DYNAMICS AND BUCKLING OF THIN PRE-TWISTED BEAMS UNDER AXIAL LOAD AND TORQUE

2010 ◽  
Vol 10 (05) ◽  
pp. 957-981 ◽  
Author(s):  
A. Y. T. LEUNG
Keyword(s):  
Strain Energy ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Rotor Blades ◽  
Initial Stresses ◽  
Beam Columns ◽  
Nonlinear Part ◽  
Straight Beam ◽  
Natural Boundary Conditions ◽  
Twisted Beam

Free vibration and buckling of pre-twisted beams exhibit interesting coupling phenomena between compression, shears, moments and torque and have been the subject of extensive research due to their importance as models of wind turbines and helicopter rotor blades. The paper investigates the influence of axial compression and torque on the natural vibration of pre-twisted straight beam based on the Euler-Bernoulli theory. The derivation begins with the three-dimensional Green strain tensor. The nonlinear part of the strain tensor is expressed as a product of displacement gradient to derive the strain energy due to initial stresses. The Frenet formulae in differential geometry are employed to treat the pre-twist. The strain energy due to elasticity and the linear kinetic energy are obtained in classical sense. From the variational principle, the governing equations and the associated natural boundary conditions are derived. To the best knowledge of the author, the buckling of pre-twisted beam due to initial torque has not been studied in details. The major contribution of the paper is in the consideration of the influence of initial stresses caused by initial shears, moments and torques for pre-twisted beam-columns by means of the Frenet formulae and second order strains. A number of numerical examples are given. Some particular cases are compared with existing results. It is noted that the first mode increases together with the rate of twist but the second decreases seeming to close the first two modes together. The gaps close monotonically as the rate of twist increases for natural frequencies and buckling compressions.

THREE-DIMENSIONAL NON-PRISMATIC BEAM-COLUMNS

2002 ◽  
Vol 02 (03) ◽  
pp. 395-408 ◽  
Author(s):  
R. LEVY ◽  
E. GAL
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Beam Columns ◽  
Straight Beam ◽  
Prismatic Beam ◽  
Principal Directions ◽  
The Finite Difference Method ◽  
Uniform Manner

This paper is concerned with three-dimensional straight beam-columns with no warping whose cross sections vary along the axis in a uniform manner with respect to the principal directions. The basic four coupled differential equations governing the behavior of 3D beam-columns are first rederived using the method of perturbations. These equations are reformulated to include varying cross sections. Finally, a 6 × 6 stiffness matrix (which is sufficient to describe 3D behavior) is computed by solving the equations 6 times for a sequence of appropriate discontinuities. The finite difference method is employed for that purpose. Timoshenko's closed form solution for the buckling load of a tapered column is chosen for comparison with that obtained by the proposed formulation. Effects of twist are also presented.

Mechanical Contribution of Fiber Angular Distribution in Connective Tissue: Comparison of Two Modeling Approaches

2010 ◽  
Author(s):  
Daniel H. Cortes ◽  
Spencer P. Lake ◽  
Jennifer A. Kadlowec ◽  
Louis J. Soslowsky ◽  
Dawn M. Elliott
Keyword(s):  
Angular Distribution ◽  
Strain Energy ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Numerical Algorithms ◽  
Collagen Fibers ◽  
Numerical Comparison ◽  
Computationally Efficient ◽  
Modeling Approaches ◽  
Dimensional Distribution

Collagen fibers and their structural arrangement influence tissue tensile stiffness and strength. While a variety of modeling approaches incorporate collagen fibers explicitly as one of the components, due to the complexity of the fiber organization, aligned fibers are usually considered. In the pioneering work of Lanir [1], a constitutive relation for continuous fiber distributions was proposed, where the strain energy and stresses are obtained by angular integration (AI) of infinitesimal fractions of fibers aligned in a given direction. Lanir’s formulation has been successfully used to describe the mechanical behavior of a variety of tissues. In particular, Ateshian et al. [2] showed that large values of the tensile Poisson’s ratio for articular cartilage in tension and the low values observed in compression can be explained using a continuous angular distribution for the fibers. A disadvantage of the AI formulation is the large number of calculations required to evaluate the strains and stresses. On the other hand, Generalized Structure Tensors (GST) have been proposed to model tissues with continuously distributed collagen fibers [3,4]. These tensors are assumed to represent the three-dimensional distribution of the fibers. Once the tensor has been defined, the strain in the fibers can be readily obtained by multiplication with a strain tensor. The advantage of this approach is the small number of calculations required to obtain the strain energy and stresses of the fibers. As a result, this formulation can be efficiently implemented in numerical algorithms like finite elements. However, this approach is limited, as it is valid only when all of the fibers are in tension and when the fiber distribution is small [5]. A numerical comparison is required to quantify when an angular distribution can be considered acceptably small to justify using this more computationally efficient approach. The objective of this study is to numerically compare the AI and GST formulations to determine the range of values of angular distribution for which the GST approach can be accurately used.

Three-Dimensional Strains on Twisted and Swept Composite Rotor Blades in Vacuum

2020 ◽  
pp. 1-16
Author(s):  
Cheng Chi ◽  
Anubhav Datta ◽  
Inderjit Chopra ◽  
Renliang Chen
Keyword(s):  
Three Dimensional ◽  
Rotor Blades

An isotropic elasto-damage-plastic micromechanical framework of innovative pothole patching materials featuring high-toughness, low-viscosity nanomolecular resin

2021 ◽  
pp. 105678952110286
Author(s):  
H Zhang ◽  
J Woody Ju ◽  
WL Zhu ◽  
KY Yuan
Keyword(s):  
Strain Energy ◽  
Three Dimensional ◽  
Elastic Strain Energy ◽  
Companion Paper ◽  
Computational Algorithms ◽  
Mechanical Behaviors ◽  
High Toughness ◽  
Low Viscosity ◽  
Innovative Materials ◽  
Continuum Thermodynamic

In a recent companion paper, a three-dimensional isotropic elastic micromechanical framework was developed to predict the mechanical behaviors of the innovative asphalt patching materials reinforced with a high-toughness, low-viscosity nanomolecular resin, dicyclopentadiene (DCPD), under the splitting tension test (ASTM D6931). By taking advantage of the previously proposed isotropic elastic-damage framework and considering the plastic behaviors of asphalt mastic, a class of elasto-damage-plastic model, based on a continuum thermodynamic framework, is proposed within an initial elastic strain energy-based formulation to predict the behaviors of the innovative materials more accurately. Specifically, the governing damage evolution is characterized through the effective stress concept in conjunction with the hypothesis of strain equivalence; the plastic flow is introduced by means of an additive split of the stress tensor. Corresponding computational algorithms are implemented into three-dimensional finite elements numerical simulations, and the outcomes are systemically compared with suitably designed experimental results.

scholarly journals Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Energy Density ◽  
Constitutive Model ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
General Method

AbstractIn this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

Effect Of Phosphorus On Ge/Si(001) Island Formation

2002 ◽  
Vol 737 ◽  
Author(s):  
Theodore I. Kamins ◽  
Gilberto Medeiros-Ribeiro ◽  
Douglas A. A. Ohlberg ◽  
R. Stanley Williams
Keyword(s):  
Deposition Rate ◽  
Strain Energy ◽  
Competitive Adsorption ◽  
Lattice Mismatch ◽  
Three Dimensional ◽  
Deposition Process ◽  
Thin Layers ◽  
Island Size ◽  
Partial Pressures ◽  
Intermediate Size

ABSTRACTWhen Ge is deposited epitaxially on Si, the strain energy from the lattice mismatch causes the Ge in layers thicker than about four monolayers to form distinctive, three-dimensional islands. The shape of the islands is determined by the energies of the surface facets, facet edges, and interfaces. When phosphorus is added during the deposition, the surface energies change, modifying the island shapes and sizes, as well as the deposition process. When phosphine is introduced to the germane/hydrogen ambient during Ge deposition, the deposition rate decreases because of competitive adsorption. The steady-state deposition rate is not reached for thin layers. The deposited, doped layers contain three different island shapes, as do undoped layers; however, the island size for each shape is smaller for the doped layers than for the corresponding undoped layers. The intermediate-size islands are the most significant; the intermediate-size doped islands are of the same family as the undoped, multifaceted “dome” structures, but are considerably smaller. The largest doped islands appear to be related to the defective “superdomes” discussed for undoped islands. The distribution between the different island shapes depends on the phosphine partial pressure. At higher partial pressures, the smaller structures are absent. Phosphorus appears to act as a mild surfactant, suppressing small islands.

scholarly journals Origin and Application of the Twinned Calcite Strain Gauge

Geosciences ◽  
2021 ◽  
Vol 11 (7) ◽  
pp. 296
Author(s):  
Richard H. Groshong
Keyword(s):  
Stress Analysis ◽  
Strain Gauge ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Constant Strain Rate ◽  
Axis Orientation ◽  
Order Of Magnitude ◽  
Rate Experiment ◽  
Strain Axis ◽  
Expected Values

This paper is a personal account of the origin and development of the twinned-calcite strain gauge, its experimental verification, and its relationship to stress analysis. The method allows the calculation of the three-dimensional deviatoric strain tensor based on five or more twin sets. A minimum of about 25 twin sets should provide a reasonably accurate result for the magnitude and orientation of the strain tensor. The opposite-signed strain axis orientation is the most accurately located. Where one strain axis is appreciably different from the other two, that axis is generally within about 10° of the correct value. Experiments confirm a magnitude accuracy of 1% strain over the range of 1–12% axial shortening and that samples with more than 40% negative expected values imply multiple or rotational deformations. If two deformations are at a high angle to one another, the strain calculated from the positive and negative expected values separately provides a good estimate of both deformations. Most stress analysis techniques do not provide useful magnitudes, although most provide a good estimate of the principal strain axis directions. Stress analysis based on the number of twin sets per grain provides a better than order-of-magnitude approximation to the differential stress magnitude in a constant strain rate experiment.

scholarly journals Mesh size sensitivity analysis for interlaminar fracture of the fiber-reinforced laminated composites

2019 ◽  
Vol 14 ◽  
pp. 155892501988346 ◽  
Author(s):  
Fatih Daricik
Keyword(s):  
Crack Closure ◽  
Strain Energy ◽  
Three Dimensional ◽  
Laminated Composite ◽  
Mesh Size ◽  
Strain Energy Release ◽  
Virtual Crack Closure Technique ◽  
Closure Technique ◽  
Interlaminar Fracture ◽  
Element Size

The virtual crack closure technique is a well-known finite element–based numerical method used to simulate fractures and it suits well to both of two-dimensional and three-dimensional interlaminar fracture analysis. In particular, strain energy release rate during a three-dimensional interlaminar fracture of laminated composite materials can successfully be computed using the virtual crack closure technique. However, the element size of a numerical model is an important concern for the success of the computation. The virtual crack closure technique analysis with a finer mesh converges the numerical results to experimental ones although such a model may need excessive modeling and computing times. Since, the finer element size through a crack path causes oscillation of the stresses at the free ends of the model, the plies in the delaminated zone may overlap. To eliminate this problem, the element size for the virtual crack closure technique should be adjusted to ascertain converged yet not oscillating results with an admissible processing time. In this study, mesh size sensitivity of the virtual crack closure technique is widely investigated for mode I and mode II interlaminar fracture analyses of laminated composite material models by considering experimental force and displacement responses of the specimens. Optimum sizes of the finite elements are determined in terms of the force, the displacement, and the strain energy release rate distribution along the width of the model.

Three-Dimensional Stress Distribution in Arteries

1983 ◽  
Vol 105 (3) ◽  
pp. 268-274 ◽  
Author(s):  
C. J. Chuong ◽  
Y. C. Fung
Keyword(s):  
Experimental Data ◽  
Stress Distribution ◽  
Vessel Wall ◽  
Strain Energy ◽  
Arterial Wall ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Exponential Form ◽  
Strain Relationship ◽  
Longitudinal Stretch

A three-dimensional stress-strain relationship derived from a strain energy function of the exponential form is proposed for the arterial wall. The material constants are identified from experimental data on rabbit arteries subjected to inflation and longitudinal stretch in the physiological range. The objectives are: 1) to show that such a procedure is feasible and practical, and 2) to call attention to the very large variations in stresses and strains across the vessel wall under the assumptions that the tissue is incompressible and stress-free when all external load is removed.

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