Notes on the foundations of the theory of small displacements of orthotropic shells
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THEORIES - PLATES↗
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Measurements of mixed-mode crack surface displacements and comparison with theory
A theoretical and an experimental technique is used to determine crack surface displacements under mixed-mode conditions. Crack surface displacements proved to be quite useful in mode 1 fracture analysis in that they are directly related to strain energy release rate and stress intensity factor. It is felt that similar relationships can be developed for the mixed-mode case. A boundary-integral method was developed for application to two-dimensional fracture mechanics problems. This technique was applied to the mixed-mode problem. A laser interferometry technique, for measurement of crack surface displacements under mixed-mode conditions, is presented. The experimental measurements are reported and the results of the two approaches are compared and discussed.
Altiero, N. J., Jr.↗
Measurements of mixed-mode crack surface displacements and comparison with theory
The problem of a finite-width tension specimen containing a crack oriented at various angles to the load axis is attacked from experimental and theoretical viewpoints. Displacements of an electro-machined slot, 12.5 mm long and oriented at angles of 0, 15, 30, 45, 60, and 75 deg, are measured using a laser-based in-plane measuring technique. Various width specimens, ranging from a crack-length/width ratio of 0.167 to 0.794, are tested. A boundary-integral equation method is extended to deal with the presence of a sharp crack. Agreement between the two approaches is generally good except near the tips of the cracks.
Sharpe, W. N., Jr.↗
On refined theories of composite laminates
A review of the developments in the displacement-based theories of plates is presented. It is shown that many of the third-order theories reported in the literature are based on the same or equivalent displacement field, and thus lead to the same results. A strain-consistent third-order theory is presented, which contains most existing third-order plate theories as special cases. The layer-wise laminate plate theory proposed is also reviewed. The accuracy of the theories is evaluated by comparing the numerical solutions with those of the three-dimensional elasticity theory.
Reddy, J. N.↗
Pressure broadening of rotational bands. I - A statistical theory
Absorption of electromagnetic waves by rotational transitions of molecules is formulated for the case in which the wave frequency is displaced from resonance by an amount large compared to the reciprocal duration of a typical binary collision, and also large compared to the differences between frequencies of the strong resonances of the gas. In this far-wing limit, Fano's relaxation operator is reduced to a scalar parameter which depends on the frequency displacement. This relaxation parameter is not symmetric with respect to reflection about resonance, but becomes symmetric when multiplied by the factor exp (h/2pi) (omega sub d)/2kT where omega sub d is the frequency displacement. The theory applies to dipolar molecules of any shape, in collisions with either dipolar or quadrupolar molecules.
Rosenkranz, P. W.↗
The surface and through crack problems in layered orthotropic plates
An analytical method is developed for a relatively accurate calculation of Stress Intensity Factors in a laminated orthotropic plate containing a through or part-through crack. The laminated plate is assumed to be under bending or membrane loading and the mode 1 problem is considered. First three transverse shear deformation plate theories (Mindlin's displacement based first-order theory, Reissner's stress-based first-order theory, and a simple-higher order theory due to Reddy) are reviewed and examined for homogeneous, laminated and heterogeneous orthotropic plates. Based on a general linear laminated plate theory, a method by which the stress intensity factors can be obtained in orthotropic laminated and heterogeneous plates with a through crack is developed. Examples are given for both symmetrically and unsymmetrically laminated plates and the effects of various material properties on the stress intensity factors are studied. In order to implement the line-spring model which is used later to study the surface crack problem, the corresponding plane elasticity problem of a two-bonded orthotropic plated containing a crack perpendicular to the interface is also considered. Three different crack profiles: an internal crack, an edge crack, and a crack terminating at the interface are considered. The effect of the different material combinations, geometries, and material orthotropy on the stress intensity factors and on the power of stress singularity for a crack terminating at the interface is fully examined. The Line Spring model of Rice and Levy is used for the part-through crack problem. The surface crack is assumed to lie in one of the two-layered laminated orthotropic plates due to the limitation of the available plane strain results. All problems considered are of the mixed boundary value type and are reduced to Cauchy type of singular integral equations which are then solved numerically.
Erdogan, Fazil↗
Analysis of laminated, composite, circular cylindrical shells with general boundary conditions
This report develops: (1) a refined approximate theory for the static and dynamic analyses of finite, laminated, composite, circular cylindrical shells with general boundary conditions; (2) an exact three-dimensional analysis of simply supported, laminated, composite, circular cylindrical shells, and (3) a thin-shell theory for laminated, composite, circular cylindrical shells. In the refined approximate theory the displacements are assumed piecewise linear across the thickness and the effects of transverse shear deformations and transverse normal stress are included. A variational approach is followed to obtain the governing differential equations and boundary conditions. A general solution of the governing differential equations is also presented. The results obtained by using the refined approximate theory and the thin-shell theory are compared with the exact results for the case of free vibrations of simply supported, laminated, composite, circular cylindrical shells. The refined approximate theory is very accurate, even for thick shells with short nodal distances, whereas thin-shell theory is reasonably accurate only for thin shells at moderate nodal distances and wave number less than 2.
Srinivas, S.↗
:Theoretical investigation of the effect of the ailerons on the wing of an airplane
The present work investigates, on the basis of Prandtl's wing theory, the form of the lift distribution when the ailerons are deflected in opposite directions. An ideal fluid and a wing with a rectangular form are assumed. The moments must not cause any rotation of the wing or any deviation from the rectilinear motion.
AILERONS - DISPLACEMENT↗
Non-Classical Stress Concentration Behavior in a Radically Stretched Hyperelastic Sheet Containing a Circular Hole
Non-classical stress concentration behavior in a stretched circular hyperelastic sheet (outer radius b = 10 in., thickness t = 0.0625 in.) containing a central hole (radius a = 0.5 in.) was analyzed. The hyperelastic sheet was subjected to different levels of remote radial stretchings. Nastran large-strain large-deformation analysis and the Blatz-Ko large deformation theory were used to calculate the equal-biaxial stress concentration factors K. The results show that the values of K calculated from the Blatz-Ko theory and Nastran are extremely close. Unlike the classical linear elasticity theory, which gives the constant K = 2 for the equal-biaxial stress field, the hyperelastic K values were found to increase with increased stretching and can exceed the value K = 6 at a remote radial extension ratio of 2.35. The present K-values compare fairly well with the K-values obtained by previous works. The effect of the hole-size on K-values was investigated. The values of K start to decrease from a hole radius a = 0.125 in. down to K = 1 (no stress concentration) as a shrinks to a = 0 in. (no hole). Also, the newly introduced stretch and strain magnification factors {K(sub λ),K(sub ε) } are also material- and deformation-dependent, and can increase from linear levels of {1.0, 4.0} and reaching {3.07, 4.61}, respectively at a remote radial extension ratio of 2.35.
stress concentration↗
Monitoring Fracture Hydromechanical Evolution in the Lab and Field Using Unsupervised Metric Learning
Fractures evolve in time through thermal‐hydraulic‐mechanical‐chemical (THMC) processes that alter their long‐range hydraulic transport properties and modify subsurface behavior and activities. The location of subsurface fractures makes it necessary to use remote sensing techniques such as passive or active seismic monitoring for fracture characterization. In this paper, we develop a machine learning approach to monitor the evolution of fracture properties using passive seismic sources in a laboratory setting and using active seismic monitoring from the Sanford Underground Research Facility in Lead, South Dakota, at a depth of 1.25 km in amphibolite rock during stimulation of natural fractures as well as during induced fracturing. The unsupervised metric learning technique applies tandem neural networks (twin (Siamese) or triplet) with contrastive loss and adaptive margins to track slowly varying systems for which class or similarity labels are not available. The approach adopts locality‐sensitive hashing to divide time‐ordered contiguous data into an arbitrary number of pseudo‐classes. Contrastive‐loss training with many hash bins generates an evolving latent‐space trajectory. This approach enables unsupervised metric learning for seismic data stacks under the condition of contiguous state sampling and slowly varying fracture properties. The displacement discontinuity theory provides a mechanistic foundation for the fracture‐dependent trajectories that are related to relaxation of fractures with time‐dependent specific stiffness responding to changes in stress or fluid saturation.
02 PETROLEUM↗
Using ion beam irradiation as a surrogate for neutron displacement damage in microelectronics: Simple theory and equivalency considerations
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Young, Joshua Michael↗
Postseismic viscoelastic deformation and stress. Part 2: Stress theory and computation; dependence of displacement, strain, and stress on fault parameters
A viscoelastic model for deformation and stress associated with earthquakes is reported. The model consists of a rectangular dislocation (strike slip fault) in a viscoelastic layer (lithosphere) lying over a viscoelastic half space (asthenosphere). The time dependent surface stresses are analyzed. The model predicts that near the fault a significant fraction of the stress that was reduced during the earthquake is recovered by viscoelastic softening of the lithosphere. By contrast, the strain shows very little change near the fault. The model also predicts that the stress changes associated with asthenospheric flow extend over a broader region than those associated with lithospheric relaxation even though the peak value is less. The dependence of the displacements, stresses on fault parameters studied. Peak values of strain and stress drop increase with increasing fault height and decrease with fault depth. Under many circumstances postseismic strains and stresses show an increase with decreasing depth to the lithosphere-asthenosphere boundary. Values of the strain and stress at distant points from the fault increase with fault area but are relatively insensitive to fault depth.
Cohen, S. C.↗
Modified Displacement Transfer Functions for Deformed Shape Predictions of Slender Curved Structures with Varying Curvatives
To eliminate the need to use finite-element modeling for structure shape predictions, a new method was invented. This method is to use the Displacement Transfer Functions to transform the measured surface strains into deflections for mapping out overall structural deformed shapes. The Displacement Transfer Functions are expressed in terms of rectilinearly distributed surface strains, and contain no material properties. This report is to apply the patented method to the shape predictions of non-symmetrically loaded slender curved structures with different curvatures up to a full circle. Because the measured surface strains are not available, finite-element analysis had to be used to analytically generate the surface strains. Previously formulated straight-beam Displacement Transfer Functions were modified by introducing the curvature-effect correction terms. Through single-point or dual-point collocations with finite-elementgenerated deflection curves, functional forms of the curvature-effect correction terms were empirically established. The resulting modified Displacement Transfer Functions can then provide quite accurate shape predictions. Also, the uniform straight-beam Displacement Transfer Function was applied to the shape predictions of a section-cut of a generic capsule (GC) outer curved sandwich wall. The resulting GC shape predictions are quite accurate in partial regions where the radius of curvature does not change sharply.
displacement transfer functions↗
Large-Deformation Displacement Transfer Functions for Shape Predictions of Highly Flexible Slender Aerospace Structures
Large deformation displacement transfer functions were formulated for deformed shape predictions of highly flexible slender structures like aircraft wings. In the formulation, the embedded beam (depth wise cross section of structure along the surface strain sensing line) was first evenly discretized into multiple small domains, with surface strain sensing stations located at the domain junctures. Thus, the surface strain (bending strains) variation within each domain could be expressed with linear of nonlinear function. Such piecewise approach enabled piecewise integrations of the embedded beam curvature equations [classical (Eulerian), physical (Lagrangian), and shifted curvature equations] to yield closed form slope and deflection equations in recursive forms.
Deflection equations↗
Variable-Domain Displacement Transfer Functions for Converting Surface Strains into Deflections for Structural Deformed Shape Predictions
Variable-Domain Displacement Transfer Functions were formulated for shape predictions of complex wing structures, for which surface strain-sensing stations must be properly distributed to avoid jointed junctures, and must be increased in the high strain gradient region. Each embedded beam (depth-wise cross section of structure along a surface strain-sensing line) was discretized into small variable domains. Thus, the surface strain distribution can be described with a piecewise linear or a piecewise nonlinear function. Through discretization, the embedded beam curvature equation can be piece-wisely integrated to obtain the Variable-Domain Displacement Transfer Functions (for each embedded beam), which are expressed in terms of geometrical parameters of the embedded beam and the surface strains along the strain-sensing line. By inputting the surface strain data into the Displacement Transfer Functions, slopes and deflections along each embedded beam can be calculated for mapping out overall structural deformed shapes. A long tapered cantilever tubular beam was chosen for shape prediction analysis. The input surface strains were analytically generated from finite-element analysis. The shape prediction accuracies of the Variable- Domain Displacement Transfer Functions were then determined in light of the finite-element generated slopes and deflections, and were fofound to be comparable to the accuracies of the constant-domain Displacement Transfer Functions
displacement transfer functions↗
Applications of Displacement Transfer Functions to Deformed Shape Predictions of the G-III Swept-Wing Structure
In support of the Adaptive Compliant Trailing Edge [ACTE] project at the NASA Armstrong Flight Research Center, displacement transfer functions were applied to the swept wing of a Gulfstream G-III airplane (Gulfstream Aerospace Corporation, Savannah, Georgia) to obtain deformed shape predictions. Four strainsensing lines (two on the lower surface, two on the upper surface) were used to calculate the deformed shape of the G III wing under bending and torsion. There being an insufficient number of surface strain sensors, the existing G III wing box finite element model was used to generate simulated surface strains for input to the displacement transfer functions. The resulting predicted deflections have good correlation with the finite-element generated deflections as well as the measured deflections from the ground load calibration test. The convergence study showed that the displacement prediction error at the G III wing tip can be reduced by increasing the number of strain stations (for each strain-sensing line) down to a minimum error of l.6 percent at 17 strain stations; using more than 17 strain stations yielded no benefit because the error slightly increased to 1.9% when 32 strain stations were used.
displacement theory↗
Applications of Displacement Transfer Functions to Deformed Shape Predictions of the GIII Swept-Wing Structure
The displacement transfer functions (DTFs) were applied to the GIII swept wing for the deformed shape prediction. The calculated deformed shapes are very close to the correlated finite element results as well as the measured data. The convergence study showed that using 17 strain stations, the wing-tip displacement prediction error was 1.6 percent, and that there is no need to use a large number of strain stations for G-III wing shape predictions.
on board string sensing system↗
On mixed and displacement finite element models of a refined shear deformation theory for laminated anisotropic plates
An improved plate theory that accounts for the transverse shear deformation is presented, and mixed and displacement finite element models of the theory are developed. The theory is based on an assumed displacement field in which the inplane displacements are expanded in terms of the thickness coordinate up to the cubic term and the transverse deflection is assumed to be independent of the thickness coordinate. The governing equations of motion for the theory are derived from the Hamilton's principle. The theory eliminates the need for shear correction factors because the transverse shear stresses are represented parabolically. A mixed finite element model that uses independent approximations of the displacements and moments, and a displacement model that uses only displacements as degrees of freedom are developed. A comparison of the numerical results for bending with the exact solutions of the new theory and the three-dimensional elasticity theory shows that the present theory (and hence the finite element models) is more accurate than other plate-theories of the same order.
Reddy, J. N.↗