dependency, unphysical singularities, and selection of a suitable failure criterion. Moreover, the inherent characteristic of peridynamics permits crack propagation in a natural way. Hence, Turner (2013) extended the original form of peridynamics by including the fluid pore pressure. However, the pore pressure was calculated by using a local approach a priori. On the other hand, this study presents a fully coupled bond based peridynamics approach to simultaneously simulate both deformation and porous flow fields. The current peridynamic formulation has a potential to be used for the analysis of more sophisticated poroelastic problems including fluid-filled rock fractures as in hydraulic fracturing and presented in Zhou and Burbey (2014), Li. et. al. (2014) and Helmons et al. (2016). Moreover, the formulation can be extended to be applicable for more complex material behavior by following a similar procedure explained in Oterkus and Madenci (2014).
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In the first case, a spherical energy storage particle with a pre-existing penny-shaped crack was considered. The materials of this particle, including pure silicon and lithiated silicon, were regarded as brittle materials. In addition, the concentration values were normalized by the maximum concentration as shown in Table 1. Before the charging started, the anode material remained as pure silicon. During the charging process, lithium ions at their maximum concentration were applied to the entire outer surface. Since the particle structure was free from any displacement constraints, it experienced free expansion during the charging process. For the three-dimensional bond-based peridynamic theory, the Poisson’s ratio is limited to 1/4. However, the Poisson’s ratio values of pure silicon and lithiated silicon, given in Table 1, are very close to the constrained value. Therefore, the limitation of bond-based peridynamic theory regarding Poisson’s ratio would not be effective in this case. Other Poisson’s ratio values can be specified using the ordinary-state based peridynamic formulation. Since the geometry of the spherical energy storage particle was symmetric and the penny-shaped crack was horizontally oriented, the sample planes were selected along the longitude and latitude of the sphere, shown in red and black, respectively, in Figure 3 to show the lithium ion concentration and mechanical deformation clearly inside the particle.
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Surface oxidation degrades the durability of polymer marix composites operating at high temperatures due to the presence of strong coupling between the thermal oxidation and structural damage evolution. The mechanism of oxidation in polymer matrix composites leads to shrinkage and damage growth. The thermo-oxidative behavior of composites introduces changes in diffusion behavior and mechanical response of the material. This study presents the derivation of peridynamic formulation for the thermo-oxidative behavior of the polymer matrix composites. As a demonstration purposes, isothermal aging of a unidirectional composite lamina is presented by using peridynamics. Oxidation contributed to the damage growth and its propagation.
As an alternative approach, peridynamics, can be utilized. Peridynamics [4-7] is a new continuum mechanics formulation originally introduced for problems including discontinuities. Peridynamic formulation can also be extended to other fields including temperature [8,9], moisture [10,11], etc. As the field variable, wetness field can also be chosen. Hence, this study presents a new approach for solution of the governing wetness equation with time dependent saturated concentration. The resulting equations can be solved by using the concept of peridynamics. It is computationally efficient as well as easy to implement without any iterations in each time step. Furthermore, the implementation is achieved by using the traditional elements and solvers available in ANSYS. Numerical results illustrate the accuracy and robustness of this approach for absorption and desorption with multi-material systems representative of electronic packages.
In this paper, a novel peridynamic formulation for cubic crystals has been introduced and all the relevant derivations have been provided. Static analyses have been carried out by considering different grain orientations, different loading conditions and different configurations (plane stress and plane strain). In all cases, a good quantitative agreement has been found between PD and FEM results. Dynamic analyses have also been carried out for different specimen configurations and loading conditions with the aim to investigate the effect of grain size, grain orientation, grain boundary strength, plane stress/strain configuration and fracture toughness on crack speed, time-to-failure, fracture behaviour and fracture morphology. Complex fracture phenomena such as crack nucleation and crack branching have been modelled without using any external fracture criterion and qualitative comparison with other numerical results has been provided. The findings of this study can be summarized as follow:
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This study presents a new state-based peridynamic formulation for Kirchhoff plate theory. The constitutive equation is obtained by utilizing the strain energy density of a material point in the form of curvatures and twist. Taylor series expansion up to the order of two is utilized to obtain PD form of strain energy density function. Due to the nonlocal characteristic of peridynamic theory, the boundary condition implementation needs extra care. Thus, implementation of two different type boundary conditions, clamped and simply supported, in the current formulation is presented. Two different numerical cases incorporating such boundary conditions are considered and very good agreement is observed between peridynamic and finite element analysis results.
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As opposed to partial differential equations that traditional approaches are based on, peridynamics utilizes integro-differential equations without containing any spatial derivatives. Hence, these equations are always applicable regardless of discontinuities such as cracks. Peridynamics has been used for the fracture analysis of many different types of materials and material behaviours [12–19]. It has also been applied for the analysis of polycrystalline materials [20–22]. However, these studies used either original bond-based formulation (BB)  or non-ordinary state-based (NOSB)  formulations. Bond-based formulation has limitations on material constants whereas non-ordinary state-based formulations may encounter the zero-energy mode problem. In order to overcome all these issues, an ordinary state-based (OSB) peridynamic formulation [23,24] can be utilized. The numerical solution of peridynamics is generally obtained by using a meshless scheme. Therefore, the formulation does not
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where is the mass density of the medium, is the stress tensor, is a vector of body force density, is the acceleration vector field of the material point within the medium at time . The key challenge in using Eq 1 to model discontinuous system behaviour is the presence of the divergence operator, which implies the existence of the spatial derivative of the stress field and consequently the displacement field within the domain of interest. However, since these field variables are not continuous over features such as crack tip and crack surfaces, the derivatives in such instance are undefined. In the peridynamic formulation, the equation of motion was casted such that the integral operators replaced the derivatives on the right hand side of Eq 1. A “bond-based” peridynamic equation of motion was originally proposed by Silling  as
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The governing equations of classical continuum mechanics (CCM) are based on partial differential equations (PDEs) and its mathematical formulation breaks down in the presence of discontinuities such as cracks. This limitation is partially overcome with the adoption of external crack growth criteria based on fracture mechanics. However, this approach presents its own limitations. In light of the limiting assumptions and difficulties of the current approaches, a new mathematical formulation of continuum mechanics was developed by Silling (2000), which is called “peridynamics”.
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The aim of this study is to investigate the effects of horizon selection on the elastic behaviour of plate type structures in the micropolar peridynamic theory. Plates with various lengths and widths have been investi- gated using micropolar peridynamic model for different horizon selections. The mathematical model of plates has been provided applying the micropolar peridynamic theory and solution of this model has been obtained by finite element methods. The displacement fields have been computed for the different horizons and dimension ratios of plates. To compute the displacement field a program code has been developed by using the software package MATHEMATICA. The results obtained have been compared with the analytical solution of the classical elasticity theory and with the solution of displacement based finite element methods. For displacement based finite element method solution the software package ANSYS has been used. Ac- cording to results it has been observed that the displacement fields of the plates are strongly affected by ho- rizon selection. Therefore a question raises that which horizon length should be used with the problem in hand or is there any method to find the appropriate/best horizon length.
This study presents a wetness approach to predict moisture concentration in electronic packages by using peridynamics. It enables the imposition of interface continuity conditions in a natural way because the peridynamic form of the moisture diffusion equation does not contain any spatial derivatives. Also, it provides correct results without a need of any iteration even in the presence of time dependent saturated concentrations. The capability of the current approach is demonstrated by considering simple benchmark problems, and a three-dimensional electronic package configuration with many different material layers.
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behaviours observed in experimental tests. Among these approaches, the extended finite element method (XFEM) is probably the most popular one (Gravouil et al., 2002) (Moës et al., 2002) (Moës et al. 1999). However, due to factors such as the complexity of the method, the difficulty in handling highly distorted meshes (e.g. problems with large deformations and fragmentation) and innacurate prediction of stress intensity factors, meshless methods are sometimes preferred with respect to XFEM, despite the generally higher computational cost (Rabczuk et al., 2010). In this regard, cracking particles method (CPM) that does not require an explicit crack representation has been successfully applied to solve static, dynamic, impact and explosive problems both in 2-D and 3-D domains (Rabczuk and Belytschko, 2004) (Rabczuk and Belytschko, 2007). Another meshless method worth of note is the extended element free Galerkin method (XEFGM) which can achieve satisfactory results with a significant reduction in computational time with respect to the CPM (Rabczuk and Zi, 2007). Finally, CCM does not have a length scale parameter and, thus, it is not readily usable for multi-scale analyses. Fracture mechanics crack growth criteria are not always available, especially in the case of new and complex engineering materials. Complex behaviours, such as crack nucleation, crack branching, coalescence of multiple cracks and crack arrest, are unlikely to be fairly predicted due to accuracy and numerical convergence issues (Madenci and Oterkus, 2014). In this study, a new continuum mechanics formulation with a length scale parameter that is suitable for failure prediction, peridynamics (PD), is utilized for modelling stress-corrosion cracking.
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The main purpose of this study is to present finite element implementation of peridynamic Timoshenko and Mindlin plate formulations. The advantage of this approach is that only one single row of material points along the thickness is required, which not only decreases the memory consumption by reducing the number of the nodes and elements, but also brings efficiency on processing speed. The feasibility and accuracy of the current approach is verified by considering various benchmark problems and comparing peridynamic results against classical finite element solutions in bending and buckling cases. A good agreement is obtained between peridynamic and finite element analyses results. Moreover, crack growth in a plate subjected to bending loading case is studied to demonstrate the failure prediction capability of the current approach. As a future study, impact analysis will be considered to extend the usage of the current approach. Developed framework can also be used in other applications such as bone mechanics . Moreover, utilizing variational approach as presented in dell’Isola and Placidi  and Placidi et al. , the current formulation will be extended to represent the boundary conditions without utilizing fictitious regions.
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In this work, we consider finite linear elastic peridynamic bars being pulled at the ends and having different micromoduli that were either introduced or adapted from expressions proposed by the aforementioned authors. Differently from Chen et al. (2016), the boundary conditions are imposed on extended parts of the bar (and not as an extension of the classical solution). Depending on the micromodulus, numerical results indicate that the displacement field is discontinuous, as in the case of the constant micromodulus mentioned above, or, has unbounded derivatives at the ends, which is in contrast to the homogeneous deformation of the bar predicted by the classical linear theory. Nevertheless, in all cases, the numerical results converge to results obtained from the classical linear theory as the horizon tends to zero.
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As an alternative approach to finite element method (FEM), peridynamics (PD) can be utilised. Peridynamic theory is a new continuum mechanics formulation introduced by Silling  to overcome the problems that Classical Continuum Mechanics encountered especially for predicting crack initiation and propagation. Peridynamic theory is based on integro-diffential equations and these equations do not contain any spatial derivatives. Since its introduction, there has been a rapid progress on peridynamics. Several novel approaches have been proposed for efficient numerical solution of peridynamic equations such as dual-horizon concept  and adaptive refinement . The technique has been applied to many different material systems [22, 23] and extended for the analysis of multifield problems [24,25]. An extensive review on peridynamics can be found in Madenci and Oterkus .
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Peridinamik (Peridynamic) teorisi ilk olarak Amerika’nın Sandia Ulusal Laboratuvarı’ nda araştırmacı Dr. S. A. Silling tarafından 2000 yılında ortaya atılmış oldukça yeni ve gelecek vaat eden bir metottur . Peridinamik teorisinde, A. L. Cauchy tarafından yaklaşık 200 sene önce ortaya atılan klasik sürekli ortamlar mekaniğinin hareket denklemleri tekrar formüle edilmiş ve denklemlerin yapısında bulunan konuma bağlı türevler kaldırılarak sadece hacimsel integraller kullanılmıştır (Bkz. Şekil 1).
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This study presents the ordinary state-based peridynamic (PD) constitutive relations for viscoelastic deformation under mechanical and thermal loads. The behavior of the viscous material is modeled in terms of Prony series. The constitutive constants are the same as those of the classical history- integral model, and they are also readily available from relaxation tests. The state variables are conjugate to the PD elastic stretch measures; hence, they are consistent with the kinematic assumptions of the elastic deformation. The PD viscoelastic deformation analysis successfully captures the relaxation behavior of the material. The numerical results concern first the verification problems, and subsequently, a double-lap joint with a viscoelastic adhesive where failure nucleates and grows.
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The aim of this paper is to develop new computational tools to study fatigue crack propagation in structural materials. In particular we compare the performance of different degradation strategies to study fatigue crack propagation phenomena adopting peridynamic based computational methods. Three fatigue degradation laws are proposed. Two of them are original. Initially a cylinder model is used to compare the computational performance of the three fatigue laws and to study their robustness with respect to variations of discretization parameters. Then the fatigue degradation strategies are implemented in a peridynamic framework for fatigue crack propagation analyses. Both cylinder model and peridynamic simulations show that the third proposed degradation law is unique in its combination of high accuracy, high stability and low computational cost.
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Abstract: This study develops an ordinary state-based peridynamic coarse-graining (OSPD-CG) model for the investigation of fracture in single layer graphene sheets (SLGS), in which the peridynamic (PD) parameters are derived through combining the PD model and molecular dynamics (MD) simulations from the fully atomistic system via energy conservation. The fracture failure of pre-cracked SLGS under uniaxial tension is studied using the proposed PD model. And the PD simulation results agree well with those from MD simulations, including the stress-strain relations, the crack propagation patterns and the average crack propagation velocities. The interaction effect between cracks located at the center and the edge on the crack propagation of the pre-cracked SLGS is discussed in detail. This work shows that the proposed PD model is much more efficient than the MD simulations and, thus, indicates that the PD-based method is applicable to study larger nanoscale systems.
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As mentioned in Section 2, in an attempt to solve the issues due to discontinuities, peridynamics was introduced by Silling in 2000 . Peridynamics is originally a continuum mechanics formulation utilizing the same continuity approximation as some other continuum mechanics formulations including Cauchy’s classical continuum mechanics (CCM) formulation. Therefore, the analysis domain is assumed to be composed of infinitely small volumes called material points. In peridynamics, all material points can interact with each other within a domain of influence named Horizon, H x , as opposed to CCM in which the range of interactions is limited to the
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