Multilayered folding with constraints

  • Timothy Dodwell

Student thesis: Doctoral ThesisPhD


In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between layers, while close packing leads to geometrically induced curvature singularities. When creation of voids is penalized by external pressure, the system trades off these competing effects, leading to various accommodating formations.Three two dimensional energy based nonlinear models are presented to describe the formation of voids at areas of intense geological folding. For each model the layers are assumed to be flexible elastic beams under hard unilateral contact constraint; which are solved as quasi-static obstacle problems with a free boundary. In each case an application of Kuhn-Tucker theory leads to representation as a nonlinear fourth order differential equation.Firstly a single layered model for voiding is presented. An elastic layer is forced into a V-shaped singularity by a uniform overburden pressure, where the fourth order free boundary problem is shown to have a unique, convex, symmetric solution. Drawing parallels with the Kuhn-Tucker theory, virtual work and ideas of duality, the physical significance of this differential equation is emphasised. Finally, appropriate scaling of either the potential energy or the differential equation shows the solutions scale to a single parametric group, for which the size of the void scales inversely with the ratio of overburden pressure to bending stiffness of the layer.Common to structural geology, one or several especially thick layers can dominate the deformation process. As a result, the remaining weak layers must accommodate into the geometry imposed by these competent layers. The second model, extends the first by introducing a plastic hinge to replicate the geometry imposed by the competent layer, and also axial springs to resist the slip over the limbs. The equilibrium equations for the system are investigated using the mathematical techniques developed for the first model. Under rigid loading the system may snap from an initially flat state to a convex voiding solution, as seen in the first model. However, if resistance to slip is high, the slightest imperfection causes the system to jump to a convoluted up-buckled solution, following a de-stiffened path to a point of self contact. These solutions have similarities with the delamination of carbon fibre composites.Finally, we extend the two single layered models to a simple multilayered model, which describes the periodic formation of voids in a chevron fold. The model shows that in the limit of high overburden pressures solutions form voids every layer, producing straight limbs punctured by sharp corners. This analysis shows good agreement when compared with recent experiments. This work provides the basis for future work on the buckling of thin multilayer assemblies in which voids may develop, and emphasizes the importance of the intricate nonlinear constraints of layers fitting together in multilayered folds.
Date of Award1 Sept 2011
Original languageEnglish
Awarding Institution
  • University of Bath
SupervisorGiles Hunt (Supervisor) & Chris Budd (Supervisor)


  • buckling
  • folding
  • bifurcation
  • geometric instabilities

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