In-situ Health Monitoring of Objects Consisting of
Visco-elastic Materials
K.-H. Laermann
FB D, Dept. of Civil-Engineering
Bergische Universität Wuppertal
D-42285 Wuppertal
ABSTRACT
To ensure their reliability, stability, functionality and safety controlling and monitoring of structures during the time of use have
become inalienable. Experimental methods have been proved as useful tools for these tasks, enabling in-situ measurements
of deformations, making it possible to draw conclusions on the actual state of structures at a time. If the characteristic material
parameters are known the stress state can be derived from the measured data. However in health monitoring of structures in
use since a more or less long time the actual parameters are unknown. In such cases identification of the actual material
parameters turns out to be the main objective in providing reliable knowledge on the actual internal parameters like the
strength of materials, on the internal stress state as well as on their changes over time caused e.g. by aging, fatigue and
environmental influences. Especially in case of visco-elastic response of materials it is quite important to dispose at any time
on reliable information and to pursue the effects of parameter changes on the response of structures in order to assess their
functionality and appraise their life-time. However generally the observed phenomena and measured data do not meet the
finally wanted information, but mathematical / numerical methods are to apply to proper evaluate the measured data in order to
get the decisive information for the structure assessment. Following it will be described how to proceed considering the twodimensional problem of plates-in-bending consisting of visco-elastic material. Based on the theory of linear visco-elasticity the
fundamental equations will be presented to evaluate the measured data of the deflection surface. For performing the
measurements the method of electronic speckle interferometry has been applied. The application of the described process will
be demonstrated by the example of a clamped plate. Finally the case will be discussed, that the measurements are performed
an arbitrarily long time after initial loading and use respectively, because then the problem of unknown history of the creep
effects must be solved.
Introduction
Considering structures and structural elements respectively, consisting of visco-elastic materials, it will be described in the
following, how the internal parameters of such materials like the relaxation bulk-modulus, the relaxation shear-modulus and
Poisson’s ratio can be determined by proper experimental/numerical processes. This is an inverse problem belonging to the
class of parameter-identification as the characteristic material parameters as time-functions shall be derived from measured
quantities.
Based on Boltzmann’s principle of superposition [1] the stress-strain relations for visco-elastic material response hold in the
Laplace-transform [2]
σ ij ( p ) = p ⋅ 2G ( p ) ⋅ eij ( p ) + p ⋅ K ( p ) ⋅ e( p ) ⋅ δ ij
(1)
where p denotes the Laplace-variable, K(p) the relaxation bulk modulus, G(p) the relaxation shear modulus, δij the KroneckerDelta, eij(p) the strain deviator, e(p) the volume strain. Re-transformation of eq. (1) into the time-space yields the Volterra+
+
nd
+
integral equation of the 2 kind with the initial values G(0 ) and K(0 ) at time t = 0 .
∂
∂
2G (t − τ ) ⋅ eij (τ )dτ + K (0 + ) ⋅ e(t ) ⋅ δ ij − ∫ K (t − τ ) ⋅ e(τ ) ⋅ δ ij dτ
∂τ
∂τ
0+
0+
σ ij (t ) = 2G (0 + ) ⋅ eij (t ) − ∫
t
t
nd
(2)
According to [3] Volterra’s integral equations of the 2 kind are well-posed and have a unique and stable solution always in an
appropriate setting.
The output data e(t) and eij (t) are taken by measurements in discrete time-intervals ∆t, which means, that the data are not
available as functions unless they are expanded in proper series. Therefore eq. (2) is transformed into a finite
difference/summation representation (eq. 3)
As to error propagation it can be anticipated, that the selected particular finite-difference/summation representation should be
extremely stable. The curves of the strains run steadily and smooth; with progressive time the increments in time intervals ∆t
are decreasing. Obviously the errors in data do not grow in size in subsequent stages of computation because of the generally
fading memory of most of the visco-elastic materials.
σ ij (t n ) = G (0 + )⋅ [eij (t n ) − eij (t n −1 )] − ∑ G (t n − tν ) ⋅ [eij (tν −1 ) − eij (tν +1 )] + G (t n ) ⋅ [eij (t1 ) + eij (0 + )] +
n −1
ν =1
( )
( )
n −1
1

+  K 0 + ⋅ [e(t n ) − e(t n −1 )] − ∑ K (t n − tν ) ⋅ [e(tν −1 ) − e(tν +1 )] + K (t n ) ⋅ [e(t1 )] + e 0 +  ⋅ δ ij .
2
ν =1

(3)
The equilibrium conditions for a three-dimensional stress-state hold
σ ij , j (t ) + X i (t ) = 0 , i , j ∈ [1 3]
(4)
The derivatives of eq. (3) in direction of the axes of a Carthesian co-ordinate system are to introduce into the equilibrium
conditions eq. (4), yielding three Volterra-integral equations of the 2nd kind in discrete formulation, substituting the strain
deviator and the volume dilatation by their derivatives eij,j and e,j. In these equations the bulk modulus K(t) and the shear
modulus G(t) as well as Poisson’s ratio are unknown, whereas the elements of the strain deviator and the volume strain and
their derivatives respectively are obtained from measured data of displacements and/or their gradients respectively, provided
these quantities can be measured for all i,j .
Basic relations for plates-in-bending
The following considerations refer to the shear-elastic plate theory of the 1st order (Mindlin-theory [4]). In contradiction to the
classical plate theory according to Kirchhoff and the Bernoulli-hypothesis the normal to the “neutral plane” does not remain
perpendicular to the neutral plane in the deformed state (Fig. 1). Under the presuppositions i) the material to be homogeneous
and isotropic, ii) the central plane of the plate to be the “neutral plane” for deformations due to bending, iii) the deflections to be
small compared with the plate thickness, which itself should be small with reference to the other geometric dimensions, iv) σ33
to be negligible small compared to the other components and set equal to zero, the shear stresses in the planes ( x1 , xβ ) and
their effects on the rotation of the plate normal are taken into account.
h/2
x1
h/2
u3,1
x3
û3
û1
u3
u3,1
Figure 1: Kinematics of the shear-elastic plate, section x2=const.
Let
uˆα ( x1 , x2 ), uˆ3 ( x1 , x2 ), α ∈ [1 2]
be given by measurements on the plate surface. Then the cinematic relations hold
u α ( x1 , x 2 , x 3 ) =
With the respective derivatives
uˆα , β
2
(5)
uˆα ( x1 , x 2 ) ⋅ x 3 , u 3 ( x1 , x 2 ) ≈ uˆ 3 ( x1 , x 2 ),
h
, uˆ 3, β , assuming σ 33 ≈ 0 , the elements of the strain tensor are determined, which
enable the formulation of the volume dilatation and the elements of the deviatoric strain tensor. Note: For lucidity the reference
to the co-ordinates (x1,x2) will be omitted in the following.
2
e = λ ⋅ uˆα ,α ⋅ x3
h
2
2
(κ ⋅ uˆ1,1 − λ ⋅ uˆ2,2 ) ⋅ x3 , e22 = (κ ⋅ uˆ2,2 − λ ⋅ uˆ1,1 ) ⋅ x3 , e12 = 1 ( uˆ1,2 + uˆ2,1 ) ⋅ x3 .
e11 =
3h
3h
h
1
2
1
2


e31 = ε 31 =  uˆ3,1 + ⋅ uˆ1  , e32 = ε 32 =  uˆ3,2 + ⋅ uˆ 2  .
2
2
h
h


(6)
(7a)
(7b)
The denotations κ and λ in eq.s (7 a/b) stand for
(8)
κ = (2 − µ ) ⋅ (1 − µ ) ; λ = (1 − 2 µ ) ⋅ (1 − µ ) .
It must be regarded, that Poisson’s ratio µ and subsequently κ and λ in principle are depending on time t. This fact will be
−1
especially considered in a later section.
The eq.s. (6) and (7a) are inserted into eq. (2), yielding the stress-strain relations for shear-elastic plates consisting of viscoelastic material. The internal forces mαβ, are obtained by integrating the respective stresses over the plate thickness h.
m11 (t ) =
h2
9
t


∂
+
ˆ
ˆ
⋅
⋅
−
⋅
−
[
(
)
(
)
]
0
G
κ
u
t
λ
u
t
G (t − τ ) ⋅ [κ ⋅ uˆ1,1 (τ ) − λ ⋅ uˆ 2, 2 (τ )]dτ 

1,1
2, 2
∫
∂τ


0+
( )
t

∂
h 
+
ˆ
ˆ
+
⋅
⋅
+
−
0
K
λ
[
u
(
t
)
u
(
t
)
]
K (t − τ ) ⋅ λ ⋅ [uˆ1,1 (τ ) + uˆ 2, 2 (τ )]dτ 

1,1
2, 2
∫
∂τ
6 

0+
û 2, 2 the respective equation for the bending moment m22 will be obtained.
( )
2
Interchanging
û1,1
and
(9)
t


∂
+
(10)
G 0 ⋅ [uˆ1, 2 (t ) + uˆ 2,1 (t )] − ∫ G (t − τ ) ⋅ [uˆ1, 2 (τ ) + uˆ 2,1 (τ )]dτ 
∂τ


0+
To determine the shear-forces qα , the shear-stresses σ3α are to integrate over h. The shear-stresses are independent of x3 ,
i.e. they are constant over the plate thickness. However in fact they are distributed parabolic and at x3=± h/2 the boundary
conditions require σ31=σ32=0. This discrepancy shows the character of approximation of Mindlin’s plate model, which violates
m12 (t ) = m21 (t ) =
h2
6
( )
the local equilibrium conditions. For correction the deformation energy will be determined, considering parabolic distribution of
the shear stresses over h on one hand, on the other constant distribution over a corrected plate thickness hs [5].
The deformation energy WI is set equal to WII ,then the corrected thickness hs amounts to 0,833 h. This consideration is true
at any time t, therefore the shear forces depending on time t can be formulated as follows:
2
2
∂




qα (t ) = G (0 ) ⋅ hs uˆ3,α (t ) + uˆα (t ) − hs ∫ G (t − τ )uˆ3,α (τ ) + uˆα (τ ) dτ ,
h
h
∂τ




0+
t
+
(11)
The respective derivatives of the internal moments (eq. 10) and the eq.s. (11) are inserted into the equilibrium conditions,
which are related to a plate loaded by a concentrated load constant over time.
mαβ , β (t ) − qα (t ) = 0
(12)
After some intermediate transformations, considering the discrete solution of Volterra’s-integral equations , (see eq. (3)), two
equations will be obtained enabling the calculation of the bulk-modulus K(tn) and the shear-modulus G(tn) based on
measured quantities of the displacements
uˆ1 (t ), uˆ 2 (t ), uˆ3 (t ) and/or their gradients respectively:
[
]
G (t n ) ⋅ Σ α∗ 0 + K (t n ) ⋅ Σα 0 = ∑ G (t n −ν ) ⋅ ∆∗αν + K (t n −ν ) ⋅ ∆ αν + G (0 + ) ⋅ ∆∗αn + K (0 + ) ⋅ ∆ αn
The abbreviations denote
(13)
( )
Σ α∗ 0 = U α∗ 0 + + U α∗ (t1 ) , ∆∗αν = U α∗ (tν −1 ) − U α∗ (tν +1 ) , ∆∗αn = U α∗ (t n −1 ) − U α∗ (t n ) ; Σα 0 , ∆ αν , ∆ αn
3h
1
1
1 1 +ν
U α∗ (t ) = κ ⋅ uˆα ,αα (t ) + uˆα , ββ (t ) + ⋅
uˆ β , βα (t ) − 3s
3
2
6 1 −ν
h
respectively.
2


uˆ 3,α (t ) + h uˆα (t ) , α ≠ β ,
(14)
(not to sum up over α and β!)
1
U α (t ) = λ ⋅ uˆ β ,βα (t )
2
(acc. to Einstein’s convention)
(15)
Neglecting the rotation of the plate-normal in case of small h a modified Mindlin–theory may be considered. Then the
components of the strain tensor hold
εαβ = u3,αβ ⋅ x3 ,
and the terms
U α∗ (t ) , U α (t ); α ∈ [1 / 2]
ε 33 = −
ν
u3,αα ⋅ x3 , ε 3α = u3,α ⋅ x3
1 −ν
(16)
in eq.s (13) stand for
12h
1
1
U α∗ (t ) = κ ⋅ u 3,αββ (t ) − 3 s ⋅ u 3,α (t ), U α (t ) = λ ⋅ u 3,αββ (t )
3
2
h
(17)
Determination of Poisson’s ratio
As linear material response can be described by only two characteristic parameters Poisson’s ratio can be substituted by the
bulk-modulus and the shear-modulus. In the Laplace-transform the relation holds
µ( p) =
With the denotations
time-space
1
−1
(18)
⋅ (3K ( p ) − 2G ( p )) ⋅ (6 K ( p ) + 2G ( p ))
p
Z ( p ) = 3K ( p ) − 2G( p ); N ( p ) = 6 K ( p ) + 2G ( p ) eq. (18) runs after re-transformation into the
t
µ (t ) = ∫ Z (t − τ )
0+
∂ −1
N (τ )dτ + N −1 0 + ⋅ Z (t )
∂τ
Discrete integration of eq. (19), taking into account ∆N (tν
as a quite fair approach of Poisson’s ratio
( )
(19)
) to be very small and therefore ∆N (tν ) ⋅ N (tν )−2 ≈ 0
µ (tν ) ≈ Z (tν ) ⋅ N (0 + )
−1
and of the factors, eq.s (8)
yields
(19a)
κ (tν ) = 1 + N (0 + ) ⋅ [N (0 + ) − Z (tν )] ; λ (tν ) = 1 − Z (tν ) ⋅ [N (0 + ) − Z (tν )]
−1
−1
(8a)
With this approach the eq.s (19) are becoming non-linear of grade two.
Due to numerous experiments in material testing and experiences of several researchers and myself it has been proved that
the changes over time of Poisson’s ratio of most of linear visco-elastic materials are negligible small and the approach
µ (tν ) ≈ µ (0 + ) = [K (0 + ) − 2G (0 + )]⋅ [6 K (0 + ) + 2G (0 + )]
−1
1
+
might be permissible. To determine the initial value of µ(0 ) the moduli
improved value
1
µ (0 + )
(19b)
( ) ( ) are to calculate on the basis of the
1
K0 ,G0
+
t(0 ) = 0 for an estimated Poisson’s ratio µ(estim). Introduced into eq. (19b) an
2
+ 2
+
yields improved values K 0 , G 0 . This iterative process may be continued until the difference
measured data of displacements at time
+
+
n
( ) ( )
∆µ (0 ) = µ (0 ) −
+
n
+
µ (0 + ) ≈ 0
n −1
.
(20)
Technique of measurement and example
Different optical methods are available to measure the displacements and / or their gradients on the surface of the objects
under consideration [6] because optical methods yield data of high precision as field information, which are necessary to
determine derivatives in the course of evaluation of the measured data. For plates-in-bending two methods have proved to be
quite practical, i.e. the electronic speckle interferometry (ESPI), [7], [8], and the electronic speckle shearing interferometry
(ESPSI), [9[, [10]. The ESPI-method provides discrete data of the displacements
uˆ1 , uˆ 2 , uˆ 3
on the plate surface in the nodal
points of an evaluation grid. The measured quantities have to undergo an adjustment and spline approximation process before
nd
determining the 2 -order derivatives, which are to insert in the eq.s (14), (15). With concern to the modified Mindlin-plate even
the 3rd-order derivative is necessary (see eq.(17)) On the other hand the ESPSI-method provides 1st derivatives of the
displacements on the plate surface and thus the elements of the strain tensor already.
The measurements are to perform in time-intervals ∆t. Assuming the material response to be independent of (x1, x2) it is
sufficient to determine the parameters K(t), G(t) in one nodal point only. However in practice a few other points may be taken
into consideration for control.
As an example of application a clamped plate has been chosen, dimensions and loading- and boundary conditions are shown
in fig. 2. The specimen has been made of epoxy resin Araldite F with hardener HY951 and plastiziser Araldite CY208.
F= 10 N
x1
x2
●
x3
50 mm
5
50 mm
Figure. 2. Specimen of the clamped plate
By means of ESPSI the gradients
û 3,α
have been measured in time-intervals ∆t = 10 sec. starting at t ≈ 0 and proceeding till
tE = 1000 sec. For data recording and processing the program ISTRA [11] has been used. The data û 3,α in the nodal points
(i,j) of the evaluation grid have been subjected an adjustment and smoothing process and the derivatives have been
calculated with concern to the modified Mindlin-plate. The Poisson’s ratio has been found by the iteration procedure described
above to about µ (0+) ≈ 0.40. The relaxation parameters K(t) and G(t) have been calculated in the centre of the plate model
according to eq.s (13). Considering the relations between the material parameters the Young’s modulus has been calculated
also (fig. 3.
1800
1600
1400
N/mm2
1200
K(t)
1000
800
600
E(t)
G(t)
400
200
0
1
10
time [sec]
100
1000
Figure.3. Evolution of the relaxation parameters (Note: half-log.-scale)
Identification in case of unknown history
The presented method is based on the assumption that the measurements start at the time of initial loading of the structure.
However in reality the more interesting case must be considered, that the state of structure and the material response are to
determine more or less long time after initial loading and use of the object and no information about the structural history
including the material characteristic parameters is available.
Assuming that only creep-effects are to consider and the unknown initial loading is constant over time a proposal will be
presented how to get approximate values of the relaxation moduli at least [12]. A one-dimensional problem is taken as a basis.
At time
tA the measurements are started yielding the increase of the strain ε (t n ) related to ε (t A ) in time-intervals
∆t , t A ≤ t n ≤ t N
which corresponds to
( )
(fig. 4). At time tN a test-load is applied. To begin with the elastic strain is to determine.
meas.
εˆ(t N )test
= ε (t N )
− ε (t N )
elastic
εˆ (0 + )elastic
test
Ê 0 + can be calculated.
For tµ > tN the unknown
related to the
values
(21)
(tˆ, εˆ )-co-ordinate system. With given σˆ (0 )
+ test
ε (t µ ), t µ = t N + µ ⋅ ∆t
the relaxation modulus
are calculated by means of a balancing polynomial e.g.
f (t ) = ∑ aκ ⋅ t κ , proceeding stepwise from tµ-1 to tµ thus yielding an approximate course of ε (t µ )
k
calc .
κ =0
the better the smaller the gradient of
ε (t µ ). In
most cases it can be anticipated the gradient to be small, approaching to
constant or even to zero. Continuing the measurements for
according to eq. (22).
εˆ (tˆµ )
test
Subsequently the relaxation modulus
Eˆ (tˆµ )
. The approach is
tˆµ
> tˆ
= ε (tˆµ )
(0 ) the strain related to the test load will be obtained
meas.
+
− ε (t µ )
calc.
(22)
can be calculated according to the respective solution of the Volterra’s integral
equation for a one-dimensional problem. And because of the time-shift invariance conclusions on the history of the timedepending response of the visco-elastic material are possible.
εˆ
test
εˆ(t N )elast . = εˆ (0 + )elast .
test
εˆ (tˆµ )
test
ε
0+
?
tˆµ
meas
ε (tˆµ )
ε (t µ )calc
tˆ
ε (t n )
∆t
tA
tn
tN
time t
tµ
start test load
Figure 4: Determination of the strain-state under test load.
Analogue to the afore described procedure the bulk-modulus as well as the shear-modulus for two- and three-dimensional
problems can be determined from in-situ measured displacements and/or their gradients enabling conclusions on the á-priori
unknown history.
To demonstrate the application of this procedure a specimen of a beam (fig. 5.) of carbon-fibre-reinforced polymer has been
tested under four-point-bending assuming the material parameters to be unknown. The strain on the upper and lower surface
have been measured by strain-gages, yielding the mean value of the elastic strain at time
( )
( )
t 0 + : ε 11 0 + = 209 µm .
strain gages
1N
1N
5 mm
85
85
145
40
145
Figure 5: Specimen of a beam under four-point-loading
( )
With σ 11 0 = M W = 0.036 kN/cm the Young’s modulus amounts to E(0 ) =172 kN/cm . The course of the strain
has been recorded every minute, after 20 minutes the loads have been doubled as test load and the measurements have
been continued for another 2o minutes, the results are plotted in fig. 6.
Assuming the history of the deformation after initial loading to be unknown up to time tA the changes of ε 11 t n in the interval
+
t A to t N
2
+
2
are taken as the basis of the extrapolation process as described above yielding finally εˆ11
(tˆ )
test
µ
( )
.
600
500
ε 11 (t )
εˆ11 (tˆµ )
test
400
unknown history
300
200
100
0
1
3
5
7
9
11
13
15
17
tA
19
21
23
tN
25
27
29
31
33
35
37
39
41
43
time [min.]
Figure.6: Determination of the test relevant strain.
On the basis of these strain data the time-depending material parameter E(t) has been calculated according eq. (23), (fig. 7)
σ 11 (t n ) = σ 11 (0 + ) = E (0 + ) ⋅ ε 11 (t n ) −
1 n
∑ [E (t n − tν ) − E (t n − tκ −1 )] ⋅ [ε 11 (tν −1 ) + ε 11 (tν )]
2 ν =1
(23)
180
80
70
E(t)
160
60
kN/cm2
50
140 40
30
120 20
10
0
0
20
0
25
5
30
10
35
15
40
20
45
25
time [min]
Figure 7: Evolution of E(t)
Conclusion
Monitoring and supervising structures and structural elements respectively consisting of polymer materials becomes
increasingly important as the actual strength of materials defines the functionality, the bearing capacity and safety of
structures. It is the objective of the preceding explanations presenting a method to identify the characteristic parameters of
linear-elastic materials on the basis of in-situ measured deformations. Optical measurement techniques like ESPI and ESPSI
yield the necessary data-sets to determine these parameters. To begin with the presented method is based on the assumption
that the supervising process starts already at the time of initial loading. Under this presupposition the applicability of the
method has been proved exemplarily examining a clamped plate. However the more interesting case must be considered that
the process of health monitoring starts at a more or less long time after initial loading and use and therefore the history of the
structural and material response till the date of monitoring is unknown. For this case an approach has been presented and
proved by the example of beam. The result may be accepted to be satisfactory. In conclusion it can be stated that the
identification of characteristic parameters of linear visco-elastic material based on measured data of deformations is possible
at any time over the life- time of structures.
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2.
3.
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