Dry cooling with night cool storage to enhance solar power plants performance in extreme conditions areas

Solar thermal power plants are usually installed in locations with high yearly average solar radiation, often deserts. In such conditions, cooling water required for thermodynamic cycles is rarely available. Moreover, when solar radiation is high, ambient temperature is very high as well; this leads to excessive condensation temperature, especially when air-condensers are used, and decreases the plant efficiency. However, temperature variation in deserts is often very high, which drives to relatively low temperatures during the night. This fact can be exploited with the use of a closed cooling system, so that the coolant (water) is chilled during the night and store. Chilled water is then used during peak temperature hours to cool the condenser (dry cooling), thus enhancing power output and efficiency. The present work analyzes the performance improvement achieved by night thermal cool storage, compared to its equivalent air cooled power plant. Dry cooling is proved to be energy-effective for moderately high day–night temperature differences (20 °C), often found in desert locations. The storage volume requirement for different power plant efficiencies has also been studied, resulting on an asymptotic tendency.

Development of a Uniform Theory of Diffraction for Scattered and Surface Electromagnetic Field Analysis on a Cylinder

Con esta tesis ”Desarrollo de una Teoría Uniforme de la Difracción para el Análisis de los Campos Electromagnéticos Dispersados y Superficiales sobre un Cilindro” hemos iniciado una nueva línea de investigación que trata de responder a la siguiente pregunta: ¿cuál es la impedancia de superficie que describe una estructura de conductor eléctrico perfecto (PEC) convexa recubierta por un material no conductor? Este tipo de estudios tienen interés hoy en día porque ayudan a predecir el campo electromagnético incidente, radiado o que se propaga sobre estructuras metálicas y localmente convexas que se encuentran recubiertas de algún material dieléctrico, o sobre estructuras metálicas con pérdidas, como por ejemplo se necesita en determinadas aplicaciones aeroespaciales, marítimas o automovilísticas. Además, desde un punto de vista teórico, la caracterización de la impedancia de superficie de una estructura PEC recubierta o no por un dieléctrico es una generalización de varias soluciones que tratan ambos tipos de problemas por separado. En esta tesis se desarrolla una teoría uniforme de la difracción (UTD) para analizar el problema canónico del campo electromagnético dispersado y superficial en un cilindro circular eléctricamente grande con una condición de contorno de impedancia (IBC) para frecuencias altas. Construir una solución basada en UTD para este problema canónico es crucial en el desarrollo de un método UTD para el caso más general de una superficie arbitrariamente convexa, mediante el uso del principio de localización de los campos electromagnéticos a altas frecuencias. Esta tesis doctoral se ha llevado a cabo a través de una serie de hitos que se enumeran a continuación, enfatizando las contribuciones a las que ha dado lugar. Inicialmente se realiza una revisión en profundidad del estado del arte de los métodos asintóticos con numerosas referencias. As í, cualquier lector novel puede llegar a conocer la historia de la óptica geométrica (GO) y la teoría geométrica de la difracción (GTD), que dieron lugar al desarrollo de la UTD. Después, se investiga ampliamente la UTD y los trabajos más importantes que pueden encontrarse en la literatura. As í, este capítulo, nos coloca en la posición de afirmar que, hasta donde nosotros conocemos, nadie ha intentado antes llevar a cabo una investigación rigurosa sobre la caracterización de la impedancia de superficie de una estructura PEC recubierta por un material dieléctrico, utilizando para ello la UTD. Primero, se desarrolla una UTD para el problema canónico de la dispersión electromagnética de un cilindro circular eléctricamente grande con una IBC uniforme, cuando es iluminado por una onda plana con incidencia oblicua a frecuencias altas. La solución a este problema canónico se construye a partir de una solución exacta mediante una expansión de autofunciones de propagación radial. Entonces, ésta se convierte en una nueva expansión de autofunciones de propagación circunferencial muy apropiada para cilindros grandes, a través de la transformación de Watson. De esta forma, la expresión del campo se reduce a una integral que se evalúa asintóticamente, para altas frecuencias, de manera uniforme. El resultado se expresa según el trazado de rayos descrito en la UTD. La solución es uniforme porque tiene la importante propiedad de mantenerse continua a lo largo de la región de transición, a ambos lados de la superficie del contorno de sombra. Fuera de la región de transición la solución se reduce al campo incidente y reflejado puramente ópticos en la región iluminada del cilindro, y al campo superficial difractado en la región de sombra. Debido a la IBC el campo dispersado contiene una componente contrapolar a causa de un acoplamiento entre las ondas TEz y TMz (donde z es el eje del cilindro). Esta componente contrapolar desaparece cuando la incidencia es normal al cilindro, y también en la región iluminada cuando la incidencia es oblicua donde el campo se reduce a la solución de GO. La solución UTD presenta una muy buena exactitud cuando se compara numéricamente con una solución de referencia exacta. A continuación, se desarrolla una IBC efectiva para el cálculo del campo electromagnético dispersado en un cilindro circular PEC recubierto por un dieléctrico e iluminado por una onda plana incidiendo oblicuamente. Para ello se derivan dos impedancias de superficie en relación directa con las ondas creeping y de superficie TM y TE que se excitan en un cilindro recubierto por un material no conductor. Las impedancias de superficie TM y TE están acopladas cuando la incidencia es oblicua, y dependen de la geometría del problema y de los números de onda. Además, se ha derivado una impedancia de superficie constante, aunque con diferente valor cuando el observador se encuentra en la zona iluminada o en la zona de sombra. Después, se presenta una solución UTD para el cálculo de la dispersión de una onda plana con incidencia oblicua sobre un cilindro eléctricamente grande y convexo, mediante la generalización del problema canónico correspondiente al cilindro circular. La solución asintótica es uniforme porque se mantiene continua a lo largo de la región de transición, en las inmediaciones del contorno de sombra, y se reduce a la solución de rayos ópticos en la zona iluminada y a la contribución de las ondas de superficie dentro de la zona de sombra, lejos de la región de transición. Cuando se usa cualquier material no conductor se excita una componente contrapolar que tiende a desaparecer cuando la incidencia es normal al cilindro y en la región iluminada. Se discuten ampliamente las limitaciones de las fórmulas para la impedancia de superficie efectiva, y se compara la solución UTD con otras soluciones de referencia, donde se observa una muy buena concordancia. Y en tercer lugar, se presenta una aproximación para una impedancia de superficie efectiva para el cálculo de los campos superficiales en un cilindro circular conductor recubierto por un dieléctrico. Se discuten las principales diferencias que existen entre un cilindro PEC recubierto por un dieléctrico desde un punto de vista riguroso y un cilindro con una IBC. Mientras para un cilindro de impedancia se considera una impedancia de superficie constante o uniforme, para un cilindro conductor recubierto por un dieléctrico se derivan dos impedancias de superficie. Estas impedancias de superficie están asociadas a los modos de ondas creeping TM y TE excitadas en un cilindro, y dependen de la posición y de la orientación del observador y de la fuente. Con esto en mente, se deriva una solución UTD con IBC para los campos superficiales teniendo en cuenta las dependencias de la impedancia de superficie. La expansión asintótica se realiza, mediante la transformación de Watson, sobre la representación en serie de las funciones de Green correspondientes, evitando as í calcular las derivadas de orden superior de las integrales de tipo Fock, y dando lugar a una solución rápida y precisa. En los ejemplos numéricos realizados se observa una muy buena precisión cuando el cilindro y la separación entre el observador y la fuente son grandes. Esta solución, junto con el método de los momentos (MoM), se puede aplicar para el cálculo eficiente del acoplamiento mutuo de grandes arrays conformados de antenas de parches. Los métodos propuestos basados en UTD para el cálculo del campo electromagnético dispersado y superficial sobre un cilindro PEC recubierto de dieléctrico con una IBC efectiva suponen un primer paso hacia la generalización de una solución UTD para superficies metálicas convexas arbitrarias cubiertas por un material no conductor e iluminadas por una fuente electromagnética arbitraria. ABSTRACT With this thesis ”Development of a Uniform Theory of Diffraction for Scattered and Surface Electromagnetic Field Analysis on a Cylinder” we have initiated a line of investigation whose goal is to answer the following question: what is the surface impedance which describes a perfect electric conductor (PEC) convex structure covered by a material coating? These studies are of current and future interest for predicting the electromagnetic (EM) fields incident, radiating or propagating on locally smooth convex parts of highly metallic structures with a material coating, or by a lossy metallic surfaces, as for example in aerospace, maritime and automotive applications. Moreover, from a theoretical point of view, the surface impedance characterization of PEC surfaces with or without a material coating represents a generalization of independent solutions for both type of problems. A uniform geometrical theory of diffraction (UTD) is developed in this thesis for analyzing the canonical problem of EM scattered and surface field by an electrically large circular cylinder with an impedance boundary condition (IBC) in the high frequency regime, by means of a surface impedance characterization. The construction of a UTD solution for this canonical problem is crucial for the development of the corresponding UTD solution for the more general case of an arbitrary smooth convex surface, via the principle of the localization of high frequency EM fields. The development of the present doctoral thesis has been carried out through a series of landmarks that are enumerated as follows, emphasizing the main contributions that this work has given rise to. Initially, a profound revision is made in the state of art of asymptotic methods where numerous references are given. Thus, any reader may know the history of geometrical optics (GO) and geometrical theory of diffraction (GTD), which led to the development of UTD. Then, the UTD is deeply investigated and the main studies which are found in the literature are shown. This chapter situates us in the position to state that, as far as we know, nobody has attempted before to perform a rigorous research about the surface impedance characterization for material-coated PEC convex structures via UTD. First, a UTD solution is developed for the canonical problem of the EM scattering by an electrically large circular cylinder with a uniform IBC, when it is illuminated by an obliquely incident high frequency plane wave. A solution to this canonical problem is first constructed in terms of an exact formulation involving a radially propagating eigenfunction expansion. The latter is converted into a circumferentially propagating eigenfunction expansion suited for large cylinders, via the Watson transformation, which is expressed as an integral that is subsequently evaluated asymptotically, for high frequencies, in a uniform manner. The resulting solution is then expressed in the desired UTD ray form. This solution is uniform in the sense that it has the important property that it remains continuous across the transition region on either side of the surface shadow boundary. Outside the shadow boundary transition region it recovers the purely ray optical incident and reflected ray fields on the deep lit side of the shadow boundary and to the modal surface diffracted ray fields on the deep shadow side. The scattered field is seen to have a cross-polarized component due to the coupling between the TEz and TMz waves (where z is the cylinder axis) resulting from the IBC. Such cross-polarization vanishes for normal incidence on the cylinder, and also in the deep lit region for oblique incidence where it properly reduces to the GO or ray optical solution. This UTD solution is shown to be very accurate by a numerical comparison with an exact reference solution. Then, an effective IBC is developed for the EM scattered field on a coated PEC circular cylinder illuminated by an obliquely incident plane wave. Two surface impedances are derived in a direct relation with the TM and TE surface and creeping wave modes excited on a coated cylinder. The TM and TE surface impedances are coupled at oblique incidence, and depend on the geometry of the problem and the wave numbers. Nevertheless, a constant surface impedance is found, although with a different value when the observation point lays in the lit or in the shadow region. Then, a UTD solution for the scattering of an obliquely incident plane wave on an electrically large smooth convex coated PEC cylinder is introduced, via a generalization of the canonical circular cylinder problem. The asymptotic solution is uniform because it remains continuous across the transition region, in the vicinity of the shadow boundary, and it recovers the ray optical solution in the deep lit region and the creeping wave formulation within the deep shadow region. When a coating is present a cross-polar field term is excited, which vanishes at normal incidence and in the deep lit region. The limitations of the effective surface impedance formulas are discussed, and the UTD solution is compared with some reference solutions where a very good agreement is met. And in third place, an effective surface impedance approach is introduced for determining surface fields on an electrically large coated metallic circular cylinder. Differences in analysis of rigorouslytreated coated metallic cylinders and cylinders with an IBC are discussed. While for the impedance cylinder case a single constant or uniform surface impedance is considered, for the coated metallic cylinder case two surface impedances are derived. These are associated with the TM and TE creeping wave modes excited on a cylinder and depend on observation and source positions and orientations. With this in mind, a UTD based method with IBC is derived for the surface fields by taking into account the surface impedance variation. The asymptotic expansion is performed, via the Watson transformation, over the appropriate series representation of the Green’s functions, thus avoiding higher-order derivatives of Fock-type integrals, and yielding a fast and an accurate solution. Numerical examples reveal a very good accuracy for large cylinders when the separation between the observation and the source point is large. Thus, this solution could be efficiently applied in mutual coupling analysis, along with the method of moments (MoM), of large conformal microstrip array antennas. The proposed UTD methods for scattered and surface EM field analysis on a coated PEC cylinder with an effective IBC are considered the first steps toward the generalization of a UTD solution for large arbitrarily convex smooth metallic surfaces covered by a material coating and illuminated by an arbitrary EM source.

Accurate analytical approximation of asteroid deflection with constant tangential thrust

We present analytical formulas to estimate the variation of achieved deflection for an Earth-impacting asteroid following a continuous tangential low-thrust deflection strategy. Relatively simple analytical expressions are obtained with the aid of asymptotic theory and the use of Peláez orbital elements set, an approach that is particularly suitable to the asteroid deflection problem and is not limited to small eccentricities. The accuracy of the proposed formulas is evaluated numerically showing negligible error for both early and late deflection campaigns. The results will be of aid in planning future low-thrust asteroid deflection missions

Overal reaction rate and stoichiometry for thermal decomposition of hydrazine

An asymptotic analysîs of the Eberstein-Glassman kinetic mechanlsm for the thermal décomposition of hydrazine is carried out. It is shown that at températures near 800°K and near 1000°K,and for hydrazine molar fractions of the order of unity, 10-2 the entire kinetics reduces to a single, overall reaction. Characteristic times for the chemical relaxation of ail active, intermediate species produced in the décomposition, and for the overall reaction, are obtained. Explicit expressions for the overall reaction rate and stoichiometry are given as functions of température, total molar concentration (or pressure)and hydrazine molar fraction. Approximate, patched expressions can then be obtained for values of température and hydrazine molar fraction between 750 and 1000°K, and 1 and 10-3 respectively.

B.I.E.M. y estudio de singularidades en problemas de potencial

El objeto del presente artículo es el estudio de singularidades en problemas de Potencial mediante el uso del Método de las Ecuaciones Integrales sobre el contorno del dominio en estudio. Frente a soluciones basadas en la mejora de la discretización, análisis asintótico o introducción de funciones de forma que representen mejor la evolución de la función, una nueva hipótesis es presentada: el término responsable de la singularidad es incluido en la integral sobre el contorno de la función auxiliar. Los resultados obtenidos mejoran los de soluciones anteriores simplificando también el tiempo de cálculo = The subject of this paper is the modelling of singularities in potential problems, using the Boundary Integral Equation Method. As a logical alternative to classical methods (discretization refinement, asymptotic analysis, high order interpolatory functions) a new hypothesis is presented: the singularity responsible term is included in the interpolatory shape function. As shown by several exemples results are splendid and computer time radically shortened.

The theory of electrostatic probes in strong magnetic fields : Thesis submitted to the Faculty of The Graduate School of the University of Colorado for the Degree Doctor of Philosophy Department of Aerospace Engineering Science

A theory is developed of an electrostatic probe in a fully-ionized plasma in the presence of a strong magnetic field. The ratio of electron Larmor radius to probe transverse dimension is assumed to be small. Poisson's equation, together with kinetic equations for ions and electrons are considered. An asymptotic perturbation method of multiple scales is used by considering the characteristic lengths appearing in the problem. The leading behavior of the solution is found. The results obtained appear to apply to weaker fields also, agreeing with the solutions known in the limit of no magnetic field. The range of potentials for wich results are presented is limited. The basic effects produced by the field are a depletion of the plasma near the probe and a non-monotonic potential surrounding the probe. The ion saturation current is not changed but changes appear in both the floating potential Vf and the slope of the current-voltage diagram at Vf. The transition region extends beyond the space potential Vs,at wich point the current is largely reduced. The diagram does not have an exponential form in this region as commonly assumed. There exists saturation in electron collection. The extent to which the plasma is disturbed is determined. A cylindrical probe has no solution because of a logarithmic singularity at infinity. Extensions of the theory are considered.

Theory of a Probe in a Strong Magnetic Field. AT(30-1)-1238-MATT 599

A kinetic approach is used to develop a theory of electrostatic probes in a fully ionized plasma in the presence of a magnetic field. A consistent asymptotic expansion is obtained assuming that the electron Larmor radius is small compared to the radius of the probe. The order of magnitude of neglected terms is given. It is found that the electric potential within the tube of force defined by the cross section of the probe decays non-mono tonic ally from the probe; this bump disappears at a certain probe voltage and the theory is valid up to this voltage. The transition region, which extends beyond plasma potential, is not exponential. The possible saturation of the electron current is discussed. Restricted numerical results are given; they seem to be useful for weaker magnetic fields down to the zero-field limit. Extensions of the theory a r e considered.

Optimal polygonal L1 linearization and fast interpolation of nonlinear systems

The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using polygonal (continuous piecewise linear) models under the L1 norm. A thorough error analysis is developed to guide an optimal design of two kinds of polygonal approximations in the asymptotic case of a large budget of evaluation subintervals N. The method allows the user to obtain the level of linearization (N) for a target approximation error and vice versa. It is suitable for, but not limited to, an efficient implementation in modern Graphics Processing Units (GPUs), allowing real-time performance of computationally demanding applications. The quality and efficiency of the technique has been measured in detail on two nonlinear functions that are widely used in many areas of scientific computing and are expensive to evaluate.

Analysis of a rate-adaptive reconciliation protocol and the effect of leakage on the secret key rate

Quantum key distribution performs the trick of growing a secret key in two distant places connected by a quantum channel. The main reason is so that the legitimate users can bound the information gathered by the eavesdropper. In practical systems, whether because of finite resources or external conditions, the quantum channel is subject to fluctuations. A rate-adaptive information reconciliation protocol, which adapts to the changes in the communication channel, is then required to minimize the leakage of information in the classical postprocessing. We consider here the leakage of a rate-adaptive information reconciliation protocol. The length of the exchanged messages is larger than that of an optimal protocol; however, we prove that the min-entropy reduction is limited. The simulation results, both in the asymptotic and in the finite-length regime, show that this protocol allows to increase the amount of a distillable secret key.

Sophisticated security verification on routing repaired balanced cell-based dual-rail logic against side channel analysis

Conventional dual-rail precharge logic suffers from difficult implementations of dual-rail structure for obtaining strict compensation between the counterpart rails. As a light-weight and high-speed dual-rail style, balanced cell-based dual-rail logic (BCDL) uses synchronised compound gates with global precharge signal to provide high resistance against differential power or electromagnetic analyses. BCDL can be realised from generic field programmable gate array (FPGA) design flows with constraints. However, routings still exist as concerns because of the deficient flexibility on routing control, which unfavourably results in bias between complementary nets in security-sensitive parts. In this article, based on a routing repair technique, novel verifications towards routing effect are presented. An 8 bit simplified advanced encryption processing (AES)-co-processor is executed that is constructed on block random access memory (RAM)-based BCDL in Xilinx Virtex-5 FPGAs. Since imbalanced routing are major defects in BCDL, the authors can rule out other influences and fairly quantify the security variants. A series of asymptotic correlation electromagnetic (EM) analyses are launched towards a group of circuits with consecutive routing schemes to be able to verify routing impact on side channel analyses. After repairing the non-identical routings, Mutual information analyses are executed to further validate the concrete security increase obtained from identical routing pairs in BCDL.

A review of electrodynamic tethers for science applications

A bare electrodynamic tether (EDT) is a conductive thin wire or tape tens of kilometres long, which is kept taut in space by gravity gradient or spinning, and is left bare of insulation to collect (and carry) current as a cylindrical Langmuir probe in an ambient magnetized plasma. An EDT is a probe in mesothermal flow at highly positive (or negative) bias, with a large or extremely large 2D sheath, which may show effects from the magnetic self-field of its current and have electrons adiabatically trapped in its ram front. Beyond technical applications ranging from propellantless propulsion to power generation in orbit, EDTs allow broad scientific uses such as generating electron beams and artificial auroras; exciting Alfven waves and whistlers; odifying the radiation belts; and exploring interplanetary space and the Jovian magnetosphere. Asymptotic analysis, numerical simulations, laboratory tests, and planned missions on EDTs are reviewed

Sheath vaporization of a monodisperse fuel-spray jet

The group vaporization of a monodisperse fuel-spray jet discharging into a hot coflowing gaseous stream is investigated for steady flow by numerical and asymptotic methods with a two-continua formulation used for the description of the gas and liquid phases. The jet is assumed to be slender and laminar, as occurs when the Reynolds number is moderately large, so that the boundary-layer form of the conservation equations can be employed in the analysis. Two dimensionless parameters are found to control the flow structure, namely the spray dilution parameter 1, defined as the mass of liquid fuel per unit mass of gas in the spray stream, and the group vaporization parameter e, defined as the ratio of the characteristic time of spray evolution due to droplet vaporization to the characteristic diffusion time across the jet. It is observed that, for the small values of e often encountered in applications, vaporization occurs only in a thin layer separating the spray from the outer droplet-free stream. This regime of sheath vaporization, which is controlled by heat conduction, is amenable to a simplified asymptotic description, independent of ε,in which the location of the vaporization layer is determined numerically as a free boundary in a parabolic problem involving matching of the separate solutions in the external streams, with appropriate jump conditions obtained from analysis of the quasi-steady vaporization front. Separate consideration of dilute and dense sprays, corresponding, respectively, to the asymptotic limits λ<<1 and λ>>1, enables simplified descriptions to be obtained for the different flow variables, including explicit analytic expressions for the spray penetration distance.

Exact solution for the conjugate fluid-fluid problem in the thermal entrance region of laminar counterflow heat exchangers

A generalized Lévêque solution is presented for the conjugate fluid–fluid problem that arises in the thermal entrance region of laminar counterflow heat exchangers. The analysis, carried out for constant property fluids, assumes that the Prandtl and Peclet numbers are both large compared to unity, and neglects axial conduction both in the fluids and in the plate, assumed to be thermally thin. Under these conditions, the thermal entrance region admits an asymptotic self-similar description where the temperature varies as a power ϳ of the axial distance, with the particularity that the self-similarity exponent must be determined as an eigenvalue by solving a transcendental equation arising from the requirement of continuity of heat fluxes at the heat conducting wall. Specifically, the analysis reveals that j depends only on the lumped parameter ƙ = (A2/A1)1/3 (α1/α2)1/3(k2/k1), defined in terms of the ratios of the wall velocity gradients, A, thermal diffusivities, α i, and thermal conductivities,k i, of the fluids entering, 1, and exiting, 2, the heat exchanger. Moreover, it is shown that for large (small) values of K solution reduces to the classical first (second) Lévêque solution. Closed-form analytical expressions for the asymptotic temperature distributions and local heat-transfer rate in the thermal entrance region are given and compared with numerical results in the counterflow parallel-plate configuration, showing very good agreement in all cases.

The structure of lean hydrogen-air flame balls

An analysis of the structure of flame balls encountered under microgravity conditions, which are stable due to radiant energy losses from H₂O, is carried out for fuel-lean hydrogen-air mixtures. It is seen that, because of radiation losses, in stable flame balls the maximum flame temperature remains close to the crossover temperature, at which the rate of the branching step H + O₂ -> OH + O equals that of the recombination step H + O₂ + M -> HO₂ + M. Under those conditions, all chemical intermediates have very small concentrations and follow the steady-state approximation, while the main species react according to the overall step 2H₂ + O₂-> 2H₂O; so that a one-step chemical-kinetic description, recently derived by asymptotic analysis for near-limit fuel-lean deflagrations, can be used with excellent accuracy to describe the whole branch of stable flame balls. Besides molecular diffusion in a binary-diffusion approximation, Soret diffusion is included, since this exerts a nonnegligible effect to extend the flammability range. When the large value of the activation energy of the overall reaction is taken into account, the leading-order analysis in the reaction-sheet approximation is seen to determine the flame ball radius as that required for radiant heat losses to remove enough of the heat released by chemical reaction at the flame to keep the flame temperature at a value close to crossover. The results are relevant to burning velocities at lean equivalent ratios and may influence fire-safety issues associated with hydrogen utilization.

Some theoretical aspects in computational analysis of adhesive lap joints

This paper is devoted to the numerical analysis of bidimensional bonded lap joints. For this purpose, the stress singularities occurring at the intersections of the adherend-adhesive interfaces with the free edges are first investigated and a method for computing both the order and the intensity factor of these singularities is described briefly. After that, a simplified model, in which the adhesive domain is reduced to a line, is derived by using an asymptotic expansion method. Then, assuming that the assembly debonding is produced by a macro-crack propagation in the adhesive, the associated energy release rate is computed. Finally, a homogenization technique is used in order to take into account a preliminary adhesive damage consisting of periodic micro-cracks. Some numerical results are presented.