Mathematical Modelling of Crack Problems in Bi-Material Structures Containing Imperfect Interfaces Using the Weight Function Technique

Student thesis: Doctoral ThesisDoctor of Philosophy


The main aim of this thesis is to generalise weight function techniques to tackle crack problems in bi-material linearly elastic and isotropic solids with imperfect interfaces.
Our approach makes extensive use of weight functions which are special solutions to homogeneous boundary value problems that aid in the evaluation of constants in asymptotic expressions describing the behaviour of physical fields near crack tips.
We use newly derived weight functions and respective techniques to tackle various aspects of a number of problems. The first major application is the use of the new weight functions to aid in the analysis of Bloch–Floquet waves; results include the derivation of a low dimensional model including junction conditions and the evaluation of a fracture criterion in the form of a constant in the asymptotic expansion of physical fields near crack tips. The second major application uses the new weight functions to assist in perturbation analysis. In particular, Betti’s formula is applied in an imperfect interface setting, which introduces new conditions and asymptotic behaviour in comparison to previously studied perfect interface cases.
We first derive a weight function by employing the Wiener-Hopf technique in a bi-material strip containing a semi-infinite crack and an imperfect interface. We then present an asymptotic algorithm that uses the new weight function to evaluate coefficients in the asymptotics of solutions to problems of wave propagation in a thin bi-material strip containing a periodic array of finite-length cracks situated along an imperfect interface between two materials. We introduce and solve a low dimensional model and give relationships between its solution’s behaviour at junction points and the behaviour of physical fields near the crack tip in the full original model problem.
The low dimensional model is then used to estimate eigenfrequencies of the periodic structure. We will find via comparisons against finite element simulations that the model gives excellent estimates in most cases for the frequencies of waves propagating through the strip; however, a small discrepancy is found for standing wave eigenfrequencies.
We address this discrepancy by suggesting an improvement to the asymptotic model and perform computations which demonstrate a greatly improved accuracy for standing wave eigenfrequencies in both the imperfect and ideal interface problems.
We then move on to consider our second major problem which concerns out-of-plane shear in an infinite domain containing a semi-infinite crack situated on an imperfect interface. We derive a weight function for this geometry and use Betti’s identity to relate the behaviour of physical fields near the crack tip to that of the weight function and prescribed loadings on the crack faces. In particular, the method presented allows for the prescribed tractions to be point forces, as well as continuous loadings.
Having obtained the weight function, we then conduct perturbation analysis to determine how small linear defects such as elliptic inclusions influence the forces near the crack tip. Computations are performed which demonstrate how the unperturbed solution depends upon the parameter of interface imperfection, and how the location of defects may shield or amplify the stresses near the crack tip.
Date of Award10 Sept 2013
Original languageEnglish
Awarding Institution
  • Aberystwyth University
SupervisorGennady Mishuris (Supervisor) & Robert Douglas (Supervisor)

Cite this