Thermo-Mechanical Micropolar Formulation

• Thermo-mechanical micropolar formulation is a continuum theory that integrates thermal, translational, and rotational effects through explicit inclusion of couple-stresses and body couples.
• It employs diverse methodologies such as Irving–Kirkwood homogenization, small-strain Cosserat elastoplasticity, and finite-strain models to capture microstructural dynamics and polarity.
• Computational frameworks like FFT-based spectral methods and level set topology optimization demonstrate its practical value in designing advanced, size-dependent materials.

Searching arXiv for recent and foundational papers on thermo-mechanical micropolar formulations. arXiv search query: "thermomechanical micropolar formulation Irving Kirkwood homogenization micropolar thermoelasticity finite strain topology optimization" Thermo-mechanical micropolar formulation denotes a class of continuum theories in which translational motion is coupled to rotational or polar effects, so that force-stresses, couple-stresses, body couples, curvature-like measures, and thermal fields enter the governing equations and energetic structure. In the arXiv literature, this designation spans several related but non-identical constructions: a homogenized polar medium obtained by continuum-on-continuum Irving–Kirkwood averaging, in which body couples and couple stresses emerge from microscopic fields and velocity fluctuations (Mandadapu et al., 2018); a small-strain Cosserat formulation with independent microrotation, generally non-symmetric force-stress, curvature, and thermodynamics-based elastoplasticity (Francis et al., 2024); a linear thermoelastic micropolar model for topology optimization under steady heat conduction and thermal expansion (Shekhar et al., 9 Jan 2026); and a finite-strain polar thermomechanical setting in which couple-stresses and body couples appear in the balances without introducing an independent bulk microrotation field, while shell moments are conjugate to curvature change (Ghaffari et al., 2019).

1. Conceptual variants of micropolar thermomechanics

The literature does not use micropolar in a single, uniform sense. In the continuum-on-continuum Irving–Kirkwood formulation, the homogenized macroscale is polar because angular momentum and energy balances contain body couples and couple stresses defined from the microscopic state, even though the macroscopic Cauchy stress is symmetric (Mandadapu et al., 2018). In the small-strain Cosserat elastoplastic formulation and in the thermoelastic topology-optimization model, the kinematics include an independent microrotation field, and the force-stress is generally non-symmetric or explicitly decomposed into symmetric and antisymmetric parts (Francis et al., 2024). In the finite-strain polar framework for anisotropic continua and shells, the paper explicitly states that no independent microrotation field is introduced in the bulk; polar effects instead enter through couple-stresses and body couples in the angular momentum balance (Ghaffari et al., 2019).

A recurring misconception is that a micropolar formulation necessarily requires an antisymmetric Cauchy stress. The Irving–Kirkwood homogenized theory contradicts that identification: the macroscopic stress $T^M$ is symmetric, yet the medium remains polar because the angular momentum balance contains a body couple density $\rho^M g^M$, a couple stress $M^M$, and an inertia-flux contribution involving $J^M$ (Mandadapu et al., 2018). By contrast, the linear Cosserat models use non-symmetric force-stress or an explicit antisymmetric coupling stress, so micropolarity is realized directly at the constitutive level (Francis et al., 2024).

Formulation	Rotational description	Stress character
Irving–Kirkwood homogenized polar medium	$w^M$, $L_s^M$, $I^M$, $J^M$	$T^M$ symmetric; couple stress $M^M$
Small-strain Cosserat elastoplasticity	microrotation $\rho^M g^M$0	$\rho^M g^M$1 generally non-symmetric
Thermoelastic micropolar topology optimization	microrotation $\rho^M g^M$2 or $\rho^M g^M$3 in 2D	$\rho^M g^M$4
Finite-strain polar bulk and shells	no independent bulk microrotation; shell curvature change $\rho^M g^M$5	bulk $\rho^M g^M$6 may be non-symmetric; shell moments from curvature

This comparison suggests that thermo-mechanical micropolar formulation is best understood as a family of polar continuum descriptions rather than a single canonical model.

2. Kinematics, state variables, and strain measures

In the homogenized Irving–Kirkwood construction, the microscopic fields are $\rho^M g^M$7, $\rho^M g^M$8, $\rho^M g^M$9, $M^M$0, $M^M$1, $M^M$2, $M^M$3, and $M^M$4, while the macroscopic fields are $M^M$5, $M^M$6, $M^M$7, angular velocity $M^M$8, spin angular momentum per unit mass $M^M$9, moment-of-inertia tensor $J^M$0, moment-of-inertia flux $J^M$1, internal energy $J^M$2, total internal energy $J^M$3, heat flux $J^M$4, and heat supply $J^M$5 (Mandadapu et al., 2018). The coarse-graining kernel satisfies

$J^M$6

with $J^M$7 for $J^M$8. The convected microscopic velocity is

$J^M$9

and the weighted center-of-mass condition is

$w^M$0

The same framework defines $w^M$1 and $w^M$2 through weighted moments of $w^M$3 and velocity fluctuations $w^M$4 (Mandadapu et al., 2018).

In the small-strain Cosserat elastoplastic setting, the primary fields are displacement $w^M$5 and microrotation $w^M$6, with linearized measures

$w^M$7

Here $w^M$8 is the micropolar relative deformation tensor and $w^M$9 is the microcurvature tensor. The thermodynamic conjugate pairs are $L_s^M$0 and $L_s^M$1 (Francis et al., 2024).

In the thermoelastic topology-optimization formulation, the primary fields are displacement $L_s^M$2, microrotation $L_s^M$3, and temperature $L_s^M$4. The generalized strain and curvature are

$L_s^M$5

with additive thermal strain

$L_s^M$6

In two dimensions, only the out-of-plane microrotation is retained, so $L_s^M$7 becomes the scalar $L_s^M$8 and $L_s^M$9 (Shekhar et al., 9 Jan 2026).

The finite-strain polar framework is kinematically distinct. In the bulk, the formulation uses the deformation map $I^M$0, deformation gradient $I^M$1, velocity $I^M$2, and temperature $I^M$3, together with the multiplicative split

$I^M$4

The paper states that no independent microrotation field $I^M$5 is introduced in the bulk; instead, the angular-velocity vector $I^M$6 is the axial of the macroscopic spin. In the shell specialization, curvature change $I^M$7 is derived from the evolving normal and acts as the energetic conjugate of the surface moment tensor $I^M$8 (Ghaffari et al., 2019).

3. Balance laws and energetic structure

The homogenized Irving–Kirkwood theory produces a micropolar system of balance laws. The mass and linear momentum balances are

$I^M$9

Angular momentum in spin form is

$J^M$0

and, using the microinertia balance,

$J^M$1

The moment-of-inertia balance is

$J^M$2

The macroscopic Cauchy stress is symmetric in this theory, so there is no $J^M$3 contribution in the angular momentum balance (Mandadapu et al., 2018).

The corresponding energy balance is explicitly polar:

$J^M$4

and the reduced internal-energy balance contains the power of couple stresses and an inertia-flux correction,

$J^M$5

This structure is one of the distinctive features of the formulation (Mandadapu et al., 2018).

In the linear Cosserat setting, the static balance equations are

$J^M$6

The isothermal free energy density is

$J^M$7

and the reduced dissipation inequality is

$J^M$8

The thermal extension synthesized alongside this framework introduces

$J^M$9

and thermal expansion $T^M$0, while noting that this extension is not implemented in the paper itself (Francis et al., 2024).

In the topology-optimization formulation, the strong forms are the static micropolar balances

$T^M$1

together with steady-state heat conduction

$T^M$2

Thermal effects enter the mechanical problem through $T^M$3 in the constitutive law; there are no additional thermo-mechanical coupling terms in the balances beyond thermal expansion (Shekhar et al., 9 Jan 2026).

At finite strain, the polar energy balance is expressed in two equivalent current forms, one of which is

$T^M$4

and the current-form Clausius–Duhem inequality is

$T^M$5

This explicitly displays the polar contribution of couple-stresses and body couples to mechanical power and entropy production (Ghaffari et al., 2019).

4. Homogenization, objectivity, and thermodynamic restrictions

The continuum-on-continuum Irving–Kirkwood formulation derives all macroscopic fields from extensivity relations. Mass, momentum, angular momentum, and total internal energy are averaged as

$T^M$6

$T^M$7

The macroscopic stress, body couple, couple stress, heat supply, and heat flux are defined directly in terms of microscopic fields and velocity fluctuations. For example,

$T^M$8

and

$T^M$9

Choosing $M^M$0 through the homogenized spin–inertia relation yields the additive decomposition

$M^M$1

This decomposition separates non-inertial internal energy from translational and rotational kinetic contributions (Mandadapu et al., 2018).

A central issue in the paper is invariance under superposed rigid motions. The kernel must satisfy

$M^M$2

and the center-of-mass condition is required for form-invariance of momentum extensivity. Under superposed motion, $M^M$3, $M^M$4, and $M^M$5. However, the macroscopic stress transforms with additional $M^M$6-dependent terms, and the paper states that $M^M$7 is not strictly objective when inertial fluctuation effects are present, even though those extra terms are divergence-free so that the linear momentum balance remains form-invariant. The body couple density $M^M$8 and couple stress $M^M$9 are likewise not objective in general, and the heat flux $\rho^M g^M$00 is not objective, with non-objective parts that are not divergence-free (Mandadapu et al., 2018).

The thermodynamic status of these theories differs across formulations. The Irving–Kirkwood paper gives the energy balance but does not derive an entropy balance or Clausius–Duhem inequality; it explicitly notes that no explicit thermodynamic inequality is provided (Mandadapu et al., 2018). The small-strain elastoplastic formulation is built around the reduced dissipation inequality and associative flow rules, with equivalent plastic strain rates chosen so that $\rho^M g^M$01 and $\rho^M g^M$02 (Francis et al., 2024). The finite-strain anisotropic theory goes further by deriving constitutive restrictions from the Clausius–Duhem inequality, using $\rho^M g^M$03 in the bulk and $\rho^M g^M$04 on the surface, and by adopting Fourier heat conduction $\rho^M g^M$05 or, alternatively, a Lord–Shulman form $\rho^M g^M$06 (Ghaffari et al., 2019).

A plausible implication is that thermo-mechanical micropolar formulation contains a genuine spectrum of thermodynamic closure: from rigorous balance-law homogenization without entropy production, to reduced-dissipation plasticity, to finite-strain constitutive restrictions derived directly from the second law.

5. Constitutive realizations and computational frameworks

For isotropic small-strain micropolar elastoplasticity, the constitutive equations take the form

$\rho^M g^M$07

with isotropic tensors

$\rho^M g^M$08

$\rho^M g^M$09

The model uses multi-criteria yield functions

$\rho^M g^M$10

with macro equivalent stress $\rho^M g^M$11 and micro equivalent couple-stress $\rho^M g^M$12, linear hardening radii $\rho^M g^M$13 and $\rho^M g^M$14, associative flow rules, and a closed-form radial-return mapping under the restrictions $\rho^M g^M$15, $\rho^M g^M$16, and $\rho^M g^M$17 (Francis et al., 2024). The paper also provides algorithmic consistent tangents $\rho^M g^M$18 and $\rho^M g^M$19 and verifies the implementation using the Method of Numerically Manufactured Solutions, with errors bounded by FFT tolerance such as $\rho^M g^M$20 (Francis et al., 2024).

The associated solver is an FFT-based micromechanical spectral method. A constant reference medium $\rho^M g^M$21 is chosen, polarization stresses are formed, and the periodic problem is solved in Fourier space through Green operators $\rho^M g^M$22 and $\rho^M g^M$23. The iterative Moulinec–Suquet basic scheme is combined with time stepping to treat elastoplastic history dependence. The method handles spatially varying elasticity, couple elasticity, and yield parameters on voxelized domains with periodic boundary conditions, and its core computational complexity is $\rho^M g^M$24 (Francis et al., 2024). The formulation explicitly identifies elastic and plastic intrinsic lengths,

$\rho^M g^M$25

which govern size dependence and the transition between macro and micro stress measures (Francis et al., 2024).

For thermoelastic micropolar solids in topology optimization, the constitutive equations are

$\rho^M g^M$26

with

$\rho^M g^M$27

and

$\rho^M g^M$28

In the reported two-dimensional benchmarks, $\rho^M g^M$29 and $\rho^M g^M$30 (Shekhar et al., 9 Jan 2026). The optimization problem uses a level set function $\rho^M g^M$31, a regularized Heaviside function, ersatz interpolation

$\rho^M g^M$32

and a density function $\rho^M g^M$33 to restrict internal heat sources to material. Sensitivities are computed by an adjoint equation,

$\rho^M g^M$34

and the level set evolves through the Hamilton–Jacobi equation

$\rho^M g^M$35

with signed-distance reinitialization (Shekhar et al., 9 Jan 2026). The numerical implementation uses \texttt{Gridap.jl}, \texttt{GridapTopOpt.jl}, and \texttt{ForwardDiff.jl}, together with Q4 elements and first-order Godunov upwind discretization (Shekhar et al., 9 Jan 2026).

The finite-strain anisotropic framework emphasizes tensorial representation and weak forms in curvilinear or Cartesian coordinates. In the bulk, the weak forms are given for three-dimensional non-polar mechanics and thermal conduction, whereas on the Kirchhoff–Love shell the mechanical weak form includes both membrane and bending contributions through $\rho^M g^M$36 and $\rho^M g^M$37 (Ghaffari et al., 2019). This distinction is significant: the paper’s shell theory is kinematically rotation-free, but it is statically polar because bending moments enter the angular momentum balance and the weak form as energetic conjugates of curvature change (Ghaffari et al., 2019).

6. Classical limits, applications, and limitations

All four formulations contain explicit classical limits. In the Irving–Kirkwood theory, if angular velocity is suppressed, $\rho^M g^M$38, and velocity fluctuation contributions are negligible so that $\rho^M g^M$39, $\rho^M g^M$40, and the fluctuation terms in $\rho^M g^M$41 and $\rho^M g^M$42 vanish, the theory reduces to a classical non-polar continuum with standard mass, momentum, and energy balances (Mandadapu et al., 2018). In the FFT-based elastoplastic model, the Cauchy limit is recovered when $\rho^M g^M$43 and $\rho^M g^M$44, in which case microstructural effects are negligible and stresses become nearly symmetric (Francis et al., 2024). In the topology-optimization study, micropolar predictions converge to classical Cauchy results for $\rho^M g^M$45, that is, for large specimens or small $\rho^M g^M$46 (Shekhar et al., 9 Jan 2026). The finite-strain framework likewise recovers the standard non-polar continuum by setting $\rho^M g^M$47 and $\rho^M g^M$48, which enforces $\rho^M g^M$49 (Ghaffari et al., 2019).

The practical regimes of interest are those in which internal lengths and rotational effects are non-negligible. The Irving–Kirkwood paper identifies high-frequency wave propagation in heterogeneous solids and composites with pronounced micro-dynamics as use cases where polar effects matter (Mandadapu et al., 2018). The FFT-based model emphasizes chiral and architected materials, polycrystals with size effects, nanocomposites, and media where skew-symmetric stresses or couple-stresses are active (Francis et al., 2024). The topology-optimization results show that increasing micropolarity through the coupling number $\rho^M g^M$50 and bending length $\rho^M g^M$51 changes optimized layouts from classical truss-like designs to more frame-like or thermally efficient connected topologies, while thermal gradients drive material away from hot regions (Shekhar et al., 9 Jan 2026).

The limitations are equally explicit. The Irving–Kirkwood derivation assumes extensivity, a continuum at both scales, a kernel invariant under superposed rigid motions, the weighted center-of-mass condition, symmetric microscopic stress $\rho^M g^M$52, no intrinsic microscopic couple stresses, and sufficient scale separation and regularity to justify transport theorems and interchange of differentiation and integration (Mandadapu et al., 2018). The elastoplastic FFT framework assumes small strains, small microrotations, rate-independent plasticity, isotropic centrosymmetric elasticity in most derivations, and a closed-form return mapping that relies on $\rho^M g^M$53, the relations among $\rho^M g^M$54, and linear hardening (Francis et al., 2024). The topology-optimization implementation assumes steady-state heat conduction, linear elasticity, temperature-independent mechanical moduli, and neglects $\rho^M g^M$55 and $\rho^M g^M$56 in the two-dimensional benchmarks (Shekhar et al., 9 Jan 2026). The finite-strain anisotropic paper does not introduce an independent bulk microrotation field, so it is polar in the balance-law sense rather than a full Cosserat theory with independent rotational degrees of freedom (Ghaffari et al., 2019).

Taken together, these formulations show that thermo-mechanical micropolar theory is not a single constitutive ansatz but a hierarchy of polar continuum descriptions. Some versions derive polarity by homogenization of extensive quantities and velocity fluctuations; others posit independent microrotation and curvature at the constitutive level; and still others encode polarity through couple-stresses and curvature change at finite strain or on surfaces. What unifies them is the systematic inclusion of rotational measures, couple interactions, and thermal effects in the continuum balance structure.