PSI - Issue 42

Jürgen Bär et al. / Procedia Structural Integrity 42 (2022) 1061–1068 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

1062

2

1. Introduction The temperature change in a cyclic loaded material can be described according to the thermoelastic law by Thomson (1853). The actual temperature of a specimen loaded with a stress amplitude  a at a loading frequency f L at the mean temperature T m is given by Eq. (1): ( ) = − ∙ 0 ∙ ∙ ⁡(2 ∙ ) (1) whereby K 0 represents the thermoelastic constant. The thermoelastic constant can be calculated from the density  the coefficient of thermal expansion  , and the specific heat capacity c p of the corresponding material. To increase the emissivity, the samples must be coated, whereby the coating layer must have a defined thickness for accurate measurements (Robinson et al (2010)). This layer also influences the measured temperature change; therefore, it is better to replace the thermoelastic constant K 0 in equation (1) by a constant K c determined in experiments with defined stress amplitudes (Urbanek and Bär (2017)). At higher loading levels in metallic materials plastic deformation takes place. The plastic deformation leads to a dissipation of energy and causes a heating of the specimen. The amount of dissipated energy per cycle increases with the loading level. This effect is often used to determine the fatigue limit of metallic materials. (Luong (1995), Fargione et al (2002), Meneghetti (2007),). For these investigations, the measurement of a global temperature of the specimen is sufficient. To analyze the dissipated energies in the vicinity of a crack spatially and temporally resolved measurements are necessary. Because of the propagating crack, it is necessary to determine the temperatures within one or a few cycles. For such analyses, the measured temperature signal is dissected into sine waves coupled with the loading frequency using an incomplete Discrete Fourier Transformation (DFT). In a specimen loaded sinusoidal with a force the temperature change due to the thermoelastic effect results in a sine wave coupled with the loading frequency (E-Mode). According to Sakagami et al (2005) and Brémond (2007) temperature changes caused by dissipated energies are assigned to a sine wave with twice the loading frequency (D-Mode). The corresponding evaluation bases on an incomplete DFT as shown in equation (2). ( ) = + ∙ 2 ( ∙ + ) ⏟ ℎ − + ∙ 2 (2 ∙ + ) ⏟ ⁡2 − + ⏟( ) ሺʹሻ Urbanek and Bär (2017 and 2017a) showed that also amplitudes of higher harmonics can be found in the vicinity of a crack and extended the approach to higher harmonic frequencies, resulting in additional D k Modes: ( ) = + ∙ 2 ( ∙ + ) ⏟ ℎ − + ∙ 2 (2 ∙ + ) ⏟ ⁡2 − + ∑ ∙ 2 (( +2) ∙ + ) =1 ⏟ ℎ ℎ ⁡ℎ + ⏟( ) ሺ͵ሻ The appearance of higher harmonics proves, that a description of the temperature changes in elastic and plastic deformed materials with equation 2 is incomplete and therefore no quantitative correct results can be achieved with this evaluation. In addition, previous investigations by Bär and Urbanek (2019) showed that the temperature changes caused by dissipated energies cannot be described with a sine function with the double loading frequency. Therefore, the appearance of D-Mode amplitudes can just be used as a qualitative criterion for non-linear effects caused by dissipative energies and asymmetric local stress e.g. in tension and compression. In this work detailed examinations of the temperature changes due to elastic-plastic loading were undertaken and an alternative evaluation method is suggested that allows to determine the temperature changes caused by dissipative effects.