Construction of piezoelectric and flexoelectric models of composites by asymptotic homogenization and application to laminates

2021 ◽  
pp. 108128652110303
Author(s):  
Maryam Nasimsobhan ◽  
Jean-François Ganghoffer ◽  
Mahnaz Shamshirsaz
Keyword(s):  
Electric Fields ◽  
Plate Thickness ◽  
Bending Moment ◽  
Expansion Method ◽  
Localization Operators ◽  
Pure Bending ◽  
Full Generality ◽  
Asymptotic Models ◽  
Solid Bodies ◽  
Piezoelectric Behavior

The effective piezoelectric and flexoelectric properties of heterogeneous solid bodies with constituents obeying a piezoelectric behavior are evaluated in full generality, based on the asymptotic expansion method. The successive situations of materials obeying a piezoelectric and flexoelectric behavior at the macroscale is envisaged in the present work. Closed-form expressions for the effective flexoelectric properties are obtained for stratified materials. A general theory for laminated piezoelectric plates is formulated on the basis of the formulated asymptotic models, and the response of the homogeneous substitution plate is evaluated for a loading consisting of a pure bending moment, triggering electric fields and strain and electric fields gradients within the plate thickness. The local mechanical and electric fields at the microscopic level within the initial heterogeneous stratified domain are evaluated by solving unit cell boundary value problems for the localization operators. An effective flexoelectric plate model for a stratified composite is constructed, showing the generation of the gradient of an electric field under application of a pure bending moment.

scholarly journals Near-Fault Seismic Response Analysis of Bridges Considering Girder Impact and Pier Size

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 704
Author(s):  
Wenjun An ◽  
Guquan Song ◽  
Shutong Chen
Keyword(s):  
Seismic Response ◽  
Bending Moment ◽  
Expansion Method ◽  
Response Analysis ◽  
Seismic Action ◽  
Transient Wave ◽  
Cross Sectional ◽  
Displacement Response ◽  
Near Fault ◽  
Continuous Girder Bridge

Given the influence of near-fault vertical seismic action, we established a girder-spring-damping-rod model of a double-span continuous girder bridge and used the transient wave function expansion method and indirect modal function method to calculate the seismic response of the bridge. We deduced the theoretical solution for the vertical and longitudinal contact force and displacement response of the bridge structure under the action of the near-fault vertical seismic excitation, and we analyzed the influence of the vertical separation of the bridge on the bending failure of the pier. Our results show that under the action of a near-fault vertical earthquake, pier-girder separation will significantly alter the bridge’s longitudinal displacement response, and that neglecting this separation may lead to the underestimation of the pier’s bending damage. Calculations of the bending moment at the bottom of the pier under different pier heights and cross-sectional diameters showed that the separation of the pier and the girder increases the bending moment at the pier’s base. Therefore, the reasonable design of the pier size and tensile support bearing in near-fault areas may help to reduce longitudinal damage to bridges.

Dynamic Fracture of a Beam or Plate in Plane Bending

1976 ◽  
Vol 43 (1) ◽  
pp. 112-116 ◽  
Author(s):  
L. B. Freund ◽  
G. Herrmann
Keyword(s):  
Crack Length ◽  
Dynamic Fracture ◽  
Crack Tip ◽  
Bending Moment ◽  
Fracture Initiation ◽  
Fracture Plane ◽  
Multiple Reflections ◽  
Pure Bending ◽  
Strain Fracture ◽  
Beam Thickness

The dynamic fracture response of a long beam of brittle elastic material subjected to pure bending is studied. If the magnitude of the applied bending moment is increased to a critical value, a crack will propagate from the tensile side of the beam across a cross section. An analysis is presented by means of which the crack length and bending moment at the fracturing section are determined as functions of time after fracture initiation. The main assumption on which the analysis rests is that, due to multiple reflections of stress waves across the thickness of the beam, the stress distribution on the prospective fracture plane ahead of the crack may be adequately approximated by the static distribution appropriate for the instantaneous crack length and net section bending moment. The results of numerical calculations are shown in graphs of crack length, crack tip speed, and fracturing section bending moment versus time. It is found that the crack tip accelerates very quickly to a speed near the characteristic terminal speed for the material, travels at this speed through most of the beam thickness, and then rapidly decelerates in the final stage of the process. The results also apply for plane strain fracture of a plate in pure bending provided that the value of the elastic modulus is appropriately modified.

scholarly journals Bi-harmonic analysis in a perforated strip

1933 ◽  
Vol 232 (707-720) ◽  
pp. 155-222 ◽  
Keyword(s):  
Viscous Fluid ◽  
Harmonic Analysis ◽  
Biharmonic Equation ◽  
Bending Moment ◽  
Slow Motion ◽  
Special Problem ◽  
Pure Bending ◽  
Special Stress ◽  
Straight Lines

A method of solving the biharmonic equation in a region bounded externally by two parallel straight lines and internally by a circle was given by one of the authors in a recent paper. General formulae were developed, but these were restricted to solutions symmetrical about both co-ordinate axes, and were applied to only one special problem of elasticity. In the present paper the analysis is generalized to include unsymmetrical solutions, and the formulae are developed to a point at which it becomes possible to solve any problem of stress within the specified boundaries. Two important special stress-systems—that corresponding to pure bending-moment, and that giving bending-moment with shear—are worked out in detail. A number of other interesting systems may be discussed by the aid of the results given. In addition, only slight modifications are needed to make the equations applicable to the slow motion of a viscous fluid.

State of Stress of Castellated Beams with Circular Apertures under Distributed Load and Pure Bending

2019 ◽  
Vol 974 ◽  
pp. 521-528
Author(s):  
Alexej I. Pritykin
Keyword(s):  
Empirical Relation ◽  
Bending Moment ◽  
Empirical Dependence ◽  
Ansys Software ◽  
Pure Bending ◽  
Equivalent Stresses ◽  
Distributed Load ◽  
State Of Stress ◽  
Von Mises ◽  
Von Mises Stresses

The regularities of stress distribution in perforated beams with circular apertures under distributed load and pure bending. Such beams are made of different materials: carbon fiber is used in aircraft for these purposes, and steel is used in construction. Beams with different perforation parameters were considered and an empirical relation was obtained for equivalent von Mises stresses in castellated beams near the apertures’ outlines based on the analysis of FEM calculations. In this paper, beams made of C345 steel were considered. It is established that the maximum values ​​of equivalent stresses near apertures under different loading types vary along the beam length in proportion to the values ​​of the bending moment. The values ​​of stress concentration coefficients for the pure bending are determined depending on the perforation parameters. The acceptability of the obtained empirical dependencies for equivalent stresses was verified using the FEM calculations based on the ANSYS software package. There is a good correlation between the results of FEM calculations and empirical dependence.

scholarly journals The behaviour of three single crystals of aluminium in fatigue under complex stresses

1935 ◽  
Vol 149 (867) ◽  
pp. 312-326 ◽  
Keyword(s):  
Single Crystals ◽  
Test Piece ◽  
Testing Machine ◽  
Bending Moment ◽  
Stress System ◽  
Pure Bending ◽  
Bending Strain ◽  
Bending Stresses ◽  
Ordinary Type ◽  
New Type

In the ordinary type of Wöhler machine used for testing materials in fatigue under reversed bending stresses, the load system is stationary in space, and variation of the stress system with respect to the test piece is obtained by rotating the test piece. It is, of course, essential to the success of the test that the system of displacements caused by the application of the load system to the test piece should remain stationary in space; but, since the test piece rotates, this requirement can only be fulfilled if the material of the test piece is isotropic. Thus, if an attempt were made to test a single crystal in a Wöhler machine it might be anticipated that either actual elastic antisotropy or the virtual anisotropy due to restricted slip movement would cause the deformation to vary with the orientation of the stress system relative to the axes of the crystal and that "whipping" of the specimen would occur. Three such attempts have indeed been made: but in spite of great care exercised in setting up the specimens and in applying the loads, only in one case, in which the orientation of the crystal was such as to provide effective symmetry about the axis of the specimen, was the test successful. A new type of testing machine recently developed at the N. P. L. for testing specimens in fatigue under systems of combined bending and torsional stresses, differs in principle from the Wöhler machine in that the variation of stress is produced by actual variation of load. In this machine both me test piece and the orientation of the stress system remain stationary, only the magnitude of the stresses being varied. The deformation of the test piece is therefore only that due to one type of stress system fixed in relation to the orientation of the test piece and varying only in magnitude. Moreover, the construction of the machine is such that the strain of the test piece is not required to be of the same type as the stress system applied, e. g ., the application of pure bending moment does not restrict the test piece to pure bending strain and the test piece remains free to twist also if necessary. These conditions render this type of machine perfectly suitable for test on single crystals. Accordingly, tests have been carried out in this machine on three single crystals of aluminium; the first was tested under reversed flexural stresses, the second under reversed torsional stresses and the third under a combination of reversed flexural and reversed torsional stresses.

Forced Vibrations of Viscoelastic Timoshenko Beams

1976 ◽  
Vol 98 (3) ◽  
pp. 820-826 ◽  
Author(s):  
C. C. Huang ◽  
T. C. Huang
Keyword(s):  
Boundary Conditions ◽  
Laplace Transform ◽  
Attenuation Factor ◽  
Equations Of Motion ◽  
Bending Moment ◽  
Expansion Method ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Mode Shapes ◽  
Timoshenko Beams

In a previous paper, the correspondence principle has been applied to derive the differential equations of motion of viscoelastic Timoshenko beams with or without external viscous damping. To study free vibrations these equations are solved by Laplace transform and boundary conditions are applied to obtain the attenuation factor and the frequency of the damped free vibrations and mode shapes. The present paper continues to analyze this subject and deals with the responses in deflection, bending slope, bending moment and shear for forced vibrations. Laplace transform and appropriate boundary conditions have been applied. Examples are given and results are plotted. The solution of forced vibrations of elastic Timoshenko beams obtained as a result of reduction from viscoelastic case and by eigenfunction expansion method concludes the paper.

On the Linear Buckling of Circular Cylindrical Shells Under Asymmetric Axial Compressive Stress Distributions

1966 ◽  
Vol 70 (672) ◽  
pp. 1095-1097 ◽  
Author(s):  
D. J. Johns
Keyword(s):  
Compressive Stress ◽  
Cylindrical Shells ◽  
Bending Moment ◽  
Transverse Load ◽  
Stress Distributions ◽  
Pure Bending ◽  
Linear Buckling ◽  
Circular Cylindrical Shells ◽  
Axial Compressive Stress

The linear buckling of circular cylindrical shells is considered with particular attention to the cantilever shell subjected to either a pure bending moment (M) or transverse load (P)—see Fig. 1. It is believed that the conclusions reached have wider application to more general loading cases.

The Vibration Based on Non-Zero Point Force Moment Elasticity Theory

2014 ◽  
Vol 590 ◽  
pp. 27-31 ◽  
Author(s):  
Wen Ba Han ◽  
Shuang Hua Huang ◽  
Jie Liu ◽  
Jin Kun Sun
Keyword(s):  
Shear Stress ◽  
Normal Stress ◽  
Elasticity Theory ◽  
Damage Mechanics ◽  
Natural Frequencies ◽  
Bending Moment ◽  
Zero Point ◽  
Pure Bending ◽  
Force Moment ◽  
The Moment

The traditional elastic theory believes that there exists normal stress in pure bending body (PBB) and shear stress in pure torsion body (PTB). However, the author proved that there is no normal stress but ‘Bent Point Moment’ (BPM) in PBB. And it also concluded that there is no shear stress but ‘Shear Point Moment’ (SPM) in PTB. This article overturns the preliminary theorems of the Elasticity Theory, which believes that the value of the moment (Bending moment & Torsion moment) on a unit area converges to zero. Just as the completely different natural frequencies of the forced vibration can lead to completely different resonant conditions. Besides, this theory has also been validated in the Damage Mechanics National Key Laboratory of Tsinghua University. Therefore, it is significant to avoid destruction produced by resonance.

Stress Concentration Factors in Shouldered Shafts Subjected to Combinations of Flexure and Torsion

1968 ◽  
Vol 90 (2) ◽  
pp. 301-307 ◽  
Author(s):  
H. G. Rylander ◽  
P. M. A. daRocha ◽  
L. F. Kreisle ◽  
G. J. Vaughn
Keyword(s):  
Stress Concentration ◽  
Maximum Stress ◽  
Small Diameter ◽  
Bending Moment ◽  
Pure Bending ◽  
Principal Stresses ◽  
Torsional Loads ◽  
Concentration Factors ◽  
Stress Concentration Factors ◽  
Experimental Values

Geometric stress concentration factors were determined experimentally for shouldered aluminum shafts subjected to combinations of flexural and torsional loads. Diameter ratios were varied from 0.42 to 0.83, and fillet radius to small diameter ratios were varied from 0.1 to 0.7 with bending moment to torque ratios varying over a range from 1:4 to 4:1. Experimental values for the stress concentration factors were obtained by using birefringent coatings and a reflection polariscope. Strain gage measurements and torsional relaxation solutions were used to verify some of the polariscope data. For the cases considered, the static geometric stress concentration factor was between 1.11 and 1:50 for pure torsion, between 1.08 and 1.46 for pure bending, and between 1.09 and 1.50 for combined torsion and bending. The directions of the principal stresses on the surface of the shouldered shafts do not change due to the presence of the discontinuity for a particular specimen and type of loading. Also, the location of the maximum stress in the fillet of a particular specimen under a certain type of loading does not change as the magnitude of the load is varied, but it does vary with the type of loading.

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