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PhD Studentship: Theoretical and computational peridynamics for fracture of materials and structures

Engineering & the Environment

Location:  Highfield Campus
Closing Date:  Wednesday 07 February 2018
Reference:  837417F2

Project Reference: NGCM-105

Background:Predicting and simulating the fracture and failure of materials and structures remains one of the greatest challenges of computational engineering. This stems from the multiscale (and often multiphysics) nature of fracture mechanics and the inherent limitations of continuum mechanics (CM) theories and associated finite element techniques to capture and predict such phenomena.

Local theories of matter assume that a material point only exchanges mass, momentum and energy with its closest neighbours. As a result, the stress state at a point depends on the deformation at that point only. In addition, within classical continuum mechanics a body remains continuous as it deforms and the mathematical formulation governed by PDEs breaks down when a discontinuity appears (e.g. a crack). In addition, limitations of current finite element (FE) methods for fracture mechanics include: (1) Cumbersome and complicated numerical implementation. (2) Assumption of pre-existing cracks rather than the nucleation of new cracks. (3) Need for mesh adaptation. (4) Inefficient and/or simply infeasible methods for multiple cracks, crack branching and crack coalescence. (5) Need to provide a kinetic relation for crack growth: how a crack evolves based on local conditions.

Although first enunciated in principle by Piola in the 19th century, peridynamics (PD) theory was practically formulated by Silling in 2000 [1]. This non-local theory of matter can address all the shortcomings mentioned above is ideally suited for fracture phenomena.


The project:

Although similar in principles to molecular dynamics (MD), peridynamics is not restricted to the very short length and time scales associated with MD.  That means that practical engineering applications operating at the macroscopic scale can be practically simulated using PD. PD can be implemented in a MD code or, especially relevant to this project, within a FE code and benefits from the more reasonable computing times of FE methods prior to crack initiation.


The overall aim of this project is to develop a novel finite element-based peridynamics computational environment to study crack propagation and stability conditions in maritime engineering structures.


A state-based PD will be used so that arbitrary constitutive models could be implemented. Departing from the original PD approach which goes from a PD continuum (theory) to its discretised version (computational implementation) we propose to follow the ideas of Gerstle [2] and introduce  a PD lattice

model which is a discretised version of matter a priori, where  lattice spacing is dictated by the internal material length scale (i.e. characteristic size of microstructure). This avoids the problem of defining/storing mass density or other internal variable fields, continuous or discontinuous (e.g. plastic strain). Instead they are defined/stored at each particle of the lattice.

Although PD address many shortcomings of traditional theories and numerical methods, one challenge where research efforts are urgently needed is concerned with the formulation of constitutive models and their calibration from experimental data which differ from CM approaches.


In this project, an efficient and modular workflow to automatically generate optimised code for new material constitutive models as well as their calibration from experimental data will be devised: the mathematical models will be formulated in symbolic form in the mature symbolic-numeric environment AceGen/AceFEM integrated within Mathematica®. From these formulations optimised numerical code (e.g. Fortran, C, Matlab or Mathematica) will be generated and integrated within a FE environment (e.g Abaqus/Implicit and /Explicit, AceFEM) for large scale simulations. The use of HPC capabilities will be essential.


[1] Silling, S. A., 2000, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Sol., 48:175-209.

[2] Gerstle, W. H., Introduction to practical peridynamics, World Scientific, London, First edition, 2016, 410 pages.


We are looking for an applicant with a background in physics, engineering mechanics, applied mathematics or computer science with strong interest and/or skills in programming, and an appetite to learn and research across conventional discipline boundaries.


The stipend is at the standard EPSRC levels. More details on facilities and computing equipment are available


The successful candidate will work in a stimulating research environment, supported by world-leading organisations such as Procter & Gamble, Rolls Royce, Lloyd’s Register, Shell and the US Air Force and will be encouraged to work with our international academic and industrial collaborators in Europe, Singapore, New Zealand and the USA.

If you wish to discuss any details of the project informally, please contact Georges Limbert, Email: Tel: +44 (0) 2380 592381
This project is run through participation in the EPSRC Centre for Doctoral Training in Next Generation Computational Modelling ( For details of our 4 Year PhD programme, please see

For a details of available projects click here

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The closing date for this job opportunity has now passed, and applications are no longer being accepted for this position

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