The Polymer Electrolyte Membrane (PEM) fuel cell has a great potential in leading the future energy generation due to its advantages of zero emissions, higher power density and efficiency. For a PEM fuel cell, the Membrane-Electrode Assembly (MEA) is the key component which consists of a membrane, two catalyst layers and two gas diffusion layers (GDL). The success of optimum PEM fuel cell power output relies on the mass transport to the electrode especially on the cathode side. The carbon based GDL is one of the most important components in the fuel cell since it has one of the basic roles of providing path ways for reactant gases transport to the catalyst layer as well as excess water removal. A detailed understanding and visualization of the GDL from micro-scale level is limited by traditional numerical tool such as CFD and experimental methods due to the complex geometry of the porous GDL structural. In order to take the actual geometry information of the porous GDL into consideration, the x-ray tomography technique is employed which is able to reconstructed the actual structure of the carbon paper or carbon cloth GDLs to three-dimensional digital binary image which can be read directly by the LB model to carry out the simulation. This research work contributes to develop the combined methodology of x-ray tomography based the three-dimensional single phase Lattice Boltzmann (LB) simulation. This newly developed methodology demonstrates its capacity of simulating the flow characteristics and transport phenomena in the porous media by dealing with collision of the particles at pore-scale. The results reveal the heterogeneous nature of the GDL structures which influence the transportation of the reactants in terms of physical parameters of the GDLs such as porosity, permeability and tortuosity. The compression effects on the carbon cloth GDLs have been investigated. The results show that the c applied compression pressure on the GDLs will have negative effects on average pore size, porosity as well as through-plane permeability. A compression pressure range is suggested by the results which gives optimum in-plane permeability to through-plane permeability. The compression effects on one-dimensional water and oxygen partial pressures in the main flow direction have been studied at low, medium and high current densities. It s been observed that the water and oxygen pressure drop across the GDL increase with increasing the compression pressure. Key Words: PEM fuel cell, GDL, LB simulation, SPSC, SPMC, x-ray tomography, carbon paper, carbon cloth, porosity, permeability, degree of anisotropy, tortuosity, flow transport.

Modifying substrate specificity of β-glucuronidase (GUS) would be helpful in various enzyme prodrug systems in delivering drug dose to the site of action in the cancer treatment. Due to the presence of endogenous enzyme in human tissues, GUS-based Antibody-Directed Enzyme Prodrug Therapy (ADEPT) requires a novel substrate to avoid undesirable systemic activation. GUS is a glycosyl hydrolase, highly specific towards the glucuronide derivatives. It catalyzes the glycosidic cleavage of β-D-glucuronides to β-D-glucuronic acid and aglycone moiety. In order to gain insight on the substrate specificity of GUS, C6 carboxyl group of glucuronic acid was modified to C6 carboxamide (amide derivative). We have examined amide derivatized substrates with a variety of different aglycone groups including p-nitrophenyl, phenyl and 4-methylumbelliferone to further probe the activity profile of GUS. In an effort to optimize GUS activity, docking studies have been performed which indicated that amino acid point mutations near C6 carboxyl group of glucuronic acid could improve binding of the derivatized substrates. As a result point mutations to Arg-562 and Lys-568 which make the active site less positively charged either by glutamine or glutamate lead to an enzyme with much lower native substrate activity but abolished activity for the amide-derivatized substrate. This research study showed that there is still a further need of finding appropriate mutations required to make glucuronamide a better substrate for the mutated version of GUS.

Kinetic inactivation of factor XIIIa and MTG were performed in a pressure range from 0.1 to 400 MPa at 40°C within a time from 0 to 60 min in a TRIS-acetate buffer at pH 6.0. The inactivation of both enzymes at these conditions followed a first order reaction model. The high inactivation rate constant of 26.6 x10-3/min-1 for factor XIIIa at low pressure (50 MP) indicated that this enzyme is much easier to inactivate than MTG, which achieved an inactivation rate constant value of 9.7 x10-3/min at higher pressure (200 MPa). An inactivation volume of –10.17±0.5 cm3/mol confirmed that MTG is very stable under high pressure. The stability of MTG under high pressure and thermal treatment was related to its conformational changes. Enzyme inactivation was accompanied by secondary and tertiary structure changes until an irreversible protein precipitation is achieved. The tertiary structure, represented by circular dichroism spectra in the aromatic region showed differences among native and MTG samples treated under high pressure, as well as at elevated temperature. Tyrosine bands, indicating protein unfolding, increased proportionally with increasing pressure treatment above 400 MPa. Nevertheless, compared to pressure, a maximal enhancement could be observed after thermal treatment at 0.1 MPa at 80°C. That demonstrated the exposure of hydrophobic groups to the protein surface with a concomitant protein unfolding. The spectra in the far ultraviolet region showed that increasing high pressure and high temperature lead to alterations in the secondary structure. The mathematical algorithms CONTIN used to calculate secondary structures stated that the 24.5% of alpha-helix of native MTG decreased to 17.2% after a treatment at 400 MPa at 40°C for 60 min and to 6.5% after a treatment at 0.1 MPa at 80°C for 2 min. However, beta-strand structures remained relatively stable after these several treatments. MTG is arranged in a way that the active site is located between beta-strand domains that are surrounded by alpha-helices, the results of this investigation suggested that MTG activity is related with the relative stability of alpha-helix and the outstanding stability of the central beta-strand structure. The irreversible precipitated protein observed at 600 MPa at 40°C for 60 min and 0.1 MPa at 80°C for 2 min was caused principally by the formation of disulfides bonds, because high pressure and high thermal treatment lead to the exposition of the Cys64 residue towards the solvent with the subsequent ability to react with neighbouring cysteine residues. Furthermore, the reaction between protein and reducing sugars resulted in the formation of Maillard products. Furosine, as an indicator of the early stages of Maillard reaction was measured. Concentration values of 261.0 mg/g protein from samples treated at 600 MPa and 40°C and 238.5 mg/g protein from samples treated at and 0.1 MPa and 80°C for 2 min were obtained. Pentosidine a subsequent product observed in the advanced Maillard reaction was also present. Concentrations of 13.7 and 6.7 mg/g protein were obtained in the samples treated at 600 MPa and 40°C for 60 min and 0.1 MPa and 80°C for 2 min, respectively. Kinetic inactivation studies of MTG in a pressure range from 0.1 to 600 MPa at 10, 30, 40, and 50°C within a long time range from 0 to 140 h were performed in order to study MTG stability under the simultaneous effect of pressure and temperature. The inactivation kinetic showed a first and very fast step and a second very slow step suggesting irreversible inactivation behaviour. Activation energy and entropy difference decreased with increasing pressure. Thereby, the inactivation rate constants of enzyme were less temperature dependent at high pressure. The effect of pressure and temperature on MTG inactivation had a synergistic behaviour. At temperatures of 10, 30, and 40°C, increasing pressure leads to increasing inactivation rate constants. However at 50°C a tendency change occurred. Negative activation volumes of –16.2±0.5, -13.6±0.1, -11.2±0.3 cm3/mol were obtained for 10, 30 and 40°C respectively and for treatment at 50°C a positive value of about +3.0±2.0 cm3/mol in a pressure range from 0.1 to 300 and a negative volume of –11.0±0.4 cm3/mol MPa from 300 to 600 MPa were calculated. A pressure/temperature diagram from inactivation rate constants was performed to represent MTG stability. The diagram shows that in a pressure and temperature range from 0.1 to 550 MPa and 10 to 40°C, pressure induces MTG stabilization against heat denaturation. At 50°C in range from 0.1 to 300 MPa, pressure induces also enzyme stabilization again heat denaturation, but at the same temperature and above 300 MPa the enzyme was inactivated. After MTG stability analysis, reaction kinetics from MTG with individual caseins in a TRIS-acetate buffer pH 6.0 were performed under atmospheric pressure (0.1 MPa) and high pressure (400 MPa) at 40°C. The reaction was monitored by gel permeation chromatography under in three assumptions: 1) The initial velocity kinetics was obtained from a non-progressive enzymatic reactions with the products. 2) The substrate concentration exceeded enzyme concentration. 3) The sum of the individual catalytic constants of the reactive glutamine residues inside caseins are represented by a single MTG-monomeric casein complex. Enzyme reaction kinetics of MTG with the individual caseins carried out at 0.1 MPa at 40°C showed Michaelis-Menten-Henri behaviour with maximal velocities of 2.7 x 10-3, 0.8 x 10-3, and 1.3 x 10-3 mmol/L∙min and Km values of 59 x 10-3, 64 x 10-3 and 50 x 10-3 mmol/L of beta-, alpha-s1-, and whole-casein, respectively. This suggested that MTG achieved a maximal velocity with ß-casein, but had the best affinity with acid casein followed by beta- casein and finally alpha-s1-casein. Enzyme reaction kinetics of beta-casein carried out at 400 MPa and 40°C also showed a Michaelis-Menten-Henri behaviour with a similar maximal velocity of 2.6 x 10-3 mmol/L×min, but the Km value of 144 x 10-3 mmol/L showing kinetical similarity to a non-competitive inhibition. The reaction of MTG with alpha-s1-casein under high pressure did not fit in to Henri-Michaelis-Menten kinetics. Kinetic parameters showed that the affinity of MTG to beta- and alpha-s1-casein under atmospheric pressure is higher than the affinity of MTG to these caseins under high pressure. This loss of affinity can be explained by a constant number of reactive glutamine residues of casein, although the protein is unfolding at high pressure, a decrease of enzyme activity of MTG to 74% after treatment at 400 MPa at 40°C for 15 min and self association of casein under thermal and high pressure treatment. Fur technological application, the formation of acid milk gels was studied under the influence of MTG within its range of pH stability. Simultaneous addition of MTG and different concentrations of glucono-delta-lactone (Gdl) to casein solutions (5% w/v) at 40°C was analysed. Gels firmness was accessed by oscillation rheometry and gel permeation chromatography. Oscillation rheometry data showed that the time of gelation decreased with an increasing Gdl concentration added to the system, however higher concentrations of Gdl caused the formation of weaker gels. Addition of 1 g Gdl/g protein without MTG caused gelation within 5 min and a storage module value G´ of 48.9 Pa. With the simultaneous addition of 1 g Gdl/g protein and 6 U MTG/ g protein the gelation time was 4 min and the reached storage modulus was 63.7 Pa. However, the addition of 0.21 g Gdl/g protein and 6 U/g protein MTG increase the gelation time to about 69 min, but, a higher module value G´ of 111.0 Pa was achieved. Addition of high Gdl concentration caused a rapid drop of pH below 5 leading to a fast enzyme inactivation. However addition of very low Gdl concentrations was also not optimal. The simultaneous influence of MTG and Gdl concentration on the gelation time and elastic properties was evaluated by a central composite rotatable design (CCRD). The resulting quadratic storage modulus model showed that, MTG concentration had a significant influence on storage modulus G´ and, that the firmness of the gels increase in direct proportion with MTG activity with the existence of a optimum Gdl concentration, whereas the resulting linear model of the gelation time stated that Gdl concentration has a significant influence on the gelation time, while it is independent of the MTG activity. A maximal firmness of 136 ± 2 Pa was reached between a range of 0.24 - 0.27 g Gdl/g protein and 5.8 U MTG/g within a time from 49 to 59 min. Gel permeation chromatography analysis demonstrated that acid gels induced by Gdl were formed by reversible cross-linking like electrostatic interactions and hydrogen bonds as well as disulfide bonds caused by temperature treatment. Whereas, the addition of MTG proved the formation of non-reversible cross-linking like oligomers based on Ne-(g-glutamyl)- lysine, which gave more firmness and stabilization on the casein gel network.

Kinetic inactivation of factor XIIIa and MTG were performed in a pressure range from 0.1 to 400 MPa at 40°C within a time from 0 to 60 min in a TRIS-acetate buffer at pH 6.0. The inactivation of both enzymes at these conditions followed a first order reaction model. The high inactivation rate constant of 26.6 x10-3/min-1 for factor XIIIa at low pressure (50 MP) indicated that this enzyme is much easier to inactivate than MTG, which achieved an inactivation rate constant value of 9.7 x10-3/min at higher pressure (200 MPa). An inactivation volume of –10.17±0.5 cm3/mol confirmed that MTG is very stable under high pressure. The stability of MTG under high pressure and thermal treatment was related to its conformational changes. Enzyme inactivation was accompanied by secondary and tertiary structure changes until an irreversible protein precipitation is achieved. The tertiary structure, represented by circular dichroism spectra in the aromatic region showed differences among native and MTG samples treated under high pressure, as well as at elevated temperature. Tyrosine bands, indicating protein unfolding, increased proportionally with increasing pressure treatment above 400 MPa. Nevertheless, compared to pressure, a maximal enhancement could be observed after thermal treatment at 0.1 MPa at 80°C. That demonstrated the exposure of hydrophobic groups to the protein surface with a concomitant protein unfolding. The spectra in the far ultraviolet region showed that increasing high pressure and high temperature lead to alterations in the secondary structure. The mathematical algorithms CONTIN used to calculate secondary structures stated that the 24.5% of alpha-helix of native MTG decreased to 17.2% after a treatment at 400 MPa at 40°C for 60 min and to 6.5% after a treatment at 0.1 MPa at 80°C for 2 min. However, beta-strand structures remained relatively stable after these several treatments. MTG is arranged in a way that the active site is located between beta-strand domains that are surrounded by alpha-helices, the results of this investigation suggested that MTG activity is related with the relative stability of alpha-helix and the outstanding stability of the central beta-strand structure. The irreversible precipitated protein observed at 600 MPa at 40°C for 60 min and 0.1 MPa at 80°C for 2 min was caused principally by the formation of disulfides bonds, because high pressure and high thermal treatment lead to the exposition of the Cys64 residue towards the solvent with the subsequent ability to react with neighbouring cysteine residues. Furthermore, the reaction between protein and reducing sugars resulted in the formation of Maillard products. Furosine, as an indicator of the early stages of Maillard reaction was measured. Concentration values of 261.0 mg/g protein from samples treated at 600 MPa and 40°C and 238.5 mg/g protein from samples treated at and 0.1 MPa and 80°C for 2 min were obtained. Pentosidine a subsequent product observed in the advanced Maillard reaction was also present. Concentrations of 13.7 and 6.7 mg/g protein were obtained in the samples treated at 600 MPa and 40°C for 60 min and 0.1 MPa and 80°C for 2 min, respectively. Kinetic inactivation studies of MTG in a pressure range from 0.1 to 600 MPa at 10, 30, 40, and 50°C within a long time range from 0 to 140 h were performed in order to study MTG stability under the simultaneous effect of pressure and temperature. The inactivation kinetic showed a first and very fast step and a second very slow step suggesting irreversible inactivation behaviour. Activation energy and entropy difference decreased with increasing pressure. Thereby, the inactivation rate constants of enzyme were less temperature dependent at high pressure. The effect of pressure and temperature on MTG inactivation had a synergistic behaviour. At temperatures of 10, 30, and 40°C, increasing pressure leads to increasing inactivation rate constants. However at 50°C a tendency change occurred. Negative activation volumes of –16.2±0.5, -13.6±0.1, -11.2±0.3 cm3/mol were obtained for 10, 30 and 40°C respectively and for treatment at 50°C a positive value of about +3.0±2.0 cm3/mol in a pressure range from 0.1 to 300 and a negative volume of –11.0±0.4 cm3/mol MPa from 300 to 600 MPa were calculated. A pressure/temperature diagram from inactivation rate constants was performed to represent MTG stability. The diagram shows that in a pressure and temperature range from 0.1 to 550 MPa and 10 to 40°C, pressure induces MTG stabilization against heat denaturation. At 50°C in range from 0.1 to 300 MPa, pressure induces also enzyme stabilization again heat denaturation, but at the same temperature and above 300 MPa the enzyme was inactivated. After MTG stability analysis, reaction kinetics from MTG with individual caseins in a TRIS-acetate buffer pH 6.0 were performed under atmospheric pressure (0.1 MPa) and high pressure (400 MPa) at 40°C. The reaction was monitored by gel permeation chromatography under in three assumptions: 1) The initial velocity kinetics was obtained from a non-progressive enzymatic reactions with the products. 2) The substrate concentration exceeded enzyme concentration. 3) The sum of the individual catalytic constants of the reactive glutamine residues inside caseins are represented by a single MTG-monomeric casein complex. Enzyme reaction kinetics of MTG with the individual caseins carried out at 0.1 MPa at 40°C showed Michaelis-Menten-Henri behaviour with maximal velocities of 2.7 x 10-3, 0.8 x 10-3, and 1.3 x 10-3 mmol/L∙min and Km values of 59 x 10-3, 64 x 10-3 and 50 x 10-3 mmol/L of beta-, alpha-s1-, and whole-casein, respectively. This suggested that MTG achieved a maximal velocity with ß-casein, but had the best affinity with acid casein followed by beta- casein and finally alpha-s1-casein. Enzyme reaction kinetics of beta-casein carried out at 400 MPa and 40°C also showed a Michaelis-Menten-Henri behaviour with a similar maximal velocity of 2.6 x 10-3 mmol/L×min, but the Km value of 144 x 10-3 mmol/L showing kinetical similarity to a non-competitive inhibition. The reaction of MTG with alpha-s1-casein under high pressure did not fit in to Henri-Michaelis-Menten kinetics. Kinetic parameters showed that the affinity of MTG to beta- and alpha-s1-casein under atmospheric pressure is higher than the affinity of MTG to these caseins under high pressure. This loss of affinity can be explained by a constant number of reactive glutamine residues of casein, although the protein is unfolding at high pressure, a decrease of enzyme activity of MTG to 74% after treatment at 400 MPa at 40°C for 15 min and self association of casein under thermal and high pressure treatment. Fur technological application, the formation of acid milk gels was studied under the influence of MTG within its range of pH stability. Simultaneous addition of MTG and different concentrations of glucono-delta-lactone (Gdl) to casein solutions (5% w/v) at 40°C was analysed. Gels firmness was accessed by oscillation rheometry and gel permeation chromatography. Oscillation rheometry data showed that the time of gelation decreased with an increasing Gdl concentration added to the system, however higher concentrations of Gdl caused the formation of weaker gels. Addition of 1 g Gdl/g protein without MTG caused gelation within 5 min and a storage module value G´ of 48.9 Pa. With the simultaneous addition of 1 g Gdl/g protein and 6 U MTG/ g protein the gelation time was 4 min and the reached storage modulus was 63.7 Pa. However, the addition of 0.21 g Gdl/g protein and 6 U/g protein MTG increase the gelation time to about 69 min, but, a higher module value G´ of 111.0 Pa was achieved. Addition of high Gdl concentration caused a rapid drop of pH below 5 leading to a fast enzyme inactivation. However addition of very low Gdl concentrations was also not optimal. The simultaneous influence of MTG and Gdl concentration on the gelation time and elastic properties was evaluated by a central composite rotatable design (CCRD). The resulting quadratic storage modulus model showed that, MTG concentration had a significant influence on storage modulus G´ and, that the firmness of the gels increase in direct proportion with MTG activity with the existence of a optimum Gdl concentration, whereas the resulting linear model of the gelation time stated that Gdl concentration has a significant influence on the gelation time, while it is independent of the MTG activity. A maximal firmness of 136 ± 2 Pa was reached between a range of 0.24 - 0.27 g Gdl/g protein and 5.8 U MTG/g within a time from 49 to 59 min. Gel permeation chromatography analysis demonstrated that acid gels induced by Gdl were formed by reversible cross-linking like electrostatic interactions and hydrogen bonds as well as disulfide bonds caused by temperature treatment. Whereas, the addition of MTG proved the formation of non-reversible cross-linking like oligomers based on Ne-(g-glutamyl)- lysine, which gave more firmness and stabilization on the casein gel network.

In der vorliegenden Arbeit wird eine Synthese-Vorauslegungsmethode beschrieben, mit der für individuelle Anforderungsprofile in einem frühen Entwicklungsprozess eine methodische Generierung von Lösungsvarianten für Polymerelektrolytmembran-Brennstoffzellen-Stacks erfolgt. Hierzu werden Parametervariationen auf Basis Kanalquerschnitts- und Flussfeldmodellen durchgeführt. Weiterhin werden Berechnungsmodule genutzt, um individuelle konstruktive und verfahrenstechnische Kennwerte der jeweiligen Lösungsvarianten zu bestimmen, unter anderem hinsichtlich GDL-Intrusionseffekten, hydraulischer Druckabfälle, der Umformschädigung metallischer Kanalstrukturen sowie der Berechnung der Flussfeldüberdeckungsbereiche zweier kombinierter Flussfelder. Auf Basis der Variantenkennwerte erfolgen wichtungsbasierte Bewertungsprozesse, aus denen gesamtheitliche Vorzugslösungen resultieren. In einem Beispielszenario wird die Anwendung der präsentierten Synthese-Vorauslegungsmethode exemplarisch dargestellt sowie die experimentellen Ergebnisse zur Charakterisierung der real umgesetzten Vorzugslösung aufgezeigt.:1 Einleitung und Motivation 2 Stand der Wissenschaft und Technik 3 Grundlagen von PEM-BZ 4 Synthese-Auslegungsmethode 5 Berechnungsmodule 6 Anwendung der Synthese-Methode 7 Schlussbetrachtungen

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[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.