Understanding the first-order inhomogeneous linear elasticity through local gauge transformations

Abstract

It is well known that classical linear elasticity equations are not form-invariant under local transformations. This is intrinsically related to the inhomogeneity of elastic media. However, the reported new linear elasticity equations for inhomogeneous media may appear in different forms. This paper tries to clarify this issue by investigating the form-invariance of the Lagrangian under local temporal or spatial gauge transformations. In this way, these new equations in different forms can be easily understood as the results from different choices of gauge fixing schemes. It recommends to choose appropriate gauges with clear physical meanings to simplify calculations.

Appendices

Appendix A: The classical Willis equations

An inhomogeneous medium with elasticity tensor C and volume density ρ is regarded as the perturbed medium compared to a homogeneous medium with properties of $\overline {\small{C}}$ and $\overline{\rho }$:

$$ {C} = \overline {{C}} + \delta { C} {\mkern 5mu} \text {and} {\mkern 5mu} \rho = \bar{\rho } + \delta \rho ,$$

(40)

where $\delta { {C}}$ and $\delta \rho$ are corresponding perturbations. Thus, the incremental stress σ and momentum p of the inhomogeneous medium are written as:

$${{\varvec{\sigma}}} = { C}:{ e} = \left( {{\overline{ C}} + \delta { C}} \right):{ e} = {\overline{ C}}:{ e} + {{varvec \tau}},$$

(41)

$${ p} = \rho {\dot{ u}} = \left( {\overline{\rho } + \delta \rho } \right){\dot{ u}} = \overline{\rho }{\dot{ u}} + {{ \pi}},$$

(42)

where ${{\varvec \tau}} = \delta { C}:{ e}$ and ${{ \pi}} = \delta \rho {\dot{ u}}$ are polarization terms. It should be noted that we do not distinguish σ from the Cauchy stress and the second Piola–Kirchhoff stress in classical linear elasticity.

Substituting Eqs. 41 and 42 into Eq. 23, yields:

$$\nabla \cdot \left( {{ C}:{ e}} \right) + { {f} + \nabla \cdot {\varvec \tau} - {\dot{ \pi}}} = \overline{\rho } \ddot { {u}}.$$

(43)

Since $\nabla \cdot {{\varvec{\tau}}} - {\dot{ \varvec{\pi }}}$ serves as an effective body force, the solution to Eq. 43 at temporal-spatial position x can be written as:

$${ u}\left( { x} \right) = {\overline{ u}}\left( { x} \right) + \int_{ \Omega^{\prime} } {{\overline{ G}}\left( {{ x},{ { x^{\prime}}}} \right)\left[ {\nabla \cdot {{\varvec \tau}}\left( {{ x^{\prime}}} \right) - {\dot{ {\varvec \pi }}}\left( {{ x^{\prime}}} \right)} \right]{\text{d}} \Omega^{\prime}} ,$$

(44)

where ${\overline{\small {G}}}$ is the dynamic Green’s function of the comparison medium, and ${\overline{ {u}}}$ is the corresponding displacement.

Applying the integration by parts on Eq. 44 gives:

$${ {u}} = {\overline{ {u}}} - { {S}} \circ {{\varvec \tau}} - { M} \circ {{\varvec \pi}}.$$

(45)

Then, the corresponding linear strain and velocity can be obtained as:

$${ e} = {\overline{ e}} - { S}_{{ {x}}} \circ {{varvec\tau}} - { M}_{{ {x}}} \circ {{\varvec \pi}}\,, \quad {\dot{ u}} = { {\dot{\overline{ u}}}} - { S}_{t} \circ {{\varvec\tau}} - { M}_{t} \circ {{\varvec\pi}}.$$

(46)

In above equations, S, S_x, S_t, M, M_x, and M_t are integral operators related to ${\overline{\small G}}$ and “◦” denotes the temporal-spatial convolution as that in Eq. 44.

With Eq. 44, we can obtain:

$$\begin{aligned}{{varvec \tau}} + \delta { C}:\left( {{ S}_{x} \circ {{\varvec \tau}} + {\ M}_{x} \circ {{\varvec \pi}}} \right) &= \delta {\ C}:{\overline{\ e}},\\ {{\varvec \pi}} + \delta \rho \left( {{\ S}_{t} \circ {{\varvec \tau}} + {\ M}_{t} \circ {{\varvec \pi}}} \right) &= \delta \rho { {\dot{\overline{\ u}}}}.\end{aligned}$$

(47)

After applying the ensemble averaging on Eq. 47 and eliminating ${\overline{\ e}}$ and ${ {\dot{\overline{\ u}}}}$, we finally obtain the homogenized elastodynamic equations (the classical Willis Equations):

$$\left\langle {{\varvec \sigma}} \right\rangle = {\ {C}}^{{{\text{eff}}}} * \left\langle {\ {e}} \right\rangle + { \ {S}}^{{{\text{eff}}}} \circ \left\langle {{\dot{\ {u}}}} \right\rangle ,$$

(48)

$$\nabla \cdot \left\langle {{\varvec \sigma}} \right\rangle + {\ {f}} = {\hat{\ {S}}}^{{{\text{eff}}}} \circ \left\langle {{\dot{ \ {e}}}} \right\rangle + {{\ \rho}}^{{{\text{eff}}}} \odot \left\langle {{ {\ddot{\ u}}}} \right\rangle ,$$

(49)

where

$${\ {C}}^{{{\text{eff}}}} \!*\! \left\langle {\ {e}} \right\rangle {\!=\! }\left(\! {\overline {\ C} \!+\! \left\langle {\delta {\ {C}}} \right\rangle }\! \right) \!:\! \left\langle {\ {e}} \right\rangle \!-\! \left\langle {\left( {\delta {\ {C}} \!-\! \left\langle {\delta {\ {C}}} \right\rangle } \right)\!:\!{\ {S}}_{{ {x}}} \!:\! \left( {\delta {\ {C}} \!-\! \left\langle {\delta {\ {C}}} \right\rangle } \right)} \right\rangle \!\circ\! \left\langle {\ {e}} \right\rangle ,$$

(50a)

$${\rho}^{\text{eff}} \cdot \left\langle { {\ddot{\ u}}} \right\rangle { = }\left( {\overline{\rho } + \left\langle {\delta \rho } \right\rangle } \right)\left\langle { {\ddot{u}}} \right\rangle - \left\langle {\left( {\delta \rho - \left\langle {\delta \rho } \right\rangle } \right){\ {M}}_{t} \left( {\delta \rho - \left\langle {\delta \rho } \right\rangle } \right)} \right\rangle \circ \left\langle { {\ddot{\ u}}} \right\rangle ,$$

(50b)

$${\ {S}}^{{{\text{eff}}}} { = } - \left\langle {\left( {\delta {\ {C}} - \left\langle {\delta {\ {C}}} \right\rangle } \right):{\ {M}}_{{ {x}}} \left( {\delta \rho - \left\langle {\delta \rho } \right\rangle } \right)} \right\rangle ,$$

(50c)

$${\hat{\ {S}}}^{{{\text{eff}}}} { = } - \left\langle {\left( {\delta \rho - \left\langle {\delta \rho } \right\rangle } \right){\ {S}}_{t} :\left( {\delta {\ {C}} - \left\langle {\delta {\ {C}}} \right\rangle } \right)} \right\rangle .$$

(50d)

It should be noted that ${\ {S}}^{{{\text{eff}}}}$ is a third-order tensor with the first two symmetric indices, i.e., $S_{{\left( {ij} \right)k}}^{{{\text{eff}}}}$. ${\hat{\ {S}}}^{{{\text{eff}}}}$ is the adjoint of ${\ {S}}^{{{\text{eff}}}}$ with the last two symmetric indices, i.e., $\hat{S}_{{i\left( {jk} \right)}}^{{{\text{eff}}}}$.

Appendix B: The Willis-form equations

As Fig. 1 shows, if the medium in the initial configuration B⁰ is inhomogeneous with pre-stresses σ⁰, the strain energy density in B⁰ and the deformed configuration B¹ are explicit functions of the spatial position [26, 62], which are denoted as $W\left( {x_{i}^{0} ,0} \right)$ and $W\left( {x_{i}^{1} ,u_{i,j} } \right)$, respectively. In linear elasticity, $W\left( {x_{i}^{1} ,u_{i,j} } \right)$ should be represented in B⁰ by using Taylor expansion as:

$$\begin{aligned}W\left( {x_{i}^{1} ,u_{i,j} } \right) &\approx W\left( {x_{i}^{0} ,0} \right) + \left. {\frac{\partial W}{{\partial u_{i,j} }}} \right|_{{x_{i} = x_{i}^{0} ,u_{i,j}\, = 0}} u_{i,j} + \left. {\frac{\partial W}{{\partial x_{i} }}} \right|_{{x_{i} = x_{i}^{0} ,u_{i,j}\, = 0}} u_{i} +\\ &\frac{1}{2}\left. {\left( {\frac{{\partial^{2} W}}{{\partial u_{i,j} \partial u_{k,l} }}u_{i,j} u_{k,l} + 2\frac{{\partial^{2} W}}{{\partial u_{i,j} \partial x_{k} }}u_{i,j} u_{k} \!+\! \frac{{\partial^{2} W}}{{\partial x_{i} \partial x_{j} }}u_{i} u_{j} } \right)} \right|_{{x_{i} = x_{i}^{0} ,u_{i,j}\, = 0}} \\ &\!\equiv\! W\!\!\left({x_{i}^{0} \!,\!0\!} \right) \!+\! \sigma_{ij}^{0} u_{i,j} \!+\! W_{,i}^{0} u_{i} \!+\! \frac{1}{2}\!\left(\! {C_{ijkl} u_{i,j} u_{k,l} \!+\! 2\sigma_{ij,k}^{0} u_{i,j} u_{k} \!+\! W_{,ij}^{0} u_{i} u_{j} } \!\right)\!,\! \end{aligned}$$

(51)

where $\sigma_{ij}^{0} = \left. {\frac{\partial W}{{\partial u_{i,j} }}} \right|_{{x_{i} = x_{i}^{0} ,u_{i,j}\, = 0}}$, $W_{,i}^{0} \equiv \left. {\frac{\partial W}{{\partial x_{i} }}} \right|_{{x_{i} = x_{i}^{0} ,u_{i,j}\, = 0}}$ and $C_{ijkl} \equiv \left. {\frac{{\partial^{2} W}}{{\partial u_{i,j} \,\,\partial u_{k,l} }}} \,\right|_{{x_{i} = x_{i}^{0} ,u_{i,j}\, = 0}}$.

To construct the Lagrangian defined in Eq. 6, we take the same T in Eq. 21, but W and Φ should consider inhomogeneity:

$$ \begin{aligned} W &= W\left( {x_{i}^{1} ,u_{i,j} } \right) - W\left( {x_{i}^{0} ,0} \right) \\ &= \sigma_{ij}^{0} u_{i,j} + W_{,i}^{0} u_{i} + \frac{1}{2}\left( {C_{ijkl} u_{i,j} u_{k,l} + 2\sigma_{ij,k}^{0} u_{i,j} u_{k} + W_{,ij}^{0} u_{i} u_{j} } \right), \\ \end{aligned} $$

(52)

$${ \Phi} = - \left( {f_{i}^{0} + \frac{1}{2}f_{i,j}^{0} u_{j} + f_{i} } \right)u_{i} ,$$

(53)

where $f_{i}^{0}$ is the external body force in configuration B⁰. Since $f_{i}^{0}$ could be inhomogeneous, its gradient is included in Eq.  53.

According to Eq. 8, the effective body force in B¹ but represented in B⁰ can be obtained as:

$$\overline{f}_{i}^{1} = - \frac{\partial L}{{\partial u_{i} }} = f_{i}^{0} + f_{i,j}^{0} u_{j} + f_{i} - \sigma_{kl,i}^{0} u_{k,l} - W_{,i}^{0} - W_{,ij}^{0} u_{j} .$$

(54)

These terms in Eq.  54 have clear physical meanings. The first three terms are the contributions from the inhomogeneous external body force. The last three are due to the inhomogeneity of W, so that they are configurational forces according to Eshelby’s definition [63, 64].

The effective body force in B⁰ can be obtained by setting $f_{i}$, $u_{j}$ and $u_{k,l}$ to zeros in Eq.  54:

$$\overline{f}_{i}^{0} = f_{i}^{0} - W_{,i}^{0} ,$$

(55)

which also accounts for the inhomogeneity of W⁰.

Also according to Eq. 8, the incremental stress represented at position x⁰ can be obtained as:

$$\sigma_{ij} = \overline{\sigma }_{ij} - \sigma_{ij}^{0} = \frac{\partial L}{{\partial u_{i,j} }} - \sigma_{ij}^{0} = C_{ijkl} u_{k,l} + \sigma_{ij,k}^{0} u_{k} .$$

(56)

Based on Eqs. 22, 55 and 56, the equation of motion for $\sigma_{ij}$ can be obtained according to the Euler–Lagrange equation given in Eq. 7:

$$\sigma_{ij,j} + f_{i} = \sigma_{kl,i}^{0} u_{k,l} + \sigma_{ij,jk}^{0} u_{k} + \rho_{ij} u_{j,tt} ,$$

(57)

where $- \sigma_{kl,i}^{0} u_{k,l} - \sigma_{ij,jk}^{0} u_{k}$ can be regarded as a configurational force.

With Eqs. 56 and 57, it is clear that $S_{{\left( {ij} \right)k}}^{0} = \sigma_{ij,k}^{0}$ and $K_{ik}^{0} = \sigma_{ij,jk}^{0}$ in Eqs. 3 and 4. Since $\sigma_{kl}^{0} = \sigma_{lk}^{0}$ and $C_{ijkl} = C_{ijlk}$, we can also replace $u_{k,l}$ with $e_{kl}$.

In addition, we should distinguish the incremental second Piola–Kirchhoff stress from the incremental Cauchy stress for inhomogeneous media with pre-stresses. For example, $\sigma_{ij}$ in Eq. 56 is actually the incremental second Piola–Kirchhoff stresses, which is nonzero even under rigid translations. This is well known in finite deformation theory [67]. However, the incremental Cauchy stress should be zero under rigid movements.