Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3715 
 

https://internationalpubls.com 

Effect of Linearly Distributed Sources in Normal Loading of a Dual-

Phase-Lag Thermoelastic Diffusion with Non-Local Formulation  

Rajneesh Kumar1, Pawan Kumari2, Kavita Rani3, Seetu Rana4 

1Department of Mathematics, Kurukshetra University, Kurukshetra 136119, India 

email: rajneeshkuk@rediffmail.com 
2 Associate Professor, Department of Mathematics, Govt. College, Hisar, Haryana,125001 India 

email:  pawankumarigawarya@gmail.com 
3Associate Professor, Department of Mathematics, CMG GCW, Bhodia Khera, Fatehabad, Haryana,125050, India 

email: karya4@gmail.com 
4 Assistant Professor, Department of Mathematics, Govt. College, Hisar, Haryana,125001 India 

email: rana.seetu@gmail.com 

 

Article History: 

Received:  04-10-2025 

Revised:  23-11-2025 

Accepted: 14-12-2025 

 

Abstract: 

The present study aims at exploring the effects of non-locality and dual-phase-lags 

in a thermoelastic half-space subjected to normal loading. The governing equations 

are solved using harmonic vibration analysis coupled with Fourier transform 

techniques. Linearly distributed sources are considered to demonstrate practical 

relevance for mechanical excitations. Numerical simulations are performed to 

evaluate displacements, stresses, temperature variations and chemical potential, 

with results presented graphically. The study is applicable to predicting the 

mechanical responses in electronics devices, aerospace components, and energy 

systems subjected to time-harmonic loads. 

Keywords: Thermoelasticity, Non-local theory, Dual-Phase-Lag, Fourier 

transform, Time Harmonic Sources. 

 

1. Introduction 

In non-local elasticity, the stress at a point is affected by both the local strain and the strain 

distribution throughout the body. This is different from classical (local) elasticity, which says that 

the stress at a point is only affected by the strain at that point. Researchers like Edelen and Laws 

[1], Eringen and Edelen [2], and others built on these ideas to find the basics of non-local elasticity. 

Later, Eringen [3-8], McCay and Narsimhan [9], and Narsimhan and McCay [10] added to these 

ideas. Eringen's monograph [11] goes into a lot of detail about the subject. 

Non-local response and lagging response are related ideas. Non-local response is about effects in 

space and lagging response is about delays in time. The results were compared with the models of 

Cao and Guo [13] and Guo and Hou [14] by Tzou [12]. He used the non-local formulation and 

single-phase-lag heat conduction. Later, Tzou and Gao [15] built on this idea by adding non-local 

elasticity to the dual-phase-lag model that Tzou [16,17] had first suggested. This created a single 

mailto:pawankumarigawarya@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3716 
 

https://internationalpubls.com 

framework that included both effects. At the same time, Sherief et al. [18] created a thermoelastic 

diffusion theory with a single relaxation time, and Sherief and Saleh [19] looked at the half-space 

problem in this context. 

Later additions to these theories made them more general. In his work [20,21], Sharma looked into 

boundary-value problems in generalized thermodiffusive elastic media and wave reflection in 

thermodiffusive elastic half-spaces with holes. Sharma et al. [22] looked at how viscosity affects 

wave propagation in Green–Naghdi type-II and type-III thermoelastic solids that are not all the 

same. Sharma and Marin [23] studied the problem of reflection and transmission of plane waves 

from imperfect boundary between two heat conducting micropolar thermoelastic solids.  

Abouelregal and Zenkour [24] looked at phase-lag effects in functionally graded thermoelastic 

microbeams that were heated in a ramp-type way. Sharma and Sharma [25] looked at how heat 

sources and relaxation time affect the way temperatures are distributed in living things. Marin et 

al. [26] looked at the Saint-Venant principle in micropolar thermoelastic diffusion when it is 

relaxed. Yu et al. [27] suggested a size-dependent thermoelastic model for more complex 

materials. This model takes into account size effects in both heat conduction and elasticity. It does 

this by using generalized free energy and extended irreversible thermodynamics. 

There are also studies by Kumar and Abbas [28], who looked into how thermomechanical sources 

can cause disturbances in poro-thermoelastic media, and Kumar et al. [29], who looked into how 

deformation happens in a modified couple stress thermoelastic rotating medium when Hall current 

and magnetic field influences happen, using the Lord-Shulman and Green-Lindsay theories with a 

ramp-type thermal source. 

A study by Kumar et al. [30] looked at a non-local microstretch thermoelastic thick circular plate 

that was subject to phase lags. Kumar et al. [31] looked at how thermal and chemical potential 

sources affected a thin beam in a modified couple stress thermoelasticity (MCT) model with a 

three-phase-lag diffusion model in a different study.  

Marin et al. [32] used the Moore-Gibson-Thompson (MGT) heat equation to come up with a basic 

solution and Green's function for a semi-infinite orthotropic photo-thermoelastic medium whose 

properties change with temperature. They used the operator theory and the superposition principle 

to come up with a general solution using harmonic functions. This solution was then used to find 

specific solutions for sources of steady point heat at the medium's surface and inside. 

Sharma et al. [33] looked into how the Moore-Gibson-Thompson equation controls 

thermomechanical deformation in a micropolar thermoviscoelastic solid. They found that the two-

temperature effects were not local and were hyperbolic. In their study [34], Abouelregal et al. 

created a thermoviscoelastic model by combining non-local elasticity with the Kelvin-Voigt 

viscoelastic framework and a Klein–Gordon-type non-local elasticity formulation. They then 

looked at a one-dimensional half-space problem that was affected by an instantaneous in-line heat 

source. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3717 
 

https://internationalpubls.com 

We look at how non-locality and phase lags affect thermoelastic diffusion when sources are spread 

out in a time-harmonic way in this study. The integral transform method is used to write the 

problem which takes into account different kinds of excitations such as normal load, thermal 

sources and chemical potential sources.  

2. Basic Equations   

 Based on the work by Tzou & Gao [15], Sherief et al. [18], and Yu et al. [27], the equations that 

describe thermoelastic diffusion with dual-phase-lag and non-local effects are : 

(i) Constitutive Relations 

𝑡𝑖𝑗 = 2µ𝑒𝑖𝑗 + 𝛿𝑖𝑗 [𝜆𝑜𝑒𝑘𝑘 - 𝛾1T - 𝛾2P]         (1)                                                                                                                    

(ii) Equation of motion 

 (𝜆𝑜+ µ) ∇ (∇.u) + µ∆u - 𝛾1∇T - 𝛾2∇P = 𝜌(1- 𝜉2∆) 
𝜕²𝒖

𝜕𝑡²
       (2) 

(iii) Equation of heat conduction 

(1 −ζ2∆ + 𝜏𝑞
𝜕

𝜕𝑡
 + 

1

2
𝜏𝑞
2  

𝜕²

𝜕𝑡²
) (𝛾1𝑇𝑜�̇� + 𝑙1𝑇𝑜�̇� + 𝑇𝑜�̇� d) = K(1+ 𝜏𝑡

𝜕

𝜕𝑡
 ) ∆T    (3) 

(iv) Equation of mass diffusion 

 (1 - ς2∆ + 𝜏𝑢
𝜕

𝜕𝑡
 +

1

2
𝜏𝑢
2  

𝜕²

𝜕𝑡²
) (𝛾2�̇� + 𝑑�̇� + 𝑛�̇� ) = D(1+𝜏𝑝

𝜕

𝜕𝑡
 )∆P    (4) 

Where: 

𝜆𝑜= 𝜆 - 𝛽2
2 /b,     𝛾1 = 𝛽1+

𝑎

𝑏
 𝛽2,      𝛾2 = 

𝛽2

𝑏
 ,     𝑙1 = 𝜌 𝐶𝑒/𝑇𝑂 + a2/b,     d = 

𝑎

𝑏
 ,      n = 

1

𝑏
            (5) 

where u is the displacement vector, T is the temperature, P is the chemical potential, 𝑒𝑖𝑗 is the  

strain tensor, 𝑡𝑖𝑗 is the stress tensor, and 𝜇 ,  𝜆0 , 𝛾1, , 𝛾2,  𝑙1 , d, n, K, D, 𝜏𝑞, 𝜏𝑡, 𝜏𝑢, 𝜏𝑝,  ξ, ζ, ς 

are material and model parameters defined as per the cited references. 

𝜉, 𝜁, 𝜍 -are non-local parameters in equations (1)-(4). 𝜏𝑞 & 𝜏𝑡 are the thermal relaxation times with 

𝜏𝑞, 𝜏𝑡 ≥ 0 and  𝜏𝑢 & 𝜏𝑝 are the diffusion relaxation times with 𝜏𝑢, 𝜏𝑝 ≥ 0. 𝛽1 = (3λ+2µ)𝛼𝑡 ,  𝛽2 = 

(3λ+2µ)𝛼𝑐 . In this case,  𝛼𝑡  , 𝛼𝑐  are the coefficient of linear thermal expansion and diffusion 

expansion respectively. ∆ is the Laplacian operator, ∇ is nabla operator which is also known as the 

Laplacian operator. Other signs mean what they normally do.                

3. Problem Description 

We look at a half-space that is uniform, isotropic, non-local, thermoelastic, and diffusive. This 

half-space has dual-phase-lag effects and is in the region 𝑥3 ≥ 0. A rectangular set of Cartesian 

coordinate system is chosen with (𝑥1, 𝑥2, 𝑥3) as its points of reference. The origin is placed on the 

boundary plane with  𝑥3 = 0. The study only looks at deformations in a plane, and all the field 

variables are functions of (𝑥1, 𝑥3, 𝑡). The half-space is loaded mechanically with a normal force, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3718 
 

https://internationalpubls.com 

heated up, and given a chemical potential source at the boundary plane 𝑥3 = 0. We assume that the 

variations are limited to the  𝑥1 - 𝑥3  plane for two-dimensional modeling. 

𝑢1(𝑥1, 𝑥3, 𝑡) and 𝑢3(𝑥1, 𝑥3, 𝑡) can be written in terms of 𝜑(𝑥1, 𝑥3, 𝑡) and ψ (𝑥1, 𝑥3, 𝑡) in a form that 

doesn't depend on the dimensions. 

For two dimensional problems, we take:  

u = (𝑢1(𝑥1, 𝑥3, 𝑡), 0, 𝑢3(𝑥1, 𝑥3, 𝑡)), T (𝑥1, 𝑥3, 𝑡), P (𝑥1, 𝑥3, 𝑡)                                                         (6) 

The following dimensionless quantities are introduced for normalization: 

ξˈ= 
𝜔∗

𝑐₁
ξ,      ζˈ= 

𝜔∗

𝑐₁
ζ,      ςˈ= 

𝜔∗

𝑐₁
ς,     𝑥𝑖ˈ=  

𝜔∗

𝑐₁
𝑥𝑖,     𝑢𝑖ˈ=  

𝜔∗

𝑐₁
 𝑢𝑖,      tˈ= ω*t,     𝜏𝑡ˈ= ω*𝜏𝑡,             

𝜏𝑞ˈ=ω*𝜏𝑞,     𝜏𝑢ˈ = ω*𝜏𝑢,     𝜏𝑝ˈ = ω*𝜏𝑝,      𝑡𝑖𝑗ˈ = 
1

𝛾₁𝑇₀
𝑡𝑖𝑗,       𝑚𝑖𝑗ˈ = 

1

𝛾₁𝑇₀

𝜔∗

𝑐₁
𝑚𝑖𝑗 ,                      (7)         

Tˈ=
𝛾₁

𝜌𝑐₁²
T,         Pˈ=

1

𝑏𝛾₂
P      

where:  

 ω* = 
𝜌𝐶𝑒𝑐1

2

𝐾
,    𝑐1

2  = 
𝜆𝑜+2µ

𝜌
 

 Here ω* denotes the characteristic frequency and 𝑐1  is the longitudinal wave velocity in the 

medium. 

Using equations (2)-(4) along with (6)- (7) and suppressing the primes, the system reduces to the 

following dimensionless form 

(𝜆𝑜+ µ)

𝜌𝑐₁²
 
𝜕𝑒

𝜕𝑥1
 + 

µ

𝜌𝑐₁²
 ∆𝑢1 -  

∂T

𝜕𝑥1
 - 

𝑏𝛾₂²

𝜌𝑐₁²

∂P

∂𝑥1
 = (1- 𝜉2∆)

𝜕²𝑢₁

𝜕𝑡²
 ,                                                                    (8) 

(𝜆𝑜+ µ)

𝜌𝑐₁²
 
𝜕𝑒

𝜕𝑥3
 + 

µ

𝜌𝑐₁²
 ∆𝑢3 -  

∂T

𝜕𝑥3
 - 

𝑏𝛾₂²

𝜌𝑐₁²

∂P

∂𝑥3
 = (1- 𝜉2∆)

𝜕²𝑢₃

𝜕𝑡²
 ,                                                                    (9) 

(1- ζ2∆ + 𝜏𝑞
𝜕

𝜕𝑡
 + 

1

2
𝜏𝑞
2  

𝜕²

𝜕𝑡²
) (

𝛾₁²

𝐾𝜌𝜔∗
𝑇𝑜�̇� + 

𝑙₁𝑐₁²

𝐾𝜔∗
𝑇𝑜�̇� + 

𝑏𝛾₂𝛾₁

𝐾𝜌𝜔∗
𝑇𝑜�̇� d) = (1+𝜏𝑡

𝜕

𝜕𝑡
 ) ∆T,                          (10) 

(1- ς2∆ + 𝜏𝑢
𝜕

𝜕𝑡
 + 

1

2
𝜏𝑢
2 𝜕²

𝜕𝑡²
 ) (

𝑐₁²

𝐷𝑏𝜔∗
�̇� + 

𝑐₁⁴𝑑𝜌

𝐷𝑏𝜔∗𝛾₁𝛾₂
�̇� + 

𝑛𝑐₁²

𝐷𝜔∗
�̇� ) = (1+𝜏𝑝

𝜕

𝜕𝑡
 ) ∆P                                  (11) 

Where: 

 ∆ =  
𝜕²

𝜕𝑥₁²
 + 

𝜕²

𝜕𝑥₃²
  ,   e = 

𝜕𝑢₁

𝜕𝑥₁
 + 

𝜕𝑢₃

𝜕𝑥₃
 

4. Solution Procedure 

The displacement components 𝑢1(𝑥1, 𝑥3, 𝑡) and 𝑢3 (𝑥1, 𝑥3, 𝑡) can be expressed in terms of the 

scalar potentials 𝜑(𝑥1, 𝑥3, 𝑡) and ψ (𝑥1, 𝑥3, 𝑡) in dimensionless form as 

𝑢1 = 
𝜕𝜑

𝜕𝑥₁
 - 

𝜕𝜓

𝜕𝑥₃
,           𝑢3 = 

𝜕𝜑

𝜕𝑥₃
 + 

𝜕𝜓

𝜕𝑥₁
                                                                                           (12) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3719 
 

https://internationalpubls.com 

Substituting (12)  into equations (8)-(11) the governing system leads to the following set of coupled 

equations: 

(𝑏₁ +𝑏₂) ∆φ – T – 𝑏₃P - (1- 𝜉2∆)
𝜕²𝜑

𝜕𝑡²
  = 0, (13)        

𝑏2∆ψ - (1- 𝜉2∆)
𝜕²𝜓

𝜕𝑡²
  = 0, (14)                               

(1- ζ2∆ + 𝜏𝑞
𝜕

𝜕𝑡
 + 

1

2
𝜏𝑞
2  

𝜕²

𝜕𝑡²
) ( 𝑏₄∆�̇�+ 𝑏₅ �̇� + 𝑏₆�̇� ) = (1+𝜏𝑡

𝜕

𝜕𝑡
 ) ∆T,         (15) 

(1- ς2∆ + 𝜏𝑢
𝜕

𝜕𝑡
 + 

1

2
𝜏𝑢
2 𝜕²

𝜕𝑡²
 )(𝑏₇∆�̇�+ 𝑏₈�̇� + 𝑏₉�̇�) = (1+𝜏𝑝

𝜕

𝜕𝑡
) ∆P                                                     (16) 

where:  

𝑏₁ = 
(𝜆𝑜+ µ)

𝜌𝑐₁²
,    𝑏₂ = 

µ

𝜌𝑐₁²
      𝑏₃ = 

𝑏𝛾₂²

𝜌𝑐₁²
 ,    𝑏₄ = 

𝛾₁²

𝐾𝜌𝜔∗
𝑇𝑜 ,    𝑏₅ = 

𝑙₁𝑐₁²

𝐾𝜔∗
𝑇𝑜 ,                                             

𝑏₆ = 
𝑏𝛾₂𝛾₁

𝐾𝜌𝜔∗
𝑇𝑜𝑑,    𝑏₇ = 

𝑐₁²

𝐷𝑏𝜔∗
 ,     𝑏₈ = 

𝑐₁⁴𝑑𝜌

𝐷𝑏𝜔∗𝛾₁𝛾₂
 ,    𝑏₉ = 

𝑛𝑐₁²

𝐷𝜔∗
                                                           (17) 

We assume 

(�̅�, �̅�, �̅�, �̅�) = (φ, ψ, T, P)𝑒𝑖𝜔𝑡                                                                                                     (18) 

We define Fourier transform as: 

𝑓 (𝜉1, 𝑥3, 𝜔) =∫ 𝑓(̅𝑥1, 𝑥3, 𝜔
∞

−∞
)𝑒𝑖𝜉₁𝑥₁ d𝑥1                                                                                    (19) 

By substituting equations (18) and (19) into (13)-(16) and simplifying, the system reduces to 

(𝐾₁𝐷1
6 + 𝐾₂𝐷1

4 + 𝐾₃𝐷1
2 +𝐾₄)(𝜑 ̂, 𝑇,̂ �̂�) = 0                                                                                 (20) 

(𝐷1
2 - 𝑚4

2) �̂� = 0                                                                                                                           (21) 

 Where: 

𝐾₁ = 𝐾₁₁𝐾₁₃𝐾₁₅ - 𝐾₃₃𝐾₃₅𝜁2𝜍2𝐾₁₁ + 𝐾₃₁𝜁2𝐾₁₅ − 𝐾₃₃𝐾₃₄𝜁2𝜍2 − 𝐾₃₁𝐾₃₅𝜁2𝜍2𝑏₄ + 𝐾₃₄𝑏₄𝐾₁₃𝜍2, 

𝐾₂ = 𝐾₀₁ − 3𝜉₁²𝐾₁,                                                                                                                    

𝐾₃ = 3𝜉₁⁴𝐾₁ − 2𝜉₁²𝐾₀₁ + 𝐾₀₂ ,                                                                                                     

𝐾₄ = 𝐾₀₁𝜉₁⁴ − 𝜉₁⁶𝐾₁ − 𝜉₁²𝐾₀₂ − 𝐾₀₃ ,     

  𝑚4
2 = 𝜉1

2 −
𝜔2

𝑏2−𝜉
2𝜔2    

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                

𝐷1 =
𝑑

𝑑𝑥3
 

and 

𝐾01=−𝐾₁₁𝐾₂₂𝐾₁₅ − 𝐾₁₁𝐾₁₃𝐾₂₆ + 𝐾₁₃𝐾₁₅𝜔2 + 𝐾₃₃𝐾₁₁𝜁2𝐾₂₅ + 𝐾₃₅𝜍2𝐾₂₃𝐾₁₁ −

𝐾₃₃𝐾₃₅𝜁2𝜍2𝜔2 − 𝐾₃₁𝜁2𝐾₂₆ − 𝐾₂₁𝐾₁₅ + 𝐾₃₃𝜁2𝐾₂₄ + 𝐾₃₄𝜍2𝐾₂₃ + 𝐾₃₁𝜁2𝐾₂₄𝑏₃ +

𝐾₃₅𝜍2𝐾₂₁𝑏₃ − 𝐾₁₃𝐾₂₄𝑏₃ − 𝑏₃𝐾₃₄𝜍²𝐾₂₂,     



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3720 
 

https://internationalpubls.com 

𝐾₀₂ = 𝐾₁₁𝐾₂₂𝐾₂₆ − 𝐾₁₃𝐾₂₆𝜔2 − 𝐾₂₂𝐾₁₅𝜔² − 𝐾₁₁𝐾₂₃𝐾₂₅ + 𝐾₃₃𝜁²𝐾₂₅𝜔² + 𝐾₃₅𝜍²𝐾₂₃𝜔² +

𝐾₂₁𝐾₂₆ − 𝐾₂₃𝐾₂₄ − 𝑏₃𝐾₂₁𝐾₂₅ + 𝑏₃𝐾₂₂𝐾₂₄ ,   

𝐾₀₃ = 𝜔2𝐾₂₆𝐾₂₂ − 𝜔²𝐾₂₃𝐾₂₅ ,  

𝐾₁₁ = 𝑏₁ + 𝑏₂ − 𝜉2𝜔2,   𝐾₁₂ = 1 + 𝑖𝜔𝜏𝑞 −
𝜔²

2
𝜏𝑞
2,   𝐾₁₃ = 𝑏₅𝑖𝜔𝜁² + 1 + 𝑖𝜔𝜏𝑡,  𝐾₁₄ = 1 +

𝑖𝜔𝜏𝑢 −
𝜔²

2
𝜏𝑢
2,   𝐾₁₅ =1+𝑖𝜔𝜏𝑝 + 𝑖𝜔𝜁²𝑏₉ ,  𝐾₂₁ = 𝐾₃₁𝐾₁₂ ,  𝐾22= 𝐾₃₂𝐾₁₂ ,  𝐾₂₃ = 𝐾₃₃𝐾₁₂,   

𝐾₂₄ = 𝐾₃₄𝐾₁₄, 𝐾₂₅ = 𝐾₃₅𝐾₁₄ ,   𝐾₂₆ = 𝐾₃₆𝐾₁₄ ,   𝐾₃₁ = 𝑏₄𝑖𝜔 ,  𝐾₃₂ =𝑏₅𝑖𝜔, 𝐾₃₃ = 𝑏₆𝑖𝜔 ,  

𝐾₃₄ = 𝑏₇𝑖𝜔 , 𝐾₃₅ = 𝑏₈𝑖𝜔 ,  𝐾₃₆ = 𝑏₉𝑖𝜔 

To solve equations (20) and (21) with �̂�, ψ̂, T̂ and �̂�  disappear as 𝑥₃  approaches infinity, we 

 write  

(�̂�, T̂, P̂ )(𝑥₃, 𝜉1, 𝑠)  = ∑ (1, 𝑅𝑖
∗, 𝑆𝑖

∗)3
𝑖=1 𝐴𝑖𝑒

−𝑚ᵢ𝑥₃                                                                                  (22)                                                                            

�̂� (𝑥₃, 𝜉1, 𝑠) = 𝐴₄𝑒−𝑚₄𝑥₃                                                                                                             (23) 

The roots of the characteristic equations (20), (21) are mᵢ (i=1,2,3,4)   

and  𝐴𝑖(i=1,2,3,4) are the corresponding amplitude coefficients determined from boundary 

conditions and 𝑅𝑖
∗ and 𝑆𝑖

∗ are derived as follows: 

𝑅𝑖
∗ = 

(𝑚ᵢ2 − 𝜉₁²)³(𝐾₃₃𝐾₃₄𝜁²𝜍² − 𝐾₃₁𝜁²𝐾₁₅) + (𝑚ᵢ2 − 𝜉₁²)²(𝐾₃₁𝜁2𝐾₂₆ + 𝐾₂₁𝐾₁₅ 

−𝐾₃₃𝜁2𝐾₂₄ − 𝐾₃₅𝜍²𝐾₂₁) + (𝑚ᵢ2 − 𝜉₁²)(𝐾₂₁𝐾₂₆ − 𝐾₂₃𝐾₂₄)

(𝑚ᵢ2 − 𝜉₁²)² (𝐾₁₃𝐾₁₅ − 𝐾₃₃𝐾₃₅𝜁²𝜍²) + (𝑚ᵢ2 − 𝜉₁²)(−𝐾₂₂𝐾₁₅ − 𝐾₁₃𝐾₂₆ + 𝐾₃₃𝜁²𝐾₂₅

+𝐾₃₅𝜍²𝐾₂₃) + (𝐾₂₂𝐾₂₆ − 𝐾₂₃𝐾₂₅)

 

      

𝑆𝑖
∗ =

(𝑚ᵢ2 − 𝜉₁²)³(𝐾₃₁𝐾₃₅𝜁²𝜍² − 𝐾₃₄𝐾₁₃𝜍²) + (𝑚ᵢ2 − 𝜉₁²)²(𝐾₁₃𝐾₂₄ + 𝐾₃₄𝜍²𝐾₂₂ − 𝐾₃₁𝜁²𝐾₂₅

−𝐾₃₅𝜍²𝐾₂₁) + (𝑚ᵢ2 − 𝜉₁²)(𝐾₂₁𝐾₂₅ − 𝐾₂₂𝐾₂₄)

(𝑚ᵢ2 − 𝜉₁²)² (𝐾₁₃𝐾₁₅ − 𝐾₃₃𝐾₃₅𝜁²𝜍²) + (𝑚ᵢ2 − 𝜉₁²)(−𝐾₂₂𝐾₁₅ − 𝐾₁₃𝐾₂₆ + 𝐾₃₃𝜁²𝐾₂₅

+𝐾₃₅𝜍²𝐾₂₃) + (𝐾₂₂𝐾₂₆ − 𝐾₂₃𝐾₂₅)

 

                                                                                                                                             i=1,2,3      

5. Boundary Conditions   

At the point where  𝑥3 = 0, on the plane boundary, the half-space is excited from the outside by 

a normal mechanical load, a thermal input, and a chemical potential source. So, the boundary 

conditions at 𝑥3 = 0 are set up to show these three types of applied fields.  

𝑡33 = −𝐹1(𝑥1)𝑒
𝑖𝜔𝑡,    𝑡31 = 0,   𝑇 = 𝐹2(𝑥1)𝑒

𝑖𝜔𝑡, 𝑃 = 𝐹3(𝑥1)𝑒
𝑖𝜔𝑡                                           (24)          

The non dimensional stress components are expressed by  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3721 
 

https://internationalpubls.com 

𝑡33 = 
2𝜇

𝛾₁𝑇₀
(
𝜕𝑢₃

𝜕𝑥₃
) + 

𝜆₀

𝛾₁𝑇₀
(
𝜕𝑢₁

𝜕𝑥₃
+

𝜕𝑢₃

𝜕𝑥₁
) −  

𝜌𝑐₁²

𝛾₁𝑇₀
𝑇 −

𝛾₂²𝑏

𝛾₁𝑇₀
 P          (25) 

𝑡₃₁ =  
𝜇

𝛾₁𝑇₀
(
𝜕𝑢₁

𝜕𝑥₁
+

𝜕𝑢₃

𝜕𝑥₃
)              (26) 

Applying Fourier transform on (25) and (26),we get 

𝑡 33= −�̂�1 (𝜉₁)𝑒𝑖𝜔𝑡,  𝑡 31 = 0,   T̂ = �̂�2 (𝜉₁)𝑒𝑖𝜔𝑡,    P̂ = �̂�3(𝜉₁) 𝑒𝑖𝜔𝑡                                              (27) 

Displacement, stress, temperature variation, and chemical potential are determined by applying 

equations (22) and (23) to the boundary conditions (27), taking into account equations (12), (22) 

and (23), as follows: 

𝑢 ₁ = −
𝑖𝜉₁

∆
∑ ∆ᵢ𝑒−𝑚ᵢ𝑥₃3
𝑖=1  + 

∆₄𝑚₄𝑒
−𝑚₄𝑥₃

∆
                                                                                          (28) 

𝑢 ₃ = −
1

∆
∑ 𝑚3
𝑖=1 ᵢ∆ᵢ𝑒−𝑚ᵢ𝑥₃ − 

𝑖𝜉₁∆₄𝑒−𝑚₄𝑥₃

∆
                                                                                        (29) 

𝑡 33= 
1

∆
∑ 𝑏1𝑖
3
𝑖=1 ∆ᵢ𝑒−𝑚ᵢ𝑥₃ + 

∆₄𝑏14𝑒
−𝑚₄𝑥₃

∆
                                                                                          (30) 

𝑡 31 = 
1

∆
∑ 𝑏2𝑖
3
𝑖=1 ∆ᵢ𝑒−𝑚ᵢ𝑥₃ + 

∆₄𝑏24𝑒
−𝑚₄𝑥₃

∆
                                                                                         (31) 

T̂ = 
1

∆
∑ 𝑅𝑖

∗3
𝑖=1 ∆ᵢ𝑒−𝑚ᵢ𝑥₃                                                                                                                 (32) 

P̂ = 
1

∆
∑ 𝑆𝑖

∗3
𝑖=1 ∆ᵢ𝑒−𝑚ᵢ𝑥₃                                                                                                                 (33) 

Here   

∆= (𝑆2
∗𝑅1

∗ − 𝑆1
∗𝑅2

∗)𝑛1 + (𝑆1
∗𝑅3

∗ − 𝑆3
∗𝑅1

∗)𝑛2+(𝑆3
∗𝑅2

∗ − 𝑆2
∗𝑅3

∗)𝑛3, 

𝑛1 = 𝑏₁₃𝑏₂₄ − 𝑏₁₄𝑏₂₃,   

𝑛2 = 𝑏₁₂𝑏₂₄ − 𝑏₁₄𝑏₂₂,   

𝑛3 = 𝑏₁₁𝑏₂₄ − 𝑏₁₄𝑏₂₁,   

and  

𝑏1𝑖 = (2𝑟₁ + 𝑟₂)𝑚𝑖² − 𝜉₁²𝑟₂ − 𝑟₃𝑅𝑖
∗ − 𝑟₄𝑆𝑖

∗,   𝑏14 = 2𝑖𝜉₁𝑟₁𝑚4,   

𝑏2𝑖 = 2𝑖𝜉₁𝑟₁𝑚𝑖 ,   𝑏24 = −𝑟₁(𝑚4
2 + 𝜉₁²),   

𝑟₁ = 
𝜇

𝛾₁𝑇₀
 ,  𝑟₂ = 

𝜆₀

𝛾₁𝑇₀
 ,  𝑟₃ = 

𝜌𝑐₁²

𝛾₁𝑇₀
 , 𝑟₄ = 

𝛾₂²𝑏

𝛾₁𝑇₀
 ,                                                 i=1, 2, 3                                                                          

∆ᵢ (=  1,2,3,4) are  determined by changing the first, second, third, and fourth columns of ∆ to 

[-�̂�1(𝜉₁)𝑒𝑖𝜔𝑡,0 , �̂�2(𝜉₁)𝑒𝑖𝜔𝑡, �̂�3(𝜉₁) 𝑒𝑖𝜔𝑡]Tr  

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3722 
 

https://internationalpubls.com 

6. Applications  

Linearly distributed source  

The linearly varying source is defined as: 

[𝐹1(𝑥1), 𝐹2(𝑥1), 𝐹3(𝑥1)] = {
1 −

|𝑥1|

𝑎₀
                 𝑖𝑓 |𝑥1| ≤ 𝑎₀

0                           𝑖𝑓 |𝑥1| > 𝑎₀
 

Applying the Fourier transform at the plane boundary 𝑥₃ = 0 in dimensionless form gives  

[�̂�1(𝜉₁), �̂�2(𝜉₁),  �̂�3(𝜉₁)] = 
2[1−cos(𝜉₁𝑎₀)]

𝜉₁²𝑎₀
                                                                                      (34)             

Here, 𝑎₀ represents the dimensionless width of the source strip. Substituting these expressions into 

equations (28)-(33) yields the corresponding solutions for the field variables. 

7. Validation 

 Setting 𝐹2 = 𝐹3 = 0 in equations (28)-(33) yields the field quantities corresponding to a normal 

force. 

Special Cases 

• Setting ξ = ζ = ς = 0 in equations (28)-(33) invoke the case of thermoelastic diffusion 

described by the dual-phase-lag model. 

• Setting 𝜏𝑡 = 𝜏𝑞 = 𝜏𝑢 = 𝜏𝑝  = 0 in equations (28)-(33) invoke the case of non-local 

thermoelastic diffusion. 

8. Inversion of the Transformation 

The transforms in equations (28)-(33) are inverted following the procedure outlined in [29]. 

9. Numerical Implementation and Explanation                                                       

In the numerical analysis, copper is employed as the representative thermoelastic diffusion 

material, in line with the procedure outlined in [19]. 

λ = 7.76× 1010Kg𝑚−1𝑠−2, 𝜇 = 3.86 × 1010 Kg𝑚−1𝑠−2, 𝑇0= 0.293× 103𝐾, 𝐶𝑒 = 0.3891× 103 

J𝐾𝑔−1𝐾−1, 𝛼𝑡= 1.78× 10−5𝐾−1, 𝛼𝑐= 1.98× 10−4𝑚3𝐾𝑔−1, 𝑎 = 1.02 × 104𝑚2𝑠−2𝐾−1, 𝑏 =

9 × 105𝐾𝑔−1𝑚5𝑠−2, 𝐷 = 0.85 × 10−8Kgs𝑚−3, 𝜌 = 8.954 × 103𝐾𝑔𝑚−3,  K= 

0.386× 103𝑊𝑚−1𝐾−1,  𝑡 = 0.01𝑠, 𝑡0= 0.2s, 𝜏𝑡 = 0.6𝑠, 𝜏𝑞= 0.7s, 𝜏𝑝 = 0.8𝑠, 𝜏𝑢=0.9s, 𝜉 =

0.395 × 10−9m,  𝜁 = 0.2 × 10−9𝑚, 𝜍 = 0.15 × 10−9m  

We use MATLAB (R2016a) to carry out numerical simulations that evaluate normal stress, 

tangential stress, tangential couple stress, temperature changes, and chemical potential under the 

following conditions: 

1. Thermoelastic diffusion incorporating both non-local effects and dual-phase-lag. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3723 
 

https://internationalpubls.com 

2. Dual-phase-lag thermoelastic diffusion without non-local effects. 

3. Non-local thermoelastic diffusion without diffusion phase-lags. 

4. Non-local thermoelastic diffusion without thermal phase-lags. 

5. Non-local thermoelastic diffusion with no phase-lags. 

In all figures: 

• Solid line (──) represents thermoelastic diffusion with non-local effects and dual-phase-

lag (TNP). 

• Small dashed line (- - -) denotes thermoelastic diffusion with dual-phase-lag but without 

non-local effects (WNTP). 

• Large dashed line (─ ─) corresponds to non-local thermoelastic diffusion without 

diffusion phase-lag (TNWDP). 

• Solid line with central marker (─*─) indicates non-local thermoelastic diffusion without 

thermal phase-lag (TNWTP). 

• Large dashed line with central marker (─₀─) represents non-local thermoelastic diffusion 

without any phase-lags (TN).  

Figure 1 displays the distribution of   𝑡33  when the boundary is subjected to a normal load. Near 

the loading zone, 𝑡33  decreases in magnitude, reflecting stress release in the immediate vicinity of 

the applied force. Beyond this region, the stress response becomes oscillatory for all the cases 

considered. Interestingly, the oscillation patterns are nearly identical in amplitude and frequency 

across all cases. This indicates a strong dominance of the applied mechanical source over model-

specific parameters. 

The variation of shear stress 𝑡31  , illustrated in Figure 2, exhibits alternating increasing and 

decreasing trends within the bounded region close to the source. This indicates that the shear 

response induced by normal loading oscillates locally before gradually stabilizing further from the 

boundary. The similarity of 𝑡31 across all cases shows that, like the normal stress response, shear 

stress is primarily governed by the direct mechanical loading and is less sensitive to the differences 

in non-locality or phase-lag formulations. 

Figure 3 describe T with 𝑥1. For all the considered cases, the temperature decreases monotonically 

with distance from the boundary, showing a consistent thermal diffusion process. The rate of 

decrease is almost identical across all models, suggesting that under normal loading, the thermal 

response is influenced by the presence or absence of non-local or phase-lag effects.  

The chemical potential P, shown in Figure 4, also decreases steadily with increasing 𝑥1 for all 

cases. The trend is smooth and nearly identical in each case, reflecting uniform diffusion of 

chemical potential away from the loaded boundary. The fact that the decrement is the same across 

all models indicates that, similar to the thermal field, chemical potential diffusion under normal 

loading is affected by variations in non-local elasticity and phase-lag formulations. 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3724 
 

https://internationalpubls.com 

                                                
Fig.1: Distance-dependent normal stress distribution. 

                                           
Fig. 2: Distance-dependent tangential stress distribution. 

                                                 
Fig.3: Distance-dependent temperature change distribution.  

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3725 
 

https://internationalpubls.com 

                                              
Fig4: Distance-dependent chemical potential distribution. 

10.Conclusion  

This study investigates the combined effects of non-locality and phase-lag phenomena in a 

modified couple stress thermoelastic half-space exposed to mechanical loading. The mathematical 

model was developed and solved using integral transform methods.  

For linearly distributed normal loading, the normal stress and chemical potential decreased in 

magnitude, while other stress components and the temperature field displayed oscillatory 

fluctuations. 

The problem find application in semiconductor technologies, thermal barrier coatings, and 

composite materials, due to coupled thermo-diffusive loading. Furthermore, the results are useful 

in biomechanics and nanotechnology. 

There is no conflict of interest involved in this study. 

References 

[1] Edelen,D. G. B. and Laws, N.: On the thermodynamics of systems with nonlocality, Archive 

for Rational Mechanics and Analysis. 43(1) (1971); 24-35. 

https://doi.org/10.1007/BF00251543 

[2] Eringen, A.C. and Edelen, D. G. B.: On nonlocal elasticity, International Journal of 

Engineering Science, 10 (1972); 233-24.http://dx.doi.org/10.1016/0020-7225(72)90039-0 

[3] Eringen, A.C.: On nonlocal fluid mechanics, International Journal of Engineering Science, 10 

(6) (1972) 561-575. https://doi.org/10.1016/0020-7225(72)90098-5 

[4] Eringen, A.C.: Nonlocal polar elastic continua, International Journal of Engineering Science, 

10(1) (1972) ;1-16. https://doi.org/10.1016/0020-7225(72)90070-5 

[5] Eringen, A.C.: Nonlocal continuum theory of liquid crystals, Molecular Crystals and Liquid 

Crystals. 75(1) (1981); 321-343. https://doi.org/10.1080/00268948108073623 

[6] Eringen, A.C.: Nonlocal inviscid magneto-hydrodynamics and dispersion of Alfven waves, 

Bull. Tech. Univ. Istanbul. 39 (1986); 393-408.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3726 
 

https://internationalpubls.com 

[7] Eringen, A.C.: Memory dependent nonlocal electrodynamics, mechanical modelling of new 

electromagnetic materials, Proceedings of IUTAM Symposium (Hsieh R.K.T.,ed.) Elsevier, 

Amsterdam, (1990); 45-49.  

[8] Eringen, A.C.: Memory dependent nonlocal electromagnetic elastic solids and super conduct 

ivity. Journal of mathematical physics, 32(3), (1991);787-796. 

https://doi.org/10.1063/1.529372   

[9] McCay, B. M. and Narsimhan, M. L. N.: Theory of nonlocal electromagnetic fluids, Archives 

of Mechanics. 33 (3) (1981);365-384. WA727_89260_P.262b-McCay-Theory 

[10] Narsimhan, M. L. N. and McCay, B. M.: Dispersion of surface waves in nonlocal dielectric 

fluids, Archives of Mechanics, 33 (3) (1981) 385-400.WA727_89260_P.262b-McCay-Theory 

[11]Eringen, A.C.: Nonlocal continuum field theories, Springer (2002) 

https://doi.org/10.1007/b97697. 

[12] Tzou, D. Y.: Thermal shock phenomena under high rate response in solids, Annual review of 

heat transfer, 4 (1992) 111-185. doi: 10.1615/AnnualRevHeatTransfer.v4.50 

[13] Cao, B.Y. and Guo, Z. Y.: Equation of motion of a phonon gas and non-Fourier heat 

conduction, Journal of Applied Physics, 102(5) (2007) 

053503.https://doi.org/10.1063/1.2775215  

[14] Guo, Z. Y. and Hou, Q.W.: Thermal wave based on the thermomass  model, Journal of  Heat 

Transfer. 132(7) (2010); 072403. https://doi.org/10.1115/1.4000987 

[15] Tzou, D.Y. and Guo, Z.Y.: Nonlocal behavior in thermal lagging, International Journal of 

Thermal Sciences, 49(7) (2010);1133-1137. 

https://doi.org/10.1016/j.ijthermalsci.2010.01.022 

[16] Tzou, D.Y.: A unified field approach for heat conduction from macro to micro scales, Journal 

of Heat Transfer, 117(1) (1995); 8-16. https://doi.org/10.1115/1.2822329 

[17] Tzou, D.Y.: The generalised lagging response in small scale and high rate heating, 

International Journal of Heat and Mass Transfer, 38(17) (1995); 3231-3240. 

https://doi.org/10.1016/0017-9310(95)00052-B 

[18] Sherief, H.H., Hamza, F.A. and Saleh, H. A.: The theory of generalised thermoelastic 

diffusion, international journal of engineering science. 42(5-6) (2004) 591-608. 

https://doi.org/10.1016/j.ijengsci.2003.05.001 

[19] Sherief, H.H. and Saleh, H.: A half space problem in the theory of generalised thermoelastic 

diffusion, international journal of solids and structures, 42(15) (2005); 4484-4493.  

https://doi.org/10.1016/j.ijsolstr.2005.01.001 

[20] Sharma, K.: Boundary value problem in generalized thermodiffusive elastic medium, Journal 

of Solid Mechanics, 2 (5) (2010); 348-362, https://oiccpress.com/jsm/article/view/12348 

[21] Sharma, K.: Reflection of plane waves in thermodiffusive elastic half space with voids, 

Multidiscipline Modeling in Materials and Structures. 8(3) (2012); 269-296. 

https://doi.org/10.1108/15736101211269113 

https://rcin.org.pl/ippt/Content/70834/PDF/WA727_89260_P.262b-McCay-Theory.pdf
https://rcin.org.pl/ippt/Content/70834/PDF/WA727_89260_P.262b-McCay-Theory.pdf
https://doi.org/10.1007/b97697
https://doi.org/10.1115/1.4000987


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3727 
 

https://internationalpubls.com 

[22] Sharma, S., Sharma, K. and Bhargava, R.: Effect of viscosity on wave propagation in 

anisotropic thermoelastic with Green-Naghdi theory type-II and type-III, Material Physics 

Mechanics. 16 (2013); 144-158. 

[23] Sharma, K., Marin, M.: Reflection and transmission of waves from imperfect boundary 

between two heat conducting micropolar thermoelastic solids, An. St. Univ. Ovidius 

Constanta, 22(2),(2014);151-175. DOI: 10.2478/auom-2014-0040 

[24] Abouelregal, A. E. and Zenkour, A. M.: Effect of phase lags on thermoelastic functionally 

graded microbeams subjected to ramp-type heating, Iranian Journal of Science and 

Technology, Transactions of Mechanical Engineering. 38(M2) (2014);321-335. 

[25] Sharma, S. and Sharma, K.: Influence of heat sources and relaxation time on temperature 

distribution in tissues, International Journal of Applied Mechanical Engineering, 19(2) (2014) 

427-433. doi: 10.2478/ijame-2014-0029 

[26] Marin, M., Abbas, I. A. and Kumar, R.: Relaxed Saint-Venant principle for thermoelastic 

micropolar diffusion, Structural Engineering and Mechanics, 51(4) (2014); 651-662. 

https://doi.org/10.12989/sem.2014.51.4.651 

[27] Yu, Y.J., Tian, X.G. and Xiong, Q. L.: Nonlocal thermoelasticity based on nonlocal heat 

conduction and nonlocal elasticity, European Journal of Mechanics-/ A Solids. 60 (2016);238-

253. 

https://doi.org/10.1016/j.euromechsol.2016.08.004 

[28] Kumar, R., Abbas, I.A.: Disturbance due to thermomechanical sources in porothermoelastic 

medium, Strength of materials 48(2) (2016); 315-332.https://doi.org/10.1007/s11223-016-

9767-y 

[29] Kumar, R., Devi, S. and Sharma, V.: Effect of hall current and rotation in modified couple 

stress generalised thermoelastic half space due to ramp type heating, Journal of solid 

mechanics, 9(3) (2017); 527-542.  

[30] Kumar, R., Miglani, A., Rani, R.: Transient analysis of nonlocal microstretch thermoelastic 

thick circular plate with phase lags, Mediterranean Journal of Modeling and Simulation. 09 

(2018) 025-042.  1983-1522252589.pdf 

[31] Kumar, R., Devi, S. and  Sharma, V.: Resonance of nanoscale beam due to various sources in 

modified couple stress thermoelastic diffusion with phase lags, Mechanics and Mechanical 

Engineering. 23(1) (2019); 36-49. doi:10.2478/mme-2019-0006 

[32] Marin, M., Sharma, S., Kumar, R and Vlase S.: Fundamental solution and Green’s function 

in orthotropic photothermoelastic media with temperature-dependent properties under the 

Moore-Gibson-Thompson model. ZAMM-J. of Appl. Math. Mech.105(6) (2025) e70124. 

https://doi.org/10.1002/zamm.70124 

[33] Sharma, K. Marin, M. and Kumar, R.: Thermomechanical deformation in a micropolar 

thermoviscoelastic solid under the Moore-Gibson-Thompson heat equation with non-local 

https://oaji.net/articles/2017/1983-1522252589.pdf
http://dx.doi.org/10.2478/mme-2019-0006
https://doi.org/10.1002/zamm.70124


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

 

3728 
 

https://internationalpubls.com 

and hyperbolic two-temperature effects, Journal of Computational Applied Mechanics, 56(4) 

(2025) 720-736. doi: 10.22059/jcamech (2025).397650.1525 

[34] Abouelregal, A E, Marin M, Alhassan, Y and Atta, D.: A Novel Space-Time Nonlocal 

Thermo-Viscoelastic Model with Two-Phase-Lags for Analysing Heat Diffusion in a Half 

space Subjected to a Heat Source, Iranian Journal of Science and Technology, Transactions 

of Mechanical Engineering 49(2025); 1315-1332. https://doi.org/10.1007/s40997-025-

00835-9