Understanding the thermal problem of variable gradient functionally graded plate based on hybrid numerical method under linear heat source

Abstract

The thermal problem of functionally graded materials (FGM) under linear heat source is studied by a hybrid numerical method. The accuracy of the analytical method and the efficiency of the finite element method are taken into account. The volume fraction of FGM in the thickness direction can be changed by changing the gradient parameters. Based on the weighted residual method, the heat conduction equation under the third boundary condition is established. The temperature distribution of FGM under the action of linear heat source is obtained by Fourier transform. The results show that the closer to the heat source it is, the greater the influence of the heat source is and the influence of the heat source is local. The temperature change trend of the observation points is consistent with the heat source, showing a linear change. The results also show that the higher the value of gradient parameter is, the higher the temperature of location point is. The temperature distribution of observation points is positively correlated with gradient parameter. When the gradient parameter value exceeds a certain value, it has a little effect on the temperature change in the model and the heat conduction in the model tends to be pure metal heat conduction, the optimal gradient parameters combined the thermal insulation property of ceramics and the high strength toughness of metals are obtained.

The theory of thermoelasticity with a memory-dependent dynamic response for a thermo-piezoelectric functionally graded rotating rod

“…The thermo-piezoelectricity problem without body force and free charges is controlled by three additional equations given by 𝐷 𝑖,𝑖 = 0, 𝐸 𝑖 = −𝜑 ,𝑖 (10) It is possible to express the equations of motion as 𝐶 𝑖𝑗𝑘𝑙 𝑢 𝑘,𝑙𝑖 + 𝑒 𝑘𝑖𝑗 𝜑 ,𝑘𝑖 − 𝛽 𝑖𝑗 𝜃 ,𝑖 =𝜌 𝑢̈𝑖 (11) Also, introducing Gauss's divergence equation (5) into equation (10), gives 𝑒 𝑘𝑖𝑗 𝑢 𝑖,𝑗𝑘 −∈ 𝑖𝑗 𝜑 ,𝑖𝑗 + 𝑝 𝑖 𝜃 ,𝑖 = 0 (12) Assuming zero piezoelectric influences (𝑒 𝑖𝑗𝑘 , ∈ 𝑖𝑗 𝑝 𝑖 → 0), the fundamental system field equations simplify to modified thermoelasticity with a relaxation time. Moreover, the equations of the coupled thermo-piezoelectricity framework are obtained by ignoring the thermal relaxation time (𝜏 0 → 0).…”

“…In addition, ferroelectric composites are inherently anisotropic and exhibit correlated behaviors between the interactions of mechanical, electrical, and electromagnetic interactions. A number of researchers have taken advantage of the generalized theories of thermoelasticity in order to investigate some issues related to the motion of waves through magneto-thermoelastic and thermoelastic materials [9][10][11][12][13][14][15].…”