Flexural modes

The stress-relaxation behavior of a material is normally determined in either the tensile or the flexural mode. In these experiments, a material specimen is rapidly elongated or compressed to produce a specified strain level and the load exerted by the specimen on the test apparatus is measured as a function of time. Specimens of certain plastics may fail during tensile or flexural stress-relaxation experiments. 

Flexural modulus is the force required to deform a material in the elastic bending region. It is essentially a way to characterize stiffness. Urethane elastomers and rigid foams are usually tested in flexural mode via three-point bending and tite flexural (or flex ) modulus is obtained from the initial, linear portion of the resultant stress-strain curve. 

A piezoelectric mass sensor is a device that measures the amount of material adsorbed on its surface by the effect of the adsorbed material on the propagation of acoustic waves. Piezoelectric devices work by converting electrical energy to mechanical energy. There are a number of different piezoelectric mass sensors. Thickness shear mode sensors measure the resonant frequency of a quartz crystal. Surface acoustic wave mode sensors measure the amplitude or time delay. Flexure mode devices measure the resonant frequency of a thin Si3N4 membrane. In shear horizontal acoustic plate mode sensors, the resonant frequency of a quartz crystal is measured. 

For the measurements, the single cantilever flexure mode was chosen. For this, rectangular polymer samples of 32 mm x 6 mm x 2 mm had to be produced. This was done by injection molding of the original polymers. Fig. 8 shows the setup of a DMA-experiment in the single cantilever configuration. 

Figure 3.39 (page 114) shows the phase velocities of the waves as a function of the product k4, where k, = 27t/A, A, is the wavelength of the bulk transverse (shear) wave in the medium of which the plate is made, and d is the plate thickness. The waves divide naturally into two sets symmetric waves (denoted by So, S],. ..) whose particle displacements are symmetric about the neutral plane of the plate, and antisymmetric waves (Aq, A, . ..), whose displacements have odd symmetry about the neutral plane. Figure 3.38 shows that for sufficiently thin plates (M < 1-6), only two waves exist — the lowest-order symmetric mode (Sq) and the lowest-order antisymmetric mode (Aq). These are the modes shown earlier in Figure 2.0d. The plate mode that we will emphasize here is the Ao mode, in which the elements of the plate undergo flexure as the wave propagates. The shape of a plate during propagation of this flexural mode has been likened to that of a flag waving in the wind.

We will now consider the response of the lowest-order flexural mode to various perturbations. For generality, we assume that there is some initial tension, Tx, in the plate. For simplicity, we assume that the plate is quite thin (d/A < 1), and so the approximate phase velocity expression of Equation 3.72 can be used as a basis for discussion. 

One Flexural Mode "AO Lamb Mode Two Flexural Modes A0 Lamb Mode — Always Lossy Scholte Mode -- Vp < Vp ... 

There are a great number of techniques for the experimental determination of viscoelastic functions. The techniques most frequently found in the literature are devoted to measuring the relaxation modulus, the creep compliance function, and the components of the complex modulus in either shear, elongational, or flexural mode (1-4). Although the relaxation modulus and creep compliance functions are defined in the time domain, whereas the complex viscoelastic functions are given in the frequency domain, it is possible, in principle, by using Fourier transform, to pass from the time domain to the frequency domain, or vice versa, as discussed earlier. 

For a solid rectangular board in a flexural mode (that is, about its neutral axis), the moment of inertia is... 

The ASTM procedure indicates that a support span-to-depth ratio should be 16 1. Decreasing the ratio much below 16 1 would move the test from a flexural mode to a shear mode and would effectively increase the apparent flexural strength of the material to very high values. The procedure does not recommend to use the ratio below 14 1. However, for some highly anisotropic composites, the procedure recommends to avoid shear effects as much as possible, particularly when flexural modulus data are required, and increase the span-to-depth ratio to 20 1, 32 1,40 1, and even to 60 1. 

It was mentioned earlier that if the support span-to-depth ratio is noticeably less than 16 1, this would move the test from a flexural mode to a shear mode and would effectively decrease the apparent flexural modulus of the material to much lower values. Data in Table 27 show, though, that for this effect to be quite noticeable, the span-to-depth ratio should not necessarily be much lower than 16 1. Even a move of the ratio from 16 1 to 11.2 1 results in 28% (on average) decrease of flexural modulus, such as from 300,000 to 234,000 psi, as an example. 

Figure 8.7 shows the relaxation behaviour of a sodium-chloride-doped polycrystalline specimen in a flexural mode, as determined by Kuroiwa (1964) for four diiferent vibration frequencies. It is immediately clear that there are three different loss processes which become dominant in different temperature ranges. The loss peak, which varies with frequency over the temperature range — 50 to - 80 °C for the vibrations studied, is simply the intrinsic... 

The curvature/strain modes method hold the promise of detecting damage from the processing of experimental data without a structural model. The premise of the method is that damage in thin-wall structures induces local discontinuities that affect strongly the curvature of the flexural modes of the structure [71]. The curvamre of the modes (i.e. the curvature modes) can be determined through space-wise double differentiation of the measured displacement/velocity/acceleration modes. They can be also measured directly with surface mounted strain gauges. 

A work targeted specifically to civil infrastructure application has reported mechanical data on freeze-thaw tests conducted on isophthalic polyester and vinyl ester pultruded/glass fiber RPs (Chapter 3). Specimens were aged in accordance with ASTM C666 (namely, 40F to OF followed by a hold at OF and a ramp up to 40F followed by a hold) while submerged in 2% sodium chloride and water. Specimens were removed after every 50 cycles and tested in ASTM 3-point flexure mode. The results clearly indicated a reduction in flexure strength and modulus after 300 cycles. 

The complex response of the material is resolved into the elastic or storage modulus (G ) and the viscous or loss modulus (G") if the deformation is in shear mode. If the deformation is in the tensile or flexural mode the E and E" are used. 

The graphs at the bottom of Fig. 4.147 display results gained in the flexure mode under conditions that satisfy the ASTM (American Society for Testing and Materials). The deflection temperature is taken where the sample has been deformed by 0.010 in... 

Figure 2. Resonant frequency shift of the second flexural mode ofPEMC sensor upon binding of Bacillus anthracis spores at various sample concentrations to antibody functionalized cantilever. The results showed that the binding rate strongly depends on concentration. The control was an antibody-immobilized cantilever immersed in PBS. Adapted from reference(24).

Another composite structure we have investigated recently is a particulate-filled epoxy. The epoxy is EPON 828 (Z-hardener) filled with irregularly shaped alumina particles with an average diameter of approximately 10/xm. This material is quite strong and brittle so we fractured most of the samples in a three-point flexure mode. The cross section of the sample was 2 mm x 6 mm. A typical EE curve plotted on a log scale is shown in Fig. 9, where t = 0 corresponds to the instant of failure. The material for this emission curve is filled at an Al203/epoxy ratio a of... 

Figure 3 Second low load flexural mode portal frame test (a) Load versus mid-span...

Ultbnate load flexural mode frame test... 

For a simple rectangular pinned base frame made of shear deformable members joined together by semi-rigid connections with a linear moment-rotation characteristic, it is possible to develop simple closed-form expressions for the deformations at points of particular interest, e.g. midspan deflection (wj and joint rotations. Such expressions have been derived assuming that the beam (span=b) and columns (height=a) are made of the same sections. For example, the expression for the mid-span deflection of the beam when the frame is loaded in a flexural mode is as follows ... 

A comparison of the measured and calculated sway mode deflections is presented for different frame models. In the semi-rigid model the P-value determined from the flexural mode back analysis is used. Table 6 shows that the predicted deflections are all much less than the measured values— ranging from 16% to 45% of the actual value. 

Table 5 Pultruded glass FRP portal frame tested in flexural mode (W =...

Dynamic mechanical analyzer n. An instrument that can test in an oscillating-flexural mode over a range of temperature and frequency to provide estimates of the real , i.e., in-phase, and imaginary , i.e., out-of-phase parts of the complex modulus. The real part is the elastic component, the imaginary part is the loss component. The square root of the sum of their squares is the complex modulus. With polymers, the components and the modulus are usually dependent on both temperature and frequency. ASTM D 4065 spells out the standard practice for reporting dynamic mechanical properties of plastics. An example of a DMA thermogram of different Perkin-Elmer Inc., manufactures the Diamond DMA instrument. Polymer films is shown. Sepe MP (1998) Dynamic mechanical analysis. Plastics Design Library, Norwich, New York. 

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