Thermomechanical interactions in nonlocal thermoelastic medium with double porosity structure

Abstract

The main objective of this work is to create a new thermoelastic model for hyperbolic thermoelasticity in the context of double porosity structure based on nonlocal elasticity theory and the dual-phase-lag model. Nonlocal elasticity theory is used to construct new constitutive relations and equations. In a homogeneous, isotropic thermoelastic material, thermomechanical interactions are studied using normal mode analysis. A time-dependent thermal shock is applied on the boundary surface. This study also produces a few unique situations, which are compared with previous results of other researchers. The normal and tangential stresses, temperature, displacement components, change in void volume fractions, and equilibrated stress vectors concerning distances and time intervals are all calculated numerically. The physical quantities mentioned above are also visually displayed for various thermoelastic models to compare and illustrate the theoretical results. A comparative analysis and graphical presentation of the effects of nonlocal parameters and porosity on various physical characteristics have been performed. The figures show that most of the physical variables decrease with the increase in distance and show oscillatory behavior with the increase in time. The behavior of the void volume fraction field of the first kind is opposite to the behavior of the void volume fraction field of the second kind with the increase in distance. It is also noticed that the behavior of equilibrated stress of the first kind is opposite to the behavior of the second kind.

Appendices

Appendix A

$$\begin{aligned} &N_{1}=\frac{p_{30}+p_{56}}{p_{17}}, N_{2}=\frac{p_{31}+p_{57}+p_{71}+s_{10}+s_{24}}{p_{17}}, N_{3}=\frac{p_{32}+p_{58}+p_{72}+s_{11}+s_{25}}{p_{17}}, \\ &N_{4}=\frac{p_{33}+p_{59}+p_{73}+s_{12}+s_{26}}{p_{17}}, N_{5}=\frac{p_{34}+p_{74}+s_{13}+s_{27}}{p_{17}},\\ &s_{1}=a_{12}a_{17}-a_{20}, s_{2}=a_{12}p_{5}-p_{9}-a_{16}p_{11}-a_{11}a_{20}, s_{3}=a_{12}p_{6}-a_{11}p_{9}+a_{16}p_{12}, \\ &s_{4}=a_{15}a_{17}-a_{19}, s_{5}=a_{15}p_{5}-p_{35}-a_{11}a_{19}-a_{16}p_{43}, s_{6}=a_{15}p_{6}-a_{11}p_{35}+a_{16}p_{37}, \\ &s_{7}=a_{15}a_{20}-a_{12}a_{19}, s_{8}=a_{12}a_{17}a_{24}-a_{15}a_{23}+p_{60}, s_{9}=a_{15}p_{12}-a_{12}p_{37}+a_{11}p_{60}, \\ &s_{10}=a_{4}s_{4}-a_{4}a_{7}s_{1}, s_{11}=a_{4}s_{5}+a_{4}a_{6}s_{4}-a_{4}a_{7}s_{2}-a_{4}a_{8}s_{7}+a_{4}a_{10}s_{8}, \\ &s_{12}=a_{4}s_{6}+a_{4}a_{6}s_{5}-a_{4}a_{7}s_{3}-a_{4}a_{8}p_{68}+a_{4}a_{10}s_{10}, s_{13}=a_{4}a_{6}s_{6}, s_{14}=-a_{23}-a_{13}p_{11}, \\ &s_{15}=a_{12}p_{7}+p_{10}-a_{11}a_{23}+a_{13}p_{12}-a_{14}p_{11}, s_{16}=a_{12}p_{8}+a_{11}p_{10}+a_{14}p_{12}, \\ &s_{17}=a_{13}p_{42}+a_{24}, s_{18}=a_{15}p_{7}-p_{36}+a_{11}a_{24}+a_{13}p_{37}+a_{14}p_{42}, \\ &s_{19}=a_{15}p_{8}-a_{11}p_{36}+a_{14}p_{37}, s_{20}=a_{12}a_{24}-a_{15}a_{23}+a_{13}p_{60}, \\ &s_{21}=a_{15}p_{10}-a_{12}p_{36}+a_{14}p_{60}, s_{22}=p_{60}-a_{15}p_{11}-a_{12}p_{42}, \\ &s_{23}=a_{15}p_{12}-a_{12}p_{37}+a_{11}p_{60}, s_{24}=a_{5}a_{7}s_{14}-a_{5}s_{17}, \\ &s_{25}=a_{5}a_{7}s_{15}-a_{5}s_{18}-a_{6}s_{17}+a_{5}a_{8}s_{20}-a_{5}a_{9}s_{22}, \\ &s_{26}=a_{5}a_{7}s_{16}-a_{5}s_{19}-a_{5}a_{6}s_{18}+a_{5}a_{8}s_{21}-a_{5}a_{9}s_{23}, s_{27}=-a_{5}a_{6}s_{19},\\ &p_{1}=a_{1}+a_{6}, p_{2}=a_{1}a_{6} , p_{3}=a_{21}+a_{25}, p_{4}=a_{21}a_{25}-a_{22}a_{27}, p_{5}=a_{17}a_{25}+a_{18}, \\ & p_{6}=a_{18}a_{25}-a_{22}a_{26}, p_{7}=a_{17}a_{27}-a_{26}, p_{8}=a_{18}a_{27}-a_{26}a_{21}, p_{9}=a_{20}a_{25}-a_{22}a_{23} ,\\ &p_{10}=a_{20}a_{27}-a_{21}a_{23} , p_{11}=a_{23}a_{17}, p_{12}=a_{20}a_{26}-a_{23}a_{18}, p_{13}=a_{1}a_{8}, p_{14}=a_{1}a_{9} , \\ &p_{15}=a_{1}a_{10}, p_{16}=-a_{23}a_{17}, p_{17}=1-a_{13}a_{17}, p_{18}=p_{3}+a_{11}-p_{5}a_{13}-a_{14}a_{17}, \\ & p_{19}=p_{4}+p_{3}a_{11}-p_{6}a_{13}-p_{5}a_{14}+p_{7}a_{16}, p_{20}=a_{11}p_{4}-a_{14}p_{6}+a_{16}p_{8}, \\ &p_{21}=a_{12}-a_{20}p_{13} , p_{22}=a_{12}p_{3}-p_{9}p_{13}-a_{14}a_{20}-a_{16}a_{23}, p_{23}=a_{12}p_{4}-a_{14}p_{9}+a_{16}p_{10}, \\ &p_{24}=a_{12}a_{17}-a_{20}, p_{25}=a_{12}p_{5}-a_{11}a_{20}+a_{16}p_{16}-p_{9},p_{26}=a_{12}p_{6}-a_{11}p_{9}+a_{16}p_{12}, \\ & p_{27}=a_{23}+a_{13}p_{16}, p_{28}=a_{12}p_{7}-p_{10}+a_{11}a_{23}+a_{13}p_{12}+a_{14}p_{16}, \\ & p_{29}=a_{12}p_{8}-a_{11}p_{10}+a_{14}p_{12}, p_{30}=p_{18}+p_{1}p_{17}-a_{8}p_{21}+a_{9}p_{24}-a_{10}p_{27}, \\ & p_{31}=p_{2}p_{17}+p_{19}+p_{1}p_{18}-a_{8}p_{22}-p_{13}p_{21}+a_{9}p_{25}+p_{14}p_{24}-a_{10}p_{28}-p_{15}p_{27}, \\ & p_{32}=p_{20}+p_{1}p_{19}+p_{2}p_{18}-a_{8}p_{23}-p_{13}p_{22}+a_{9}p_{26}+p_{14}p_{25}-a_{10}p_{29}-p_{15}p_{28}, \\ & p_{33}=p_{1}p_{20}+p_{2}p_{19}-p_{13}p_{23}+p_{14}p_{26}-p_{15}p_{29}, p_{34}=p_{2}p_{20}, p_{35}=a_{19}a_{25}-a_{22}a_{24}, \\ & p_{36}=a_{19}a_{27}-a_{21}a_{24}, p_{37}=a_{19}a_{26}-a_{18}a_{24}, p_{38}=-a_{2}a_{7}, p_{39}=a_{2}a_{8}, p_{40}=-a_{2}a_{9}, \\ & p_{41}=a_{2}a_{10}, p_{42}=-a_{17}a_{24}, p_{43}=a_{17}a_{24}, p_{44}=1-a_{13}a_{17}, \\ & p_{45}=a_{11}+p_{3}-a_{13}p_{5}-a_{14}a_{17}, p_{46}=a_{11}p_{3}-a_{13}p_{6}-a_{14}p_{5}+a_{16}p_{7}, \\ &p_{47}=a_{11}p_{4}-a_{14}p_{6}+a_{16}p_{8}, p_{48}=a_{15}-a_{13}a_{19}, \\ &p_{49}=a_{15}p_{3}-a_{13}p_{35}-a_{14}a_{19}-a_{16}a_{24}, p_{50}=a_{15}p_{4}-a_{14}p_{35}+a_{16}p_{36}, \\ &p_{51}=a_{15}a_{17}-a_{19}, p_{52}=a_{15}p_{5}-p_{35}-a_{11}a_{19}+a_{16}p_{42}, p_{53}=a_{15}p_{6}-a_{11}p_{35}+a_{16}p_{37}, \\ &p_{54}=a_{15}p_{7}-p_{36}+a_{11}a_{24}+a_{13}p_{37}-a_{14}p_{43}, p_{55}=a_{15}p_{8}-a_{11}p_{36}+a_{14}p_{37}, \\ &p_{56}=p_{38}p_{44}, p_{57}=p_{38}p_{45}+p_{39}p_{48}+p_{40}p_{51}-a_{13}p_{41}p_{43}+a_{24}p_{41}, \\ &p_{58}=p_{38}p_{46}+p_{39}p_{49}+p_{40}p_{52}+p_{41}p_{54}, p_{59}=p_{38}p_{47}+p_{39}p_{50}+p_{40}p_{53}+p_{41}p_{55}, \\ &p_{60}=a_{19}a_{23}-a_{20}a_{24}, p_{61}=a_{12}-a_{13}a_{20}, p_{62}=a_{12}p_{3}-a_{13}p_{9}-a_{14}a_{20}-a_{16}a_{23}, \\ &p_{63}=a_{12}p_{4}-a_{14}p_{9}+a_{16}p_{9}, p_{64}=a_{15}-a_{13}a_{19}, p_{65}=a_{15}p_{3}-a_{13}p_{35}-a_{14}a_{19}-a_{16}a_{24}, \\ & p_{66}=a_{15}p_{4}-a_{14}p_{35}+a_{16}p_{36}, p_{67}=a_{15}a_{20}-a_{12}a_{19}, p_{68}=a_{15}p_{9}-a_{12}p_{35}+a_{16}p_{60}, \\ &p_{69}=a_{12}a_{24}+a_{13}p_{60}-a_{15}a_{23}, p_{70}=a_{15}p_{10}-a_{12}p_{36}+a_{14}p_{60}, p_{71}=a_{3}a_{7}p_{61}-a_{3}p_{64}, \\ &p_{72}=a_{3}a_{7}p_{62}-a_{3}p_{65}-a_{3}a_{6}p_{64}+a_{3}a_{9}p_{67}-a_{3}a_{10}p_{69}, \\ &p_{73}=a_{3}a_{7}p_{63}-a_{3}p_{66}-a_{3}a_{6}p_{65}+a_{3}a_{9}p_{68}-a_{3}a_{10}p_{70}, p_{74}=-a_{3}a_{6}p_{66}. \end{aligned}$$

Appendix B

$$\begin{aligned} \Delta = \begin{vmatrix} e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ g_{1} & g_{2} & g_{3} & g_{4} & g_{5} \\ f_{1} & f_{2} & f_{3} & f_{4} & f_{5} \\ l_{1} & l_{2} & l_{3} & l_{4} & l_{5} \\ m_{1} & m_{2} & m_{3} & m_{4} & m_{5} \end{vmatrix} , \Delta _{1}= \begin{vmatrix} G & e_{2} & e_{3} & e_{4} & e_{5} \\ 0 & g_{2} & g_{3} & g_{4} & g_{5} \\ 0 & f_{2} & f_{3} & f_{4} & f_{5} \\ 0 & l_{2} & l_{3} & l_{4} & l_{5} \\ 0 & m_{2} & m_{3} & m_{4} & m_{5} \end{vmatrix} ,\\ \Delta _{2}= \begin{vmatrix} e_{1} & G & e_{3} & e_{4} & e_{5} \\ g_{1} & 0 & g_{3} & g_{4} & g_{5} \\ f_{1} & 0 & f_{3} & f_{4} & f_{5} \\ l_{1} & 0 & l_{3} & l_{4} & l_{5} \\ m_{1} & 0 & m_{3} & m_{4} & m_{5} \end{vmatrix} , \Delta _{3}= \begin{vmatrix} e_{1} & e_{2} & G & e_{4} & e_{5} \\ g_{1} & g_{2} & 0 & g_{4} & g_{5} \\ f_{1} & f_{2} & 0 & f_{4} & f_{5} \\ l_{1} & l_{2} & 0 & l_{4} & l_{5} \\ m_{1} & m_{2} & 0 & m_{4} & m_{5} \end{vmatrix} ,\\ \Delta _{4}= \begin{vmatrix} e_{1} & e_{2} & e_{3} & G & e_{5} \\ g_{1} & g_{2} & g_{3} & 0 & g_{5} \\ f_{1} & f_{2} & f_{3} & 0 & f_{5} \\ l_{1} & l_{2} & l_{3} & 0 & l_{5} \\ m_{1} & m_{2} & m_{3} & 0 & m_{5} \end{vmatrix} , \Delta _{5}= \begin{vmatrix} e_{1} & e_{2} & e_{3} & e_{4} & G \\ g_{1} & g_{2} & g_{3} & g_{4} & 0 \\ f_{1} & f_{2} & f_{3} & f_{4} & 0 \\ l_{1} & l_{2} & l_{3} & l_{4} & 0 \\ m_{1} & m_{2} & m_{3} & m_{4} & 0 \end{vmatrix}. \end{aligned}$$