[pt] O problema tratado nesta dissertação é o da modelagem da excitação, propagação e sensoriamento de ondas acústicas de alta frequência em compósitos piezoelétricos laminados. Empregou-se uma formulação de vigas baseada na teoria discreta de Reddy para compósitos laminados. Esta teoria tem como característica o emprego de funções de interpolação de ordem arbitrária para descrever a variação dos campos de deslocamento e do potencial elétrico ao longo da espessura do laminado. A utilização deste tipo de teoria possibilita a representação de campos de deslocamento que variam rapidamente ao longo da espessura, e que na faixa de altas frequências, são representativos daqueles associados com ondas guiadas em compósitos laminados. No domínio da frequência, o sistema de equações que descreve a resposta eletromecânica do laminado piezoelétrico é colocado na forma de uma equação de estado. A técnica de solução utilizada é baseada no método da imersão invariante, também conhecido como método da varredura ou método de Riccati. Para avaliar os limites da faixa de frequências onde diferentes aproximações são válidas, empregou-se uma técnica baseada na comparação entre o espectro de dispersão exato e aquele previsto pela teoria aproximada. Os resultados destas comparações, mostram que utilizando-se o número de funções de interpolação apropriado, a teoria discreta de Reddy é capaz de representar ondas de comprimento até quatro vezes menores que a espessura da viga. Na faixa de baixas frequências, os resultados são comparados com soluções numéricas fornecidas pelo método dos elementos finitos. Também são apresentados resultados que mostram a variação na resposta eletro-mecânica da viga quando descontinuidades são introduzidas nesta, bem como resultados de experimentos que envolvem a utilização de piezoelétricos como atuadores e sensores. / [en] The proposed contribution adresses the problem of modeling high- frequency excitation, propagation, and sensing, of structural waves in composites containing piezoelectric sensors and actuators. The model for the active composite beam proposed here is based on Reddy’s discrete layerwise laminate theory. The displacement and electric potential are interpolated in the through-thickness direction using piecewise linear functions. In the frequency domain, the governing equations are written in a state space form. Comparisons between approximate and exact wave dispersion spectra are performed in order to assess the efficiency of the layerwise theory in a given frequency range. It is shown that by employing the proper interpolation, Reddy s theory is able to describe the dynamic response of the composite at frequencies where the associated wavelengths are of the same order or even shorter than the thickness of the piezoelectric layers. Next, the solution for the state space equation is obtained by employing an algorithm based on a discrete version of the Riccati transformation, It is shown that the algorithm is stable over a wide range of frequencies. In the low frequency range, the analytical results are compared with finite element results. Also are shown the change in the electrical and mechanical response when discontinuites are introduced in the beam, as well as basic experiments involving piezoelectric sensors and actuators.

A análise de compósitos laminados apresenta grandes desafios, pois, diferentemente dos materiais isotrópicos homogêneos, os compósitos laminados são constituídos de materiais heterogêneos e anisotrópicos. Além disso, as distribuições de tensões interlaminares obtidas com as formulações convencionais são descontínuas e imprecisas. Sua melhoria, portanto, é imprescindível para buscar e modelar critérios de falha relacionados às estruturas formadas por compósitos laminados. Diante disso, este trabalho se concentrou no desenvolvimento e implementação computacional de um elemento finito posicional de pórtico plano laminado cuja cinemática é descrita ao longo da espessura do laminado de acordo com a teoria Layerwise. A formulação do elemento considera a não linearidade geométrica, originada pela ocorrência de grandes deslocamentos e rotações, e admite deformações moderadas, em função da lei constitutiva de Saint-Venant-Kirchhoff. O desenvolvimento deste trabalho se iniciou com uma preparação teórica sobre mecânica dos sólidos deformáveis e métodos numéricos para que fossem adquiridos os subsídios teóricos necessários ao desenvolvimento de códigos computacionais, à interpretação dos resultados e à tomada de decisões quando das análises numéricas. A formulação desenvolvida é Lagrangiana total com emprego do método dos elementos finitos baseado em posições. Inicialmente o elemento finito posicional de pórtico plano homogêneo é proposto, uma vez que sua cinemática possibilita uma expansão natural para o caso laminado. Os graus de liberdade são compostos por posições nodais e por vetores generalizados que representam o giro e a variação na altura da seção transversal. A eficiência do elemento é constatada através de análises realizadas em problemas de pórtico sujeitos a grandes deslocamentos e rotações. Os resultados obtidos apresentaram excelente concordância com soluções numéricas e analíticas disponíveis na literatura. Uma expansão natural da cinemática é empregada na formulação do elemento laminado. Os graus de liberdade do elemento são as posições nodais e as componentes de vetores generalizados associados às seções transversais de cada lâmina. Dessa forma, as lâminas têm liberdade para variação de espessura e giro independente das demais, mas com as posições compatibilizadas nas interfaces. Os resultados de análises numéricas realizadas em vários exemplos demonstram a eficiência da formulação proposta, pois as distribuições de deslocamentos e tensões ao longo da espessura do laminado apresentaram excelente concordância com as obtidas a partir de análises numéricas utilizando um elemento finito bidimensional em uma discretização bastante refinada. Os exemplos analisados contemplam problemas com seção laminada fina ou espessa. / The analysis of laminated composites presents challenges because, unlike homogeneous isotropic materials, the laminated composites are made up of heterogeneous and anisotropic materials. Moreover, the distribution of interlaminar stresses obtained with conventional formulations are discontinuous and inaccurate. His improvement is therefore essential to check and modeling failure criteria related to structures formed by laminates. Thus, this work focused on developing and computational implementation of a positional finite element of laminated plane frame whose kinematics is described throughout the thickness of the laminate according to Layerwise theory. The formulation element considers the geometric nonlinearity, caused by the occurrence of large displacements and rotations, and admits moderate deformation, in the constitutive law function of Saint-Venant-Kirchhoff. The development of this work began with a theoretical preparation on mechanics of deformable solids and numerical methods for the acquired of the theoretical support needed for the development of computational codes, interpretation of results and decision-making when of the numerical analyzes. The developed formulation is total Lagrangian with use of the finite element method based on positions. Initially the positional finite element of homogeneous plane frame is proposed, since their kinematic enables a natural expansion for the laminate case. The degrees of freedom are composed of nodal positions and generalized vectors representing the spin and the variation in the height of the cross section. The efficiency of the element is verified through analyzes performed in frame problems subject to large displacements and rotations. The results showed excellent agreement with numerical and analytical solutions available in the literature. A natural expansion of the kinematics is used in the formulation of the laminate element. The degrees of freedom of the element are the nodal positions and components of the generalized vectors associated to cross-sections of each lamina. Thus, the laminas are free for the thickness variation and for independent spin, but with the positions matched in the interfaces. The results of numerical analysis performed in various examples show the effectiveness of the proposed formulation, since the distributions of displacements and stresses through the thickness of the laminate agreed well with those obtained from numerical analysis using a discretization with two-dimensional finite elements in a very refined. The examples discussed include problems with thin or thick laminated section.

A análise de compósitos laminados apresenta grandes desafios, pois, diferentemente dos materiais isotrópicos homogêneos, os compósitos laminados são constituídos de materiais heterogêneos e anisotrópicos. Além disso, as distribuições de tensões interlaminares obtidas com as formulações convencionais são descontínuas e imprecisas. Sua melhoria, portanto, é imprescindível para buscar e modelar critérios de falha relacionados às estruturas formadas por compósitos laminados. Diante disso, este trabalho se concentrou no desenvolvimento e implementação computacional de um elemento finito posicional de pórtico plano laminado cuja cinemática é descrita ao longo da espessura do laminado de acordo com a teoria Layerwise. A formulação do elemento considera a não linearidade geométrica, originada pela ocorrência de grandes deslocamentos e rotações, e admite deformações moderadas, em função da lei constitutiva de Saint-Venant-Kirchhoff. O desenvolvimento deste trabalho se iniciou com uma preparação teórica sobre mecânica dos sólidos deformáveis e métodos numéricos para que fossem adquiridos os subsídios teóricos necessários ao desenvolvimento de códigos computacionais, à interpretação dos resultados e à tomada de decisões quando das análises numéricas. A formulação desenvolvida é Lagrangiana total com emprego do método dos elementos finitos baseado em posições. Inicialmente o elemento finito posicional de pórtico plano homogêneo é proposto, uma vez que sua cinemática possibilita uma expansão natural para o caso laminado. Os graus de liberdade são compostos por posições nodais e por vetores generalizados que representam o giro e a variação na altura da seção transversal. A eficiência do elemento é constatada através de análises realizadas em problemas de pórtico sujeitos a grandes deslocamentos e rotações. Os resultados obtidos apresentaram excelente concordância com soluções numéricas e analíticas disponíveis na literatura. Uma expansão natural da cinemática é empregada na formulação do elemento laminado. Os graus de liberdade do elemento são as posições nodais e as componentes de vetores generalizados associados às seções transversais de cada lâmina. Dessa forma, as lâminas têm liberdade para variação de espessura e giro independente das demais, mas com as posições compatibilizadas nas interfaces. Os resultados de análises numéricas realizadas em vários exemplos demonstram a eficiência da formulação proposta, pois as distribuições de deslocamentos e tensões ao longo da espessura do laminado apresentaram excelente concordância com as obtidas a partir de análises numéricas utilizando um elemento finito bidimensional em uma discretização bastante refinada. Os exemplos analisados contemplam problemas com seção laminada fina ou espessa. / The analysis of laminated composites presents challenges because, unlike homogeneous isotropic materials, the laminated composites are made up of heterogeneous and anisotropic materials. Moreover, the distribution of interlaminar stresses obtained with conventional formulations are discontinuous and inaccurate. His improvement is therefore essential to check and modeling failure criteria related to structures formed by laminates. Thus, this work focused on developing and computational implementation of a positional finite element of laminated plane frame whose kinematics is described throughout the thickness of the laminate according to Layerwise theory. The formulation element considers the geometric nonlinearity, caused by the occurrence of large displacements and rotations, and admits moderate deformation, in the constitutive law function of Saint-Venant-Kirchhoff. The development of this work began with a theoretical preparation on mechanics of deformable solids and numerical methods for the acquired of the theoretical support needed for the development of computational codes, interpretation of results and decision-making when of the numerical analyzes. The developed formulation is total Lagrangian with use of the finite element method based on positions. Initially the positional finite element of homogeneous plane frame is proposed, since their kinematic enables a natural expansion for the laminate case. The degrees of freedom are composed of nodal positions and generalized vectors representing the spin and the variation in the height of the cross section. The efficiency of the element is verified through analyzes performed in frame problems subject to large displacements and rotations. The results showed excellent agreement with numerical and analytical solutions available in the literature. A natural expansion of the kinematics is used in the formulation of the laminate element. The degrees of freedom of the element are the nodal positions and components of the generalized vectors associated to cross-sections of each lamina. Thus, the laminas are free for the thickness variation and for independent spin, but with the positions matched in the interfaces. The results of numerical analysis performed in various examples show the effectiveness of the proposed formulation, since the distributions of displacements and stresses through the thickness of the laminate agreed well with those obtained from numerical analysis using a discretization with two-dimensional finite elements in a very refined. The examples discussed include problems with thin or thick laminated section.

Tese de mestrado integrado. Engenharia Mecânica. Faculdade de Engenharia. Universidade do Porto. 2012

Composite laminates are being more employed as fundamental structures due to its low weight and high stiffness. An example of this innovation is the primary structures of modern aircraft, which are lighter than the material formerly used. To predict the material response as load gradually increases can be quite demanding due to composite\'s complex failure mechanism. Hence superior computational models should be further investigated to precisely predict the mechanical behavior of composite media. This dissertation addresses an extended finite element procedure based on the layerwise displacement theory to simulate purely mode I delamination failure in composite laminates. The present model has the potential to perform structural analyzes in a pre-delaminated structure and also considering progressive failure. The type of element to be employed at the discretion of the model is the rectangular 4-node iso-parametric homogeneous element whose displacement field is approximated based in the layerwise theory. There are four types of degrees of freedom, one displacement in each direction, and one degree of freedom associated to the strong discontinuity. Numerical examples already solved in the bibliography are suggested in this dissertation, which demonstrate the potential of the model developed to accurately simulate pure mode I delamination in case of the investigation here is further elaborated. In addition, one possibility of future development of this dissertation is the modeling of fracture mode I without the need to discretize the cohesive planes as realized in traditional Cohesive Zone Methods. / Compósitos laminados estão sendo mais empregados como estruturas fundamentais devido ao seu baixo peso e alta rigidez. Um exemplo dessa inovação são as estruturas primárias das aeronaves modernas, que são mais leves do que as os materiais empregados antigamente. Prever a resposta do material à medida que a carga aumenta gradualmente pode ser difícil devido ao complexo mecanismo de falha dos compósitos. Portanto, modelos computacionais mais refinados devem ser investigados a fim de se prever um comportamento mecânico mais preciso. Esta dissertação aborda um procedimento de elementos finitos estendido baseado na teoria de deslocamento layerwise para simular falhas de delaminação modo I em laminados compósitos. O modelo abordado tem potencial para realizar análises em uma estrutura prédelaminada além de falha progressiva. O tipo de elemento a ser empregado na discrição do modelo é o isoparamétrico, homogêneo de 4 nós, retangular, e o campo de deslocamento é baseado na teoria layerwise. Existem quatro tipos de graus de liberdade, um deslocamento em cada direção, e um grau de liberdade associado à forte. Sugere-se nesse trabalho, exemplos, que são comparados com a bibliografia, e que apontam que o modelo desenvolvido nesta dissertação tem o potencial de simular o fenômeno de delaminação em modo I com acurácia, caso o estudo nessa dissertação seja estendido. Além disso, uma possibilidade de desenvolvimento futuro desse trabalho é a modelagem da fratura modo I sem a necessidade de discretizar os planos coesivos entre as lâminas, como realizado em métodos coesivos tradicionais.

Composite laminates are being more employed as fundamental structures due to its low weight and high stiffness. An example of this innovation is the primary structures of modern aircraft, which are lighter than the material formerly used. To predict the material response as load gradually increases can be quite demanding due to composite\'s complex failure mechanism. Hence superior computational models should be further investigated to precisely predict the mechanical behavior of composite media. This dissertation addresses an extended finite element procedure based on the layerwise displacement theory to simulate purely mode I delamination failure in composite laminates. The present model has the potential to perform structural analyzes in a pre-delaminated structure and also considering progressive failure. The type of element to be employed at the discretion of the model is the rectangular 4-node iso-parametric homogeneous element whose displacement field is approximated based in the layerwise theory. There are four types of degrees of freedom, one displacement in each direction, and one degree of freedom associated to the strong discontinuity. Numerical examples already solved in the bibliography are suggested in this dissertation, which demonstrate the potential of the model developed to accurately simulate pure mode I delamination in case of the investigation here is further elaborated. In addition, one possibility of future development of this dissertation is the modeling of fracture mode I without the need to discretize the cohesive planes as realized in traditional Cohesive Zone Methods. / Compósitos laminados estão sendo mais empregados como estruturas fundamentais devido ao seu baixo peso e alta rigidez. Um exemplo dessa inovação são as estruturas primárias das aeronaves modernas, que são mais leves do que as os materiais empregados antigamente. Prever a resposta do material à medida que a carga aumenta gradualmente pode ser difícil devido ao complexo mecanismo de falha dos compósitos. Portanto, modelos computacionais mais refinados devem ser investigados a fim de se prever um comportamento mecânico mais preciso. Esta dissertação aborda um procedimento de elementos finitos estendido baseado na teoria de deslocamento layerwise para simular falhas de delaminação modo I em laminados compósitos. O modelo abordado tem potencial para realizar análises em uma estrutura prédelaminada além de falha progressiva. O tipo de elemento a ser empregado na discrição do modelo é o isoparamétrico, homogêneo de 4 nós, retangular, e o campo de deslocamento é baseado na teoria layerwise. Existem quatro tipos de graus de liberdade, um deslocamento em cada direção, e um grau de liberdade associado à forte. Sugere-se nesse trabalho, exemplos, que são comparados com a bibliografia, e que apontam que o modelo desenvolvido nesta dissertação tem o potencial de simular o fenômeno de delaminação em modo I com acurácia, caso o estudo nessa dissertação seja estendido. Além disso, uma possibilidade de desenvolvimento futuro desse trabalho é a modelagem da fratura modo I sem a necessidade de discretizar os planos coesivos entre as lâminas, como realizado em métodos coesivos tradicionais.

L’utilisation des multicouches prend de plus en plus d’ampleur dans le domaine des sciences de l’ingénieur, tout d’abord dans l’industrie, et plus récemment de plus en plus en Génie Civil. Qu’il s’agisse de complexes mêlant des polymères, du bois ou du béton, des efforts importants sont nécessaires pour la modélisation fine de ce type de matériaux. En effet, des phénomènes induits par l’anisotropie et l’hétérogénéité sont associés à ces multi-matériaux : effets de bords, dilatations thermiques différentielles, délaminages/décollements ou non linéarités de type viscosité, endommagement, plasticité dans les couches ou aux interfaces. Parmi les modèles proposés dans la littérature, on trouve par exemple des modèles monocouche équivalente ou de type "Layerwise" (une cinématique par couche). Appartenant à cette deuxième catégorie, des modèles ont été développés depuis quelques années dans le laboratoire Navier et permettent une description suffisamment fine pour aborder les problématiques spécifiques citées plus haut tout en conservant un caractère opératoire certain. En introduisant des efforts d’interfaces comme des efforts généralisés du modèle, ces approches ont montré leur efficacité vis-à-vis de la représentation des détails au niveau inter- et intra-couches. Il est alors aisé de proposer des comportements et des critères d’interfaces et d’être efficace pour la modélisation du délaminage ou décollement, phénomène très présent dans les composites multicouches assemblés et collés. Par conséquent, un programme éléments finis MPFEAP a été développé dans le laboratoire Navier. Le modèle a également été introduit sous la forme d’un User Element dans ABAQUS, dans sa forme la plus simple (interfaces parfaites).Un nouveau model layerwise est proposé dans ce mémoire pour les plaques multicouches, appelé "Statically Compatible Layerwise Stresses with first-order membrane stress approximations per layer in thickness direction" SCLS1. Le modèle est conforme aux équations d’équilibre 3D ainsi qu’aux conditions aux limites de bord libre. En outre, une version raffinée du nouveau modèle est obtenu en introduisant plusieurs couches mathématiques par couche physique. Le nouveau modèle a été mis en œuvre dans une nouvelle version du code éléments finis MPFEAP.En parallèle, un programme d’éléments finis basé sur la théorie Bending-Gradient développée dans le laboratoire Navier est proposé ici. Le modèle est une nouvelle théorie de plaque épaisse chargée hors-plan où les inconnues statiques sont celles de la théorie Love-Kirchhoff, à laquelle six composantes sont ajoutées représentant le gradient du moment de flexion. La théorie Bending- Gradient est obtenue à partir de la théorie Generalized-Reissner: cette dernière implique quinze degrés de liberté cinématiques, huit d’entre eux étant lié uniquement à la déformation de Poisson hors-plan, et donc l’idée principale de la théorie de plaque Bending-Gradient est de simplifier la théorie Generalized-Reissner en réglant ces huit d.o.f. à zéro et de négliger la contribution de la contrainte normale σ33 dans l’équation constitutive du modèle de plaque. Un programme éléments finis appelé BGFEAP a été développé pour la mise en œuvre de l’élément de Bending-Gradient. Un User Element dans Abaqus a été aussi développé pour la théorie Bending-Gradient / The use of multilayer is becoming increasingly important in the field of engineering, first in the industry, and more recently more and more in Civil Engineering. Whether complex blend of polymers, wood or concrete, significant efforts are required for accurate modeling of such materials. Indeed, phenomena induced anisotropy and heterogeneity are associated with these multi-material: edge effects, differential thermal expansion, delamination/detachment or nonlinearities viscosity type damage, plasticity in layers or interfaces. Among the models proposed in the literature, we found for example equivalent monolayer model or of "LayerWise" type (a kinematic per layer). Belonging to the second category, models have been developed in recent years in Navier allow a sufficiently detailed description to address specific issues mentioned above while maintaining a surgical nature. By introducing interface forces as generalized forces of the model, these approaches have demonstrated their effectiveness vis-à-vis the representation of details at inter- and intra-layers. It is then easy to offer behaviors and interfaces criteria and to be effective for modeling delamination or detachment, phenomenom very present in multilayered composites assembled and glued together. Therefore, a finite element program MPFEAP was developed in Navier laboratory. The model was also introduced as a User Element in ABAQUS, in its simplest form (perfect interfaces).A new layerwise model for multilayered plates is proposed in this dissertation, named Statically Compatible Layerwise Stresses with first-order membrane stress approximations per layer in thickness direction SCLS1. The model complies exactly with the 3D equilibrium equations and the free-edge boundary conditions. Also, a refined version of the new model is obtained by introducing several mathematical layers per physical layer. The new model has been implemented in a new version of the in-house finite element code MPFEAP.In parallel, a finite element program based on the Bending-Gradient theory which was developed in Navier laboratory, is proposed here. The model is a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Love-Kirchhoff theory, to which six components are added representing the gradient of the bending moment. The Bending-Gradient theory is obtained from the Generalized-Reissner theory: the Generalized-Reissner theory involves fifteen kinematic degrees of freedom, eight of them being related only to out-of-plane Poisson’s distortion and thus, the main idea of the Bending-Gradient plate theory is to simplify the Generalized-Reissner theory by setting these eight d.o.f. to zero and to neglect the contribution of the normal stress σ33 in the plate model constitutive equation. A finite element program called BGFEAP has been developed for the implementation of the Bending-Gradient element. A User Element in Abaqus was also developed for the Bending-Gradient theory

A hybrid method is presented for the analysis of layers, plates, and multi-layered systems consisting of isotropic and linear elastic materials. The problem is formulated for the general case of a multi-layered system using a total potential energy formulation and employing the layerwise laminate theory of Reddy. A one-dimensional finite element model is used for the analysis of the multi-layered system through its thickness, and integral Fourier transforms are used to obtain the exact solution for the in-plane problem. Explicit expressions are obtained for the fundamental solution of the typical infinite layer, which are applied in the two-dimensional boundary integral equation model to produce the integral representation of the solution. The boundary integral equation model is two-dimensional, displacement-based and assumes piecewise continuous distribution of the displacement components through the system's thickness. The developed model describes the three-dimensional displacement field, the stress field, the strains and the interlaminar stresses over the entire domain of the problem as continuous functions of the position. This detailed three-dimensional analysis is achieved by incorporating only contour integrals. The boundary integral equations are discretized using the boundary element method and a numerical model is developed for the single numerical layer (element). This model is extended to the case of a multilayered system by introducing appropriate continuity conditions at the interfaces between the layers (firmly bonded layers, or separation, slip and friction between the layers). Assembly of the element matrices yields the global system of equations, which can be solved via iterative techniques. In addition, numerical techniques are developed for the evaluation of the boundary and domain integrals involved in the construction of the element matrices. The singular boundary integrals are computed using a special coordinate transformation, along with a subdivision of the boundary element and a transformation of the Gauss points. The domain integrals (regular, singular or near-singular) are transformed to regular definite integrals along the boundary through a semi-analytical approach. The proposed method provides a simple, efficient, and versatile model for a three-dimensional analysis of thick plates or multilayered systems. It can also be used to study plates resting on elastic foundations or plates with internal supports. The proposed method can be applied in an obvious manner to anisotropic materials and vibration problems. / Ph. D.

Σκοπός της παρούσας διδακτορικής διατριβής είναι η ανάπτυξη προτύπων πεπερασμένων στοιχείων ικανών να προσομοιώσουν την μορφική και δυναμική απόκριση ευφυών δοκών με διαστρωματική αποκόλληση. Έμφαση δίνεται στη χρήση πιεζοηλεκτρικών διεγερτών και αισθητήρων. Στη πρώτη ενότητα πραγματοποιούνται πειραματικές μετρήσεις σε δοκούς με αποκόλληση χρησιμοποιώντας διάφορες μεθοδολογίες διέγερσης και δειγματοληψίας, οι οποίες περιλαμβάνουν πιεζοηλεκτρικά στοιχεία (actuators,sensors) εκτός των κλασικών μεθόδων (shaker, accelerometer). Από τις πειραματικές μετρήσεις διαπιστώνονται οι δυνατότητες χρήσης πιεζοηλεκτρικών στοιχείων ενώ ανιχνεύεται και ποσοτικοποιείται η επίδραση της ύπαρξης και έκτασης αποκόλλησης στα μορφικά χαρακτηριστικά των δοκών. Στη συνέχεια διατυπώνονται μοντέλα μεσομηχανικής αποσκοπώντας στη προσομοίωση της ηλεκτρομηχανικής συμπεριφοράς ευφυών δοκών με αποκόλληση. Τα πρότυπα αυτά είναι συζευγμένα ηλεκτρομηχανικά και με τη χρήση κατάλληλου πεπερασμένου στοιχείου δοκού οδηγούν στο ηλεκτρομηχανικό σύστημα το οποίο χρησιμοποιείται για να προβλέψει τη στατική και μορφική απόκριση δοκών με διαστρωματική αποκόλληση. Στην επόμενη ενότητα τα πρότυπα μεσομηχανικής καθώς και το πεπερασμένο στοιχείο επεκτείνονται, έτσι ώστε να επιβάλλονται με φυσικό τρόπο συνθήκες επαφής και περιορισμού της οριζόντιας κίνησης (τριβής) μεταξύ των δύο μερών της αποκόλλησης. Το συζευγμένο μη‐γραμμικό ηλεκτρομηχανικό σύστημα που προκύπτει χρησιμοποιείται για να προσδιορίσει τη μεταβατική απόκριση δοκών με αποκόλληση και επαληθεύεται με πειραματικές μετρήσεις. Τέλος παρουσιάζεται ένα δισδιάστατο πρότυπο πεπερασμένο στοιχείο ευφυούς δοκού με αποκόλληση. Στο πρότυπο αυτό τα μοντέλα μεσομηχανικής επεκτείνονται περιλαμβάνοντας μεταβλητό πεδίο επίπεδων και εγκάρσιων αξονικών μετατοπίσεων, αποτελώντας τις δύο διαστάσεις του μοντέλου ώστε να προβλέψει τη στατική, μορφική και δυναμική απόκριση δοκών μεγάλου πάχους με αποκόλληση σε υψηλές συχνότητες. Χρησιμοποιείται για να προσομοιάσει συμμετρικές κατά το πάχος ιδιομορφές και την κυματική διάδοση υψηλών συχνοτήτων ευφυών δοκών με αποκόλληση. Τέλος τα μοντέλα εφαρμόζονται για τον περαιτέρω προσδιορισμό της βλάβης μέσα από την απόκριση χρόνου‐συχνότητας. / -