ABSTRACT
Thermoelastic stress analysis (TSA) is a well established tool for non-destructive full-field experimental stress analysis. In TSA the change in the sum of the principal stresses is derived and it is generally accepted that the TSA relationship does not allow the evaluation of residual stress, which is essentially a mean stress. However, modifications to the linear form of the thermoelastic equation that incorporate the mean stress have enabled estimations of residual stresses. It has also been shown that the application of plastic strain modifies the thermoelastic constant, K, in some materials, causing a change in thermoelastic response, which can also be related to the residual stress. The changes in response due to plastic strain and mean stress are of the order of a few mK, and are significantly less than those expected to be resolved in standard TSA. Recent developments in infra-red detector technology have enabled these small variations in the thermoelastic response to be identified, leading to renewed interest in the use of TSA for residual stress analysis. The residual stress distribution around cold expanded holes is relatively well defined, and hence provides an opportunity to examine any changes in thermoelastic response caused by residual stress. In the present paper, the thermoelastic response around cold expanded holes in aluminium plates is investigated, and the feasibility of applying a TSA based approach for residual stress analysis to components containing realistic residual stress levels is assessed.
Key Words: Thermoelastic stress analysis, Residual stress, Cold expanded hole, Non-destructive evaluation
INTRODUCTION
Residual stresses may be introduced into a component throughout the entire manufacturing process, so it is highly unlikely that an in-service component is entirely free of residual stresses. Since residual stresses are an almost unavoidable bi-product of manufacture, it is important to understand how residual stresses are distributed in a component; at present, there are several techniques available for measuring residual stresses. Destructive methods are not always practical for an in-service industrial environment, while the non-destructive methods are typically expensive and time consuming. Therefore, demand for a cheaper and quicker non-destructive, non-contact, full-field residual stress evaluation technique is increasing.
Thermoelastic stress analysis (TSA) [1] has been identified as a possible solution for a robust and portable means of nondestructive residual stress assessment. TSA is a well established non-contacting analysis technique that provides full-field stress data over the surface of a cyclically loaded component. It is based on the small temperature changes that occur when a material is subject to a change in elastic strain, generally referred to as the ‘thermoelastic effect’. When a material is subjected to a cyclic load, the induced strain produces a cyclic variation in temperature. The temperature change (AT) can be related to the change in the ‘first stress invariant’,
, or the sum of the principal stresses [1]. An infra-red detector is used to measure the small temperature change, which can then be related to the stress using the following equation:
where T0 is the absolute temperature and K is the thermoelastic constant,
are the material constants of the coefficient of thermal expansion, mass density and the specific heat at constant pressure, of the material respectively.
As residual stress is essentially a mean stress, it is accepted that the linear form of the TSA relationship given in Eqn. 1 does not allow its evaluation. However, there are situations where this linear relationship is not valid so that a means of establishing the residual stresses from the thermoelastic response has been developed. The changes in the thermoelastic response resulting from the inclusion of residual stress produce temperature change differences of a few mK, which are significantly less than those expected to be resolved in standard TSA. At present, three approaches have been investigated as potential candidates for residual stress measurement using the thermoelastic response [2]. Two are based on the mean stress effect and the revised higher order thermoelastic effect [3]. One utilises the thermoelastic response at the second harmonic of the loading frequency [4], and the other directly relates the change in the thermoelastic response to the principal stresses [5]. The major limitation of these two approaches is that they are not suitable for steel components since the temperature dependence of the elastic properties of steel are negligible at room temperature. The third approach [6] is based on Eqn. 1 and the change in the thermoelastic constant, K, resulting from plastic deformation during manufacture or assembly. In the third approach the main disadvantage is that plastic deformation must have taken place, but it has the advantage that it may be valid for a larger range of materials, not just those with temperature dependent elastic properties. A significant disadvantage common to all three approaches is that any change in the thermoelastic response resulting from either the mean stress, am, or from the modification of K will be small. In actual components the changes in the response are around the noise threshold of the detectors. Success in detecting these changes has been achieved by applying very large residual stress or plastic strain, by using materials that are very sensitive to the mean stress effect, or from investigation of specifically designed specimens. Recently the sensitivity of infra-red detectors has improved to the extent where it may be possible to accurately measure changes representative of those in actual components, hence leading to a renewed interest using TSA for residual stress analysis. Since the variations in thermoelastic response are small, it is important to minimise sources of signal attenuation and understand the possible sources of error within the measurement. The major factors known to influence the change in thermoelastic response (and subsequently K) are the high emissivity coating, background temperature, applied stress and the infra-red detector settings; these effects must always be considered in parallel to any evaluations of the significance of changes in response due to mean stress or from the modification to K.
Typical manufacturing residual stresses may be detrimental to the components performance; however it is possible to enhance a component by introducing beneficial residual stresses. An example widely used in the aerospace industry is the cold expansion of holes to prevent fatigue crack growth and initiation. The residual stress distribution around cold expanded holes is relatively well defined, providing an opportunity to investigate the variations in thermoelastic response due to residual stress, and assess the feasibility of applying a TSA based approach for residual stress assessment to actual components containing realistic levels of plastic strain. The present paper focuses on examining small changes in thermoelastic response obtained from the region around cold expanded holes where residual stress is present. TSA is conducted on aluminium plate (AA2024-T351 and AA7085-T7651) containing holes with different levels of cold expansion, which are similar to the levels found in service. Direct point by point comparison of the thermoelastic response is made between specimens to attempt to identify the areas affected by the cold expansion process. The effect of plastic strain on the thermoelastic constant for each alloy is investigated using tensile dogbone-type specimens loaded uniaxially; similarly, the effects of material directionality (due to cold rolling) and strain hardening on the thermoelastic response is also examined.
BACKGROUND AND APPROACH
The cold expansion process is an established means of generating a beneficial compressive residual stress distribution around a hole, preventing fatigue crack growth and initiation; it has also been shown to inhibit growth in existing cracks. The cold expansion process involves plastically deforming the material directly adjacent to the hole to develop compressive residual stress [7]. The split-sleeve expansion method is most commonly used, and involves pulling an oversized tapered mandrel through an internally lubricated split-sleeve that has been positioned inside the hole, as shown in Fig. 1. The combined thickness of the sleeve and mandrel relative to the size of the hole controls the amount of expansion. The hole is expanded to cause plastic deformation, and as the mandrel is removed the material undergoes partial, but not full elastic recovery. The material further from the hole, which is only elastically deformed springs back from the expanded state and forces the plastically deformed material, closer to the hole, into compression. Consequently a large compressive residual stress is formed close to the hole, and reduces with distance from the hole. Fig. 2 shows the typical residual stress field following cold expansion; the maximum compressive stress is typically similar in magnitude to the yield stress of the material. As the mandrel is removed from the hole, the material on the entry side of the hole begins to relax, however the material on the exit face does not, since it is still constrained by the presence of the mandrel. As a result, the final residual stress distribution is three-dimensional with a different magnitude of residual stress on the entry and exit sides of the hole.
Fig. 1 Schematic of the split sleeve cold expansion method
Fig. 2 Typical tangential residual stress profile around a cold expanded hole
There are many features that influence the residual stress field, and although the shape is well defined, it can be difficult to quantify the final distribution; some of the influencing factors include: method of expansion, level of expansion, material, plate thickness, and the holes’ position relative to other holes. In this study, each specimen consists of a 10 mm thick aluminium plate containing a single central hole that has been cold expanded using the split-sleeve method. Holes containing three different levels of cold expansion are inspected, and both the exit and entry sides of the plate are examined due to the three-dimensional residual stress profile. It is estimated that the region of compressive residual stress extends approximately 5 mm from the hole edge, and the reverse yielding zone approximately 1 – 2 mm from the hole edge. The variations in thermoelastic response are investigated, and the possible causes discussed, which include the mean stress effect and the effects of plastic strain on the thermoelastic constant.
The mean stress effect was first observed by Belgen [8] and later confirmed by Machin et al [9]; the revised higher order theory of the thermoelastic effect was proposed by Wong et al [3], which accounted for the temperature dependence of elastic properties, and it was shown that the temperature response is dependent on the stress state, i.e. the mean stress, as well as the change in stress, i.e. the stress amplitude. Since the presence of residual stress within a component would form a contribution to the mean stress, there would be a small difference in the thermoelastic response from that measured in the same component with zero residual stress, subjected to the same loading conditions. In the simplest case of a uniaxial stress field, if (a) the change in thermoelastic response between a specimen containing unknown residual stress and a specimen with zero residual stress was measured, (b) the effect of changing the mean stress on the thermoelastic response was known for the given material, and (c) known adiabatic loading conditions were prescribed, there is potential for the amount of residual stress within the specimen to be estimated [4]. However, this process of residual stress assessment becomes considerably more difficult in specimens containing non-uniform stress fields, i.e. that seen around a cold expanded hole, or in situations of multi-axial loading. While the mean stress effect is very small, and often considered negligible at room temperature in many engineering materials [10], is has been shown to be measurable in materials with temperature dependent elastic properties, including titanium (Ti-6Al-4V), Inconel 718, and some grades of aluminium [11]. Thus, it is possible that any variation in thermoelastic response between holes that have experienced different amounts of expansion could be a result of a change in the effective mean stress, due to the presence of residual stress.
The other potential cause of variations in thermoelastic response around cold expanded holes is the effect of plastic deformation on the thermoelastic constant. It has been shown [6], that the introduction of plastic deformation caused a modification of the thermoelastic constant in some metals due to a change of the material properties contained in K. Rosenholtz et al [12], and Rosenfield et al [13] have both demonstrated that in steel and aluminium, an application of plastic strain will cause a change in the material property, a, the coefficient of linear thermal expansion. Rosenfield et al [13] also noted that this change in a increases significantly when subjected to compressive strains, and less with tensile plastic straining. It has also been suggested that the change in a is affected by the strain hardening capability of the material [6], i.e. the change in K for a material that does strain harden will be different than for a material that does not. Since the area immediately adjacent to the hole has undergone significant plastic deformation, is it possible that this effect will contribute to a change in thermoelastic response. Plates were made from two aluminium alloys (AA2024-T351 or AA7085-T7651) that have different strain hardening characteristics; enabling any contribution to variations in thermoelastic response due to the effect of plastic deformation on K to be established.
To supplement the investigation of cold expanded holes, it is necessary to perform calibration tests to characterise the materials under investigation; this involves loading dog-bone type specimens in uniaxial tension to: (a) determine the effect of plastic deformation on the thermoelastic constant, K, for each alloy, (b) establish the different strain hardening characteristics and the potential affect on the thermoelastic response around the hole, and (c) examine any material directionality that is present due to cold rolling and investigate what affect this has on the thermoelastic response.
In a dog-bone type specimen loaded in uniaxial tension, the stress can be calculated in a straightforward manner, given that the loading conditions are defined, thus allowing the thermoelastic constant, K, to be computed from the applied stress and measured AT and T values using Eqn. 1. This methodology has been used for determining the effect of plastic strain on the thermoelastic constant for the following three reasons: (i) If this type of specimen is loaded beyond the material’s yield point and then unloaded, it will result in a residual strain; however, there will be no residual stress as the stress can be fully relaxed by the elastic unloading. Without a residual stress that would result in an increase in am when loaded, any change in the thermoelastic response would be due to a change in one of the material properties, a, p or Cp, and not due to the mean stress effect. (ii) In a dynamically loaded tensile specimen, non-adiabatic conditions cannot occur because there is no stress gradient, and therefore there is no heat transfer within the specimen. (iii) Since the stress in a specimen loaded uniaxially can be calculated, the change in thermoelastic constant due to plastic deformation can be obtained by calculating thermoelastic constant, KP, for a specimen containing a known amount of plastic strain, and comparing it to K0, which is calculated from a reference specimen containing zero plastic strain. Similarly, by comparing tensile specimens loaded in uniaxial tension, any variations in the measured thermoelastic response due to strain hardening, material directionality or the mean stress effect will become apparent, assuming prescribed loading conditions are used and the background temperature does not change significantly.
Having ascertained the mechanical and thermoelastic behaviour from tensile specimens, a catalogue of information is available that characterises the materials under consideration. Subsequently, the variations in thermoelastic response around the cold expanded holes can be exhaustively investigated to provide a complete view of the material behaviour. Hence revealing the potential for TSA to be used to detect or measure residual stresses in real components, and provide a focus for further work on the topic.
