Deformation based temperature rise: a review K.N. Pandey a , Satish Chand b, * a Department of Mechanical Engineering, K.N. Institute of Technology, Sultanpur 228 118, India b Department of Mechanical Engineering, M.N. National Institute of Technology, Allahabad 211004, India Revised 15 July 2003; accepted 26 July 2003 Abstract The thermomechanical response of material under both elastic and plastic conditions has been used to determine various material properties under monotonic and cyclic loading conditions. The change in temperature at any point/zone of the material indicates the deformation and damage. This paper presents a review concerned with the application of the deformation based temperature rise, mainly in mechanics, and the different temperature measuring techniques employed. The concept is very useful, particularly in installed structures such as pressure vessels and piping, to determine the response in service. q 2003 Elsevier Ltd. All rights reserved. Keywords: Thermomechanical response; Source; Sink; Heat generation; Heat dissipation 1. Introduction Deformation, either elastic or plastic, always changes the temperature of a body. In a metallic material tensile elastic deformation results in an increase in volume and a decrease in temperature, while compressive elastic deformation results in a decrease in volume and an increase in temperature. Thus according to the type of loading, during elastic deformation, a body may become a heat sink or a heat source _ Q e ; which may be expressed in the form _ Q e ¼ 2 a rC ð3l þ 2mÞT 0 _ 1 e kk ð1Þ where a is the coefficient of thermal expansion, T 0 is the initial temperature, l and m are elastic constants, _ 1 e kk is the elastic strain rate tensor, with k indicating x; y; z co-ordinates. C is the specific heat, and r is the density. From Eq. (1) it is clear that the effective thermoelastic source, _ Q e ; will be positive or negative depending on the nature of _ 1 e kk : If deformation is plastic, most of the irreversible work done on the material is converted to heat and results in an increase in the material temperature, converting it to a positive heat source as given by _ Q p ¼ b rC s ij _ 1 p ij ð2Þ where _ Q p is the effective plastic heat source, b is the fraction of plastic work converted to heat, s ij and _ 1 p ij are stress and plastic strain rate tensor components, respectively. Fig. 1 shows the thermomechanical response of a material under elastic – plastic loading. Loading is shown in Fig. 1a and the thermomechanical response of the material during loading along path 1 is shown in three parts (i), (ii) and (iii) in Fig. 1b. Part (i) is thermoelasic cooling during the elastic tensile loading part of path 1; (ii) is the non-linear part due to localized yielding; and (iii) shows an increase in temperature, i.e. heating due to plastic deformation beyond yielding. Thermomechanical response of unloading during path 1 is not shown in Fig. 1b. Thus, the net temperature rise will be the result of thermoelastic heating or cooling and heating due to conversion of some fraction of plastic work into heat. The results of Zehnder [1] and Zehnder et al. [2] indicate that the fraction of plastic energy converted into heat depends on plastic strain, yield strain and strain hardening of the material and decreases with increase in strain hardening. However, for metals, Zehnder et al. [3,4], Kallivayalil et al. [5] and Weichert and Schonert [6] have taken the value of b 0308-0161/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2003.07.001 International Journal of Pressure Vessels and Piping 80 (2003) 673–687 www.elsevier.com/locate/ijpvp * Corresponding author. Tel.: þ 91-532-2540212/þ 91-532- 2445103/04x1110; fax: þ 91-532-2445106/07. E-mail addresses: chandsatish@indiatimes.com, satishchand@ rediffmail.com (S. Chand); knpandey123@indiatimes.com (K.N. Pandey).