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Consider an annular disk initially at rest and reference temperature Ty. Now suppose that the disk is exposed to an axisymmetric thermal shock load applied into its inner surface. The coupled thermoelastic equations of the disk, with plane-stress assumption and based on the Lord—Shulman model of thermoelasticity theory, is obtained from Eqs. (8), (10) and (12) as Under the plane stress assumption, the von Mises stress in an annular disk is

Figure 1 Consider an annular disk initially at rest and reference temperature Ty. Now suppose that the disk is exposed to an axisymmetric thermal shock load applied into its inner surface. The coupled thermoelastic equations of the disk, with plane-stress assumption and based on the Lord—Shulman model of thermoelasticity theory, is obtained from Eqs. (8), (10) and (12) as Under the plane stress assumption, the von Mises stress in an annular disk is

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The subscript “‘m”’ denotes the metal property and Ty, is a characteristic temperature defined to normalize the tem- perature. Using the introduced dimensionless terms, Eqs. (14)(17) are written as In uncoupled case, thermal field is not affected by dis- placement field and thus those terms related to displace- ment in Eq. (20) are vanished. The uncoupled system of Eqs. (19) and (20) may be obtained when C = 0. In the pre- ceding equations, it is to be noted that the FGM’s proper- ties are position-dependent functions, and thus the elastic and thermal waves propagation speeds vary as functions of position in an FGM.
The subscript “‘m”’ denotes the metal property and Ty, is a characteristic temperature defined to normalize the tem- perature. Using the introduced dimensionless terms, Eqs. (14)(17) are written as In uncoupled case, thermal field is not affected by dis- placement field and thus those terms related to displace- ment in Eq. (20) are vanished. The uncoupled system of Eqs. (19) and (20) may be obtained when C = 0. In the pre- ceding equations, it is to be noted that the FGM’s proper- ties are position-dependent functions, and thus the elastic and thermal waves propagation speeds vary as functions of position in an FGM.
Fig. 1. Power law index effect on metal volume fraction.
Fig. 1. Power law index effect on metal volume fraction.
where L is the length of element along the radius. The terms on the right-hand side of Eqs. (35) and (36) cancel each other between any two adjacent elements, except the nodes located on the boundaries of the disk. These bound- ary conditions in term of nodal temperature and displace- ment are where U ; and U i, are the radial displacements of the inner and outer boundaries of the disk, respectively. Also, the subscript | and M are referred to the first and last nodes of the solution domain that are located at the inner and outer radii of the disk, respectively. Due to the assumed
where L is the length of element along the radius. The terms on the right-hand side of Eqs. (35) and (36) cancel each other between any two adjacent elements, except the nodes located on the boundaries of the disk. These bound- ary conditions in term of nodal temperature and displace- ment are where U ; and U i, are the radial displacements of the inner and outer boundaries of the disk, respectively. Also, the subscript | and M are referred to the first and last nodes of the solution domain that are located at the inner and outer radii of the disk, respectively. Due to the assumed
Here, V is the volume of the base element (e). With the aid of weak formulation and local coordinates y = ? — 7+, where 7; is the radius of the first node of the base element, Eqs. (33) and (34) may be appeared in the form
Here, V is the volume of the base element (e). With the aid of weak formulation and local coordinates y = ? — 7+, where 7; is the radius of the first node of the base element, Eqs. (33) and (34) may be appeared in the form
Now the system of equations may be solved to find the displacement and temperature fields in the Laplace domain.
Now the system of equations may be solved to find the displacement and temperature fields in the Laplace domain.
Fig. 3. Radial displacement distribution at three different times for pure Aluminum (n = 0) with similar boundary conditions of Ref. [10]. Fig. 2. The base element through the radius direction and the correspond Lagrangian linear polynomial shape functions.
Fig. 3. Radial displacement distribution at three different times for pure Aluminum (n = 0) with similar boundary conditions of Ref. [10]. Fig. 2. The base element through the radius direction and the correspond Lagrangian linear polynomial shape functions.
Fig. 4. Distribution of the nondimensional temperature along the nondimensional radius of the disk for n= 1. cases. These include investigating the waves propagation in FG disk and the power law index effects on variation of the temperature, displacement, radial stress and hoop stress. Also, the effect of reference temperature and relaxation time on the variation of the temperature, displacement and stresses are investigated. Aluminum and Alumina are considered as metal and ceramic constituents of the func- tionally graded annular disk. Since there is no experimental data for the relaxation time of specific materials, often some approximate values are used for the Lord—Shulman theory. The numerical values of the relaxation times are of the range 107'°s to 10~'*s [8]. Ceramics are known as thermal barrier materials, where their thermal wave speed of propagations are lower than the thermal wave speed of propagations of metals. Also, the thermal wave speed is proportional to the square of reciprocity of relaxation time [13]. Thus, a greater value for the relaxation time for cera- mic is considered in comparison to the value of relaxation time for metal. In this paper, the nondimensional values
Fig. 4. Distribution of the nondimensional temperature along the nondimensional radius of the disk for n= 1. cases. These include investigating the waves propagation in FG disk and the power law index effects on variation of the temperature, displacement, radial stress and hoop stress. Also, the effect of reference temperature and relaxation time on the variation of the temperature, displacement and stresses are investigated. Aluminum and Alumina are considered as metal and ceramic constituents of the func- tionally graded annular disk. Since there is no experimental data for the relaxation time of specific materials, often some approximate values are used for the Lord—Shulman theory. The numerical values of the relaxation times are of the range 107'°s to 10~'*s [8]. Ceramics are known as thermal barrier materials, where their thermal wave speed of propagations are lower than the thermal wave speed of propagations of metals. Also, the thermal wave speed is proportional to the square of reciprocity of relaxation time [13]. Thus, a greater value for the relaxation time for cera- mic is considered in comparison to the value of relaxation time for metal. In this paper, the nondimensional values
Fig. 5. Distribution of the nondimensional radial displacement along the nondimensional radius of the disk for n = 1.
Fig. 5. Distribution of the nondimensional radial displacement along the nondimensional radius of the disk for n = 1.
Nondimensional r Fig. 7. Distribution of the nondimensional hoop stress along the nondimensional radius of the disk for n = 1. Fig. 6. Distribution of the nondimensional radial stress along the nondimensional radius of the disk for n= 1.
Nondimensional r Fig. 7. Distribution of the nondimensional hoop stress along the nondimensional radius of the disk for n = 1. Fig. 6. Distribution of the nondimensional radial stress along the nondimensional radius of the disk for n= 1.
Fig. 8. Variations of the nondimensional temperature for points at the middle radius of the disk versus the nondimensional time for different values of power law index. The effects of power law index on temperature, displace- ment, radial stress and hoop stress at middle point of the disk is shown in Figs. 8-11, respectively. In this case, the value of relaxation time and reference temperature are assumed as previously. It is seen that the maximum of the temperature, displacement and stresses variations reduce when n is increased. Also, the amplitude of oscilla- tions of these waves are reduced when v has larger values.
Fig. 8. Variations of the nondimensional temperature for points at the middle radius of the disk versus the nondimensional time for different values of power law index. The effects of power law index on temperature, displace- ment, radial stress and hoop stress at middle point of the disk is shown in Figs. 8-11, respectively. In this case, the value of relaxation time and reference temperature are assumed as previously. It is seen that the maximum of the temperature, displacement and stresses variations reduce when n is increased. Also, the amplitude of oscilla- tions of these waves are reduced when v has larger values.
Fig. 9. Variations of the nondimensional radial displacement for points at the middle radius of the disk versus the nondimensional time for different values of power law index.
Fig. 9. Variations of the nondimensional radial displacement for points at the middle radius of the disk versus the nondimensional time for different values of power law index.
Fig. 10. Variations of the nondimensional radial stress for points at the middle radius of the disk versus the nondimensional time for different values of power law index.
Fig. 10. Variations of the nondimensional radial stress for points at the middle radius of the disk versus the nondimensional time for different values of power law index.
Fig. 11. Variations of the nondimensional hoop stress for points at the middle radius of the disk versus the nondimensional time for different values of power index.
Fig. 11. Variations of the nondimensional hoop stress for points at the middle radius of the disk versus the nondimensional time for different values of power index.
Fig. 12. Variations of the nondimensional stresses at the inner, middle, and outer points of the disk for (m = 1).
Fig. 12. Variations of the nondimensional stresses at the inner, middle, and outer points of the disk for (m = 1).
Fig. 13. Variations of the nondimensional von Mises stress for points at the inner radius of the disk versus the nondimensional time for different values of power index.
Fig. 13. Variations of the nondimensional von Mises stress for points at the inner radius of the disk versus the nondimensional time for different values of power index.
Fig. 16. Reference temperature (coupling term) effect on nondimensional hoop stress of points at the middle radius of the disk versus the nondimensional time. Fig. 15. Reference temperature (coupling term) effect on nondimensional radial stress of points at the middle radius of the disk versus the nondimensional time.
Fig. 16. Reference temperature (coupling term) effect on nondimensional hoop stress of points at the middle radius of the disk versus the nondimensional time. Fig. 15. Reference temperature (coupling term) effect on nondimensional radial stress of points at the middle radius of the disk versus the nondimensional time.
Fig. 14. Reference temperature (coupling term) effect on nondimensional temperature and displacement of points at the middle radius of the disk versus the nondimensional time.
Fig. 14. Reference temperature (coupling term) effect on nondimensional temperature and displacement of points at the middle radius of the disk versus the nondimensional time.
Fig. 17. Variations of the nondimensional temperature of points at the middle radius of the disk versus the nondimensional time for different values of relaxation times.
Fig. 17. Variations of the nondimensional temperature of points at the middle radius of the disk versus the nondimensional time for different values of relaxation times.
Fig. 18. Variations of the nondimensional displacement of points at the middle radius of the disk versus the nondimensional time for different values of relaxation times.
Fig. 18. Variations of the nondimensional displacement of points at the middle radius of the disk versus the nondimensional time for different values of relaxation times.
References Fig. 20. Variations of the nondimensional hoop stress of points at the middle radius of the disk versus the nondimensional time for different values of relaxation times. Fig. 19. Variations of the nondimensional radial stress of points at the middle radius of the disk versus the nondimensional time for different values of relaxation times.
References Fig. 20. Variations of the nondimensional hoop stress of points at the middle radius of the disk versus the nondimensional time for different values of relaxation times. Fig. 19. Variations of the nondimensional radial stress of points at the middle radius of the disk versus the nondimensional time for different values of relaxation times.
Related topics:
EngineeringLaplace TransformComposite structuresFunctionally GradedLaplace transformationHoop StressNumerical InversionPower Law Distribution
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