Generalized thermo-elastic waves propagating in bars with a rectangular cross-section

Abstract

With the rapid development and application of the laser ultrasonic technology in nondestructive testing in recent years, thermo-elastic waves in diverse waveguides have captured a multitude of attention. However, they are mainly focused on one-dimensional and half-space structures. In engineering, there are also a lot of two-dimensional structures, such as joist steel, straight bars and rings. However, rare attention is paid on thermo-elastic waves in these structures. Accordingly, in the context of Green–Lindsay (G–L) generalized thermo-elasticity theory, a modified double orthogonal polynomial approach is exploited to investigate thermo-elastic waves in bars with a rectangular cross-section. The dispersion, attenuation and displacement curves of thermo-elastic waves are illustrated. Subsequently, influences of the cross-section size and relaxation time on wave characteristics are analyzed. Results indicate that the cross-section size and relaxation time have a significant influence on thermo-elastic waves. The phase velocity and attenuation values of thermal wave modes decrease as the relaxation time increases. These results obtained can be utilized to guide the laser ultrasonic nondestructive testing for this kind of structures.

Appendix

The explicit expressions for matrix elements are detailed as follows:

$$ \begin{gathered} A_{{11}}^{{n,p,j,m}} = - u[n,p,j,m,0,0], \hfill \\ A_{{12}}^{{n,p,j,m}} = A_{{13}}^{{n,p,j,m}} = A_{{14}}^{{n,p,j,m}} = A_{{21}}^{{n,p,j,m}} = A_{{23}}^{{n,p,j,m}} = A_{{24}}^{{n,p,j,m}} = 0, \hfill \\ \end{gathered} $$

$$ \begin{gathered} A_{{22}}^{{n,p,j,m}} {\text{ }}\; = \; - \;c_{8} \times u[n,p,j,m,0,0],\;A_{{31}}^{{n,p,j,m}} = A_{{32}}^{{n,p,j,m}} = A_{{34}}^{{n,p,j,m}} = A_{{41}}^{{n,p,j,m}} = A_{{42}}^{{n,p,j,m}} = A_{{43}}^{{n,p,j,m}} = 0, \hfill \\ A_{{33}}^{{n,p,j,m}} \; = \; - \;c_{7} \times u[n,p,j,m,0,0],\;A_{{44}}^{{n,p,j,m}} \; = \; - \;u[n,p,j,m,0,0], \hfill \\ \end{gathered} $$

$$ B_{11}^{n,p,j,m}\,=\,B_{22}^{n,p,j,m}\,=\,B_{23}^{n,p,j,m}\,=\,B_{24}^{n,p,j,m} \,= \,B_{32}^{n,p,j,m}\,=\,B_{33}^{n,p,j,m}\, =\, B_{34}^{n,p,j,m} \,=\,0, $$

$$ \begin{gathered} B_{{42}}^{{n,p,j,m}} = B_{{43}}^{{n,p,j,m}} = B_{{44}}^{{n,p,j,m}} = 0, \hfill \\ B_{{12}}^{{n,p,j,m}} {\text{ }} = i \times \left( {c_{1} + c_{8} } \right) \times u[n,p,j,m,1,0] + i \times c_{8} \times K_{y} [n,p,j,m,0,0], \hfill \\ B_{{13}}^{{n,p,j,m}} {\text{ }} = i \times \left( {c_{3} + c_{7} } \right) \times u[n,p,j,m,0,1] + i \times c_{7} \times K_{z} [n,p,j,m,0,0], \hfill \\ B_{{14}}^{{n,p,j,m}} {\text{ }} = - \left( {i + t_{1}^{*} \times \omega } \right) \times u[n,p,j,m,0,0], \hfill \\ B_{{21}}^{{n,p,j,m}} {\text{ }} = i \times \left( {c_{1} + c_{8} } \right) \times u[n,p,j,m,1,0] + i \times c_{1} \times K_{y} [n,p,j,m,0,0], \hfill \\ B_{{31}}^{{n,p,j,m}} {\text{ }} = i \times \left( {c_{3} + c_{7} } \right) \times u[n,p,j,m,0,1] + i \times c_{3} \times K_{z} [n,p,j,m,0,0], \hfill \\ B_{{41}}^{{n,p,j,m}} {\text{ }} = - \eta \times \omega \times u[n,p,j,m,0,0], \hfill \\ \end{gathered} $$

$$ \begin{gathered} C_{11}^{n,p,j,m} { = }c_{8} \times u[n,p,j,m,2,0] + c_{7} \times u[n,p,j,m,0,2] \hfill \\ + c_{8} \times K_{y} [n,p,j,m,1,0] + c_{7} \times K_{z} [n,p,j,m,0,1], \hfill \\ \end{gathered} $$

$$ \begin{gathered} C_{{12}}^{{n,p,j,m}} {\text{ }} = C_{{13}}^{{n,p,j,m}} {\text{ }} = C_{{14}}^{{n,p,j,m}} C_{{22}}^{{n,p,j,m}} {\text{ }} = c_{2} \times u[n,p,j,m,2,0] + c_{6} \times u[n,p,j,m,0,2] \hfill \\ + c_{2} \times K_{y} [n,p,j,m,1,0] + c_{6} \times K_{z} [n,p,j,m,0,1],{\text{ }} = C_{{21}}^{{n,p,m,j}} {\text{ }} = C_{{31}}^{{n,p,m,j}} = C_{{41}}^{{n,p,m,j}} = 0, \hfill \\ C_{{23}}^{{n,p,j,m}} {\text{ }} = \left( {c_{4} + c_{6} } \right) \times u[n,p,j,m,1,1] + c_{4} \times K_{y} [n,p,j,m,0,1] + c_{6} \times K_{z} [n,p,j,m,1,0], \hfill \\ C_{{24}}^{{n,p,j,m}} {\text{ }} = - \lambda _{1} \times \left( {1 - i \times t_{1}^{*} \times \omega } \right) \times u[n,p,j,m,1,0] - \lambda _{1} \times \left( {1 - i \times t_{1}^{*} \times \omega } \right) \times K_{y} [n,p,j,m,0,0], \hfill \\ C_{{32}}^{{n,p,j,m}} {\text{ }} = \left( {c_{4} + c_{6} } \right) \times u[n,p,j,m,1,1] + c_{6} \times K_{y} [n,p,j,m,0,1] + c_{4} \times K_{z} [n,p,j,m,1,0], \hfill \\ C_{{33}}^{{n,p,j,m}} {\text{ }} = c_{6} \times u[n,p,j,m,2,0] + c_{5} \times u[n,p,j,m,0,2] + c_{6} \times K_{y} [n,p,j,m,1,0] + c_{5} \times K_{z} [n,p,j,m,0,1], \hfill \\ C_{{34}}^{{n,p,j,m}} {\text{ }} = - \;\lambda _{2} \times \left( {1 - i \times t_{1}^{*} \times \omega } \right) \times u[n,p,j,m,1,0] - \;\lambda _{2} \times \left( {1 - i \times t_{1}^{*} \times \omega } \right) \times K_{z} [n,p,j,m,0,0], \hfill \\ C_{{42}}^{{n,p,j,m}} {\text{ }} = \eta \times \lambda _{1} \times i \times \omega \times u[n,p,j,m,1,0],\;C_{{43}}^{{n,p,j,m}} {\text{ }} = \eta \times \lambda _{2} \times i \times \omega \times u[n,p,j,m,0,1], \hfill \\ C_{{44}}^{{n,p,j,m}} {\text{ }} = \alpha _{1} \times u[n,p,j,m,2,0] + \alpha _{2} \times u[n,p,j,m,0,2]\end{gathered} $$

$$ M_{n,p,j,m} = -\omega^{2} \times u[n,p,j,m,0,0], \quad T_{n,p,j,m} = - \left( {i \times \omega + t_{0}^{*} \times \omega^{2} } \right) \times u[n,p,j,m,0,0], $$

where

$$ u[n,p,j,m,g,q]{\text{ = }}\int_{0}^{h} {\int_{0}^{d} {I(y,z) \times Q_{n} (z) \times Q_{p} (y) \times } } \frac{{\partial ^{g} Q_{j} (y)}}{{\partial y^{g} }} \times \frac{{\partial ^{q} Q_{m} (z)}}{{\partial z^{q} }}dydz,$$

$$ K_{y} [n,p,j,m,g,q]{\text{ = }}\int_{0}^{h} {\int_{0}^{d} {\frac{{\partial I(y,z)}}{{\partial y}} \times Q_{p} (y) \times Q_{n} (z) \times } } \frac{{\partial ^{g} Q_{j} (y)}}{{\partial y^{g} }} \times \frac{{\partial ^{q} Q_{m} (z)}}{{\partial z^{q} }}dydz, $$

$$ K_{z} [n,p,j,m,g,q]{\text{ = }}\int_{0}^{h} {\int_{0}^{d} {\frac{{\partial I(y,z)}}{{\partial z}} \times Q_{n} (z) \times Q_{p} (y) \times } } \frac{{\partial ^{g} Q_{j} (y)}}{{\partial y^{g} }} \times \frac{{\partial ^{q} Q_{m} (z)}}{{\partial z^{q} }}dydz. $$