A numerical study on the application of the Weibull theory to brittle materials

Abstract

In this paper, the effects of assuming that a brittle material can be modelled by the two-parameter Weibull distribution, if the true representation of the material behaviour is actually a three-parameter distribution, are examined. It is shown that if data from a material characterisation test is used to predict the failure of a component and the stressed surface areas are very different, then this can lead to major discrepancies in the predicted failure stress. It is also shown that if the three-parameter model is a true representation of the material behaviour, then the Weibull modulus for the two-parameter approximation will vary with surface area.

Introduction

When analysing the failure of brittle materials, for example ceramic components, the Weibull theory is normally used. This theory was developed, by Weibull, in 1939 [1] using the idea that when one link in a chain fails, the whole chain fails. Weibull also considered multi-axial stress problems and assumed that the normal stress acting on a crack causes the failure of a component.

The Weibull theory for a uniaxial stress state can be written as [2]:

$$\text{P}{}_{\mathrm{<mtext>f</mtext>}}\text{=1−}\text{exp}\text{∫{−[(σ−σ}{}_{\mathrm{<mtext>th</mtext>}}\text{)/σ}{}_0\text{]}{}^{\mathrm{m}}\text{}(}\text{d}\text{A/a)}$$

In this equation, P_f is the probability of failure, A is the surface area and a is a unit area which is introduced for consistency of units. It can be seen that there are three parameters which control the failure probability; σ₀, σ_th and m. These are, respectively, a value related to the mean strength of the brittle material, a threshold stress below which the brittle material will not fail and a measure of the scatter in the failure strengths of nominally identical components. σ is the failure stress and, in this case, the equation is integrated over the surface area. However, most analyses ignore the threshold stress and assume that it is zero. This is done because it gives a conservative prediction [3] and because it is much simpler to find the remaining two parameters. Examples of analyses using two parameters are given in the papers by Swank and Williams [4], Smart and Fok [5] and Bruckner-Foit et al. [6]. This list is very incomplete but these papers do give an indication of how the theory is used. Occasionally the three-parameter version of the theory is used, for example, by Cooper and Margetson [7] and Duffy et al. [8]. In Ref. [7] the authors also show how the three parameters may be extracted from test data. Typically, the material will be characterised in a bending test (three- or four-point) and the parameters will then be used to predict the failure of a component.

As well as the problem of whether to use the two- or three-parameter Weibull theory, there are further problems. For example, Dortmans et al. [9] reference three different local failure criteria to account for a three-dimensional stress field which can be evaluated either over the volume or the surface area. The variations in the predictions from the different criteria depend on how much account is taken of the shear stresses in a given stress field, what contribution a negative stress makes to a crack failing and whether the integrals should be evaluated over the surface of the component or the total volume. Smart and Fok [5] examined this problem and suggested a testing regime that will allow the theoretical predictions to be separated to help decide which is the most relevant. Fok and Smart [10] also suggested a test configuration which produces a stress state similar to pure shear (i.e. one where the maximum and minimum principal stresses are of equal magnitude but of opposite sign). This helps to separate the predictions from the various criteria.

Because of the statistical nature of brittle failure, there is considerable uncertainty in the estimated parameters and this leads to the question as to how many specimens need to be tested. This problem was addressed by Ritter et al. [11] who showed that there can be considerable variation in the value for the Weibull modulus, m, and that the standard deviation of m divided by its mean is given by 1/√n where n is the number of samples. This possible variation in m, caused by a finite sample size, can then lead to errors in the prediction of component failure stresses.

A further problem arises as to the failure origin. It was mentioned earlier that failure theories can be based on an integral over the surface or over the volume. The surface integral can account for surface damage due to machining and for pores and cracks which penetrate the surface. The volume integral can account, in addition, for internal cracks and porosity. It is possible that both surface and volume flaws are present. Specimen failure will depend on whether the surface or volume flaw is the most critical and this will vary with loading condition and specimen geometry. This is known as having a bimodal distribution [12].

Thus it can be seen that there is considerable uncertainty in any failure prediction for brittle materials. In this paper, a further uncertainty will be considered. It is assumed that the correct failure theory for the material is the three-parameter Weibull theory but that the two-parameter Weibull theory is used to characterise the material and then predict component failure. It is also assumed that the failure is caused by surface cracks but the results will also be indicative of failure caused by volume flaws and that the material is characterised in four-point bending with a given surface area and predictions then made for a component with a different stressed area. This is realistic as, for example, a four-point bend specimen has a constant stress between the central support points and this may be a relatively large area whilst a component under load may have the largest stress concentrated over a small area such as a notch. Alternatively, the specimen used to find the material properties may be small but the data obtained is then used to predict the failure of a much larger component. Thus, the failure predictions using two- and three-parameter theories are compared and shown to give potentially large discrepancies as the area of the component is varied.