Self-similar voiding solutions of a single layered model of folding rocks

In this paper we derive an obstacle problem with a free boundary to describe the formation of voids at areas of intense geological folding. An elastic layer is forced by overburden pressure against a V-shaped rigid obstacle. Energy minimization leads to representation as a nonlinear fourth-order ordinary differential equation, for which we prove their exists a unique solution. Drawing parallels with the Kuhn-Tucker theory, virtual work, and ideas of duality, we highlight the physical significance of this differential equation. Finally we show this equation scales to a single parametric group, revealing a scaling law connecting the size of the void with the pressure/stiffness ratio. This paper is seen as the first step towards a full multilayered model with the possibility of voiding

A finite deformation Cosserat continuum model for uncured carbon fibre composites

A new three-dimensional, finite deformation Cosserat continuum model for the elastic response of uncured carbon fibre composites is presented. The new composite process model captures the bending contribution of bundles of fibres at the microscale within a mesoscale continuum description of a composite ply. This is achieved by introducing higher-order, independent rotational degrees of freedom into the continuum formulation. This paper demonstrates the inclusion of such mechanics is essential to accurately model various bending responses induced during typical composite manufacturing processes. This includes large deformation forming, finite strain consolidation and wrinkling (the formation of an unwanted defect). If such mechanics are not included, the literature demonstrates the resulting finite element solutions have a pathological dependence on the mesh size. As a result, simulations require users to fit mesh-dependent material parameters, which limits confidence in their predictive capabilities. The Cosserat continuum, which can be seen as a form of the regularised continuum model, overcomes these challenges. In particular, this paper presents details of the finite element formulation of the new continuum model within a nonlinear Taylor–Hood Cosserat Element. Implementation details of embedding this new element within the commercial code Abaqus are given, alongside a series of increasingly complex validation simulations. Notably, the examples include modelling the formation of internal fibre wrinkles and large deformation forming, which involves complex ply-to-ply and tool-to-ply contact. The paper concludes by describing: (1) how the elastic Cosserat model can be integrated into existing large deformation process models in the literature. The approach set out readily allows researchers to include the important effects of resin flow, cure kinetics and temperature distribution, not considered in this contribution, and (2) how it is envisaged that the ply scale model can be naturally scaled up to large laminate scale simulation using mathematical upscaling techniques

Optimum fibre-steering of composite plates for buckling and manufacturability

This paper reveals the potential weight saving in creating lightweight composite panels using the novel Continous Tow Shearing technique. The method provides a unique capability to simultaneously manipulate fibre angles and local thicknesses, whilst achieving defect-free, high curvature fibre paths. Considering the buckling of a typical aircraft wing skin, the nonlinear optimisation problem presented, shows theoretical designs with significant weights savings if compared to panels made from conventional straight fibre angles (38%) or steered using Automated Fibre Placement techniques (20%). The implications due to constraints on panelstrength, damage tolerance and manufacturing costs are then considered

Resin treatment of free edges to aid certification of through thickness laminate strength

Large aerospace parts are typically certified by testing narrow specimens, such as curved laminates, which have exposed free edges. These edges (not present in the production part) have been found to reduce the 3D strength of curved laminates by over 20%, showing this certification method is unreasonably conservative. The free edges also create a singularity, such that finite element (FE) modelling is challenging, which is typically approximated using non-linear analysis of cohesive interlaminar zones. A new treatment process is developed whereby a layer of resin is applied to the free edges of curved laminates. This significantly reduces the edge effect and delays failure. The resin edge treatment increases the strength of the curved laminate test specimens by 16%. The treatment also simplifies FE modelling by allowing for non-zero stresses normal to the laminate edge, removing the singularity. This enables use of linear FE models, which converge at the laminate edge. A linear FE method developed in this paper is conservative and predicts the strength of treated curved laminates to within 5% of the average test value. Hence it is shown that the resin edge treatment can be used to improve reliability of both certification tests and FE models

Resin treatment of free edges to aid certification of through thickness laminate strength

Large aerospace parts are typically certified by testing narrow specimens, such as curved laminates, which have exposed free edges. These edges (not present in the production part) have been found to reduce the 3D strength of curved laminates by over 20%, showing this certification method is unreasonably conservative. The free edges also create a singularity, such that finite element (FE) modelling is challenging, which is typically approximated using non-linear analysis of cohesive interlaminar zones. A new treatment process is developed whereby a layer of resin is applied to the free edges of curved laminates. This significantly reduces the edge effect and delays failure. The resin edge treatment increases the strength of the curved laminate test specimens by 16%. The treatment also simplifies FE modelling by allowing for non-zero stresses normal to the laminate edge, removing the singularity. This enables use of linear FE models, which converge at the laminate edge. A linear FE method developed in this paper is conservative and predicts the strength of treated curved laminates to within 5% of the average test value. Hence it is shown that the resin edge treatment can be used to improve reliability of both certification tests and FE models

Chevron folding patterns and heteroclinic orbits

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.We present a model of multilayer folding in which layers with bending stiffness EI are separated by a very stiff elastic medium of elasticity k2 and subject to a horizontal load P. By using a dynamical systems analysis of the resulting fourth order equation, we show that as the end shortening per unit length E is increased, then if k2 is large there is a smooth transition from small amplitude sinusoidal solutions at moderate values of P to larger amplitude chevron folds, with straight limbs separated by regions of high curvature when P is large. The chevron solutions take the form of near heteroclinic connections in the phase-plane. By means of this analysis, values for P and the slope of the limbs are calculated in terms of E and k2.We would like to acknowledge the support of the FP7 Marie-Curie ITN FIRST, the Pacific Institute for Mathematics Sciences (PIMS) and the NSERC Discovery Grant for the funding of the research described in this pape