Festkörpermechanik

Due to the capability of mold injecting manufacturing short fiber-reinforced composites are increasingly in use in the aeronautical and automotive industries. However, a crucial aspect is their spatially distributed material properties induced by the probabilistic characteristics of the microstructure. To predict the structural response of components made of short fiber-reinforced composites by numerical simulation correctly the probabilistic information must be included in the modeling approach. Furthermore, commonly used matrix material is characterized by a distinct plastic deformation even at low stress levels. Therefore, in this work, a modeling approach is proposed that utilizes second-order Gaussian random fields for the representation of the spatially distributed material properties on the component level in the elastic and plastic domain. The modeling approach comprises the cross-correlation analysis of the material parameters describing the elastic-ideal plastic material behavior and a subsequent representation of the parameters by second-order Gaussian random fields. The analysis reveals a complex cross-correlation structure of the parameters, which depends on the window size on the mesoscale and requires the use of suitable numerical methods like the multiple correlated Karhunen-Loève expansion to synthesize the representation of the material parameters. The numerical simulations of tensile test specimens in the elastic and plastic domain predict the structural response under uniaxial loading accurately. The localized plastic deformation of the specimen is observable and meets the experimental validation by tensile tests until failure. Furthermore, the experimental data is used to determine the correlation length. Besides this, the modeling approach is validated by nanoindentation tests on the mesoscale, which reveal the spatial distribution of the material properties. Furthermore, it is shown that the area characterized by nanoindentation tests is 25 times larger than the projected area of the used Berkovich tip. In conclusion, the proposed modeling approach utilizing random fields is capable of representing the localized deformation of short fiber-reinforced composites induced by the probabilistic characteristics of the microstructure. Furthermore, the correlation structure can be derived by numerical simulation on the mesoscale, which can be experimentally analyzed by nanoindentation tests. Finally, the correlation length is an independent material parameter, which can be derived from experimental data.

In der vorliegenden Arbeit werden experimentelle und numerische Untersuchungen zum Bruchverhalten von interlaminaren Sensoren durchgeführt. Diese sollen für die strukturelle Zustandsüberwachung von Faser-Kunststoff-Verbunden nutzbar sein und mithilfe eines piezo-gesteuerten Strahldosierers hergestellt werden. Das Ziel ist die Verbesserung der kohäsiven Eigenschaften, der Geometrie und der Position der Sensoren, sodass deren Schädigungswirkung auf das Bauteil minimiert wird. Grundlage für die Herstellung der Sensoren bildet eine Tinte auf Basis von Kohlenstoffnanoröhren und Epoxidharz, wodurch gute elektrische und mechanische Eigenschaften ermöglicht werden sollen. Die Zusammensetzung der Tinte wird unter Nutzung eines statistischen Versuchsplanes optimiert. Bei einem Füllgrad von 0.25 wt% Kohlenstoffnanoröhren wird die Reproduzierbarkeit des Druckprozesses sowie ein geringer elektrischer Schichtwiderstand des entstehenden Komposits sichergestellt. In Doppelkragträger- und endgeschlitzten Dreipunkt-Biegeversuchen werden kohäsive Eigenschaften des Komposits nachgewiesen, welche vergleichbar mit denen von reinen Glasfaser-Kunststoff-Verbunden sind. Numerische Modelle dieser Bruchversuche mit homogenen Kohäsivschichten werden anschließend erstellt. Sie nutzen bewusst grob gewählte finite Elemente sowie Kohäsivzonenmodelle, welche das lokale Risswachstum beschreiben. Zur Validierung werden die numerischen Lösungen mit den analytischen Lösungen der Bruchversuche verglichen. Zu diesem Zweck wird ein Kalibrierungsverfahren für die kohäsive Festigkeit erarbeitet und angewandt. Die Erweiterung der numerischen Modelle auf heterogene Kohäsivschichten ermöglicht in der Folge die Untersuchung des Einflusses von rechteckigen, interlaminaren Sensoren auf das Bruchverhalten. Durch die Variation der kohäsiven Eigenschaften, der Geometrie und der Position der Sensoren werden Designaspekte für eine geringe Schädigungswirkung abgeleitet. Der kumulative Schaden wird als Optimierungskriterium eingeführt. Insbesondere die Fläche und die Orientierung nicht-quadratischer Sensoren unter Beachtung des wirkenden Rissbeanspruchungsmodus bieten großes Optimierungspotenzial. Die abgeleiteten Designaspekte werden anschließend auf interlaminare Sensoren in einem numerischen Faser-Kunststoff-Stringer übertragen und bestätigt. Die Ergebnisse zeigen, dass bereits 87% der Schädigungswirkung durch die Optimierung der Geometrie vermieden werden können. Dies kann zukünftig die interlaminare Integration von Sensormaterialien mit schwächeren kohäsiven Eigenschaften im Vergleich zum Faser-Kunststoff-Verbund ermöglichen.

For the numerical simulation of short fiber-reinforced composites and the correct analysis of the deformation, information about the plastic behavior and its spatial distribution is essential. When using purely deterministic modeling approaches information of the probabilistic microstructure is not included in the simulation process. One possible approach for the integration of stochastic information is the use of random fields, which requires information about the correlation structure of all material input parameters. In this study the correlation structure for finite strain elasto-plastic material behavior of short fiber-reinforced composites is analyzed. This approach combines the use of already established procedures for linear-elastic material behavior with a homogenization method for plasticity. The obtained results reveal a complex correlation structure, which is approximated with triangle and exponential correlation functions influenced by the window size. Due to the dependence of the hyperelastic and plastic material parameters on the fiber mass fraction, the strain-energy density function coefficients are cross-correlated with the yield strength of the composite. With this knowledge at hand, in a subsequent work numerical simulations of tensile tests are conducted that cover the elastic and plastic domain and include spatially distributed material properties.

For the numerical simulation of components made of short fiber-reinforced composites, the correct prediction of the deformation including the elastic and plastic behavior and its spatial distribution is essential. When using purely deterministic modeling approaches, the information of the probabilistic microstructure is not included in the simulation process. One possible approach for the integration of stochastic information is the use of random fields. In this study, numerical simulations of tensile test specimens were conducted utilizing a finite deformation elastic–ideal plastic material model. A selection of the material parameters covering the elastic and plastic domain are represented by cross-correlated second-order Gaussian random fields to incorporate the probabilistic nature of the material parameters. To validate the modeling approach, tensile tests until failure were carried out experimentally, which confirmed the assumption of the spatially distributed material behavior in both the elastic and plastic domain. Since the correlation lengths of the random fields cannot be determined by pure analytic treatments, additionally numerical simulations were performed for different values of the correlation length. The numerical simulations endorsed the influence of the correlation length on the overall behavior. For a correlation length of 5 (Formula presented.) (Formula presented.), a good conformity with the experimental results was obtained. Therefore, it was concluded that the presented modeling approach was suitable to predict the elastic and plastic deformation of a set of tensile test specimens made of short fiber-reinforced composite sufficiently.

Indentation tests are widely used to characterize the material properties of heterogeneous materials. So far there is no explicit analysis of the spatially distributed material properties for short fiber-reinforced composites on the mesoscale as well as a determination of the effective cross-section that is characterized by the obtained measurement results. Hence, the primary objective of this study is the characterization of short fiber-reinforced composites on the mesoscale. Furthermore, it is of interest to determine the corresponding area for which the obtained material parameters are valid. For the experimental investigation of local material properties of short fiber-reinforced composites, the Young’s modulus is obtained by indentation tests. The measured values of the Young’s modulus are compared to results gained by numerical simulation. The numerical model represents an actual microstructure derived from a micrograph of the used material. The analysis of the short fiber-reinforced material by indentation tests reveals the layered structure of the specimen induced by the injection molding process and the oriented material properties of the reinforced material are observed. In addition, the experimentally obtained values for Young’s modulus meet the results of a corresponding numerical analysis. Finally, it is shown, that the area characterized by the indentation test is 25 times larger than the actual projected area of the indentation tip. This leads to the conclusion that indentation tests are an appropriate tool to characterize short fiber-reinforced material on the mesoscale.

In this study a modeling approach for short fiber-reinforced composites is presented which allows one to consider information from the microstructure of the compound while modeling on the component level. The proposed technique is based on the determination of correlation functions by the moving window method. Using these correlation functions random fields are generated by the Karhunen–Loève expansion. Linear elastic numerical simulations are conducted on the mesoscale and component level based on the probabilistic characteristics of the microstructure derived from a two-dimensional micrograph. The experimental validation by nanoindentation on the mesoscale shows good conformity with the numerical simulations. For the numerical modeling on the component level the comparison of experimentally obtained Young’s modulus by tensile tests with numerical simulations indicate that the presented approach requires three-dimensional information of the probabilistic characteristics of the microstructure. Using this information not only the overall material properties are approximated sufficiently, but also the local distribution of the material properties shows the same trend as the results of conducted tensile tests.

The present work provides a profound analytical and numerical analysis of the material properties of SFRC on the mesoscale as well as the resulting correlation structure taking into account the probabilistic characteristics of the fiber geometry. This is done by calculating the engineering constants using the analytical framework given by Tandon and Weng as well as Halpin and Tsai. The input parameters like fiber length, diameter and orientation are chosen with respect to their probability density function. It is shown, that they are significantly influenced by the fiber length, the fiber orientation and the fiber volume fraction. The verification of the analytically obtained values is done on a numerical basis. Therefore, a two-dimensional microstructure is generated and transferred to a numerical model. The advantage of this procedure is, that there are several fibers with different geometrical properties placed in a preset area. The results of the numerical analysis meet the analytically obtained conclusions. Furthermore, the results of the numerical simulations are independent of the assumption of a plane strain and plane stress state, respectively. Finally, the correlation structure of the elasticity tensor is investigated. Not only the symmetry properties of the elasticity tensor characterize the correlation structure, but also the overall transversely-isotropic material behavior is confirmed. In contrast to the influencing parameters, the correlation functions vary for a plane strain and a plane stress state.

Im Rahmen dieser Arbeit wird die Anwendbarkeit der nichtlinearen Ausbreitung von Lamb-Wellen auf Grundlage höherharmonischer Generierung unter Verwendung nichtlinearer Stoffgesetze zur Quantifizierung der Werkstoffdegradation infolge mikrostruktureller Schädigungen in geschichteten Faserverbundstrukturen numerisch untersucht. Basierend auf Vorarbeiten werden drei nichtlineare Werkstoffmodelle entwickelt, die gegenüber dem bekanntesten Ansatz nach Murnaghan statt vierzehn nur sechs Parameter benötigen und gegenüber dem Ansatz aus der Biomechanik keiner modifizierten Invarianten bedürfen und Kompressibilität berücksichtigen. Vorgestellte Werkstoffmodelle werden dabei hinsichtlich der notwendigen Materialstabilität für die Verwendung in FEM-Berechnungen untersucht und im Bezug auf die Bedingungen zur Entstehung kumulativ bzw. quasi-kumulativ zweit- und drittharmonischer Wellenmoden in geschichteten Faserverbundstrukturen mit und ohne Entkopplungseffekt der Lamb- und horizontalen Scherwellen theoretisch analysiert. Außerdem werden eine verbesserte Berechnungsmethode der für die Untersuchung höherharmonischer Generierung zu Grunde zu legenden Dispersionsdiagramme und ein erweiterter Ansatz zur Abschätzung des Nichtlinearitätsgrads des verwendeten Werkstoffmodells mittels quasi-kumulativ höherharmonischer Generierung präsentiert. Numerische Ergebnisse der 2D-Simulationen für den entkoppelten Fall sowie die der 3D-Simulationen für den gekoppelten Fall zeigen, dass mittels der vorgestellten Werkstoffmodelle quasi-kumulativ höherharmonische Generierung gemäß der theoretischen Analyse möglich ist und dass der Nichtlinearitätsgrad des verwendeten Werkstoffmodells und somit die dadurch numerisch abgebildete Werkstoffdegradation durch Erzeugung und Analyse quasi-kumulativ zweit- und drittharmonischer Wellenmoden quantifiziert werden kann. Die vorgestellten Werkstoffmodelle bzw. deren Entwicklungskonzept stellen sich als gute Alternative zu bekannten Werkstoffmodellen zur numerischen Abbildung nichtlinearer Wellenausbreitung in geschichteten Kompositen dar und bieten Potential zur Erweiterung.

Estimation of damage position and extent in the inspected structure is among the emerging problems of active structural health monitoring (SHM) with elastic guided waves. Unlike conventional non-destructive testing (NDT) techniques, which assume continuous surface scanning, SHM systems operate only with data from a limited number of sensors. Nevertheless, in the case of isotropic (metal) plate-like structures, computational time-reversal techniques have proved to be effective for estimating in situ the location and size of wave sources and/or local scatterers within the SHM concept. With composite plates, the reconstruction procedure faces additional difficulties associated with the complexity of wave phenomena caused by their lamination and anisotropy. In this article, we present an extension of the time reversal technique for the case of composite laminate plate-like structures. This technique relies on the simulation of the reversed guided waves generated by reciprocal surface loads applied at a sparse set of measurement points of a real sensor network. The proposed implementation is based on the far-field asymptotics for guided waves generated in arbitrarily anisotropic laminate waveguides by a prescribed source, which have been derived from the path integral representations in terms of Green’s matrix for the structures considered. The performance of this approach has been experimentally tested on cross-ply carbon fiber-reinforced plastic plates showing reliable and adequate results for both active (piezoactuators) and passive (artificial defects) source characterization. © The Author(s) 2019.