Jorn S. Hansen

A general approach forcing convexity of ply angle optimization in composite laminates

This paper deals with optimization of laminated composite structures in which the ply angles are taken as design variables. One of the major problems when using ply-angles as design variables, is the lack of convexity of the objective function and thus the existence of local optima, which implies that usual gradient based optimization procedures may not be effective. Therefore, a new general approach that avoids the abovementioned problems of nonconvexity when ply-angles are used as design variables is proposed. The methodology is based upon the fact that the design space for an optimization problem formulated in lamination parameters [introduced by Tsai and Pagano (1968)] is proven to be convex, because the laminate stiffnesses are expressed linearly in terms of the lamination parameters. However, lamination parameters have at least two major shortcomings: as yet, for the general case involving membrane-bending coupling, the constraints between the lamination parameters are not completely defined; also, for a prescribed set of lamination parameters physically realizable composite laminates (e.g. laminates with equal thickness plies) may not exist. The approach here, uses both lamination parameters and ply-angles and thereby uses the advantages of both and eliminates the shortcomings of both.In order to illustrate this approach, several stiffness optimization examples are provided.

The effect of cooling rate and annealing on residual stress development in graphite fibre reinforced PEEK laminates

The effect of cooling rate from the molten state and annealing from the amorphous state on thermal and crystallization residual stress development in graphite fibre reinforced polyetheretherketone (APC-2) laminates is studied experimentally by measuring the radius of curvature developed in (0/90) unsymmetric laminates. Curvature is measured for cooling rates ranging from 1.5°C/min to 8000°C/min, and for annealing temperatures ranging from 175°C to 250°C. Stress relaxation effects are also quantified.

The Effect of Cooling Rate on Free-Edge Stress Development in Semi-Crystalline Thermoplastic Laminates:

A numerical model is developed for prediction of the process-induced thermal residual stresses in thermoplastic composite laminates. The model addresses the development of the three-dimensional residual stress state in fracture-critical free-edge regions as well as through-thickness stress variations. The current approach and analysis provide a unique capability for the investigation of the influence of thermal processing and structural parameters on the resulting buildup of residual stresses during manufacturing. Therefore, it can assist in the design and analysis of thermoplastic composites in terms of tailoring of mechanical and strength characteristics. The applied surface cooling rate from the melt has a strong effect on the free-edge stress levels obtained at room temperature, and in particular the interlaminar stresses. Results are shown for the case of typical cross-ply and quasi-isotropic APC-2 (graphite/polyether-ether-ketone) laminates.

Enhanced Elastic Buckling Loads of Composite Plates With Tailored Thermal Residual Stresses

Thermal residual stresses introduced during the manufacturing process and their effect on the buckling load of stringer reinforced composite plates is investigated. The principal idea is to include stiffeners on the perimeter of the plate and thereby, during manufacture, induce a favorable thermal residual-stress state in the structure; these stresses arise by considering the difference in thermal expansion coefficients and elastic properties of the plate and the stiffeners. In this manner, it is shown that thermal residual stresses can be tailored to significantly enhance the performance of the structure. The analysis is taken within the context of an enhanced Reissner-Mindlin plate theory and the finite element technique is used to analyze the problem. A 16 node bi-cubic Lagrange element is implemented in a FORTRAN code to determine the buckling load of the composite plate in the presence of thermal residual stresses. Three different plate-stiffener geometries are used as illustrations. The analyses indicate that buckling loads can be significantly increased by properly tailoring the thermal residual stresses. Therefore it may be concluded that an evaluation of these stresses and a judicious analysis of their effects must be included in the design procedure for this class of composite structure.

Stress Stiffening Effects in Laminated Beams with Piezoelectric Actuators

The effectiveness of using piezoelectric elements to enhance the mechanical performance of laminated structures by inducing favorable in-plane stresses is investigated. A finite element formulation is presented for the analysis of laminated Timoshenko beams with an arbitrary number of piezoelectric actuators and/or sensors placed along the length of the beam. Von Kdrman nonlinear strain-displacement relations are used and ideal linear behavior is assumed for the piezoelectric actuation. It is shown that if the beam is longitudinally restrained at both ends the piezolectric actuators induce in-plane stresses that significantly affect the mechanical behavior of the beam. It is also shown that stress stiffening is more pronounced for slender beams. A configuration with a piezoelectric actuator at the top of a slender beam and a sensor at the bottom is analyzed. It is demonstrated that neglecting the stress stiffening effects may lead to large errors in both the static and dynamic analyses of this widely used configuration for slender beams. Further, the stiffness of the beam can be piezoelectrically tuned to significantly change the natural frequency; this has interesting applications in active vibration and acoustics control or smart structures.

The Effect of Thermal Processing on Residual Strain Development in Unidirectional Graphite Fibre Reinforced PEEK

The effect of annealing on residual strain development in amorphous graphite fibre reinforced PEEK (APC-2) is studied experimentally. A method is presented for embedding strain gauges in a unidirectional laminate at the time of manufacture and then quenching the laminate into the amorphous state. With this method, thermal and crys tallization strains are measured directly during quasi-isothermal heating of the laminate. Both longitudinal and transverse lamina strains resulting from contraction during crystallization are measured. Longitudinal crystallization strains are found to be negligible due to fibre stiffness. Transverse contraction due to crystallization is large and increases with anneal temperature. A coefficient of crystalline expansion (CCE) is defined which is analogous to the CTE. This coefficient describes the permanent crystallization strain as a function of anneal tem perature and is determined experimentally.

Structural topology optimization for multiple load cases using a dynamic aggregation technique

A series of techniques is presented for overcoming some of the numerical instabilities associated with SIMP materials. These techniques are combined to create a robust topology optimization algorithm designed to be able to accommodate a large suite of problems that more closely resemble those found in industry applications. A variant of the Kreisselmeier–Steinhauser (KS) function in which the aggregation parameter is dynamically increased over the course of the optimization is used to handle multi-load problems. Results from this method are compared with those obtained using the bound formulation. It is shown that the KS aggregation method produces results superior to those of the bound formulation, which can be highly susceptible to local minima. Adaptive mesh-refinement is presented as a means of addressing the mesh-dependency problem. It is shown that successive mesh-refinement cycles can generate smooth, well-defined structures, and when used in combination with nine-node elements, virtually eliminate checkerboarding and flexural hinges.

Sandwich-Plate Vibration Analysis: Three-Layer Quasi-Three-Dimensional Finite Element Model

A three-layer finite element model for the vibration analysis of sandwich plates with laminated composite face sheets is evaluated. In the model the face sheets are represented as Reissner-Mindlin plates, and the core is modeled as a three-dimensional continuum. This representation allows accurate modeling for a wide range of core types. The three-dimensional problem is reduced to two dimensions by analytical through-thickness integration of the energy expressions for the evaluation of mass and stiffness matrices. The results from this model are compared to finite element results based on solid elements, classical sandwich analysis, and classical plate theory. The objective in the work is to compare natural frequency and mode shape predictions using these models for a broad range of core stiffness. When large differences between face sheet and core stiffness are present, it is illustrated that traditional laminate theories yield significant inaccuracy. Moreover, unlike plate models, the present theory is also capable of representing a variety of three-dimensional boundary conditions. Furthermore, compared to solid models, the present laminated model avoids numerical problems as a result of three-dimensional element aspect ratio. Therefore, the present model provides a powerful general tool for the analysis of natural modes and frequencies of sandwich plates.

A Method to Predict the Effect of Thermal Residual Stresses on the Free-Edge Delamination Behavior of Fibre Reinforced Composite Laminates

A method is presented which accounts for the effect of process-induced thermal residual stresses on the free-edge delamination behavior of fibre reinforced laminates. A strain energy release rate fracture mechanics formulation is derived based upon the Irwin crack closure integral. The finite element method is then used to calculate Mode I and Mode II crack opening displacements and stiffnesses for free-edge delamination cracks originating at a specified ply interface from which strain energy release rates due to thermal and mechanical loading are then obtained. The effect of processing temperature on strain energy development is included in the calculation of thermal energy release rates by using experimentally measured material property data. To verify the model, (+352/-352/02/902)s graphite-reinforced PEEK tensile test specimens were manufactured and tested to failure. Measured delamination strains agree well with predictions when the effect of thermal residual stresses is included. Neglecting residual stresses over-estimates the tensile delamination strength by approximately 32%.

Some two-mode buckling problems and their relation to catastrophe theory

The asymptotic buckling analysis of semisymmetric, two-degree-of-freedom static systems is presented. The types of behavior which can occur for this class of system are determined and categorized in accordance with the results of catastrophe theory. The expressions for the critical load-initial imperfection surfaces are determined in closed form. Within the context of an asymptotic analysis all possible secondary bifurcations cases are isolated. (The term bifurcation is used here in the sense which is usual in elastic stability analyses and not in the broader sense which is common in catastrophe theory.) In addition, the physical significance of the different criticaS load-initial imperfection surfaces is determined, and the primary surface is identified for each case. The general results are demonstrated in the example of the two-mode buckling of an axially loaded beam resting on a nonlinear elastic foundation.