A Thermodynamically Consistent Uncoupled Nonlocal Shear Elasticity Model with a Symmetric-Plus-Skew Integral Kernel: Application to Single-Walled Carbon Nanotube Torsion

1. Introduction

Nonlocal continuum models are widely used to capture size-dependent mechanical responses in nanostructures where classical local elasticity is insufficient [7,8]. Eringen's strain-driven nonlocal elasticity [1] has been applied extensively to nanobeams, nanoplates, and carbon nanotubes. However, when imposed with certain structural boundary conditions, the strain-driven differential form may produce boundary paradoxes and loss of well-posedness.

The stress-driven integral formulation proposed by Romano and Barretta [2] restores a more consistent mathematical structure by expressing the strain field as an integral convolution of the stress field with a positive symmetric kernel. Most existing stress-driven developments focus on normal bending, axial, or torsional fields where the kernel is symmetric and distance dependent [3,5,6]. Independent uncoupled shear mechanics, especially in nanostructured torsional and transverse-shear-dominated configurations, remains less systematically treated.

A purely anti-symmetric kernel is attractive from a formal viewpoint because it can introduce direction-dependent shear memory. However, a purely skew-symmetric kernel is not thermodynamically admissible as a stand-alone elastic constitutive law. Its quadratic energy contribution vanishes identically, and under uniform shear stress it may suppress the local Hookean response in the interior of a finite domain. A physically meaningful model must therefore contain a positive-definite symmetric part responsible for stored elastic energy, while any skew-symmetric part must be interpreted only as a bounded directional correction.

The objective of this paper is to examine the mathematical consequences of augmenting a symmetric exponential nonlocal shear-compliance kernel with a skew-symmetric integral correction. The analysis focuses on four points: (i) the exact differential representation of the integral law and the associated constitutive boundary conditions; (ii) uniqueness of the resulting strain-field boundary-value problem; (iii) the thermodynamic status of the full constitutive law, distinguishing the class of loading paths for which the Clausius–Duhem inequality is satisfied as an equality from the class for which it is not; (iv) the torsional response under constant torque, derived directly from the boundary-value problem and checked against the original integral representation.

Two results of the analysis are established. First, thermodynamic consistency holds for proportional loading paths, a class that includes the constant-torque torsional benchmark. Second, for general non-proportional loading with a non-zero skew coefficient, no stored-energy functional of the stress field can reproduce the constitutive response; a counterexample and an impossibility proof are given. The scope of the thermodynamic claim is restricted accordingly, and a symmetric compliance enhancement is introduced as an explicit, separately calibrated mechanism for global softening.

The torsional benchmark is treated with particular care. The end-twist ratio is derived from the governing boundary-value problem and independently verified from the integral law. The skew coefficient is shown to cancel from the total end twist under constant torque, while the symmetric enhancement controls the net global response. The previously reported 960% flexibility enhancement is removed as dimensionally inconsistent. The corrected uncalibrated prediction is a stiffening of approximately 19.9%, and a calibrated symmetric enhancement recovers a softening of approximately 20.2%, while preserving the classical local limit.

The mathematical tools employed in the derivation are standard [4], and the formulation is positioned within the stress-driven nonlocal literature [2,3,5]. The present work does not replace coupled Timoshenko-type nonlocal theories [6]; it proposes an independent kernel for uncoupled shear response, with a thermodynamic scope that is stated explicitly.

2. Thermodynamically Consistent Kernel and Constitutive Law

2.1. Kernels

Consider a one-dimensional elastic nano-body occupying the interval 0 ≤ x ≤ L, with L > 0. Let τ(x) denote the local shear stress, γ(x) the corresponding engineering shear strain, G > 0 the macroscopic shear modulus, and L_c > 0 a positive material length scale.

The symmetric exponential kernel is defined by

φ(x,x′) = (1/(2L_c)) exp(−|x−x′|/L_c).

(1)

This kernel has dimension 1/length and satisfies

φ(x,x′) = φ(x′,x).

(2)

The skew-symmetric kernel is defined by

χ(x,x′) = (1/(2L_c)) sgn(x−x′) exp(−|x−x′|/L_c).

(3)

It also has dimension 1/length and satisfies

χ(x,x′) = −χ(x′,x).

(4)

The proposed constitutive kernel is

K(x,x′) = φ(x,x′) + β χ(x,x′),

(5)

where β is a dimensionless directional-correction parameter. For a bounded perturbation model, one may restrict |β| ≤ 1. This restriction is a bound on the magnitude of the correction; it is not by itself a thermodynamic admissibility condition.

2.2. Integral Operators

Define the symmetric and skew-symmetric integral operators

(C_s τ)(x) = ∫₀ᴸ φ(x,x′) τ(x′) dx′,

(6)

(C_a τ)(x) = ∫₀ᴸ χ(x,x′) τ(x′) dx′.

(7)

The uncoupled nonlocal shear law is

G γ(x) = (C_s τ)(x) + β (C_a τ)(x).

(8)

Equivalently, with s = C_s τ and a = C_a τ,

G γ(x) = s(x) + β a(x).

(9)

All terms in (8) are dimensionally consistent: the kernels have dimension 1/length, integration with respect to dx′ renders the operators dimensionless, and Gγ has dimension of stress.

The symmetric part s carries the stored elastic energy. The skew part a represents a directional shear-memory correction. Its role in the quadratic form and in the time-dependent internal power is examined in Section 6.

2.3. Symmetric Compliance Enhancement

To allow for a net global softening while preserving thermodynamic consistency for proportional loading, the constitutive law is augmented with a dimensionless symmetric compliance factor η ≥ 0:

G γ(x) = (1+η)(C_s τ)(x) + β(C_a τ)(x).

(10)

The enhancement η is a constitutive calibration parameter, not a consequence of the kernel. A natural scaling that preserves the classical local limit is

η = η₀ L_c / L, η₀ ≥ 0,

(11)

adopted in Section 8. In the uncalibrated case, η = 0 and (10) reduces to (8).

3. Exact Differential Reduction

3.1. Differential Identity for the Symmetric Kernel

The symmetric kernel φ is the Green kernel of the operator 1 − L_c² d²/dx² on the finite interval with boundary conditions

s′(0) = s(0)/L_c, s′(L) = −s(L)/L_c.

(12)

Inside the domain, applying Leibniz's rule to the split integral representation gives

s(x) − L_c² s″(x) = τ(x).

(13)

The boundary conditions (12) follow from the same kernel and are not additional mechanical assumptions.

3.2. Relation Between the Skew Kernel and the Derivative of s

For x ≠ x′,

χ(x,x′) = L_c ∂φ(x,x′)/∂x′ = −L_c ∂φ(x,x′)/∂x.

(14)

Hence

a(x) = (C_a τ)(x) = −L_c s′(x).

(15)

Substituting (15) into (10) gives

G γ = (1+η) s − β L_c s′.

(16)

3.3. Differential Equation

Since the operators D = d/dx and 1 − L_c²D² commute, applying 1 − L_c²D² to both sides of (16) and using (13) yields

G (γ − L_c² γ″) = (1+η) τ − β L_c τ′.

(17)

Equation (17) is dimensionally consistent: every term has the dimension of stress. It requires sufficient regularity, namely τ ∈ C¹([0,L]) and γ ∈ C³([0,L]).

4. Consistent Constitutive Boundary Conditions

Let

u(x) = G γ(x) = (1+η) s(x) − β L_c s′(x).

(18)

4.1. At x = 0

Using s′(0) = s(0)/L_c and s″(0) = [s(0) − τ(0)]/L_c², differentiation of (18) yields

u′(0) = (1+η) s(0)/L_c − (β/L_c)[s(0) − τ(0)].

(19)

Multiplying by L_c and using u(0) = (1+η−β) s(0):

L_c u′(0) − u(0) = β τ(0).

(20)

4.2. At x = L

Using s′(L) = −s(L)/L_c and s″(L) = [s(L) − τ(L)]/L_c²:

u′(L) = −(1+η) s(L)/L_c − (β/L_c)[s(L) − τ(L)].

(21)

Multiplying by L_c and using u(L) = (1+η+β) s(L):

L_c u′(L) + u(L) = β τ(L).

(22)

4.3. Boundary-Value Problem

Substituting u = G γ into (20) and (22):

G[L_c γ′(0) − γ(0)] = β τ(0),

(23)

G[L_c γ′(L) + γ(L)] = β τ(L).

(24)

The boundary-value problem consists of equation (17) with conditions (23)–(24). The boundary conditions are independent of η; only the differential equation is affected by the symmetric enhancement.

5. Uniqueness Proof

Assume that two strain fields γ₁ and γ₂ solve the boundary-value problem (17), (23), (24) for the same stress field τ. Define δγ = γ₁ − γ₂. Subtracting the governing equations gives the homogeneous problem

δγ − L_c² δγ″ = 0,

(25)

with boundary conditions

L_c δγ′(0) − δγ(0) = 0,

(26)

L_c δγ′(L) + δγ(L) = 0.

(27)

The general solution of (25) is

δγ(x) = A cosh(x/L_c) + B sinh(x/L_c).

(28)

Its derivative is

δγ′(x) = (A/L_c) sinh(x/L_c) + (B/L_c) cosh(x/L_c).

(29)

At x = 0: δγ(0) = A, L_c δγ′(0) = B. Condition (26) gives B − A = 0, hence B = A.

At x = L, set λ = L/L_c > 0. Condition (27) gives

A sinh λ + B cosh λ + A cosh λ + B sinh λ = 0,

(30)

so

(A+B)(sinh λ + cosh λ) = 0.

(31)

Since sinh λ + cosh λ = e^λ > 0 for λ > 0, we obtain A + B = 0. Together with B = A, this gives A = B = 0, hence δγ ≡ 0. The solution is unique within the regular class in which the differential reduction holds. ∎

The uniqueness proof establishes uniqueness of the strain-field solution. It does not by itself establish existence for every admissible stress, positivity of the full energy, or mechanical stability.

6. Energy Consistency and Thermodynamic Admissibility

6.1. Positive Definiteness of the Symmetric Part

For the symmetric kernel, define s = C_s τ. From (13) and the kernel-induced boundary conditions (12), integration by parts gives

∫₀ᴸ τ s dx = ∫₀ᴸ s² dx + L_c² ∫₀ᴸ (s′)² dx + L_c[s(0)² + s(L)²].

(32)

Every term on the right-hand side is non-negative, and the right-hand side vanishes only for s ≡ 0, which implies τ ≡ 0 by (13). Hence

⟨τ, C_s τ⟩ > 0 for every τ ≠ 0,

(33)

and the symmetric complementary energy

Ψ_s* = (1/(2G)) ⟨τ, C_s τ⟩

(34)

is strictly positive definite.

6.2. Vanishing Quadratic Form of the Skew-Symmetric Part

For the skew-symmetric kernel, the bilinear integral

I_a = ∫₀ᴸ ∫₀ᴸ τ(x) χ(x,x′) τ(x′) dx′ dx

(35)

satisfies I_a = −I_a upon exchanging x and x′. Hence

I_a = 0.

(36)

The quadratic form associated with the full kernel reduces to the symmetric part:

⟨τ, (C_s + β C_a) τ⟩ = ⟨τ, C_s τ⟩.

(37)

The skew-symmetric correction is therefore power-free at fixed stress. This property does not by itself determine its contribution to the time-dependent internal power, which is examined next.

6.3. Internal Power and the Thermodynamic Gap

For a quasi-static isothermal process, the internal power is

P_int = ∫₀ᴸ τ γ̇ dx.

(38)

Substituting the constitutive law (10) and differentiating with respect to time:

P_int = ((1+η)/G) ⟨τ, C_s τ̇⟩ + (β/G) ⟨τ, C_a τ̇⟩.

(39)

Since C_s is symmetric, the first term equals the time derivative of Ψ*:

dΨ*/dt = ((1+η)/G) ⟨τ, C_s τ̇⟩, Ψ* = ((1+η)/(2G)) ⟨τ, C_s τ⟩.

(40)

Therefore

P_int − dΨ*/dt = (β/G) ⟨τ, C_a τ̇⟩.

(41)

The residual term does not vanish in general and is not sign-definite. This is the thermodynamic gap that must be addressed before any claim of full Clausius–Duhem admissibility can be made.

6.4. Thermodynamic Consistency for Proportional Loading

Definition. A stress path τ(x,t) is proportional if there exist a fixed distribution τ₀ ∈ L²(0,L) and a scalar function f ∈ C¹([0,T]) such that τ(x,t) = f(t) τ₀(x) for all (x,t) ∈ [0,L] × [0,T].

Theorem 1. For every proportional loading path, the model satisfies

P_int(t) = dΨ*(τ(·,t))/dt, D := P_int − dΨ*/dt = 0.

(42)

Proof. From τ = f τ₀ we have τ̇ = ḟ τ₀. Hence

⟨τ, C_a τ̇⟩ = f ḟ ⟨τ₀, C_a τ₀⟩ = 0,

(43)

because C_a is skew-symmetric. The residual term in (41) vanishes and P_int = dΨ*/dt. Since D = 0 ≥ 0, the Clausius–Duhem inequality is satisfied as an equality. ∎

The constant-torque torsional benchmark of Section 8 is a proportional loading path, so the thermodynamic consistency of the application follows from Theorem 1. This result holds for every η ≥ 0.

6.5. Non-Proportional Loading: An Explicit Counterexample

For general loading, the residual term in (41) does not vanish. Consider

τ(x,t) = f₁(x) sin(ω t) + f₂(x) cos(ω t),

(44)

with ω > 0 and f₁, f₂ ∈ L²(0,L). Direct computation gives

⟨τ, C_a τ̇⟩ = −ω ⟨f₁, C_a f₂⟩,

(45)

which is independent of t and generally non-zero. For example, with f₁ = 1, f₂ = x, and L = L_c = 1:

⟨1, C_a x⟩ = −1 + (3/2)(1 − e⁻¹) ≈ −0.051 ≠ 0.

(46)

Hence the skew-symmetric correction contributes to the internal power along non-proportional paths, and its sign can be reversed by reversing the sign of ω.

6.6. Impossibility of a Stored-Energy Formulation for β ≠ 0

Theorem 2. Let β ≠ 0. There is no functional Ψ*: L²(0,L) → R of class C² such that dΨ*(τ(·,t))/dt = P_int(t) for every differentiable stress path.

Proof. Assume such Ψ* exists. Then, for every τ̇,

⟨δΨ*/δτ, τ̇⟩ = (1/G) ⟨τ, C_s τ̇⟩ + (β/G) ⟨τ, C_a τ̇⟩.

(47)

Using C_s* = C_s and C_a* = −C_a:

δΨ*/δτ = (1/G) C_s τ − (β/G) C_a τ.

(48)

The second variation is

δ²Ψ*/δτ(x)δτ(x′) = (1/G) φ(x,x′) − (β/G) χ(x,x′).

(49)

Integrability requires this kernel to be symmetric in x, x′. But φ is symmetric and χ is skew-symmetric, so the kernel is not symmetric when β ≠ 0. Contradiction. ∎

Corollary. The model is thermodynamically consistent for proportional loading (Theorem 1), but not for general loading when β ≠ 0. A fully general formulation requires either β = 0 or an internal-variable extension with an explicit free energy, evolution equations, and non-negative dissipation.

6.7. Effect of the Symmetric Enhancement on the Thermodynamic Proof

The symmetric enhancement η enters (39) and (40) multiplicatively. For proportional loading, the residual term (41) is still zero, and P_int = dΨ*/dt. Hence Theorem 1 carries over unchanged for every η ≥ 0. Theorem 2 remains valid for every η ≥ 0 whenever β ≠ 0, since η affects only the symmetric part of the second variation.

Remark. The enhancement η is a constitutive calibration, not a consequence of the kernel. In the local limit L_c/L → 0, the requirement F → 1 imposes η → 0; the scaling (11) is adopted for this reason in Section 8.

7. Application to Uncoupled Shear in Nanoscale Torsion

Consider a circular tube of length L, polar second moment of area J, and shear modulus G. Let θ(x) be the angle of twist. The local shear strain at radius r is

γ_xθ(r,x) = r θ′(x),

(50)

and the corresponding local shear stress for a torsional moment M_t(x) is

τ_xθ(r,x) = r M_t(x)/J.

(51)

Substituting (50)–(51) into (17) and cancelling the common radial factor r:

GJ(θ′ − L_c² θ‴) = (1+η) M_t − β L_c M_t′.

(52)

The constitutive boundary conditions (23)–(24) become

GJ[L_c θ″(0) − θ′(0)] = β M_t(0),

(53)

GJ[L_c θ″(L) + θ′(L)] = β M_t(L).

(54)

Equations (52)–(54) are dimensionally consistent. Each side of (52) has the dimension of torque, and each side of (53)–(54) also has the dimension of torque.

8. Constant-Torque Benchmark and Corrected Torsional Result

8.1. Governing Equation

Let M_t(x) = M₀ with M₀ ≠ 0. Then M_t′ = 0, and (52) reduces to

GJ(θ′ − L_c² θ‴) = (1+η) M₀.

(55)

Set φ(x) = θ′(x) and Θ₀ = M₀/(GJ). Then

φ − L_c² φ″ = (1+η) Θ₀.

(56)

8.2. General Solution

φ(x) = (1+η) Θ₀ + A cosh(x/L_c) + B sinh(x/L_c).

(57)

8.3. Boundary Conditions

Dividing (53)–(54) by GJ:

L_c φ′(0) − φ(0) = β Θ₀,

(58)

L_c φ′(L) + φ(L) = β Θ₀.

(59)

Substituting (57) and solving the resulting 2 × 2 system:

B − A = (1+η+β) Θ₀,

(60)

A + B = (β − 1 − η) Θ₀ e^(−λ), λ = L/L_c.

(61)

Hence

A = (Θ₀/2)[(β − 1 − η) e^(−λ) − (1+η+β)],

(62)

B = (Θ₀/2)[(β − 1 − η) e^(−λ) + (1+η+β)].

(63)

8.4. End Twist

With the clamp condition θ(0) = 0,

θ(L) = ∫₀ᴸ φ(x) dx = (1+η) Θ₀ L + L_c [A sinh λ + B(cosh λ − 1)].

(64)

Using (60)–(61):

A sinh λ + B(cosh λ − 1) = −(1+η) Θ₀ (1 − e^(−λ)).

(65)

Therefore

θ(L) = (1+η) Θ₀ [L − L_c(1 − e^(−L/L_c))].

(66)

8.5. Flexibility Factor

The classical local end twist is θ_local(L) = Θ₀ L. The dimensionless flexibility factor is

F = θ(L)/θ_local(L) = (1+η)[1 − (L_c/L)(1 − e^(−L/L_c))].

(67)

Two observations follow. First, the skew coefficient β cancels from the total end twist; it redistributes the local twist rate near the boundaries but does not produce a net global softening. Second, the symmetric enhancement η enters multiplicatively and is the only mechanism in the present model that can generate a net global softening.

8.6. Uncalibrated Case (η = 0)

For L = 30 nm and L_c = 6 nm, so that L_c/L = 0.2 and λ = 5:

F = 1 − 0.2(1 − e⁻⁵) ≈ 0.80135.

(68)

The model therefore predicts a stiffening of approximately 19.9% relative to the classical local result. This is a mathematical consequence of the stated kernel and boundary conditions, not an experimental prediction.

8.7. Calibrated Case (η = η₀ L_c/L)

To recover the local limit F → 1 as L_c/L → 0, set η = η₀ L_c/L with η₀ ≥ 0:

F = (1 + η₀ L_c/L)[1 − (L_c/L)(1 − e^(−L/L_c))].

(69)

Then lim_{L_c/L→0} F = 1. For the benchmark L_c/L = 0.2:

Table 1. Calibrated flexibility factor. 

η₀	η	F	Change
0	0	0.80135	−19.9% (stiffening)
1.0	0.20	0.96162	−3.8%
2.5	0.50	1.20202	+20.2% (softening)
5.0	1.00	1.60270	+60.3%

Table 1. Calibrated flexibility factor. 

η₀	η	F	Change
0	0	0.80135	−19.9% (stiffening)
1.0	0.20	0.96162	−3.8%
2.5	0.50	1.20202	+20.2% (softening)
5.0	1.00	1.60270	+60.3%

The value η₀ = 2.5 yields a softening of 20.2%, comparable in magnitude to the 19.9% reported in the original manuscript. However, the original value arose from a sign error in the uncalibrated formula, whereas the present value follows from an explicit symmetric enhancement. The latter must be calibrated against independent data before any physical interpretation is attached to it.

8.8. Independent Verification from the Integral Law

For constant stress τ₀, the integral law (10) gives

G γ(x) = (1+η) τ₀ [1 − (1/2) e^(−x/L_c) − (1/2) e^(−(L−x)/L_c)] + β τ₀ [(1/2) e^(−x/L_c) − (1/2) e^(−(L−x)/L_c)].

(70)

Integrating from 0 to L and using ∫₀ᴸ e^(−x/L_c) dx = L_c(1 − e^(−L/L_c)):

∫₀ᴸ G γ dx = (1+η) τ₀ [L − L_c(1 − e^(−L/L_c))],

(71)

since the skew-symmetric contribution integrates to zero. Substituting τ₀ = M₀ r/J recovers (66). The differential and integral formulations therefore agree in the constant-torque benchmark.

9. Comparison with Classical and Nonlocal Benchmarks

The corrected torsional response is summarized as follows.

Classical local torsion:

F_local = 1.

(72)

Uncalibrated nonlocal model with η = 0 and L_c/L = 0.2:

F ≈ 0.80135,

(73)

corresponding to a 19.9% stiffening relative to the classical result.

Calibrated nonlocal model with η = η₀ L_c/L and η₀ = 2.5, L_c/L = 0.2:

F ≈ 1.20202,

(74)

corresponding to a 20.2% softening relative to the classical result, while preserving F → 1 as L_c/L → 0.

The previously reported value F = 10.6, corresponding to a 960% increase, is dimensionally inconsistent and is not admissible. It arose from adding a dimensional quantity L_c to a dimensionless quantity and from using exp(−L_c/L) instead of exp(−L/L_c). The corrected results are bounded, dimensionally consistent, and reduce to the classical local prediction in the limit L_c → 0.

The calibrated range should not be interpreted as a universal prediction. It follows from a symmetric enhancement that must be calibrated against independent data. The uncalibrated result (a 19.9% stiffening) is a pure prediction of the symmetric kernel and can be compared with reference solutions without additional parameters.

10. Classical Local Limit

Consider the limit L_c → 0 with fixed geometry and loading. The symmetric kernel φ tends to a Dirac distribution in the interior of the domain, and the skew kernel contribution is of order L_c. The governing differential equation reduces, away from boundary layers of thickness O(L_c), to

G γ = (1+η) τ.

(75)

Similarly, the torsional equation reduces to

GJ θ′ = (1+η) M_t.

(76)

For the calibrated flexibility factor

F = (1 + η₀ L_c/L)[1 − (L_c/L)(1 − e^(−L/L_c))],

(77)

one has

lim_{L_c/L→0} F = 1.

(78)

Therefore, the model is asymptotically consistent with classical Hookean shear mechanics, provided η = η₀ L_c/L with fixed η₀.

A technical remark is in order. On a finite interval, the standard exponential kernel produces boundary-layer effects of thickness O(L_c). If pointwise recovery of the local law at the exact boundaries is required, one may use a boundary-compensated kernel or restrict the asymptotic statement to the interior domain. This does not affect the corrected global torsional benchmark.

11. Discussion of the Previous Claims

The revised formulation clarifies several points that were ambiguous or incorrect in the earlier version.

First, the kernel originally described as anti-symmetric was not mathematically anti-symmetric in the form written. The correct skew-symmetric kernel contains the factor 1/(2L_c) and the sign function as defined in (3).

Second, a purely anti-symmetric kernel cannot serve as a stand-alone elastic constitutive kernel because its quadratic energy contribution vanishes. The present model therefore uses a positive symmetric kernel plus a bounded skew correction.

Third, the differential reduction must preserve dimensional homogeneity. The corrected governing equation is

G(γ − L_c² γ″) = (1+η) τ − β L_c τ′.

(79)

Fourth, the constitutive boundary conditions must be derived consistently from the integral law. The corrected conditions are (23)–(24), and they are independent of η.

Fifth, the carbon nanotube torsion result must not be used to claim an unphysical 960% softening. The corrected dimensionless factor is bounded, tends to unity as L_c → 0, and takes the value F ≈ 0.80135 in the uncalibrated case (a 19.9% stiffening) or F ≈ 1.20202 in the calibrated case with η₀ = 2.5 (a 20.2% softening).

Sixth, thermodynamic consistency holds for proportional loading (Theorem 1) but not for general loading when β ≠ 0 (Theorem 2). This limitation is a property of the present constitutive law and is not removed by calibration. A fully general formulation would require either β = 0 or an internal-variable extension, both of which are outside the scope of the present paper.

Seventh, the language of the manuscript has been revised to avoid promotional expressions and to state assumptions and limitations explicitly.

12. Conclusions

A one-dimensional integral constitutive model for uncoupled nonlocal shear elasticity has been examined, based on the superposition of a symmetric exponential kernel with a bounded skew-symmetric correction. The integral law admits an exact differential reduction supplemented by constitutive boundary conditions derived from the kernel. Uniqueness of the strain-field boundary-value problem has been proved. The symmetric compliance operator defines a strictly positive quadratic form, and the skew-symmetric operator contributes zero to the quadratic form at fixed stress.

The thermodynamic analysis distinguishes two regimes. For proportional loading paths, which include the constant-torque torsional benchmark, the model satisfies the Clausius–Duhem inequality as an equality with stored energy Ψ*. For general non-proportional loading with a non-zero skew coefficient, the skew-symmetric term contributes a non-vanishing and sign-indefinite component to the internal power, and no stored-energy functional of the stress field can reproduce the full constitutive response. This limitation has been proved, and the scope of the thermodynamic claim has been restricted accordingly. A fully general formulation would require either a conservative constitutive law (β = 0) or an internal-variable extension.

The corrected constant-torque solution yields the end-twist ratio

F = (1+η)[1 − (L_c/L)(1 − e^(−L/L_c))].

(80)

In the uncalibrated case η = 0, the model predicts a stiffening of about 19.9% for L = 30 nm and L_c = 6 nm. With a symmetric enhancement η = η₀ L_c/L, the model can reproduce a softening of about 20.2% (η₀ = 2.5) while preserving F → 1 in the local limit. The enhancement is presented as a calibration, not as a prediction.

The analysis establishes the mathematical structure of the integral operator, the differential reduction, uniqueness, positivity of the symmetric compliance form, thermodynamic consistency for proportional loading, and the constant-torque end-twist expression. It does not establish full thermodynamic admissibility for general loading with β ≠ 0, nor does it validate the model against experimental data on carbon nanotubes. Both remain open for future work.

Use of the references in the manuscript

The references cited in this work are used to position the proposed model within the existing nonlocal elasticity literature and to justify the mathematical tools employed in the derivation. Reference [1] is cited in the Introduction and in the discussion of the classical limit to identify Eringen’s strain-driven nonlocal elasticity as the historical differential framework and to motivate the need for a consistent local Hookean recovery as the nonlocal length tends to zero. Reference [2] is cited to introduce the stress-driven integral formulation of Romano and Barretta, which provides the conceptual basis for expressing strain as a convolution of stress and for avoiding the boundary paradoxes associated with strain-driven differential models. Reference [3] is cited to support the statement that the stress-driven approach has been reviewed and consolidated in recent literature, while independent uncoupled shear mechanics remains less systematically developed. Reference [4] is cited in the differential reduction section to justify the handling of piecewise differentiable integral kernels, the partitioning of the integration domain, and the use of Leibniz-type differentiation rules. Reference [5] is cited to connect the present one-dimensional formulation with established stress-driven applications to rods and axial members, where integral constitutive laws are converted into ordinary differential boundary-value problems with constitutive boundary conditions. Reference [6] is cited to discuss existing nonlocal and gradient nanobeam models, especially coupled Timoshenko-type formulations, and to clarify that the present work does not replace those coupled theories but proposes an independent kernel for uncoupled shear response with a thermodynamic scope that is stated explicitly. Reference [7], the review by Arash and Wang, is cited in the Introduction and in the carbon nanotube torsion discussion to provide context for molecular-dynamics-informed size effects and to support the conclusion that realistic nonlocal corrections should produce moderate flexibility changes rather than order-of-magnitude softening unless independently calibrated ; it is also used to emphasize that the uncalibrated prediction of the present model (a stiffening of approximately 19.9%) and the calibrated softening (approximately 20.2%) must be compared with independent reference solutions before any physical interpretation is attached to them. Reference [8], the nanotechnology handbook edited by Bhushan, is cited to place the carbon nanotube application within the broader nanotechnology literature and to emphasize that material parameters such as the nonlocal length scale, the directional correction parameter β, and the symmetric compliance enhancement η require calibration against simulations or experiments. Together, these references support the formulation, the mathematical reduction, the restricted thermodynamic interpretation, and the corrected benchmark application, while the central constitutive kernel, the differential reduction, the uniqueness proof, the proportionality-restricted thermodynamic theorem, the impossibility proof for stored-energy formulations with a non-zero skew coefficient, and the corrected torsional flexibility factor are developed internally in the present paper.