1 Introduction
2 Integral Equation Formulation
2.1 The Potential Problem
2.2 Elastostatics
3 Uniformity of the Hill and Eshelby Tensors
3.1 The Potential Problem
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In the host region \(V_{0}\) the temperature gradient is generally not uniform.
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For non-homogeneous temperature gradient conditions in the far-field, the temperature gradient field inside an ellipsoidal inhomogeneity is generally not uniform. However if the prescribed temperature gradient is a polynomial of order \(n\), then so is the field inside an ellipsoidal inhomogeneity, see [6]. This is known as the polynomial conservation property for ellipsoids.
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Generally for non-ellipsoidal inhomogeneities in unbounded domains and general shaped inhomogeneities in bounded host domains \(V\), the temperature gradient inside the inhomogeneities is not uniform, although interacting E-inclusions [71] can lead to uniform interior temperature gradients and for specific loadings, non-ellipsoidal inhomogeneities can yield uniform interior temperature gradients, e.g., the counterexample of the Strong Eshelby conjecture given by Liu [71].
3.2 Elastostatics
3.3 The Potential Gradient Tensor and Strain Concentration Tensor
4 The Potential Problem: Specific Cases
4.1 Isotropic Host Phase
4.1.1 Sphere in an Isotropic Host Phase
4.1.2 Circular Cylinder in an Isotropic Host Phase
4.1.3 Ellipsoid in an Isotropic Host Phase
4.1.4 Spheroid in an Isotropic Host Phase
4.1.5 Limiting Case of an Elliptical Cylinder
4.1.6 Limiting Cases of a Cavity, Penny-Shaped Crack and Ribbon-Crack
4.2 Anisotropic Host Phase
4.2.1 Spheroid in a Transversely Isotropic Host Phase
4.2.2 Circular Cylinder in a Transversely Isotropic Host Phase
4.2.3 Ellipsoid in an Orthotropic Host Phase
Host anisotropy | Inhomogeneity shape |
P-tensor |
---|---|---|
Isotropic | Ellipsoid
\(a_{1}\neq a_{2}\neq a_{3}\)
\(\varepsilon _{n}=a_{3}/a_{n}\)
| Use potential theory:
\(P_{ij} = \frac{1}{k_{0}}\sum_{n=1}^{3}\mathcal{E}(\varepsilon _{n};\varepsilon _{1},\varepsilon _{2})\delta_{in}\delta_{jn}\)
|
Spheroid
\(a_{1}=a_{2}=a\neq a_{3}\)
\(\varepsilon =a_{3}/a\)
| Use potential theory:
\(P_{ij} = \frac{1}{k_{0}} (\gamma \varTheta_{ij}+\gamma _{3}\delta_{i3}\delta _{j3} )\)
\(\gamma =\frac{1}{2}(1-\gamma _{3}), \gamma _{3} = \mathcal{S}(\varepsilon )\)
| |
Sphere | Use symmetry:
\(P_{ij} = \frac{1}{3k_{0}}\delta_{ij}\)
| |
Transversely isotropic
\(\upsilon_{1}=\upsilon_{2}=1\neq\upsilon_{3}=\upsilon\)
| Ellipsoid
\(a_{1}\neq a_{2}\neq a_{3}\)
\(\varepsilon _{n}=a_{3}/a_{n}\)
| Use scalings and potential theory:
\(P_{ij} = \frac{1}{k_{0}}\sum_{n=1}^{3}\frac{1}{\upsilon_{n}}\mathcal {E}(\hat{\varepsilon }_{n}; \hat{\varepsilon }_{1},\hat{\varepsilon }_{2})\delta_{in}\delta_{jn}\)
\(\hat{\varepsilon }_{n} = \hat{a}_{3}/\hat{a}_{n}\) and \(\hat{a}_{n} = a_{n}/\sqrt {\upsilon_{n}}\)
|
Spheroid \(a_{1}=a_{2}=a\neq a_{3}\) and \(a_{3}\) is aligned with axis \(x_{3}\) of transverse isotropy | Use scalings and potential theory:
\(P_{ij} = \frac{1}{k_{0}} (\gamma \varTheta _{ij}+\gamma _{3}\delta_{i3}\delta_{j3} )\)
\(\gamma =\frac{1}{2}(1-\upsilon \gamma _{3})\), \(\gamma _{3} = \frac{1}{\upsilon }\mathcal{S} (\frac{\varepsilon }{\sqrt {\upsilon}} )\)
| |
Sphere | Special case of spheroid result above:
\(P_{ij} = \frac{1}{k_{0}} (\gamma \varTheta_{ij}+\gamma _{3}\delta_{i3}\delta _{j3} )\)
\(\gamma =\frac{1}{2}(1-\upsilon \gamma _{3}), \gamma _{3} = \frac{1}{\upsilon }\mathcal{S} (\frac{1}{\sqrt{\upsilon}} )\)
| |
Orthotropic
\(\upsilon_{1}=1\neq\upsilon_{2}\neq\upsilon_{3}\)
| Ellipsoid
\(a_{1}\neq a_{2}\neq a_{3}\)
| Use scalings and potential theory:
\(P_{ij} = \frac{1}{k_{0}}\sum_{n=1}^{3}\frac{1}{\upsilon_{n}}\mathcal {E}(\hat {\varepsilon }_{n};\hat{\varepsilon }_{1},\hat{\varepsilon }_{2})\delta_{in}\delta _{jn}\)
\(\hat{\varepsilon }_{n} = \hat{a}_{3}/\hat{a}_{n}\) and \(\hat{a}_{n} = a_{n}/\sqrt {\upsilon_{n}}\)
|
Worse than orthotropic or semi-axes of ellipsoids not aligned with axes of anisotropy | Use general integral form:
\(P_{ij}=\frac{\mathrm {det}(\mathbf{a})}{4\pi} \int _{S^{2}}\frac {\varPhi_{ij}\,dS(\overline{\boldsymbol{\xi }})}{(\overline{\xi}_{k} a_{k\ell}a_{\ell m}\overline{\xi }_{m})^{3/2}}\)
\(\varPhi_{ij}= \overline{\xi}_{i}\overline{\xi}_{j}/(C_{k\ell }^{0}\overline{\xi}_{k}\overline{\xi}_{\ell})\)
|
5 Elastostatics: Specific Cases
5.1 Isotropic Host Phase
5.1.1 Sphere in an Isotropic Host Phase
5.1.2 Circular Cylinder in an Isotropic Host Phase
\(\mathcal{H}^{1}\)
|
\(\mathcal{H}^{2}\)
|
\(\mathcal{H}^{3}\)
|
\(\mathcal{H}^{4}\)
|
\(\mathcal{H}^{5}\)
|
\(\mathcal{H}^{6}\)
| |
---|---|---|---|---|---|---|
\(\mathcal{H}^{1}\)
|
\(\mathcal{H}^{1}\)
|
\(\mathcal{H}^{2}\)
| 0 | 0 | 0 | 0 |
\(\mathcal{H}^{2}\)
| 0 | 0 | 2\(\mathcal{H}^{1}\)
|
\(\mathcal {H}^{2}\)
| 0 | 0 |
\(\mathcal{H}^{3}\)
|
\(\mathcal{H}^{3}\)
| 2\(\mathcal{H}^{4}\)
| 0 | 0 | 0 | 0 |
\(\mathcal{H}^{4}\)
| 0 | 0 |
\(\mathcal{H}^{3}\)
|
\(\mathcal{H}^{4}\)
| 0 | 0 |
\(\mathcal{H}^{5}\)
| 0 | 0 | 0 | 0 |
\(\mathcal{H}^{5}\)
| 0 |
\(\mathcal{H}^{6}\)
| 0 | 0 | 0 | 0 | 0 |
\(\mathcal{H}^{6}\)
|