1 Introduction
2 Non-local MGTE thermoelasticity
3 Formulation of the problem
4 Solution in the Laplace transform domain
5 Zakian’s method
j | Material properties | Value |
---|---|---|
1 | 12.83767675 + 666063445i | − 36902.08210 + 196990.4257i |
2 | 2 12.22613209 + 5.012718792i | 61277.02524 − 95408.62551i |
3 | 3 10.93430308 + 8.409673116i | − 28916.56288 + 18169.18531i |
4 | 4 8.776434715 + 11.92185389i | 4655.361138 − 1.901528642i |
5 | 5 5.225453361 + 15.72952905i | − 118.7414011 − 141.3036911i |
6 Special cases
6.1 Generalized theories of thermoelasticity
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The coupled thermoelasticity theory (CTE) can be obtained by setting \({\tau _{0}=K=K}^{{*}}=0\).
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The generalized theory of thermoelasticity with relaxation time (LS) can be achieved when \(\tau _{0}\mathrm {>0}\) and taking \(K^{{*}}=0\).
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The generalized Green–Naghdi theory of type II (GN-II) can obtained by assuming \(\tau _{0}\mathrm {=0}\) and putting \(K\mathrm {=0}\).
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The generalized Green–Naghdi theory of type III (GN-III) is accessible by adopting \(\tau _{0}\mathrm {=0}\) and \({K,K}^{{*}}>0\).
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The generalized Moore–Gibson–Thompson thermoelasticity (MGTE) is available when \({\tau _{0},K,K}^{{*}}>0\).
6.2 Non-local theories of thermoelasticity
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The non-local classical theory of thermoelasticity (NCTE) can be obtained by setting \({\tau _{0}=K=K}^{{*}}=0\).
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The non-local generalized thermoelasticity theory with relaxation time (NLS) can be obtained when \(\tau _{0}\mathrm {>0}\) and taking into account \(K^{{*}}=0\).
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The non-local generalized Green–Naghdi theory of type II (NGN-II) is available by assuming \(\tau _{0}\mathrm {=0}\) and putting \(K\mathrm {=0}\).
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The non-local generalized Green–Naghdi theory of type III (NGN-III) is valid by assuming \(\tau _{0}\mathrm {=0}\) and putting \({K,K}^{{*}}>0\).
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The non-local Moore–Gibson–Thompson thermoelasticity (NMGTE) is accessible when \({\tau _{0},K,K}^{{*}}>0\).
7 Numerical results
7.1 The effect of non-local parameter
7.2 Comparison of different thermoelasticity models
x | NCTE | NLS | NGN-II | NGN-III | NMGTE |
---|---|---|---|---|---|
0.0 | 0.913099 | 0.546272 | 0.471496 | 0.712558 | 0.49124 |
0.5 | 0.464366 | 0.339959 | 0.309865 | 0.395407 | 0.318432 |
1.0 | 0.0726438 | 0.0449211 | 0.0403095 | 0.0553329 | 0.0423645 |
1.5 | 0.0326925 | 0.0190259 | 0.0164673 | 0.0246476 | 0.016466 |
2.0 | 0.0136876 | 0.00956508 | 0.00864791 | 0.011342 | 0.0088852 |
2.5 | 0.00298238 | 0.00189095 | 0.00169282 | 0.00232006 | 0.0017586 |
3.0 | 0.00115897 | 0.000708719 | 0.000622444 | 0.000894157 | 0.000629663 |
3.5 | 0.000436966 | 0.000296909 | 0.000267179 | 0.000356188 | 0.000274229 |
4.0 | 0.000112863 | 7.22195E–05 | 6.45E–05 | 8.85542E–05 | 0.00006661 |
4.5 | 4.05916E–05 | 2.54472E–05 | 2.25E–05 | 3.16869E–05 | 2.29003E–05 |
5.5 | 1.44099E–05 | 9.61E–06 | 8.62E–06 | 0.000011618 | 8.84E–06 |
x | NCTE | NLS | NGN-II | NGN-III | NMGTE |
---|---|---|---|---|---|
0.0 | 0 | 0 | 0 | 0 | 0 |
0.5 | − 0.171014 | − 0.113854 | − 0.0935007 | − 0.157913 | − 0.0976409 |
1.0 | − 0.0540906 | − 0.0531456 | − 0.0520725 | − 0.0536953 | − 0.0546321 |
1.5 | − 0.0137213 | − 0.019453 | − 0.0209205 | − 0.0155647 | − 0.020463 |
2.0 | − 0.00397472 | − 0.00767204 | − 0.00821823 | − 0.00470507 | − 0.00728732 |
2.5 | − 0.00137457 | − 0.00378378 | − 0.00427237 | − 0.0015657 | − 0.00402967 |
3.0 | − 0.00046814 | − 0.00199585 | − 0.00263436 | − 0.00054170 | − 0.00265626 |
3.5 | − 0.00014580 | − 0.000991854 | − 0.00155047 | − 0.00018394 | − 0.00154523 |
4.0 | − 4.3216E–05 | − 0.000461038 | − 8.2600E–04 | − 6.0461E–05 | − 0.000777992 |
4.5 | − 1.2900E–05 | − 0.000208825 | − 4.1300E–04 | − 1.9489E–05 | − 0.000364552 |
5.5 | − 3.9700E–06 | − 9.58772E–05 | − 2.0500E–04 | − 6.2725E–06 | − 0.00017627 |
x | CTE | LS | GN-II | GN-III | MGTE |
---|---|---|---|---|---|
0.0 | − 0.408868 | − 0.155985 | − 0.145709 | − 0.333855 | − 0.214435 |
0.5 | − 0.184129 | − 0.0988131 | − 0.0623638 | − 0.184029 | − 0.0578964 |
1.0 | − 0.0976559 | − 0.123225 | − 0.126269 | − 0.0831271 | − 0.13072 |
1.5 | − 0.00191019 | − 0.0641543 | − 0.0828664 | − 0.0327721 | − 0.0890927 |
2.0 | − 0.00140958 | − 0.022701 | − 0.0332923 | − 0.0062077 | − 0.0300153 |
2.5 | − 0.00109538 | − 0.00748039 | − 0.0107562 | − 0.00129558 | − 0.00763243 |
3.0 | − 0.00049598 | − 0.00339493 | − 0.00475315 | − 0.00052566 | − 0.00395257 |
3.5 | − 0.00011148 | − 0.00189069 | − 0.00310482 | − 0.00019944 | − 0.00314844 |
4.0 | − 1.8570E–05 | − 0.000998984 | − 0.00200886 | − 6.6049E–05 | − 0.002033 |
4.5 | − 7.1008E–06 | − 0.000476063 | − 1.1300E–03 | − 1.9381E–05 | − 0.00105175 |
5.5 | − 2.2116E–06 | − 0.000213832 | − 0.000564072 | − 5.4503E–06 | − 0.000473434 |
x | CTE | LS | GN-II | GN-III | MGTE |
---|---|---|---|---|---|
0.0 | 0.154293 | 0.0985745 | 0.0811364 | 0.129721 | 0.0864024 |
0.5 | 0.0353649 | 0.049002 | 0.0493561 | 0.0494183 | 0.0549301 |
1.0 | 0.0121599 | 0.0198227 | 0.0217495 | 0.0178378 | 0.0218605 |
1.5 | 0.00561491 | 0.00810607 | 0.0082742 | 0.00663544 | 0.00675295 |
2.0 | 0.00242289 | 0.00421452 | 0.00380838 | 0.00268781 | 0.00298421 |
2.5 | 0.000895646 | 0.00263724 | 0.00255834 | 0.0011538 | 0.00245018 |
3.0 | 0.000238747 | 0.0016596 | 0.00194611 | 0.000499064 | 0.00206128 |
3.5 | 5.90576E–05 | 0.000973708 | 0.00137519 | 0.000211311 | 0.00145486 |
4.0 | 2.87545E–05 | 0.000533152 | 0.000871106 | 8.71337E–05 | 0.000875979 |
4.5 | 1.18972E–05 | 0.000280434 | 0.000505564 | 3.52647E–05 | 0.000469975 |
5.5 | 4.32000E–06 | 0.000146736 | 0.000280442 | 1.41755E–05 | 0.000241428 |