## 1 Introduction  
Elastic materials consisting of small pores are defined as linear elastic materials with voids. In classical elastic theory, the volumes of these pores are neglected; they play an important role because independent kinematic variables must consider the volume of these pores. Recently, the problems of heat transfer in porous media, essentially with short thermal periods and small-time scales, have been addressed by many researchers. The effective thermal conductivity generated by weakly conducting pores is used to characterize heat conduction for a volume of material with hundreds of holes. Furthermore, the interactions between thermo-elastic liquids and solids, such as water, a domain of poro-thermal elasticity, are receiving more attention because of their multiple applications in related domains, including geophysics. The poro-elastic theory was developed by [PERSON] [1]. [PERSON] [2; 3] developed the theoretical foundation for propagating iso-thermal waves in fluid-saturated, poro-elastic mediums for high and low-frequency instances. Many researchers have looked at the challenges of plane and surface wave propagation in liquid-saturated poro-elastic media using [PERSON]'s model. For example, three forms of theories were proposed by [PERSON] [4; 5; 6] (GN-I, GNII, and GN-III). The constitutive equations of the GN models have been linearized. The classical coupled thermo-elasticity theory is similar to type I, but it has a different name (GN-I). [PERSON] [7] used the dual-phase-lag models to study the interaction between photons and electrons on a microscopical level. At a microscopical scale, the relaxing source caused various relaxed responses. [PERSON] [8] confirmed the physical concepts and practicality of the dual-phase-lag model in their experimental results. The elastic theory of a material with a void is one of the most serious generalizations of the classical elastic material model. This model applies to elastic material composed of distributions of tiny holes, with the empty volume included among the kinematic variables. This model has been rationally constructed to generate a perfectly consistent model that can logically accommodate thermal impulse transmissions. [PERSON] [9] presented a three-phase lag thermo-elasticity model (TPL) that encompassed the generalizations. Many researchers, including [10; 11; 12; 13], have contributed to various difficulties in poro-thermo-elastic media after the work of [3]. [PERSON] et al. [14] used a generalized thermo-elasticity model to explore the effects of the initial stress, rotation, magnetic field, and voids on plane waves. The natural frequency of a poro-elastic circular cylinder was investigated by [PERSON] [15]. [PERSON] and [PERSON] [16] investigated the propagations of the transient waves in a one-dimension poro-elastic column. [PERSON] et al. [17] investigated the impacts of thermal dispersions on free convections in a fluid-saturated porous medium. [PERSON] et al. [18] used the finite element method to study the thermo-elastic interaction in a poro-elastic medium under the GL model. [PERSON] et al. [19] looked into the influence of memory on the propagation of thermal waves in an elastic material containing voids. The analytical solutions for time-dependent problems for coupled and nonlinear/linear systems exist only for simple boundary conditions with initial conditions. The finite element scheme is preferable to the time domain problem in order to avert this complication. Procedures for solving deformations concerning this problem using finite element methods have been presented. It is a powerful scheme and has been expanded to solve complex problems in the structural mechanics. [PERSON] et al. [20] studied the physics and simulations of wave propagation in linear poro-thermo-elastic mediums. [PERSON] et al. [21] investigated the propagations of thermo-elastic waves in an unsaturated poro-thermo-elastic medium. With fractional thermo-elastic theory, [PERSON] et al.