Material Parameter Identification using Finite Elements and Digital Image Correlation

In der Natur gibt es viele Materialien, die ein unterschiedliches Verhalten aufweisen. Im Bereich der Festkörpermechanik wird dieses Materialverhalten mit Hilfe von konstitutiven Modellen wie Elastizität, Plastizität usw. beschrieben. Heutzutage helfen computergestützte Simulationen bei der Analyse von realen Prozessen, indem sie komplizierte Probleme numerisch lösen. Im Rahmen der Festkörpermechanik werden diese komplizierten Probleme im Allgemeinen mit den weithin bekannten Finite-Elemente-Methoden gelöst. Um die verschiedenen Prozesse korrekt vorhersagen zu können, ist es wichtig, die Materialparameter zu kennen, die zur Charakterisierung des konstitutiven Modells verwendet werden, das zur Modellierung der physikalischen Eigenschaften wie Elastizitätsmodul, Poisson-Zahl usw. dient. In diesem Zusammenhang wird das Verfahren der Materialparameteridentifikation eingesetzt. Das Ziel dieser Arbeit ist es, verschiedene Aspekte der Identifizierung von Materialparametern zu diskutieren. Um das Verhalten eines Materials unter verschiedenen Belastungsbedingungen vorhersagen zu können, ist es wichtig, Materialparameter mit einer gewissen Sicherheit zu identifizieren. Dies ist die Grundlage für diese Forschungsarbeit. Es werden die Grundlagen der Kontinuumsmechanik und die Methode der finiten Elemente zur Lösung der partiellen Differentialgleichung formuliert. Zur Diskretisierung des Problems im zeitlichen Bereich werden Zeitintegratoren hoher Ordnung verwendet. Dies hat den Vorteil höherer Genauigkeit und größerer Flexibilität, so dass das Konzept der Zeitadaptivität genutzt werden kann. Das durch die räumliche und zeitliche Diskretisierung erhaltene nichtlineare Gleichungssystem wird mit dem klassischen Newton-Raphson-Verfahren oder dem Mehrebene-Newton-Verfahren (MLNA) gelöst. Das Grundproblem der Identifikation wird zusammen mit dem Konzept der lokalen Identifizierbarkeit formuliert. Das Konzept der lokalen Identifizierbarkeit ist ein sehr wichtiger Aspekt der Festkörpermechanik, der von Forschern meist ignoriert wird. Der Grund dafür ist unbekannt. Die Vorhersage des Materialverhaltens kann zu sehr fehlerhaften Ergebnissen führen, wenn die identifizierten Materialparameter nicht genau sind. Dies ist einer der Hauptschwerpunkte dieser Arbeit. Bei inhomogenen Verformungen muss die Finite-Elemente-Methode zur Ermittlung der Materialparameter verwendet werden. Wenn zusätzlich zur Anwendung der Finite-Elemente-Methode auch Vollfelddaten mit einem Digital Image Correlation (DIC)-System während der Experimente gemessen werden können, liefert dies eine Vielzahl von Informationen zur genauen Bestimmung der Materialparameter. Die Bestimmung der für die Identifizierung erforderlichen Empfindlichkeiten kann ein langwieriger Prozess sein. Üblicherweise werden Finite-Differenzen-Schemata (auch bekannt als External Numerical Differentiation (END)) verwendet, was ein zeitaufwändiger Prozess ist, wenn viele Materialparameter identifiziert werden müssen. Alternativ können die Empfindlichkeiten mit Hilfe der Internal Numerical Differentiation (IND) bestimmt werden. Dieses Konzept wird anhand von MLNA im Detail erläutert. In dieser Arbeit werden verschiedene Aspekte der Parameteridentifikation anhand mehrerer Beispiele diskutiert. Mehrere einfache Beispiele werden analysiert, um die grundlegenden Probleme bei der Parameteridentifikation zu verstehen. Es kann festgestellt werden, dass bestimmte Qualitätsmaße analysiert werden müssen, um sicherzustellen, dass die identifizierten Parameter innerhalb eines bestimmten Vertrauensbereichs liegen. Schließlich wird der Identifizierungsprozess durchgeführt, um Parameter eines viskoelastischen Modells mit große Deformation des Überspannungstyps zu identifizieren, das für eine Gummiprobe modelliert wurde. An der Gummiprobe wurden verschiedene ratenabhängige biaxiale Experimente und ein biaxiales Multistep-Relaxations-Experiment mit vollständigen Felddaten unter Verwendung eines DIC-Systems durchgeführt. Die Sensitivitäten für Dehnungsmaße und Reaktionskräfte werden dem Optimierer explizit zur Verfügung gestellt. Dies ermöglichte den Vergleich der Berechnungszeit für die Ermittlung der Parameter mit END und IND. Aus den Ergebnissen lässt sich schließen, dass IND schneller ist als END. Diese Forschungsarbeit verdeutlicht die Bedeutung einer korrekten Identifizierung der Materialparameter.In nature, there exists many materials exhibiting different behavior. In the field of Solid Mechanics, these material behaviors are characterized with the help of constitutive models like elasticity, plasticity etc. Nowadays, computer aided simulations assist in analyzing real processes by solving complicated problems numerically. Within the context of Solid Mechanics, these complicated problems are generally solved using the widely known Finite Element Methods. In order to predict correctly the different processes, it is essential to know the material parameters that are used to characterize the constitutive model used in modeling the physical properties like Young’s modulus, Poisson’s ratio etc. To this extent, the process of material parameter identification is used. The aim of the thesis is to discuss several aspects of material parameter identification. In order to predict the behavior of a material under different loading conditions, it is essential to identify material parameters with a certain confidence. This is the foundation of this research work. The basics of continuum mechanics and the method of finite elements to solve the partial differential equation are formulated. High-order time integrators are used to discretize the problem in temporal domain. It has the advantage of higher accuracy and greater flexibility so that concept of time adaptivity can be used. The non-linear system of equations obtained by the spatial and temporal discretization is solved using classical Newton-Raphson method or Multilevel-Newton Algorithm (MLNA). The basic problem of identification along with the concept of local identifiability is formulated. The concept of local identifiability is a very important aspect of Solid Mechanics which is mostly ignored by researchers. The reason for this is unknown. The prediction of material behavior might lead to highly erratic results, if the identified material parameters are not accurate. This is one of the primary focus of this thesis work. For inhomogeneous deformations, finite element method has to be used to identify material parameters. In addition to the usage of finite elements, if full-field data by using a Digital Image Correlation (DIC) system can also be measured during experiments, it provides a great deal of information to identify the material parameters accurately. The determination of sensitivities required for the identification can be a tedious process. Typically, finite difference schemes (also known as External Numerical Differentiation (END)) are used, which is a time-consuming process if there are many material parameters to be identified. Alternatively, the sensitivities can be determined using the Internal Numerical Differentiation (IND). This concept is explained in detail with the help of MLNA. In this thesis, different aspects of parameter identification are discussed with the help of several examples. Several simple examples are analyzed to understand the basic problems in parameter identification. It can be concluded that certain quality measures must be analyzed to ensure that the identified parameters are within a certain confidence. Finally, the identification process is performed to identify parameters of an overstress-type finite strain viscoelastic model, modeled for a rubber specimen. Different rate-dependent biaxial experiments and multistep relaxation biaxial experiment with full field data using a DIC-system were performed on the rubber specimen. The sensitivities for strain measures and reaction forces are explicitly provided to the optimizer. This enabled the comparison of the computational time to identify the parameters using END and IND. From the results, it can be concluded that IND is faster than END. This research work points out the significance of proper material parameter identification

Material parameter identification using finite elements with time-adaptive higher-order time integration and experimental full-field strain information

In this article, we follow a thorough matrix presentation of material parameter identification using a least-square approach, where the model is given by non-linear finite elements, and the experimental data is provided by both force data as well as full-field strain measurement data based on digital image correlation. First, the rigorous concept of semi-discretization for the direct problem is chosen, where—in the first step—the spatial discretization yields a large system of differential-algebraic equation (DAE-system). This is solved using a time-adaptive, high-order, singly diagonally-implicit Runge–Kutta method. Second, to study the fully analytical versus fully numerical determination of the sensitivities, required in a gradient-based optimization scheme, the force determination using the Lagrange-multiplier method and the strain computation must be provided explicitly. The consideration of the strains is necessary to circumvent the influence of rigid body motions occurring in the experimental data. This is done by applying an external strain determination tool which is based on the nodal displacements of the finite element program. Third, we apply the concept of local identifiability on the entire parameter identification procedure and show its influence on the choice of the parameters of the rate-type constitutive model. As a test example, a finite strain viscoelasticity model and biaxial tensile tests applied to a rubber-like material are chosen

Material parameter identification of unidirectional fiber-reinforced composites

In this article, several aspects of material parameter identification are addressed. We compare several methods to identify material parameters of a constitutive model for small strain, linear elastic transverse isotropy based on experimental data of specimens made from composite plates. These approaches range from identifying the five material parameters from purely analytical considerations to the fully numerical identification on the basis of finite elements and various data provided by digital image correlation (DIC). The underlying experimental tests range from purely uniaxial tensile tests with varying fiber orientation to shear and compression tests. A specific measuring instrument has been developed for the latter tests to obtain unique material parameters—motivated by the concept of local identifiability. Besides, we compare the numerical differentiation, which is the common procedure in parameter identification, with the fully analytical derivation of sensitivities within the DIC/FEM approach

Parameter identification of the passive response in arteries

This article discusses the passive response of arteries, with a particular focus on the material parameter identification process of constitutive model of anisotropic hyperelasticity. The arterial wall is composed of three layers: tunica intima, tunica media and tunica adventitia. However, only the media and the adventitia are assumed to be mechanically relevant tissue layers. Thus, it is necessary to determine a set of material parameters for each contributing layer, based on inhomogeneous stress-strain state in an experimental setup. In these tests, tensile and internal pressure loading paths are applied on a human mammary artery, which is embedded in a tank filled with Krebs solution. The artery was proved, in previous works, to be slightly compressible and anisotropic. We draw on the model of Nolan et al. (2014) to identify the material parameters, based on the experimental data provided by contour lines and using digital imaging analysis. The experimental protocol is explained in detail. From the experiments, the axial reaction force and displacement in the radial direction are used to determine the material parameters by using finite element simulations. A particular focus lies on the highly correlated solution between material parameters in the layer, emphasizing the extreme difficulties of a “unique” identification

Parameter identification of the passive response in arteries

This article discusses the passive response of arteries, with a particular focus on the material parameter identification process of constitutive model of anisotropic hyperelasticity. The arterial wall is composed of three layers: tunica intima, tunica media and tunica adventitia. However, only the media and the adventitia are assumed to be mechanically relevant tissue layers. Thus, it is necessary to determine a set of material parameters for each contributing layer, based on inhomogeneous stress-strain state in an experimental setup. In these tests, tensile and internal pressure loading paths are applied on a human mammary artery, which is embedded in a tank filled with Krebs solution. The artery was proved, in previous works, to be slightly compressible and anisotropic. We draw on the model of Nolan et al. (2014) to identify the material parameters, based on the experimental data provided by contour lines and using digital imaging analysis. The experimental protocol is explained in detail. From the experiments, the axial reaction force and displacement in the radial direction are used to determine the material parameters by using finite element simulations. A particular focus lies on the highly correlated solution between material parameters in the layer, emphasizing the extreme difficulties of a “unique” identification