relaxation method based on the work of Underwood [6]. The method is based on introduction of damping to the equations of motion, towards finding the equilibrium state much quicker than solving the problem dynamically. The adaptive dynamic relaxation method used by Kilic [5] leads to calculation of damping coefficients for each material point at each time step.

5.4 Applications

The peridynamic theory has been used for simulation of numerous problems of applied mechanics. A few applications are presented in this section: First, fracture performance of an advanced ceramic layer under impact by a sand particle is discussed. Next, investigation of fracture patterns in glass is presented followed by anodized aluminum thin film cracking. Finally, preliminary work on three-dimensional fracture observed in polycrystalline materials is shown.

5.4.1 Damage in a Ceramic Layer Due to Small Particle Impact

Materials that transmit Long Wave InfraRed (LWIR) are inherently softer and weaker than their Mid-Wave Infrared (MWIR) counterparts because the price paid for extending transparency to longer wavelengths is a reduction in lattice bond strength. Zinc selenide has excellent transmittance in the LWIR but is too soft for use in mechanically harsh environments, as are the many chalcogenide-based LWIR transmitting glasses. The semiconducting materials, Ge and Si, are mechanically strong and hard but do not transmit into the visible. Diamond has excellent optical and mechanical properties but the cost for diamond optics will likely remain prohibitive for many years. Consequently, most LWIR systems exposed to mechanical stresses use Zinc Sulfide (ZnS) for windows and domes, as it represents the best compromise between strength, hardness , and transparency. Over the last three decades much effort has gone into developing coating materials to improve the modest erosion resistance of ZnS to both rain and sand impacts damage, including sputtered ZnS with an over-layer of Y2O3 (Raytheon’s DAR-REP system), amorphous boron phosphide based glasses, diamond-like carbon, and germanium (see [7] for thorough review through 1999). Much of this effort is empirical in nature due to the lack of predictive tools for characterizing impact damage.

To date, most theoretical models of particle impact damage in brittle materials seek to define the stress fields caused by the elastic/plastic nature of an impact event so that crack propagation can be predicted (see [8–12]). Unfortunately, the inherent continuum nature of these models prevents them from properly accounting for crack initiation/introduction. In addition, most of the models focus on characterizing hardness indentation damage as a surrogate for particle impact damage, addressing impact damage largely by generalization, in a semi-quantitative manner. Also, most of the current models are based on an idealized geometry of a single impact event that has difficulty accommodating complexities such as irregularly shaped, frangible, tumbling particles, non-normal incidence, multiple impacts, and the use of protective coatings with variable surface adhesion and internal stresses.

In this subsection, we present an application of the peridynamic theory for accurate quantification of damage/failure in a ceramic layer. A new methodology for extraction of critical stretch parameter and yield strength based on indentation experiments is also given. These material parameters are subsequently used in simulation of sand impact of ZnS.

Vickers indentation testing produces an impression that is square in cross-section with cracks visible on the surface emanating from the corners (see Fig. 5.2). The lengths of these cracks, c, are measured from the center of the indent out to the crack tip. However, it is common to report crack lengths in terms of 2c, the distance from one crack tip through the impression to the tip of the crack emanating from the opposite corner. The measurements in this study were made on multispectral grade ZnS using a Leco model M-400-H1 Micro Hardness Tester. Dead weight loads of 100, 200, 300 and 500 g (equivalent to about 1, 2, 3 and 5 N) were applied using a dashpot-controlled descent taking about 5 s. Dwell time was 10 s. The resulting indents were characterized in the SEM and the values of 2c are reported in Table 5.1.

Fig. 5.2

Schematic of Vickers indentation and definition of 2c

Table 5.1 Hardness test data for multispectral grade ZnS

Peridynamic simulations of Vickers indentation tests were performed next. The PD model is comprised of two material regions corresponding to the indenter and the ceramic substrate. The indenter is modeled as the pyramid shape of a standard Vickers hardness apparatus while for the substrate a cylindrical shape having a 120 μm height and radius is considered. Total number of material points in the PD model including both the indenter and the substrate is 148,550. Figure 5.3 shows the PD models of the indenter and ceramic substrate prior to indentation. The bottom surface of the substrate is constrained from displacement in all directions. The magnitude of the indenter velocity is held constant during the entire simulation. During the “loading” phase, it is directed downward, into the substrate, and during the “unloading” phase it is directed upward or away from the substrate. The constant velocity condition imposed on the indenter implies that no damage occurs in the indenter as it moves down (or up) as a rigid body. Specified velocity conditions combined with specified loading/unloading times dictate the indentation depth for each simulation; this is in contrast to test conditions where maximum force is controlled. A preliminary study was conducted to identify an approximate relationship between the indentation depth versus the maximum force. This relationship is in turn utilized to decide on four distinct cases to simulate with specified indentation depths. The damage developed in the ceramic substrate is analyzed and the crack length 2c is measured for each simulated case. An example of the damage pattern visible on the substrate surface is shown in Fig. 5.4. The shape of the high damage region matches the surface cracks observed in the experiments. However, PD simulations do not produce clearly defined crack faces, rather concentration of high damage at sites cracks are expected to form. This is likely due to the PD grid density and that a more refined grid is expected to lead to a better defined fracture ; in this study, the damage pattern is used as approximation to fracture. The total force in the vertical direction, $F_{3}$ is also calculated in PD simulations. A typical indentation force versus indentation depth response calculated by the PD simulations is shown in Fig. 5.5. Several simulations were performed with a range of values for $f_{y}$ and $s_{0}$ in search of the values which produce the best match of $F_{3}$ and 2c to the data in Table 5.1. In this study, for ZnS, the optimum material parameters are found to be $\sigma_{y} = 525$ MPa and $s_{0} = 0.0007$. Figure 5.6 shows the good agreement between the PD simulated force versus crack length response of the four cases when the optimal parameters are used and the experimental measurements.

Fig. 5.3

The PD model of the indenter and the ceramic substrate used in Vickers indentation simulations

Fig. 5.4

Damage pattern consistent with hardness test fracture visible on the surface of the substrate

Fig. 5.5

A typical force versus indentation response simulated by PD

Fig. 5.6

PD simulated force versus crack length response compared against experimental measurements

The dynamic fracture of a zinc sulfide layer due to a rigid impactor is considered in order to demonstrate the effectiveness of the use of peridynamic theory for high-fidelity simulation of impact events experienced by electromagnetic windows, The experimental guidelines listed in Harris [7] for testing IR windows and domes under sand impact are taken as the starting point for peridynamic simulation model parameters. The sand particle is represented by a rigid impactor having a cylindrical geometry with 160 μm diameter and 60 μm height; the mass of the impactor is calculated based on a density of 3300 kg/m³. The impactor is given an impact velocity of 75 m/s. The ZnS layer is a cylinder with 3 mm diameter and 1.5 mm height. The peridynamic model parameters governing ZnS behavior (deformation and failure ) are calculated based on the following properties: density of 4070 kg/m³, bulk modulus of 87.38 GPa and critical stretch value of 0.0007 as extracted from hardness tests. The model uses a grid spacing of 8 μm, and the time-step size is 0.68 ns.

Figure 5.7 shows the evolution of damage due to sand impact as predicted by the peridynamic model; only half of the solution domain is shown so that the damage through the thickness can be clearly observed. In these plots, the damage varies from zero (purple, no damage) to unity (red, maximum damage). The impactor is not show in order to provide a better view of the surfaces. The damage progresses straight down early in the process (Fig. 5.7a, b) with increasing impact load. As unloading from the impact begins, lateral cracks below the surface and shallow radial cracks start to emanate from the center radially (Fig. 5.7c) curving upwards towards the free surface (Fig. 5.7d). Simultaneous with lateral and radial crack initiation, completely damaged material directly below the impactor rebounds upward and separates from the ZnS layer as it forms a conical impact impression, clearly visible in Fig. 5.7f. The crack formation visible on the top surface of the substrate exhibits a combination of short and long radial cracks. The overall fracture pattern from the peridynamic simulation matches well the morphology observed during experimental testing of sand impact damage in ZnS. Such damage is shown in Fig. 5.8a, where the lateral and radial cracks caused by impact of a 149–177 μm sand particle traveling at 75 m/s are clearly evident. For comparison, a top view of the PD simulated crack pattern is shown in Fig. 5.8b, which exhibits striking similarity.

Fig. 5.7

Damage evolution in ZnS substrate at times, a 27 ns, b 47 ns, c 68 ns, d 99 ns, e 102 ns and f 136 ns

Fig. 5.8

a Top view of experimentally observed fracture pattern on multi-spectral ZnS caused by impact of 144–177 μm sand particle at velocity of 75 m/s, b simulated fracture pattern of ZnS under impact by 160 μm diameter sand particle at velocity of 75 m/s

As noted earlier, many researchers have recognized the strong correlation in fracture behavior between particle impact damage and indentation damage. Figure 5.9, taken from Evans [8] documents crack formation and growth in ZnS during the loading and unloading stages of an elastic/plastic spherical indentation test. While the sequence of events, as outlined in Fig. 5.9, do not exactly match those observed from the PD simulation, where radial crack formation lags that of lateral cracks, the similarity in the phenomenology of the two fracture events is striking, despite differences in load conditions. In both cases, radial cracks, lateral cracks, extensive damage below the impact site (median crack in Fig. 5.9) and impact crater formation occurs.

Fig. 5.9

Schematic showing the sequence of crack formation and growth during spherical indentation studies of ZnS (after evans [8])

In order to further highlight PD theory ’s ability to simulate 3-D, non-planar multiple fracture surfaces, the results are presented in Fig. 5.10, for material points with damage greater than 0.2; i.e., material points with less damage is removed from view. Four different views (left to right: oblique, side, top and front) at five different times (top to bottom: 47, 61, 88, 95 and 102 ns) are shown. The selection of 0.2 as the cut-off damage parameter for these figures ensures viewing of all regions with any significant damage, including localized regions that do not have fully developed cracks .

Fig. 5.10

PD simulated damage evolution from different angles with material points with damage less than 0.2 removed from view

These results clearly demonstrate the utility of peridynamic theory to model mechanical damage, especially for those situations where continuum mechanics has difficulty capturing the complexities of the evolving stress fields and crack formation. Future work to improve and expand the capabilities of this model include investigation of various coating materials with varying thicknesses, simulations involving deformable and friable impactors to better understand the physics of “soft” particle impact on “hard” substrates, and investigation of raindrop impact.

5.4.2 Fracture Patterns in Anodized Aluminum

Plasma etching is an important step of wafer manufacturing in integrated circuit industry. The process aims to precisely engineer the surface features of the wafer. It involves generation of plasma, which requires a sealed chamber. The plasma is exposed to the target (wafer) and any other material surface inside the chamber including the plasma chamber walls. Anodized aluminum is a common plasma chamber and component material in the IC industry; an oxide coating (alumina) is produced on the aluminum surface leading to strong anodic polarization, significantly reducing reactivity. A common problem is the landing of foreign particles on the wafer while it is being etched rendering part of the wafer with zero yielding die. These particles are believed to originate from the plasma chamber components surrounding the wafers. One of the models is mechanical failure in the form of fracturing of the coating and flaking off. There are two main reasons for the coating failure [13]: (i) the difference between the coefficients of thermal expansion of the coating and the underlying metal, and (ii) change of surface chemistry due to plasma environment. These two mechanisms may or may not work together. Very few studies have been conducted to address the problems associated with integrity of anodized aluminum parts used in plasma chambers; most focus on cleaning and conditioning of the surface [14, 15]. A physics based understanding of the failure mechanisms at play is essential for improving the surface integrity of anodized aluminum parts.

This study aims to investigate the underlying physics of the failure modes observed in anodized aluminum parts used in plasma chambers. Experimental investigation involves mechanical property characterization and clear identification of failure modes examination through SEM. Simulations of aluminum substrate with alumina coating under expansion loading conditions are performed. Computational investigation also explores the potential effect of an additional thin layer between the coating and the substrate.

Ground electrodes encase wafers processed in a plasma chamber. These electrodes are exposed to plasma and corrosive gases used during the etching process. The chamber temperature varies a few tens of degrees from room temperature during the etch process. Commercially available ground electrode after use was cross sectioned for this study. The images were taken under bright field conditions at 15 kV with both the interior and exterior surfaces showing similar sized crazing (Fig. 5.11). The mean crack -free area for the inside surface was 3952 μm² while that for the exterior was 3521 μm². The thicknesses of exterior and interior coatings were approximately 22 and 19 μm, respectively (Fig. 5.12). EDS mapping of the interior surface showed the same elements common in the top-down spectra but without the F signal. No morphological or compositional layering was evident.

Fig. 5.11

Fracture pattern observed along the inside surface of the plasma chamber

Fig. 5.12

Cross-sectional view of the alumina coating and aluminum substrate. Fracturing of alumina coating is observed

The fracture due to coefficient of thermal expansion mismatch between the alumina coating and the aluminum substrate is simulated using the peridynamic theory . Results for the material system with a single alumina layer are presented first, followed by the investigation of effect of adding a compliant layer between the aluminum and the alumina.

The geometry of the material system (two-layer system) is shown in Fig. 5.13. The problem domain is a rectangular prism with length l = 400 μm, width w = 400 μm, and a total height of h _t  = 220 μm. The thickness of the alumina film is taken as h _f  = 20 μm. The grid spacing in the model is taken as 3.33 μm leading to approximately 950,000 grid points.

Fig. 5.13

Geometry of the anodized aluminum material system

The material properties for the aluminum substrate are: elastic modulus of 68 GPa and density of 2,700 kg/m³. Similarly, for the alumina film, the elastic modulus is taken as 370 GPa while its density is kg/m³. The critical stretch for the thin alumina coating is calculated to be around 5 % while aluminum is about 20 % owing to its ductile nature.

In order to simulate the deformation field due to the mismatch between coefficients of thermal expansion values of the constituent materials, a large portion of the substrate in the depth direction (80 %) was subjected to isotropic expansion in the planar directions (length and width as defined earlier). The expansion is then transferred to the film through deformation. The loading is applied in a ramped fashion gradually so as to prevent premature fracture near the boundary regions.

Figure 5.14 shows the top view of the damage progression along the surface of the coating. Four different time steps are shown, with time increasing from top to bottom. Fracture starts at one edge and propagates as the expansion of the substrate continues. Due to the nature of the isotropic extension, a number of branches emerge leading to the final fracture configuration, shown in the bottom segment of Fig. 5.14. However, there is further fracture beneath the surface that is of interest. In order to examine the fracture morphology inside the coating, damage contours are plotted such that only those material points with damage values ranging between 0.2 and 1.0 are shown in Fig. 5.15 from top view. Same set of results are shown from an oblique angle in Fig. 5.16 in order to clarify the extent and geometric distribution of the damage in 3-D. Examination of Figs. 5.15 and 5.16 and their comparison to Fig. 5.14 reveal that a considerable damage is accumulated along the interface between the aluminum substrate and the alumina coating. This is consistent with the failure mode observed in experiments, where part of the crazed coating peels/flakes off, suggesting interface delamination. Peridynamic simulations appear to capture the correct failure modes observed in experimental setup.

Fig. 5.14

Damage progression along the top surface of the alumina layer predicted by peridynamic theory for the two-layer system

Fig. 5.15

Damage progression in the alumina layer through the thickness from top view predicted by peridynamic theory for the two-layer system

Fig. 5.16

Damage progression in the alumina layer through the thickness from oblique view predicted by peridynamic theory for the two-layer system

Additionally, the top surface fracture pattern predicted by peridynamic theory is compared to the micrographs of the crazed coating inside the plasma chamber in Fig. 5.17. A scale bar of 300 μm is included; it applies to both the SEM micrographs (left) and the peridynamic results (right). The size and shape of the surface fracture predicted by peridynamic theory closely resemble those observed experimentally. The “cellular” characteristic of the fracture is prominent on both sides. Peridynamic theory is able to predict the shape and size of the fractured-coating pieces with satisfactory accuracy.

Fig. 5.17

Comparison of experimentally observed fracture patterns (left) against those predicted by peridynamic theory for the two-layer system

Modification of the anodization process may allow a layered coating system with intention of designing a layer sequence to alleviate or eliminate cracking in the exposed surfaces. The anodized coatings are inherently porous; depending on the durations and concentrations of acid bath process, different porosity values are attained. Therefore, there may be configurations of layered anodized coatings that might lead to better fracture performance of the chamber interior coating. With this hypothesis, the same problem is considered with an added thin layer between the alumina coating and the aluminum substrate (leasing to a three-material system) as sketched in Fig. 5.18. This intermediate layer has the same thickness as the alumina coating but with different material properties. The relationship between the mechanical properties and porosity is well-documented for alumina [16–18]. As the porosity increases, elastic modulus decreases, which makes the material more compliant allowing the material to deform more before fracturing. By making the intermediate layer compliant, the effect of the mismatch between the substrate and the original coating will be lessened, leading to less cracks and flaking off. This added layer will serve as a buffer layer.

Fig. 5.18

Geometry of the anodized aluminum material system with the compliant intermediate layer

Therefore, in this hypothetical test case, the intermediate layer elastic modulus was decreased to half of the original value while the critical stretch was increased to about 7.5 % (compared to 5 %). The remaining parameters of the model is identical to the previous problem.

The damage progression along the top surface of the exposed coating is shown in Fig. 5.19. The amount and distribution of the damage in the three-material system are less than those of the two-layer system. Further, similar to the previous case, the damage contours for the current configuration are plotted for values between 0.2 and 1.0 in order to examine the damages in the thickness direction as shown in Fig. 5.20. In order to make the comparison easier, side-by-side comparison of damage patterns for the two-layer and three-layer systems are shown in Fig. 5.21. The two-layer system results are shown in the left column while the right column is shows the three-layer system damages. Top row in Fig. 5.21 shows only the top surface while bottom row shows damages greater than 0.2 through the thickness. It is clear that the interface delamination problem is significantly reduced; the predicted cracks have vertical faces. This prediction suggests that the peeling/flaking off phenomena could be significantly reduced or eliminated by carefully engineering an anodization process leading to a desired layered system.

Fig. 5.19

Damage progression along the top surface of the alumina layer predicted by peridynamic theory for the three-layer system

Fig. 5.20

Damage progression in the alumina layer through the thickness from top view predicted by peridynamic theory for the three-layer system

Fig. 5.21

Side-by-side comparison of damage patterns for the two-layer (left) and three-layer (right) systems. Top row shows only the top surface while bottom row shows damages greater than 0.2 through the thickness

In this study, a common problem in IC manufacturing, which involves peeling/flaking off of fractured coating of anodized aluminum plasma chamber walls, is described. A computational approach, peridynamic theory , is used to simulate the surface fracturing due to uniform expansion is demonstrated. It was shown that the peridynamic theory captures the correct failure modes observed in experiments. Also, it is able to predict the shape and size of the fractured-coating pieces with satisfactory accuracy.

In order to reduce or eliminate the peeling/flaking off problem, incorporation of an intermediate layer between the coating and substrate is considered. Peridynamic simulation of this hypothetical structure under the identical expansion conditions suggests that a compliant intermediate layer between the coating and substrate has potential to reduce the problem through eliminating interface fracture (delamination) and reducing the surface fracture.

Further study would explore the effects a range of elastic properties and critical stretch values as well as geometry parameters (e.g. thickness).

5.4.3 Dynamic Fracture of Glass

Glass has been used as engineering material over a long period of time. Due to their amorphous micro/nano structure, their behavior under mechanical loading has not been fully understood. Currently, glass materials are ubiquitous in electronic, automotive and aerospace industries; methods with high fidelity fracture predictions would pave the way to components with increased reliability and durability.

In this study, we demonstrate use of peridynamic theory to simulate mechanical response of brittle glass materials under impact loading. The material system involve polymethyl methacrylate (PMMA) plates of 18 mm radius with different thicknesses: 0.5, 1.0, and 3.0 mm. The plates are impacted by a steel ball of radius 1.8 mm and a mass of 16 g. The experimental results are taken from [19]. The impact velocity varies between 10 and 120 m/s.

Peridynamic models for the PMMA plates were generated to have approximately 2.2, 2.6 and 3.2 million material points for thicknesses 0.5, 1.0 and 3.0 mm, respectively. Elastic moduli of 2.42 and 180 GPa were used for PMMA and stainless steel, respectively. In all cases, the peridynamic grid was generated using arbitrarily oriented concentric rings layered orthogonally to the direction of impact in order to eliminate the potential bias in fracture morphology that might be introduced by using a regular rectangular grid. Figure 5.22 demonstrates such a bias introduced when using a rectangular grid (Fig. 5.22a), and a more realistic fracture morphology when circular grid is used (Fig. 5.22b) for a 1.0 mm plate impacted by the projectile with velocity of 80 m/s.

Fig. 5.22

Comparison of fracture morphologies in a 1.0 mm thick plate impacted at 80 m/s, modeled with a rectangular grid, and b circular grid

The experimental observations focused on correlating the number of cracks and impact velocity [19]. The cracks counted mostly consisted of radial cracks emanating from the impact site, but the circumferential cracks were also counted when present. For example, Fig. 5.23 shows the progression of the fracture pattern predicted by the peridynamic simulation for a 1 mm thick plate impacted at 20 m/s. Four radial cracks emanate from the impact point and continue to grow as time progresses. As the velocity of the impactor increases, the number of radial cracks that form increases. After a critical velocity is reached, circumferential cracks form allowing the impactor to break through the plate. Figure 5.24 shows formation of a circumferential crack for a 1 mm plate is impacted at 80 m/s.

Fig. 5.23

Time progression of fracture for a 1 mm thick plate impacted at 20 m/s

Fig. 5.24

Time progression of fracture for a 1 mm thick plate impacted at 80 m/s

Comparisons to the experimental work was done both qualitatively and quantitatively. Figures 5.25 and 5.26 compare the peridynamic simulations performed at 20 and 60 m/s impact velocities against the experiments for 1 mm thick PMMA plate with impact velocities measured at 22.2 and 66.2 m/s, respectively. In the case of 22.2 m/s impact velocity, in experiments four radial cracks are observed with no circumferential cracking. The peridynamic simulation with 20 m/s captures the morphology and the number of radial cracks exactly (Fig. 5.25). Similarly, the comparison of higher velocity impact response is shown in Fig. 5.26; it is worth noting that peridynamic simulation accurately captures both the radial and circumferential cracks.

Fig. 5.25

Comparison of peridynamic simulation fracture morphology against experimental observations for a plate with a thickness of 1 mm impacted at 20 m/s

Fig. 5.26

Comparison of peridynamic simulation fracture morphology against experimental observations for a plate with a thickness of 1 mm impacted at 60 m/s

The quantitative study involved 10 distinct peridynamic simulations involving the three different plate thicknesses and impact velocities ranging from 20 to 80 m/s. Results of the peridynamic simulations are compared against the experimental results in Figs. 5.27 through 5.29. In these figures, the circular symbols are the experimental observations while the square symbols are the peridynamic simulation results. In both experimental and peridynamic results presentations, solid symbols indicate that only radial cracks formed and open symbols indicate that circumferential cracks were present. The lines in the plots are least-squares linear fit of the experimental observations to guide the eye. In all three thickness cases peridynamic predictions of number of radial cracks are close to experimentally observed ones.

Fig. 5.27

Quantitative comparison of number of radial cracks observed experimentally against captured through peridynamic simulations for a plate with a thickness of 0.5 mm impacted at various velocities

The qualitative (Figs. 5.25 and 5.26) and quantitative (Figs. 5.27, 5.28 and 5.29) comparisons clearly illustrate that peridynamic theory is able to capture specific fracture morphology (radial vs. circumferential) as well as the number of radial cracks under dynamic fracture conditions due to impact in PMMA.

Fig. 5.28

Quantitative comparison of number of radial cracks observed experimentally against captured through peridynamic simulations for a plate with a thickness of 1.0 mm impacted at various velocities

Fig. 5.29

Quantitative comparison of number of radial cracks observed experimentally against captured through peridynamic simulations for a plate with a thickness of 3.0 mm impacted at various velocities