Peridynamic wetness approach for moisture concentration analysis in electronic packages

Within the finite element framework, a commonly accepted indirect approach employs the concept of normalized concentration to compute moisture concentration. It is referred to as “wetness” approach. If the saturated concentration value is not dependent on temperature or time, the wetness equation is analogous to the standard diffusion equation whose solution can be constructed by using any commercial finite element analysis software such as ANSYS. However, the time dependency of saturated concentration requires special treatment under temperature dependent environmental conditions such as reflow process. As a result, the wetness equation is not directly analogous to the standard diffusion equation. This study presents the peridynamic wetness modeling for time dependent saturated concentration for computation of moisture concentration in electronic packages. It is computationally efficient as well as easy to implement without any iterations in each time step. Numerical results concerning the one-dimensional analysis illustrate the accuracy of this approach. Moisture concentration calculation in a three-dimensional electronic package configuration with many different material layers demonstrates its robustness

Peridynamic shell membrane formulation

Peridynamics (PD) is a non-local continuum theory that enables failure prediction. It enables both crack initiation and propagation as well as crack branching. Also, it has been utilized to model simplified structures such as beams, plates and shells. In this study, a new peridynamic shell membrane formulation is presented. The equations of motion are obtained by using Euler-Lagrange equations. The bond constant is determined by comparing peridynamic and classical equations of motion for shell membranes for a special condition of peridynamic internal length parameter, horizon, approaching zero. Comparison of peridynamic results with analytical results for a benchmark problem confirms the validity of the present shell membrane formulation

Peridynamic modelling of periodic microstructured materials

With the enhancement in additive manufacturing technology, microstructured materials has attracted significant attention during the last few years. Although these materials can show homogenised properties at the macroscopic scale, their microstructural properties can be very influential on the overall material behaviour especially on the fracture strength of the material since defects such as microcracks and voids can exist. Analysing each and every detail of the microstructure can be computationally expensive. Therefore, homogenisation approaches are widely used especially for periodic microstructured materials including composites. However, some of the existing homogenisation approaches can have limitations if defects exist since displacements become discontinuous if cracks occur in the structure which requires extra attention. As an alternative approach, peridynamics can be utilised since peridynamic equations are based on integro-differential equations and do not contain any spatial derivatives. Hence, in this study peridynamic modelling of periodic microstructured will be presented and the capability of the approach will be demonstrated with several numerical examples with and without defects

Peridynamic formulation for Timoshenko beam

It is common to encounter structures with complex geometries including aerospace and ship structures. Moreover, in these structures one dimension can be either much smaller or bigger than other two dimensions. In such cases, special formulations were developed such as beam, plate and shell formulations. These formulations are available both in classical and non-classical frameworks. In this study, a new peridynamic formulation is presented for Timoshenko beams. Peridynamic equations are obtained by using Euler-Lagrange equations and Taylor’s expansion. To validate the newly developed peridynamic formulation, a Timoshenko beam subjected to central point load under simply supported, clamped and mixed (clamped-simply supported) boundary conditions is considered. Peridynamic results are compared against finite element analysis solutions and a very good agreement is observed between the two solutions

Investigation of the effect of porosity on intergranular brittle fracture using peridynamics

Brittle materials are widely used and their fracture behaviour can be significantly influenced by their microstructures and microstructural defects including pores, microcracks, grains and grain boundaries. In this study, the effect of porosity on intergranular fracture behaviour of polycrystalline materials is investigated by using peridynamics. Different number of cases are considered for different number of grains, different porosity ratios, different locations of pores and different grain boundary strengths. It is concluded that the severity of the cracks, especially the newly created cracks, are influenced by the number of grains and porosity. Moreover, with the increase of grain boundary strength, the effect of porosity dramatically decreases and the fracture pattern in microscale becomes identical to the macroscale crack pattern

Effect of horizon shape in peridynamics

As a new continuum mechanics formulation, peridynamics has a non-local character by having an internal length scale parameter called horizon. Although the effect of the size of the horizon has been studied earlier, the shape of the horizon can also be influential. In this study, the effect of horizon shape is investigated for both ordinary state-based and non-ordinary state-based peridynamics. Three different horizon shapes are considered including circle, irregular and square. Both static and dynamic analyses are studied by considering plate under tension and vibration of a plate problems. For both static and dynamic conditions, square shape could not capture accurate vertical displacements for ordinary-state based peridynamics. On the other hand, results obtained for all three horizon shapes agree very well with finite element analysis results for non-ordinary state-based peridynamics

Peridynamic modelling of Hertzian indentation fracture

Hertzian indentation technique is a method that is widely used in the investigation of the fracture toughness and the Young’s modulus of a brittle material. Thus, it has been studied in various research studies during the past century. The problem describes axisymmetric fracture behaviour of a flaw-free brittle solid under compression due to the impact of a stiff indenter. During the fracture process, the evolution of the crack is divided into two stages. Initially, a ring crack forms spontaneously outside the contact region under a critical load. Then, it propagates for a small distance perpendicular to the free surface of the brittle solid. With the increase in load applied on the indenter, a cone-shaped crack occurs at the bottom of the ring flaw and it grows at a certain angle. Hence, in this study a new numerical technique, peridynamics, is utilised to analyse the historical complex fracture problem. To reduce the computational time, the problem is simplified by considering an axisymmetric model. The effect of Poisson’s ratio on the orientation and size of the cone-shaped crack are investigated

Peridynamic surface elasticity formulation based on modified core-shell model

Continuum mechanics is widely used to analyse the response of materials and structures to external loading conditions. Without paying attention to atomistic details, continuum mechanics can provide us very accurate predictions as long as continuum approximation is valid. There are various continuum mechanics formulations available in the literature. The most common formulation was proposed by Cauchy two hundred years ago and the equation of motion for a material point is described by using partial differential equations. Although these equations have been successfully utilised for the analysis of many different challenging problems of solid mechanics, they encounter difficulties when dealing with problems including discontinuities such as cracks. In such cases, a new continuum mechanics formulation, peridynamics can be more suitable since the equations of motion in peridynamics are in integro-differential equation form and do not contain any spatial derivatives. In nano-materials, material properties close to the surfaces can be different than bulk properties. This variation causes surface stresses. In this study, modified core-shell model is utilised to define the variation of material properties in the surface region by considering surface effects. Moreover, directional effective material properties are obtained by utilising analytical and peridynamic solutions

Double horizon peridynamics

In this study, a new double horizon peridynamics formulation was introduced by utilising two horizons instead of one as in traditional peridynamics. The new approach allows utilisation of large horizon sizes by controlling the size of the inner horizon size. To demonstrate the capability of the current approach, four different numerical cases were examined by considering static and dynamic conditions, different boundary and initial conditions, and different outer and inner horizon size values. For both static and dynamic cases, it was observed that as the inner horizon decreases, peridynamic solutions converge to classical continuum mechanics solutions even by using a larger horizon size value. Therefore, the proposed approach can serve as an alternative approach to improve computational efficiency of peridynamic simulations by obtaining accurate results with larger horizon sizes

Ordinary state-based peridynamic homogenization of periodic micro-structured materials

In this work, an ordinary state-based peridynamic homogenization method is presented, in which periodic boundary condition is enforced by coupling the displacement of periodic point pairs. Effective material properties are obtained from the peridynamic displacement gradient tensor. The governing equation of peridynamics is in integro-differential form instead of more common spatial differential form, which grants it unique advantage in performing homogenization analysis involving defects. Recently, with the rapid advancement in additive manufacturing technology, micro-structured materials have attracted significant attention. Microscopic defects occurring during manufacturing process can have a noticeable impact on the overall material behavior especially on the fracture strength of the material. The current study provides a new approach to obtain the effective properties of periodic micro-structured materials with defects