In this section, we consider field theories with the form \({\bar{\mathcal {L}}}({\bar{\mathbf{x}}},t;\mathbf{x},{d_{t}}{\mathbf{x}},\mathbf{F})\), where the independent material coordinates are represented by a time-independent reference configuration \({\bar{\mathbf{x}}}\), and first derivatives of position are represented by a “deformation gradient” [58] \(\mathbf{F}\equiv {\bar{\nabla }}\mathbf{x}\equiv \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}{\bar{\mathbf{x}}}} = \nabla _i\mathbf{x}{\bar{\nabla }}^i{\bar{\mathbf{x}}}\) that applies the gradient \({\bar{\nabla }}\) of the reference space to the position vector \(\mathbf{x}\) of the present configuration (note that \(\mathbf{F}\) also serves to transform between bases; for example \(d_i\mathbf{x}\delta {\eta }^i = \mathbf{F}\cdot d_j {\bar{\mathbf{x}}}\delta {\eta }^j\), and \({{\bar{\nabla }}}() = {\nabla }() \cdot \mathbf{F}\) if () has no free indices). This description is appropriate, and indeed quite traditional in solid mechanics, for the description of simple elastic bodies that fill some portion of three-dimensional space (that is, \(i \in \{1,2,3\}\)), but is not suitable for the description of incompatible elastic systems or lower-dimensional bodies such as the elastic surfaces we will consider later in Sect. 6.
We will identify the balance and conservation laws that arise from variation of the dependent and independent variables. The temporal and material boundary terms from (9) can be computed as \(\mathcal {E}_{(\mathcal {Q})}= \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot {\tilde{\delta }}\mathbf{x}\) and \({\varvec{\mathcal {E}}}_{(\mathcal {J})}={J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot \,{\tilde{\delta }}\mathbf{x}\). The Euler–Lagrange, charge, and current terms corresponding to the variation (10) are
$$\begin{aligned} {\varvec{\mathcal {E}}}&= {J^{-1}}\frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{x}} - {J^{-1}}{d_{t}}\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\right) -\nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right) , \end{aligned}$$
(13)
$$\begin{aligned} \mathcal {Q}&= \left( {\bar{\mathcal {L}}}\,\delta t + \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot \tilde{\delta }\mathbf{x}\right) , \end{aligned}$$
(14)
$$\begin{aligned} {\varvec{\mathcal {J}}}&= {J^{-1}}\left( {\bar{\mathcal {L}}}\,\mathbf{F}\cdot \delta {\bar{\mathbf{x}}} + \mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot \tilde{\delta }\mathbf{x}\right) . \end{aligned}$$
(15)
However, to clearly identify the terms power-conjugate to each of the different variational quantities, we rewrite the variations at a fixed material label appearing in (14, 15), to express the charge and current in terms of the total variation \(\delta \mathbf{x}\) at a fixed material point as well as variations with respect to the independent material and temporal variables. Using the relation (5) to substitute for \(\tilde{\delta }\mathbf{x}\), we rearrange Eqs. (14, 15) to obtain
$$\begin{aligned} \mathcal {Q}&=\frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot \delta \mathbf{x}+\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot \mathbf{F}\right) \cdot \left( -\delta {\bar{\mathbf{x}}}\right) + \left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot {d_{t}}{\mathbf{x}}-{\bar{\mathcal {L}}}\right) \left( -\delta t\right) , \end{aligned}$$
(16)
$$\begin{aligned} {\varvec{\mathcal {J}}}&= \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right) \cdot \delta \mathbf{x}+\left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot \mathbf{F}- {J^{-1}}{\bar{\mathcal {L}}}\,\mathbf{F}\right) \left( -\delta {\bar{\mathbf{x}}}\right) \nonumber \\&\quad +\left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot {d_{t}}{\mathbf{x}}\right) \left( -\delta t\right) . \end{aligned}$$
(17)
In this form, we may identify several familiar quantities. In (16), the charges conjugate to \(\delta \mathbf{x}\), \(-\delta {\bar{\mathbf{x}}}\), and \(-\delta t\) are, respectively, the (spatial) momentum, pseudomomentumFootnote 5, and Hamiltonian density [57]. In (17), the currents conjugate to \(\delta \mathbf{x}\), \(-\delta {\bar{\mathbf{x}}}\), and \(-\delta t\) are, respectively, the (spatial) stress, pseudostressFootnote 6, and power expended by the stress. As we are working in the present configuration, the stress in question is that of Cauchy [58], \(-{J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\). This is the Piola transform of the first Piola-Kirchhoff stress \(-\frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\) whose transpose will appear naturally in the referential frame in “Appendix A”. Similarly, the Eshelby tensor conjugate to \(-\delta {\bar{\mathbf{x}}}\) in (17) is a transformed version of the usual referential form of this tensor.
We now proceed to integrate (10) by parts to obtain bulk balance laws as well as boundary and jump conditions. The latter are singular balance laws that hold at an internal non-material surface of discontinuity \(\mathcal {S}(t)\). This surface is assumed to move with some “velocity” through the coordinates, whose normal component is denoted \(\bar{U}\). Let \({\varvec{{\hat{n}}}}\) and \({\varvec{{\hat{N}}}}\) be the unit normals to the external boundary and internal surface of discontinuity. The relevant forms of the divergence and transport theorems for a piecewise continuous tensorial quantity \(\mathbf{A}\) are [58, 59]
$$\begin{aligned} \int \limits _{\mathcal {B}}\mathrm{d}V\,{\nabla }\cdot \mathbf{A}&= \int \limits _{\partial \mathcal {B}}\!\! \mathrm{d}A\, {\varvec{{\hat{n}}}}\cdot \mathbf{A} - \int \limits _{\mathcal {S}(t)}\!\! \mathrm{d}A{\llbracket {\varvec{{\hat{N}}}}\cdot \mathbf{A}\rrbracket }, \end{aligned}$$
(18)
$$\begin{aligned} {d_{t}}{\int \limits _{\mathcal {B}}\mathrm{d}V\, {J^{-1}}\mathbf{A}}&= \int \limits _{\mathcal {B}}\mathrm{d}V\,{J^{-1}}{d_{t}}\mathbf{A} - \int \limits _{\mathcal {S}(t)}\!\! \mathrm{d}A{\llbracket {J^{-1}}U\mathbf{A}\rrbracket }, \end{aligned}$$
(19)
where \(\llbracket \rrbracket \) denotes the jump in the enclosed quantity across the discontinuity. The quantity \({J^{-1}}U = {J_S^{-1}}\bar{U}\), where \({J_S^{-1}}\) is the areal Jacobian at the surface, is in general continuous and so can be moved outside the brackets. In the present consideration of space-filling bodies, \({\varvec{{\hat{N}}}}\) will be continuous as well. Using (18–19) and (5), we obtain
where we have used \(\delta \mathbf{x}= \mathbf{F}\cdot \delta {\bar{\mathbf{x}}}\). These three integrals provide the (free) boundary conditions, jump conditions, and bulk field equations. We now consider separately the balance laws conjugate to the variation in the current configuration, material coordinates, and time.
Pure variations \(\delta \mathbf{x}\) of the current configuration, with \(\delta {\bar{\mathbf{x}}}= {\varvec{0}}\) and \(\delta t = 0\), provide the bulk equation, the boundary condition, and the jump condition for momentum,
$$\begin{aligned} {\varvec{\mathcal {E}}}({\bar{\mathcal {L}}}) = {J^{-1}}\frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{x}} - {J^{-1}}{d_{t}}{\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\right) } - \nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right)&={\varvec{0}}\quad \text {on}\;\mathcal {B}, \end{aligned}$$
(21)
$$\begin{aligned} {\varvec{{\hat{n}}}}\cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right)&= {\varvec{0}}\quad \text {on}\;\partial \mathcal {B}, \end{aligned}$$
(22)
$$\begin{aligned} {\llbracket -{\mathbf {\hat{N}}}\cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right) + {J^{-1}}U \frac{\partial \mathcal {L}}{\partial {d_{t}}{\mathbf{x}}}\rrbracket }&= {\varvec{0}}\quad \text {on}\;\mathcal {S}(t). \end{aligned}$$
(23)
Pure variations \(\delta {\bar{\mathbf{x}}}\) of the reference configuration, with \(\delta \mathbf{x}= {\varvec{0}}\) and \(\delta t = 0\), provide balance laws for pseudomomentum,
$$\begin{aligned} {\varvec{\mathcal {E}}}({\bar{\mathcal {L}}})\cdot \mathbf{F}= \left[ {J^{-1}}\frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{x}} - {J^{-1}}{d_{t}}{\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\right) } - \nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right) \right] \cdot \mathbf{F}&={\varvec{0}}\quad \text {on}\;\mathcal {B}, \end{aligned}$$
(24)
$$\begin{aligned} {\varvec{{\hat{n}}}}\cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot \mathbf{F}- {J^{-1}}{\bar{\mathcal {L}}}\,\mathbf{F}\right)&={\varvec{0}}\quad \text {on}\;\partial \mathcal {B}, \end{aligned}$$
(25)
$$\begin{aligned} {\llbracket -{\mathbf {\hat{N}}}\cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot \mathbf{F}- {J^{-1}}{\bar{\mathcal {L}}}\,\mathbf{F}\right) + {J^{-1}}U\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot \mathbf{F}\right) \rrbracket }&={\varvec{0}}\quad \text {on}\;\mathcal {S}(t). \end{aligned}$$
(26)
Finally, purely temporal variations \(\delta t\), with \(\delta \mathbf{x}= \delta {\bar{\mathbf{x}}} = {\varvec{0}}\)), provide balance laws for energy,
$$\begin{aligned} {\varvec{\mathcal {E}}}({\bar{\mathcal {L}}})\cdot {d_{t}}{\mathbf{x}} = \left[ {J^{-1}}\frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{x}} - {J^{-1}}{d_{t}}{\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\right) } - \nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right) \right] \cdot {d_{t}}{\mathbf{x}}&=0\quad \text {on}\;\mathcal {B}, \end{aligned}$$
(27)
$$\begin{aligned} {\varvec{{\hat{n}}}}\cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot {d_{t}}{\mathbf{x}}\right)&=0\quad \text {on}\;\partial \mathcal {B}, \end{aligned}$$
(28)
$$\begin{aligned} {\llbracket -{\mathbf {\hat{N}}}\cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot {d_{t}}{\mathbf{x}}\right) + {J^{-1}}U\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot {d_{t}}{\mathbf{x}}-{\bar{\mathcal {L}}}\right) \rrbracket }&= 0\quad \text {on}\;\mathcal {S}(t). \end{aligned}$$
(29)
While the bulk balances for pseudomomentum and energy are simply projections of the momentum balance onto the deformation gradient and velocity, respectively, the boundary and jump conditions are distinct. Note also that the two sets of vector equations correspond to different “legs” of the two-point tensorial quantities they contain; the extant leg in the momentum equations corresponds to the present configuration while that in the pseudomomentum equations corresponds to the reference configuration. Referential forms of these balance laws are presented in “Appendix A”. Because Eqs. (24–26) arise from a continuous shift in material coordinates made possible by a continuum description of a body, they have no analogue in a discrete set of particles [11, 13].
Although the simple relationship between the bulk balance laws in our system implies that satisfaction of the balance of momentum (21) means that the other balances (24) and (27) hold as well, this obscures a crucial point, namely that the conserved quantities associated with the corresponding symmetries are not identical. In the following section, we will rearrange these equations to illustrate that the source terms arising from broken spatial, temporal, and material symmetries are mutually independent quantities.
Forces and material forces
Following A. Golebiewska Herrmann [11] and Maugin [15], we recast the balances of energy (27) and pseudomomentum (24) into a standard form that clearly reveals the form of the source terms. Referential expressions that follow a strict conservation law form are presented in “Appendix A”.
First note that the balance of momentum (21) can be easily written with a source term on the right hand side,
$$\begin{aligned} {J^{-1}}{d_{t}}{\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\right) } + \nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right) = {J^{-1}}\frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{x}}. \end{aligned}$$
(30)
Any explicit dependence of the Lagrangian density on the position \(\mathbf{x}\), such as the presence of a gravitational potential, breaks the symmetry of the embedding space and provides a source of momentum.
The balance of energy (27) can be rearranged by employing the chain rule
$$\begin{aligned} {d_{t}}{{\bar{\mathcal {L}}}} = \frac{\partial {\bar{\mathcal {L}}}}{\partial t} + \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{x}}\cdot {d_{t}}{\mathbf{x}} + \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot {d_{t}}{{d_{t}}{\mathbf{x}}} + \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}:(\nabla {d_{t}}{\mathbf{x}}\cdot \mathbf{F}), \end{aligned}$$
(31)
where the final term involves the material time derivative of \(\mathbf{F}\); the notation means that the \(\nabla \) leg is contracted with the present leg of \(\mathbf{F}\), and double contraction associates present and referential legs with their respective counterparts. After some integration by parts, (27) becomes
$$\begin{aligned} {J^{-1}}{d_{t}}{\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot {d_{t}}{\mathbf{x}} - {\bar{\mathcal {L}}}\right) } + \nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot {d_{t}}{\mathbf{x}}\right) = -{J^{-1}}\frac{\partial {\bar{\mathcal {L}}}}{\partial t}. \end{aligned}$$
(32)
As one might expect, any explicit dependence of the Lagrangian density on the time t manifests as a source term in the energy balance.
Similarly, the balance of pseudomomentum (24) can be rearranged with the help of the chain rule
$$\begin{aligned} \nabla {\bar{\mathcal {L}}}\cdot \mathbf{F}= \frac{\partial {\bar{\mathcal {L}}}}{\partial {\bar{\mathbf{x}}}} + \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{x}}\cdot \mathbf{F}+ \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot {d_{t}}{\mathbf{F}} + \mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}:\nabla \mathbf{F}, \end{aligned}$$
(33)
where the double contraction involves the two present legs of the gradient of \(\mathbf{F}\). After some integration by parts and use of the Piola identity \({\nabla }\cdot \left( {J^{-1}}\mathbf{F}\right) = {{\varvec{0}}}\), (24) becomes
$$\begin{aligned} {J^{-1}}{d_{t}}{\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot \mathbf{F}\right) } + \nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot \mathbf{F}- {J^{-1}}{\bar{\mathcal {L}}}\, \mathbf{F}\right) = - {J^{-1}}\frac{\partial {\bar{\mathcal {L}}}}{\partial {\bar{\mathbf{x}}}}. \end{aligned}$$
(34)
Any explicit dependence of the Lagrangian density on the reference configuration \({\bar{\mathbf{x}}}\) breaks the symmetry of the material continuum and provides a source of pseudomomentum. Forms of the balance law (34) in present or referential form, with or without the source term, can be found in [2, 11,12,13,14,15,16, 31, 32, 35, 60].
The source terms in the three balances (30), (32), and (34) are entirely independent. In particular, the balance of pseudomomentum is related to the symmetry of the material continuum, a feature independent of any properties of the embedding space. Just as the source term in (21) is often interpreted as a body force, we may interpret the source term in (24) as a “material body force”. However, it is important to note that these forces are not of the same type. While spatial (Newtonian) forces are vectors associated with the embedding space, material (Eshelbian) forces are associated with a material space. Although in the example under consideration, the material space can be thought of as a reference configuration embedded in the same space as the present configuration, with material forces associated with vectors in the reference configuration, and although the present configuration of a space-filling body is often associated with the embedding space itself, this does not mean that spatial and material forces can be conflated or added together in any meaningful way. They pertain, respectively, to the motion of material bodies in space and the motion of non-material objects within a material.
A few symmetries and conservation laws
Here we apply Noether’s theorem to obtain conservation lawsFootnote 7 for momentum and pseudomomentum. We insert variations corresponding to spatial and material symmetries of the action into the general expression
$$\begin{aligned} {J^{-1}}{d_{t}}{\mathcal {Q}} + \nabla \cdot {\varvec{\mathcal {J}}} = 0, \end{aligned}$$
(35)
where \(\mathcal {Q}\) and \({\varvec{\mathcal {J}}}\) are given by (16) and (17). A static version of this general statement in elasticity can be found in Edelen [25].
The embedding space is symmetric under translations \(\delta \mathbf{x}= \mathbf{D}\) and rotations \(\delta \mathbf{x}= \mathbf{D}\times \mathbf{x}\), where \(\mathbf{D}\) is a (small) constant vector. With \(\delta {\bar{\mathbf{x}}}={\varvec{0}}\) and \(\delta t=0\), we obtain linear and angular momentum conservation laws,
$$\begin{aligned} {J^{-1}}{d_{t}}{\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\right) } + \nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right)&= {\varvec{0}}, \end{aligned}$$
(36)
$$\begin{aligned} {J^{-1}}{d_{t}}{\left( \mathbf{x}\times \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\right) } + \nabla \cdot \left( \mathbf{x}\times {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\right)&={\varvec{0}}. \end{aligned}$$
(37)
By comparing [24] the linear momentum conservation law (36) with the balance law (30), we can see that conservation implies that the Lagrangian density cannot depend explicitly on position \(\mathbf{x}\).
If the material is uniform, “translations” in material coordinates \(\delta {\bar{\mathbf{x}}}\) produce the linear pseudomomentum conservation law
$$\begin{aligned} {J^{-1}}{d_{t}}{\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot \mathbf{F}\right) } + \nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot \mathbf{F}- {J^{-1}}{\bar{\mathcal {L}}}\,\mathbf{F}\right)&= {\varvec{0}}. \end{aligned}$$
(38)
In the present context, this means [24] that the Lagrangian density cannot depend explicitly on the reference configuration \({\bar{\mathbf{x}}}\), as can be seen by comparing (38) with (34). Material “rotational” symmetry and angular pseudomomentum conservation will be exploited in Sect. 6. Pseudomomentum conservation laws can be found in [2, 11,12,13,14,15,16, 31, 32, 35].
J-integral
As demonstrated by A. Golebiewska Herrmann [12, 13], conservation laws arising from invariance of material space are intimately related to well-known path independent integrals of hyperelastic fracture mechanics [8, 9]. Consider the conservation of “translational” material momentum (38) integrated over an arbitrary volume V with boundary \(\partial V\) and unit normal \({\varvec{{\hat{\nu }}}}\),
$$\begin{aligned} \int \limits _{V}\!\mathrm{d}V\left[ {J^{-1}}{d_{t}}\left( \frac{\partial {\bar{\mathcal {L}}}}{\partial {d_{t}}{\mathbf{x}}}\cdot \mathbf{F}\right) +\nabla \cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot \mathbf{F}- {J^{-1}}{\bar{\mathcal {L}}}\,\mathbf{F}\right) \right] = {\varvec{0}}. \end{aligned}$$
(39)
If V encapsulates a defect such as an inclusion or crack tip—a point where the conservation law fails to hold—the right hand side of the above equation need not be zero. Such a source term would represent the total material force on the defect that seeks to drive it through the material rather than through space. Equation (39) is known as the dynamic generalization of the J-integral [17, 61, 62]. Markenscoff [61] also discusses the corresponding “rotational” L-integral. For the static case, the time derivative vanishes and the divergence term may be written as a surface integral
$$\begin{aligned} \int \limits _{\partial V}\!\! \mathrm{d}A\,{\varvec{{\hat{\nu }}}}\cdot \left( {J^{-1}}\mathbf{F}\cdot \left[ \frac{\partial {\bar{\mathcal {L}}}}{\partial \mathbf{F}}\right] ^\mathrm {T}\cdot \mathbf{F}- {J^{-1}}{\bar{\mathcal {L}}}\,\mathbf{F}\right) = {\varvec{0}}, \end{aligned}$$
(40)
the original J-integral of Rice [8] and Cherepanov [9].