Abstract

This paper reports a unified analogy-based computation methodology, together with a concept of multifield, multifunctional sensing, from elasticity to electromagnetic-chemical-thermal fields, via utilizing the similarities of mechanical-electromagnetic-chemical-thermal (MEMCT) field variables, governing equations, and the material properties pertaining to each individual field. Two equivalences are systemized, which are the field-formulation equivalence and surface-value equivalence. Due to similarity, a number of thermal, electromagnetic, or chemical solutions can be obtained from the direct degeneration of existing mechanical solutions by making specified equivalences of 2Gk0ϖ0μ0β0 with G for shear modulus, k0 for heat conductivity, ϖ0 for dielectric permittivity, μ0 for magnetic permeability, and β0 for chemical diffusivity, as well as by setting Poisson’s ratio ν → 0.5. These specified equivalences enable quick solutions to other fields directly from mechanics formulations, such as those in the forms of the Galerkin vectors and Papkovich-Neuber potentials, and field coupling, by means of analogy. Several examples are given, one is used to demonstrate that the field solutions of a layered half-space with imperfect thermal, electromagnetic, or chemical interfaces can be readily obtained from the elastic solutions involving interfacial imperfections via the obtained formulation equivalence. A set of simple equations are derived to relate surface behaviors of different fields via the obtained surface-value equivalence, on which a concept of multifield sensing is proposed.

1 Introduction

In nature, the macroscopic energy of a material system is commonly in the form of mechanical, thermal, electromagnetic, and chemical (MEMCT) energies [14]. 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 [79] and functionally graded [10,11] materials using analytical or numerical methods, micromechanics of materials involving inhomogeneities or inclusions [1215] by the phase-field model or the equivalent inclusion method, and nanomechanics of thin films and nano/microstructures [1621] 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 [2325] and functionally graded [26,27] materials, as well as in materials involving inhomogeneities or inclusions [2830]. 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 [3337].

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 [3740]. 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.

2 MEMCT Analogy From the Mechanical Point of View

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 ui leads to the elastic strain tensor ɛij = (ui,j + uj,i)/2. The elastic strain tensor, ɛij, and elastic stress tensor, σij, are linked by the constitutive laws of σij = cijklɛkl, with cijkl for the elastic property tensor. The elastic stress tensor, σij, should satisfy the mechanical equilibrium of σij,i+fi=ρ2ui/t2, where fi 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 θi satisfying the equation of θi = −T,i. The Fourier’s law for heat conduction defines heat flux qi as qi = kijθj, with kij for the heat conductivity tensor. The first law of thermodynamics is written in the form of qi,i+q0=ρCpT/t, where q0 is the strength of a volumetric heat source, Cp is the specific heat capacity.

In the electromagnetic field, the gradient of electric potential φ is the strength of the electric field, expressed as Ei = −φ,i, while the gradient of magnetic potential ϕ is the strength of the magnetic field with Hi = −ϕ,i. The constitutive relationships for electricity without polarization is Di=ϖijEj, while for magnetism without magnetization is Bi = μijHj, where ϖij is the dielectric permittivity tensor, μij is the magnetic permeability tensor, Di is the electric displacement vector, and Bi is the magnetic induction vector. The electric displacement should satisfy Gauss’ law for electric field of −Di,i + ρf = 0, with ρf for free charge density, and the magnetic induction should satisfy Gauss’ law for magnetism of Bi,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, ξi, can be given by Fick’s first law of ξi=βijϑj, with βij for the diffusivity tensor. The mass balance in a diffusion process is ξi,i+W0=ρNt, 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.

2.1 Similarities.

Defining a generalized field collection Φ, satisfying Φ = (uiTφϕN), a generalized dual variable collection Ψ, satisfying Ψ = (σij ↔ −qi ↔ −Di ↔ −Bi ↔ −ξi), and a generalized material property CM, yielding CM=(cijklkijϖijμijβij), with “↔” for equivalence, results in the generalized constitutive law of Ψ = CM · 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 Ψi + Fi = Ω(Φ, t), where Fi = (fiq0ρf ↔ 0 ↔ W0) is the strength of a generalized volumetric source.

2.2 Differences.

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 = (uiTφϕN), elastic displacement ui 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=(cijklkijϖijμijβij), elastic constant cijkl is a fourth-order tensor, while heat conductivity kij, dielectric permittivity ϖij, magnetic permeability μij, and chemical diffusivity βij are second-order tensors. In the dual variable collection of Ψ = (σij ↔ −qi ↔ −Di ↔ −Bi ↔ −ξi), elastic stress σij is a second-order tensor, while heat flux qi, electric displacement Di, magnetic induction Bi, and diffusion flux ξi are vectors.

The time scales in the MEMCT fields are different. In elasticity, the time-related term is Ω=ρ2ui/t2, while it is Ω=ρCpT/t in the thermal field, and Ω=ρN/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.

2.3 Mechanical-Thermal Analogy.

Consider a homogeneous and isotropic half-space B, subjected to a time-varied normal force, p¯(x,y,t), on its boundary along the z direction, which is set as the direction of the surface normal, shear transtions, s¯1(x,y,t) and s¯2(x,y,t) along the x and y directions or a time-varied heat flux, q¯(x,y,t), normal to the surface. For any material point (x, y, z) inside B, the initial condition at time t = 0 is
ux(x,y,z,0)=ux0(x,y,z)uy(x,y,z,0)=uy0(x,y,z)uz(x,y,z,0)=uz0(x,y,z)
(1)
for elasticity, and
T(x,y,z,0)=T0(x,y,z)
(2)
for temperature, where ui0 is the initial displacement, T0 is the initial temperature. For time t > 0, the boundary condition is
σzx(x,y,0,t)=s¯1(x,y,t)σzy(x,y,0,t)=s¯2(x,y,t)σzz(x,y,0,t)=p¯(x,y,t)
(3)
for elasticity, and
k0T(x,y,0,t)z=q¯(x,y,t)
(4)
for heat conduction, where σij = σji is the stress component, k0 is an isotropic heat conductivity.
By comparing Eqs. (1) with (2), and Eqs. (3) with (4), and considering that shear tractions have no influence in system degeneration, to be set to zero, there is a complete analogy between the variables in elasticity and those in heat conduction:
displacementuztemperatureT
(5)
pressurep¯surfaceheatfluxq¯
(6)
stressσzzk0Tz
(7)
where stress σzz can be written as
σzz=λΘ+2Guz,z
(8)
with λ=νE[(1+ν)(12ν)] and G=E[2(1+ν)] for the Lamé constants, E for Young’s modulus, ν for Poisson’s ratio, Θ = ux,x + uy,y + uz,z for the volumetric strain. In view of Eqs. (7) and (8), complete equivalences of uzT and σzzk0T/z lead to (i) the volumetric strain Θ = 0, which is corresponding to incompressible elasticity with Poisson’s ratio ν → 0.5; and (ii) 2Gk0. Such equivalences are partly based on the pioneer works reported in Refs. [29,5356].
Furthermore, in order to compare the transient behaviors in elasticity and heat conduction, one of the Lamé-Navier equations is
(λ+μ)Θ,z+G2uz+fz=ρ2uzt2
(9)
and the heat-conduction equation is
k02T+q0=ρCpTt
(10)
where fz is the body force along the z-direction. Again, there is a complete equivalence between body force and volumetric heat source: fzq0.

The complete equivalence of Eqs. (9) and (10) is resulted from (i) Θ,z = 0; (ii) ρG2uzt2ρCpk0Tt. 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 uz is defined as a harmonic function Fe with elastic wave frequency ωe, and temperature T is defined as another harmonic function Ft with thermal wave frequency ωt, condition ρG2uzt2ρCpk0Tt yields ωe2iCp2ωt. For example, if an elastic displacement has a harmonic solution of uz(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)e2ω2Cpt.

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 zuz, z2uz, 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.

2.3.1 Solutions Using Papkovich-Neuber Potentials

  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, uz, can be found in Ref. [6]
    uz(x,y,z)=P4πG[z2r3+2(1ν)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, 2Gk0, PQ, 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 uz can be found in Ref. [58], which are
    Fr(uz)=12Gr[α(DreαzrD¯reαzr)(34νr)(Creαzr+C¯reαzr)zrα(CreαzrC¯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 Cr, C¯r, Dr 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 2Grkr, νr = 0.5, and removing the term with z and h multiplication, because h resembles z, which means to remove the last term, zrα(CreαzrC¯reαzr), in Eq. (13), and the term multiplied by 2αh in Cr, C¯r, Dr and D¯r. Therefore, the FRFs for temperature T can be obtained from Eq. (13)
    Fr(T)=1kr(Creαzr+C¯reαzr),r=1,2
    (14)
    where
    C1=1α(1k3e2αh)
    C¯1=k3e2αhα(1k3e2αh)
    C2=(1k3)eαhα(1k3e2αh)
    C¯2=Dr=D¯r=0
    with h for layer thickness, and k3 = (k1k2)/(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+y21.0mm. 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].

2.3.2 Solutions Built on the Galerkin Vectors.

Consider an isotropic and homogeneous media (e.g., full-space or half-space) with or without a volumetric irregular region subject to eigenstrains distributed arbitrarily. The elastic displacement, ui, caused by forces or eigenstrains, can be expressed in terms of the Galerkin vectors, Fi [60]
2Gui=2(1v)Fi,jjFj,ji
(15)
Here and throughout this section, the standard index notation is used, where comma represents differentiation with respect to the suffix coordinate, and the Einstein summation convention is used for repeated index. Subscripts 1, 2, and 3 of variables indicate x, y, and z, respectively.
The Galerkin vector for elasticity has three components, but in most cases of heat conduction, only one of them is needed, which is the one for the z-direction to consider a half-space with its depth in z. The complete equivalence of uzT with the condition of Poisson’s ratio ν → 0.5 leads to the temperature field in terms of the Galerkin vector for heat conduction, FT = Fi = F3, starting from Eq. (15)
2Gui=(Fi,jjFj,ji)v0.5k0T=(F3,jjF3,33)v0.5=(F,jjTF,33T)
(16)
Furthermore, substituting Eq. (16) into Eq. (9) results in T=1/k0D1F,jjT, with D1 for a constant, meaning that only one term of the FT differentiation is sufficient. However, Eq. (15), or (16), is still used in the following to show the details of the equivalence approach.

It should be mentioned that, even though FT 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(1v), where RI=(xx)2+(yy2+(zz)2. We can also define R=(xx)2+(yy)2+(z+z)2 for use later.

    The z-direction elastic displacement at (x, y, z), uz, given in Eq. (15), is the following, subjected to the equivalence treatment
    2Guz=2(1v)F,jjF,33=18π(1v)(4(1v)ϕIϕI(zz)ϕ,3I)removezterm34v8π(1v)1RI
    (17)
    where φI = 1/RI.
    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 34v8π(1v)(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 uz ↔ 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(1v) 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(1v)[00RI+(34υ)R2zzϕ4Dzψ4HDθ]T
    (19)
    where D = 1 − 2ν, θ = Rzln[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 = RI when z’ = 0, Eq. (19) becomes
    F=14π(1v)[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(1v)[00RI+(34υ)R2zzϕ4Dzψ4HDθ]T
    (21)
    The “displacement,” uz, 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,jjTF,33T)=Q4πk0(1RI+1R)
    (23)
    Equation (23) is equivalent to Eq. (22) by letting 2Gk0, 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 eij 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(1ν)Ω(2νekkgcejkgjk)dx=Ωfdx. Corresponding to eigenstrains, eigen-temperature is expected for heat-conduction problems. Letting Poisson’s ratio ν → 0.5, which also means ekk = 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 e3j is the equivalence to eigen-temperature gradient ΔTj*, 2Gk0, 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,jjTf,33T)14πe3j(R,jkkIR,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 2Gk0 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)z000002R,1]T
    (27)
    gi2=G2π[000(RI+R)x(RI+R)y(RI+R)z002R,2]T
    (28)
    gi3=G2π[000000(RI+R)x(RI+R)y(RI+R2zzϕ)z]T
    (29)
    Therefore, after converting the differentiations to x, y, z
    f=ejkgjk=G2π[e1j(RI+R),j+2e13R,3e2j(RI+R),j+2e23R,3e3j(R,jIR,j)]T
    (30)
    Note that eij = eji. Only the third component of the above should be used, due to half-space symmetry, as the core of the Galerkin vector, and again, e3j is the equivalence to eigen-temperature gradient ΔTj*, 2Gk0, 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,jIR,j)(30)
    (30’)
    Thus, the integral core of temperature, from Eq. (16), is
    T=1k0(f,jjTf,33T)14πΔTj*(ϕ,jIϕ,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 Fi 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.

2.4 Thermal-Chemical Analogy.

Again, consider a chemical species diffusing into a homogeneous and isotropic half-space B cross its surface, with a time-varied surface diffusive flux ξ¯(x,y,t). For any material point (x, y, z) inside B, the initial condition for mass concentration of the diffusive species at time t = 0 is
N(x,y,z,0)=N0(x,y,z)
(32)
with N0 for the initial mass concentration of the diffusive species. For time t > 0, the boundary condition is
β0N(x,y,0,t)z=ξ¯(x,y,t)
(33)
where β0 is the diffusivity.
It has been shown in Fig. 1, the chemical variables are in a complete analogy with the heat-conduction variables, i.e.
massconcentrationNtemperatureT
(34)
surfacediffusivefluxξ¯surfaceheatfluxq¯
(35)
diffusivityβ0conductivityk0
(36)
masssourceW0heatsourceq0
(37)
NtCpTt
(38)
It is found from Eq. (38) that the time scales in chemical and thermal fields are different, but they are proportional. If mass concentration N is defined as a harmonic function, FN, of diffusion wave frequency ωN, and temperature T is defined as Ft of thermal wave frequency ωt, mentioned in Sec. 2.3, condition NtCpTt result in ωNCpωt. The equivalence between heat conduction and chemical diffusion also affects the equivalence between elasticity and chemical diffusion.

2.5 Thermal-Electromagnetic Analogy.

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.

Consider a homogeneous and isotropic half-space, B, subjected to static surface electrical flux ie(x, y) and surface magnetic flux im(x, y). For any material point (x, y, z) inside B, the boundary conditions are
D(x,y,0,t)=ie(x,y,t)
(39)
B(x,y,0,t)=im(x,y,t)
(40)
The electromagnetic variables can make a complete analogy set with the thermal variables as illustrated in Fig. 1, i.e.
electricpotentialφmagneticpotentialϕtemperatureT
(41)
electricalfluxiemagneticfluximheatfluxq¯
(42)
permittivityϖ0permeabilityμ0conductivityk0
(43)
freechargedensityρfheatsourceq0
(44)
Likewise, this affects the equivalence between elasticity and static electromagnetics.

2.6 Analogy in MEMCT Field Coupling.

The generalized constitutive equations can be written as follows to complement the MEMCT coupling theory [2]
Ψ=MΦ,i
(45)
where the generalized dual variable collection is Ψ = (σij ↔ −qi ↔ −Di ↔ −Bi ↔ −ξi), the state variable collection is Φ,i=(εijθiEiHiϑi), and the generalized material matrix, [M], includes the coupling properties is
M=[cijklλijkeijkdijkaijkkijhijmijbijϖijgijtijsym.μijsijβij]
(46)

The diagonal terms in [M] are the material properties, C, in each individual field, and CM=(cijklkijϖijμijβij), and the other elements in [M] denote the field coupling parameters, A, with A = (λijkeijkdijkaijkhijmijbij ↔ gijtijsij). For example, λijk, eijk, dijk, and aijk are the elastic-thermal tensor, piezoelectric tensor, piezomagnetic tensor, and elastic-chemical tensor, respectively, where λij = cijklαkl with αkl denoting thermal expansion coefficients. hij, mij, bij, gij, tij, and sij 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 CM pertaining to each individual field, in the field coupling parameters A = (λijkeijkdijkaijkhijmijbij ↔ gijtijsij), the mechanics-related coupling parameters, λijkeijkdijkaijk, 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 (qiBi), (ii) state variables (θiHi), (iii) individual parameters (kijμij), and (iv) coupling parameters (λijkdijk) and (hij ↔ gij), where hij are the pyroelectric coefficients and gij the magnetoelectric coefficients.

3 An Application Case

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 [6870]. 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].

As described in Ref. [74], set imperfect mechanical interface conditions, known as the spring-like conditions of layered materials, can be written as
κc[uiL(x,y,h)uiS(x,y,0)]=σ3iL(x,y,h)σ3iL(x,y,h)=σ3iS(x,y,0)κc[0,);i=1,2,3
(47)
where κc is a spring-imperfection index, superscript S or L is used to distinguish variables associated with the layer or the substrate half-space, and h denotes the layer thickness. The spring-like interface conditions in Eq. (47) show that the interfacial stresses are continuous, but the interfacial displacements are discontinuous, governed by the spring index κc. When κc is infinite, uiL(x,y,h) approaches uiS(x,y,0), which is referred to as the perfectly bonded interface condition. A debonding interface occurs when κc approaches zero.
As described in Refs. [71,77], the imperfect thermal impedance conditions of the layered materials can be written as
rt[TL(x,y,h)TS(x,y,0)]=q3L(x,y,h)q3L(x,y,h)=q3S(x,y,0)rt[0,)
(48)
where rt is the thermal conductance, and q3 is the normal heat flux. When rt = 0, the interface is a complete thermal barrier, while heat conduction across the interface is perfect as rt approaches infinity.
As mentioned in Refs. [7880], the imperfect magnetoelectric interface conditions are given by
re[φL(x,y,h)φS(x,y,0)]=D3L(x,y,h)D3L(x,y,h)=D3S(x,y,0)re[0,)
(49)
and
rm[ϕL(x,y,h)ϕS(x,y,0)]=B3L(x,y,h)B3L(x,y,h)=B3S(x,y,0)rm[0,)
(50)
where re is the electric capacitor parameter, rm is the magnetic inductor parameter, D3 denotes the normal electric displacement, B3 is the normal magnetic induction. When re = 0, the interface is a complete electric barrier, while the electric conduction across the interface is perfect as re approaches infinity. Likewise, when rm = 0, the interface becomes a complete magnetic barrier, while the interface is magnetically perfect as rm approaches infinity.
The mass diffusion imperfect interface conditions are given by [81,82]
rc[NL(x,y,h)NS(x,y,0)]=ξ3L(x,y,h)ξ3L(x,y,h)=ξ3S(x,y,0)rc[0,)
(51)
where rc is the transmission parameter, N is the mass concentration, ξ3 is the normal diffusion flux. Similarly, if rc = 0, the interface is a complete diffusion barrier, while the solution transmission is perfect as rc approaches infinity.

Based on the analogies of field collection Φ = (u3TφϕN), and dual variable collection Ψ = (σ3j ↔ −q3 ↔ −D3 ⍰⍰ −B3 ↔ −ξ3), the imperfect conditions in Eqs. (31)(34) can be equivalent by letting κcrtrermrc, for the problem illustrated in Fig. 3. Figures 4 and 5 show the thermal and electric behaviors of this layered half-space with imperfect thermal and electric interfaces, respectively, solved by the current equivalence method based on the elastic solutions in Ref. [74] using the DC-FFT algorithm [59] with a 256 × 256 × 100 grid for the domain of 6.0 mm × 6.0 mm × 4.0 mm. The layer thickness is h = 1.0 mm. Figure 4(a) plots the interfacial heat flux q3L(x,y,h), as well as the interfacial temperature difference, with thermal conductance rt = 100 W/mm2 · K, for the problem of a uniform surface heat input of density 2000 W/mm2 on the region x2+y21.0mm. The substrate conductivity is k2 = 25W/mm · K, and the layer conductivity is k1 = 0.5k2 or 2.0k2. The value of rt[TL(x, y, h) − TS(x, y, 0)] well agrees with q3L(x,y,h), which is identical to the imperfectly thermal boundary condition (Eq. (48)). Furthermore, Fig. 4(b) shows the subsurface temperature distributions along the z-direction for heat conductivities of k1 = 0.5k2 and 2.0k2. The temperature is discontinuous in the layer-substrate interface (z = h = 1.0 mm) due to the imperfect thermal boundary condition.

Figure 5(a) plots the interfacial electric flux D3L(x,y,h), as well as the interfacial electric potential difference, solved by using the current equivalence method, with electric capacitor parameter re = 1.5 × 10−2 C2/Nmm3, for the problem of a uniform surface electric flux of density 2 × 10−2 C/mm2 on region x2+y21.0mm. The substrate dielectric permittivity is ϖ2=1×102C2/Nmm2, and the layer permittivity is ϖ1=0.5ϖ2 or 2.0ϖ2. The value of re[φL(x, y, h) − φS(x, y, 0)] well agrees with D3L(x,y,h), which is identical to the imperfect electric boundary condition given in Eq. (32). Figure 5(b) shows the subsurface electric potential distributions along the z-direction for dielectric permittivities of ϖ1=0.5ϖ2 and 2.0ϖ2. The electric potential is discontinuous in the layer-substrate interface (z = h = 1.0 mm) due to the imperfect electric boundary condition. The distribution behaviors of temperature T and electric potential φ are similar, so are magnetic potential ϕ and mass concentration N, all obeying the analogies of field collection Φ = (u3TφϕN).

4. Discussion and a Concept of Multifield Sensing

4.1 Concept Description.

As shown in Sec. 2, a systematic analogy and unified computation methodology from elasticity to electromagnetic-chemical-thermal fields are developed by setting Poisson’s ratio ν → 0.5 and removing terms multiplied by z with power n. It leads to the identical surface-value relation of the elastic and thermal (or chemical, electric, magnetic) fields if the material is incompressible (ν = 0.5) because the zn-multiplied terms automatically disappear at z = 0. These existing surface-value equivalences indicate that if the surface elastic field of an incompressible material is measured under a unit surface force, the surface thermal (or chemical, electric, magnetic) field under a unit specific flux can be immediately obtained by using parameter conversion without additional thermal (or chemical, electric, magnetic) measurement. For example, the surface-value equivalence for displacement and temperature at any meaningful surface location is uz0(v0.5)=k0T02G, caused by unit surface point fluxes, where uz0 = uz(x, y, z = 0) and T0 = T(x, y, z = 0).

For practical materials which is not incompressible, the following relationship can be obtained from Eq. (11)
uz0uz0(v0.5)=2(1v)
(52)
Equation (52) indicates that the equivalence from surface elasticity to the thermal (or chemical, electric, magnetic) field can be achieved by dividing 2(1 − ν), which leads to the following
β0N0ϖ0φ0μ0ϕ0k0T02Guz0(v0.5)=G(1v)uz0
(53)
where N0 is the surface mass concentration, φ0 is the surface electric potential, ϕ0 is the surface magnetic potential, and T0 is the surface potential. Equation (53) manifests that if the surface elastic field of a material is measured under a unit force, the surface thermal (or chemical, electric, magnetic) field under a unit-specific flux can be immediately obtained through dividing the elastic response by 2(1 − ν).

In fact, if one of these surface field values can be measured under a certain flux strength, the other four under the same shapes of flux distributions can be immediately obtained through Eq. (53). This means that a multifield-multifunctional sensor can be built with one measurement system. Further, with the boundary values known, the entire distribution of one field can be conveniently predicted by the corresponding model. This suggests a multifield model-based monitoring system supported with the multifield-multifunctional surface sensor.

4.2 Case of Using a Thermal Couple for Displacement Measurement.

The concept of multifield sensing can be implemented to measure the surface elastic field via temperature monitoring, or sense the thermal field via surface normal displacement monitoring at one location, or certain locations, by means of the analogy methodology and the surface-value equivalence approach presented in Sec. 4.1. Consider the upper body shown by the left plot of Fig. 6, which is under a steady sliding, or its mating surface is sliding; it receives a steady-state surface heat flux due to friction. The surface radius of this body is R, loaded by surface pressure and heat flux of distribution radius r. The normal displacement or temperature sensors can be located anywhere at the surface from 0 to 10r from the centerline of the body and r < R/10 to satisfy the half-space assumption.

If the Coulomb friction is considered, the frictional heat flux and pressure are in the same shapes of distributions but proportional in values, i.e.,
Heatflux=heatcoefficient×pressure=μT×pressure
(54)
where heat coefficient μT equals to friction coefficient × velocity × heat partition coefficient.
When the thermal couple at location X measures surface temperature T0(X), surface normal displacement uz0(X) is immediately obtained based on the equivalence equation Eq. (53) without using a displacement sensor. The calculation expression is
uz0(X)=(1ν)k0T0(X)GμT
(55)
Again, μT is the heat coefficient in Eq. (54), k0 is the heat conductivity, G is the shear modulus, ν is the Poisson’s ratio. Note that convolution calculation should have been involved; however, the same shape of pressure and frictional heating distributions cancels the convolution operation. Note here, shear traction induced normal displacement is not included.
Equation (55) shows another exciting result; no matter the surface sources are concentrated or distributed, mechanical or others, the relationship of the responses at X still obeys that for concentrated sources as long as the corresponding distribution shapes are the same. This offers a great convenience for sensing, and one sensor, one calculation, applies to all types of loading with simple parameter conversions. The equivalence equations for other fields are given as follows:
uz0(X)=(1ν)k0T0(X)GμT=(1ν)ϖ0φ0(X)GμE=(1ν)μ0ϕ0(X)GμM=(1ν)β0N0(X)GμD
(56)
where μT, μE, μM, and μD are the heat coefficient, electric-mechanical ratio, magnetic-mechanical ratio, and diffusion-mechanical ratio, respectively. Those coefficients evaluate the proportional values between the flux and pressure, along with their same shapes of distributions, as discussed below Eq. (53) and above Eq. (54). If these coefficients and ratios are set to unit, Eq. (56) becomes Eq. (53).

However, caution should be given to material property variations. The rate of degradation of the material parameters pertaining to one field may be different from that of the parameters of another field. In-depth material studies are needed.

5 Conclusions

A systematic analogy and unified computation methodology from elasticity to electromagnetic-chemical-thermal fields have been developed based on the similarities of mechanical-electromagnetic-chemical-thermal (MEMCT) field variables, governing equations, and the material properties pertaining to each individual field. For an homogeneous material, or layered material set, subjected to different surface flux, the thermal, electromagnetic, or chemical solutions can be obtained from the direct degeneration of the corresponding elastic solutions in the same way, instead of solving individual problems in each field, by making

  1. Poisson’s ratio ν → 0.5;

  2. equivalences of 2Gk0ϖ0μ0β0 with G for shear modulus, k0 for heat conductivity, ϖ0 for dielectric permittivity, μ0 for magnetic permeability, and β0 for chemical diffusivity;

  3. removal of the terms that involve z (and its equivalent, h), or zn, as a multiplier in elastic solutions if excited along the z-direction; and likewise, removal of terms with x − or y − multiplication if the excitation source is along the x- or y-direction;

  4. equivalences of ωe2iCp2ωti2ωc, with ωe for elastic wave frequency, ωt for thermal wave frequency, and ωc for diffusion frequency. These equivalences also mark the time-scale differences in different fields.

For static fields, or steady-state elastic, thermal, and chemical fields, measuring one at the surface excited by one type flux is sufficient to sense the others at the same location due to their corresponding flux, and thus a multifunctional sensing system can be developed using one sensor. Furthermore, no matter the surface sources are concentrated or distributed, mechanical or others, the relationship of the responses at the same surface location obeys that for concentrated sources if the corresponding distribution shapes are the same.

The thermal, electromagnetic, and chemical responses of a layered half-space material system with imperfect thermal, electromagnetic, or chemical interfaces have been explored, and the results further confirm that the solutions can be directly obtained by applying the proposed analogy methodology to the elastic solutions.

However, for inhomogeneous materials, the analogy should be done at the stage of the fundamental solutions, for example, the Galerkin-vector stage.

Acknowledgment

The authors would like to thank the support of the Center for Surface Engineering and Tribology at Northwestern University, USA. Zhang X. would also like to acknowledge the supports by the National Natural Science Foundation of China (12102085), the Postdoctoral Science Foundation of China (2020M683278, 2021T140091), and the Sichuan Science and Technology Program (2021YFG0217).

Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. This article does not include any research in which animal participants were involved.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

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