Model for Interlaminar Normal Stresses in Doubly Curved Laminates
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1 DISS. ETH NO Model for Interlaminar Normal Stresses in Doubly Curved Laminates A dissertation submitted to the Swiss Federal Institute of Technology Zurich for the degree of Dr. sc. ETH Zurich presented by René Roos Dipl. Masch. Ing. ETH born February 27, 1979 citizen of Egolzwil (LU) accepted on recommendation of Prof. Dr. P. Ermanni, examiner Prof. Dr. J. Botsis, co-examiner Prof. Dr. K. Rohwer, co-examiner Dr. G. Kress, co-examiner 2008
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3 Abstract This thesis investigates several aspects of interlaminar normal stresses in thick-walled curved laminates resulting from the need of accurate and numerical efficient calculation tools for composite structures. New mechanical models for the interlaminar normal stress calculation in thick-walled singly and doubly curved laminates are presented. The analytical models provide accurate results in consideration of the severe simplifications. Moreover, the new model and solid finite element (FE) simulations are used to assess the maximum delamination load which is provided by delamination strength tests. A few concepts for modelling interlaminar shear and normal stresses in laminated structures, caused by holding equilibrium to internal forces, edge and curvature effects, are explained. Especially the through-thethickness distribution of the interlaminar normal stress is discussed and modelling rules for solid finite element models (FEM) are defined so that accurate stress results are achieved. The new model for the interlaminar normal stress calculation in singly curved thick-walled laminates requires the in-plane strain information which is calculated by using the classical laminate theory (CLT). Further, the model has to map the kinematical effects caused by the curved laminate shape where all shear effects are neglected. The interlaminar normal stress distributions, obtained from the new model, are compared with FEM results for unidirectional and crossply laminates. Based on the existing model, the influence of the interlaminar shear is modelled by the first-order shear deformation theory and the improvement of accuracy is validated. The new model for the interlaminar normal stress calculation in thickwalled doubly curved laminates is derived from that for singly curved laminates. It is based on the through-the-thickness equilibrium condition for arbitrary doubly curved laminates and shares with the CLT the assumption that the cross-sections remain plane. The agreement between the mechanical model and solid FEM results is investigated for different laminates and materials.
4 The model is developed for element-level post-processing and is implemented in an in-house FE program with linear and quadratic shell element types. The required curvature radius evaluation is based on the FE mesh. It is assumed that the reference surface of the shell FEs follows a parabolic function. A generic model assessment of the post-processing method investigates the accuracy and robustness for both element types, different laminate shapes, material properties, and angel-ply laminates. In addition to the post-processing method, the interlaminar normal strength of a unidirectional laminate is investigated within this thesis. Failure is predicted by using Hashin s failure criteria and therefore a thick-walled singly curved specimen is designed which produces smoothly distributed interlaminar stresses. The mechanical tests provided the maximum load of a unidirectional laminate and the failure location was observed by acoustic emission measurements. The displacement, measured at onset of delamination, is applied as load in a solid FEM which provides the interlaminar normal stresses. The statistic spread of the interlaminar strength tests is discussed on the basis of (embedded) voids at the maximum interlaminar stress location. Moreover, the solid FE results are compared with performance data such as reaction force and strain. Finally, a sensitivity and error analysis investigates the accuracy of the interlaminar strength evaluation.
5 Zusammenfassung Diese Dissertation befasst sich mit verschiedenen Aspekten der interlaminaren Normalspannung in dickwandigen und gekrümmten Laminaten, motiviert durch das Bedürfnis an genauen und numerisch effizienten Programmen für die Berechnung von Kompositstrukturen. Es werden neue mechanische Modelle für die Berechnung der interlaminaren Normalspannung in dickwandigen einfach und zweifach gekrümmten Laminaten vorgestellt. Die analytischen Modelle ergeben genaue Resultate unter Berücksichtigung der starken Vereinfachungen. Ausserdem werden Finite-Elemente-Simulationen und das neue Modell verwendet um die maximale Delaminationsbeanspruchung zu untersuchen, die durch Festigkeitstest bestimmt wurde. Einige Konzepte für die Modellierung von interlaminaren Schub- und Normalspannungen, verursacht durch das Erfüllen des Gleichgewichts der Schnittlasten oder durch Rand- und Krümmungseffekte, werden erläutert. Die Dickenverteilung der interlaminaren Normalspannung wird dabei ausführlicher beschrieben und Modellierungsrichtlinien für die Finite-Element-Modelle (FEM) werden aufgestellt, damit genaue Spannungsresultate erreicht werden. Das neue Modell für die Berechnung der interlaminaren Normalspannung in einfach gekrümmten dickwandigen Laminaten benötigt die Laminatdehnungen (in-plane), die durch die klassische Laminattheorie (CLT) berechnet werden. Das Modell muss zudem die kinematischen Effekte abbilden können, welche durch die gekrümmte Laminatform verursacht werden, wobei alle Schubeinflüsse vernachlässigt werden. Die mit dem neuen Modell berechneten interlaminaren Normalspannungsverteilungen für Kreuz- und unidirektionale Laminate werden mit FEM Resultaten verglichen. Basierend auf dem vorhandenen Modell wird der interlaminare Schubeinfluss mit der Schubtheorie erster Ordnung modelliert und die Genauigkeitsverbesserung bestimmt. Das Modell für die Berechnung der interlaminaren Normalspannung in dickwandigen zweifach gekrümmten Laminaten ist vom demjenigen für einfach gekrümmte Laminate abgeleitet. Es basiert auf der
6 Gleichgewichtsbedingung in Dickenrichtung für beliebig zweifach gekrümmte Geometrien und geht wie die CLT von der Annahme aus, dass die Querschnitte eben bleiben. Die Übereinstimmung zwischen den Ergebnissen des mechanischen Modells und der Berechnung mit 3D finiten Elementen wird für verschiedene Laminate und Materialien untersucht. Das neue Modell ist für die elementbasierte Nachlaufrechnung entwickelt worden und ist in einem institutseigenen FE-Programm mit linearen und quadratischen Schalenelementen implementiert. Die dazu notwendige Berechnung der Krümmungsradien basiert auf dem FE-Netz. Es wird angenommen, dass die Referenzfläche der finiten Schalenelementen einer parabolischen Funktion entspricht. Eine allgemeine Beurteilung der Nachlaufrechnungsmethode untersucht die Genauigkeit und Robustheit für beide Elementtypen, verschiedene Laminatformen, Materialeigenschaften und Winkellaminate. Neben der Berechnung der interlaminaren Normalspannung wird im Rahmen dieser Dissertation die interlaminare Normalfestigkeit eines unidirektionalen Laminates untersucht. Das Versagen wird mittels Hashins Versagenskriterien vorhergesagt und deshalb ist ein einfach gekrümmter Probenkörper entwickelt worden, welcher kontinuierlich verteilte interlaminare Spannungen erzeugt. Die maximale Beanspruchung wurde durch mechanische Versuche ermittelt und anhand von akustischen Emissionsmessungen konnte der Versagensort festgestellt werden. Die statistische Streuung der interlaminaren Festigkeitsversuche wird mittels (eingebetteten) Fehlstellen am maximalen interlaminaren Anstrengungsort besprochen. Ausserdem werden die 3D FE-Resultate mit den Messergebnissen wie Kraft und Dehnung verglichen. Schlussendlich wird die Genauigkeit der interlaminaren Festigkeitsberechnung anhand einer Sensitivitäts- und Fehlerrechnung untersucht.
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