Eshelby-type inclusion problems of an infinite or a finite homogeneous isotropic elastic body containing an arbitrary-shape inclusion prescribed with an eigenstrain and an eigenstrain gradient are analytically solved. The solutions are based on a simplified strain gradient elasticity theory (SSGET) that includes one material length scale parameter in addition to two classical elastic constants. For the infinite-domain inclusion problem, the Eshelby tensor is derived in a general form by using the Green’s function in the SSGET. This Eshelby tensor captures the inclusion size effect and recovers the classical Eshelby tensor when the strain gradient effect is ignored. By applying the general form, the explicit expressions of the Eshelby tensor for the special cases of a spherical inclusion, a cylindrical inclusion of infinite length and an ellipsoidal inclusion are obtained. Also, the volume average of the new Eshelby tensor over the inclusion in each case is analytically derived. It is quantitatively shown that the new Eshelby tensor and its average can explain the inclusion size effect, unlike its counterpart based on classical elasticity. To solve the finite-domain inclusion problem, an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET are proposed and utilized. The solution for the disturbed displacement field incorporates the boundary effect and recovers that for the infinite-domain inclusion problem. The problem of a spherical inclusion embedded concentrically in a finite spherical body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. It is demonstrated through numerical results that the newly obtained Eshelby tensor can capture the inclusion size and boundary effects, unlike existing ones. Finally, a homogenization method is developed to predict the effective elastic properties of a heterogeneous material using the SSGET. An effective elastic stiffness tensor is analytically derived for the heterogeneous material by applying the Mori-Tanaka and Eshelby’s equivalent inclusion methods. This tensor depends on the inhomogeneity size, unlike what is predicted by existing homogenization methods based on classical elasticity. Numerical results for a two-phase composite reveal that the composite becomes stiffer when the inhomogeneities get smaller.

Σκοπός της παρούσας διδακτορικής διατριβής είναι η ανάπτυξη μεθοδολογίας συνοριακών στοιχείων για την αριθμητική επίλυση τρισδιάστατων (3-D) στατικών προβλημάτων στα πλαίσια μιας θεωρίας βαθμοελαστικότητας, που στηρίζεται σε μια απλουστευμένης μορφής της θεωρίας του Mindlin και διατυπώθηκε από τους Vardoulakis and Sulem, η οποία λαμβάνει υπόψη και την επιφανειακή ενέργεια, και από τους Aifantis και συνεργάτες. Η διδακτορική διατριβή αποτελείται από δύο ενότητες. Στην πρώτη ενότητα (κεφάλαια 1 και 2) γίνεται μία πλήρης ανασκόπηση της βιβλιογραφίας ως προς τις θεωρίες βαθμοελαστικότητας και στη συνέχεια, περιγράφεται διεξοδικά η παρούσα θεωρία βαθμοελαστικότητας με επιφανειακή ενέργεια. Στη δεύτερη ενότητα παρουσιάζεται η μέθοδος των Συνοριακών Στοιχείων (ΜΣΣ) όπως αυτή εφαρμόζεται για την επίλυση τρισδιάστατων και αξονοσυμμετρικών βαθμοελαστικών προβλημάτων, αντίστοιχα. Η ΜΣΣ βασίζεται στη διατύπωση των ολοκληρωτικών εξισώσεων των βαθμοελαστικών προβλημάτων. Οι άγνωστοι των ολοκληρωτικών εξισώσεων είναι οι συνοριακές τιμές του βασικού πεδίου των μεταβλητών και οι παράγωγοί τους, που για τη βαθμοελαστικότητα είναι τα διανύσματα των μετατοπίσεων, των βαθμίδων τω μετατοπίσεων και τα διανύσματα των επιφανειακών τάσεων. Η προσέγγιση των συναρτήσεων αυτών πάνω στο σύνορο γίνεται με τη βοήθεια συναρτήσεων παρεμβολής από τις αντίστοιχες τιμές τους σε έναν επιλεγμένο αριθμό κόμβων. Η ταχύτητα και η ακρίβεια της ΜΣΣ κατά την εφαρμογή της επηρεάζεται σημαντικά από την ταχύτητα και την ακρίβεια του υπολογισμού των ιδιόμορφων και υπερ-ιδιόμορφων ολοκληρωμάτων. Στην παρούσα διατριβή τα ιδιόμορφα και υπερ-ιδιόμορφα ολοκληρώματα υπολογίζονται με τη χρήση τεχνικών ιδιόμορφης και υπερ-ιδιόμορφης ολοκλήρωσης (Guiggiani (1992) και Huber et al. (1993)) αντίστοιχα. Στα πλαίσια της παρούσας διδακτορικής διατριβής κατασκευάστηκε αλγόριθμος που επιλύει τρισδιάστατα στατικά προβλήματα βαθμοελαστικότητας καθώς και αλγόριθμος που επιλύει στατικά βαθμοελαστικά προβλήματα με αξονική συμμετρία. Στο τέλος κάθε κεφαλαίου, επιλύονται αντίστοιχα στατικά βαθμοελαστικά προβλήματα με ή χωρίς να λαμβάνεται υπόψη η επιφανειακή ενέργεια και με γνωστές αναλυτικές λύσεις. Τα αριθμητικά αποτελέσματα των παραπάνω προβλημάτων συγκρίνονται με τα αντίστοιχα αναλυτικά. Τέλος, γίνεται μία ανακεφαλαίωση της διδακτορικής διατριβής και διατυπώνονται προτάσεις για μελλοντική έρευνα. / In the present Doctoral Thesis a boundary element methodology (BEM) is developed in order to solve numerically 3-D and axis-symmetric static gradient elastic problems. Microstructural effects on the macroscopic behavior of the considered materials have been taken into account by means of a simple strain gradient theory with surface energy obtained as a special case of the general one due to Mindlin, proposed by Vardoulakis and Sulem. All possible boundary conditions (classical and non-classical) have been determined with the aid of a variational statement of the problem. The fundamental solution of the gradient elastic with surface energy has been explicitly determined and used to establish the boundary integral representation of the solution of the problem with the aid of the reciprocal identity, specifically constructed for this gradient elastic with surface energy case. The boundary integral representation consists of one equation for the dispalcement and another one for its normal derivative. Also, the integral forms of the gradient of displacement as well as the Cauchy, relative, double and total stresses in the interior of the gradient elastic body have been derived and presented. The numerical implementation of the integral equations is accomplished with the aid of quadratic isoparametric line (axis-symmetry case) and surface (3-D case) boundary elements. The computation of the singular and hyper-singular integrals involved is done with the aid of highly accurate advanced algorithms.

Size-dependent material responses observed at fine length scales are receiving growing attention due to the need in the modeling of very small sized mechanical structures. The conventional continuum theories do not suffice for accurate descriptions of the exact material behaviors in the fine-scale regime due to the lack of inherent material lengths. A number of new theories/models have been propounded so far to interpret such novel phenomena. In this dissertation a few enriched-continuum theories - the adhesive contact mechanics, surface elasticity and strain gradient elasticity - are employed to study the mechanical behaviors of a semi-infinite solid induced by the boundary forces. A unified treatment of axisymmetric adhesive contact problems is developed using the harmonic functions. The generalized solution applies to the adhesive contact problems involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile, and it links existing solutions/models for axisymmetric non-adhesive and adhesive contact problems like the Hertz solution, Sneddon's solution, the JKR model, the DMT model and the M-D model. The generalized Boussinesq and Flamant problems are examined in the context of the surface elasticity of Gurtin and Murdoch (1975, 1978), which treats the surface as a negligibly thin membrane with material properties differing from those of the bulk. Analytical solution is derived based on integral transforms and use of potential functions. The newly derived solution applies to the problems of an elastic half-space (half-plane as well) subjected to prescribed surface tractions with consideration of surface effects. The newly derived results exhibit substantial deviations from the classical predictions near the loading points and converge to the classical ones at a distance far away from those points. The size-dependency of material responses is clearly demonstrated and material hardening effects are predicted. The half-space contact problems are also studied using the simplified strain gradient elasticity theory which incorporates material microstructural effects. The solution is obtained by taking advantage of the displacement functions of Mindlin (1964) and integral transforms. Significant discrepancy between the current and the classical solutions is seen to exist in the immediate vicinity of the loading area. The discontinuity and singularity exist in classical solution are removed, and the stress and displacement components change smoothly through the solid body.

[pt] A modelagem matemática de microdispositivos, em que estrutura e microestrutura têm aproximadamente a mesma escala de magnitude, assim como de macroestruturas de natureza predominantemente granular ou cristalina, requer uma abordagem não-local de deformações e tensões. Há mais de cem anos os irmãos Cosserat já tinham desenvolvido uma teoria de grãos rígidos. No entanto, e sem detrimento de desenvolvimentos devidos a Toupin e outros pesquisadores, os trabalhos de Mindlin na década de 1960 podem ser considerados a base da chamada teoria gradiente de deformações, que se tornou recentemente objeto de um grande número de investigações analíticas e experimentais, motivadas pelo desenvolvimento de novos materiais estruturais e do crescente uso de dispositivos micro- e nanomecânicos na indústria. Mais recentemente, Aifantis e colaboradores conseguiram desenvolver uma teoria gradiente de deformações mais simplificada, com base somente em duas constantes elásticas adicionais, representativas de comprimentos característicos relacionados às energias de deformação superficial e volumétrica. Uma série de trabalhos recentes desenvolvidos por Beskos e colaboradores estendeu o campo de aplicações da proposta inicial de Aifantis e introduziu uma solução fundamental que de fato remonta aos trabalhos de Mindlin. A equipe de pesquisa de Beskos propôs as primeiras implementações 2D e 3D de elementos de contorno para análises de elasticidade gradiente tanto estáticas quanto no domínio da freqüência, inclusive para problemas da mecânica da fratura. Desde o tempo de Toupin e Mindlin procura-se estabelecer uma base variacional da teoria e uma formulação consistente das condições de contorno cinemáticas e de equilíbrio, o que parece ter tido êxito com os recentes trabalhos de Amanatidou e Aravas. Esta dissertação faz uma revisão da teoria gradiente da deformações e apresenta um estudo didático do problema mais simples que se possa conceber, que é o de uma barra sob diferentes tipos de ações axiais (Aifantis, Beskos). A solução fundamental para problemas 2D e 3D também é apresentada e estudada, tanto em termos de forças pontuais aplicadas, para uma implementação em termos de elementos de contorno, quanto de desenvolvimentos polinomiais (no caso estático), para implementação em termos de elementos finitos. Mostra-se que a teoria gradiente de deformação de Aifantis é adequada a uma formulação no contexto do potencial de Hellinger-Reissner, o que possibilita implementações híbridas de elementos finitos e de contorno. O presente trabalho de pesquisa objetiva o estudo do estado da arte no tema, com uma abordagem dos principais problemas de implementação computacional, inclusive em termos das integrais singulares que surgem. O desenvolvimento completo de programas de análise de elementos híbridos finitos e de contorno, para problemas estáticos e dinâmicos, está planejado para uma tese de doutorado em futuro próximo. / [en] The mathematical modeling of micro-devices in which structure and the microstructure are about the same scale of magnitude, as well as of macrostructure of markedly granular or crystal nature (microcomposites), demands a nonlocal approach for strains and stresses. More than one hundred years ago the Cosserat brothers had already developed a theory for rigid grains. However, and in no detriment due to Toupin and other researchers, Mindlin s work in the 1960s may be accounted the basis of the so-called strain gradient theory, which has recently become the subject of a large number of analytical and experimental investigations motivated by the development of news structural materials together with the increasing use of micro and nano-mechanical devices in the industry. More recently, Aifantis and coworkers managed to develop a simplified strain gradient theory based only on two additional elasticity constants that are representative of material lengths related to surface and volumetric strain energy. A series of very recent works done by Beskos and collaborators extended the field of applications of Aifantis propositions and introduced a fundamental solution that actually remounts to developments already laid down by Mindlin. Beskos workgroup may be regarded as the proponent of the first of the first boundary element 2D and 3D implementations on the subject for both statics and frequency-domain analyses, also including crack problems. Since Toupin and Mindlin`s time, investigations have been under development to establish the variational basis of the theory and to consistently formulate equilibrium and kinematic boundary conditions established by Amanatidou and Aravas. This dissertation makes a revision of the gradient strain elasticity theory and presents a didactic study of the simplest problem that can be conceived, i.e., a bar under different axial actions (Aifantis, Beskos). The fundamental solution for 2D and 3D problems is also presented and studied for an elastic medium submitted to a point force, for boundary methods developments, as well as submitted to polynomial stress fields (for static problems), as in the hybrid finite element method. It is shown that Aifantis strain gradient theory may be developed in the context of the Hellinger-Reissner potential, for the sake of hybrid finite and boundary element implementations. Goal of the present research work is as a detailed study of state art of the theme, which comprises an investigation of the singular integrals one must deal with in a computational implementation. The complete computational development for static and dynamic hybrid boundary/finite analyses is planned for a future doctoral thesis.

Through enrichment of the elastic potential by the second-order gradient of deformation, gradient elasticity formulations are capable of taking nonlocal effects into account. Moreover, geometry-induced singularities, which may appear when using classical elasticity formulations, disappear due to the higher regularity of the solution. In this contribution, a mixed finite element discretization for finite strain gradient elasticity is investigated, in which instead of the displacements, the first-order gradient of the displacements is the solution variable. Thus, the C1 continuity condition of displacement-based finite elements for gradient elasticity is relaxed to C0. Contrary to existing mixed approaches, the proposed approach incorporates a rot-free constraint, through which the displacements are decoupled from the problem. This has the advantage of a reduction of the number of solution variables. Furthermore, the fulfillment of mathematical stability conditions is shown for the corresponding small strain setting. Numerical examples verify convergence in two and three dimensions and reveal a reduced computing cost compared to competitive formulations. Additionally, the gradient elasticity features of avoiding singularities and modeling size effects are demonstrated.

During this thesis, a Boundary Element Method (BEM) has been developed for the solution of static linear elastic problems with microstructural effects in two (2D) and three dimensions (3D).The second simplified form of Mindlin's Generalized Gradient Elasticity Theory (Mindlin's Form II)has been employed. The fundamental solution of the 4th order partial differential equation, that describes the aforementioned theory, has been derived and the integral equations that govern Mindlin's Form II Gradient Elasticity Theory have been obtained. Furthermore, a BEM formulation has been developed and specific Boundary Value Problems (BVPs) were solved numerically and compared with the corresponding analytical solutions to verify the correctness of the formulation and demonstrate its accuracy. Moreover, two new partially discontinuous boundary elements with variable order of singularity, a line and a quadrilateral element, have been developed for the solution of fracture mechanics problems. The calculation of the unknown fields near the crack tip (or front) demanded the use of elements that could interpolate abruptly varying fields. The new elements were created in a way that their interpolation functions were no longer quadratic but their behavior depended on the order of singularity of each field. Finally, the Stress Intensity Factor (SIF) of the crack has been calculated with high accuracy, based on the element's nodal traction values. Static fracture mechanics problems for Mode I and Mixed Mode (I & II) cracks, have been solved in 2D and 3D and the corresponding SIFs have been obtained, in the context of both classical and Form II Gradient Elasticity theories. / Κατά τη διάρκεια της παρούσας διδακτορικής διατριβής, αναπτύχθηκε Μέθοδος Συνοριακών Στοιχείων (ΜΣΣ) για την επίλυση στατικών προβλημάτων ελαστικότητας με επιδράσεις μικροδομής σε δύο και τρεις διαστάσεις. Η θεωρία στην οποία εφαρμόστηκε η ΜΣΣ είναι η δεύτερη απλοποιημένη μορφή της γενικευμένης θεωρίας ελαστικότητας του Mindlin. Για τη συγκεκριμένη θεωρία ευρέθη η θεμελιώδης της μερικής διαφορικής εξίσωσης 4ης τάξης που περιγράφει τη συμπεριφορά των συγκεκριμένων υλικών και κατασκευών. Επίσης διατυπώθηκε η ολοκληρωτική εξίσωση των αντίστοιχων προβλημάτων και έγινε η αριθμητική εφαρμογή μέσω της ΜΣΣ. Επιλύθηκα αριθμητικά συγκεκριμένα προβλήματα συνοριακών τιμών και έγινε σύγκριση των αποτελεσμάτων με τα αντίστοιχα θεωρητικά. Στη συνέχεια αναπτύχθηκαν δύο νέα ασυνεχή στοιχεία μεταβλητής τάξης ιδιομορφίας με σκοπό την επίλυση προβλημάτων θραυστομηχανικής, ένα για δισδιάστατα και ένα για τρισδιάστατα προβλήματα. Συγκεκριμένα, επειδή τα πεδία των τάσεων απειρίζονται στην κορυφή μιας ρωγμής και περιέχουν συγκεκριμένων τύπων ιδιομορφίες δεν ήταν δυνατός ο ακριβής υπολογισμός των πεδίων αυτών κοντά στη ρωγμή με τα συνήθη τετραγωνικά συνοριακά στοιχεία. Ως εκ τούτου τα νέα στοιχεία κατασκευάστηκαν με τέτοιο τρόπο ώστε οι συναρτήσεις παρεμβολής τους να μην είναι τετραγωνικες, αλλά να εξαρτώνται από τον τύπο ιδιομορφίας του κάθε πεδίου. Έπειτα, έγινε ακριβής υπολογισμός του συντελεστή έντασης τάσης της ρωγμής με βάση τις τιμές του πεδίου των τάσεων κοντά σε αυτή. Τέλος επιλύθηκαν στατικά προβλήματα θραυστομηχανικής σε δύο και τρεις διαστάσεις και υπολογίστηκαν οι συντελεστές έντασης τάσης για ρωγμές σε υλικά με επίδραση μικροδομής.

The thesis presents a novel computational method for analysing the static and dynamic behaviour of a multi-damaged beam using local and non-local elasticity theories. Most of the lumped damage beam models proposed to date are based on slender beam theory in classical (local) elasticity and are limited by inaccuracies caused by the implicit assumption of the Euler-Bernoulli beam model and by the spring model itself, which simplifies the real beam behaviour around the crack. In addition, size effects and material heterogeneity cannot be taken into account using the classical elasticity theory due to the absence of any microstructural parameter. The proposed work is based on the inhomogeneous Euler-Bernoulli beam theory in which a Dirac's delta function is added to the bending flexibility at the position of each crack: that is, the severer the damage, the larger is the resulting impulsive term. The crack is assumed to be always open, resulting in a linear system (i.e. nonlinear phenomena associated with breathing cracks are not considered). In order to provide an accurate representation of the structure's behaviour, a new multi-cracked beam element including shear effects and rotatory inertia is developed using the flexibility approach for the concentrated damage. The resulting stiffness matrix and load vector terms are evaluated by the unit-displacement method, employing the closed-form solutions for the multi-cracked beam problem. The same deformed shapes are used to derive the consistent mass matrix, also including the rotatory inertia terms. The two-node multi-damaged beam model has been validated through comparison of the results of static and dynamic analyses for two numerical examples against those provided by a commercial finite element code. The proposed model is shown to improve the computational efficiency as well as the accuracy, thanks to the inclusion of both shear deformations and rotatory inertia. The inaccuracy of the spring model, where for example for a rotational spring a finite jump appears on the rotations' profile, has been tackled by the enrichment of the elastic constitutive law with higher order stress and strain gradients. In particular, a new phenomenological approach based upon a convenient form of non-local elasticity beam theory has been presented. This hybrid non-local beam model is able to take into account the distortion on the stress/strain field around the crack as well as to include the microstructure of the material, without introducing any additional crack related parameters. The Laplace's transform method applied to the differential equation of the problem allowed deriving the static closed-form solution for the multi-cracked Euler-Bernoulli beams with hybrid non-local elasticity. The dynamic analysis has been performed using a new computational meshless method, where the equation of motions are discretised by a Galerkin-type approximation, with convenient shape functions able to ensure the same grade of approximation as the beam element for the classical elasticity. The importance of the inclusion of microstructural parameters is addressed and their effects are quantified also in comparison with those obtained using the classical elasticity theory.

The presented master’s thesis deals with the application of the gradient elasticity in fracture mechanics problems. Specifically, the displacement and stress field around the crack tip is a matter of interest. The influence of a material microstructure is considered. Introductory chapters are devoted to a brief historical overview of gradient models and definition of basic equations of dipolar gradient elasticity derived from Mindlin gradient theory form II. For comparison, relations of classical elasticity are introduced. Then a derivation of asymptotic displacement field using the Williams asymptotic technique follows. In the case of gradient elasticity, also the calculation of the J-integral is included. The mathematical formulation is reduced due to the singular nature of the problem to singular integral equations. The methods for solving integral equations in Cauchy principal value and Hadamard finite part sense are briefly introduced. For the evaluation of regular kernel, a Gauss-Chebyshev quadrature is used. There also mentioned approximate methods for solving systems of integral equations such as the weighted residual method, especially the least square method with collocation points. In the main part of the thesis the system of integral equations is derived using the Fourier transform for straight crack in an infinite body. This system is then solved numerically in the software Mathematica and the results are compared with the finite element model of ceramic foam.

[pt] Atualmente está bem difundido o uso de novas modelagens matemáticas para o estudo do comportamento de micro e nano sistemas mecânicos e eléctricos. O problema de escala é notável quando o tamanho das moléculas, partículas, grãos ou cristais de um sólido é relativamente considerável em relação ao comprimento do microdispositivo. Nesses casos a teoria clássica dos meios contínuos não descreve apropriadamente a solicitação estrutural e é necessária uma abordagem mais geral através de teorias generalizadas não-clássicas que contém a elasticidade clássica como um caso particular delas, onde os parâmetros constitutivos que representam às partículas são desprezíveis. Quando os efeitos microestruturais são importantes, o comportamento não responde como um material homogêneo se não como um material homogêneo. Cem anos atrás os irmãos Cosserat desenvolveram uma teoria de grãos rígidos imersos dentro de um macromeio elástico; posteriormente Toupin, Mindlin e outros pesquisadores na década de 60 formularam a chamada teoria gradiente de deformações, que recentemente é um objeto de muitas investigações analíticas e experimentais. Na década de oitenta, Aifantis e colaboradores conseguiram desenvolver uma teoria de gradiente de deformações simplificada, baseada em só uma constante elástica adicional não-clássica representativa da energia de deformação volumétrica para caracterizar satisfatoriamente os padrões dos fenômenos não-clássicos. Beskos e colaboradores estenderam o campo de aplicações da proposta inicial de Aifantis e fizeram as primeira implementações de elementos de contorno 2D e 3D para análises de elasticidade gradiente estática, no domínio da frequência e a mecânica da fratura. Desde o tempo de Toupin e Mindlin, procura-se estabelecer uma base variacional da teoria e uma formulação consistente das condições de contorno cinemáticas e de equilíbrio, o que parece ter tido êxito com os recentes trabalhos de Amanatidou e Aravas. Esta tese apresenta a formulação do método híbrido de elementos de contorno e finitos na elasticidade gradiente desenvolvida por Dumont e Huamán decompondo o potencial de Hellinger-Reissner em dois princípios de trabalhos virtuais: o primeiro em deslocamentos virtuais e o segundo em forças virtuais. Com esta finalidade é considerado além dos parâmetros clássicos, o trabalho realizado pelas tensões, deformações, forças e deslocamentos não-clássicos. É apresentado o desenvoltimento das soluções fundamentais singulares e polinomiais atráves das equações diferenciais de sexta ordem obtidas da equação de equilíbrio em termos de deslocamento na elasticidade gradiente. É apresentada também a aplicaçõ do método híbrido de contorno para problemas de tensão axial unidimensional e flexão bidimensional de vigas. Finalmente mostra-se a aplicação numérica do método em elementos finitos, é verificado o patch test de elementos finitos de diferentes ordem e mostra-se também análises de convergência. / [en] The use of new mathematical modeling in the study of micro and Nano electro mechanical systems is currently becoming widespread. The scaling problem is apparent when the length of molecules, particles or grains immersed in the material is relatively important compared with the whole micro device dimension. Under this approach the classical theories of mechanics cannot describe suitably the structural requirement and it is necessary a more general outlook through non classical generalized theories which enclose the classical elasticity as a particular case where the non-classical constitutive parameters are negligible. When the microstructural effects are important, the material does not respond as a homogeneous but as a non-homogeneous one. A hundred years ago Cosserat brothers formulated a new theory of rigid grains which were embedded in an elastic macro medium; later Toupin, Mindlin along others researchers in 1960s developed a gradient strain theory which has been recently the source of many analystics and experimental investigations. In 1980s Ainfantis et al could develop a simplified strain gradient theory with just one additional non classical elastic constant which represents the volumetric elastic strain energy and characterized successfully the whole non classical pattern phenomenon. Beskos et al extended the treatment proposed initially by Aifantis and developed the first numerical applications for 2D and 3D boundary element methods and solved static as dynamic and crack problems. Since the times of Toupin and Mindlin it is looking for to establish a variational theory with a consistent cinematic and equilibrium boundary conditions, which seemed to have had success in the recent works of Amanatiodou and Aravas. This work presents the formulation of the hybrid boundary and finite element methods under the strain gradient scope which were developed by Dumont and Huamán through the versatile decomposition of the Hellinger-Reissner potential in two work principles: the displacements virtual work and the forces virtual work; both principles contain the virtual work performed by the non-classical magnitudes. Following, it is presented the complete development of singular and polynominal fundamental solutions abtained through the sixth order strain gradient differential equilibrium equations in terms of displacements. Next it is shown an application of the method to unidimensional truss element and bidimensional beam. Finally, it is presented a numerical application to strain gradient finite element, it is checked the patch tests to different elements orders and it is also shown a series of convergence analysis.