InternationalJournalof Fracture41: R71-R76, 1989.
© 1989KluwerAcademicPublishers.Printedin theNetherlands.
R7 ]
ON THE DETERMINATION OF THE CONSTITUTIVE PROPERTIES OF THIN
INTERPHASE LAYERS - AN EXACT INVERSE SOLUTION*
Peter Olsson
Division of Mechanics, Chalmers University of Technology
S-412 96 Gi~teborg, Sweden
tel: +46 31 721525
UlfStigh
Division of Solid Mechanics, Chalmers University of Technology
S-412 96 G~teborg, Sweden
tel: +46 31 721509
The double cantilever beam DB(T) specimen is one of the most well defined
specimens for evaluation of the constitutive, peel, properties of adhesives and
solders. It consists of two parallel beams joined together and tom apart by
concentrated loads applied at the end, cf. Fig. 1. Here L indicates the length of the
specimen, which is assumed to be sufficiently long, as will be discussed in the
sequel.
The traditional way of analysing this specimen is to assume that the behaviour
of the adhesive is known, e.g. rigid [1], linear elastic [2], ideal plastic [3], or
softening [4,5], and then try to fit the results of the analysis to the experimental
results. This procedure is not very satisfying since just limited families of assumed
constitutive behaviour can be analysed and no real deduction of the correct model
can be made.
In the present paper a method is presented whereby the constitutive properties
of the joining material are measured with only rather weak a priori assumptions of
its behaviour.
In the following it will be assumed that the beams may be approximated as
Euler-Bernoulli beams and that the adhesive layer will act as a continuous
distribution of nonlinear springs. With w denoting the total relative deflection
between the beams and with EI denoting the equivalent bending stiffness of the two
beams, given by
1~El = (1/EI)~e, r + (1/EI)j....
(1)
*This paper is a corrected version of "Mechanics of the Double Cantilever Beam
Specimen" by Ulf Stigh, International Journal of Fracture 40 (1989) which was
published in error. It had been the author's wish that the manuscript be withdrawn
prior to publication. The editorial staff apologizes for any inconvenience this has
caused.
Int Journ of Fracture 41 (1989)
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the deflection w is given by
d4w
EI--'~T+ q(w)=O
(2)
where q(w) is the load/unit length exerted by the adhesive on the beams. This load
is determined by w, the local elongation of the adhesive, and it is this relation
between q and w that is to be determined. We must assume the existence of a unique
solution to the differential equation under the boundary conditions discussed below.
That this restricts the range of possible functions q is obvious, cf. the case q-- 0.
Two things are assumed a priori about q(w). First, we assume that a
deformation energy density function, W(w), corresponding to q exists, i.e.
W(w) -- ~0wq(w)dw
(3)
Secondly we assume that there exists an (arbitrarily small) open interval containing
w = 0, in which q is proportional to w.
The boundary conditions are
x=0
w=A
andd2W_0
d~2(4a)
x --+~
w =0
dw
and--= 0
dx
(4b)
In fact only the vanishing of w needs to be assumed at infinity, since this, together
with the second assumption on q above, implies the vanishing of all drivatives of w
at infinity.
In practice, a "sufficiently long" specimen will of course suffice, due to the
exponential decay of the deflection.
The inverse problem to be solved here is then to find q(w) from measurements
on the specimen. The method proposed here is to measure the deflection A and slope
O at the point of load application together with the load F. The constitutive
properties are then found from the formula
q(A) = d ~ ( F O )
Int Journ of Fracture 41 (1989)
(5)
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Since both F and O are assumed to be uniquely determined by A, given a certain
relation q = q(w), there exists a relation between the two. This indicates that (5) may
be further simplified, at least in principle.
Equation (5) is derived by multiplying (2) by dw/dx and then integrating it
along the beam, i.e.
fo E1 daWdw
(6)
The left hand side of (6) is then integrated by parts and the fight hand side is
identified as the deformation energy density of the interphase layer, i.e. (3),
evaluated at w = A. Thus
[ ! d wdw']**
z
2
(7)
Evaluating the left hand side by means of the boundary conditions, we obtain
W(A) = F O
(8)
where F is the negative of the shear force E1 d~w/dx3 at x = 0 and O = - dw/dx at x
=0.
Differentiating, we get (5).
To obtain the shape of q(w) over an interval, the fight hand side of (5) need
only be evaluated for A in that interval.
As afirst elementary example consider the solution of (2) when q is
proportional to w, i.e. q(w) = kw. This corresponds to two well known problems, i.e.
the bending of a beam on a Winkler foundation and the bending of a cylindrical
shell, cf. e.g. [6]. With the boundary conditions, (4), the solution is given by
F
lk A
=~
(9a)
O=~A
(9b)
Here ~¢ = k/(4EI). By use of (5) one obtains q(A) = kA as expected.
Int Journ of Fracture 41 (1989)
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The second example involves a more complex situation a q(w) relation as
shown in Fig. 2. With wo/w,. = 0 (Wo>0) this behaviour corresponds to an ideally
plastic interphase and with wo/w~ = 1 one obtains a response corresponding to a
brittle interphase.
Surprisingly, this problem has an analytical solution, cf. [5]. The solution is
divided into three regions. The first one corresponds to small A<Wo. In this case w
will be smaller than Wofor all x and the response of the specimen will be linear as in
the first example, i.e. the inverse solution gives the correct result.
The second region corresponds to the growth of a damage zone, from x=0,
where the interphase layer softens. Denoting the.length of the damage zone by L v in
agreement with the notation in [5], one finds the force, F, load point deflection, A,
and load point slope, @, as functions of L v Due to the complexity of the expressions
involved we have contented ourselves with a numerical evaluation of the relevant
functions.
It is found that A increases monotonically with L 2, thus (5) may be written as
q - ~ "
(10)
Evaluating (10) and the expression for A, given in [5], for increasing L 2 yields the
correct result.
The final third region corresponds to growth of a crack, i.e. a zone emanating
from x=0 where the interphase layer has fractured. Also in this case one finds F, A,
and 0. as functions of L v Perhaps counter intuitively one observes that A increases
monotonically with decreasing L~, thus (I0) may still be used.
Evaluation of (10) gives q=0 as expected.
The formula for q in (5) immediately suggests an experimental procedure for
evaluation of the consititive properties of thin interphase layers such as adhesives
and solders.
Notably, if the product o f F and O is found to be strictly concave in some
interval, this indicates softening of the interphase layer.
As already noted the functions F and O are not independent of each other.
Hence, (5) may be further simplified if an explicit expression may be found that
gives one in terms of the other.
It perhaps seems questionable that in (5) the parameter EI does not appear
explicitly. This is however not so surprising in view of the fact that q is a function
describing the properties of the interphse layer, and not the properties of the beam.
Int Journ of .Fracture 41 (1989)
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REFERENCES
[ 1] S. Mostovoy, P.B. Crosley and E.J. Ripling, Journal of Materials 2 (1967)
661-681.
[2] M.F. Kanninen, International Journal of Fracture 9 (1973) 83-92.
[3] S.E. Yamada, Engineering Fracture Mechanics 27 (1987) 315-328.
[4] T. Ungsuwarungsri and W.G. Knauss, International Journal of Fracture 35
(1987) 221-241.
[5] U. Stigh, International Journal of Fracture 37 (1988) R13-R18.
[6] A.P. Boresi and OIM. Sidebottom, Advanced Mechanics of Materials, 4th ed.,
John Wiley & Sons, New York (1985).
31 May 1989
IF
//-- (EI)upper
ii
III
I ~iil
i
......
ii r~i iilllliil ..... ,,i,i,,ii, ii,,~iii
. . . . .
i~F
~ ....
. . . . . . . . . .
~--(£1)'°wer
L ~--adhesive
,i, I
I
,]
Figure i. Double cantilever beam specimen.
Int Journ of Fracture 41 (1989)
R76
q
qo
i
Y
/
W0
W L
W
Figure 2. Constitutive behavour of adhesive assumed in the second verification example.
Int Journ of Fracture 41 (1989)