Peridynamics : a novel approach for material and structural modelling

Solid mechanics is an interdisciplinary subject mainly deals with deformation of materials and structures under external loading conditions. In many applications, safety is an important concern. For instance, a small crack can cause catastrophic damage to an airplane which can result in loss of many lives. Moreover, such incidents can cause significant damage to the environment which might take years for complete recovery process such as the effect of oil spill due to a damaged tanker, corroded pipeline, etc. In order to prevent from such undesired incidents, several approaches are commonly followed within the solid mechanics framework. One common approach is to perform experiments to test the durability of materials and structures. Although such an approach can provide us realistic data, such experiments are not always possible and they are mostly expensive. Today, engineers and researchers commonly use theoretical or computational approaches as an alternative to analyse the problems that they are working on. Theoretical approaches mostly depend on various assumptions to simplify the calculations. Moreover, they are restricted to specific geometries and loading conditions. Hence, it is essential to use a technique which does not have such limitations and is also economically feasible. Computational techniques are very good candidates for this purpose and they are commonly used both in industry and academia

In-plane and out-of plane failure of an ice sheet using peridynamics

When dealing with ice structure interaction modeling, such as designs for offshore structures/icebreakers or predicting ice cover's bearing capacity for transportation, it is essential to determine the most important failure modes of ice. Structural properties, ice material properties, ice-structure interaction processes, and ice sheet geometries have significant effect on failure modes. In this paper two most frequently observed failure modes are studied; splitting failure mode for in-plane failure of finite ice sheet and out-of-plane failure of semi-infinite ice sheet. Peridynamic theory was used to determine the load necessary for in-plane failure of a finite ice sheet. Moreover, the relationship between radial crack initiation load and measured out-of-plane failure load for a semi-infinite ice sheet is established. To achieve this, two peridynamic models are developed. First model is a 2 dimensional bond based peridynamic model of a plate with initial crack used for the in-plane case. Second model is based on a Mindlin plate resting on a Winkler elastic foundation formulation for out-of-plane case. Numerical results obtained using peridynamics are compared against experimental results and a good agreement between the two approaches is obtained confirming capability of peridynamics for predicting in-plane and out-of-plane failure of ice sheets

An Euler-Bernoulli beam formulation in ordinary state-based peridynamic framework

Every object in the world has a 3-Dimensional geometrical shape and it is usually possible to model structures in a 3-Dimensional fashion although this approach can be computationally expensive. In order to reduce computational time, the 3-Dimensional geometry can be simplified as a beam, plate or shell type of structure depending on the geometry and loading. This simplification should also be accurately reflected in the formulation which is used for the analysis. In this study, such an approach is presented by developing an Euler-Bernoulli beam formulation within ordinary-state based peridynamic framework. The equation of motion is obtained by utilizing Euler-Lagrange equations. The accuracy of the formulation is validated by considering various benchmark problems subjected to different loading and displacement/rotation boundary conditions

Peridynamic mindlin plate formulation for functionally graded materials

In this study, a new peridynamic Mindlin plate formulation is presented which is suitable for the analysis of functionally graded materials. The governing equations of peridynamic formulation are obtained by using Euler-Lagrange equations in conjunction with Taylor’s expansion. To validate the new formulation, three different numerical benchmark problems are considered for a Mindlin plate subjected to simply supported, fully clamped and mixed (clamped-simply supported) boundary conditions. Peridynamic results are compared against results from finite element analysis and a good agreement is observed between the two methods

A state-based peridynamic formulation for functionally graded Kirchhoff plates

Functionally graded materials are a potential alternative to traditional fibre-reinforced composite materials as they have continuously varying material properties which do not cause stress concentrations. In this study, a state-based peridynamic model is presented for functionally graded Kirchhoff plates. Equations of motion of the new formulation are obtained using the Euler–Lagrange equation and Taylor’s expansion. The formulation is verified by considering several benchmark problems including a clamped plate subjected to transverse loading and a simply supported plate subjected to transverse loading and inclined loading. The material properties are chosen such that Young’s modulus is assumed to be varied linearly through the thickness direction and Poisson’s ratio is constant. Peridynamic results are compared against finite element analysis results, and a very good agreement is obtained between the two approaches

Peridynamic model for a Mindlin plate resting on a Winkler elastic foundation

In this study, a peridynamic model is presented for a Mindlin plate resting on a Winkler elastic foundation. In order to achievestatic and quasi-static loading conditions, direct solution of the peridynamic equations are utilised by directly assigning inertia terms to zero rather than using widely adapted Adaptive Dynamic Relaxationapproach. The formulation is verified by comparing againsta finite elementsolutionfor transverse loading conditionwithout considering damageand comparing against a previous study for pure bending of a Mindlin plate with a central crack made of PMMA material havingnegligibly small elastic foundation stiffness. Finally, the fracture behaviour of a pre-cracked Minlin platerested on a Winkler foundation subjected to transverse loadingrepresenting a floating ice floe interacting with sloping structures. Similar fracture patternsobserved in field observations weresuccessfully captured by peridynamics

Influence of different types of small-size defects on propagation of macro-cracks in brittle materials

The presence of defects in the structure requires noticeable attention and understanding of fracture mechanisms in brittle materials has to be established. Defects in the form of holes, macro- and micro-cracks are the main interest of this paper. This work investigates the dual role of holes and micro-crack arrays on toughening and degradation mechanisms in concrete structures. An ordinary state-based peridynamics (PD) model is utilized to analyse the fracture problem at the micro-level. The application of PD shows its advantage in crack-hole, macro- and micro-crack interaction problems since PD can accurately predict the contribution of defects on structural behaviour. The study of the three-point bending problem with five types of holes existing in the structure showed the crack arrest phenomena at the hole boundary and the “attraction” of the crack to propagate towards the hole. For the study of the macro- and micro-cracks interaction problem, various cases of the micro-crack distribution and inclination angles are considered and validated with analytical studies. The PD quasi-static simulations show good agreement with analytical solutions. Moreover, PD dynamic solutions show the capability of PD to capture complex crack propagation paths. It is observed that the presence of micro-cracks and holes ahead of the main crack can suppress its further propagation as well as have an influence on the crack propagation direction. The numerical results demonstrate the efficiency of the PD modelling of multiple crack interaction problems

Model order reduction of linear peridynamic systems using static condensation

Static condensation is widely used as a model order reduction technique to reduce the computational effort and complexity of classical continuum-based computational models, such as finite-element models. Peridynamic theory is a nonlocal theory developed primarily to overcome the shortcoming of classical continuum-based models in handling discontinuous system responses. In this study, a model order reduction algorithm is developed based on the static condensation technique to reduce the order of peridynamic models. Numerical examples are considered to demonstrate the robustness of the proposed reduction algorithm in reproducing the static and dynamic response and the eigenresponse of the full peridynamic models

Family member search algorithms for peridynamic analysis

Peridynamic equation of motion is usually solved numerically by using meshless approaches, Family search process is one of the most time consuming parts of a peridynamic analysis. Especially for problems which require continuous update of family members inside the hurizoli of a material point, the time spent to search for family members becomes crucial. Hence, efficient algorithms are required to reduce the computational time. In this study, various family member search algorithms suitable for peridynamic simulations are presented including brute-force search, region partitioning and tree data structures. By considering problem cases for different number of material points, computational time between different algorithms is compared and the most efficient algorithm is determined

Peridynamic modeling of diffusion by using finite element analysis

Diffusion modeling is essential in understanding many physical phenomena such as heat transfer, moisture concentration, electrical conductivity, etc. In the presence of material and geometric discontinuities, and non-local effects, a non-local continuum approach, named as peridynamics, can be advantageous over the traditional local approaches. Peridynamics is based on integro-differential equations without including any spatial derivatives. In general, these equations are solved numerically by employing meshless discretization techniques. Although fundamentally different, commercial finite element software can be a suitable platform for peridynamic simulations which may result in several computational benefits. Hence, this study presents the peridynamic diffusion modeling and implementation procedure in a widely used commercial finite element analysis software, ANSYS. The accuracy and capability of this approach is demonstrated by considering several benchmark problems