The Weibull analysis provides a probability of failure at a given stress, and is often expressed by the empirical relationship:
Equation 1
to as the position parameter (where σo where 37% of samples survive).14
mally taken to equal zero), X is the strength limiting dimen- sion of the material, and σo
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, referred Since the elongation at
= (1-1/e), the value
failure is more sensitive to microstructural changes than the tensile strength, Weibull modulus is optimally evaluated for this quantity as well.
Therefore, for specimens of constant geometry, Equation 2 The simplest method of obtaining σo and m from a series of
data is to rank the stress (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:
the linear least squares method,13 The best estimates of σo Equation 4
the cáceres-selling model; influence of Defects on mechanical Properties of Aluminium castings
Cáceres and Selling11 posed by Surappa et al.,16
in elongation to fracture Ef
expanded the earlier model pro- and thereby related the decrease , in a tensile test, to the pres-
ence of any types of defects present on the fracture surface of a test specimen. The Cáceres-Selling model11
proposes defect, Ai
a relationship between a cross sectional area not contain- ing a defect, Ao
tion covered by the defect. In this case, load equilibrium is maintained if:
, such that Ai
σi(1-f)Aoe-εi =σh Where σi
Ao respectively, outside the defect. inside the defect, and σh and εi e-εh
, and the cross sectional area containing a =Ao
(1-f), where f is the area frac- Equation 6
are the true stress and strain respectively, and εh
are the true stress and strain
If during stressing, the material follows the Ludwik-Hollo- man equation: σ=Kεn
Equation 7
where σ is true stress, ε is true plastic strain, K is a constant, known as the strength coefficient, and n is the strain harden- ing exponent;
then combining Eqns 6 and 7 leads to (1-f)e-εi
εi n = e-εh εh n Equation 8
and m can then be obtained using but in practice, m is the
slope of Eqn 4, and is determined quickly.
A physical background to the Weibull distribution has been es- tablished,15
sizes within the material, with f(a) being approximated by: Equation 5
Where n in Eqn 5 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 equation 5 is the same notation as the original reference 15
and is not the same
as the strain hardening exponent, n, used later in equations 7 and 8). Therefore, the scatter in data and hence the value of the Weibull modulus, is directly related to the flaw size distribution and casting quality.
International Journal of Metalcasting/Fall 2011 which relates to the probability density f(a), of flaw
This equation relates the strain inside the defect contain- ing region to the strain outside the defect containing region. When solving Eqn 8, as the defect area present on the frac- ture surface f, increases, for a fixed value of strain outside the defect (εh (εi
), the strain inside the defect containing region ) increases more rapidly.
Quality index for Aluminium castings Cáceres9,10
showed that for any aluminium alloy, deriva-
tion of the model flow curve of Eqn 7 from experimen- tal data could also provide the basis for the assignment of iso-quality (q) indices based on proportions of what would constitute a defect-free casting; that is, where the true strain ε=n. This follows since n can be shown ex- perimentally to be equivalent to the uniform elongation, or the true strain at the onset of necking. This means that a defect-free casting will continue to elongate until the maximum stress is achieved and necking begins, leading to failure. The presence of defects in the material means that the metal fails at a lower level of strength and ductil- ity where ε<n, the level being proportionate to the frac- tion of defects present in the material.
49
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