Academic literature on the topic 'Anisotropic Elastic Members'

The Bakken Formation consists of three members: The Upper Bakken and Lower Bakken are dark marine shales with high organic content, whereas the Middle Bakken consists of mixed carbonates and clastics and is the main reservoir unit, despite having low porosity and permeability. Dipole S-wave data acquired in a lateral well in the Middle Bakken Formation revealed this formation to be anisotropic. Backus upscaling of logs acquired in a nearby vertical pilot well in the same layers sampled by the lateral well gave estimates of the anisotropy that were too small to explain the S-wave anisotropy measured in the lateral well. The observed anisotropy was interpreted in terms of bedding-parallel compliant discontinuities such as microcracks and low-aspect-ratio pores. The presence of bedding-parallel microcracks and low-aspect-ratio pores may contribute to the permeability of the tight Middle Bakken reservoir, and the sensitivity of P- and S-wave velocities to the presence of microcracks and low aspect ratio pores suggested the use of sonic and seismic measurements for identifying the productive zones in the low-permeability Middle Bakken reservoir.

We consider a periodic lattice structure in d =2 or 3 dimensions with unit cell comprising Z thin elastic members emanating from a similarly situated central node. A general theoretical approach provides an algebraic formula for the effective elasticity of such frameworks. The method yields the effective cubic elastic constants for three-dimensional space-filling lattices with Z =4, 6, 8, 12 and 14, the last being the ‘stiffest’ lattice proposed by Gurtner & Durand (Gurtner & Durand 2014 Proc. R. Soc. A 470 , 20130611. ( doi:10.1098/rspa.2013.0611 )). The analytical expressions provide explicit formulae for the effective properties of pentamode materials, both isotropic and anisotropic, obtained from the general formulation in the stretch-dominated limit for Z = d +1.

Geodesic gridshells are shell structures made of continuous elements following geodesic lines. Their properties ease the use of beams with anisotropic cross-sections by avoiding bending about their strong axis. However, such bending may arise when flattening arbitrary geodesic grids, which forbids their initial assembly on the ground. This study provides a process to design elastic geodesic gridshells, that is, gridshells that minimise bending moments in both formed and near-flat configurations. The generation process first brings a target geodesic network onto a plane by maintaining arc lengths. The flat mesh is then relaxed to minimise its main curvatures and hence bending moments in its members. The result is an elastic geodesic gridshell that can be assembled flat on the ground and then lifted up into its target surface. The method is applied to the design of six geodesic gridshells made of reclaimed skis.

This paper presents a methodology and some results on the dynamic stability of an elastic rotating system consisting of one- and twodimensional members. These parts may contain different kinds of unsymmetries: either from mass- or stiffness imperfections or from anisotropic especially hydrodynamic bearings. The equations of motion are formulated using virtual work and an Finite Element approach. Special attention is paid to a kinematically consistent coupling of the elastic shell and disc. The eigenvalue extraction is based upon the method of Lanczos including a modal reduction and a correction process in order to ensure true diagonal system matrices. Some typical results for a shaft-disc-shell system with different bearings and imperfections are presented in detail.

An incremental three-dimensional constitutive relation for concrete has been developed. The linear anisotropic and path-dependent behavior is modeled by updating the stiffness matrix at each load increment. The material is assumed incrementally elastic and the six elastic moduli E11, E12 .... E33 are expressed in terms of both the tangential hydrostatic and deviatoric stiffness whereas the three tangential shear moduli are expressed in terms of the deviatoric stiffness only. The hydrostatic and deviatoric stiffness are determined from uniaxial stress-strain relationships by employing the space truss concept. The unaxial stress-strain relationships are in a sense the stress-strain relationships of the members of the truss, and they were based on a rheological stochastic model developed earlier. The predictions of the model compare favorably with experimental data reported by various investigators. Complex loading paths are reproduced with acceptable accuracy as is demonstrated in the second part of this paper.

Proteins represent one of the most important building blocks for most biological processes. Their biological mechanisms have been found to correlate significantly with their dynamics, which is commonly investigated through molecular dynamics (MD) simulations. However, important insights on protein dynamics and biological mechanisms have also been obtained via much simpler and computationally efficient calculations based on elastic lattice models (ELMs). The application of structural mechanics approaches, such as modal analysis, to the protein ELMs has allowed to find impressive results in terms of protein dynamics and vibrations. The low-frequency vibrations extracted from the protein ELM are usually found to occur within the terahertz (THz) frequency range and correlate fairly accurately with the observed functional motions. In this contribution, the global vibrations of lysozyme will be investigated by means of a finite element (FE) truss model, and we will show that there exists complete consistency between the proposed FE approach and one of the more well-known ELMs for protein dynamics, the anisotropic network model (ANM). The proposed truss model can consequently be seen as a simple method, easily accessible to the structural mechanics community members, to analyze protein vibrations and their connections with the biological activity.

Full-field numerical solutions for a crack which lies along the interface of an elastic-plastic medium and a rigid substrate are presented. The solutions are obtained using a small strain version of the J2-deformation theory with power-law strain hardening. In the present article, results for loading causing only small scale yielding at the crack tip are described; in subsequent articles the mathematical structure of the crack-tip fields under small scale yielding and results for contained yielding and fully plastic behavior will be presented. We find that although the near-tip fields do not appear to have a separable singular form, of the HRR-type fields as in homogeneous media, they do, however, bear interesting similarities to certain mixed-mode HRR fields. Under small scale yielding, where the remote elastic fields are specified by a complex stress-concentration vector Q = |Q|eiφ with φ being the phase angle between the two in-plane stress modes, we find that the plastic fields are members of a family parameterized by a new phase angle ξ, ≡ φ + εln(QQ/σ02L), and the fields nearly scale with the well-defined energy release rate as evaluated by the J-integral. Here σ0 is the reference yield stress and L is the total crack length (or a relevant length of the crack geometry). Numerical procedures appropriate for solving a general class of interface crack problems are also presented. A description of a numerical method for extracting the mixed mode stress intensities for cracks at interfaces and in homogeneous isotropic or anisotropic media, is included.

Understanding the controls on the elastic properties of reservoir rocks is crucial for exploration and successful production from hydrocarbon reservoirs. We studied the static and dynamic elastic properties of shale gas reservoir rocks from Barnett, Haynesville, Eagle Ford, and Fort St. John shales through laboratory experiments. The elastic properties of these rocks vary significantly between reservoirs (and within a reservoir) due to the wide variety of material composition and microstructures exhibited by these organic-rich shales. The static (Young’s modulus) and dynamic (P- and S-wave moduli) elastic parameters generally decrease monotonically with the clay plus kerogen content. The variation of the elastic moduli can be explained in terms of the Voigt and Reuss limits predicted by end-member components. However, the elastic properties of the shales are strongly anisotropic and the degree of anisotropy was found to correlate with the amount of clay and organic content as well as the shale fabric. We also found that the first-loading static modulus was, on average, approximately 20% lower than the unloading/reloading static modulus. Because the unloading/reloading static modulus compares quite well to the dynamic modulus in the rocks studied, comparing static and dynamic moduli can vary considerably depending on which static modulus is used.

Source-rock reservoirs are fine-grained petroleum source rocks (“shales” or “mudstones”) having geomechanical properties that allow those rocks to produce hydrocarbons at economic rates after stimulation by hydraulic fracturing. Many of the assumptions commonly adopted by geophysicists to characterize shales cannot be applied to source-rock reservoirs. For example, the mineralogies of many source-rock reservoirs are not dominated by clay minerals and so mathematical and/or conceptual models developed for clay-dominated mudstones are not appropriate and cannot be applied to them. Instead, mudstones of shale plays are generally dominated by biogenic calcite and/or quartz. We use terminology of sedimentary geology to show that anisotropy is scale-dependent in source-rock reservoirs, and we discuss the depositional and diagenetic processes that control these and other geophysical properties of interest. The mudstones of source-rock reservoirs may or may not be anisotropic at the lamination scale (i.e., millimeters), the scale commonly used to measure anisotropic parameters via core plugs, but they are nearly always anisotropic at the bedset (centimeters to several meters) and member (tens of meters) scales. Because of the anisotropic nature of mudstones, elastic properties are not scalars at the length/thickness scales that can be defined using seismic methods. Properties of interest are likely to be different parallel to bedding compared to perpendicular to bedding. Because of the subseismic scale of much of this variability, thin-bed effects are likely to influence the AVO behavior of source-rock reservoirs.

AbstractA survey of various methods to measure the elastic constants Ki of nematic liquid crystals is given. To determine K1, K3 and the anisotropy of the diamagnetic susceptibility Δx of two members of the 5n-hexyl-2-[4n-alkyloxy-phenyl]-pyrimidines and a mixture of both, we used a combined electromagneto- optical method, consisting in independent measurements of the optical phase difference in electric and magnetic fields acting on the same cell. The temperature dependence of the K1- and K3-values for these phenylpyrimidines can be explained by common theories. The Δx data show the same temperature dependence as the values of the orientational order parameter S obtained by 1H-NMR.

The goal of this work is to develop and demonstrate a comprehensive analysis of single and multi-body composite strip-beam systems using an asymptotically-correct geometrically nonlinear theory. The comprehensiveness refers to the two distinguishing features of this work, namely the unified framework for the analysis and the inclusion of the usually ignored cross-sectional nonlinearities in thin-beam and multi-beam analyses. The first part of this work stitches together an approach to analyse generally anisotropic composite beams. Based on geometrically exact nonlinear elasticity theory, the nonlinear 3-D beam problem splits into either a linear (conventionally considered) or nonlinear (considered in this work) 2-D analysis of the beam cross-section and a nonlinear 1-D analysis along the beam reference curve. The two sub-tasks of this work (viz. nonlinear analysis of the beam cross-section and nonlinear beam analysis) are accomplished on a single platform using an object-oriented framework. First, two established nonlinear cross-sectional analyses (numerical and analytical), both based on the Variational-Asymptotic Method (VAM), are invoked. The numerical analysis is capable of treating cross-sections of arbitrary geometry and material distributions and can capture certain nonlinear effects such as the trapeze effect. The closed-form analytical analysis is restricted to thin rectangular cross-sections for generally anisotropic composites but captures ALL cross-sectional nonlinearities, and not just the well-known Brazier and trapeze effects. Second, the well-established geometrically-exact nonlinear 1-D governing equations along the beam reference curve, after being generalized to utilize the expressions for nonlinear stiffness matrix, are solved using the mixed variational finite element method. Finally, local 3-D stress, strain and displacement fields for representative sections in the beam are recovered, based on the stress resultants from the 1-D global beam analysis. This part of the work is then validated by applying it to an initially twisted cantilevered laminated composite strip under axial force. The second part is concerned with the dynamic analysis of nonlinear multi-body systems involving elastic strip-like beams made of laminated, anisotropic composite materials using an object-oriented framework. In this work, unconditionally stable time-integration schemes presenting high-frequency numerical dissipation are used to solve the ensuing governing equations. The codes developed based on such time-integration schemes are first validated with the literature for two standard test cases: non-linear spring mass oscillator and pendulum. In order to apply the comprehensive analysis code thus developed to a multi-body system, the four-bar mechanism is chosen as an example. All component bars of the mechanism have thin rectangular cross-sections and are made of fiber reinforced laminates of various types of layups. They could, in general, be pre-twisted and/or possess initial curvature, either by design or by defect. They are linked to each other by means of revolute joints. Each component of the mechanism is modeled as a beam based on the first part of this work. Results from this analysis are compared with those available in the literature, both theoretical and experimental. The margins between the linear and non-linear results are evaluated specifically due to the cross-sectional nonlinearities and shown to vary with stacking sequences. This work thus demonstrates the importance of geometrically nonlinear cross-sectional analysis of certain composite beam-based four-bar mechanisms in predicting system dynamic characteristics. To enable graphical visualization, the behavior of the four-bar mechanism is also observed by using commercial software (I-DEAS + NASTRAN + ADAMS). Finally, the component-laminate load-carrying capacity is estimated using the Tsai-Wu-Hahn failure criterion for various layups and the same criterion is used to predict the first-ply-failure and the mechanism as a whole.

The goal of this work is to develop and demonstrate a comprehensive analysis of single and multi-body composite strip-beam systems using an asymptotically-correct geometrically nonlinear theory. The comprehensiveness refers to the two distinguishing features of this work, namely the unified framework for the analysis and the inclusion of the usually ignored cross-sectional nonlinearities in thin-beam and multi-beam analyses. The first part of this work stitches together an approach to analyse generally anisotropic composite beams. Based on geometrically exact nonlinear elasticity theory, the nonlinear 3-D beam problem splits into either a linear (conventionally considered) or nonlinear (considered in this work) 2-D analysis of the beam cross-section and a nonlinear 1-D analysis along the beam reference curve. The two sub-tasks of this work (viz. nonlinear analysis of the beam cross-section and nonlinear beam analysis) are accomplished on a single platform using an object-oriented framework. First, two established nonlinear cross-sectional analyses (numerical and analytical), both based on the Variational-Asymptotic Method (VAM), are invoked. The numerical analysis is capable of treating cross-sections of arbitrary geometry and material distributions and can capture certain nonlinear effects such as the trapeze effect. The closed-form analytical analysis is restricted to thin rectangular cross-sections for generally anisotropic composites but captures ALL cross-sectional nonlinearities, and not just the well-known Brazier and trapeze effects. Second, the well-established geometrically-exact nonlinear 1-D governing equations along the beam reference curve, after being generalized to utilize the expressions for nonlinear stiffness matrix, are solved using the mixed variational finite element method. Finally, local 3-D stress, strain and displacement fields for representative sections in the beam are recovered, based on the stress resultants from the 1-D global beam analysis. This part of the work is then validated by applying it to an initially twisted cantilevered laminated composite strip under axial force. The second part is concerned with the dynamic analysis of nonlinear multi-body systems involving elastic strip-like beams made of laminated, anisotropic composite materials using an object-oriented framework. In this work, unconditionally stable time-integration schemes presenting high-frequency numerical dissipation are used to solve the ensuing governing equations. The codes developed based on such time-integration schemes are first validated with the literature for two standard test cases: non-linear spring mass oscillator and pendulum. In order to apply the comprehensive analysis code thus developed to a multi-body system, the four-bar mechanism is chosen as an example. All component bars of the mechanism have thin rectangular cross-sections and are made of fiber reinforced laminates of various types of layups. They could, in general, be pre-twisted and/or possess initial curvature, either by design or by defect. They are linked to each other by means of revolute joints. Each component of the mechanism is modeled as a beam based on the first part of this work. Results from this analysis are compared with those available in the literature, both theoretical and experimental. The margins between the linear and non-linear results are evaluated specifically due to the cross-sectional nonlinearities and shown to vary with stacking sequences. This work thus demonstrates the importance of geometrically nonlinear cross-sectional analysis of certain composite beam-based four-bar mechanisms in predicting system dynamic characteristics. To enable graphical visualization, the behavior of the four-bar mechanism is also observed by using commercial software (I-DEAS + NASTRAN + ADAMS). Finally, the component-laminate load-carrying capacity is estimated using the Tsai-Wu-Hahn failure criterion for various layups and the same criterion is used to predict the first-ply-failure and the mechanism as a whole.

This paper describes a comprehensive approach to analyse anisotropic composite beams. Based on geometrically non-linear elasticity theory, the non-linear 3-D beam problem splits into either a linear or non-linear 2-D analysis of the beam cross-section and a non-linear 1-D analysis along the beam reference line. Usually cross-sectional analyses are linear, but there are a few exceptions, like the “trapeze effect” and “Brazier effect”. The two sub-tasks of this work (viz. non-linear analysis of the beam cross-section and non-linear beam analysis) are to be accomplished on a single platform using object-oriented framework. First, we perform a non-linear numerical cross-sectional analysis, based on the Variational-Asymptotic Method (VAM). It is capable of treating cross-sections of arbitrary geometry and generally anisotropic material. Second, we formulate the comprehensively non-linear 1-D governing equations along the beam reference line using the mixed variational method and the expressions for non-linear stiffness matrix. The dynamic response of non-linear, flexible multibody systems is thus simulated within the framework of energy-preserving and energy-decaying time integration schemes that provide unconditional stability for non-linear systems. Finally, local 3-D stress, strain and displacement fields for representative sections in the component beams are recovered, based on the stress resultants from a 1-D global beam analysis. Results from this analysis are compared with those available in the literature, both theoretical and experimental, and focus on the behavior of multi-body systems involving members with elastic couplings.