Rozprawy doktorskie na temat „Mechanical constitutive law”

Constitutive laws commonly used to model friction stir welding have been evaluated, both qualitatively and quantitatively, and a new application of a constitutive law which can be extended to materials commonly used in FSW is presented. Existing constitutive laws have been classified as path-dependent or path-independent. Path-independent laws have been further classified according to the physical phenomena they capture: strain hardening, strain rate hardening, and/or thermal softening. Path-dependent laws can track gradients in temperature and strain rate characteristic to friction stir welding; however, path-independent laws cannot. None of the path-independent constitutive laws evaluated has been validated over the full range of strain, strain rate, and temperature in friction stir welding. Holding all parameters other than constitutive law constant in a friction stir weld model resulted in temperature differences of up to 21%. Varying locations for maximum temperature difference indicate that the constitutive laws resulted in different temperature profiles. The Sheppard and Wright law is capable of capturing saturation but incapable of capturing strain hardening with errors as large as 57% near yield. The Johnson-Cook law is capable of capturing strain hardening; however, its inability to capture saturation causes over-predictions of stress at large strains with errors as large as 37% near saturation. The Kocks and Mecking model is capable of capturing strain hardening and saturation with errors less than 5% over the entire range of plastic strain. The Sheppard and Wright and Johnson-Cook laws are incapable of capturing transients characteristic of material behavior under interrupted temperature or strain rate. The use of a state variable in the Kocks and Mecking law allows it to predict such transients. Constants for the Kocks and Mecking model for AA 5083, AA 3004, and Inconel 600 were determined from Atlas of Formability data. Constants for AA 5083 and AA 3004 were determined with the traditional Kocks and Mecking model; however, constants for Inconel 600 could not be determined without modification to the model. The temperature and strain rate combinations for Inconel 600 fell into two hardening domains: low temperatures and high strain rates exhibited twinning while high temperatures and low strain rates exhibited slip. An additional master curve was added to the Kocks and Mecking model to account for two hardening mechanisms. The errors for the Kocks and Mecking model predictions are generally within 10% for all materials analyzed.

This thesis deals with determination of representative constitutive model for describing atherosclerotic carotid artery behavior. The first part of the thesis provides brief summary of medical knowledge needed as well as detailed describtion of current experimental methods for determination of mechanical properties of atherosclerotic arteries. The main part is focused on mechanical testing of atheroslecotic carotid arteries. The description of sample preparation, testing device and the experiment itself is involved. Statistical analysis of measured data is done, focusing on comparison of factors potentially influencing mechanical behavior. In conclusion, the FEA analysis on simplified geometry of carotid artery with atheroma is performed. Emphasis is placed on the impact of material model obtained from experiments on equivalent stress in fibrous cap.

Environment assisted fatigue involves plastic deformation and degrading chemical reactions, which occur in a localized region ahead of the crack tip. Basically, transgranular crack growth proceeds by alternating slip processes. In this study, a transgranular fatigue crack growth rate model is developed on the basis of restricted slip reversibility (RSR), where transgranular fatigue crack growth rate is related to plastic deformation accommodation ahead of the crack tip (the product of the cyclic plastic strain range and the plastic zone size). The model is shown to take the form of the Paris equation with a power law exponent of 3 at positive R values. Fatigue crack growth behavior of a 8090 aluminum-lithium alloy has been examined by a series of tests using compact tension (C(T)) specimens with the load axis (a) parallel to the rolling direction (LT specimen), (b) inclined at 15$\sp\circ$ (L + 15$\sp\circ$), (c) inclined at 30$\sp\circ$ (L + 30$\sp\circ$), (d) inclined at 45$\sp\circ$ (L + 45$\sp\circ$) to the rolling direction. It has been found that in the LT, L + 15$\sp\circ$ and L + 30$\sp\circ$ specimens macroscopic cracks propagate along the plane normal to the rolling direction regardless of the deviation of loading directions and the fatigue crack in the L + 45$\sp\circ$ specimen propagates along the plane of specimen symmetry. Fatigue crack growth rate has been found to vary with the specimen orientation with the LT direction exhibiting the best fatigue crack growth resistance. These phenomena are discussed in terms of the crystallographic texture and the highly planar slip behavior of this ally. A revised RSR model is developed for the description of transgranular fatigue crack growth in aluminum-lithium alloys, where the effect of texture is related to a geometric factor for the favorable slip planes. Extension of the RSR model to environment assisted fatigue is also discussed. It is recognized that environmental effects contribute to crack propagation by the formation and rupture of an embrittlement zone in front of the crack tip. By incorporating a corrosion damage zone into the RSR model, fatigue crack growth rate in a deleterious environment is shown to be consists of two components: (i) mechanical fatigue which occurs by partially reversible slip and (ii) environmental enhancement of crack growth that results from the rupture of the embrittlement zone and is directly related to the characteristic dimension of this corrosion damage zone. In addition, fracture kinetics analysis is extended to crack growth behavior which exhibits the positive-negative temperature dependence. A constitutive law is derived from the general rate equation for a two-barrier consecutive system which represents stress corrosion cracking. The transition condition of the positive-negative temperature dependence is discussed and defined in terms of microstructural characteristic quantities (activation energy and work factor) and loading constraints (stress intensity factor and temperature). For the description of plastic deformation, a set of evolutionary rate equations is developed from deformation kinetics theory. Corresponding constitutive equations are derived for the dislocation glide mechanism, glide-plus-climb mechanism and diffusional flow. The operational equations are solved from the governing differential rate equation to determine deformation responses under different loading constraints.

Compared with other mature engineering disciplines, fracture mechanics of concrete is still a developing field and very important for structures like bridges subject to dynamic loading. An historical point of view of what done in the field is provided and then the project is presented. The project presents an application of the Digital Image Correlation (DIC) technique for the detection of cracks at the surface of concrete prisms (500mmx100mmx100mm) subject to flexural loading conditions (Four Point Bending test). The technique provide displacement measurements of the region of interest and from this displacement field information about crack mouth opening (CMOD) are obtained and related to the applied load. The evolution of the fracture process is shown through graphs and graphical maps of the displacement at some step of the loading process. The study shows that it is possible with the DIC system to detect the appearance and evolution of cracks, even before the cracks become visually detectable.

Gegenstand der Dissertation ist die Entwicklung eines Stoffmodells für duktile Salzgesteine, mit dem alle drei Kriechphasen, abhängig von einem inneren Zustandsparameter, beschrieben werden können. Die Modellierung der Schädigungs- bzw. Dilatanzentwicklung und deren Rückkopplung auf das Kriechverhalten ist das Kernstück der Stoffmodellentwicklung. Es wird eine Beziehung abgeleitet, die die Dilatanzentwicklung abhängig vom Manteldruck und der spezifischen Formänderungsarbeit beschreibt. Durch diese Formulierung und deren Verknüpfung mit dem o. g. Zustandsparameter lassen sich tertiäres Kriechen, Kriechbruch, Nachbruchverhalten und Restfestigkeit geschwindigkeitsabhängig beschreiben. Es erfolgte eine Validierung des Modells anhand von Laboruntersuchungen. Weiter wurden Parametersätze für zwei unterschiedliche Steinsalztypen abgeleitet. Anhand von drei realen Problemstellungen konnte gezeigt werden, dass das gemessene In-situ-Verhalten mit dem Modell berechnet werden kann und Prognoseberechnungen zu plausiblen Ergebnissen führen.

Gegenstand der Dissertation ist die Entwicklung eines Stoffmodells für duktile Salzgesteine, mit dem alle drei Kriechphasen, abhängig von einem inneren Zustandsparameter, beschrieben werden können. Die Modellierung der Schädigungs- bzw. Dilatanzentwicklung und deren Rückkopplung auf das Kriechverhalten ist das Kernstück der Stoffmodellentwicklung. Es wird eine Beziehung abgeleitet, die die Dilatanzentwicklung abhängig vom Manteldruck und der spezifischen Formänderungsarbeit beschreibt. Durch diese Formulierung und deren Verknüpfung mit dem o. g. Zustandsparameter lassen sich tertiäres Kriechen, Kriechbruch, Nachbruchverhalten und Restfestigkeit geschwindigkeitsabhängig beschreiben. Es erfolgte eine Validierung des Modells anhand von Laboruntersuchungen. Weiter wurden Parametersätze für zwei unterschiedliche Steinsalztypen abgeleitet. Anhand von drei realen Problemstellungen konnte gezeigt werden, dass das gemessene In-situ-Verhalten mit dem Modell berechnet werden kann und Prognoseberechnungen zu plausiblen Ergebnissen führen.

Due to the advent of varied types of masonry systems a comprehensive failure mechanism of masonry essential for the understanding of its behaviour is impossible to be determined from experimental testing. As masonry is predominantly used in wall structures a biaxial stress state dominates its failure mechanism. Biaxial testing will therefore be necessary for each type of masonry, which is expensive and time consuming. A computational method would be advantageous; however masonry is complex to model which requires advanced computational modelling methods. This thesis has formulated a damage mechanics inspired modelling method and has shown that the method effectively determines the failure mechanisms and deformation characteristics of masonry under biaxial states of loading.

Les méthodes d'apprentissage automatique basées sur les réseaux de neurones offrent de nouvelles perspectives pour la modélisation du comportement mécanique des matériaux. Grâce à leur capacité à servir d'interpolateurs universels de fonctions, ces réseaux sont capables de capturer une grande variété de comportements complexes. Cependant, leur mise en œuvre requiert des ensembles de données volumineux, souvent difficilement accessibles par des moyens expérimentaux. Afin de pallier cette limitation, ce manuscrit introduit différents biais physiques permettant de modéliser le comportement mécanique de matériau, et plus spécifiquement le comportement viscoélastique non linéaire, à partir de données expérimentales restreintes.Les deux principes fondamentaux de la thermodynamique constituent un cadre contraignant pour la formulation de lois de comportement. Il permet de réduire le nombre de données nécessaires à l'entraînement des modèles, tout en renforçant leur robustesse face aux erreurs de mesure.Les réseaux de neurones récurrents, quant à eux, sont particulièrement bien adaptés pour modéliser des comportements dépendant de l'histoire du chargement. Leurs mémoires cachées offrent une analogie intéressante avec les variables internes introduites par le principe de l'état local en mécanique. Cependant, ces réseaux posent des défis en matière d'entraînement et de généralisation. Pour surmonter ces difficultés, un modèle de réseau de neurones à encodage mécanique est proposé. Ce modèle tire parti des variables internes d'un modèle de viscoélasticité linéaire pour encoder l'histoire du matériau, ce qui s'avère suffisant pour modéliser le comportement mécanique non linéaire du matériau.Un des défis majeurs de la modélisation en trois dimensions à partir de données expérimentales réside dans la prise en compte des symétries matérielles pour éviter des essais redondants. Dans le cas des matériaux isotropes, une méthode d'augmentation de données par rotation aléatoire des essais, combinée à un apprentissage par transfert latéral, permet de développer une loi de comportement 3D en utilisant seulement deux types d'essais uniaxiaux. Une formulation thermodynamique respectant intrinsèquement l'isotropie du matériau est proposée, bien que des défis d'entraînement restent à surmonter pour optimiser cette approche
The application of machine learning techniques based on neural networks provides novel insights into the modelling of the mechanical behaviour of materials. These networks are capable of capturing a wide variety of complex behaviours due to their ability to act as universal function approximators. However, the deployment of these techniques requires large datasets, which are often difficult to obtain experimentally. This manuscript introduces various physical biases that enable the modelling of mechanical behaviour, specifically non-linear viscoelastic behaviour, using limited experimental data, thereby addressing this limitation.The two fundamental principles of thermodynamics provide a robust framework for constraining the formulation of constitutive laws. This approach reduces the quantity of data required for model training, while simultaneously improving the models' resilience to measurement errors.Recurrent neural networks, on the other hand, are particularly well-suited for modelling behaviour that depends on the loading history. Their hidden memories mirror the internal variables introduced in mechanics by the local state principle. However, these networks present challenges in terms of training and generalisation. To overcome these difficulties, a neural network model with mechanical encoding is proposed. This model employs the internal variables of a linear viscoelasticity model to encode the material's history, which proves to be sufficient for modelling its non-linear mechanical behaviour.One of the most significant challenges in three-dimensional modelling from experimental data is the incorporation of material symmetries in order to avoid the need for superfluous testing. For isotropic materials, a method of increasing the data set by randomly rotating the tests, combined with lateral transfer learning, enables the development of a three-dimensional constitutive law using only two types of uniaxial test. A thermodynamically consistent formulation that inherently preserves the material's isotropy is proposed; however, challenges related to training remain to be addressed in order to optimise this approach

Thesis (M. S.)--Mechanical Engineering, Georgia Institute of Technology, 2005.
David N. Ku, Committee Chair ; Alexander Rachev, Committee Co-Chair ; Elliot L. Chaikof, Committee Member. Includes bibliographical references.

In this study, two major internal erosion initiation processes, suffusion and concentrated leak mechanisms, which lead to both defect formation in a dam’s body and its foundation and high throughflow in dams subjected to internal erosion were studied. This understanding has the potential to facilitate numerical modelling and expedite dam safety assessment studies.  The throughflow properties of coarse rockfill material were studied by; analysing filed pump test data, performing extensive laboratory experiments with a large-scale apparatus and numerically simulating the three-dimensional flow through coarse rock materials, replicating the material used in the laboratory experiments. Results from the tests demonstrate that the parameters of the nonlinear momentum equation of the flow depend on the Reynolds number for pore Reynolds numbers lower than 60000.  Numerical studies were also carried out to conduct numerical experiments. By applying a Lagrangian particle tracking method, a model for estimating the lengths of the flow channels in the porous media was developed.  The shear forces exerted on the coarse particles in the porous media were found to be significantly dependent on the inertial forces of the flow. Suffusion and concentrated leak mechanisms were also studied by means of laboratory experiments to develop a theoretical framework for continuum-based numerical modelling. An erosion apparatus was designed and constructed with the capability of applying hydraulic and mechanical loading. Results were then used to develop constitutive laws of the soil erosion as a function of the applied hydromechanical load for both suffusion and concentrated leak mechanisms. Both the initiation and mass removal rate of were found to be dependent on the soil in-situ stresses. A three-dimensional electrical-resistivity-based tomography method was also adopted for the internal erosion apparatus and was found to be successful in visualising the porosity evolution due to suffusion.

QC 20161006

In heavily settled areas, deformations induced by the tunnel excavation may cause serious damage to nearby structures. In this study it is aimed to model ground deformations induced by main tunnels and connection tunnels excavations as well as groundwater drainage. Therefore, it is necessary to study effective means of controlling tunnel induced deformations. The main parameters affecting the failure and deformation state of the soil around a circular underground opening are the physical characteristics of the soil, the diameter of the opening, and the support pressure. During the construction stage of Necatibey Station of KizilayÇ
ayyolu metro line (Ankara, Turkey), challenging ground conditions involving highly heterogeneous and locally water saturated foundation soils have been encountered. Possibility of damage at the surface and/or on the underground structures can be estimated using finite difference method (FDM) of analysis. In this study, two geophysical methods namely Electrical Resistivity Imaging (ERI) and Ground Penetrating Radar (GPR) were utilized to distinguish soil types at the study area. By correlating these geophysical survey results with the boring v logs, 3-Dimensional soil profile was revealed at the study area to build up a basis for numerical models. 3-Dimensional (3D) FDM analyses were conducted to assess tunneling induced deformations, along with movements around shallow soft ground main tunnels and connection tunnels. During sequential excavations, temporary and permanent shotcrete lining was also simulated. The soil behavior is assumed to be governed by an elastic-perfectly plastic constitutive relation based on the Mohr&ndash
Coulomb criterion. The computed deformations around these openings have been compared with the in-situ measurements. The results of the study revealed that the 3-D elasto-plastic analyses yield comparably good correlation with the in-situ measurements. Also, in this study, the effects of main tunnels excavations on each other and the effects of connection tunnels excavations on main tunnels were identified in terms of ground deformations. In order to simulate induced surface settlement due to groundwater withdrawal at the site 3-D fully coupled (fluidmechanical) numerical models were run using different time durations. The model studies revealed that deformations monitored at the ground surface are directly related with the tunnel construction practice. Pumping groundwater has very little or no effect on the measured deformations.

Nous présentons dans ce mémoire l'influence que peuvent avoir les écoulements de fluide au sein de la matrice rocheuse fracturée, laquelle est sujette aux variations locales ou régionales de l'état de contrainte in situ. Du fait de l'augmentation de la pression de pore, la longueur et l'ouverture de la (les) fracture(s) peuvent subir des variations significatives et conduire à la formation de chemins préférentiels pour l'écoulement du fluide dans le milieu géologique. Les modèles théorique et numérique évoqués ici sont des modèles de comportement hydro-mécanique pour le milieu poreux saturé en présence d'une seule phase fluide. La méthode des éléments finis étendue (XFEM) est utilisée afin de modéliser la dynamique des fractures ainsi que les écoulements de fluide dans la matrice rocheuse fracturée, sans être tributaire de la dépendance au maillage. Ainsi, nous considérons: (i) qu'il existe une pression fluide induite par l'écoulement au sein de la fracture, (ii) que la dynamique de la fracture est gérée grâce à un modèle de zone cohésive en supposant un chemin de propagation prédéfini, et (iii) que des échanges entre la fracture et la matrice poreuse peuvent se produire. Ce dernier aspect sera pris en compte en introduisant, dans la formulation du problème couplé, un champ de multiplicateur de Lagrange. Ce champ résulte de la dualisation de la condition d'égalité entre la pression de pore et de la pression de fluide au niveau des parois de la fracture. Afin de respecter les contraintes liées à XFEM, nous avons choisi d'introduire dans la formulation une loi cohésive non-régularisée de type Talon-Curnier. Ce type de loi est capable de gérer la propagation et/ou la refermeture de la fracture. Le modèle HM-XFEM a été validé à partir des solutions analytiques du modèle 2D de fracture KGD, et ce, pour différents régimes de propagation. Nous avons ensuite appliqué le modèle HM-XFEM au cas d'un réseau de fractures non connectées entre elles et évoluant sur des chemins de propagation prédéfinis, afin d'analyser comment les fractures d'un réseau peuvent influer les unes sur les autres lorsqu'elles sont soumises à un écoulement. En particulier, une étude paramétrique a été menée afin de montrer l'influence que peuvent avoir la viscosité, le débit d'injection et l'écartement entre les fractures sur leur propagation. Une attention particulière sera porté à l'évolution du stress-shadowing effect (i.e. modification de l'état de contrainte due à l'effet d'interaction entre les fractures)
In the present work, we address the issue of groundwater flow in the fractured porous media submitted to local or regional stress-state variations. Due to the increasing pore fluid pressure, the length and aperture distribution of the fractures are modified resulting in the formation of preferential flow channels within the geological formation. The numerical approach proposed is a fully coupled hydro-poro-mechanical model in saturated conditions involving single-phase flow both in fractures and in the porous matrix. The extended finite element method (XFEM) is employed for modeling fracture dynamics and flow calculation for fracture which do not lie on the mesh but cross through the elements. In this study: (i) we consider the pressure build up generated by fluid flow inside and through the fracture, (ii) the fracture dynamics by using a cohesive zone model (CZM) on pre-existing propagation path and (iii) fluid exchanges may occur in between fractures and porous medium. The last specification of the HM-XFEM model is taken into account through the introduction of a Lagrange multiplier field along the fracture path. These fields are the result of the dualised condition of pressure continuity between the pore pressure and the fluid pressure inside the fracture. As a function of the Lagrange multiplier value, both permeable and impervious fractures can be considered. The cohesive law employed is a non-regularized-type cohesive law to ensure propagation and eventually closure of the fracture. Validation of the model has been conducted by means of the well-known KGD fracture model when different propagation regimes are considered. We applied the HM-XFEM model to the case of multi-stage fracture network stimulated by the injection of incompressible fluid at constant rate. Fractures are not connected to each other and evolve on pre-existing propagation paths. We aim at appreciating the influence of the fluid viscosity, the injection rate and spacing between each fracture, on the fracture propagation. A peculiar attention is paid to the stress-shadowing effect (i.e. interaction between fractures)

La présente étude vise à comprendre le comportement chimico-hydro-mécanique des argiles raides à travers deux formations géologiques, l'argile de Boom et les argiles yprésiennes, qui sont censées être des formations hôtes potentielles pour le stockage de déchets radioactifs en Belgique. Le comportement volumique a été étudié tant à l'état intact qu'à l'état reconstitué, et sous différentes conditions : K0 et isotrope, sous des boucles de chargement - déchargement. Les résultats obtenus montrent que le comportement volumique de ces argiles est gouverné par la compétition entre l'effet physico-chimique et l'effet mécanique, caractérisée par une contrainte seuil qui correspond à la contrainte de gonflement en termes de changements de structure. Une loi de comportement volumique a été ainsi développée afin de décrire cet aspect. La perméabilité a été déterminée, comparée avec les résultats dans la littérature et corrélée avec les paramètres comme l'indice des vides. La variation de la perméabilité avec la profondeur ont mis en évidence le rôle déterminant des macro-pores dans le transfert des fluides. Le comportement volumique et la perméabilité des argiles de Boom et yprésiennes intactes sont aussi influencés par la variation de la composition chimique de l'eau de pore, qui modifie la double couche diffuse et favorise l'agrégation des particules argileuses. Les caractéristiques élastiques, la surface de charge et l'enveloppe de rupture ont été identifiées pour le comportement déviatorique des argiles de Boom et yprésiennes. Un modèle élasto-plastique conceptuel a été développé permettant de tenir compte des effets du gonflement et de la compétition entre l'effet mécanique et l'effet physico-chimique

Les formations argileuses profondes telles que celles du Callovo-Oxfordien (COx) sont une des options actuellement envisagées pour le stockage des déchets radioactifs en France par l’Agence nationale pour la gestion des déchets radioactifs. Les observations (déformation, pression interstitielle, …) menées à l’échelle des ouvrages in situ du laboratoire de recherche souterrain de Meuse/Haute-Marne ont, cependant, révélé un comportement hydromécanique complexe avec une fracturation induite fortement anisotrope que les modèles constitutifs classiques ont du mal à appréhender. Le travail de cette thèse s’est focalisé sur la formation des zones de fractures connectées et discrètes induites par l’excavation et leurs conséquences sur les propriétés de transports. Sur la base des résultats expérimentaux et des modèles rhéologiques récents pour cette roche, il a été proposé un modèle macroscopique anisotrope, élastoplastique et endommageable qui décrit les mécanismes clés du comportement à court terme (pré et post-pic) de l’argilite du COx. Ce modèle a été implémenté dans Comsol Multiphysics®. L’application du modèle proposé a été menée sur la galerie GCS du LS M/HM par de simulations hydromécaniques en conditions saturées. Les résultats numériques ont été confrontés avec succès aux observations et mesures in situ en termes, notamment, de distribution de pression de pore et d’extension de ZFC/ZFD (approchée par les zones plastiques en post-pic et pré-pic, respectivement) et de convergences horizontale et verticale. Enfin, le caractère opérationnel du modèle pour des applications de modélisation thermo-hydro-mécanique d’alvéoles HA (projet Cigéo) a été étudié sur une géométrie réduite à un micro-tunnel d’alvéoles
Deep clay formations such as Callovo-Oxfordian (COx) are one of the options currently being considered for the storage of radioactive waste disposal in France by the National Radioactive Waste Management Agency. Observations (deformation, pore pressure, ...) carried out on the scale of in situ structures at the Meuse/Haute-Marne underground research laboratory have, however, revealed complex hydromechanical behaviour with highly anisotropic induced fracturing that is difficult to capture and to reproduce using classical constitutive models. The work of this thesis focused on the formation of the connected fractured zone and discrete fractured zone induced by excavation and their consequences on transport properties. Based on experimental results and the recently developed rheological models of this rock, a macroscopic anisotropic, elastoplastic and damage constitutive model was proposed that describes the key mechanisms of the short-term (pre- and post-peak) behaviour of COx claystone. This model was implemented in Comsol Multiphysics®. The application of the proposed model was carried out on the GCS gallery of the LS M/HM by hydromechanical simulations under saturated conditions. The numerical results were successfully compared with in situ observations and measurements in terms of, in particular, pore pressure distribution, ZFC/ZFD extension (approached by the plastic zones in post-peak and pre-peak, respectively) and horizontal and vertical convergences. Finally, the operational character of the model for thermo-hydro-mechanical modelling applications of HA cells (Cigéo project) was investigated on a geometry reduced to a micro-tunnel of cells

This study investigated the performance of orthotropic constitutive laws describing the passive mechanical behaviour of the myocardium. The performance was validated against simple shear experiments of pig hearts which were available from earlier studies. First, a homogeneous deformation model was developed which captured the main features of the deformation process. This served as the basis for a comparative study between three phenomenological material laws that had been published in the literature. Two of these laws exhibited certain limitations and two further constitutive laws were therefore developed that removed these limitations. Thus, five material laws were investigated in terms of their performance to fit the given experimental data by reducing a least-square objective function between the experimental and model data. Furthermore the consistency of the material parameters amongst experiments was investigated. As part of this study, a modified least-squares objective function was developed that decreased the computational time involved by about two orders of magnitude with comparable error. Second, the assumption of a homogeneous deformation of simple shear was removed and the parameters were estimated using a finite element environment using an inverse estimation technique and therefore fulfilling the equations of motion that underpin continuum mechanics. It was found that the material parameters of all laws were in the same range compared to those obtained from the homogeneous study. Relaxing the homogeneous assumptions slightly reduced the objective function error although the computational time increased by three orders of magnitude. Third, the experimental protocol of six simple shear modes was supplemented with three uniaxial deformations modes. The material parameters for the same constitutive relations were estimated. It was possible to show that the material parameters that were associated with shear strain were very similar to those obtained from the simple shear study. The axial material parameters, however, were considerably different. Finally, since it is recognised that phenomenological material laws do not provide insight into the underlying micro-structural mechanisms, the framework for a multi-scale constitutive relation was developed. This is based on multi-scale images of rat myocardium.