METHOD AND SYSTEM FOR MULTIFIELD, MULTIFUNCTIONAL SENSING

Abstract

A sensing system includes a sensor mounted on a surface that is subjected to a load. The sensor is configured to detect a first field value related to the surface. A memory is configured to store information regarding the surface. A processor is coupled to the memory and in communication with the sensor. The processor is configured to determine a second field value based on the detected first field value and the information regarding the surface.

Description

BACKGROUND

In nature, the macroscopic energy of a material system is commonly in the forms of mechanical, thermal, electromagnetic, and chemical (MEMCT) energies. Therefore, many research efforts have mainly been focused on basic issues of individual mechanical, thermal, electromagnetic, or chemical fields. Recently, problems of multidisciplinary and multifield features have drawn increasingly strong attention.

SUMMARY

An illustrative system includes a sensor mounted on or proximate to a surface that is subjected to a load. The sensor is configured to detect a first field value related to the surface. A memory is configured to store information regarding the surface. A processor is coupled to the memory and in communication with the sensor. The processor is configured to determine a second field value, or additional field values, based on the detected first field value and the information regarding the surface.

In one embodiment, the sensor is a thermal couple or infrared detector such that the first field value comprises a temperature of the surface. The second field value can be electric field potential of the surface or magnetic field potential of the surface, and the processor can determine the electric field potential of the surface or magnetic field potential based at least in part on the temperature. The second field value can also be an amount of displacement of the surface, and the processor can determine the amount of displacement of the surface based at least in part on the temperature. The second field value can also be diffused mass density across the surface, and the processor can determine the diffused mass density of the surface based at least in part on the temperature. In addition, the first field value can be any of the MEMCT fields (e.g., mechanical, thermal, electromagnetic, or chemical), such that any of the other MEMCT fields become the second field. For example, the first field can be displacement detected by an optical sensor, capacitance sensor, etc.

In an illustrative embodiment, the information regarding the surface includes one or more of an elastic modulus of the surface, thermal conductivity of the surface, dielectric permittivity of the surface, magnetic permeability of the surface, and chemical diffusivity of the surface. In another embodiment, the sensor comprises a first sensor in the form of a temperature sensor, and the system also includes a second sensor in the form of a displacement sensor. In such an embodiment, the processor uses measurements from the displacement sensor to adjust a heat-flux coefficient of the surface, and the processor determines the second field value based in part on the adjusted heat-flux coefficient.

In another embodiment, the sensor is located at a distance X from a center of the load that is applied to the surface, and the distance X is less than or equal to ten times a radius of load distribution of the load applied to the surface. In one embodiment, the processor determines the shape of a load distribution of the load, and the processor determines a shape of a heat flux of the surface. The processor compares the shape of the load distribution of the load to the shape of the heat flux, and, responsive to a determination that the shape of the load distribution and the shape of the heat flux are the same, the processor adjusts a heat-flux coefficient for the surface. In another embodiment, the sensor is located at a first location on the surface, and the processor determines one or more field values at a second location on the surface based on a symmetry of the load.

An illustrative method of performing multifield, multifunctional sensing includes sensing, by a sensor mounted on a surface that is subjected to a load, a first field value related to the surface. The method includes storing, in a memory, information regarding the surface. The method also includes determining, by a processor coupled to the memory and in communication with the sensor, a second field value based on the detected first field value and the information regarding the surface.

In one embodiment, the sensor comprises a first sensor in the form of a temperature sensor, and the method further includes sensing, by a second sensor in the form of a displacement sensor, a displacement of the surface in response to the load. The method can also include adjusting, by the processor, a heat-flux coefficient of the surface based at least in part on the sensed displacement of the surface. In such an embodiment, the processor can determine the second field value based in part on the adjusted heat-flux coefficient. In another embodiment, the sensor comprises a temperature sensor that detects a temperature of the surface, and the second field value comprises electric field potential of the surface, magnetic field potential of the surface, an amount of displacement of the surface, or a diffused mass density of the surface. In such an embodiment, the processor determines the second field value based at least in part on the temperature. In another embodiment, the sensor is located at a first location on the surface, and the method further includes determining, by the processor, one or more field values at a second location on the surface based on a symmetry of the load.

Other principal features and advantages of the invention will become apparent to those skilled in the art upon review of the following drawings, the detailed description, and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Illustrative embodiments of the invention will hereafter be described with reference to the accompanying drawings, wherein like numerals denote like elements.

FIG. 1 is a diagram that depicts the main variables and governing equations for mechanical, thermal, electromagnetic, and chemical fields of a material in accordance with an illustrative embodiment.

FIG. 2A depicts surface temperature of a layered half-space solved by the current equivalence method the heat condition solutions, for the problem of a uniform surface heat input in accordance with an illustrative embodiment.

FIG. 2B depicts sub-surface heat flux of the layered half-space solved by the current equivalence method the heat condition solutions, for the problem of a uniform surface heat input in accordance with an illustrative embodiment.

FIG. 3 is a table that summarizes the Galerkin vectors for elastic and temperature problems in accordance with an illustrative embodiment.

FIG. 4 illustrates an example of a layered half-space with imperfect layer-substrate interfaces, where the layer thickness is h, and a uniform force, heat flux, electric/magnetic flux, or diffusion flux is applied on the top surface of the layer in accordance with an illustrative embodiment.

FIG. 5A is a plot of the interfacial heat flux

${\text{q}}_3^L(x,y,h),$

as well as the interfacial temperature difference, with thermal conductance r_t = 100 W/mm² • K, for the problem of a uniform surface heat input of density 2000 W/mm² on region

$\sqrt{x^2+y^2}\le 1.0\text{mm}$

in accordance with an illustrative embodiment

FIG. 5B shows the subsurface temperature distributions along the z direction for heat conductivities of k₁ = 0.5k₂ and 2.0k₂ in accordance with an illustrative embodiment.

FIG. 6A is a plot of the interfacial electric flux

$D_3^L(x,y,h),$

as well as the interfacial electric potential difference, solved by using the current equivalence method, with electric capacitor parameter r_e = 1.5 × 10^-2 C²/Nmm³, for the problem of a uniform surface electric flux of density 2×10^-2 C/mm² on region

$\sqrt{x^2+y^2}\le 1.0\text{mm}$

in accordance with an illustrative embodiment.

FIG. 6B shows the subsurface electric potential distributions along the z direction for dielectric permittivities of ω̅₁ = 0.5ω̅₂ and 2.0ω̅₂ in accordance with an illustrative embodiment.

FIG. 7 depicts various machining tools that can be used in conjunction with multifield, multifunctional sensors in accordance with an illustrative embodiment.

FIG. 8 depicts load distribution on tools surfaces in accordance with an illustrative embodiment.

FIG. 9 depicts sensor location relative to a tool surface in accordance with an illustrative embodiment.

FIG. 10 depicts load distribution of a tool in contact with a known material in accordance with an illustrative embodiment.

FIG. 11 depicts load distribution and tool surfaces of different radii of curvature R in accordance with an illustrative embodiment.

FIG. 12 depicts a sensing system in accordance with an illustrative embodiment.

FIG. 13 depicts a sensing system for performing multifield, multifunctional sensing in accordance with an illustrative embodiment.

DETAILED DESCRIPTION

In many cases of technology developments, the mechanical effects, such as contact of various material systems, have been studied in great detail. For example, indentation mechanics of layered and functionally graded materials have been studied using analytical or numerical methods, micromechanics of materials involving inhomogeneities or inclusions have been studied by the phase-field model or the equivalent inclusion method, and nanomechanics of thin films and nano/micro structures have been studied 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 and the studies of heat conduction in layered and functionally graded materials, as well as in materials involving inhomogeneities or inclusions. Likewise, the Maxwell equations describe how the electric and magnetic fields are generated and mutually influenced. 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.

Similarities in some aspects of mechanical, thermal, electromagnetic, and chemical fields have been used for dimension analyses in surface/interfacial mechanics, nano/mesoscopic mechanics, and continuum mechanics. 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, from Fourier’s law for heat conduction and Ohm’s law for electricity, etc. Based on the concept analogy, for example, between heat flow and mass diffusion, the solution methods used in the thermal or thermomechanical solutions have been applied to analyze diffusion, poroelastic, or chemo-mechanical problems. These methods include the numerical manifold method for transient moisture diffusion in graded materials in analogy to the corresponding transient thermal analysis, the equivalent inclusion method for frictional heating in inhomogeneous half-spaces in analogy to the corresponding inclusion problem in elasticity, the unified solution approach for poroelasticity and thermoelasticity, the prediction method for the conversion between the thermal-energy flow and mass transport, the analogy between shear stress in twisted bar and soap film, and the analogy on material properties among electric, elastic, and piezoelectric parameters. These investigations have intensively explored different facets of the MEMCT fields by analogies of their concepts and solution methods, 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, etc.). Described herein is a methodology for multi-theory analogy across the MEMCT fields for wide ranges of material systems including full-space, half-space, and layered materials loaded by surface or internal sources, and with or without eigenstrains. The methodology and solution framework are based on the concept analogy and similarities of a number of MEMCT field variables, governing equations, and their own material properties, which can help utilizing existing mechanics results to solve other complicated problems. The proposed MEMCT analogy methodology expands the electromagnetic-elastic analogy theory and adds to previous MEMCT coupling theory for better understanding of multi-field interactions, actions, and conversions.

More specifically, described herein is a unified computation methodology, together with a concept of multifield, multifunctional sensing, from elasticity to electromagnetic-chemical-thermal fields based on the similarities of mechanical-electromagnetic-chemical-thermal (MEMCT) field variables, governing equations, and the material properties pertaining to each individual field. Due to similarity, a number of thermal, electromagnetic, or chemical solutions can be obtained from the direct degradation of existing mechanical solutions by making specified equivalences of 2G ←→ k₀ ←→ ω̅₀ ←→ µ₀ ←→ β₀ with G for shear modulus, k₀ for heat conductivity, ω̅₀ for dielectric permittivity, µ₀ for magnetic permeability, and β₀ for chemical diffusivity, as well as setting Poisson’s ratio v = 0.5. These specified equivalences enable quick solutions to other fields directly from mechanical formulations and field coupling directly from analogy, for both the elastic solutions in the forms of Galerkin vectors and Papkovich-Neuber potentials. An example is analyzed to demonstrate that the field solutions of a layered half-space with imperfect thermal, electromagnetic, or chemical interfaces can be directly obtained from the elastic solutions involving interfacial imperfections via the obtained equivalence approach.

The description below includes various formulas and equations that are used to explain the proposed system. The following nomenclature is used for the variables in these equations and formulas. Specifically, B_i is magnetic induction vector, c_ijkl is elastic tensor, C_M is generalized material property, C_M= (c_ijkl ←→ k_ij ←→ ω̅_ij ←→ µ_ij ←→ ƒ_ij), C_p is specific heat capacity, D_i is electric displacement vector, E is Young’s modulus, E_i is electric field strength, ƒ_i is body force component, F_i is generalized volumetric source,

$F_i=\left(f_i\leftrightarrow {\text{q}}_0\leftrightarrow {\rho }_f\leftrightarrow 0\leftrightarrow {\text{W}}_0\right),$

${\stackrel{\widetilde{~}}{F}}_r^{(T)}$

is frequency response functions for T,

${\stackrel{\widetilde{~}}{F}}_r^{(u_z)}$

is frequency response functions for u_z, h is layer thickness, H_i is magnetic field strength, i_e is surface electrical flux, i_m is surface magnetic flux, J_ƒ is current density, k_ij is heat conductivity tensor, m, n are Fourier-transformed variables with respect to x, y directions, N is mass concentration, p̅ is surface normal force, P is surface point load, q₀ is strength of volumetric heat source, q_i is heat flux, q̅ is surface heat flux, Q is surface point heat source, r_c is transmission parameter, r_e is electric capacitor parameter, r_m is magnetic inductor parameter, r_t is thermal conductance, t is time, T is temperature rise, u_i is displacement vector , and W₀ is intensity of local mass source.

Additionally, the following Greek letters are used herein. The letter α is distance of a node, (m, n), to the origin of the frequency domain, β_ij is diffusivity tensor, B is homogeneous and isotropic half-space, ε_ij is elastic strain tensor, θ_i is gradient of temperature rise, Θ is volumetric strain, Θ = u_x,x + u_y,y + u_z,z′ ϑ_i is gradient of mass concentration, κ_c is spring imperfect index, λ, G are Lamé constants,

$\lambda =\frac{vE}{\left[\left(1+v\right)\left(1-2v\right)\right]},$

and

$G=\frac{E}{\left[2\left(1+v\right)\right]},{\mu }_{ij}$

is magnetic permeability tensor, v is Poisson’s ratio, ξ_i is a diffusion flux vector, ω_e is elastic wave frequency, ω_N is diffusion wave frequency, ω_t is thermal wave frequency, ω̅_ij is dielectric permittivity tensor, ρ is mass density, ρ_ƒ is free charge density, σ_ij is elastic stress, φ is electric potential, Ψ refers to generalized dual collection, Ψ = (σ_ij ←→ -q_i ←→ -D_i ←→ -B_i ←→ —ξ_i), ϕ is magnetic potential, Φ refers to generalized field collection, Φ = (u_i ←→ T ←→ φ ←→ ϕ ←→ N), Φ_,i is the gradient of field collection, Φ_,i = (ε_ij ←→ -θ_i ←→ —E_i ←→ —H_i ←→ -ϑ_i), and Ω is a generalized time-related term

FIG. 1 is a diagram that depicts the main variables and governing equations for mechanical, thermal, electromagnetic, and chemical fields of a material in accordance with an illustrative embodiment. In the mechanical field, the gradient of displacement vector u_i leads to the elastic strain tensor ε_ij = (u_i,j + u_j,i)/2 . The elastic strain tensor, ε_ij, and elastic stress tensor, σ_ij, are linked by the constitutive laws of σ_ij = c_ijklε_kl, with c_ijkl for the elastic property tensor. The elastic stress tensor, σ_ij, satisfies the mechanical equilibrium of

${\sigma }_{i,j,i}+f_i=\rho \frac{{\partial }^2{u}_i}{\partial t^2},$

where ƒ_i is the body-force component, ρ is the mass density, and t is time.

In the thermal field, the gradient of temperature rise T is defined as 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

$-q_{i,i}+q_0=\rho C_p\frac{\partial T}{\partial t},$

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 D_i = ω̅_ijE_j, while for magnetism without magnetization is B_i = µi_jH_j, where ω̅_ij 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_ƒ = 0, with ρ_ƒ for free charge density, and the magnetic induction should satisfy Gauss’ law for magnetism of B_i,i = 0. Furthermore, the Maxwell-Faraday equation

$\nabla \times \text{E =}-\frac{\partial \text{B}}{\partial t}$

and Ampère’s circuital law

$\nabla \times \text{H =}\frac{\partial \text{D}}{\partial t}+{\text{J}}_f$

(with J_ƒ for current density) govern the variation of the electromagnetic field.

In the chemical field, the gradient of mass concentration N is defined as vector ϑ_i, satisfying ϑ_i = —N_,i. The diffusion-flux vector, ξ_i, can be given by Fick’s first law of ξ_i = β_ijϑ_j, with β_ij for the diffusivity tensor. The mass balance in a diffusion process is

$-{\xi }_{i,i}+W_0=\rho \frac{\partial N}{\partial t},$

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 non-linear 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 H -D_i ←→ -B_i ←→ —ξ_i), and a generalized material property C_M, yielding C_M = (c_ijkl ←→ k_ij ←→ ω̅_ij ←→ µ_ij ←→ β_ij), 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, τ), with respect to time t and field collection Φ, the generalized equilibrium equation can be written as Ψ_,i + F_i = Ω(Φ,τ), where F_i = (f_i ←→ q₀ ←→ ρ_ƒ ←→ 0 ←→ W₀) is the strength of a generalized volumetric source.

The above variables can make a complete analogy, but it is noted that this analogy can only be stretched so 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 C_M = (c_ijkl ←→ k_ij ←→ ω̅_ij ←→ µ_ij ←→ β_ij), elastic constant c_ijkl is a fourth-order tensor, while heat conductivity k_ij, dielectric permittivity ω̅_ij, 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

$\text{Ω}=\rho \frac{{\partial }^2{u}_i}{\partial t^2}$

while it is

$\text{Ω}=\rho C_p\frac{\partial T}{\partial t}$

in the thermal field, and

$\text{Ω}=\rho \frac{\partial N}{\partial t}$

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 degradation of the corresponding higher-order mechanical variables.

If one considers a homogeneous and isotropic half space B, subjected to a time-varied normal force, p̅(x,y, t), on its boundary along the z direction, or a time-varied heat flux, q̅(x,y, t), normal to the surface. For any material point (x, y, z) inside B, the initial condition at time t = 0 is:

$\begin{array}{c}{u}_x\left(x,y,z,0\right)=u_x^0\left(x,y,z\right),\\ u_y\left(x,y,z,0\right)=u_y^0\left(x,y,z\right),\\ u_z\left(x,y,z,0\right)=u_z^0\left(x,y,z\right),\end{array}$

for elasticity, and

$T\left(x,y,z,0\right)=T^0\left(x,y,z\right),$

for temperature, where

$u_i^0$

is the initial displacement, T⁰ is the initial temperature. For time t > 0, the boundary condition is:

$\begin{array}{l}{\sigma }_{zx}\left(x,y,0,t\right)=0\hfill \\ {\sigma }_{zy}\left(x,y,0,t\right)=0\hfill \\ {\sigma }_{zz}\left(x,y,0,t\right)=-\overline{p}\left(x,y,t\right),\hfill \end{array}$

for elasticity, and

$k_0\frac{\partial T\left(x,y,0,t\right)}{\partial z}=-\overline{q}\left(x,y,t\right),$

for heat conduction, where σ_ij = σ_ji is the stress component, k₀ is an isotropic heat conductivity.

By comparing Equations (1) with (2), and Equations (3) with (4), there is a complete analogy between the variables in elasticity and those in heat conduction: displacement

${\text{displacement u}}_z\leftrightarrow \text{temperature T,}$

$\text{pressure}\overline{p}\leftrightarrow \text{surface heat flux}\overline{q},$

$\text{stress}{\sigma }_{zz}\leftrightarrow k_0\frac{\partial T}{\partial z},$

where stress σ_zz can be written as:

${\sigma }_{zz}=\lambda \text{Θ}+2G{u}_{z,z},$

with

$\lambda =\frac{vE}{\left[\left(1+v\right)\left(1-2v\right)\right]}\text{and}G=\frac{E}{\left[2\left(1+v\right)\right]}$

for the Lamé constants, E for Young’s modulus, v for Poisson’s ratio, and Θ = u_x,x + u_y,y + u_z,z for the volumetric strain. In view of Equations (7) and (8), complete equivalences of u_z ←→ T and

${\sigma }_{zz}\leftrightarrow k_0\frac{\partial T}{\partial z}$

lead to i) the volumetric strain Θ = 0, which is corresponding to incompressible elasticity with Poisson’s ratio v = 0.5; and ii) 2G ←→ k₀.

Furthermore, in order to compare the transient behaviors in elasticity and heat conduction, one of the Lamé-Navier equations is,

$\left(\lambda +\mu \right){\text{Θ}}_{,z}+G{\text{∇}}^2{u}_z+f_z=\rho \frac{{\partial }^2{u}_z}{\partial t^2},$

and the heat-conduction equation is,

$k_0{\nabla }^{2}T+q_0=\rho C_p\frac{\partial T}{\partial t},$

where ƒ_z is the body force along the z direction. Again, there is a complete equivalence between body force and volumetric heat source: f_z ←→ q₀.

The complete equivalence of Equations (9) and (10) results from: i) Θ,_z = 0; ii)

$\frac{\rho }{G}\frac{{\partial }^2{u}_z}{\partial t^2}\leftrightarrow \frac{\rho C_p}{k_0}\frac{\partial T}{\partial t}.$

For i) Θ,_z = 0, this is naturally satisfied due to Θ = 0 stated below Equation (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), z²ƒ(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_τ, condition

$\frac{\rho }{G}\frac{{\partial }^2{u}_z}{\partial t^2}\leftrightarrow \frac{\rho C_p}{k_0}\frac{\partial T}{\partial t}$

yields

${\omega }_e^2\leftrightarrow \frac{i{C}_p}{2}{\omega }_t.$

For example, if an elastic displacement has a harmonic solution of u_z (x,y,z, t) = Z₁(x,y,z) · e^-iωτ with ω for the wave frequency, temperature should have a harmonic solution in the form of

$T\left(x,y,z,t\right)=Z_2\left(x,y,z\right)\cdot e^{\frac{2{\omega }^2}{C_p}\cdot t}.$

In summary, such a complete equivalence between elasticity and heat conduction is obtained by setting: i) Poisson’s ratio v = 0.5; ii) shear modulus 2G ←→ heat conductivity k₀; iii) removing term with z if it is presented individually in the elastic solutions, such as zu_z, _z²u_z, etc.; and iv) elastic wave frequency

${\omega }_e^2\leftrightarrow$

thermal wave frequency

$-\frac{i{C}_p}{2}{\omega }_t.$

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.

Solutions using Papkovich-Neuber potentials are described below. Considering an isotropic and homogeneous elastic half-space, whose surface is subjected to normal force P at the origin and along the z direction, the elastic displacement, u_z, is:

$u_z\left(x,y,z\right)=\frac{P}{4\pi G}\left[\frac{z^2}{r^3}+\frac{2\left(1-v\right)}{r}\right],$

where

$r=\sqrt{x^2+y^2+z^2}.$

For the problem of an isotropic and homogeneous half-space subjected to a surface point heat source, Q, by making v = 0.5, 2G →k₀, P → Q and removing the term with individual variable z, which means to remove the first term,

$\frac{z^2}{r^3},$

in Equation (11), the temperature distribution can be obtained as follows:

$T\left(x,y,z\right)=\frac{Q}{2\pi k_0}\frac{1}{r},$

A layered half-space was also considered. 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) can be solved first. The FRFs for normal displacement u_z are:

$\begin{array}{l}{\tilde{\tilde{F}}}_r^{\left(u_z\right)}=\frac{1}{2G_r}\left[-\alpha \left(D_r{e}^{-\alpha z_r}-{\overline{D}}_r{e}^{\alpha z_r}\right)-\left(3-4v_r\right)\right)\\ \left(\left(C_r{e}^{-\alpha z_r}+{\overline{C}}_r{e}^{\alpha z_r}\right)-z_r\alpha \left(C_r{e}^{-\alpha z_r}-{\overline{C}}_r{e}^{\alpha z_r}\right)\right]\end{array}$

where

$\alpha =\sqrt{m^2+n^2}$

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 substrate half-space, respectively. Coefficients C_r, C_r, D_r and D_r (C₂ = D₂ = 0) can be found in the literature.

For the problem of a layered half-space subjected to a unit point heat source, the inventors used 2G_r ←→ k_r and v_r = 0.5, and removed the term with individual z and h, because h resembles z, which means that the last term, z_rα(C_re^-αzr- C_re^αzr) in Equation (13) was removed, and the term was multiplied by 2αh in C_r, C_r, D_r and D_r. Therefore, the FRFs for temperature T can be obtained from Equation (13),

${\tilde{\tilde{F}}}_r^{\left(T\right)}=\frac{1}{k_r}\left(C_r{e}^{-\alpha z_r}+{\overline{C}}_r{e}^{\alpha z_r}\right),$

where

$\begin{array}{l}{C}_1=\frac{1}{\alpha \left(1-k_3{e}^{-2\alpha h}\right)},{\overline{C}}_1\frac{k_3{e}^{-2\alpha h}}{\alpha \left(1-k_3{e}^{-2\alpha h}\right)},C_2=\\ \frac{\left(1-k_3\right)e^{-\alpha h}}{\alpha \left(1-k_3{e}^{-2\alpha h}\right)},\text{and}{D}_r={\overline{D}}_r=0,\end{array}$

with h for layer thickness, and k₃ = (k₁ - k₂)/(k₁ + k₂).

FIG. 2 compares the surface temperature and sub-surface heat flux distributions of a layered half-space solved by the current equivalent method via elastic solutions and the heat conduction solutions, using a discrete convolution-fast Fourier transform (DC-FFT) algorithm 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 region

$\sqrt{x^2+y^2}\le$

1.0 mm. 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. FIG. 2A depicts surface temperature of a layered half-space solved by the current equivalence method the heat condition solutions, for the problem of a uniform surface heat input in accordance with an illustrative embodiment. FIG. 2B depicts sub-surface heat flux of the layered half-space solved by the current equivalence method the heat condition solutions, for the problem of a uniform surface heat input in accordance with an illustrative embodiment. As shown, the results of the temperature and heat flux from the current equivalent method based on the elastic solutions, are identical to those from the heat conduction solutions.

Solutions built on the Galerkin vectors were explored. One can imagine an isotropic and homogeneous media (e.g. full space or half space) contains a volumetric irregular region subject to eigenstrains distributed arbitrarily. The elastic displacement u_i caused by the eigenstrains can be expressed in terms of the Galerkin vectors, F_i,

$2G{u}_i=2\left(1-v\right)F_{i,jj}-F_{j,ji}$

In Equation 15, it is noted that 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. The subscripts 1, 2 and 3 of variables indicate those along the x, y and z directions, respectively.

Complete equivalences of u_z ←→ T and Poisson’s ratio v = 0.5 lead to the temperature field in terms of the Galerkin vectors, F_i,

$u_i=\frac{1}{2G}{\left(F_{i,jj}-F_{j,ji}\right)}_{v=0.5}\to T=D_1\left(F_{3,jj}-F_{3,33}\right)=D_1\left(F_{,jj}^T-F_{,33}^T\right),$

where D₁ is a constant. Substitution of Equation (16) into the heat conduction equation (Equation (10)) and ignoring the transient term and body source, yield to the following. For a homogeneous full-space, one can consider single mechanical forces in the x, y, or z directions of the full space, such that the magnitudes of the Galerkin vectors are all the same as

$\begin{array}{l}F=C_A{R}^I=\frac{R^I}{8\pi (1-v)}{\text{, where R}}^I=\\ \sqrt{{(x-x\prime )}^2+(y-y\prime )+{(z-z\prime )}^2}.\end{array}$

According to Equation (15), the elastic displacement u_z is,

$\begin{array}{l}2G{u}_z=2\left(1-v\right)F_{,jj}-F_{,33}\\ =\frac{1}{8\pi \left(1-v\right)}\left(4\left(1-v\right){\varphi }^I-{\varphi }^I-\left(z-z\prime \right){\varphi }_{,3}^I\right)\\ \stackrel{\text{remove}z\text{-term}}{\to }\frac{3-4v}{8\pi \left(1-v\right)}\frac{1}{R}\end{array}$

where

${\varphi }^I=1/R^I{}_{\text{and}}R=\sqrt{{\left(x-x\prime \right)}^2+{\left(y-y\prime \right)}^2+{\left(z+z\prime \right)}^2}.$

By using the equivalences between Equation (17) and Equation (11), which are: i) Poisson’s ratio v = 0.5; ii) shear modulus 2G ←→ heat conductivity k₀; iii) removing term z if it is presented individually in elastic solutions, Equation (17) becomes the following:

$T=\frac{1}{4\pi k_0}{\frac{Q}{R^I}}_{,}$

where the displacement u_z ↔ temperature T. Equation (18) is exactly the full-space heat conduction solution for temperature field under a steady point source, twice that of the half-space solutions. Therefore,

$D_1={\frac{Q}{4\pi k_0}}_{,}$

and F^T = R^I by the comparison of Equations (16) and (18).

For a half space loaded by a surface force along the z direction, the Galerkin vector is:

$g_3=\frac{1}{8\pi \left(1-v\right)}{\left[\begin{array}{l}0\\ 0\\ R^I+\left(3-4\upsilon \right)R-2z\cdot z\prime \varphi -4Dz\prime \psi -4HD\theta \end{array}\right]}^T$

where D=1-2v θ = R - zln[R + z], ψ = ln[R + z]. By dropping the terms with D and z-terms, Equation (19) becomes,

$g_3=\frac{1}{4\pi \left(1-v\right)}{\left[\begin{array}{l}0\\ 0\\ R^I\end{array}\right]}^T{,}_{}$

which is twice that of the full space solution (same with the temperature field).

A half-space with inside sources was also considered. The Galerkin vector for a single force along z direction inside the half-space is,

$g_3=\frac{1}{8\pi \left(1-v\right)}{\left[\begin{array}{l}0\\ 0\\ R^I+\left(3-4\upsilon \right)R-2z\cdot z\prime \varphi -4Dz\prime \psi -4HD\theta \end{array}\right]}^T$

The displacement u_z is,

$8\pi G{u}_z=\frac{1}{R^I}+\frac{1}{R}$

While the Galerkin vector for heat conduction is

$F=g_3=\frac{1}{8\pi \left(1-v\right)}{\left[\begin{array}{l}0\\ 0\\ R^I+R\end{array}\right]}^T$

Therefore, the temperature is

$T=D_1\left(F_{,jj}^T-F_{,33}^T\right)=\frac{Q}{4\pi k_0}\left(\frac{1}{R^I}+\frac{1}{R}\right)$

Equation (22) is equivalent to Equation (24) by letting 2G ↔ k₀.

A full-space with eigenstrains was also explored. Considering an isotropic and homogeneous full space contains a volumetric irregular region subject to eigenstrains distributed arbitrarily, the full-space elastic problem for with eigenstrains is solved in part using a core function, where the core function is

$\text{F}\left(\text{x}\right)=\frac{G}{4\pi \left(1-v\right)}{{\displaystyle {\int }_{\Omega }\left(2v{e}_{kk}{g}_c-e_{jk}{g}_{jk}\right)d\text{x}\prime }}_{.}$

Letting Poisson’s ratio v = 0.5 the core function becomes,

$F=-\frac{G}{2\pi }{\left[\begin{array}{c}{e}_{1j}{R}_{,j}^I\\ e_{2j}{R}_{,j}^I\\ e_{3j}{R}_{,j}^I\end{array}\right]}^T$

By using the F₃ component, the temperature in Equation (16) can be expressed as:

$T=D_1\left(F_{,jj}^T-F_{,33}^T\right)=D_1{e}_{3j}\left(R_{,jkk}^I-R_{,j33}^I\right)\to D_1{e}_{3j}{\varphi }_{,j}^I{}_{,}$

where e_3j is the eigen-temperature gradient. By comparison of Equations (25) and (26) and considering 2G ↔ k₀, it yields to

$D_1={\frac{k_0}{4\pi }}_{,}$

which is identical to that in the literature.

For a half-space with eigenstrains, by letting the Poisson’s ratio v = 0.5 and dropping the z-containing terms, the Galerkin vectors are

$g_{i1}=-{\left[\begin{array}{ccc}\frac{\partial \left(R^I+R\right)}{\partial x\prime }& \frac{\partial \left(R^I+R\right)}{\partial y\prime }& \frac{\partial \left(R^I+R\right)}{\partial z\prime }\\ 0& 0& 0\\ 0& 0& 2R_{,1}\end{array}\right]}^T$

$g_{i2}=-{\left[\begin{array}{ccc}0& 0& 0\\ \frac{\partial \left(R^I+R\right)}{\partial x\prime }& \frac{\partial \left(R^I+R\right)}{\partial y\prime }& \frac{\partial \left(R^I+R\right)}{\partial z\prime }\\ 0& 0& 2R_{,2}\end{array}\right]}^T$

$g_{i3}=-{\left[\begin{array}{l}\\ \begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ \frac{\partial \left(R^I+R\right)}{\partial x\prime }& \frac{\partial \left(R^I+R\right)}{\partial y\prime }& \frac{\partial \left(R^I+R-2z\cdot z\prime \varphi \right)}{\partial z\prime }\end{array}\end{array}\right]}^T$

Therefore,

$g={\left[\begin{array}{c}-e_{1j}\left(R^I+R\right){,}_j+2e_{13}R{,}_3\\ -e_{2j}\left(R^I+R\right){,}_j+2e_{13}R{,}_3\\ -e_{j3}\left(R{,}_{j^{}}^I+R{,}_j\right)2e_{33}z\varphi \end{array}\right]}^T={\left[\begin{array}{c}-e_{1j}\left(R^I+R\right){,}_j+2e_{13}R{,}_3\\ -e_{2j}\left(R^I+R\right){,}_j+2e_{32}R{,}_3\\ -e_{j3}\left(R{,}_{j^{}}^I+R{,}_j\right)\end{array}\right]}^T$

By setting

$F^T=-e_{j3}\left(R_j^I-R_j\right),$

the temperature is

$T=\frac{k_0}{4\pi }\left(F_{,jj}^T-F_{,33}^T\right)=\frac{k_0}{4\pi }{e}_{3j}\left({\varphi }_{,j}^I+{\varphi }_{,j}\right)$

In summary, by making the mechanical-thermal equivalence: (i) If the results are structured from the Papkovich-Neuber solutions, which only have harmonic potentials, the analogy can be done directly at the final solutions, or at any step of the solution derivation; (ii) For solutions built on the Galerkin vectors, one can use F₃ only due to solution symmetry for half-space problems, but use of 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 are proper for the Galerkin vectors for temperature. For homogeneous materials, direct result analogy is applicable, but for inhomogeneous materials, analogy should be done at the Galerkin-vector stage. FIG. 3 is a table that summarizes the Galerkin vectors for elastic and temperature problems in accordance with an illustrative embodiment.

Thermal-chemical equivalence was also studied. Again, one can consider a chemical species diffusing into a homogeneous and isotropic half-space B across 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:

$N\left(x,y,z,0\right)=N^0\left(x,y,z\right),$

with N⁰ for the initial mass concentration of the diffusive species. For time t > 0, the boundary condition is,

${\beta }_0\frac{\partial N\left(x,y,0,t\right)}{\partial z}\overline{\xi }\left(x,y,t\right),$

where β₀ is the diffusivity.

It has been shown in FIG. 1 that the chemical variables are in a complete analogy with the heat-conduction variables, i.e.:

$\text{mass}\text{concentration}N\leftrightarrow \text{temperature}T,$

$\text{surface}\text{diffusive}\text{flux}\overline{\xi }\leftrightarrow \text{surface}\text{heat}\text{flux}\overline{q},$

$\text{diffusivity}{\beta }_0\leftrightarrow \text{conductivity}{k}_0,$

$\text{mass}\text{source}{W}_0\leftrightarrow \text{heat}\text{source}{q}_0,$

$\frac{\partial N}{\partial t}\leftrightarrow C_p\frac{\partial T}{\partial t}.$

It is found from Equation (38) that the time scales in chemical and thermal fields are different. If the mass concentration N is defined as a harmonic function F_N of diffusion wave frequency ω_N, and the temperature T is defined as F_t of thermal wave frequency ω_t, mentioned above, condition

$\frac{\partial N}{\partial t}\leftrightarrow C_p\frac{\partial T}{\partial t}$

result in ω_N ↔ C_pω_t. The equivalence between heat conduction and chemical diffusion also effects the equivalence between elasticity and chemical diffusion.

Thermal-electromagnetic equivalence was considered. 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.. Considering the fact that the frequency of magnetoelectric waves is much higher than those of the mechanical, thermal, or chemical diffusion waves, only the static electromagnetic solutions are carried out here. If the transient behaviors of electromagnetic field are ignored, the Maxwell-Faraday equation and Ampère’s circuital law vanish, and only the Gauss’ laws for electricity and magnetism remain.

One can consider a homogeneous and isotropic half-space, B, subjected to static surface electrical flux i_e(x,y) and surface magnetic flux i_m(x,y). For any material point (x, y, z) inside B, the boundary conditions are,

$D\left(x,y,0,t\right)=-i_e\left(x,y,t\right),$

$B\left(x,y,0,t\right)=-i_m\left(x,y,t\right).$

The electromagnetic variables can make a complete analogy set with the thermal variables as illustrated in FIG. 1, i.e.:

$\text{electric}\text{potential}\phi \leftrightarrow \text{magentic}\text{potential}\varphi \leftrightarrow \text{temperature}T,$

$\text{electrical}\text{flux}{i}_e\leftrightarrow \text{magnetic}\text{flux}{i}_m\leftrightarrow \text{heat}\text{flux}\overline{q},$

$\text{permittivity}{\text{ϖ}}_0\leftrightarrow \text{permeability}{\mu }_0\leftrightarrow \text{conductivity}{k}_0,$

$\text{free charge density}{\rho }_f\leftrightarrow \text{heat source}{q}_0.$

Likewise, this effects the equivalence between elasticity and static electromagnetics.

The inventors also studied the Equivalence in MEMCT field coupling. The generalized constitutive equations can be written as follows to complement the MEMCT coupled theory:

$\text{Ψ=}\text{M}\cdot \text{Φ}{}_{,i}.$

where the generalized dual variable collection is Ψ = (σ_ij ↔ -q_i ↔ -D_i ↔ -B_i ↔ -ξ_i), the state variable collection is Φ,_i = (ε_ij ↔ -θ_i ↔ -E_i ↔ -H_i ↔ -ϑ_i), and the generalized material matrix, [M], includes the coupling properties is,

$\text{M=}\left[\begin{array}{ccccc}{c}_{ijkl}& {\lambda }_{ijk}& e_{ijk}& d_{ijk}& a_{ijk}\\ & k_{ij}& h_{ij}& m_{ij}& b_{ij}\\ & & {\varpi }_{ij}& g_{ij}& t_{ij}\\ & \text{sym}\text{.}& & {\mu }_{ij}& s_{ij}\\ & & & & {\beta }_{ij}\end{array}\right].$

The diagonal terms in [M] are the material properties, C, in each individual field, and C_M = (c_ijkl ↔ k_ij ↔ ω̅_ij ↔ µ_ij ↔ β_ij), 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 analyses above shows 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 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. Due to property mismatching, crystal defects, or impurities, the material interfaces may involve a bonding issue. The imperfect interfaces would affect the transmissions of force and/or displacement in elasticity, influence heat transfer, cause the imperfect gluing and/or soldering in electronic packaging, and reduce the charge-diffusion rate in batteries. The imperfect interfaces in elasticity have been addressed most deeply, such as the spring-like interface in layered materials, coupled dislocation-like and force-like interfaces in layered materials, and in joined half-spaces. FIG. 4 illustrates an example of a layered half-space with imperfect layer-substrate interfaces, where the layer thickness is h, and a uniform force, heat flux, electric/magnetic flux, or diffusion flux is applied on the top surface of the layer in accordance with an illustrative embodiment.

A set imperfect mechanical interface conditions, known as the spring-like conditions of layered materials, can be written as,

$\begin{array}{l}{\kappa }_c\left[u_i^L\left(x,y,h\right)-u_i^S\left(x,y,0\right)\right]=-{\sigma }_{3i}^L\left(x,y,h\right),\\ {\sigma }_{3i}^L\left(x,y,h\right)={\sigma }_{3i}^S\left(x,y,0\right),\\ {\kappa }_c\in \left[0,\left(\infty \right);i=1,2,3,\right)\end{array}$

where κ_c 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 Equation (47) show that the interfacial stresses are continuous, but the interfacial displacements are discontinuous, governed by spring index κ_c. When κ_c is infinite,

$u_i^L\left(x,y,h\right)$

approaches

$u_i^S\left(x,y,0\right),$

which is referred to as the perfectly bonded interface condition. A debonding interface occurs as κ_c approaches zero.

The imperfect thermal impedance conditions of the layered materials can be written as,

$\begin{array}{l}{r}_t\left[T^L\left(x,y,h\right)-T^S\left(x,y,0\right)\right]=q_3^L\left(x,y,h\right),\\ q_3^L\left(x,y,h\right)=q_3^S\left(x,y,0\right),\\ r_t\in \left[0,\left(\infty \right)\right),\end{array}$

where r_t is the thermal conductance, and q₃ is the normal heat flux. When r_t = 0, the interface is a complete thermal barrier, while the heat conduction across the interface is perfect as r_t approaches infinity.

The imperfect magnetoelectric interface conditions are given by,

$\begin{array}{l}{r}_e\left[{\phi }^L\left(x,y,h\right)-{\phi }^S\left(x,y,0\right)\right]=D_3^L\left(x,y,h\right),\\ D_3^L\left(x,y,h\right)=D_3^S\left(x,y,0\right),\\ r_e\in \left[0,\left(\infty \right)\right),\end{array}$

and

$\begin{array}{l}{r}_m\left[{\varphi }^L\left(x,y,h\right)-{\varphi }^S\left(x,y,0\right)\right]=B_3^L\left(x,y,h\right),\\ B_3^L\left(x,y,h\right)=B_3^S\left(x,y,0\right),\\ r_m\in [0,\left(\infty \right),\end{array}$

where r_e is the electric capacitor parameter, r_m is the magnetic inductor parameter, D₃ denotes the normal electric displacement, B₃ is the normal magnetic induction. When r_e = 0, the interface is a complete electric barrier, while the electric conduction across the interface is perfect as r_e approaches infinity. Likewise, when r_m = 0, the interface has a completely magnetic barrier, while the interface is magnetically perfect as r_m approaches infinity.

The mass diffusion imperfect interface conditions are given by:

$\begin{array}{l}{r}_c\left[N^L\left(x,y,h\right)-N^S\left(x,y,0\right)\right]={\xi }_3^L\left(x,y,h\right),\\ {\xi }_3^L\left(x,y,h\right)={\xi }_3^S\left(x,y,0\right),\\ r_c\in \left[0,\left(\infty \right)\right),\end{array}$

where r_c is the transmission parameter, N is the mass concentration, ξ₃ is the normal diffusion flux. Similarly, if r_c = 0, the interface is a complete diffusion barrier, while the solution transmission is perfect as r_c approaches infinity.

Based on the analogies of field collection Φ = (u₃ ↔ T ↔ φ ↔ ϕ ↔ N), and dual variable collection Ψ = (σ_3j ↔ -q₃ ↔ -D₃ ↔ -B₃ ↔ -ξ₃), the imperfect conditions in Equations (31-34) can be equivalent by letting κ_c, ↔ r_t ↔ r_e ↔ r_m ↔ r_c, for the problem illustrated in FIG. 4. FIGS. 5 and 6 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 using the discrete convolution and fast Fourier transform (DC-FFT) algorithm 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.

Specifically, FIG. 5A is a plot of the interfacial heat flux

$q_3^L\left(x,y,h\right),$

as well as the interfacial temperature difference, with thermal conductance r_t = 100 W/mm² · K, for the problem of a uniform surface heat input of density 2000 W/mm² on region

$\sqrt{x^2+y^2}\le$

1.0 mm in accordance with an illustrative embodiment. The substrate conductivity is k₂ = 25 W/mm · K, and the layer conductivity is k₁ = 0.5k₂ or 2.0k₂. The value of r_t[T^L(x,y,h) - T^S(x,y,0)] agrees well with

$q_3^L\left(x,y,h\right),$

which is identical to the imperfectly thermal boundary condition in Equation (48). Furthermore, FIG. 5B shows the subsurface temperature distributions along the z direction for heat conductivities of k₁ = 0.5k₂ and 2.0k₂ in accordance with an illustrative embodiment. The temperature is discontinuous in the layer-substrate interface (z=h=1.0 mm) due to the imperfectly thermal boundary condition.

FIG. 6A is a plot of the interfacial electric flux

$D_3^L\left(x,y,h\right),$

as well as the interfacial electric potential difference, solved by using the current equivalence method, with electric capacitor parameter r_e = 1.5 × 10^-2 C²/Nmm³, for the problem of a uniform surface electric flux of density 2×10^-2 C/mm² on region

$\sqrt{x^2+y^2}\le 1.0\text{mm}$

in accordance with an illustrative embodiment. The substrate dielectric permittivity is ω̅₂=1×10^-2 C²/N·mm², and the layer permittivity is ω̅₁ = 0.5ω̅₂ or 2.0ω̅₂. The value of r_e[φ^L(x,y,h) - φ^s(x,y,0)] agrees well with

$D_3^L\left(x,y,h\right),$

which is identical to the imperfect electric boundary condition used. FIG. 6B shows the subsurface electric potential distributions along the z direction for dielectric permittivities of ω̅₁ = 0.5ω̅₂ and 2.0ω̅₂ in accordance with an illustrative embodiment. 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 = (u₃ ↔ T ↔ φ ↔ ϕ ↔ N).

Thus, it has been shown that for homogeneous materials or homogenized materials, a systematic analogy and unified computation methodology from elasticity to electromagnetic-chemical-thermal fields are developed by setting Poisson’s ratio v = 0.5 and removing terms multiplied by the power of z. This leads to the identical surface-value relation (z-multiplied terms automatically disappear at z = 0) of the elasticity and thermal (or chemical, electric, magnetic) fields if the material is incompressible (v = 0.5). The result indicates 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. If N₀ is the surface mass concentration, φ₀ is the surface electric potential, ϕ₀ is the surface magnetic potential, and T₀ is the surface potential, this means

$\frac{u_{z0}}{T_0}=\frac{P}{Q}\frac{k_0}{2G}$

$\frac{u_{z0}}{{\phi }_0}=\frac{P}{I_e}\frac{{\varpi }_0}{2G}$

$\frac{u_{z0}}{{\varphi }_0}=\frac{P}{I_m}\frac{{\mu }_0}{2G}$

$\frac{u_{z0}}{N_0}=\frac{P}{J_f}\frac{{\beta }_0}{2G}$

For practical materials which are not incompressible, the following relationship can be obtained from Equation (11):

$\frac{u_{z0}}{u_{z0}\left(v=0.5\right)}=2\left(1-v\right),$

where u_z0 = u_z(x,y,z = 0) is the surface elastic displacement.

Equation (56) indicates that the analogy from surface elasticity to the thermal (or chemical, electric, magnetic) field can be achieved by dividing 2(1 - v), which leads to the following:

$N_0\leftrightarrow {\phi }_0\leftrightarrow {\varphi }_0\leftrightarrow T_0\leftrightarrow u_{z0}\left(v=0.5\right)=\frac{u_{z0}}{2\left(1-v\right)},$

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 (57) 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 - v).

In fact, if one of these surface field values can be measured, the other four can be immediately obtained.

$\frac{u_{z0}}{T_0}=2\left(1-v\right)\frac{P}{Q}\frac{k_0}{2G}$

$\frac{u_{z0}}{{\phi }_0}=2\left(1-v\right)\frac{P}{I_e}\frac{{\varpi }_0}{2G}$

$\frac{u_{z0}}{{\varphi }_0}=2\left(1-v\right)\frac{P}{I_m}\frac{{\mu }_0}{2G}$

$\frac{u_{z0}}{N_0}=2\left(1-v\right)\frac{P}{J_f}\frac{{\beta }_0}{2G}$

This means that a multifunctional sensor can be built with on measurement system to measure one of the pairs of (P, u_zo), (Q, T_o), (l_e, φ₀), (I_m, ϕ₀), (J_f, N₀). This idea can be extended to other bodies of a homogeneous (or homogenized) material, as long as the boundary conditions are consistent for the five fields of MEMCT, either equations like (58-61) can be derived, or their relations can be established through calibrations of five initial measurements with different sensors. 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 multifunctional surface sensor.

Thus, a systematic analogy and unified computation methodology from elasticity to electromagnetic-chemical-thermal fields are developed herein 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 isotropic material, or layered material set, subjected to different surface flux, instead of solving individual problems in each field, the thermal, electromagnetic, or chemical solutions are obtained from the direct degeneration of the corresponding elastic solutions by making Poisson’s ratio v = 0.5, equivalences of 2G ↔ k₀ ↔ ω̅₀ ↔ µ₀ ↔ β₀ with G for shear modulus, k₀ for heat conductivity, ω̅₀ for dielectric permittivity, µ₀ for magnetic permeability, and β₀ for chemical diffusivity. The system also performs removal of the terms that involve z (and its equivalent, h), or the power of z, as a multiplier in elastic solutions if excited along the z direction; and likewise removal of x or y if the excitation source is along the x or y direction. The system also makes equivalences of

${\omega }_e^2\leftrightarrow -\frac{\text{i}{c}_p}{2}{\omega }_t\leftrightarrow -\frac{\text{i}}{2}{\omega }_c,$

with ω_e for elastic wave frequency, ω_t for thermal wave frequency, and ω_c 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. The thermal, electromagnetic, or chemical response of a layered half-space material system with imperfect thermal, electromagnetic, or chemical interfaces is explored, and the results further confirm that the solutions can be directly obtained by applying the proposed analogy method to the elastic solutions.

FIG. 7 depicts various machining tools that can be used in conjunction with multifield, multifunctional sensors in accordance with an illustrative embodiment. The machining tools include forming tools (e.g., a router bit), cutting tools (e.g., a blade or knife), drilling tools (e.g., drill bits), and polishing tools (e.g., cloth, sanding block, etc.). The tools can be made of any tool steel, ceramics, or other metal, and the workpieces can be any materials. Machining tool surface elastic/thermal/electrical/magnetic/mass diffusion fields can be determined via monitoring the surface displacement or temperature at one location, or certain locations by means of the theory of analogy as described herein.

FIG. 8 depicts load distribution on tools surfaces in accordance with an illustrative embodiment. The tool shape can be any, and the tip can be considered as a sphere of radius R, which is also radius of curvature, or an ellipsoid with one of the principal radii of curvature to be R. The load (source) can be pressure, heat flux, mass flux, electrical flux, magnetic flux, etc. centered at location O. The variable r is the radius of load distribution, and 2r is the load distribution width (if there is a 2D problem). The variable R is the radius of a tool tip, for example from 10 microns (sharp tooling) to infinite (surface tooling), i.e., half space. The sensor location X can be anywhere from X=0 to X=~10r. The response to the tooling of displacement, temperature, electrical potential, magnetic potential, and mass concentration are obtained at location X.

FIG. 9 depicts sensor location relative to a tool surface in accordance with an illustrative embodiment. The concentrated load can be pressure, heat flux, mass flux, electrical flux, or magnetic flux at location O. When r is very small, the load becomes concentrated. R represents the radius of the tool, and sensor location X can be anywhere between X=0 and X=R/10. Load flux is at a point (line) (location O), its strength can be the average of a flux distribution if the flux distribution width (radius) is less than ~⅒ of the tool-tip radius. As noted, if the fluxes are related to pressure, then heat flux = heat flux coefficient x pressure, and mass flux = diffusion coefficient x pressure. Measuring the temperature at X can directly result in displacement, electrical field strength, magnetic field strength, and diffused mass concentration at the same location, using Equations (58-61). Equations (58-61) also give the responses at the source location O using Equation 12.

Still referring to FIG. 9, in another example, the heat flux = heat flux coefficient x mechanical load (pressure) with heat flux coefficient = friction coefficient x heat partition x velocity. If a thermal couple (temperature sensor) located at X measures the temperature as T(X), then the normal displacement u_z at X is

$u_z=\frac{\left(1-v\right)k_{o}T\left(X\right)}{G{\mu }_f},$

where µ_f is heat-flux coefficient, k_o is heat conductivity, and G is shear modulus.

Again referring to FIG. 9, it is noted that the relationship between each flux and response at X can be calibrated with the methods described above (i.e., with reference to FIG. 9). The calibration can also result in the relationship between the response at X (such as T(X)) and the response at the source location (machining location, T(O)) using Equation (12), i.e., the temperature relationship, the displacement relationships, the diffusion mass relationships, etc. Then monitoring the response values at X equals monitoring the response (and the flux source) changes at the source location, i.e., the pressure, heat source, mass flux as the source fluxes, displacement, temperature, mass concentration, etc., as the responses at the source location. The rate of changes also marks the rate of changes of the electromagnetic fluxes and fields, and their field values at O and X.

FIG. 10 depicts load distribution of a tool in contact with a known material in accordance with an illustrative embodiment. In a scenario in which r (radius of load distribution) is not small, but r < R/10, the load becomes distributed. Again, 2r refers to load distribution width (if a 2D problem), R is the radius of the tool body, and sensor location can be anywhere between X=0 and X=~10r. The load can be pressure, heat flux, mass flux, electrical flux, or magnetic flux. In FIG. 10, if the fluxes are in the same shape of the distributions, but proportional in value, then Heat flux = Heat-flux coefficient x pressure, and Mass flux = diffusion coefficient x pressure. Also, if a thermal couple at X measures the temperature as T(X), then the normal displacement u_z at X is the same as discussed above with reference to FIG. 9 (i.e.,

$u_z=\frac{\left(1-v\right)k_{o}T\left(X\right)}{G{\mu }_f}$

)because the convolution expressions are cancelled.

It is also noted that Equations 58-61 are applicable for both concentrated and distributed fluxes as long as the flux distributions are in the same shape. The equation

$u_2=\frac{\left(1-v\right)k_{o}T\left(X\right)}{G{\mu }_f}$

shows that a temperature sensor becomes a multifield sensor with Equations 58-61, and that bringing u_z to Equations 59-61allows other field values to be readily obtained. If the fluxes are not in the same shape of the distribution, individual integration/convolution can be performed using Equations 58-61 (via Equation 12) to find the other field values.

Still referring to FIG. 10, the relationship between each flux and response at X can be calibrated from the method discussed above. The calibration can also result in the relationship between the response at X and the response at the source location (machining location), i.e., the temperature relationship, the displacement relationships, the diffusion mass relationships, etc. Then monitoring the response values at X equals monitoring the response (and the flux source) changes at the source location, i.e., the pressure, the heat source, mass flux as the source fluxes, and displacement, temperature, and mass concentration, etc., as the responses at the source location. The rate of changes also marks the rate of changes of the electromagnetic fluxes and fields, and the field values at O and X.

FIG. 11 depicts load distribution and tool surfaces of different radii of curvature R in accordance with an illustrative embodiment. If the load is axis-symmetry, the measurement results at each X represent the results on a circle with X as the radius. Changing X leads to the surface field distributions. If the load is mirror symmetry about the dash-dot line, the field value at -X equals that at X. If the load is not axis-symmetry, the measurement should be done at each surface location of interest.

FIG. 12 depicts a sensing system in accordance with an illustrative embodiment. A sensor is mounted relative to a tool surface, as shown. In one embodiment, the sensor can be a thermocouple or infrared temperature detector. In an illustrative embodiment, a computing device 1200 receives sensed data from the sensor. The computing device 1200 can include a processor, a memory, a transceiver (receiver and transmitter), etc. The processor can be configured to implement Equations 58-61 to perform parameter conversions, etc. For example, the sensor can be used to measure temperature, which is provided to the computing device 1200, which uses Equations 58-61 to calculate other field values. Inputs can include elastic modulus, Poisson’s ratio, thermal conductivity, dielectric permittivity, magnetic permeability, chemical diffusivity, etc. The outputs from the computing device 1200 can include temperature, displacement, electrical potential, magnetic potential, diffused mass density if inside the loading zone, or 0 if outside of the loading zone.

As another example, in the system of FIG. 12, the sensor can be a thermal couple (or infrared sensor) combined with a displacement sensor to provide a sensing system with coefficient calibration. In one embodiment, the displacement sensor can be an optical position detector that senses displacement of the tip of the thermal couple. The temperature sensor can be used to measure temperature, and the displacement sensor can be used to measure displacement at X, which is used to adjust the heat-flux coefficient and perform proportion coefficient calibration. Equations 58-61 can then be used by the computing device 1200 to compute the other values.

In an illustrative embodiment, any of the operations described herein can be performed by a computing device, as discussed with reference to FIG. 12. As another example, FIG. 13 depicts a sensing system for performing multifield, multifunctional sensing in accordance with an illustrative embodiment. In an illustrative embodiment, the computing components of the sensing system can be separate from the tool/material being tested, but can be in communication with one or more sensors mounted thereto. Alternatively, the computing components can be incorporated into the tool/material.

The sensing system includes a computing device 1302 that has a processor 1305, an operating system 1310, a memory 1315, an input/output (I/O) system 1320, a network interface 1325, and a conversion application 1330. In alternative embodiments, the computing device 1302 may include fewer, additional, and/or different components. The components of the computing device 1302 communicate with one another via one or more buses or any other interconnect system. The computing device 1302 can be any type of computing device (e.g., smartphone, tablet, laptop, desktop, etc.), including a dedicated standalone computing system that is designed to perform the calculations described herein.

The processor 1305 can be in electrical communication with and used to control any of the device components described herein. For example, the processor can be used to execute the conversion application 1330, control the sensor(s) 1340, etc. The processor 1305 can be any type of computer processor known in the art, and can include a plurality of processors and/or a plurality of processing cores. The processor 1305 can include a controller, a microcontroller, an audio processor, a graphics processing unit, a hardware accelerator, a digital signal processor, etc. Additionally, the processor 1305 may be implemented as a complex instruction set computer processor, a reduced instruction set computer processor, an x86 instruction set computer processor, etc. The processor 1305 is used to run the operating system 1310, which can be any type of operating system.

The operating system 1310 is stored in the memory 1315, which is also used to store programs, received measurements/data from the sensor(s) 1340, materials constants and properties (i.e., inputs), mathematical constants and algorithms, network and communications data, peripheral component data, the conversion application 1330, and other operating instructions. The memory 1315 can be one or more memory systems that include various types of computer memory such as flash memory, random access memory (RAM), dynamic (RAM), static (RAM), a universal serial bus (USB) drive, an optical disk drive, a tape drive, an internal storage device, a non-volatile storage device, a hard disk drive (HDD), a volatile storage device, etc. In some embodiments, at least a portion of the memory 1315 can be in the cloud to provide cloud storage for the system. Similarly, in one embodiment, any of the computing components described herein (e.g., the processor 1305, etc.) can be implemented in the cloud such that the system can be run and controlled through cloud computing.

The I/O system 1320 is the framework which enables users and peripheral devices to interact with the computing device 1302. The I/O system 420 can also include one or more speakers, one or more displays, one or more microphones, a keyboard, a mouse, one or more buttons or other controls, etc. that allow the user to interact with and control the computing device 1302. The I/O system 1320 also includes circuitry and a bus structure to interface with peripheral computing devices such as the imaging sensor, power sources, universal service bus (USB) devices, data acquisition cards, peripheral component interconnect express (PCIe) devices, serial advanced technology attachment (SATA) devices, high definition multimedia interface (HDMI) devices, proprietary connection devices, etc.

The network interface 1325 includes transceiver circuitry (e.g., a transmitter and a receiver) that allows the computing device 1302 to transmit and receive data to/from other devices such as the sensor(s) 1340, remote computing systems, servers, websites, etc. The network interface 1325 enables communication through the network 1335, which can be one or more communication networks. The network 1335 can include a cable network, a fiber network, a cellular network, a wi-fi network, a landline telephone network, a microwave network, a satellite network, etc. The network interface 1325 also includes circuitry to allow device-to-device communication such as Bluetooth® communication.

The conversion application 1330 can include software and algorithms in the form of computer-readable instructions which, upon execution by the processor 1305, performs any of the various operations described herein such as receiving information from the sensor(s) 1340, calibrating the sensor(s) 1340, solving equations, determining field values, making conversions, outputting results in the form of sensed values and calculated field values that are based on the sensed values, etc. The conversion application 1330 can utilize the processor 1305 and/or the memory 1315 as discussed above. In an alternative implementation, the conversion application 1330 can be remote or independent from the computing device 1302, but in communication therewith.

The sensor(s) 1340 can include a temperature sensor (e.g., thermal couple, infrared detector, etc.), a displacement sensor (e.g., optical position detector, accelerometer, position sensor, eddy current sensor, capacitance sensor, etc.), an electrical field strength sensor, an electrical current sensor, a voltage sensor, a magnetic field strength sensor, etc. In the tool/surface embodiment, the sensor(s) can be positioned on either tool or the workpiece, depending on the implementation. In addition to use monitoring a tool/surface interface, the proposed system can also be used in monitoring tools in die-less metal forming, monitoring cam-follower set in engines, pumps, compressors, etc. (for both cam and follower), monitoring railway rails and train wheels, monitoring rolling elements and races of rolling bearings, etc.

The word “illustrative” is used herein to mean serving as an example, instance, or illustration. Any aspect or design described herein as “illustrative” is not necessarily to be construed as preferred or advantageous over other aspects or designs. Further, for the purposes of this disclosure and unless otherwise specified, “a” or “an” means “one or more”.

The foregoing description of illustrative embodiments of the invention has been presented for purposes of illustration and of description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed, and modifications and variations are possible in light of the above teachings or may be acquired from practice of the invention. The embodiments were chosen and described in order to explain the principles of the invention and as practical applications of the invention to enable one skilled in the art to utilize the invention in various embodiments and with various modifications as suited to the particular use contemplated. It is intended that the scope of the invention be defined by the claims appended hereto and their equivalents.

Claims

1. A sensing system comprising: a sensor mounted on or proximate to a surface that is subjected to a load, wherein the sensor is configured to detect a first field value related to the surface;a memory configured to store information regarding the surface; anda processor coupled to the memory and in communication with the sensor, wherein the processor is configured to determine a second field value based on the detected first field value and the information regarding the surface.

2. The system of claim 1, wherein the sensor comprises a thermal couple or infrared detector such that the first field value comprises a temperature of the surface.

3. The system of claim 2, wherein the second field value comprises electric field potential of the surface or magnetic field potential of the surface, and wherein the processor determines the electric field potential of the surface or magnetic field potential based at least in part on the temperature.

4. The system of claim 2, wherein the second field value comprises an amount of displacement of the surface, and wherein the processor determines the amount of displacement of the surface based at least in part on the temperature.

5. The system of claim 2, wherein the second field value comprises diffused mass density of the surface, and wherein the processor determines the diffused mass density of the surface based at least in part on the temperature.

6. The system of claim 1, wherein the information regarding the surface includes one or more of an elastic modulus of the surface, thermal conductivity of the surface, dielectric permittivity of the surface, magnetic permeability of the surface, and chemical diffusivity of the surface.

7. The system of claim 1, wherein the sensor comprises a first sensor in the form of a temperature sensor, and further comprising a second sensor in the form of a displacement sensor, and wherein the processor uses measurements from the displacement sensor to adjust a heat-flux coefficient of the surface.

8. The system of claim 7, wherein the processor determines the second field value based in part on the adjusted heat-flux coefficient.

9. The system of claim 1, wherein the sensor comprises a first sensor and the first field comprises any of a mechanical field, a thermal field, an electromagnetic field, or a chemical field, and further comprising a second sensor configured to detect a second field related to the surface, wherein the second field is different from the first field and comprises one or more of the mechanical field, the thermal field, the electromagnetic field, and the chemical field.

10. The system of claim 1, wherein the sensor is located at a distance X from a center of the load that is applied to the surface.

11. The system of claim 10, wherein the distance X is less than or equal to ten times a radius of load distribution of the load applied to the surface.

12. The system of claim 1, wherein the processor determines a shape of a load distribution of the load, and wherein the processor determines a shape of a heat flux of the surface.

13. The system of claim 12, wherein the processor compares the shape of the load distribution of the load to the shape of the heat flux, and wherein, responsive to a determination that the shape of the load distribution and the shape of the heat flux are the same, the processor adjusts a heat-flux coefficient for the surface.

14. The system of claim 1, wherein the sensor is located at a first location on the surface, and wherein the processor determines one or more field values at a second location on the surface based on a symmetry of the load.

15. A method of performing multifield, multifunctional sensing, the method comprising: sensing, by a sensor mounted on a surface that is subjected to a load, a first field value related to the surface;storing, in a memory, information regarding the surface; anddetermining, by a processor coupled to the memory and in communication with the sensor, a second field value based on the detected first field value and the information regarding the surface.

16. The method of claim 15, wherein the sensor comprises a first sensor in the form of a temperature sensor, and further comprising sensing, by a second sensor in the form of a displacement sensor, a displacement of the surface in response to the load.

17. The method of claim 16, further comprising adjusting, by the processor a heat-flux coefficient of the surface based at least in part on the sensed displacement of the surface.

18. The method of claim 17, wherein the processor determines the second field value based in part on the adjusted heat-flux coefficient.

19. The method of claim 15, wherein the sensor comprises a temperature sensor that detects a temperature of the surface, wherein the second field value comprises electric field potential of the surface, magnetic field potential of the surface, an amount of displacement of the surface, or a diffused mass density of the surface, and wherein the processor determines the second field value based at least in part on the temperature.

20. The method of claim 15, wherein the sensor is located at a first location on the surface, and further comprising determining, by the processor, one or more field values at a second location on the surface based on a symmetry of the load.