Saint-Venant torsion of exponentially graded micropolar circular and annular bars

Abstract

In this paper, we consider Saint-Venant torsion of a functionally graded micropolar (Cosserat) beam, where the elastic moduli vary within the plane of the beam cross-section. We state a well-posedness result for the corresponding Neumann boundary-value problem in the weak Sobolev-space setting and show the existence and uniqueness of the solution up to the natural rigid micropolar mode. In view of symmetry, the circular and annular cases reduce to an ordinary differential equation for the radial amplitude. Numerical examples are presented for the stress and couple-stress fields and for the torsional rigidity. The characteristic micropolar length associated with the chosen material parameters is compared with the size of the cross-section, and the classical limit is recovered as a special case.

Keywords

Cosserat elasticity variable elastic moduli torsion weak solutions functionally graded materials collocation method

1. Introduction

Functionally graded materials (FGMs) rapidly attract more attention and interest of industrial practitioners because of their application in aerospace and automotive industry, defence, energy, medicine and optoelectronics [1]. FGMs are materials with a microstructure and elastic properties which gradually vary from point to point, i.e., they are characterized by the compositional gradient from one component to the other. Examples of FGMs include metal/ceramic composites and biological materials such as teeth [2]. FGMs are being used as interfacial zones to improve the bonding strength of layered composites, to reduce the residual and thermal stress in bonded dissimilar materials, as wear-resistant layers in machine and engine components, in energy conversion devices and as a penetration-resistant material in armour plates and bulletproof vests because of the ability of FGM to inhibit crack propagation (see, for example, [3 –7] for the survey of the applications). In spite of many advanced experimental and computational results obtained for FGMs over the last 20 years (e.g., [8 –12]), the investigation of the material behaviour of FGMs under the provisions of the theory of micropolar elasticity appears to have received limited attention in the literature.

The theory of micropolar elasticity was proposed by Eringen [13] and Nowacki [14] to account for the effect of material microstructure on the elastic behaviour of the solids for which the material microstructure is significant. Examples of such solids could include those manufactured of graphite, synthetic polymers and FGMs [15]. Therefore, it would be natural to assume that the models for FGMs, in theory, should be considered under the umbrella of the micropolar (Cosserat) theory.

In this paper, we consider Saint-Venant torsion of a functionally graded micropolar cylinder in the case when the elastic moduli vary from point to point throughout the plane of the cross-section. When the cross-section is functionally graded, the Neumann boundary-value problem for anti-plane Cosserat elasticity [16] reduces to the problem for the operator with variable coefficients. The system of the partial differential equations with variable coefficients cannot be treated using the boundary integral equation method as in [16 –18] because it is hard, if not impossible, to find the fundamental solution. Therefore, we prove the existence and uniqueness results in the weak Sobolev space setting as in [19] and in a companion paper and then solve a number of illustrative examples numerically for the beams of circular and annular cross-sections through the consideration of the reduced radial boundary-value problem. We consider the case of circular and annular functionally graded micropolar bars, reduce the problem to a single ordinary differential equation and obtain the corresponding stress fields and torsional rigidity using a one-dimensional collocation approach. Finally, we recover the classical limit [20] and the homogeneous micropolar reference solution [21 –24].

2. Preliminaries

In what follows, the Greek indices $α, β$ take the values $1, 2$ , while the Latin indices $i, j$ take the values $1, 2, 3$ . The convention of summation over repeated indices is understood, $M_{m \times n}$ is the space of $(m \times n)$ -matrices, $E_{n}$ is the identity element in $M_{n \times n}$ , a superscript T indicates the matrix transposition, $\partial_{α} = \partial ∕ \partial x_{α}$ and $(. . .),_{α} \equiv \partial (. . .) ∕ \partial x_{α}$ . Also, if X is a space of scalar functions and v is a matrix-valued function, $v \in X$ means that every component of v belongs to X.

Let the prismatic beam occupy volume $V = S \times (0, L)$ , where $S \subset ℝ^{2}$ is the cross-sectional domain with boundary ∂S and outward unit normal $n = (n_{1}, n_{2})$ . The domain is occupied by an isotropic micropolar elastic material with varying constitutive moduli

\begin{matrix} μ = μ (x), α = α (x), β = β (x), γ = γ (x), κ = κ (x), x = (x_{1}, x_{2}) \in S . \end{matrix}

(1)

where $μ, α, β, γ$ and κ denote the variable micropolar elastic moduli.

The beam is in equilibrium, with no body forces or couples, and its lateral surface is free of traction. This state is maintained by a twisting couple of magnitude M applied at the end plane $x_{3} = L$ , where the following conditions hold:

\begin{matrix} \begin{matrix} \int_{S} σ_{31} dσ & = 0, \int_{S} σ_{32} dσ = 0, \int_{S} σ_{33} dσ = 0, \\ \int_{S} (x_{2} σ_{33} + ϱ_{31}) dσ & = 0, \int_{S} (x_{1} σ_{33} - ϱ_{32}) dσ = 0, \\ \int_{S} (x_{1} σ_{32} - x_{2} σ_{31} + ϱ_{33}) dσ & = M . \end{matrix} \end{matrix}

(2)

where $σ_{ji}$ are the components of the stress tensor and $ϱ_{ji}$ are the components of the couple-stress tensor as in [16]. The torsion loading is characterized by the constant twist per unit length τ about the $x_{3}$ axis.

We introduce three torsion functions $ψ_{1} (x_{1}, x_{2})$ , $ψ_{2} (x_{1}, x_{2})$ and $ϕ (x_{1}, x_{2})$ [16] and assume that

\begin{matrix} u (x_{1}, x_{2}) = {(ψ_{1}, ψ_{2}, ϕ)}^{T}, x = (x_{1}, x_{2}) \in S . \end{matrix}

(3)

The system of the governing equations, following the standard anti-plane reduction as in [17] can be written in the form

\begin{matrix} L (x, \partial) u = 0 in S, \end{matrix}

(4)

where

\begin{matrix} A (x) = 2 γ (x) + β (x), B (x) = γ (x) + κ (x), C (x) = γ (x) - κ (x), D (x) = μ (x) + α (x) . \end{matrix}

(5)

and

\begin{matrix} L (x, \partial) = (\begin{matrix} \partial_{1} (A \partial_{1}) + \partial_{2} (B \partial_{2}) - 4 α & \partial_{1} (β \partial_{2}) + \partial_{2} (C \partial_{1}) & 2 α \partial_{2} \\ \partial_{1} (C \partial_{2}) + \partial_{2} (β \partial_{1}) & \partial_{1} (B \partial_{1}) + \partial_{2} (A \partial_{2}) - 4 α & - 2 α \partial_{1} \\ - 2 α \partial_{2} & 2 α \partial_{1} & \partial_{1} (D \partial_{1}) + \partial_{2} (D \partial_{2}) \end{matrix}) . \end{matrix}

(6)

The boundary stress operator associated with (6) is

\begin{matrix} T (x, \partial; n) u = f on ∂S, \end{matrix}

(7)

where

\begin{matrix} T (x, \partial; n) = (\begin{matrix} n_{1} A \partial_{1} + n_{2} B \partial_{2} & n_{1} β \partial_{2} + n_{2} C \partial_{1} & 0 \\ n_{1} C \partial_{2} + n_{2} β \partial_{1} & n_{1} B \partial_{1} + n_{2} A \partial_{2} & 0 \\ - 2 α n_{2} & 2 α n_{1} & D \partial_{n} \end{matrix}), \partial_{n} = n_{1} \partial_{1} + n_{2} \partial_{2} . \end{matrix}

(8)

and the boundary data are in the form

\begin{matrix} f = {(γ (x) n_{1,} γ (x) n_{2,} μ (x) (x_{2} n_{1} - x_{1} n_{2}))}^{T} on ∂S \end{matrix}

(9)

3. Well-posedness of the variable-coefficient torsion problem

Before proceeding to the consideration of circular and annular cross-sections, we record a well-posedness result for the general variable-coefficient anti-plane micropolar torsion problem. This result ensures that the boundary value problem (4) and (7) used in the developments that follow are mathematically meaningful under standard positivity assumptions on the elastic moduli. A detailed proof, which includes the weak formulation, continuity and coercivity estimates for the associated bilinear form and an application of the Lax–Milgram theorem, is given in our companion paper and is not repeated here in full.

Let $S \subset ℝ^{2}$ be a bounded Lipschitz domain with boundary ∂S. Assume that the elastic moduli

μ, α, β, γ, κ

are bounded in S, and that there exists a constant $m_{0} > 0$ such that

μ (x) \geq m_{0}, α (x) \geq m_{0}, γ (x) \geq m_{0}, κ (x) \geq m_{0}, 2 γ (x) + β (x) \geq m_{0}, x \in S .

In particular, in the homogeneous case with constant coefficients [16,17], the above assumption reduces to the standard micropolar positivity conditions, and we may simply choose

m_{0} = \min {μ, α, γ, κ, 2 γ + β} > 0 .

These bounds ensure uniform ellipticity of the principal second-order part of the operator $L$ , which is the requirement for the application of the Lax–Milgram theorem (see [25]).

Theorem 1 (Existence and uniqueness up to the rigid mode). Under the above assumptions, the anti-plane micropolar torsion problems (4) and (7) with Saint-Venant torsion boundary loading admit a weak solution

u = {(ψ_{1}, ψ_{2}, ϕ)}^{T} \in H^{1} {(S)}^{3} .

The solution is unique up to the rigid micropolar mode

{(0, 0, c)}^{T}, c = const .

Consequently, after imposing the normalization condition

\int_{S} ϕ dσ = 0,

the weak solution is unique.

Remark 2. A short proof sketch is as follows. One first introduces the natural bilinear form associated with (4) and (7) on the normalized space

V_{0} = {(v_{1}, v_{2}, v_{3}) \in H^{1} {(S)}^{3} : \int_{S} v_{3} dσ = 0} .

Under the positivity assumptions stated above, the principal part is uniformly elliptic, the lower-order coupling terms are bounded, and the resulting bilinear form is continuous and coercive on $V_{0}$ . The existence of a unique weak solution in $V_{0}$ then follows from the Lax–Milgram theorem. The non-uniqueness in the third component before normalization corresponds precisely to the rigid micropolar mode ${(0, 0, c)}^{T}$ . Full details are given in [25].

4. Circular and annular bars with radial exponential grading

We now reduce the general variable-coefficient torsion problem to circular and annular cross-sections with radial grading. In this setting, the rotational symmetry of the geometry and the boundary loading allow us to reduce the systems (4) and (7) to a single ordinary differential equation for the radial amplitude. This reduction is the key step that makes the subsequent stress, couple-stress and torque computations practical.

4.1. Circular bar and the first Fourier harmonic reduction

Let $S = {(r, θ) : 0 \leq r < R, 0 \leq θ < 2 π}$ be the disc of radius R in polar coordinates. For notational convenience, we denote by b the inner radius of the cross-section (so $b = 0$ for the disc and $b > 0$ for an annulus). Assume that the elastic moduli are graded in the radial direction. In what follows, we focus on the exponential law, which is a standard model for continuously graded FGMs [5 –7] and is particularly convenient in torsion problems; however, other grading laws can be considered as well as long as they satisfy the assumptions for Theorem 1. It is not very difficult to show, for example, that the proper bounds can be chosen for the linear grading law which would result in the existence of the solution. We can see that on a bounded cross-section, the exponential grading law preserves positivity of the moduli and yields bounded coefficient combinations with uniform positive lower bounds whenever the reference moduli are positive; consequently, it satisfies the hypotheses of the well-posedness theorem stated in Theorem 1. Specifically, we take

\begin{matrix} p (r) = p_{0} \exp (η \frac{r}{R}), p \in {μ, α, β, γ, κ} and p_{0} \in {μ_{0}, α_{0}, β_{0}, γ_{0}, κ_{0}}, \end{matrix}

(10)

where $p (r)$ denotes any one of the graded constitutive moduli, η is the non-dimensional grading parameter, and $p_{0}$ is the corresponding reference value of that modulus at $r = 0$ in the circular case. In the annular case, the boundary values at $r = b$ and $r = R$ are obtained from the same exponential law. Thus, the coefficient combinations $A, B, C, D$ in (5) are functions of r only.

Moreover, since $r \in [b, R]$ (with $b = 0$ in the disc case), the exponent satisfies

\exp (\min {η \frac{b}{R}, η}) \leq \exp (η \frac{r}{R}) \leq \exp (\max {η \frac{b}{R}, η}), r \in [b, R],

and hence, for each modulus p in (10), we have the explicit bounds

p_{0} \exp (\min {η \frac{b}{R}, η}) \leq p (r) \leq p_{0} \exp (\max {η \frac{b}{R}, η}) .

In particular, the assumptions of Theorem 1 are satisfied, and one may take

m_{0} = \min {μ_{0}, α_{0}, γ_{0}, κ_{0}, 2 γ_{0} + β_{0}} \exp (\min {η \frac{b}{R}, η}) > 0 .

On the circular boundary $r = R$ , the outward unit normal is $n = e_{r} = (\cos θ, \sin θ)$ and $x_{2} n_{1} - x_{1} n_{2} \equiv 0$ . Hence, the boundary data (9) reduce to

\begin{matrix} f = {(γ (R) \cos θ, γ (R) \sin θ, 0)}^{T} on r = R . \end{matrix}

(11)

Since the coefficients $μ, α, β, γ, κ$ are independent of θ and the boundary data contain only the first Fourier harmonic, we seek a solution of (4) in the form

\begin{matrix} ψ_{1} (r, θ) = U (r) \cos θ, ψ_{2} (r, θ) = U (r) \sin θ, ϕ (r, θ) = V (r) . \end{matrix}

(12)

For $r = R$ , the third boundary condition in (7) becomes

- 2 α n_{2} ψ_{1} + 2 α n_{1} ψ_{2} + D \partial_{n} ϕ = 0 .

Substituting $n_{1} = \cos θ$ , $n_{2} = \sin θ$ and (12), the coupling term cancels identically, so that

\begin{matrix} D (R) V^{'} (R) = 0 . \end{matrix}

(13)

Together with the normalization condition $\int_{S} ϕ dσ = 0$ from Theorem 1, this yields

\begin{matrix} V (r) \equiv 0 . \end{matrix}

(14)

Therefore, the circular torsion problem reduces to a scalar boundary-value problem for $U (r)$ . In the literature, the function $U (r)$ is sometimes called a radial amplitude.

4.2. Polar-coordinate identities and reduced radial equation

For further analysis, the following well-known formulae will be used for derivatives in polar coordinates [20]

\partial_{1} = \cos θ \partial_{r} - \frac{\sin θ}{r} \partial_{θ}, \partial_{2} = \sin θ \partial_{r} + \frac{\cos θ}{r} \partial_{θ} .

In view of (12), we obtain

\begin{matrix} ψ_{1, 1} = U^{'} \cos^{2} θ + \frac{U}{r} \sin^{2} θ, ψ_{2, 2} = U^{'} \sin^{2} θ + \frac{U}{r} \cos^{2} θ, \end{matrix}

(15)

\begin{matrix} ψ_{1, 2} = ψ_{2, 1} = (U^{'} - \frac{U}{r}) \sin θ \cos θ, \end{matrix}

(16)

and, specifically,

\begin{matrix} ψ_{1, 1} + ψ_{2, 2} = U^{'} + \frac{U}{r}, ψ_{2, 1} - ψ_{1, 2} = 0 . \end{matrix}

(17)

Substituting (12) and (14) into the first two equations of the component form of (4), using the radial dependence of the coefficients and the isotropic identities

B (r) + C (r) = 2 γ (r), B (r) + C (r) + β (r) = 2 γ (r) + β (r) = A (r),

the angular dependence cancels, and we obtain the scalar radial equation

\begin{matrix} \frac{d}{dr} [A (r) (U^{'} (r) + \frac{U (r)}{r})] - 4 α (r) U (r) = 0, 0 < r < R . \end{matrix}

(18)

The regularity condition at the origin implies

\begin{matrix} U (r) = O (r) as r \to 0, \end{matrix}

(19)

and, in particular, we may impose

\begin{matrix} U (0) = 0 . \end{matrix}

(20)

On $r = R$ , substitution of (12) and (14)–(16) into the first (or, equivalently, second) traction equation in (7) gives the Robin boundary condition

\begin{matrix} A (R) U^{'} (R) + β (R) \frac{U (R)}{R} = γ (R) . \end{matrix}

(21)

Hence, for the circular bar with radial grading, the coupled torsion system reduces to the scalar boundary-value problem (18), (20) and (21).

4.3. Homogeneous reference case and the graded circular-bar problem

When the coefficients are constant, $A (r) \equiv A_{0}$ and $α (r) \equiv α_{0}$ , equation (18) becomes

\begin{matrix} U^{′′} + \frac{1}{r} U^{'} - \frac{1}{r^{2}} U - λ^{2} U = 0, λ^{2} = \frac{4 α_{0}}{A_{0}}, \end{matrix}

(22)

whose general solution is

U (r) = C_{1} I_{1} (λr) + C_{2} K_{1} (λr) .

where $I_{1} (λr)$ and $K_{1} (λr)$ are the modified Bessel functions of the first and second kind of order one, and $C_{1}$ and $C_{2}$ are the arbitrary constants.

Regularity at $r = 0$ implies $C_{2} = 0$ , hence

\begin{matrix} U (r) = C_{1} I_{1} (λr), \end{matrix}

(23)

where $C_{1}$ is determined by the boundary condition (21) (with constant coefficients).

For the exponentially graded circular bar, the reduced problem is the smooth two-point boundary-value problem

\begin{matrix} \frac{d}{dr} [A (r) (U^{'} (r) + \frac{U (r)}{r})] - 4 α (r) U (r) = 0, U (0) = 0, A (R) U^{'} (R) + β (R) \frac{U (R)}{R} = γ (R) . \end{matrix}

(24)

4.4. One-dimensional collocation formulation

For the numerical computations, the reduced problem (24) is solved on the radial interval by a standard one-dimensional collocation method. Let

0 = r_{0} < r_{1} < \dots < r_{N} = R

be a partition of $[0, R]$ , and let $U_{h}$ be a piecewise-cubic approximation satisfying the essential condition $U_{h} (0) = 0$ . On each subinterval $[r_{j}, r_{j + 1}]$ , we impose the differential equation at two interior collocation points $r_{j}^{(1)}, r_{j}^{(2)}$ , namely

\begin{matrix} R [U_{h}] (r_{j}^{(ℓ)}) = 0, ℓ = 1, 2, j = 0, 1, \dots, N - 1, \end{matrix}

(25)

where the residual operator is

\begin{matrix} R [U] : = \frac{d}{dr} [A (r) (U^{'} (r) + \frac{U (r)}{r})] - 4 α (r) U (r) . \end{matrix}

(26)

The Robin condition at $r = R$ is enforced directly,

\begin{matrix} A (R) U_{h}^{'} (R) + β (R) \frac{U_{h} (R)}{R} = γ (R) . \end{matrix}

(27)

For the disc, the regularity condition at the origin is imposed through $U_{h} (0) = 0$ . For the annulus, the same collocation procedure is used on $[b, R]$ , with Robin conditions imposed at both boundaries. After the nodal values are obtained, the torsional rigidity is evaluated by numerical quadrature of the torque expression introduced in the next subsection.

4.5. Annular bar reduction and post-processing formulas

Let $S = {(r, θ) : b < r < R, 0 \leq θ < 2 π}$ be an annulus, and assume the same radial grading law we used for the circular cross-section (10). The trial form of the solution (12) remains valid and again yields $ϕ \equiv 0$ . The reduced radial equation is still (18), but now posed on $b < r < R$ :

\begin{matrix} \frac{d}{dr} [A (r) (U^{'} (r) + \frac{U (r)}{r})] - 4 α (r) U (r) = 0, b < r < R . \end{matrix}

(28)

On each circular boundary, the traction conditions reduce to Robin conditions. In the homogeneous micropolar case ( $A (r) \equiv A_{0}$ , $α (r) \equiv α_{0}$ , $β (r) \equiv β_{0}$ ), they take the form

\begin{matrix} A_{0} U^{'} (b) + \frac{β_{0}}{b} U (b) = γ_{0}, A_{0} U^{'} (R) + \frac{β_{0}}{R} U (R) = γ_{0}, \end{matrix}

(29)

and the solution on $b < r < R$ is

\begin{matrix} U (r) = C_{I} I_{1} (λr) + C_{K} K_{1} (λr), λ^{2} = \frac{4 α_{0}}{A_{0}}, \end{matrix}

(30)

where $C_{I}$ and $C_{K}$ are determined from (29) [21]. For the graded annulus, (28) is solved numerically together with the corresponding Robin conditions at $r = b, R$ .

Once $U (r)$ is known (for either the disc or the annulus), the force-stress components and the couple-stress contribution follow from the constitutive relations explicitly presented in [16]

\begin{matrix} σ_{13} & = (μ + α) Φ,_{1} + 2 α φ_{2} - (μ - α) τ x_{2}, \\ σ_{31} & = (μ - α) Φ,_{1} - 2 α φ_{2} - (μ + α) τ x_{2}, \\ σ_{23} & = (μ + α) Φ,_{2} - 2 α φ_{1} + (μ - α) τ x_{1}, \\ σ_{32} & = (μ - α) Φ,_{2} + 2 α φ_{1} + (μ + α) τ x_{1}, \\ σ_{33} & = 0, σ_{αβ} = 0, ϱ_{3 α} = ϱ_{α 3} = 0, \\ ϱ_{11} & = (2 γ + β) φ_{1, 1} + β (φ_{2, 2} + τ), \\ ϱ_{22} & = (2 γ + β) φ_{2, 2} + β (φ_{1, 1} + τ), \\ ϱ_{12} & = (γ + κ) φ_{1, 2} + (γ - κ) φ_{2, 1}, \\ ϱ_{33} & = β (φ_{1, 1} + φ_{2, 2}) + (2 γ + β) τ . \end{matrix}

Assuming the transformation [16]

Φ (x_{1}, x_{2}) = τϕ (x_{1}, x_{2}), φ_{α} (x_{1}, x_{2}) = τ (ψ_{α} (x_{1}, x_{2}) - \frac{1}{2} x_{α})

for the first-harmonic field (12), we obtain

\begin{matrix} σ_{13} = τ \sin θ (α (r) U (r) - μ (r) r), σ_{23} = τ \cos θ (μ (r) r - α (r) U (r)), \end{matrix}

(31)

and therefore, the shear magnitude is radial:

\begin{matrix} \sqrt{σ_{13}^{2} + σ_{23}^{2}} = | τ | | μ (r) r - α (r) U (r) | . \end{matrix}

(32)

The relevant couple-stress component can be written as

\begin{matrix} ϱ_{33} (r) = τ (2 γ (r) + β (r)) (U^{'} (r) + \frac{U (r)}{r}), \end{matrix}

(33)

and the transmitted torque is

\begin{matrix} M = \int_{S} (x_{1} σ_{23} - x_{2} σ_{13} + ϱ_{33}) dσ . \end{matrix}

(34)

For the annulus (and similarly for the disc when $b = 0$ ), in polar coordinates, this becomes

\begin{matrix} M = 2 π \int_{b}^{R} [r^{2} (μ (r) r - α (r) U (r)) + r ϱ_{33} (r)] dr . \end{matrix}

(35)

Equations (24)–(35) provide the computational framework used in the numerical examples and in the comparison with the homogeneous micropolar and classical limits.

5. Numerical results and discussion

5.1. Parameters and numerical scheme

For the circular and annular examples, we adopt the exponential grading law

p (r) = p_{0} \exp (η \frac{r}{R}), p \in {μ, α, β, γ, κ},

with the dimensionless reference parameters

R = 1, b = 0.35, τ = 1, μ_{0} = 1, α_{0} = 0.20, β_{0} = 0.30, γ_{0} = 0.60, κ_{0} = 0.40 .

The reduced boundary-value problem for the radial amplitude $U (r)$ is solved numerically using the collocation formulation described above on the radial interval. For each geometry, we compare four cases: (i) graded micropolar ( $η = 1.2$ ), (ii) homogeneous micropolar ( $η = 0$ ) [21,22], (iii) classical homogeneous ( $α = 0$ , $η = 0$ ) [20] and (iv) classical graded ( $α = 0$ , $η = 1.2$ ) [9]. In the torsional rigidity plots, we also display the homogeneous references as horizontal lines. In the classical cases, the torsional response is computed from the Cauchy–Saint-Venant formula with graded $μ (r)$ , i.e., $M_{cl} (η) = 2 πτ \int μ (r) r^{3} dr$ , normalized by the homogeneous classical value $M_{cl, hom}$ [20].

For the chosen micropolar parameters, the intrinsic torsion length may be estimated from the homogeneous reduced equation (22) as

ℓ_{c} = \frac{1}{λ} = \sqrt{\frac{A_{0}}{4 α_{0}}} = \sqrt{\frac{2 γ_{0} + β_{0}}{4 α_{0}}} .

With the present values $α_{0} = 0.20$ , $β_{0} = 0.30$ and $γ_{0} = 0.60$ , one obtains $A_{0} = 1.50$ , and hence, $ℓ_{c} \approx 1.37$ . Thus, for the solid disc, $ℓ_{c} ∕ R \approx 1.37$ , while for the annulus, $ℓ_{c} ∕ (R - b) \approx 2.11$ . The selected moduli are therefore intended to illustrate a regime in which micropolar effects are visible at the scale of the cross-section rather than to model a specific calibrated engineering material. This explains the pronounced difference between the classical and micropolar responses in the numerical examples.

5.2. Solid disc (circular bar)

Figure 1 compares the radial amplitude $U (r)$ for the solid disc. The graded micropolar solution deviates substantially from the homogeneous micropolar profile and from the classical reference. The spatially varying coefficients yield a smooth, non-polynomial profile $U (r)$ that captures the combined effects of microstructure and graded stiffness. In the classical elasticity limit ( $α = 0,$ $η = 0)$ , the radial amplitude for a solid disc varies linearly with the radius [20]. When exponential grading is introduced into the classical setting ( $α = 0,$ $η = 1.2)$ as in [9], the circular cross-section remains warping -free, so the classical graded and homogeneous curves coincide. In contrast, in the homogeneous micropolar case ( $α = α_{0}$ , $η = 0)$ , the intrinsic length scale modifies the solution, yielding Bessel-type profiles and producing a distinct curvature near the boundary [21].

Figure 1.

Solid disc ( $R = 1$ ): radial amplitude $U (r)$ for graded micropolar ( $η = 1.2$ ), homogeneous micropolar ( $η = 0$ ), classical homogeneous ( $α = 0$ , $η = 0$ ) and classical graded ( $α = 0$ , $η = 1.2$ ). The two classical curves overlap.

Figure 2 shows the shear-stress magnitude $\sqrt{σ_{13}^{2} + σ_{23}^{2}}$ as a function of r. In contrast to Figure 1, the graded classical curve differs from the homogeneous classical curve because $μ (r)$ enters the stress magnitude as a multiplicative factor. For the present parameter set, the homogeneous micropolar shear profile remains close to the classical linear trend, whereas functional grading produces an “exponential-like” variation that mirrors the underlying coefficient variation and the corresponding change in the solution fields. Adding the effects of exponential grading migrates the concentration of the couple stresses towards the direction of the grading. This result is consistent with the conclusion in [5], where the effect of exponential grading for a circular micropolar bar is considered along the axial direction.

Figure 2.

Solid disc ( $R = 1$ ): shear-stress magnitude profile $\sqrt{σ_{13}^{2} + σ_{23}^{2}}$ for the four comparison cases (graded/homogeneous micropolar and graded/homogeneous classical).

The normalized torsional rigidity $M ∕ M_{cl, hom}$ is plotted in Figure 3 as a function of the grading parameter η. The graded micropolar response exhibits a pronounced sensitivity to η, whereas the homogeneous micropolar reference appears as a constant level. The couple-stress component $ϱ_{33}$ exhibits a spatial distribution distinct from the force-stress shear and contributes directly to the transmitted torque. The classical graded rigidity increases monotonically with η due to the exponential increase of $μ (r)$ towards the outer boundary.

Figure 3.

Solid disc ( $R = 1$ ): normalized torsional rigidity $M ∕ M_{cl, hom}$ vs. grading parameter η. Shown are the graded micropolar curve, the graded classical curve ( $α = 0$ ), and the homogeneous micropolar and homogeneous classical references (horizontal lines).

5.3. Annular bar

Next, we consider an annulus with $b = 0.35$ and $R = 1$ . The profiles of the radial amplitude and shear-stress magnitude are shown in Figures 4 and 5. As in the disc case, the two classical $U (r)$ profiles coincide in Figure 4, while the graded classical shear magnitude differs from the homogeneous classical profile. There is approximately up to 20% difference in the amount of shear stress between the classical graded case and the micropolar graded case. This observation is consistent with the results obtained for stresses around cracks in micropolar medium [26] and for stresses around rigid inclusions [27].

Figure 4.

Annulus ( $b = 0.35$ , $R = 1$ ): radial amplitude $U (r)$ for graded micropolar ( $η = 1.2$ ), homogeneous micropolar ( $η = 0$ ), classical homogeneous ( $α = 0$ , $η = 0$ ) and classical graded ( $α = 0$ , $η = 1.2$ ). The two classical curves overlap.

Figure 5.

Annulus ( $b = 0.35$ , $R = 1$ ): shear-stress magnitude profile $\sqrt{σ_{13}^{2} + σ_{23}^{2}}$ for the four comparison cases.

Figure 6 shows the corresponding normalized torsional rigidity as a function of η. The annular geometry reduces the overall rigidity relative to the solid disc, but the qualitative grading trends remain similar: the classical graded rigidity increases monotonically with η, while the micropolar graded curve exhibits stronger sensitivity due to the combined force-stress and couple-stress contributions.

Figure 6.

Annulus ( $b = 0.35$ , $R = 1$ ): normalized torsional rigidity $M ∕ M_{cl, hom}$ vs. grading parameter η (graded micropolar and graded classical curves; homogeneous micropolar and homogeneous classical references).

5.4. Remarks

The four-case comparisons in Figures 1 –6 separate the effects of (i) grading within the micropolar model, (ii) grading within the classical (Cauchy) model and (iii) microstructure at fixed grading. For the circular and annular geometries, the classical Saint-Venant solution is warping-free; hence, the classical $U (r)$ profile is linear and independent of η, and the classical graded and classical homogeneous curves overlap in Figures 1 and 4. In contrast, the micropolar field $U (r)$ is sensitive to grading: for $η = 1.2$ , the graded micropolar amplitude is substantially larger than the homogeneous micropolar profile and exhibits noticeable curvature (Figures 1 and 4).

The grading influence becomes pronounced in the stress measures. In the homogeneous case, the micropolar shear magnitude is slightly below the classical profile (Figures 2 and 5). When the grading is introduced, the classical graded shear magnitude increases rapidly towards the outer boundary because $μ (r)$ enters the stress as a multiplicative factor. The graded micropolar shear magnitude follows the same qualitative trend but remains below the graded classical profile for the parameters considered, indicating that microstructure effects can partially relax the shear response at fixed grading.

The rigidity plots quantify these trends. For $η = 0$ , the homogeneous micropolar reference predicts a higher torsional rigidity than the classical homogeneous value (approximately a factor 2.8 for the disc and 2.6 for the annulus in the present parameter set). Both the classical graded and the micropolar graded rigidities increase monotonically with η, but the micropolar graded curve exhibits much stronger sensitivity: at $η = 2$ , $M ∕ M_{cl, hom}$ reaches $O (1 0^{1})$ for the disc and $O (1 0^{2})$ for the annulus, whereas the corresponding classical graded rigidity is $O (1)$ (Figures 3 and 6). The annular geometry reduces the baseline rigidity relative to the disc and introduces a non-zero stress level at the inner boundary $r = b$ , but the qualitative grading trends remain similar.

6. Conclusion

We studied Saint-Venant torsion of functionally graded micropolar (Cosserat) bars with circular and annular cross-sections, assuming spatial variation of the elastic moduli in the cross-sectional plane. The general anti-plane torsion problem was formulated as a variable-coefficient Neumann-type boundary value problem for a coupled system of three second-order equations. For this general formulation, a well-posedness result was stated (existence and uniqueness of the weak solution up to the natural rigid micropolar mode, with uniqueness restored by normalization), while the proof is deferred to a companion paper.

For circular and annular bars with radial grading, symmetry reduces the coupled problem to a scalar boundary-value problem for the radial amplitude. This reduction provides a practical framework for computing the displacement, stress and couple-stress fields, as well as the torsional response. The homogeneous micropolar solutions were recovered in closed form in terms of Bessel functions, and the classical elasticity limit was obtained as a special case.

The numerical results demonstrate that radial grading can significantly affect the shear-stress distribution, the couple-stress response and the effective torsional rigidity of micropolar bars. The proposed formulation therefore provides a useful mechanics model for analysing torsion in functionally graded microstructured beams and offers a natural basis for further studies involving other grading laws, parameter identification and design-oriented optimization.

Footnotes

Acknowledgements

The authors would like to thank the Natural Science and Engineering Research Council of Canada (NSERC) for the financial support of this work through the grant RGPIN-04192-2018. The authors also acknowledge that Generative AI tools were used to assist with drafting computational scripts and preparing some figures; all computations and conclusions were verified by the authors.

ORCID iD

Stanislav Potapenko

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors received financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through Discovery Grant RGPIN-04192-2018.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

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