| On the unicity of Weibull's distribution function.
G. Díaz (1), V. Martínez (2) and P. Kittl (2) (1) Departamento de Ingeniería de los
Materiales, IDIEM,
Introduction. A large number of mechanical properties whose knowledge is required
for designing structures are non-deterministic by nature and they follow
a certain distribution function of probability. Fracture stresses in brittle
materials as well as yielding stresses in ductile materials obey Weibull's
distribution function [1,2]. As a matter of fact this last function has
been widely used to explain the probabilistic distribution of said stresses
in a large collection of materials and in diverse strain-states[3-8]. The
fundamentals of the Probabilistic Strength of Materials as well as the
sundry deductions of Weibull's distribution function have been discussed
in many works [2,9-12]. In a previous work, Kittl and Dìaz [12]
have discussed five different deductions of Weibull's statistics. It is
interesting to find a more general distribution function in order to see
if anomalous behavior in fracture materials can be explained through this
manner.
Generalization of the functional equation. Consider an isotropic homogeneous body of volume V subjected to a constant state of stress, and an arbitrary division of V into volumes V1 and V2 without common elements so that V=V1 + V2. Let N be the number of similarly-fabricated samples of a material and that are to be subjected to a fracture or yielding test when the material is brittle or ductile, respectively. Three possibilities may arise in this connection, namely: volume V1 fails; volume V2 fails; and there is no failure. Let n1 be the number of fractures started within volume V1, n2 the number of fractures started within volume V2, and n the number of tests without any fracture at all, so that then N = n1 + n2 + n. All these tests are made by applying, gradually, some stress s so that 0 £s£si where i= 1,2,3,..., N. According to the Probabilistic Strength of Materials, the fundamental
functional equation originating the same is [2]:
where
where F1 and F2 are the cumulative probabilities
of fracture of the bodies whose respective volumes are V1 and
V2. Using equation (2) allows to rewrite equation (1):
and here, to simplify writing, the argument of the functions of cumulative probability of non-fracture have been omitted. Equation (3) can be generalized as follows:
where F12 is the cumulative probability of fracture of body of volume V and l is a real number. In equation (4), to the set of cases without fracture there must be added the set of cases with fracture in volume V1 only (none in V2), and the set of cases with fracture in volume V2 only (none in V1). Using again equation (2) allows to rewrite equation (4):
and this expression becomes in terms of the functions of cumulative
probabilities of fracture and of non-fracture:
Obviously functional equation (6) yields functional equation (1) in the particular instance of l =1. Functional equation (6) can be converted into:
Unicity of cumulative probabilty of non-fracture. Now it will be shown that the function of cumulative probability of
non-fracture is unique, using therefor the following boundary condition:
that is, the non-fracture probability will be approaching the value
1 in the case of some body whose very small volume V ®
0 is devoid of whatever defect liable to start some fracture. According
to equation (7), if V1=0 and V2=V:
and likewise if V2=0 and V1=V:
Consequently the function of cumulative probability of non-fracture
is unique, that is to say:
Hence the functional equation (7) can be written as follows:
Solution of the generalized functional equation. Now functional equation (12) and, therefore, functional equation (7)
will be solved. Small volume-increments yield:
Considering the following Taylor-series expansion:
its introduction into equation (13) produces:
Similarly, also
Equating equations (15) and (16) and making some rearrangements yield
the following differential equation allowing to solve the functional equation
(7):
Putting:
differential equation (17) becomes:
whose solution is:
where k is a constant determined using the boundary condition
After carrying out some transformations the solution of functional equation
(7) is:
The expression for
where V0 is the unit volume whereas f
(s ) is the Weibull’s specific risk function.
Hence equation (22) becomes:
Equation (24) complies with boundary condition
that is, non-fracture probability will be approaching the value zero
in the case of some body whose very large volume V ®¥
always includes some defect liable to start some fracture. When V ®¥
, equation (24) is reduced to:
whose value is not zero except when l = 1. Accordingly, the probability function obtained by solving functional equation (7) is only compatible with boundary conditions set forth in equations (8) and (25) when l = 1, so that instance l = 0 remains to be studied below. Extreme instances l = 0 and l
= 1 regarding functional equation (7) define two distribution functions.
Thus, if l = 1 then the following distribution
function is obtained from equation (24):
and like result is obtained by putting l
= 1 in equation (17). Using the cumulative probability of fracture function
the equation (27) yields:
which is the well-known Weibull’s distribution function [1]. On the
other hand, if l =0 then the following differential
equation is obtained from equation (17):
and its solution is:
where C1 is a constant determined using the boundary condition
defined in equation (8) and its value is 0. Hence the solution to equation
(29) is:
Employing the cumulative probability of fracture function and introducing
equation (23) into equation (31) yields:
The distribution function of cumulative probability defined by equation (32) is a Weibull statistics when the volume of the material, or the stress applied thereto, is approaching the value zero, as can be seen when equation (27) is expanded in Taylor series. In short, according to the generalization proposed herein for the functional
equation of Probabilistic Strength of Materials, only one statistics compatible
with the boundary conditions is existing, that is, the Weibull’s distribution
function when l = 1, equation (28), because
the distribution function when l =0, equation
(32), can be obtained from the above equation (28) by making (V/V0)f
(s)® 0.
Acknoledgments. The authors are grateful to the Fondo Nacional de Desarrollo Científico
y Tecnológico, FONDECYT, for the Grant of Projects N° 1931056
and 1961105.
References.
Kittl, P. and Díaz, G., 1990, "Five Deductions of Weibull’s Distribution Function in the Probabilistic Strength of Materials", Engng.Fract.Mech., Vol. 36, pp. 749-762. |
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is the cumulative
probability of non-fracture of the body of volume V while 
and 
are the cumulative probabilities
of non-fracture of the bodies of respective volumes V1 and V2
when the material is subjected to a constant stress s
. The functional equation (1) is based on the hypothesis of the independence
of the cumulative probabilities of non-fracture. Hence, considering the
mentioned experience with N tests, the following probabilities derive therefrom:
can be
expanded as a Taylor series:
.
Thus the solution to differential equation (19) is:
is
already known and can be found in [2], for example:
.
In addition, there is also another boundary condition in the ambit of Probabilistic
Strength of Materials, namely:
