The Weibull analysis provides a probability of failure at a given stress, and is often expressed by the empirical rela- tionship:
Equation 1 Equation 7
times referred to as the position parameter. Therefore, for specimens of constant geometry,
mally taken to equal zero), X is the strength limiting dimen- sion of the material, and σo
Equation 2 the simplest method of obtaining σo and m from a series of
data is to rank the σ data from smallest to largest and assign P values so that:
Equation 3
Where “i” is the rank and N the total number of specimens, Eqn 2 can then be expressed as:
y = A + Bx Where:
Equation 4
Where P is the probability of failure at a stress, σ, m is the Weibull modulus, σu
is the stress at which P = 0 (and is nor- is a normalizing factor, some-
Where n in Eqn 7 is the rate at which f(a) tends to zero for a>>c/n and c is a scaling parameter. Assuming there are a large number of randomly oriented flaws, m and n are related through m = 2n - 2. (Note that “n” used in Eqn 7 is the same notation as the original reference 4 and is not the same as the strain hardening exponent, n, defined later for Eqns 13 and 14.) Therefore, it is clear the scatter in strength data and hence the value of the Weibull modulus, is directly related to the flaw size distribution and therefore casting quality. One additional advantage to the use of Weibull modulus is in utilization of the information gained for material at different scales. For example, Eqn 1 may alternately be written as:5
Equation 8
Where; V is the volume of the component. Therefore, for the same probability of survival (or failure; note that probability of survival = 1- probability of failure) the stress at failure
varies with the volume of the component. For example, if σu = 0, and V1
> V2 , then: Equation 9 the linear least squares method, for example: The best estimates of σo Equation 5 and m can then be obtained using
and σ1
/ σ2 = (V2 / V1 )l/m Equation 10
So for an equal probability of survival (or failure), a larger volume of the component will display a lower stress to cause failure.
Similarly, the stress ratio between bending and tension for the same probability of survival (or failure) is also given by:
Equation 6 σbend / σtensile Where:
More commonly however, it is simplest to plot the xy data from Eqn 4 and determine the slope of the line which gives the value of the Weibull modulus, m. The value of position parameter, σo
, is equal to the value at which 37% (1 - 1/e) of samples survive.2 38 = {2(m + 1)2 }l/m Equation 11
the influence of Defects on mechanical Properties of Aluminium castings
A range of different defect types exist in aluminium cast- ings and all affect mechanical properties. Elongation and tensile strength are particularly sensitive to the presence of any types of casting defect. Surappa et al.6
has shown that
the decrease in elongation to fracture through the presence of porosity could be related to the projected area of pores on the fracture surface. This model was further expanded
International Journal of Metalcasting/Summer 2011
A theoretical physical background to the Weibull distribu- tion has also been established,4
related to the probability
density f(a) of flaw sizes within the material, with f(a) being approximated by:
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