Cellular core structures are the state-of-the-art technology for light weight structures in the aerospace industry. In an aerospace product, sandwich panels with cellular core represent the primary structural component as a given aerospace product may contain a large number of sandwich panels. This reveals the necessity of understanding the mechanical behavior of the cellular core and the impact of that behavior on the overall structural behavior of the sandwich panel, and hence the final aerospace product. As the final aerospace product must go through multiple qualification tests to achieve a final structure that is capable of withstanding all environments possible, analyzing the structure prior to testing is very important to avoid any possible failures and to ensure that the final design is indeed capable of withstanding the loads. To date, due to the lack of full understanding of the mechanical behavior of cellular cores and hence the sandwich panels, there still remains a significant lack of analytical capability to predict the proper behavior of the final product and failures may still occur even with significant effort spent on pre-test analyses. Analyzing cellular core to calculate the equivalent material properties of this type of structure is the only way to properly design the core for sandwich enhanced stiffness to weight ratio of the sandwich panels. A detailed literature review is first conducted to access the current state of development of this research area based on experiment and analysis. Then, one of the recently developed homogenization schemes is chosen to investigate the mechanical behavior of heavy, non-corrugated square cellular core with a potential application in marine structures. The mechanical behavior of the square cellular core is then calculated by applying the displacement approach to a representative unit cell finite element model. The mechanical behavior is then incorporated into sandwich panel finite element model and in an in-house code to test the predicted mechanical properties by comparing the center-of-panel displacement from all analyses to that of a highly detailed model. The research is then expanded to cover three cellular core shapes, hexagonal cores made of corrugated sheets, square cores made of corrugated sheets, and triangular cores. The expansion covers five different cell sizes and twenty one different core densities for each of the core shapes considering light cellular cores for space applications, for a total of 315 detailed studies. The accuracy of the calculated properties for all three core shapes is checked against highly detailed finite element models of sandwich panels. Formulas are then developed to calculate the mechanical properties of the three shapes of cellular cores studied for any core density and any of the five cell sizes. An error analysis is then performed to understand the quality of the predicted equivalent properties considering the panel size to cell size ratio as well as the facesheet thickness to core thickness ratio. The research finally expanded to understand the effect of buckling of the unit cell on the equivalent mechanical property of the cellular core. This part of the research is meant to address the impact of the local buckling that may occur due to impact of any type during the manufacturing, handling or assembly of the sandwich panels. The variation of the equivalent mechanical properties with the increase in transverse compression load, until the first folding of the unit cell is complete, is calculated for each of the three core shapes under investigation. / Ph. D.

No description available.

The behavior of jointed rock mass is very complex and is influenced by many factors such as location of joints, joint frequency, joint orientation and joint strength. A thorough review of literature on different aspects of jointed rock mass indicate that the discontinuities or planes of weakness present in rock mass significantly influence its behavior. Numerous experimental tests were conducted to study the behavior of natural as well as artificial joints in rocks. Laboratory tests are time consuming and give results applicable to specific joint fabric and confining pressure. Numerical methods are the best alternative to laboratory tests to study the behavior of jointed rock mass. With the advent of computers numerical methods of analysis have become very popular, as they are highly flexible and can represent all complex geometries and material behavior. The accuracy of a numerical model depends upon the how well constitutive relations for the jointed rock mass are defined in the analysis. Empirical relationships for describing the mechanical behavior of discontinuities obtained from scaling the laboratory data is crucial unresolved problem, which will affect the quality of results obtained. One more important aspect in the numerical model is strength criteria used for jointed rock mass. The applicability of existing strength criteria to a particular jointed rock has to be carefully examined before they are used. Equivalent continuum approach simplifies the modeling of jointed rock mass as the joints are not modeled separately. Instead in equivalent continuum approach the jointed rock mass is represented by an equivalent continuum whose properties are defined by a combination of intact rock properties and joint properties. The accuracy of this kind of modeling depends upon the relationships used to define the jointed rock mass properties as a function of intact rock properties and joint properties. In the present study, an effort has been made (i) to establish empirical relations to define the properties of jointed rock mass as a function of intact rock properties and joint factor (ii) to develop a numerical model based on equivalent continuum approach using the empirical relations derived above, for easy and efficient modeling of jointed rock mass (iii) comparison of existing strength criteria for jointed rock masses using the equivalent continuum model developed above (iv) Modeling of joints explicitly and comparing these results with the equivalent continuum model results. Empirical relationships expressing the uniaxial compressive strength and elastic modulus of jointed rock as a function of corresponding intact rock properties and joint factor have been derived based on the statistical analysis of large amount of experimental data of uniaxial and triaxial tests collected from the literature. The effect of joints in the jointed rock is taken in to account by the joint factor. A comparative study of the empirical relationships arrived by the above analysis has been made to choose the best relation for the numerical analysis. Empirical relationships thus arrived for jointed rock mass are used in the equivalent continuum approach to represent the jointed rock properties as a combination of intact rock properties and joint factor. Equivalent continuum model developed is thoroughly tested, validated and applied for single, multiple and block jointed rocks. The equivalent continuum model developed has been applied for analysis of the power cavern for Shiobara power station. Different strength criteria available for jointed rock namely Mohr-Coulomb, Hoek and Drown, Yudhbir et al. and Rarnamurthy are incorporated in the equivalent continuum model to evaluate their applicability for jointed rock masses. Ramarnurthy's strength criterion gives the best values of failure stress for almost all the test cases and hence used in the equivalent continuum model. Alternatively, the joints in jointed rock mass are represented explicitly using interface element in the nonlinear finite element analysis. The explicit finite element model has been tested and validated using the experimental stress strain curves and failure stress values. Comparison of results obtained using equivalent continuum analysis and explicit modeling of joints has been given in the form of stress strain curves and failure stress plots for jointed rock masses along with the experimental results. Some of the major conclusions from the present study are as follows. Statistical relationships arrived to express the properties of the jointed rock as a function of intact rock and joint factor give a fair estimate of jointed rock in the absence of experimental data. Equivalent continuum model developed using statistical relations arrived above simplifies the numerical modeling of jointed rock to a large extent and also gives a fair estimate of jointed rock behavior with minimum input data. From the equivalent continuum analysis of Shiobara power cavern, it can be concluded that this approach is very advantageous for modeling highly discontinuous systems provided the joint factor is estimated properly so that it represents the real fabric of the joints present in the system. Comparison of different strength criteria shows that Ramamurthy's strength criterion is the best for jointed rocks. When the rock mass has one or two major joints it is advantageous to model it explicitly so that the behavior of the joint can be studied in detail. Explicit representation of the joints in the finite element analysis gives a lair estimate of the zones most susceptible to failure in a jointed rock. From comparison of experimental values, equivalent continuum model results and the explicit joint model results, it can be concluded that results obtained using equivalent continuum model are nearest to the experimental results in almost all the cases.

<p>The equivalent continuum approach uses equivalent propertiesof rock mass as the input data for a continuum analysis. Thisis a common modeling method used in the field of rock mechanicsand hydrogeology. However, there are still unresolvedquestions; how can the equivalent properties be determined andis the equivalent continuum approach suitable for modeling thediscontinuous fractured rock mass.</p><p>The purpose of this paper is to establish a methodology todetermine the equivalent hydraulic and mechanical properties offractured rock masses by explicit representations of stochasticfracture systems, to investigate the scale-dependency of theproperties, and to investigate the conditions for theapplication of the equivalent continuum approach for thefractured rock masses. Geological data used for this study arefrom the site characterization of Sellafield, Cumbria, UK. Aprogram for the generation of stochastic Discrete FractureNetwork (DFN) is developed for the realization of fractureinformation and ten parent DFN models are constructed based onthe location, trace length, orientation and density offractures. Square models with the sizes varying from 0.25 m× 0.25 m to 10 m × 10 m are cut from the center ofthe each parent network to be used for the scale dependencyinvestigation. A series of the models in a parent network arerotated in 30 degrees interval to be used for investigation oftensor characteristic. The twodimensional distinct elementprogram, UDEC, was used to calculate the equivalentpermeability and compliance tensors based on generalizedDarcy’s law and general theory of anisotropic elasticity.Two criteria for the applicability of equivalent continuumapproach were established from the investigation: i) theexistence of properly defined REV (Representative ElementaryVolume) and ii) existence of the tensor in describing theconstitutive equation of fractured rock The equivalentcontinuum assumption cannot be accepted if any one of the abovetwo criteria is not met. Coefficient of variation and meanprediction error is suggested for the measures toquantitatively evaluate the errors involved in scale dependencyand tensor characteristic evaluation.</p><p>Equivalent permeability and mechanical properties (includingelastic modulus and Poisson’s ratios) determined onrealistic fracture network show that the presence of fracturehas a significant effect on the equivalent properties. Theresults of permeability, elastic moduli and Poisson's ratioshow that they narrow down with the increase of scale andmaintain constant range after a certain scales with someacceptable variation. Furthermore, Investigations of thepermeability tensor and compliance tensor in the rotated modelshow that their tensor characteristics are satisfied at acertain scale; this would indicate that the uses of theequivalent continuum approach is justified for the siteconsidered in this study.</p><p>The unique feature of the thesis is that it gives asystematic treatment of the homogenization and upscaling issuesfor the hydraulic and mechanical properties of fractured rockswith a unified approach. These developments established a firmfoundation for future application to large-scale performanceassessment of underground nuclear waste repository byequivalent continuum analysis.</p><p><b>Keywords :</b>Equivalent continuum approach, Equivalentproperty, Representative Elementary Volume (REV), DistinctElement Method, Discrete Fracture Network (DFN)</p>

No description available.

The equivalent continuum approach uses equivalent propertiesof rock mass as the input data for a continuum analysis. Thisis a common modeling method used in the field of rock mechanicsand hydrogeology. However, there are still unresolvedquestions; how can the equivalent properties be determined andis the equivalent continuum approach suitable for modeling thediscontinuous fractured rock mass. The purpose of this paper is to establish a methodology todetermine the equivalent hydraulic and mechanical properties offractured rock masses by explicit representations of stochasticfracture systems, to investigate the scale-dependency of theproperties, and to investigate the conditions for theapplication of the equivalent continuum approach for thefractured rock masses. Geological data used for this study arefrom the site characterization of Sellafield, Cumbria, UK. Aprogram for the generation of stochastic Discrete FractureNetwork (DFN) is developed for the realization of fractureinformation and ten parent DFN models are constructed based onthe location, trace length, orientation and density offractures. Square models with the sizes varying from 0.25 m× 0.25 m to 10 m × 10 m are cut from the center ofthe each parent network to be used for the scale dependencyinvestigation. A series of the models in a parent network arerotated in 30 degrees interval to be used for investigation oftensor characteristic. The twodimensional distinct elementprogram, UDEC, was used to calculate the equivalentpermeability and compliance tensors based on generalizedDarcy’s law and general theory of anisotropic elasticity.Two criteria for the applicability of equivalent continuumapproach were established from the investigation: i) theexistence of properly defined REV (Representative ElementaryVolume) and ii) existence of the tensor in describing theconstitutive equation of fractured rock The equivalentcontinuum assumption cannot be accepted if any one of the abovetwo criteria is not met. Coefficient of variation and meanprediction error is suggested for the measures toquantitatively evaluate the errors involved in scale dependencyand tensor characteristic evaluation. Equivalent permeability and mechanical properties (includingelastic modulus and Poisson’s ratios) determined onrealistic fracture network show that the presence of fracturehas a significant effect on the equivalent properties. Theresults of permeability, elastic moduli and Poisson's ratioshow that they narrow down with the increase of scale andmaintain constant range after a certain scales with someacceptable variation. Furthermore, Investigations of thepermeability tensor and compliance tensor in the rotated modelshow that their tensor characteristics are satisfied at acertain scale; this would indicate that the uses of theequivalent continuum approach is justified for the siteconsidered in this study. The unique feature of the thesis is that it gives asystematic treatment of the homogenization and upscaling issuesfor the hydraulic and mechanical properties of fractured rockswith a unified approach. These developments established a firmfoundation for future application to large-scale performanceassessment of underground nuclear waste repository byequivalent continuum analysis. <b>Keywords :</b>Equivalent continuum approach, Equivalentproperty, Representative Elementary Volume (REV), DistinctElement Method, Discrete Fracture Network (DFN) / NR 20140805

In this thesis, fractured rock masses are treated asequivalent continua for large-scale analyses of rockengineering projects. Systematic developments are made for thedetermination of equivalent mechanical and hydraulic propertiesof fractured rock masses using a hybrid discrete fracturenetwork - distinct element method (DFN-DEM) approach. Thedetermined equivalent properties are then used for a far-fieldfinite element analysis of the thermo-mechanical impacts on thestress, deformation and permeability of fractured rockssurrounding a hypothetical geological repository of nuclearwaste. The geological data were extracted from the results ofan extensive site investigation programme at Sellafield, UK,conducted by Nirex UK Ltd. The scale dependencies of the hydraulic and mechanicalproperties were investigated by using multiple realizations ofthe fracture system geometry with increasing model sizes untilproperly defined hydraulic and mechanical representativeelementary volumes (REVs) were reached. The validity of thesecond order permeability tensor and the fourth-ordermechanical compliance tensor were tested for continuum analysesat larger scales. The REV was determined to be around 5 m formechanical and hydraulic data in this study. Analysis of the stress-dependent mechanical and hydraulicproperties shows that the effect of rock stresses is crucial.The elastic moduli increase significantly with the increase ofstress and an empirical equation of stress-dependent elasticmodulus is suggested based on results of numerical experiments.Calculations of the Poisson's ratios suggest greater valuesthan are normally assumed in practice. Depending on the stateof stress, permeability decreases or increases with increasingcompressive stress. Stress-induced flow channeling effect iscaptured by numerical modeling for the first time and detailedmechanisms of shear dilation of fractures are provided. Basedon the numerical experiments, a set of empirical equations wassuggested for the stress-dependent permeability, consideringboth normal deformation and shear dilation of fractures. Thermo-mechanical impact on the performance of ahypothetical repository at a far-field scale (5 km by 1 km) wasinvestigated with the stress-dependent equivalent propertiesdetermined at the REV scale. This analysis shows thatmechanical responses vary significantly depending on how themechanical properties were determined. The change ofpermeability due to the thermal loading is, however, notsignificant in this particular case. The thesis provides a framework for systematic analysis oflarge-scale engineering applications in fractured rock masses,such as geological repositories of nuclear wastes. Keyword:Fractured rock masses, Equivalent Continuum,Discrete Fracture Network (DFN), Distinct Element Method (DEM),Finite Element Method (FEM), Nuclear Waste Disposal, CoupledThermo-Hydro-Mechanical Processes

Most rock masses encountered in civil and mining engineering projects contain pre-existing discontinuities. These discontinuities weaken the rock masses to an extent, which depends very much on the size of engineering structure relation to discontinuity spacing. The strength and deformability of rock mass is controlled not only by the intact portion of rock, but by the characteristic of the joints that break up the mass, particularly their pattern and their orientation with respect to the in-situ stresses. In considering the effect of joints, the discrete approach emerged as an efficient tool and advocated since 1970s (Cundall, 1971). However, the numerical approach with modelling the joints explicitly has the limitation of computational complexity for modelling large-scale problems with extremely large number of joints. As an alternative to this limitation, the equivalent continuum approach models the jointed rock masses as a continuum with the equivalent properties that represent implicitly the effects of the joints. Several numerical methods have been developed by various researchers to model jointed rock masses as equivalent continuum, using various techniques. However, the existing equivalent continuum models are complicated and need more input data from experimental or field testing in order to carry out the analysis. Present approach attempts to use statistical relations, which are simple and obtained after analyzing a large data from the literature on laboratory test results of jointed rock masses. Systematic investigations were done including laboratory experiments to develop the methodologies to determine the equivalent material properties of rock mass and their stress-strain behaviour, using a hyperbolic approach (Duncan and Chang, 1970). Present study covers the development of equivalent continuum model for rock mass right from developing statistical correlations to find out equivalent material properties based on experimental results, to the implementation of the model in FLAC3D for 3-dimensional applications and subsequently verification leading to real field application involving jointed rocks. Experimental work carried out to study the strength and deformation characteristics of jointed rock by using standard laboratory tests on cylindrical specimens of plaster of Paris by introducing artificial joints. The objective was to derive the compressive strength and elastic modulus of rock mass as a function of intact rock strength/modulus and joint factor. The obtained experimental results and developed relations were compared with the previous experimental data on jointed rocks. Further, a failure criterion as proposed by Ramamurthy (1993) has been validated from these experimental results of intact and jointed rock specimens. Empirical relationships similar to Ramamurthy’s relations are established for the prediction of rock mass strength and were compared with proposed equation by Ramamurthy (1993) and are found comparable. However, the equations by Ramamurthy were based on different variety of rocks and therefore recommended for further use and were used in numerical models. For efficient application to the field problems the equivalent continuum model is implemented in the program Fast Lagrangian analysis of continua (FLAC3D). The model was rigorously validated by simulating jointed rock specimens. Element tests were conducted for both uniaxial and triaxial cases and then compared with the respective experimental results. The numerical test program includes laboratory tested cylindrical rock specimens of different rock types, from plaster of Paris representing soft rock to granite representing very hard rock. The results of the equivalent continuum modelling were also compared with explicit modelling results where joints were incorporated in the model as interfaces. To represent highly discontinuous system, the laboratory investigation on block jointed specimens of gypsum plaster (Brown and Trollope, 1970) was modelled numerically using equivalent continuum approach. To extend the applicability of the model to field applications, investigation were done by undertaking numerical modelling of two case studies underground caverns, one Nathpa Jhakri hydroelectric power cavern in Himachal Pradesh, India, and the other one Shiobara hydroelectric power cavern in Japan. This study verifies the efficiency of the present approach in estimating ground movement and stress distribution around the excavations in jointed rock masses. The modelling results were also compared with six other computation models as presented by Horii et al. (1999) for the Shiobara power house cavern. An attempt has also been made to numerically model the support system for the cavern and investigate the efficiency of reinforcements using FLAC3D. The model was also used for analyzing large scale slope in jointed rocks using the equivalent continuum model by undertaking numerical modelling of Anji bridge abutment slopes, in Jammu and Kashmir, India. Slope stability analysis is done using equivalent continuum approach for both, the original profiles as well as with the pier loads on cut profiles. Attempt was also made to exhibit the shear strength dependency of the strain though the hyperbolic stress- strain model. The shear strain developed in the slope increases with reducing the shear strength. The relationship between the shear strength reduction ratio ‘R’ and axial strain ‘ε’, for different values of failure ratio ‘Rf’ was studied and it was observed that, the value of ‘ε’ increases, as the value of ‘R’ increases especially it increases rapidly when the value ‘R’ approaches certain critical value, which varies with the value of ‘Rf’. This critical value of R is known as the critical shear strength reduction factor Rc and is highly sensitive to the confining stress. As the value of Rf increases, representing a transition from linear elastic nature to nonlinear nature, the value of critical shear strength reduction ratio also decreases. Relationship between the critical shear strength reduction ratio and the safety factor were examined to elucidate their physical meaning. It was observed that at critical value of the shear strength reduction ratio, a well defined failure shear zone developed from the toe to the crest of the slope. Intelligent models using ANNs were also developed to predict the elastic modulus of jointed rocks as an alternative to empirical equations and without predefining a mathematical model to correlate the properties.

<p>In this thesis, fractured rock masses are treated asequivalent continua for large-scale analyses of rockengineering projects. Systematic developments are made for thedetermination of equivalent mechanical and hydraulic propertiesof fractured rock masses using a hybrid discrete fracturenetwork - distinct element method (DFN-DEM) approach. Thedetermined equivalent properties are then used for a far-fieldfinite element analysis of the thermo-mechanical impacts on thestress, deformation and permeability of fractured rockssurrounding a hypothetical geological repository of nuclearwaste. The geological data were extracted from the results ofan extensive site investigation programme at Sellafield, UK,conducted by Nirex UK Ltd.</p><p>The scale dependencies of the hydraulic and mechanicalproperties were investigated by using multiple realizations ofthe fracture system geometry with increasing model sizes untilproperly defined hydraulic and mechanical representativeelementary volumes (REVs) were reached. The validity of thesecond order permeability tensor and the fourth-ordermechanical compliance tensor were tested for continuum analysesat larger scales. The REV was determined to be around 5 m formechanical and hydraulic data in this study.</p><p>Analysis of the stress-dependent mechanical and hydraulicproperties shows that the effect of rock stresses is crucial.The elastic moduli increase significantly with the increase ofstress and an empirical equation of stress-dependent elasticmodulus is suggested based on results of numerical experiments.Calculations of the Poisson's ratios suggest greater valuesthan are normally assumed in practice. Depending on the stateof stress, permeability decreases or increases with increasingcompressive stress. Stress-induced flow channeling effect iscaptured by numerical modeling for the first time and detailedmechanisms of shear dilation of fractures are provided. Basedon the numerical experiments, a set of empirical equations wassuggested for the stress-dependent permeability, consideringboth normal deformation and shear dilation of fractures.</p><p>Thermo-mechanical impact on the performance of ahypothetical repository at a far-field scale (5 km by 1 km) wasinvestigated with the stress-dependent equivalent propertiesdetermined at the REV scale. This analysis shows thatmechanical responses vary significantly depending on how themechanical properties were determined. The change ofpermeability due to the thermal loading is, however, notsignificant in this particular case.</p><p>The thesis provides a framework for systematic analysis oflarge-scale engineering applications in fractured rock masses,such as geological repositories of nuclear wastes.</p><p><b>Keyword:</b>Fractured rock masses, Equivalent Continuum,Discrete Fracture Network (DFN), Distinct Element Method (DEM),Finite Element Method (FEM), Nuclear Waste Disposal, CoupledThermo-Hydro-Mechanical Processes</p>