A Unified Analogy-Based Computation Methodology From Elasticity to Electromagnetic-Chemical-Thermal Fields and a Concept of Multifield Sensing

In nature, the macroscopic energy of a material system is commonly in the form of mechanical, thermal, electromagnetic, and chemical (MEMCT) energies [1–4]. Therefore, many research efforts have mainly been focused on basic issues of individual mechanical, thermal, electromagnetic, or chemical fields. It is well known that these fields have certain similarities, and recently, problems of interdisciplinary and multifield features have drawn increasingly strong attention.

In many cases of technology developments, the mechanical effects, such as power transmission and contact of various material systems [5,6], have been studied in great detail, for example, indentation mechanics of layered [7–9] and functionally graded [10,11] materials using analytical or numerical methods, micromechanics of materials involving inhomogeneities or inclusions [12–15] by the phase-field model or the equivalent inclusion method, and nanomechanics of thin films and nano/microstructures [16–21] by the theories of strain-gradient plasticity or couple-stress elasticity. In heat transfer, a great deal of work has been focused on capturing the thermal behaviors of different material systems, among them are the development of the theories of heat transfer [22] and the studies of heat conduction in layered [23–25] and functionally graded [26,27] materials, as well as in materials involving inhomogeneities or inclusions [28–30]. Likewise, the Maxwell equations describe how the electric and magnetic fields are generated and mutually influenced [31,32]. On the interaction of materials, researchers have developed a mathematical framework to describe the mixture behaviors of material systems with diffusion, known as the mixture theory [33–37].

Similarities in some aspects of mechanical, thermal, electromagnetic, and chemical fields have been identified, and related quantities have been widely used for dimension analyses in surface/interfacial mechanics and structural mechanics [37–40]. Actually, the concept analogy across mechanical field, thermal, and electric/magnetic fields can be dated back to how the related theories were developed, from Einstein who imagined a “flow” of electric charge or heat in analogy to a mechanical force [41], from Fourier’s law for heat conduction and Ohm’s law for electricity, which were primarily inspired by Newton's law of cooling, especially from Thompson [42], who derived the concepts of different physical fields from their mathematical analogy. The thought of analogy, for example, between heat flow and mass diffusion, has to enable the use of the solution methods for thermal or thermomechanical systems to analyze diffusion, poroelastic, or chemo-mechanical problems, among them are the numerical manifold method for transient moisture diffusion in graded materials [43] in analogy to the corresponding transient thermal analysis [26], the equivalent inclusion method for frictional heating in inhomogeneous half-spaces [29] in analogy to the corresponding inclusion problem in elasticity [44], the unified solution approach for poroelasticity and thermoelasticity [45,46], the prediction method for the conversion between a thermal-energy flow and a mass transport [47], the analogy between the shear stress in a twisted bar and a stretched soap film [48], and the analogy on material properties among electric, elastic, and piezoelectric parameters [49]. These investigations have intensively explored different facets of the MEMCT fields by analogies of their characters, but still involve resolving and re-deriving the field equations of the material systems under consideration.

Long-time mechanics research has resulted in rich theories and solutions for wide ranges of material structures and systems (e.g., layered materials, functionally graded materials, composite materials). This work proposes a novel methodology for multi-theory analogy across the MEMCT fields for a wide range of material systems including full-space, half-space, and layered materials, with or without eigenstrains, loaded by surface or internal sources. The methodology and solution framework are based on the conceptual analogy and mathematical similarities of a number of MEMCT field variables, governing equations, and their own material properties, which can help to utilize existing mechanics results to solve other complicated problems. The proposed MEMCT analogy methodology also expands the electromagnetic-elastic analogy theory [50,51] and enriches our previous MEMCT coupling theory [2] for in-depth understanding of multifield interactions and conversions. Further, it is expected that the systematic theoretical analogy and field equivalence may stimulate certain technological inventions or engineering method integration.

In this paper, analogy is the methodology to view the similarity between physical terms in mathematical expressions, and equivalence means quantification with the same expression.

Figure 1 presents the main variables and governing equations for mechanical, thermal, electromagnetic, and chemical fields.

In the mechanical field, the gradient of the displacement vector u_i leads to the elastic strain tensor ɛ_ij = (u_i,j + u_j,i)/2. The elastic strain tensor, ɛ_ij, and elastic stress tensor, σ_ij, are linked by the constitutive laws of σ_ij = c_ijklɛ_kl, with c_ijkl for the elastic property tensor. The elastic stress tensor, σ_ij, should satisfy the mechanical equilibrium of $σij,i+fi=ρ∂2ui/∂t2$⁠, where f_i is the body force component, ρ is the mass density, and t is the time.

In the thermal field, the gradient of temperature rise T is defined as the vector θ_i satisfying the equation of θ_i = −T_,i. The Fourier’s law for heat conduction defines heat flux q_i as q_i = k_ijθ_j, with k_ij for the heat conductivity tensor. The first law of thermodynamics is written in the form of $−qi,i+q0=ρCp∂T/∂t$⁠, where q₀ is the strength of a volumetric heat source, C_p is the specific heat capacity.

In the electromagnetic field, the gradient of electric potential φ is the strength of the electric field, expressed as E_i = −φ_,i, while the gradient of magnetic potential ϕ is the strength of the magnetic field with H_i = −ϕ_,i. The constitutive relationships for electricity without polarization is $Di=ϖijEj$⁠, while for magnetism without magnetization is B_i = μ_ijH_j, where $ϖij$ is the dielectric permittivity tensor, μ_ij is the magnetic permeability tensor, D_i is the electric displacement vector, and B_i is the magnetic induction vector. The electric displacement should satisfy Gauss’ law for electric field of −D_i,i + ρ_f = 0, with ρ_f for free charge density, and the magnetic induction should satisfy Gauss’ law for magnetism of B_i,i = 0. Furthermore, the Maxwell-Faraday equation $∇×E=−∂B/∂t$ and Ampère's circuital law $∇×H=∂D/∂t+Jf$ (with $Jf$ for current density) govern the variation of the electromagnetic field.

In the chemical field, the gradient of mass concentration N is defined as the vector $ϑi$⁠, satisfying $ϑi=−N,i$⁠. The diffusion flux vector, ξ_i, can be given by Fick’s first law of $ξi=βijϑj$⁠, with β_ij for the diffusivity tensor. The mass balance in a diffusion process is $−ξi,i+W0=ρ∂N∂t$⁠, where W₀ is the intensity of a local mass source, such as the rate of a chemical reaction. Here, a linear process is assumed; however, a nonlinear diffusion law can also be implemented if a localized linearization is possible.

Defining a generalized field collection Φ, satisfying Φ = (u_i ↔ T ↔ φ ↔ ϕ ↔ N), a generalized dual variable collection Ψ, satisfying Ψ = (σ_ij ↔ −q_i ↔ −D_i ↔ −B_i ↔ −ξ_i), and a generalized material property C_M, yielding $CM=(cijkl↔kij↔ϖij↔μij↔βij)$⁠, with “↔” for equivalence, results in the generalized constitutive law of Ψ = C_M · Φ_,i. If the time-related terms of the MEMCT fields are defined by a function, Ω(Φ, t), with respect to time t and field collection Φ, the generalized equilibrium equation can be written as Ψ_i + F_i = Ω(Φ, t), where F_i = (f_i ↔ q₀ ↔ ρ_f ↔ 0 ↔ W₀) is the strength of a generalized volumetric source.

The above variables can make a complete analogy, but this analogy should not be stretched too far.

The variables in elasticity have higher orders than those in other fields. For the field collection of Φ = (u_i ↔ T ↔ φ ↔ ϕ ↔ N), elastic displacement u_i is a first-order tensor (or vector), while temperature rise T, electric potential φ, magnetic potential ϕ, and mass concentration N are zero-order tensors (or scalars). For the material properties of $CM=(cijkl↔kij↔ϖij↔μij↔βij)$⁠, elastic constant c_ijkl is a fourth-order tensor, while heat conductivity k_ij, dielectric permittivity $ϖij$⁠, magnetic permeability μ_ij, and chemical diffusivity β_ij are second-order tensors. In the dual variable collection of Ψ = (σ_ij ↔ −q_i ↔ −D_i ↔ −B_i ↔ −ξ_i), elastic stress σ_ij is a second-order tensor, while heat flux q_i, electric displacement D_i, magnetic induction B_i, and diffusion flux ξ_i are vectors.

The time scales in the MEMCT fields are different. In elasticity, the time-related term is $Ω=ρ∂2ui/∂t2$⁠, while it is $Ω=ρCp∂T/∂t$ in the thermal field, and $Ω=ρ∂N/∂t$ in the chemical field. However, the dependence of the electromagnetic field on time is governed by Maxwell’s equations, i.e., the sum of Gauss’ laws for electric field and magnetism, the Maxwell-Faraday equation, and Ampère's circuital law.

For clarity, the similarities and differences of the MEMCT field variables, governing equations, and the material properties pertaining to each individual field, are described in Fig. 1, which shows that the variables in the mechanical field have higher orders than those in the other fields. Theoretically, the lower-order variables can be obtained from the degradation of the corresponding higher-order mechanical variables. According to Thomson and Wise’s ideas [42,52], the vision from the mechanical field should foresee a reasonable path to describe the analogy.

The complete equivalence of Eqs. (9) and (10) is resulted from (i) Θ_,z = 0; (ii) $ρG∂2uz∂t2↔ρCpk0∂T∂t$⁠. For (i) Θ_,z = 0, this is naturally satisfied due to Θ = 0 stated below Eq. (8). Moreover, if z, or the power of z, is a multiplier of a term in an elastic solution, for examples, zf(x,y,z), z²f(x,y,z), variable z should be set zero, and likewise for x or y removal if the excitation source is along the x or y direction. For (ii), it is found that the time scales in elasticity and in heat conduction are different. If elastic displacement u_z is defined as a harmonic function F_e with elastic wave frequency ω_e, and temperature T is defined as another harmonic function F_t with thermal wave frequency ω_t, condition $ρG∂2uz∂t2↔ρCpk0∂T∂t$ yields $ωe2↔−iCp2ωt$⁠. For example, if an elastic displacement has a harmonic solution of u_z(x, y, z, t) = Z₁(x, y, z) · e^−iωt with ω for the wave frequency, temperature should have a harmonic solution in the form of $T(x,y,z,t)=Z2(x,y,z)⋅e2ω2Cp⋅t$⁠.

In summary, such a complete equivalence between elasticity and heat conduction is obtained by setting: (i) Poisson’s ratio ν → 0.5; (ii) shear modulus 2G ↔ heat conductivity k₀; (iii) removing terms multiplied by zⁿ in the elastic solutions, such as zu_z, z²u_z, zⁿf(x,y,z), etc.; and (iv) elastic wave frequency $ωe2↔$ thermal wave frequency $−iCp/2ωt$⁠. Two examples (the Papkovich-Neuber potentials and Galerkin vectors) are given below to explain how to make use of the equivalent method between elasticity and heat conduction.

1. Half-space

   Considering an isotropic and homogeneous elastic half-space, whose surface is subjected to the normal force P at the origin and along the z-direction, the elastic displacement, u_z, can be found in Ref. [6]

   $uz(x,y,z)=P4πG[z2r3+2(1−ν)r]$

   (11)

   where $r=x2+y2+z2$⁠.
   For the problem of an isotropic and homogeneous half-space subjected to a surface point heat source, Q, by making ν = 0.5, 2G → k₀, P → Q, and removing the term multiplied by zⁿ, which means to remove the first term, z²/r³, in Eq. (11), the temperature distribution can be obtained from Eq. (11)

   $T(x,y,z)=Q2πk01r$

   (12)

   which is identical to that in Ref. [57].

2. Layered half-space

   Because it is difficult to obtain the space-domain analytical solutions for a layered half-space subjected to a unit normal force, the frequency response functions (FRFs) are usually solved first. The FRFs for normal displacement u_z can be found in Ref. [58], which are

   $F≈r(uz)=12Gr[−α(Dre−αzr−D¯reαzr)−(3−4νr)(Cre−αzr+C¯reαzr)−zrα(Cre−αzr−C¯reαzr)]$

   (13)

   where $α=m2+n2$ denotes the distance of a node, (m, n), to the origin of the frequency domain. Subscript r = 1 or 2 is used to distinguish variables associated with the layer or the substrate half-space, respectively. Coefficients C_r, $C¯r$⁠, D_r and $D¯r$$(C¯2=D¯2=0)$ can be found in Eqs. (22a)–(22f) of Ref. [58].
   For the problem of a layered half-space subjected to a unit point heat source, by making 2G_r ↔ k_r, ν_r = 0.5, and removing the term with z and h multiplication, because h resembles z, which means to remove the last term, $zrα(Cre−αzr−C¯reαzr)$⁠, in Eq. (13), and the term multiplied by 2αh in C_r, $C¯r$⁠, D_r and $D¯r$⁠. Therefore, the FRFs for temperature T can be obtained from Eq. (13)

   $F≈r(T)=1kr(Cre−αzr+C¯reαzr),r=1,2$

   (14)

   where

   $C1=1α(1−k3e−2αh)$

   $C¯1=k3e−2αhα(1−k3e−2αh)$

   $C2=(1−k3)e−αhα(1−k3e−2αh)$

   $C¯2=Dr=D¯r=0$

   with h for layer thickness, and k₃ = (k₁ − k₂)/(k₁ + k₂). Equations in (14) are identical to the FRFs for temperature in Ref. [25].

   To be clear, Fig. 2 compares the surface temperature and subsurface heat flux distributions of a layered half-space solved by the current equivalence approach via the elastic solutions in Ref. [58] and the heat conduction solutions given in Ref. [25], using the discrete convolution-fast Fourier transform (DC-FFT) algorithm [59] with a 256 × 256 × 100 grid on a physical domain of 6.0 mm × 6.0 mm × 4.0 mm, for the problem of a uniform surface heat input of density 2000 W/mm² on the region $x2+y2≤1.0mm$⁠. The layer thickness is h = 1.0 mm, the substrate conductivity is k₂ = 25 W/mm·K, and the layer conductivity is k₁ = 0.5k₂ or 2.0k₂. As shown in Figs. 2(a) and 2(b), respectively, the results of the temperature and heat flux from the current equivalence approach based on the elastic solutions in Ref. [58] are identical to those from the heat-conduction solutions given in Ref. [25].

It should be mentioned that, even though F^T is no longer a vector, it is still called the Galerkin vector for terminology consistence.

1. Homogeneous full-space

   Consider single unit mechanical forces at (x′, y′, z′) in the x, y, or z directions of the full-space, the magnitudes of the Galerkin vectors are all the same as $F=RI8π(1−v)$⁠, where $RI=(x−x′)2+(y−y′2+(z−z′)2$⁠. We can also define $R=(x−x′)2+(y−y′)2+(z+z′)2$ for use later.

   The z-direction elastic displacement at (x, y, z), u_z, given in Eq. (15), is the following, subjected to the equivalence treatment

   $2Guz=2(1−v)F,jj−F,33=18π(1−v)(4(1−v)ϕI−ϕI−(z−z′)ϕ,3I)−−−−−−−⟶removez−term3−4v8π(1−v)1RI$

   (17)

   where φ^I = 1/R^I.
   By using the equivalence conditions between Eqs. (17) and (11), which are (i) Poisson’s ratio ν → 0.5; (ii) shear modulus 2G ↔ heat conductivity k₀, which means setting $3−4v8π(1−v)(2G)→14π(2G)∼14πk0$ for a unit heat source, and (iii) removing the z(z')-multiplication term (done above in Eq. (17)) in the elastic solution, Eq. (17) becomes the following for heat flux Q

   $T(x,y,z)=14πk0QRI$

   (18)

   where displacement u_z ↔ temperature T. Equation (18) is exactly the full-space heat-conduction solution for the temperature field under a steady point source, one half that of the half-space solution [6], to be discussed below, under the same heat source. At the same time, the Galerkin vector for heat conduction in the full-space is $FT=14πRI$ in equivalence to $F=RI8π(1−v)$ with setting Poisson’s ratio ν → 0.5.

2. Half-space with surface sources

   For a half-space loaded by a unit surface force at (x′, y′) along the z-direction, the Galerkin vector is [61]

   $F=18π(1−v)[00RI+(3−4υ)R−2z⋅z′ϕ−4Dz′ψ−4HDθ]T$

   (19)

   where D = 1 − 2ν, θ = R − zln[R + z], ψ = ln[R + z]. By dropping the terms with D and z(z’)- multiplication, making v → 0.5 inside the brackets, and noting that R = R^I when z’ = 0, Eq. (19) becomes

   $F=14π(1−v)[00RI]T$

   (20)

   The Galerkin vector for heat conduction in the half-space subjected to a surface source is from the third one of Eq. (20), $FT=12πRI$⁠, which is twice that of the full-space solution, and so is the temperature field, the same as Eq. (12).

3. Half-space with internal sources

   The Galerkin vector for a single unit force at (x′, y′, z′) along the z-direction inside the half-space is

   $F=18π(1−v)[00RI+(3−4υ)R−2z⋅z′ϕ−4Dz′ψ−4HDθ]T$

   (21)

   The “displacement,” u_z, is the following after dropping the z(z′)-multiplication terms and making v → 0.5

   $2Guz=14π(1RI+1R)$

   (22)
   Therefore, the temperature due to heat flux Q is

   $T=Qk0(F,jjT−F,33T)=Q4πk0(1RI+1R)$

   (23)

   Equation (23) is equivalent to Eq. (22) by letting 2G ↔ k₀, and the Galerkin vector for heat conduction in the half-space with internal sources is from the third component of Eq. (21)

   $FT=14π(RI+R)$

   (24)

4. Full-space materials with eigenstrains

   Considering an isotropic and homogeneous full-space with a volumetric irregular region subject to eigenstrains e_ij distributed at (x′, y′, z′) arbitrarily, the full-space elastic eigenstrain solutions are given in Refs. [62,63], where the Galerkin vector is $F(x)=G4π(1−ν)∫Ω(2νekkgc−ejkgjk)dx′=∫Ωfdx′$⁠. Corresponding to eigenstrains, eigen-temperature is expected for heat-conduction problems. Letting Poisson’s ratio ν → 0.5, which also means e_kk = 0 in mechanics, the integral core of the Galerkin vector becomes

   $f=−G2π[e1jR,jIe2jR,jIe3jR,jI]T$

   (25)

   Taking the third component of Eq. (25), and noting that e_3j is the equivalence to eigen-temperature gradient $ΔTj*$⁠, 2G ↔ k₀, with k₀ as the thermal conductivity of the eigen-temperature region, the integral core of the Galerkin vector for heat conduction in the full-space subjected to an eigen-temperature gradient is

   $fT=−k04πΔTj*R,jI$

   (25′)

   The result is identical to that in Ref. [64].
   Using Eq. (16) and dropping the z(z′)-related terms, the integral core of temperature can be expressed as

   $T=1k0(f,jjT−f,33T)→−14πe3j(R,jkkI−R,j33I)→−14πΔTj*ϕ,jI$

   (26)

   It should be mentioned that for this problem of full symmetry, any of the vectors in Eq. (25), or all, can be utilized to obtain the Galerkin-vector equivalence. But Eq. (26) cannot be directly obtained from the elastic displacement results by setting ν → 0.5 and 2G ↔ k₀ and by dropping the z(z′)- multiplication terms.

5. Half-space with eigenstrains

   For a half-space with eigenstrains, the integral cores of the Galerkin vectors for elasticity [15,62] become the following via the equivalence approach

   $gi1=−G2π[∂(RI+R)∂x′∂(RI+R)∂y′∂(RI+R)∂z′000002R,1]T$

   (27)

   $gi2=−G2π[000∂(RI+R)∂x′∂(RI+R)∂y′∂(RI+R)∂z′002R,2]T$

   (28)

   $gi3=−G2π[000000∂(RI+R)∂x′∂(RI+R)∂y′∂(RI+R−2z⋅z′ϕ)∂z′]T$

   (29)
   Therefore, after converting the differentiations to x, y, z

   $f=−ejkgjk=G2π[−e1j(RI+R),j+2e13R,3−e2j(RI+R),j+2e23R,3−e3j(R,jI−R,j)]T$

   (30)

   Note that e_ij = e_ji. Only the third component of the above should be used, due to half-space symmetry, as the core of the Galerkin vector, and again, e_3j is the equivalence to eigen-temperature gradient $ΔTj*$⁠, 2G ↔ k₀, with k₀ as the thermal conductivity of the inhomogeneity. The integral core of the Galerkin vector for heat conduction in the half-space with eigen-temperature gradient is

   $fT=−k04πΔTj*(R,jI−R,j)(30′)$

   (30’)

   Thus, the integral core of temperature, from Eq. (16), is

   $T=1k0(f,jjT−f,33T)→−14πΔTj*(ϕ,jI−ϕ,j)$

   (31)

   Equation (31) is identical to the result in Ref. [64].

   In summary, the mechanical-thermal equivalence can be established with the following considerations: (i) If the mechanical formulations are structured from the Papkovich-Neuber solutions, which only have harmonic potentials, the equivalence can be implemented directly at the final solutions, or at any step of the solution derivation; (ii) If the mechanics solutions are built on the Galerkin vectors, use F₃ only due to solution symmetry for half-space problems, but use any or all of F_i is optional for the full-space solutions; (iii) Because the Galerkin-vector solutions include biharmonic potentials, not all terms in elasticity are proper for heat-conduction solutions. For homogeneous materials, direct result equivalence is appropriate, but for eigen-temperature and inhomogeneity problems, the equivalence should be done at the Galerkin-vector stage (Table 1).

   The Galerkin vectors for elastic and heat-conduction problems are summarized below, where ψ = ln[R + (z + z′)]; θ = R − (z + z′)ψ.

The dependence of electromagnetic fields on time is governed by Maxwell’s equations, which means that the magnetic and electric fields are mutually affected. The electromagnetic wave propagation can be equivalent to the elastic wave propagation, as stated in Ref. [51]. Considering the fact that the frequency of magnetoelectric waves is much higher than those of the mechanical, thermal, or chemical diffusion waves [66], only the static electromagnetic solutions are carried out here. If the transient behaviors of the electromagnetic field are ignored, the Maxwell-Faraday equation and Ampère's circuital law vanish, and only the Gauss’ laws for electricity and magnetism remain.

The diagonal terms in $[M]$ are the material properties, C, in each individual field, and $CM=(cijkl↔kij↔ϖij↔μij↔βij)$⁠, and the other elements in $[M]$ denote the field coupling parameters, A, with A = (λ_ijk ↔ e_ijk ↔ d_ijk ↔ a_ijk ↔ h_ij ↔ m_ij ↔ b_ij ↔ g_ij ↔ t_ij ↔ s_ij). For example, λ_ijk, e_ijk, d_ijk, and a_ijk are the elastic-thermal tensor, piezoelectric tensor, piezomagnetic tensor, and elastic-chemical tensor, respectively, where λ_ij = c_ijklα_kl with α_kl denoting thermal expansion coefficients. h_ij, m_ij, b_ij, g_ij, t_ij, and s_ij are the pyroelectric coefficient, pyroelectric coefficient, pyrochemical coefficient, magnetoelectric coefficient, electric-chemical constant, and magnetic-chemical constant, respectively. The detail definitions of material parameters are given in the Appendix of [2]. The analyses in Sec. 2.5 show that the lower-order variables can be obtained from degrading the corresponding higher-order mechanical variables and the equivalating related material properties. Besides the material properties C_M pertaining to each individual field, in the field coupling parameters A = (λ_ijk ↔ e_ijk ↔ d_ijk ↔ a_ijk ↔ h_ij ↔ m_ij ↔ b_ij ↔ g_ij ↔ t_ij ↔ s_ij), the mechanics-related coupling parameters, λ_ijk ↔ e_ijk ↔ d_ijk ↔ a_ijk, are also in higher orders. As an example, the thermo-electro-mechanical solutions for multilayered materials can be obtained from the recent multilayered electromagnetic-mechanical solutions [67] by using the equivalences of (i) dual variables (q_i ↔ B_i), (ii) state variables (θ_i ↔ H_i), (iii) individual parameters (k_ij ↔ μ_ij), and (iv) coupling parameters (λ_ijk ↔ d_ijk) and (h_ij ↔ g_ij), where h_ij are the pyroelectric coefficients and g_ij the magnetoelectric coefficients.

Material interfaces between two or more constituent solids with different properties are commonly found in various material structures, such as the coating-substrate interface in layered materials and inclusion-matrix interface in composites [68–70]. Due to property mismatching, crystal defects, or impurity diffusion, the material interfaces may involve a bonding issue. The imperfect interfaces would affect the transmissions of force and/or displacement in elasticity [69], influence heat transfer [71,72], cause the imperfect gluing and/or soldering in electronic packaging [71], and reduce the charge-diffusion rate in batteries [73]. The imperfect interfaces in elasticity have been addressed most deeply, such as the spring-like interface in layered materials [74], coupled dislocation-like and force-like interfaces in layered materials [69] and in joined half-spaces [70,75]. Figure 3 illustrates an example of a layered half-space with imperfect layer-substrate interfaces, where the layer thickness is h, and a uniform force, or heat flux, or electric/magnetic flux, or diffusion flux, is applied on the top surface of the layer. Note that Fig. 3 is for the steady-state conditions, while the scheme for transient conditions for convection diffusion can be referenced to the example by the lattice Boltzmann method [76].