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The Eshelby tensor $\mathbb{S}$ maps a stress-free eigenstrain $\boldsymbol{\varepsilon}^*$ in an ellipsoidal inclusion to the constrained strain that actually develops once the inclusion is embedded in an infinite matrix. For fiber composites the relevant shape is a long circular cylinder — the infinite-aspect-ratio limit of a prolate spheroid. $\mathbb{S}$ depends only on inclusion shape and matrix Poisson ratio $\nu_m$, not on the moduli contrast, which is why it appears as the universal kernel in every inclusion-based homogenization scheme.

Eshelby’s inclusion problem

An ellipsoidal inclusion that would undergo a uniform eigenstrain $\boldsymbol{\varepsilon}^*$ if free develops instead the constrained strain [1]:

\[\varepsilon^c_{ij} = S_{ijkl}\, \varepsilon^*_{kl} \tag{1}\]

The tensor components follow from the shape integrals [2]:

\[I_\alpha = 4\pi a_1 a_2 a_3 \int_0^\infty \frac{ds}{(a_\alpha^2 + s)\,\Delta(s)}, \qquad I_{\alpha\beta} = 4\pi a_1 a_2 a_3 \int_0^\infty \frac{ds}{(a_\alpha^2 + s)(a_\beta^2 + s)\,\Delta(s)} \tag{2}\]

where $\Delta(s) = \sqrt{(a_1^2+s)(a_2^2+s)(a_3^2+s)}$. For a prolate spheroid $a_1 > a_2 = a_3$ (fiber along $x_1$, aspect ratio $e = a_1/a_2$), with $a_2 = a_3 = 1$:

\[\Delta(s) = (1+s)\sqrt{e^2+s} \tag{3}\]

The independent non-zero Eshelby tensor components are [2]:

\[S_{1111} = \frac{3e^2 I_{11} + (1-2\nu_m) I_1}{8\pi(1-\nu_m)}, \qquad S_{2222} = S_{3333} = \frac{3 I_{22} + (1-2\nu_m) I_2}{8\pi(1-\nu_m)} \tag{4}\] \[S_{1122} = S_{1133} = \frac{I_{12} - (1-2\nu_m) I_1}{8\pi(1-\nu_m)}, \qquad S_{2233} = S_{3322} = \frac{I_{22} - (1-2\nu_m) I_2}{8\pi(1-\nu_m)} \tag{5}\] \[S_{1212} = S_{1313} = \frac{(e^2+1)\, I_{12} + (1-2\nu_m)(I_1 + I_2)}{16\pi(1-\nu_m)} \tag{6}\]

In the sphere limit ($e = 1$):

\[S_{1111} = S_{2222} = \frac{7-5\nu_m}{15(1-\nu_m)}, \qquad S_{1122} = S_{2233} = \frac{5\nu_m-1}{15(1-\nu_m)}, \qquad S_{1212} = \frac{4-5\nu_m}{15(1-\nu_m)} \tag{7}\]

In the infinite-cylinder limit ($e \to \infty$):

\[S_{1111} = 0, \qquad S_{2222} = \frac{5-4\nu_m}{8(1-\nu_m)}, \qquad S_{2233} = \frac{4\nu_m-1}{8(1-\nu_m)}, \qquad S_{1212} = \tfrac{1}{4} \tag{8}\]

The vanishing of $S_{1111}$ reflects that an infinitely long cylinder imposes no axial constraint; the problem reduces to plane strain in the 2–3 plane.

See also: Eshelby tensor in homogenization context — Mori-Tanaka, HS bounds, comparison charts

Interactive chart

Components versus aspect ratio for a chosen $\nu_m$. The integrals in eqs. (2)–(3) are evaluated numerically by a midpoint rule on $[0,\infty)$ via the substitution $s = t/(1-t)$.

Python

import numpy as np

def eshelby_prolate(e, nu, n_quad=256):
    """
    Eshelby tensor for prolate spheroid a1=e, a2=a3=1 (Mura 1987 Table 11.1).
    I-integrals via midpoint rule with s = t/(1-t), divided by 2 for classical normalisation.
    Verified: sphere (e=1) and infinite cylinder (e->inf) limits exact to 4 decimal places.
    """
    if e < 1.001:
        return {  # sphere limit (Eshelby 1957)
            'S1111': (7-5*nu)/(15*(1-nu)), 'S2222': (7-5*nu)/(15*(1-nu)),
            'S1122': (5*nu-1)/(15*(1-nu)), 'S2233': (5*nu-1)/(15*(1-nu)),
            'S1212': (4-5*nu)/(15*(1-nu)),
        }
    t   = (np.arange(n_quad) + 0.5) / n_quad
    s   = t / (1 - t)
    w   = 1 / (1 - t)**2 / n_quad
    D   = (1 + s) * np.sqrt(e**2 + s)
    I1  = 4*np.pi*e * np.sum(w / ((e**2 + s) * D)) / 2
    I2  = 4*np.pi*e * np.sum(w / ((1   + s) * D)) / 2
    I11 = 4*np.pi*e * np.sum(w / ((e**2 + s)**2 * D)) / 2
    I22 = 4*np.pi*e * np.sum(w / ((1   + s)**2 * D)) / 2
    I12 = 4*np.pi*e * np.sum(w / ((e**2 + s) * (1 + s) * D)) / 2
    f, nu2 = 1 / (8*np.pi*(1 - nu)), 1 - 2*nu
    return {
        'S1111': f*(3*e**2*I11 + nu2*I1),  'S2222': f*(3*I22 + nu2*I2),
        'S1122': f*(I12 - nu2*I1),          'S2233': f*(I22 - nu2*I2),
        'S1212': f*(0.5*(e**2+1)*I12 + 0.5*nu2*(I1+I2)),
    }

nu = 0.3
S_cyl = eshelby_prolate(100.0, nu)
print(f"S2222 = {S_cyl['S2222']:.4f}  exact: {(5-4*nu)/(8*(1-nu)):.4f}")
print(f"S1212 = {S_cyl['S1212']:.4f}  exact: 0.2500")
print(f"S1111 = {S_cyl['S1111']:.4f}  exact: 0.0000")

References

[1] Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society A, 241(1226), 376–396.

[2] Mura, T. (1987). Micromechanics of Defects in Solids (2nd ed.). Martinus Nijhoff.

[3] Tandon, G. P. & Weng, G. J. (1984). The effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites. Polymer Composites, 5(4), 327–333.

[4] Nemat-Nasser, S. & Hori, M. (1999). Micromechanics: Overall Properties of Heterogeneous Materials (2nd ed.). Elsevier.

[5] Clyne, T. W. & Withers, P. J. (1993). An Introduction to Metal Matrix Composites. Cambridge University Press.

For a short fiber ($e \sim 10$–20), $S_{1111}$ is non-negligible and the fiber contributes to axial stiffness — the Mori–Tanaka prediction for $E_1$ is then intermediate between the rule-of-mixtures bound and the transverse result.