LA NECESIDAD DE SER BELLO

x= 0 (4) Symmetric thick epoxy

epoxy sample (3) Asymmetric epoxy (2) Symmetric epoxy w t

(1) Rigid constrained_faces

z

y x

Fig. 2.17:Mounting models.Mounting models used in the finite element anal- ysis. Red faces have applied displace- ment constraints, and blue volumes are epoxy. Reproduced from [34].

In order to optimise the sample mounting scheme I used finite element analysis to investigate the possible mounting scenarios. For each I will quantify once more the load transfer length and assess each in terms of the strain homogeneity and the sample bending.

Here I will discuss four models for the sample mounts, all depicted in figure 2.17. They are: (1) “Rigid:” here the sample is fixed with hard, perfectly rigid epoxy on its top and bottom surfaces at both ends. (2) “Symmetric epoxy:” a softer epoxy bonded on both the top and bottom surfaces to perfectly rigid sample plates. (3) “Asymmetric epoxy:” the same soft layer of epoxy but only the bottom surface is bonded to a sample plate. (4) “Symmetric thick epoxy:” the same as model 2 but with thicker layers of epoxy on both sides. For all the models the sample and epoxy are taken to be isotropic. We set the Young’s modulus of the sample to be 10×that

2.5New uniaxial stress cell 31

of the epoxy, and the sample and epoxy to both have a Poisson’s ratio of 0.3. We use aspect ratios close to those of the typically mounted samples: the sample’s width,w, is set to 3_×the sample’s

thickness,t, and the length (gap between the sample plates) to

3.5w. The thin layers of epoxy are 0.25t and the thick layers in

model 4 are equal to the sample’s thickness. The sample plates are not directly modelled but the boundary conditions imposed on the models are such to sufficiently capture their effect. Only the faces of the epoxy that would make contact with the sample plates, or the end portions of the sample in the rigid epoxy model, have applied displacement constraints; the constrained faces are illustrated in figure 2.17. The bulk of the epoxy and sample is not constrained. Displacement rather than force constants are used

(1)

(2)

(3)

(2)

x z x z x z x y εxx=−0.099 % εxx=−0.075 % εxx=−0.065 % edge of epoxy

A

B

x/t 2 0 −2 −4 −6 -0.0012 -0.0006 0 StrainXX −0.10 −0.05 0.00 εxx (%) strain gradient stress concentration mount model

Fig. 2.18: Fea simulations. Strainεxxfor samples mounted as in the models of figure 2.17. In all the

models the sample plates, represented by the thick red lines, were moved towards each other by 0.1 % ofL. The deformations have been exaggerated 200 times. A.Cuts through the centrexz-plane of the

32 Uniaxial Stress Technique

Fig. 2.19: Strain along the centre- line of the sample.Strainεxxalong the centreline of the sample for the mounting models in figure 2.17. The legend includes the load transfer lengths,λ, from fits to the portion of

the sample inside the mounts. Nega- tivex/tcorresponds to where there

is epoxy, the scale is the same as in figure 2.18. ∝exp−|x| λ

x/t

ε

xx

(%)

mounting model: centreline

−15

−10

−5

0

5

−0.1

0.0

(1)λ= 0.62t (2)λ= 2.0t (3)λ= 2.8t (4)λ= 3.7t

since it is the strain that is controlled by the device, not the stress. These faces have constrainedx,yandzdisplacements. Theiryand z displacements are held at zero, i.e. we assume the sample plates

are not expanding or contracting transversely. The xdisplacement

sets the applied strain, and for a model with an applied strain of

−0.1 % the constrained faces are displaced towards each other by

0.05 %×L.

For these calculations I programmed a custom FEA simulation incorporating the meshing capabilities of Gmsh [70] and matrix solvers in MATLAB® _{[37]. Each model had on the order of 10}6 elements, all linear tetrahedrons. The end portions of the sample were always made much longer than the load transfer length λ

to negate effects due to only partial transmission of the load. No effects of differential thermal expansion are included in the models presented in this thesis.

The simulation results of the models in figure 2.17 are shown in figures 2.18 and 2.19. Figure 2.18 shows the strainεxx. The thick

red lines mark the constrained faces which were moved towards each other by 0.1 % ofLand the deformations have been exaggerated

200 times. The three plots in part A of figure 2.18 show cuts in the centrexz-plane of the sample for mounting models 1, 2 and 3. The

plot in panel B shows a cut the centrexy-plane for mount model 2.

Figure 2.19 shows the εxx strain along the centreline of the

sample for all the mounting models. Here one can clearly see the exponential decay of the strain into the mount as well as the highly homogeneous region in the centre of the sample. The load transfer length,λ, is shown for each model and is taken from a fit to the

strain along the centreline of the sample in region of the sample within the mount.

The load transfer length is shortest for the rigid epoxy model and correspondingly the highest strain is achieved in this sample, but the cross-section through thexz-plane in figure 2.18 clearly shows

2.5New uniaxial stress cell 33

t/L

∆

ε

xx

/ε

xx

(%)

(a) 0.25t (b) 0.375t 0.125t (c)

0

0.1

0.2

0.3

0

20

40

60

80

Fig. 2.20: Bending induced strain inhomogeneity. The difference in strain between the top and bottom of a bent sample at the centre divided by the average strain across the central plane of the sample plotted against the sample’s aspect ratio. The three cases from top to bottom are: (a) rigid epoxy holding only the lower sample face; (b) soft epoxy on the under side of the sample with a thickness equal to 0.25t; (c) a sample mounted asymmetrically between top and bottom sample plates. The

sample is off centre by 0.125tand the total space between the two plates is 1.5t.

would be the failure point for a sample mounted with this scheme. In the models with the layers of soft epoxy the stress concentration is reduced and, we expect, higher strains can ultimately be achieved. However, the exact thickness of the epoxy leads to some uncertainty in the exact amount of strain achieved in the sample; take note of the range of strains seen at the middle of the sample for the models with soft epoxy in figure 2.19.

Table 2.1: Guide for the end portions of the sample to exclude from measure- ments.Length at the end of the sample to exclude in order to achieve a given level of strain homogeneity. Mounting models and dimensions as per figure 2.17. % Inhomogeneity Mounting model 5 % 1 % 1 0.5w 0.9w 2 0.2w 0.5w 4 0.1w 0.4w

For samples mounted in symmetrical mounts the strain ho- mogeneity is very high, see both figures 2.18B and 2.19. The inhomogeneity dies away moving towards the sample centre so measurements should be designed to be sensitive only to the central region of the sample. A guide for the length of sample to exclude from both ends of the sample is given in table 2.1. This length says that after excluding this amount at the ends of the sample, the strainεxxacross the rest of the volume of the sample does not

differ from the average strain at the centre of the sample by more than the given percentage. So to achieve a strain inhomogeneity less than 5 % for a sample mounted using model 2, a length equal to 0.2wof the sample needs to be excluded from both ends of the

sample. With suitable sample mounts it is therefore possible to achieve very high strain homogeneity over almost the entire exposed region of the sample.

Any asymmetry in mounting causes the sample to bend as shown in figure 2.18A(3) and this introduces further strain inhomogeneity; there is a clear strain gradient between the bottom and top of the sample. This inhomogeneity can be quantified by taking the

34 Uniaxial Stress Technique

difference in strain between the top and bottom divided by the average strain across the central plane of the sample. This quantity is plotted in figure 2.20 for a range of sample aspect ratios and three different asymmetric mounting models. The inhomogeneity is worst for a sample mounted with rigid epoxy from a single side. There is improvement with a softer epoxy but the inhomogeneity is still large. For example with an aspect ratio (L/t) of 20, not far below

the buckling limit, the inhomogeneity is still above 10 %. It is clear then that symmetric sample mounting should always be aimed for. The final curve in figure 2.20 shows a problem that might occur when symmetrical mounting is aimed for but the sample ends up off centre in the mount. Here the total space between the mounting plates encasing the sample is 1.5t so a symmetrically mounted

sample would have layers of epoxy 0.25tthick on each side but here

the bending induced inhomogeneity is shown for the case when the sample moves half this distance off centre. The inhomogeneity is still better than the sample mounted only from a single side, but for small aspect ratio (L/t) samples the inhomogeneity can still be

quite significant.

In document Imaginarios estético corporales de los estudiantes adolescentes y su incidencia en las relaciones interpersonal : estudiantes de grado décimo y once de las instituciones educativas distritales Rafael Uribe, Estados Unidos y Brisas del Diamante  (página 93-96)