# Thermoelastic solutions for thermal distributions moving over thin slim rod under memory-dependent three-phase lag magneto-thermoelasticity

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### Cites result from "Thermoelastic solutions for thermal..."

...It has been found that the nature of distributions in all physical field in the current study and the corresponding outcomes of generalized thermoelasticity with a single memory-dependent derivative relaxation parameter is consistent with the existence of the physical field variable distribution for both Lord and Shulman as derived from Ezzat and El-Bary [41] and Mondal and Kanoria [52]....

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...Mallik and Kanoria [21] considered the thermoelastic interaction in functionally graded an unbounded solid because of varying heat source....

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31 citations

### Cites background from "Thermoelastic solutions for thermal..."

...So, the common derivative d dt can be seen as the limit of Dx as x ! 0 [53, 54]....

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30 citations

### Cites background from "Thermoelastic solutions for thermal..."

...…have been considered by many investigations (Abbas 2015; Deswal, Kalkal, and Sheoran 2016; Mishra, Sharma, and Sharma 2017; Othman and Eraki 2017; Xiong and Niu 2017; Biswas, Mukhopadhyay, and Shaw 2019; Davydov, Zemskov, and Akhmetova 2019; Mondal, Sur, and Kanoria 2019; Mondal and Kanoria 2019)....

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##### References

5,201 citations

### "Thermoelastic solutions for thermal..." refers methods in this paper

...…Fourier law of heat conduction by using the fractional calculus by Caputo (1967), Caputo and Fabrizio (2015), Carpinteri and Mainardi (1997), Carpinteri and Mainardi (2014), Podlubny (1998), Hilfer (2000), Ignaczak and Ostoja-Starzewski (2009), Debnath and Bhatta (2007), and Abbas (2014a)....

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3,962 citations

### "Thermoelastic solutions for thermal..." refers background or methods in this paper

...One can refer to Podlubny (1998) for a survey of applications of fractional calculus....

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...…Fourier law of heat conduction by using the fractional calculus by Caputo (1967), Caputo and Fabrizio (2015), Carpinteri and Mainardi (1997), Carpinteri and Mainardi (2014), Podlubny (1998), Hilfer (2000), Ignaczak and Ostoja-Starzewski (2009), Debnath and Bhatta (2007), and Abbas (2014a)....

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3,372 citations

### "Thermoelastic solutions for thermal..." refers background in this paper

...This definition of fractional derivative appears to be appropriate for mechanical phenomena related to plasticity, fatigue and with electromagnetic hysteresis by Caputo and Fabrizio (2015). A memory-dependent derivative (MDD) has been introduced by Wang and Li (2011) and Yu, Hu, and Tian (2014). The first order (f1⁄4 1) of function f which is simply defined in an integral form of a common derivative with a kernel function Kðt nÞ (chosen arbitrarily) on a slipping interval [t – s, t] in the form...

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...This definition of fractional derivative appears to be appropriate for mechanical phenomena related to plasticity, fatigue and with electromagnetic hysteresis by Caputo and Fabrizio (2015). A memory-dependent derivative (MDD) has been introduced by Wang and Li (2011) and Yu, Hu, and Tian (2014)....

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3,266 citations

### "Thermoelastic solutions for thermal..." refers background in this paper

...Among these theories, the first is held by Lord and Shulman (1967) who acquired a wave-type heat equation by hypothesizing another law of heat equation to take the frame q x; t þ sqð Þ ¼ Krh x; tð Þ: (1) Another modified heat conduction law, namely dual-phase lag (DPL) model, given by Tzou (1995)…...

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...At the same time, it contains another constant sq that goes about as a relaxation time in generalized thermoelasticity by Lord and Shulman (1967)....

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...Very recently, several problems in generalized thermoelasticity in the context of memorydependent derivative have been reported in the following literatures by Ezzat, El-Karamany, and El-Bary (2017b), Shaw and Mukhopadhyay (2017), Lotfy and Sarkar (2017), Kant and Mukhopadhyay (2018), Sur, Pal, and Kanoria (2018), Sur and Kanoria (2017), Purokait et al....

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2,287 citations

### "Thermoelastic solutions for thermal..." refers background in this paper

...The traditional coupled dynamic thermoelasticity theories predict the infinite speed of thermal waves, which depend on the mixed parabolic-hyperbolic governing equations of Biot (1956) but the predicted infinite speed of the thermal wave is not an acceptable phenomenon....

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...The traditional coupled dynamic thermoelasticity theories predict the infinite speed of thermal waves, which depend on the mixed parabolic-hyperbolic governing equations of Biot (1956) but the predicted infinite speed of the thermal wave is not an acceptable phenomenon. The infinite speed of thermal waves results in an instantaneous effect on the body, no matter how far it is from the point of heat source applied. The classical theory of thermoelasticity has been generalized and changed into different thermoelastic models that keep running under the mark of “hyperbolic thermoelasticity.” At present, there are a few theories of the hyperbolic thermoelasticity. It ensures the finite speed of wave propagation. Among these theories, the first is held by Lord and Shulman (1967) who acquired a wave-type heat equation by hypothesizing another law of heat equation to take the frame...

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