Thermal stresses in an orthotropic hollow sphere under thermal shock: a uni�ed generalized thermoelasticity

: This paper deals with the thermoelasticity problem in an orthotropic hollow sphere. A unified governing equation are derived which includes the Classical, Lord-Shulman and Green-Lindsay coupled theories of thermoelasticity. Time-dependent thermal and mechanical boundary conditions are applied to the inner and outer surfaces of the hollow sphere and the problem is solved analytically using the finite Hankel transform. The inner surface of the sphere is subjected to a thermal shock in the form of a prescribed heat flux. Subsequently, the thermal response, radial displacement, as well as radial, tangential, and circumferential stresses of the sphere are determined. The influence of different orthotropic material properties and relaxation time values is investigated and presented graphically. The obtained results demonstrate excellent agreement with the existing literature.


Introduction
Spherical vessels are extensively utilized across diverse engineering sectors like marine, aerospace, petrochemical, and mechanical.In addition to the mechanical loads that these structures experience, the coupling phenomenon between thermal and mechanical behaviors of materials hold significant relevance in diverse fields such as acoustics, geology, geophysics, and engineering.The thermal loads associated with these phenomena can sometimes reach magnitudes that lead to structural failure.To address this issue, generalized theories of thermoelasticity have been developed in recent decades.These theories introduce modifications to the energy equation, transforming it into a hyperbolic partial differential equation that admits finite speed for the propagation of thermal waves.Unlike classical theories, generalized thermoelasticity theories offer a more realistic approach to addressing engineering problems involving high heat fluxes, short time intervals, and low temperatures.
In this regard, Lord and Shulman's theory is one of the most well-known generalized thermoelasticity theories, which incorporates a relaxation time to modify Fourier's law of heat conduction.Another thermoelasticity theory that admits the second sound effect is Green-Lindsay theory.Green and Lindsay introduced alterations to the stress-strain relationship and the energy equation by incorporating two relaxation times, which establish connections between stress, entropy, and the rate of temperature change.However, the governing equations of thermoelasticity are inherently complex due to the intricate coupling between elastic and thermal fields.This coupling leads to the introduction of additional constants and relaxation times in generalized thermoelasticity theories, further complicating the governing equations.As a result, analytical solutions, especially in the context of generalized theories, have not been extensively developed.Furthermore, the growing demand for materials with enhanced strength-to-weight ratios has prompted advances in the design of novel materials such as fiber-reinforced composites and manufacturing processes.The utilization of composite materials or diverse manufacturing techniques for spheres may lead to anisotropic mechanical properties.Consequently, it becomes imperative to analyze the response of structures composed of such materials to accurately predict their behavior and performance.
They obtained the distribution of the transient thermoelastic stresses using the Laplace transform.In their approach, stress and temperature distribution were determined simultaneously.Hata [2] investigated the thermal shock in a hollow sphere caused by rapid uniform heating.They employed Ray's theory to extract closed-form relations for dynamic stresses.However, in their approach, no generality is considered for the thermal load.Misra et al. [3] discussed the generation of thermal stresses in an aeolotropic homogeneous continuum solid with a spherical cavity.They used Laplace transform to obtain temperature, stress, and displacement distributions.Wang et al. [4] using the separation of variables, presented numerical results to show the uniformly heated hollow spheres' dynamic stress responses.They solved the problem by resolving the radial displacement into two functions.
One of these functions satisfies homogeneous mechanical boundary conditions while the other fulfills inhomogeneous boundary conditions.A method to obtain the temperature distribution is not presented in their work.Kiani and Eslami [5] solved the thermally nonlinear thermoelasticity problem of an isotropic homogeneous thick sphere subjected to a heat flux.They employed the Lord-Shulman theory of thermoelasticity and solved the problem using the generalized differential quadrature method and Newmark time marching scheme.Bagri and Eslami [6] studied the dynamic response of an isotropic annular disk using Lord and Shulman's theory of thermoelasticity.The problem is solved using the Laplace transform and Finite Element Method.They also investigated the effects of relaxation time and thermoelastic coupling coefficient.Javani et al. [7] proposed a unified formulation, that includes Lord-Shulman, Green-Lindsay, and Green-Naghdi theories, to investigate the thermo-visco-elastic response of a hollow sphere under thermal shock.They solved the problem using Newmark time marching and Finite Element methods.In the context of FGM and orthotropic materials, some new studies in thermoelasticity are recently published [8][9][10][11][12][13].
Alavi et al. [14] studied the thermo-elastic behavior of thick functionally graded hollow spheres under combined thermal and mechanical loads.Mechanical and thermal properties of FGM sphere are assumed to be functions of radial position.Lee [15] presented quasi-static thermoelastic for multi-layered spheres.The governing equation's general solutions are obtained in the transform domain using the Laplace transform.The solution is obtained using the matrix similarity transformation and inverse Laplace transform.Stampouloglou et al. [16] studied the axisymmetric thermoelastic problem for a radially nonhomogeneous equal thickness spherical shell.They solved the problem under a radially varying temperature field, with axisymmetric geometry and loading.Bagri and Eslami [17] proposed a unified formulation to investigate both cylinders and spheres made of an anisotropic heterogeneous material.The proposed formulation covers generalized theories of coupled thermoelasticity based on the Lord-Shulman, Green-Lindsay, and Green-Naghdi models.They solved the problem in a sphere and cylinder subjected to thermal shock using Laplace transform and numerically calculated the inversions.Sharifi [18] proposed a unified formulation to investigate the thermal shock problem in an orthotropic cylinder.The formulation is based on the Green-Lindsay and Lord-Shulman theories of thermoelasticity.They solved the problem using the finite Hankel transform.
Despite extensive investigations, to the best of the authors' knowledge, the problem of generalized coupled thermoelasticity in the orthotropic sphere remains unaddressed in the existing literature.This research paper aims to fill this gap by providing an analytical solution for the aforementioned problem.This study presents a unified formulation for the thermoelasticity problem based on the Classical, Lord-Shulman, and Green-Lindsay's theories of thermoelasticity for an orthotropic hollow sphere.The thermal boundary conditions entail prescribing a heat flux on the inner surface and a constant temperature on the outer surface of the sphere.Meanwhile, the mechanical boundary conditions involve constant displacement for the inner boundary while traction is prescribed on the outer boundary.The finite Hankel transform is utilized to obtain the solution for the displacement and temperature fields.As a case study, we examine an orthotropic sphere exposed to a constant heat flux on its inner surface while maintaining a zero temperature on the outer surface.The numerical outcomes of this scenario are then depicted in the form of graphical figures, illustrating the propagation and reflection of temperature and stress waves.To validate the accuracy of the obtained solution, the generalized coupled thermoelasticity problem is reduced to a special case, and a comparison is made with results obtained by other researchers, demonstrating excellent agreement.Furthermore, the effects of orthotropic material and different relaxation times on the temperature distribution, displacement, and stresses are investigated and depicted through graphical representations.

Formulation
A hollow orthotropic sphere of the inner and the outer radii  and , respectively, is considered in undisturbed condition and at the reference temperature  0 .We use a spherical coordinate (, , ) with the sphere's center as the origin.Since the sphere is subjected to spherically symmetric boundary conditions, the displacement  = [(, ), 0, 0] and temperature  are assumed to be functions of radius () and time () only [19].Thus, the relations between stress and strain components are [10,20] and: The unified heat conduction equation for a general anisotropic homogeneous solid is [21]: where   are the heat conduction coefficients,  is the specific heat,  is the internal heat generation, and i  (strain rate) reflects the thermomechanical coupling.In cases with spherically symmetric thermal boundary conditions and no internal heat generation, the temperature distribution becomes independent of θ and φ [22]; thus, Eq. ( 7) is reduced to: It can be seen that, when  1 =  2 =  3 = 0 the Eqs.( 4) and ( 8) reduce to classical dynamic theory, when  1 = 0 and  2 =  3 , Lord-Shulman theory can be deduced, and setting  3 = 0 leads to Green-Lindsay theory of thermoelasticity.
The non-vanishing stress components, namely   ,   and   are related to the displacement components and the temperature as: In terms of mechanical boundary conditions, the inner surface of the sphere experiences a constant displacement, while traction is prescribed on the outer surface: where As is seen, the mechanical boundary conditions are considered to be time-dependent and in a general form.For the initial conditions of the radial displacement field, we have; where  3 () and  4 () are defined functions of the radial position.For the energy equation, temperature is prescribed on the outer surfaces of the sphere, and the inner surface is subjected to a heat flux.Thus, we assumed the boundary and initial conditions in the following form: where  1 () and  2 () are defined time-dependent functions and  3 () and  4 () are defined functions of the radial position.

The Method of Solution
For the sake of analysis, using the following dimensionless parameters, the proceeding equations may be changed into the dimensionless form [17]: also, introducing: Dropping the hate sign for convenience, equations ( 4) and ( 8) could be rewritten as: in which: The coefficient  quantifies the impact of accounting for orthotropic material behavior, while  represents the thermomechanical coupling coefficient.It's worth noting that when material properties are equal in different directions,  and  2 equals 2 and 2.25 respectively, causing the equations for the orthotropic sphere to simplify to those of an isotropic material [17].For the mechanical boundary and initial conditions, we have: where: Similarly, the thermal boundary and initial conditions take the following form: where: To solve the coupled thermoelasticity equations (Eqs.( 18) and ( 19)), (, ) and (, ) is resolved into two components [23]: Then, the boundary value problem related to the displacement equation is resolved into the following two boundary value problems.In which, boundary conditions are allocated to the first homogeneous part of the differential equation, while initial conditions pertain to the second inhomogeneous part: and: The Energy equation is treated similarly: and: Eqs. ( 29) and ( 31) are Bessel-type equations and could be solved using the finite Hankel transform [24]: The inverse transformations are defined as [25]: where: in which: Applying the finite Hankel transform to Eqs. (29a) and (31a), yields: Eqs. ( 45) and ( 46) are a non-homogeneous ordinary differential equation, which can be solved in the following manner: where ∆ = √4 2   2 − 1.Using the inversion relations (Eqs.(39) and (40)) we have: Up to now, we have successfully solved the first set of equations.To address the second set of equations, we adopt the following forms for  2 (, ) and  2 (, ) [23]: where  is the Kronecker delta and: Multiplying Eq. ( 53) by  1 (,   ), and Eq.(54) by  2 (,   ), integrating over the interval from  to , and subsequently applying the orthogonality relation, we obtain: in which: The appropriate form for the initial conditions can be derived by substituting Eqs.(30d) and (30e) into (51): ̇2(, 0) =  (0). 1 (,   ) =  4 * () (64) Using the orthogonality relation, Eqs. ( 63) and (64) lead to: The initial conditions for the energy equation can be obtained similarly: It's evident that Eqs. ( 59) and ( 60) are coupled and can be uncoupled through some mathematical manipulations [13].Differentiating Eqs. ( 59) and ( 60) with respect to time results in: 2  (4) +  ⃛ +   2  ̈=  2 [ ⃛ +    ̅ ⃛ 1 +  3 ( (4) +    ̅ 1 )] Now substituting  ̈ from Eq. (60) in Eq. (71) leads to: Now by substituting  from Eq. (59) and  ̇ from Eq. (69) into Eq.(73) we have; It can be observed that equation ( 74) is independent of .Now similarly, substituting  (4) from Eq. (71) into Eq.(72) yields: Then substituting  ⃛ from Eq. (69) into Eq.(75) yields: +    ̅ ⃛ 1 + Solving Eqs. ( 78) and (79) gives () and () and as a result  2 (, ) and  2 (, ).
The solutions of these equations are dependent on the initial conditions, therefore, () and () are presented in the case study section.Now, both parts of (, ) and (, ) are obtained and using Eqs.(17), closed-form relations for (, ) and (, ) are as follows: The methodology employed in this study holds potential for resolving problems featuring various kinds of thermal-mechanical boundary conditions by selecting the suitable kernel of the transformation.

Case study
For numerical computations of the magneto-thermoelasticity problem of an orthotropic sphere under external loads, the following material properties have been considered for the calculations: For thermal boundary conditions, a thermal shock in the form of heat flux is prescribed on the inner surface of the sphere and the mechanical boundary conditions include a constrained inner surface, and the outer surface is considered to be traction-free: (, ) +   (, ) = 0 (83b) (, ) = 0 (84b) The mechanical and thermal initial conditions are; (, 0) = 0 (85a) Therefore, we have: using Eqs.( 25) and (46) lead to: Also, Considering Eqs. ( 23) and (45) lead to: And, as a result: Then, substituting Eqs. ( 90) and (92) into Eqs.( 49) and (50) give: where; ) ) Substituting Eq. ( 97) into (98) leads to: in which,   's are the roots of the following equation: and   's can be obtained using Eqs.( 87) and ( 88); where To validate the obtained solution for the thermoelasticity problem in an orthotropic sphere, we examine the thermoelasticity problem of an isotropic sphere experiencing a sudden and uniform temperature increase  0 across the entire sphere.While an orthotropic sphere exhibit varying mechanical and thermal properties along three orthogonal axes, when these properties are equivalent in all directions, the orthotropic sphere effectively simplifies into an isotropic sphere.In essence, the isotropic sphere serves as a particular case of the more general orthotropic sphere, characterized by uniform material properties in all directions.Figs 1 and 2 exhibit the history of radial and tangential stresses, respectively.It is observed that when the orthotropic sphere possesses identical properties in different directions and the relaxation times are disregarded, the obtained results align with those presented by Hata [2] for the isotropic sphere.In Hata's work, nondimensional time and nondimensional stress components are defined as follows: To explore the thermoelastic responses of the orthotropic sphere to a thermal shock, we've conducted a comparative analysis involving three theories: Classical theory, Lord-Shulman, and Green-Lindsay.Additionally, an abrupt jump occurs in the initial moments in the Green-Lindsay theory when the wavefront reaches a radial position.This is a consequence of the presence of a temperature gradient in the stress components (Eqs.9).While these jumps completely disappear after  = 4, because the temperature gradient diminishes across the sphere with the rise in medium's temperature.It can be inferred from the results shown in while some differences are detected, there are only minor differences for the tangential and circumferential stress histories.mid-radius of the sphere according to Green-Lindsay theory.The results demonstrate that with an increase in relaxation time, the peak temperature value escalates, however, it happens at later points in times.The velocity of propagated temperature wave can be determined utilizing the following equation: The analysis demonstrates that an increase in relaxation time leads to a lower velocity for the temperature wave and a decrease in the gradient of temperature with respect to time.This results in the temperature wave taking more time to reach its peak position.It is observed that an increase in the orthotropic coefficient leads to higher magnitudes of radial and tangential stresses.Incorporating this parameter during the design stage can be advantageous in controlling stress levels by selecting materials with specific mechanical properties.

Conclusion
This study deals with the generalized coupled thermoelasticity problem in an orthotropic sphere.A unified formulation based on the Classical, Lord-Shulman, and Green-Lindsay theories of thermoelasticity is presented for the orthotropic sphere.The problem is onedimensional and solved analytically using finite Hankel transform.The inner boundary is constrained, while the outer boundary is traction-free.For the thermal boundary conditions, the inner boundary experiences a thermal shock in the form of heat flux, while the outer boundary is subjected to a constant temperature.
Closed-form relations are presented for the displacement and temperature distributions due to using an analytical method to solve the problem.Distributions of temperature, displacement, radial, and hoop stresses at several times and along the radius of the sphere are obtained and shown in the figures for both Lord-Shulman, and the Green-Lindsay theories.
Also, a comparison between all the theories is conducted and has been presented in graphically.
From these graphs, it is possible to calculate the speed of propagation of the elastic and thermal waves.
In addition, the effects of different relaxation times and material properties on the stress waves and temperature are shown in the figures for the Green-Lindsay theory.Observations from the figures indicate that as the relaxation time increases, the peak values of the graphs for temperature, displacement, and stresses also increase, but these peaks occur at later time points.
This behavior is due to the fact that with a longer relaxation time, a thermal wave exhibits greater inertia, causing thermal disturbances to propagate at lower speeds, which in turn results in greater energy absorption within the structure.Furthermore, it is seen in the figures that increases in the orthotropic coefficient, leads to higher magnitude in the displacement, as well as radial and hoop stresses.
To validate the obtained results of this solution, the generalized coupled thermoelasticity problem is reduced to a special case, and the outcomes are compared with the findings of Hata [2] which shows excellent agreement.The method employed in this research applies to a wide range of problems in thermoelasticity.The results presented in this study have practical applications for researchers in material science, as well as designers of new materials in various domains, including mechanical engineering, acoustics, geophysics, and optics.

Fig. 1 . 5 Fig. 2 .
Fig. 1.Comparison of radial stress with Hata [2] in the case of b/a = 5 Fig 3 illustrates the history of nondimensional temperature at the midradius of the orthotropic sphere for all aforementioned theories.It's evident that in the Lord-Shulman and Green-Lindsay theories, an abrupt change occurs in temperature at the initial moments due to their incorporation of a finite speed for the thermal wave.Moreover, because both Lord-Shulman and Green-Lindsay models assume the same value for relaxation times, the histories of nondimensional temperature are coincident for both theories.

Fig. 3 .
Fig. 3. History of nondimensional temperature at mid-radius for different theories.

Fig 4
Fig 4 depicts the history of nondimensional displacement at mid-radius of the orthotropic sphere for all of the theories.Remarkably, the Green-Lindsay theory foresees higher peak

Figs 5 -
Figs 5-7 display the history of nondimensional radial, tangential, and circumferential stresses at the mid-radius of the sphere for various theories, respectively.It can be seen, in classical theory, stress starts to decrease right from the outset due to the assumption of an infinite velocity for the temperature wave.Conversely, in generalized theories, the temperature wave takes longer to propagate to radial positions, causing a delayed reduction in stress.

Fig. 5 .
Fig. 5. History of nondimensional radial stress at mid-radius for different theories.

Fig. 6 .
Fig. 6.History of nondimensional tangential stress at mid-radius for different theories.

Fig. 7 .
Fig. 7. History of nondimensional circumferential stress at mid-radius for different theories.

Fig 8
Fig 8 exhibits the impact of relaxation time on the history of nondimensional temperature at

Fig. 8 .
Fig. 8. History of nondimensional temperature at mid-radius for Green-Lindsay theory and different values of relaxation time.

Figs 9 -
Figs 9-11 show the effect of the different values of relaxation time on the nondimensional displacement, radial stress, and tangential stress components at mid-radius of the sphere based on Green-Lindsay theory.As depicted in Fig 9, with higher relaxation times, the peak value of nondimensional displacement increases and occurs at a later point in time.This phenomenon arises from the heightened energy absorption associated with an increase in relaxation time, leading to larger temperature and displacement peaks.

Figs 10 and 11
Figs 10 and 11 clearly illustrate that because the thermal load is applied to the inner boundary of the sphere, the initial stress peak is compressive and the first peak of stress is negative, resulting from the temperature rise.As observed in Fig 10, as the relaxation time increases from 0.85 to 1.73 (resulting in a decrease in the velocity of the thermal wave from 1.085 to 0.76), the amplitude of the abrupt jump escalates.Conversely, for the  2 = 3.94 the abrupt jump completely vanishes.It's important to note that with  2 = 3.94, the velocity of the

Fig. 9 .
Fig. 9. History of nondimensional radial displacement at mid-radius of the sphere for Green-Lindsay theory for different values of relaxation time.

Fig. 10 .
Fig. 10.History of nondimensional radial stress at mid-radius of the sphere for Green-Lindsay theory for different values of relaxation time.

Fig. 11 .
Fig. 11.History of nondimensional tangential stress at mid-radius of the sphere for Green-Lindsay theory for different values of relaxation time.

Figs 12 -
Figs 12-14 depict the history of nondimensional radial displacement, radial stress, and tangential stress at different radial positions of the sphere for Green-Lindsay theory for  1 =  2 = 0.85.In the prescribed thermal boundary conditions, the spherical dilatation wave propagates outward from the inner boundary.The influence of the mechanical boundary conditions on the propagation and the reflection of the stress wave can be seen in Figs 13 and 14.The thermoelastic wave begins in a compressive mode as it initiates propagation from the inner boundary.However, upon returning from the outer boundary, it shifts into a tensile mode.This behavior arises from the application of traction-free boundary conditions on the outer surface of the orthotropic sphere, whereas the displacement-type inner boundary

Fig. 12 .
Fig. 12. History of nondimensional radial displacement at different radial positions for Green-Lindsay theory.

Fig. 13 .
Fig. 13.History of nondimensional radial stress at different radial positions for Green-Lindsay theory.

Fig. 14 .
Fig. 14.History of nondimensional tangential stress at different radial positions for Green-Lindsay theory.Fig 15 shows the distribution of temperature across the thickness of the orthotropic sphere for Green-Lindsay theory.It is evident that each point's temperature gradually rises over time until reaches the steady-state condition.Unlike the parabolic form observed in classical theory, generalized theories adopt a hyperbolic energy equation, which causes the formation of the temperature wave.The wavefront of the temperature wave is depicted in Fig 15 for Time = 0.25, Time = 0.5, and Time = 0.75.Specifically, the temperature wave corresponding to r = 1 and Time = 0.25 originates at about 0.2 and diminishes to zero at about r = 1.28.Additionally, it is evident that at Time = 1, the temperature wave approximately reaches the outer boundary, aligning with the temperature wave propagation dimensionless velocity of 1.085 derived from Eq. (108).

Fig. 15 .
Fig. 15.Through-thickness variation of nondimensional temperature for Green-Lindsay theory at different times.

Figs 16 -
Figs 16-18 show the through-thickness variation of the nondimensional radial displacement, radial stress, and tangential stress at different times for Green-Lindsay theory for  1 =  2 = 0.85.As depicted in Fig 16, the fixed inner boundary condition causes the sphere to expand outward.The effect of the boundary conditions on the propagation of elastic waves is also evident in Figs 17 and 18.These figures reveal that the radial and tangential stresses are compressive at intervals smaller than Time = 0.92 and become tensile after this point.Indeed, this change indicates the onset of wave reflection, signifying that the elastic wave is now propagating back into the medium but in the opposite direction.The abrupt jumps in the stress waves can also be seen in these figures.As Fig8 demonstrates, during the initial moments of the thermal shock application, the gradient of the temperature with respect to the time is high.This leads to the higher amplitude in the abrupt jump in elastic waves, as is seen in Figs17 and 18.As time progresses and the system approaches the steady state in temperature distribution, the amplitude of the abrupt jump gradually decreases.

Fig. 16 .
Fig. 16.Through-thickness variation of nondimensional radial displacement for Green-Lindsay theory at different times.

Fig. 17 .
Fig. 17.Through-thickness variation of nondimensional radial stress for Green-Lindsay theory at different times.

Fig. 18 .
Fig. 18.Through-thickness variation of nondimensional tangential stress for Green-Lindsay theory at different times.

Figs 19 -
Figs 19-21 illustrate the history of nondimensional radial displacement, radial stress, and tangential stress at different radial positions of the sphere for Lord-Shulman theory at  2 =  3 = 0.85.Similar to Green-Lindsay theory, the spherical dilatation wave propagates outward from the inner boundary and traction-free boundary conditions on the outer surface change the mode of the thermoelastic wave, while the displacement-type inner boundary reflects the wave in the same mode that it encounters.

Fig. 21 .
Fig. 21.History of nondimensional tangential stress at different radial positions for Lord-Shulman theory.Figs 22-24 depict the through-thickness variation of the nondimensional radial displacement, radial stress, and tangential stress at different times for Lord-Shulman theory at  2 =  3 = 0.85.The onset of wave reflection can also be seen in these figures.In Fig 22, Time = 0.25, Time = 0.5, and Time = 0.75 show the displacement wave propagation, while Time = 1, Time = 1.25, and Time = 1.5 indicate the wave reflection from the outer radius of the sphere.Also, Figs 23 and 24 illustrate the presence of stress wavefronts at various time instances.Notably, at Time = 0.25, the stress wavefront is located at 1.2 and progressively advances through the medium as time passes.During the initial stages of the thermal shock application the gradient of the elastic wavefront with respect to time is larger, and gradually diminishes over time, while the magnitude of the wavefront increases with the progression of time.

Fig. 22 .
Fig. 22. Through-thickness variation of nondimensional radial displacement for Lord-Shulman theory at different times.

Fig. 23 .
Fig. 23.Through-thickness variation of nondimensional radial stress for Lord-Shulman theory at different times.

Fig. 24 .
Fig. 24.Through-thickness variation of nondimensional tangential stress for Lord-Shulman theory at different times.

Figs 25 -
Figs 25-27 show how the nondimensional displacement, radial stress, and tangential stress components are affected when orthotropic material is taken into account based on the Green-Lindsay theory.For the material that is used in this section D = 2.14 and for isotropic materials D = 2.The effect of the orthotropicity is shown by plotting two extreme cases of D.

Fig. 25 .
Fig. 25.Effect of orthotropicity on the history of nondimensional radial displacement for Green-Lindsay theory.