Rubber membrane

For the rubber membrane, the material parameter e1 in the neo-Hookean model (see Ap- pendix C, section C.1) is estimated:

w

⁼ C i ( 1 1 - 3) (4-3)

In the reference state, displacements and reaction forces are set to zero. During the increasing extension of the rubber membrane, the reaction forces at the affection point increase and the strain field becomes more inhomogeneous. The principle struin domain ^(€1 , ~ 2 ) for Experiment i at maximum extension is shown in Figure 4.9, where strains are calculated by observing marker groups according to the method described in Appendix D. It is clear that most of the principle strains are positive and cover a relatively large area, indicating a strongly inhomogeneous strain field.

Principle strains

0 1 -

o O8 O 06

~

- w

O 04

o o2 - -

O r -0 o2

-0 04

O 005 O 1 O15 O 2 O25

E1

Figure 4.9: Principle strain domain of Experiment 1 at maximum extension, determined from the displacements of 4055 markers

As stated before, an object function J will be minimized with respect to the unknown param- eter values in case of the considered experiment(s). All parameter estimates will be based on

32 Chapter

I

the confidence in the experimental output, so _W will from now on be equal t o zero. Initially, the diagonal terms of

y D

will be set to 2.5 lo5

[A]

and the diagonal terms of

y F

are set to 2.5

[hl;

These values are based on the RMS-values of the measured displacements and forces respectively, as well as the total number of z and y force components (=32) with respect to the total number of z and y displacement components (=8110), the latter ratio is settled in

y F

in this case.

Figure 4.10 gives the object function plot for forces and displacements, as a function of e1 for Experiment i, where the experimental output of several load cases is combined. The changes in the residuals

cD

of the displacement field and the residuals of the forces

C F

as a function of the material parameter change have been separately investigated by setiing respectively

- V F and

ED

to zero in Equation 2.9 and then calculate J as a function of the parameter(s) values. From Figure 4.10, several remarks can be made:

O 100 150 200 250 .300 ~ 350 400 450

Cl

1W i 5 0 200 250 300 . 350 400 450 500

Cl

Figure 4.10: Object function as a function of c1 [mN/mm2]; Left: only forces considered,

y D

⁼ 0

and the diagonal terms of

Y F

equal 2.5

[A];

Right: only displacements considered,

y F

⁼ 0 and the diagonal i e r m s of

y D

equal 2.5

.

I O 5

[A]

o The object function for force shows a clear minimum value at a cl-value of about 200-250 [mN/mm2].

o A change in e1 has a small influence on the residuals of the displacement field: the variation of J is small for all considered parameter values. Therefore, in the following part of this subsection, the measured displacement field will be left out of consideration by setting

Y D

^{= Q} and only the force residuals will be used to obtain the optimal parameter value.

The estimated values of e1 are presented in Table 4.1 for different load cases separately and a combined case. Also, the RMS-values of the force and displacement residuals,

<gMs

^and

E x p e r i m e n t s 33

(iMs

respectively, is given for each load case; these are defined as:

(4.4)

(4.5) where:

cf

^and are the k-th element of

cD ^-

^and

cF

^- respectively;

M D = 8110 and M F = 32 are the number of observed displacements and forces respectively;

The RMS-values &mgMS of the measurement inaccuracies of the displacements are calculated as described in section 4.3, where j represents the load case number. For the RMS-values ómCMs of the error in the measured force components, 4% of the RMS-value of the absolute force components

(mF)

is taken:

GmRMs = 0.2024

.

lop2

.

^j (4.6)

(4.7) The ratios cgMs/6mgMs and CiMs/Gm$Ms are determined, which can provide information about the presence of modeling errors. The values of the residuals at load case 11 of ^{E x -}

I/

^{exp. nr.}

1 1 1 1 1 1 1 2 3

load case 1 4 7 9 10 11 comb.

11 11

C1

218.9 218.6 219.4 220.0 221.0 221.2 220.4 216.3 227.9

C R M S

0.008 0.021 0.035 0.045 0.050 0.055

-

0.039 0.043

2.53 2.42 2.43 2.42 2.46

1.77 1.92

9.1 16.3 22.6 26.6 28.5 30.6

-

3.76 1.66 1.34 1.23 1.19 1.17

-

20.4

I

^1.98

24.4

I

^1.38

Table 4.1: Final parameter estimates [ m N / m m 2 ] , the values of force and displacement residuals and their ratio with the variances of the measurement inaccuracies

34 C h a p t e r

4

p e r i m e n t i are examined4 in order to determine the quality of the fit of the output data of the numerical model y(@) onto the experimental output data n ~ . The vector plot of the dis- placement residuals over the surface of the specimen is shown in Figure 4.11; each data point contains one residual vector. The elements

C,",

k = 1,

. . .

, 3 2 , of the force residual column

2"

are shown in Figure 4.12, where k = 1 corresponds to the z-component of the reaction force at affection point 1 and k ⁼ 32 corresponds to the y-component of the reaction forces at affection point 16. The following remarks can be made about the final parameter estimates,

Figure 4.11: Vector plot of the residual in Experiment 1 of the rubber membrane

displacement field ( x 26) a t the state of m a x i m u m extension

using observables of one load case:

o The estimated cl-values are almost the same for all load cases of one experiment;

o The e1 values show a minor difference between two different experiments;

o The parameter values are realistic values.

Remarks with respect to final parameter estimates, using observables of several load cases of one experiment together:

4These residuals are calculated for e1 = 220.4 [mN/mm2], which is the parameter value obtained from the combined information of several load cases of Experiment 1

E x p e r i m e n t s 35

-800

'

^I

1 5 10 15ik 20 25 3032

-400

i

-600

1 I

-800 ₁

'

_{5 10 1 5 ~ ~ 20 25 3032}

Figure 4.12: Experiment i, load case 11 of the rubber; Left: measured forces rnf ('x') and calculated forces y{ ('+

7;

Right: residual forces

CIy"

('o ')

o The final parameter estimate is some sort of average of the final estimates at individual load cases.

Remarks with respect to the residuals:

o Both variances of displacement and force residuals increase with an increasing extension of the membrane. This can be explained by the fact that the inaccuracies in both displacement and force measurements increase with increasing extension and/or the presence of a modeling error; the ratio (.RMS/6m&s is about 2.5, except for load load case 1 it is higher; the ratio C~Ms/Gm~,s decreases with an increasing load case to a value slightly larger than one. The fact that both ratios are larger than one is a sign for the presence of modeling error(s).

D

o In general, the vector plot of

CD

is randomly distributed (except local deflections, which can be a result of the minorvarying thickness of the sample, which is assumed to be constant in the model)

o The residuals

C F

of the x and y force components of E x p e r i m e n t I , load case 11, appear to be randomand relatively small with respect to measured and/or calculated forces (nf and

qF (e)

respectively).

In Figure 4.13, the calculated reaction forces of the membrane at the sixteen affection points as a function of the local elongation factors is shown together with the measured reaction forces for E x p e r i m e n t I . The local elongation factor, which is the variable on the horizontal axis of each subfigure, is defined as

F,

^{where T d and r,,f} are the distances of the affection point of the considered prescribed displacements to the center point C of the rig in the de- formed and in the reference state respectively (see Figure 4.14). The forces on the vertical axis are the a b s o h t e forces, i.e. NOT the forces with respect to the reference state. The abso- lute forces in the reference state can be read by taking the local elongation factors equal to one.

r e f

36 C h a p t e r

4

1 1.1 1.2 1 1.1 1.2 1 1.1 1.2 1 1.1 1.2

1 O00 1 O00 -1 O00

1 1.1 1.2 1 1.1 1.2 1 1.1 1.2 1 1.1 1.2

1.1 1.2 1 1.1 1.2 1 1.1 1.2 1 1.1 1.2

1

1 1.1 1.2 1 1.1 1.2 1 1.1 1.2 1 1.1 1.2

Figure 4.13: Measured

('+y

and calculated (solid line) reaction forces at the affection points as a function of the local elongation factor for the rubber (Experiment 1, load cases 1,

4,

7, 9, 10 and 11)

Figure 4.14: Illustration of r d in the deformed state and r,,f in the reference state of the specimen

Experiments 37

According to Figure 4.13, the reaction forces increase almost linearly with a n increasing local elongation factor. The goodness of the fit is quite well (i.e. the difference between the measured and calculated reaction forces is small) and the goodness of the fit is almost independent on the value of the initial forces.

In document Eindhoven University of Technology MASTER On the multi-axial testing of membranes Brouwers, H. (pagina 42-48)