UDC: 772.96,776.34 zyxwvutsrqp Introduction to thermoelastic stress analysis by J. M. Dulieu-Barton zyxwvutsrq The University of Liverpool, Department of Engineering, Brownlow Hill, Liverpool, L69 3GH, zyxw UK Abstract The theory of thermoelastic stress analysis is reviewed and the assumptions in developing the theory are assessed. The zyxwvu temperature relationship for an isotropic material under plane stress conditions equipment for thermoelastic stress analysis is based on infra-red detection systems. The commercially available zyxwvu (El +VE,)- zyxw ~ Ea AT zy 0,,= - E (1 -v2) (1 -v) calibrating the output from the detectors are also provided. systems are described and appraised. Techniques for q*= - (E2 + ve,) - __ Ea AT E (1 -v*) (1 -v) Key words: TSA, SPATE, Deltatherm, calibration Introduction Thermoelastic stress analysis is now a well established non- contacting technique that provides full-field stress data directly from the surface of an actual component’. The technique is based on the measurement of a small temperature change that occurs in a solid subjected to elastic cyclic stresses. It is readily shown that the temperature change is directly related to the stresses and it is a straightforward matter to ‘measure’ the temperature change using an infra-red detector. The purpose of this paper is to review the development of the theory of thermoelastic stress analysis and to describe the infra-red detection systems that are currently commercially available. The paper is intended to give an introduction to the technique hence providing background information for the following four papers in this issue of Strain. Theory The relationship between the small temperature change, (T, caused by the change in the stress state of a linear elastic, homogeneous material and the strain in the solid can be derived in the form2 where T is the absolute temperature of the material, Cf is the specific heat at constant strain, p is the density, oi, is the stress change tensor, E,, is the strain change tensor, and Q is the heat input. In thermoelastic studies it is standard practice to cyclically load the component at such a rate that virtually no heat conduction takes place, therefore the second term in equation (1) can be neglected. As the infra-detector takes readings from the surface and as practically adiabatic conditions exist, it is reasonable to assume that the small temperature change derived from the detector output is related to the surface stresses. Hence in developing the theory it is relevant to use the following stress-strain- where E is Young’s modulus, v is Poisson’s ratio, and a is the linear coefficient of thermal expansion of the material. By making the assumption that E and u are independent of temperature when evaluating the quantity ao,/dT, equation (I ) can be expressed as (3) where E,~ are the changes in the strains. Using the standard “Hooke’s law” relationship between stresses and strains the following expression can be derived that provides the changes in the stresses, bii, in terms of AT (4) c 0.. = - - i =1.2 To simplify equation (4) CE is related to the specific heat at constant pressure, Cp, by 2Ea2T c,=c - ~ p(1-v) so that equation (4) can be rewritten as The quantity dpCP is known as the thermoelastic constant, K, and equation (6) can be expressed as follows AT = - KT (o1 +o?) (7) where o1 and o2 are the changes in the principal stresses. ‘Strain’, May 1999 35