Fractional Strain Tensor and Fractional Elasticity

A new fractional strain tensor \(\epsilon ^{\alpha}(u)\) of order \(\alpha \) (\(0<\alpha <1\)) is introduced for a displacement \(u\) of a body occupying the entire three-dimensional space. For \(\alpha \uparrow 1\), the fractional strain tensor approaches the classical infinitesimal strain tensor of the linear elasticity. It is shown that \(\epsilon ^{\alpha}(u)\) satisfies Korn’s inequality (in a general \(L^{p}\) version, \(1< p<\infty \)) and the fractional analog of Saint-Venant’s compatibility condition. The strain \(\epsilon ^{\alpha}(u)\) is then used to formulate a three-dimensional fractional linear elasticity theory. The equilibrium of the body in an external force \(f\) is determined by the Euler-Lagrange equation of the total energy functional. The solution \(u\) is given by Green’s function \(G_{\alpha}\):

For an isotropic body the equilibrium equation reads

where \(\lambda \), \(\mu \) are the Lamé moduli of the material and \((-\Delta )^{\alpha}\), \(\nabla ^{\alpha}\) and \(\mathop{\mathrm{div}}\nolimits ^{\alpha}\) are the fractional laplacean, gradient and divergence. Green’s function can be determined explicitly in this case:

\(x\in {\mathbf{R}}^{3}\), \(x\neq 0\), where \(I\) is the identity tensor (matrix), and \(c_{\alpha}\) a normalization factor (determined below). For \(\alpha \uparrow 1\) the function \(G_{\alpha}\) approaches Green’s function of the standard linear elasticity. Similar approach applies to the equilibrium solution.

1. E.g., in thin film mechanics, fracture mechanics, the theory of dislocations etc.

2. See [26] and [15] for recent reviews.

3. This negative result is independent of the chosen type of the one-dimensional fractional derivative \(\mathop{\mathrm{D}\,}\nolimits ^{\alpha }_{x_{i}}\varphi \) (Riemann-Liouville, Grünwald-Letnikov, …).

4. See the references in Sect. 2.

5. We use complex values in view of the future use of the Fourier transformation (starting from Sect. 4).

6. Since the domain of definition of functions from various function spaces is always \(\mathbf{R}^{n}\), we indicate only the ranges: thus, e.g., the symbol \(L^{p}(\mathbf{C}^{m})\) denotes the space of \(\mathbf{C}^{m}\)-valued functions on \(\mathbf{R}^{n}\) that belong to \(L^{p}\).

7. The space of Riesz potentials is larger than the more familiar space of Bessel potentials, see (39).

8. After the completion of the present research, the author has learned of the paper by Evgrafov & Bellido [9] on nonlocal and fractional elasticity. Section 3.5 of their paper has many features in common with the present paper.

9. Here I use the form (10) given by Comi & Stefani [5, Sect. 2], which is equivalent to the formulas for \(\nabla ^{\alpha }\) given elsewhere in the literature. A similar remark applies to the fractional divergence \(\mathop{\mathrm{div}}\nolimits ^{\alpha }\), introduced originally in [30, Sect. 2] by an equivalent formula.

10. The restriction \(0<\alpha <1\) is imposed for notation simplification only; the definition can be generalized to arbitrary orders \(\alpha >0\) either by iterating the definition (1) or by admitting fractional gradients of any order in (40).

Authors and Affiliations

1. Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic

   Miroslav Šilhavý

Corresponding author

Correspondence to Miroslav Šilhavý.

Competing interests

The authors declare no competing interests.

In memory of J.L. Ericksen

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This research was supported by RVO 67985840.

Appendix A: Fourier Transformation of Homogeneous Distributions

Definition A.1

A distribution \(f\) from \({\mathscr{S}}'(Z)\) is said to be homogeneous of degree \(\lambda \in {\mathbf{C}}\) if

for every \(t>0\) and every \(g\in {\mathscr{S}}(Z)\), where \(\eta _{t}:{\mathbf{R}}^{n}\to {\mathbf{R}}^{n}\) is given by

Definition A.2

[17, Definition 3.2.5]

A distribution \(f\in {\mathscr{S}}(Z)\) is said to be locally in the space of Bessel potentials \(L^{s,2}(Z)\) if \(gf\) belongs to \(L^{s,2}(Z)\) for every infinitely differentiable function \(g:{\mathbf{R}}^{n}\to {\mathbf{C}}\) with compact support that is contained in \({\mathbf{R}}^{n}\setminus \{0\}\).

Theorem A.3

[17, Corollary 3.2.6]

The Fourier transform of a homogeneous distribution \(f\) of degree \(\lambda \in {\mathbf{C}}\) that is locally in \(L^{s,2}(Z)\) is a homogeneous distribution \(\hat{f}\) of degree \(-\lambda -n\) that is locally in \(L^{s-\mathop{\mathrm{Re}}\lambda -n/2,2}(Z)\).

Appendix B: The Space \({\mathscr{U}}(Z)\)

This appendix introduces the space of test functions whose Fourier transforms can be safely divided by \(|\xi |\) and its positive powers, as needed in the proof of Theorem 7.3.

Proposition B.1

The set

is dense in \(L^{p}(Z)\) for every \(p\in (1,\infty )\).

Proof

It suffices to prove that any \(f\in {\mathscr{S}}(Z)\) can be approximated by a sequence \(f_{k}\in {\mathscr{U}}(Z)\) in the \(L^{p}\) norm. Let \(\psi :{\mathbf{R}}\to {\mathbf{C}}\) be a function whose Fourier transform \(\hat{\psi}\) is infinitely differentiable and satisfies \(\hat{\psi}=1\) on \(B(0,1)\) and \(\hat{\psi}=0\) on \({\mathbf{R}}^{n}\setminus B(0,2)\). Let \(\psi _{t}(x)=t^{n}\psi (tx)\), \(x\in {\mathbf{R}}^{n}\), \(t>0\). One has

and thus if \(1< p<\infty \), \(|\psi _{t}|_{L^{p}} \to 0 \) for \(t\to 0 \). Since every \(f\in {\mathscr{S}}\) is integrable, Young’s convolution inequality implies that \(|f\ast \psi _{t}|_{L^{p}} \to 0 \) for \(t\to 0 \), i.e., the function \(f_{t}:=f-f\ast \psi _{t}\) satisfies

To show that \(f_{t}\in {\mathscr{U}}(Z)\), we note that the well-known rules for the Fourier transformation under scaling and convolution give

where \(\hat{\psi}_{t}(\xi )=\hat{\psi}(\xi /t)\) for every \(\xi \in {\mathbf{R}}^{n}\) and \(t>0\). One finds that \(\hat{\psi}_{t}=1\) on \(B(0,t)\) and hence on \(B(0,t)\). □

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