Residual stress measurement by Hertzian indentation
Introduction
Indentation techniques have been widely used in the analysis and characterisation of fracture and deformation properties of brittle ceramics [1], [2], [3]. They provide a convenient means to study the effects of localised loading. Indentation fracture theory has given insight into a wide range of mechanical properties such as machining damage, erosion and wear. Indentation methods are gaining wide spread popularity as a simple and inexpensive technique for quantifying fracture toughness of brittle materials [1], [2], [3], [4], [5] as well as more conventional ‘hardness’ measurements. Indentation testing is experimentally simple and widely used for quick material evaluation; the analysis may however, be complex. Indentation fracture theory concerns two basic type of contacts: sharp indenters — where the contact is essentially plastic up until fracture; and blunt indenters — where the contact is completely elastic up to the point of fracture. In this paper, we consider the use of blunt, spherical indenters — ‘Hertzian contact’.
The contact stress field for the Hertzian indentation is purely elastic and though complex, it is well-defined [6], [7]. There have been several attempts to use Hertzian indentation to determine the fracture toughness of brittle materials [8], [9], [10], [11], [12], [13], [14], [15]. The main principle depends on the interaction of the elastic stress field with a pre-existing surface flaw. The reasons why the Hertzian test has not been as popular as those using sharp indenters (e.g. Vickers indentation) may be due to: (i) the variability of the results obtained so far; (ii) the steep stress gradient which has made it difficult to obtain accurate estimates for stress–intensity factors for cracks driven by Hertzian loading; and (iii) the results of the analysis being very sensitive to the value of the Poisson’s ratio of the substrate [16]. There is, however, one significant advantage of Hertzian indentation over pointed indenters — the substrate’s deformation is entirely elastic until fracture occurs. This avoids the complications arising from the somewhat ill defined indentation residual stress always associated with pointed indenters. The material properties that may be determined by this test include: (a) fracture toughness of the near-surface material; (b) the densities and sizes of surface cracks; and (c) residual stresses in the near-surface material [15], [16]. We give below a brief introduction to the studies of fracture toughness and residual stress using Hertzian indentation.
Section snippets
Hertzian contact
When a hard sphere (radius R and elastic constants E1, ν1) is pressed with a normal load P on to a flat substrate (elastic constants E2, ν2), the contact radius is given by [16]:where
A complex but well defined stress field is set up around the contact site; the peak pressure under the sphere P0 is given by
The principal component of this stress field that is responsible for crack propagation is the radial stress. This component is tensile close to the
Fracture toughness KIC determination by Hertzian indentation
Warren [16] has shown that for every load, there is one size and position of flaw that gives maximum stress intensity factor Kmax of all possible flaws in the surface. Kmax increases monotonically with applied indentation load. Ring cracks can form only if this Kmax equals the fracture toughness KIC of the material. Hence, there is a minimum load Pmin below which no fracture occurs. The principle of the method for determining KIC is thus to find this Pmin. This is done by performing a series of
Residual stress measurement
Brittle materials, such as glass and ceramics, have low fracture toughness and tend to fail from defects that exist on the surface due to manufacturing process or handling. One of the popular practices of reinforcing them is to introduce compressive stresses at the surface. This can be done using a variety of techniques: ion-exchange or thermal tempering process in soda-lime glass [18], [19], [20], [21], [22]; phase transformation in partially stabilised ZrO2 [23] and by the formation of low
Experimental technique
To experimentally validate the technique described in Section 4, we used a ring-on-ring bending jig to induce a known compressive stress on the surface of the test materials. During loading the inner portion of the specimen is bent into a spherical curvature whereby the top surface is under compression and the bottom surface is under tension. This bending stress, which is uniform and bi-axial, is measured using strain gauges mounted on the specimen surfaces. For each test, four strain gauges
Results and discussion
For every series of 25 indentations, the fracture loads were sorted in ascending order so that probability to fracture could be assigned to each fracture load. The probability to fracture versus fracture load of the glass specimen is shown in Fig. 3. There is a clear minimum fracture load, Pmin and a progressive increment in Pmin with increasing compressive stress. The steep gradient of the probability curve represents the initiation of ring cracks over a very narrow range of loads close to the
Conclusion
We have used the Hertzian indentation technique to measure residual stress on ceramics and glass specimens. Although it needs further refining, the Hertzian method has a potential for an easy, straightforward measurement of near surface residual stress in brittle materials to a good first approximation [27], [40]. This would give a useful technique for both quality and process control of engineering ceramic components.
References (43)
- et al.
Acta Metall.
(1992) Acta Metall.
(1978)J. Eur. Ceram. Soc.
(1995)J. Non-Cryst. Solids
(1985)- et al.
Acta Mater.
(1998) - et al.
J. Am. Ceram. Soc.
(1998)- et al.
J. Mater. Sci.
(1975) Fracture of Brittle Solids, Ch. 8
(1993)- et al.
Int. J. Fract.
(1987)