The Filon construct for moving dislocations

This paper continues the study of Filon’s construct first announced at the Edinburgh meeting in 1921 of the British Association for the Advancement of Science. Filon [1] demonstrated for plane isotropic linear elasticity in a multiply connected plate that an analogy exists between dislocation theory and the difference between linear elastic solutions to the same boundary value problem but for two different values of Poisson’s ratio. Muskhelishvili [2] and Boley and Weiner [3] established a similar relationship between dislocation and thermoelastic problems in two- and three-dimensions, respectively. The unifying concept depends upon the notion of residual or initial stress and ultimately upon the subcutaneous Riemann structure of linear elasticity.

The present aim is to explore the construct in two- and three-dimensional linear theories of anisotropic non-homogeneous elastostatics and elastodynamics. Certain aspects of the analogy in statics are discussed in [4, 5]. Consequently, primary concern is with dynamic problems.

Requisite elements of plastic dislocation theory are presented in Sect. 2, while properties of linear elastodynamics are reviewed in Sect. 3. The difference between the solutions to the elastic initial boundary problem for two different sets of moduli and different body-forces, initial, and boundary data is shown in Sect. 4 to possess the same structure as the solution to the moving dislocation problem. A correspondence between the respective solutions is therefore possible which considerably extends Filon’s original construct to arrays of moving dislocations in three-dimensional regions. The general process is described in Sect. 5. The inverse Filon construct, treated in Sect. 6, generates constituent elastodynamic solutions from a given dislocation problem. Complete generality requires the introduction of an auxiliary plastic distortion tensor and retention of body-forces. Section 7, for convenience, specialises the main formulae to isotropic elasticity. Filon’s construct for equilibrium problems is illustrated in [4, 5] by application to the stationary edge and screw dislocations, and these examples are not repeated here. Nor is Filon’s construct applied to moving dislocations since the static treatment suggests the procedure is comparatively straightforward. Instead, attention is devoted to the inverse Filon construct which is considered in Sect. 8 for the stationary edge dislocation and uniformly moving screw dislocation both in a homogeneous isotropic linear elastic whole space. The edge dislocation leads to constituent plane elastic fields corresponding to a point force, while the moving screw dislocation has anti-plane shear motions as one possible constituent pair. A different pair is derived to demonstrate that such pairs are not unique. A few brief concluding remarks are contained in Sect. 9.

Vector and tensor quantities are represented by bold lower and upper case letters, respectively, although lower case bold letters are occasionally employed to also denote tensor quantities. The distinction should be obvious from the context. Both indicial and direct notations are used as convenient, with the standard conventions adopted of summation over repeated subscripts and superscripts, and a subscript comma to denote partial differentiation. Latin subscripts range over 1, 2, 3,  while Greek indices assume the values 1, 2. The spatial argument of a function is abbreviated so that, for example, \(\pmb {{ h}}(x_{i})\) indicates \(\pmb {{ h}}(x_{1},\,x_{2},\,x_{3})\). In the direct notation, the gradient, divergence, and rotation operators as applied to vectors and tensors are denoted, respectively, by \(\nabla ,\,\mathrm {grad},\,\mathrm {Grad},\, \mathrm {div},\,\mathrm {Div}\), and \(\nabla \times ,\, \mathrm {curl},\,\mathrm {Curl}\). Transposition is indicated by the superscript T, the identity tensor by \(\pmb {\textit{I}}\), and the scalar and tensor products by their usual symbols. The indicial forms of these operations become where \(e_{ijk}\) denotes components of the alternating tensor.

We assume the existence of a solution suitable to our needs.

Elements of the theory of moving dislocations required subsequently are briefly recalled. A more complete account is presented, for example, in [6, 7]. Specialisation recovers the stationary and quasi-static theories.

Consider an unbounded or bounded, simply or multiply connected region \(\Omega \subseteq \mathrm {I\!R\!}^{n},\,n=2,3\) that when bounded possesses a Lipschitz smooth boundary \(\partial \Omega \) on which the outward unit vector normal is \(\pmb {{ n}}\). The region \(\Omega \) is occupied by an inhomogeneous anisotropic compressible linear elastic material subject to prescribed time-dependent external body-force, mixed boundary conditions, and Cauchy initial data. The body, however, is self-stressed due to an array of moving dislocations that can be approximated by a continuous distribution of dislocations of prescribed density specified by the second-order non-symmetric tensor \(\pmb {\alpha }(x_{i},t),\,(\pmb {\textit{x}},t)\in \Omega \times [0,T)\). Here, and in what follows, \(x_{i},\,i=1,2,3,\) are the Cartesian components of the position vector \(\pmb {\textit{x}}\), and [0, T) is the maximal interval of existence in the initial boundary problem of moving dislocations. It is convenient to suppose that this is also the interval of existence for the elastic problems discussed in Sect. 3.

Let \(\Sigma \subset \Omega \) be any (time-varying) open surface bounded by the closed curve \(\partial \Sigma \) described in a right-handed sense. The Burgers vector \(\pmb {\textit{b}}\) corresponding to the dislocation distribution is given by (c.p., [8, 9])where \(\pmb {\mathrm{{d}}}\pmb {\textit{S}}\) denotes the vector surface area element. The sign convention is opposite to that usually adopted.

Kröner [9], amongst others, postulates the second-order non-symmetric elastic distortion tensor \(\pmb {\textit{U}}^{(|E)}(x_{i},t)\) as a second state variable and relates it to Burgers vector by where Stokes’ theorem is employed to derive the last equation, and \(\pmb {\mathrm{{d}}}\pmb {\textit{s}}\) denotes the curvilinear vector line element of \(\partial \Sigma \). Elimination of \(\pmb {\textit{b}}\) between (2.1) and (2.3) yields the fundamental field equationfrom which is deduced the conditionA compatible (non-symmetric) continuously differentiable tensor field \(\pmb {\textit{U}}(x_{i})\) is defined to satisfyWhen \(\pmb {{ U}}\) is not compatible, the tensor is incompatible. Poincaré’s lemma states that when \(\Omega \) is simply connected, a necessary and sufficient condition for compatibility is the existence of a twice continuously differentiable vector field \(\pmb {{ v}}(x_{i})\) such thatA different compatibility condition for a symmetric second-order tensor is stated in (2.17).

For non-vanishing dislocation density \(\pmb {\alpha }\), (2.4) implies that \(\pmb {\textit{U}}^{(E)}\) is an incompatible tensor. Nevertheless, the stress \(\pmb {\sigma }(x_{i},t)\) in \(\Omega \) is linearly related to \(\pmb {\textit{U}}^{(E)}(x_{i},t)\) bywhere \(\pmb {\textit{C}}\) denotes the fourth-order elastic moduli tensor whose Cartesian components possess the symmetrieswhich imply the fourth symmetry \(C_{ijkl}=C_{ijlk}\). We also suppose that the tensor \(\pmb {\textit{C}}\) is convex in the sense that there exists a constant \(C_{0}\) such thatfor all symmetric second-order tensors \(\pmb {\psi }=\pmb {\psi }^\mathrm{T}\).

The symmetries (2.9) enable the constitutive relation (2.8) to be rewritten aswhere so thatA similar notation is adopted for other second-order non-symmetric tensors.

The time evolution of the dislocation density together with time-varying boundary conditions causes the body to change shape with time. The change is measured by the compatible total displacement \(\pmb {\textit{u}}(x_{i},t)\), in terms of which the second-order incompatible plastic distortion tensor \(\pmb {\textit{U}}^{(P)}(x_{i},t)\) is defined to beThe strain tensor \(\pmb {\textit{e}}\) corresponding to \(\pmb {\textit{u}}\) is given by the symmetric expressionwhich for a spatially continuously differentiable displacement field \(\pmb {\textit{u}}(x_{i},t)\) impliesConversely, for a given symmetric second-order tensor field \(\pmb {\textit{e}}(x_{i},t)\), there exists a single-valued continuously differentiable vector field \(\pmb {\textit{u}}(x_{i},t)\) satisfying (2.16) if the region \(\Omega \) is simply connected and the symmetric tensor \(\pmb {\textit{e}}\) satisfies the Saint-Venant compatibility relations (2.17)

Note that the representations (2.7) and (2.16) are associated with different compatibility conditions (2.6) and (2.17), respectively.

It follows from (2.4) that \(\pmb {\textit{U}}^{(E)}\) is incompatible and indeed (2.4) and (2.15) yield from which we conclude that \(\pmb {\textit{U}}^{(P)}\) is incompatible. Moreover, (2.18) and (2.20) imply thatElimination of the antisymmetric tensor \(\pmb {\textit{W}}^{(P)}\) from (2.21) leads to the relationwhich may be solved for \(\pmb {\textit{E}}^{(P)}\) in terms of \(\pmb {\eta }\) (Eshelby [10]). Furthermore, the relation (2.23) implies that \(\pmb {\eta }\) vanishes with \(\pmb {\alpha }\) but the converse is false. The symmetric part \(\pmb {\textit{E}}^{(P)}\) of \(\pmb {{ U}}^{(P)}\) may vanish or be the gradient of some differentiable vector field, while the non-symmetric component \(\pmb {\textit{W}}^{(P)}\) satisfies \(\nabla \times \pmb {\textit{W}}^{(P)}\ne 0\). In consequence, it may be concluded from (2.21) that \(\pmb {\alpha }\) does not vanish with \(\pmb {\eta }\).

In the quasi-static theory, inertial effects due to the total displacement are discarded, but in the dynamic theory, the corresponding equations of motion are given bywhere \(\rho (x_{i})\) represents the mass density of the body, and \(\pmb {\textit{f}}(x_{i},t)\) is the body-force per unit mass. The importance of including body-forces will become apparent later.

The inertial term necessitates initial Cauchy data for the total displacement vector \(\pmb {\textit{u}}(x_{i},t)\) in terms of which the dislocation initial boundary value problem is given by to which are adjoined initial conditions,displacement boundary conditions,and traction boundary conditionswhere \(\partial \Omega =\partial \Omega _{1}\cup \partial \Omega _{2}\), and \(\pmb {\textit{h}}(x_{i},t),\,\pmb {\textit{g}}(x_{i},t),\,\pmb {\textit{u}}_{0}(x_{i}),\,\pmb {\textit{u}}_{1}(x_{i})\) are prescribed functions on their respective domains of definition.

Detailed discussion of the uniqueness of the solution to this initial boundary value problem is presented in [11], which also considers the relation between the total displacement gradient plasticity theory just formulated and the Volterra theory of dislocations (see [11, §3]).

We repeat that the corresponding quasi-static problem suppresses the inertial term in (2.27), while for the corresponding equilibrium boundary value problem all terms are independent of time, and initial conditions (2.28) are irrelevant.

We consider a non-homogeneous anisotropic compressible linear elastic body that occupies a bounded or unbounded region \(\Omega \subseteq \mathrm {I\!R\!}^{n},\,n=2,3\) which may be simply or multiply connected. When bounded, \(\Omega \) possesses the smooth boundary \(\partial \Omega \) with unit outward vector normal \(\pmb {\textit{n}}\).

It is well known that multiply connected bodies can be self-stressed (see, e.g., [12‐14]) and the standard Kirchhoff and Neumann uniqueness theorems of linear elasticity no longer apply.

Two different sets \(\pmb {\textit{C}}^{(\gamma )},\,\gamma =1,2,\) of the elastic modulus tensor are considered whose components with respect to a Cartesian system of orthogonal coordinates with origin in \(\Omega \) satisfy the same symmetries specified in (2.9); that is,It is further supposed that the respective elastic modulus tensors are positive-definite in the sense that for all non-zero second- order symmetric tensors \(\pmb {\psi }\), the following inequality holds for \(\gamma =1,2\):When the elastic modulus tensors \(\pmb {\textit{C}}^{(\gamma )}(x_{i})\) are continuously differentiable on \(\Omega \), the positive-definite condition (3.2) by continuity implies a convexity condition similar to (2.10).

The elastic compliance tensor \(\pmb {\textit{C}}^{(-1)(\gamma )}\) is the inverse of the elastic modulus tensor and consequently for each \(\gamma \) the components satisfy the relationwhere there is no sum over \(\gamma \), \(I_{ijkl}\) are the Cartesian components of the fourth-order unit tensor \(\pmb {\textit{I}}\) and \(\delta _{ij}\) denotes the Kronecker delta function. The elastic compliance tensor enjoys the same minor symmetries as the elastic modulus tensor.

For each \(\gamma \), components of the elastic modulus and compliance tensors for an isotropic elastic material, respectively, become where \(\lambda ^{(\gamma )},\,\mu ^{(\gamma )}\) are the Lamé moduli related to Poisson’s ratio \(\nu ^{(\gamma )}\) byIn plane strain linear isotropic elasticity, the corresponding expressions are Cartesian components of the symmetric linear strain tensor \(\pmb {\textit{e}}^{(\gamma )}\) are derived as usual from the differentiable displacement vector field \(\pmb {\textit{u}}^{(\gamma )}\) according toand as described in Sect. 2, are compatible in the sense that for \(\pmb {\textit{u}}^{(\gamma )}\in C^{3}(\Omega ,{\mathbb {R}}^{3})\) they satisfy the Saint–Venant compatibility relations analogous to (2.17):or equivalentlywhere, as already stated, \(e_{ijk}\) are the components of the usual alternating tensor. We note that the compatibility condition (3.11), necessary for the existence of a continuously differentiable single-valued displacement vector field, is also sufficient provided that the region \(\Omega \) is simply connected. (See, for example, Gurtin [15, §.14.2].) It should be remarked, however, that subsequent discussion assumes certain solutions contain point singularities for which (3.12) is satisfied only in regions external to the neighbourhood of the singularity.

Introduction of suitable cuts converts multiply connected regions into simply connected regions. Assume that (3.12) holds so that the strains \(\pmb {\textit{e}}^{(\gamma )}\) are compatible on the cut region \({\widehat{\Omega }}\). Let \(\pmb {{ u}}^{(\gamma )}\) be the corresponding continuously differentiable displacement, and let \({\mathcal {C}}\) be any continuous curve in \({\widehat{\Omega }}\) connecting the points with position vectors \(\pmb {\textit{x}}_{0}\) and \(\pmb {\textit{x}}\). The Volterra–Cesaro integration procedure (see, for example, [16] [15, §14]) then giveswherewhich is independent of \({\mathcal {C}}\) provided a cut is not traversed.

When the endpoints \(\pmb {\textit{x}}\) and \(\pmb {\textit{x}}_{0}\) are on contiguous sides of a cut, the last expression determines the jump in \(\pmb {\textit{u}}^{(\gamma )}\) across the cut as \(\pmb {\textit{x}}\rightarrow \pmb {\textit{x}}_{0}\), and consequently the corresponding Burgers vector.

Let \(\pmb {\sigma }^{(\gamma )}\) be the stress tensor which for \(\gamma =1,2\) is related to the strain \(\pmb {\textit{e}}^{(\gamma )}\) by the constitutive assumptionThe equations of motion, boundary, and initial conditions satisfied by the respective elastodynamic fields are where \(\partial \Omega =\partial \Omega _{1}\cup \partial \Omega _{2}\), and the mass density \(\rho (x_{i})\) (assumed the same for both sets of elastic moduli), body-force vector \(\pmb {\textit{f}}^{(\gamma )}(x_{i},t)\) per unit mass, surface traction \(\pmb {\textit{g}}^{(\gamma )}(x_{i},t)\), surface displacement vector \(\pmb {\textit{h}}^{(\gamma )}(x_{i},t)\), and initial data \(\pmb {\textit{u}}^{(\gamma )}_{0}(x_{i}), \pmb {\textit{u}}^{(\gamma )}_{1}(x_{i})\) are prescribed. Of later separate interest are problems for which the quantities \(\pmb {\textit{h}}^{(\gamma )} \text{ and } \pmb {\textit{g}}^{(\gamma )}\) are, respectively, the same, e.g., \(\pmb {{ h}}^{(1)}=\pmb {{ h}}^{(2)}.\) The interval of existence of a solution is given by [0, T), where T is either finite or infinite. As already indicated, this interval is assumed to coincide with that for the dislocation problem.

The elastostatic and quasi-static boundary value problems may be derived by taking, for example, the inertial term to be zero in (3.16) and ignoring the initial values (3.19).

Michell [12] (see also Gurtin [15, §§46, 47]) has proved that stresses in the traction boundary value problem of plane isotropic homogeneous linear elasticity are independent of the elastic moduli provided body-forces vanish and the region is simply connected. The result also holds for multiply connected regions when the resultant force, but not necessarily the couple, vanishes over each separate boundary. Earlier discussion is given by Maxwell [17, p. 201], and Lévy [18], while subsequent consideration is due to Timpe [19], who remarked that contravention of the result is to be expected in plane multiply connected regions since these can be self-stressed by notional cutting and welding operations. The multi-valued displacement, stress, and strain depend upon the elastic moduli. Volterra developed a general theory of such multi-valued displacement which is fundamental to his theory of dislocations.

In other two- and three-dimensional boundary value problems of linear elastostatics, and initial boundary value problems of linear elastodynamics the stress, strain, and displacement in general depend upon the elastic moduli in both simply and multiply connected regions. Conditions for independence in isotropic elasticity are discussed by several authors including Carlson [20‐22], Dundurs [23], Knops [5], and Markenscoff [24]. In particular, Carlson [22] notes that for elastodynamics there are only a few problems for which it can be expected that the solution is independent of elastic moduli. Continuous dependence is explored by Bramble and Payne [25], amongst others.

We derive equations for the differences between the quantities introduced in Sect. 3 when the moduli are varied, but when the mass density remains unaltered. Accordingly, we define the difference fields to be where \({\bar{\Omega }}\) denotes the closure of \(\Omega \).

Subtraction of the constitutive relations (3.15) leads towhere we recall that \(\pmb {\textit{I}}\) is the identity tensor, whose Cartesian components are here repeated for convenience:the tensor \(\pmb {\textit{D}}\) is given byand \(\pmb {\textit{C}}^{(-1)(1)}\) is the corresponding elastic compliance tensor satisfying (3.3). Definition (4.10) implies that \(\pmb {\textit{D}}\) possesses minor symmetries so that its Cartesian components satisfySubtraction of the respective elastodynamics equations, initial, and boundary conditions (3.16)–(3.19), together with the constitutive relations (4.8) shows that the difference fields satisfy the initial boundary value problem This system corresponds to the standard initial (mixed) boundary value problem with non-zero body-force, non-zero initial data, and non-zero surface traction and displacement.

The “strain” \(\pmb {\textit{De}}^{(2)}\) is incompatible in the sense that in general

The dislocation initial boundary value problem specified by (2.26)–(2.30) becomes identical to the variation system (4.12)–(4.15) on setting and letting \(\pmb {\textit{u}},\,\pmb {{ e}},\,\pmb {{ f}},\,\pmb {{ h}},\,\pmb {{ g}}\) retain their respective definitions assigned in Sects. 2 and  4.

The analogy may be used to derive solutions to a dislocation problem from known constituent elastodynamics solutions defined in Sect. 4. Their difference uniquely specifies the total displacement \(\pmb {\textit{u}}\), the stress \(\pmb {\sigma }\), and therefore the symmetric part \(\pmb {\textit{E}}^{(E)}\) of the elastic distortion \(\pmb {\textit{U}}^{(E)}\) for the dislocation problem in which the symmetric part of the plastic distortion \(\pmb {\textit{U}}^{(P)}\) is prescribed by (5.2)

The dislocation solution, however, when derived from the construct is restricted. Only the symmetric part of the plastic distortion tensor \(\pmb {\textit{U}}^{(P)}\) is determined by (5.2), and moreover must satisfy the conditionsince the strain \(\pmb {\textit{e}}^{(2)}\) is supposed compatible in the sense that the Saint-Venant relation (2.17) is satisfied. Relation (5.2) allows condition (5.3) to be alternatively expressed asRestrictions on \(\pmb {\textit{E}}^{(P)}\) may be partially relaxed by introduction of an auxiliary second-order symmetric tensor field \(\pmb {\textit{E}}^{*}(x_{i},t)\), which in general is not derivable from a displacement vector field. Therefore, \(\pmb {\textit{E}}^{*}\) is incompatible and (2.17) is not satisfied. For each time instant t, we take \(\pmb {\textit{E}}^{*}\) to be a particular solution to the boundary value problem where, for the moment, the vector functions \(\pmb {\textit{f}},\,\pmb {\textit{g}}\) are arbitrary. Alternatively, the tensor \(\pmb {\textit{E}}^{*}\) can be arbitrarily chosen but still incompatible with \(\pmb {\textit{f}}\) and \(\pmb {\textit{g}}\) determined from (5.5) and (5.6).

When   \(\pmb {\textit{f}}^{(1)}-\pmb {\textit{f}}^{(2)}=\pmb {\textit{f}}\), \(\pmb {\textit{g}}^{(2)}-\pmb {\textit{g}}^{(1)}=\pmb {\textit{g}}\), and \(\partial \Omega _{1}=\emptyset \), a particular solution is the trivial solution \(\pmb {\textit{E}}^{*}\equiv 0.\)

Once \(\pmb {\textit{E}}^{*}\) is determined, the symmetric part of the plastic distortion is no longer derived from relation (5.2), but instead is obtained from the formulaThe Filon construct by virtue of (5.5) and (5.6) then leads to the equations of motion for the dislocation problem. We have subject to the boundary and initial conditions where \(\pmb {\textit{f,\,g}}\) are the arbitrary vector functions appearing in (5.5) and (5.6)

The total displacement \(\pmb {\textit{u}}=\pmb {\textit{u}}^{(1)}-\pmb {\textit{u}}^{(2)}\) is the solution to the dislocation initial boundary value problem (5.11)–(5.15) for which the plastic distortion strain tensor \(\pmb {\textit{E}}^{(P)}\) is determined by (5.7) subject to body-force \(\pmb {\textit{f}}\) and traction boundary conditions (5.13). The elastic distortion strain tensor \(\pmb {\textit{E}}^{(E)}\) remains given by \(\nabla \pmb {\textit{u}}-\pmb {\textit{E}}^{(P)}\), but the dislocation stress \(\pmb {\sigma }=\pmb {\textit{C}}^{(1)}\pmb {\textit{E}}^{(E)}=\pmb {\sigma }^{(1)}-\pmb {\sigma }^{(2)}+\pmb {\textit{C}}^{(1)}\pmb {\textit{E}}^{*}\) is no longer the difference stress \(\pmb {\sigma }^{(1)}-\pmb {\sigma }^{(2)}\).

These calculations demonstrate the importance of body- and surface forces upon relaxation of the restriction on \(\pmb {\textit{E}}^{(P)}\) imposed by condition (5.2). In the absence of \(\pmb {\textit{E}}^{*}\), the body-force \(\pmb {\textit{f}}\) and traction \(\pmb {\textit{g}}\) in (5.11) and (5.13) are not arbitrary, but are given by \(\pmb {\textit{f}}=\pmb {\textit{f}}^{(1)}-\pmb {\textit{f}}^{(2)}\) and \(\pmb {\textit{g}}=\pmb {\textit{g}}^{(2)}-\pmb {\textit{g}}^{(1)}\). The stationary single edge and screw dislocation discussed in [4] require only (5.2) and are examples of when the auxiliary tensor \(\pmb {\textit{E}}^{*}\) and body-forces are not necessary.

The Filon construct in which relation (5.7) determines the symmetric part \(\pmb {\textit{E}}^{(P)}\) of \(\pmb {\textit{U}}^{(P)}\) fails to uniquely determine the dislocation density from (2.23). The Volterra theory of dislocations suggests that results involving the Filon construct will be of interest when the region \(\Omega \) is multiply connected or when the constituent solutions become singular in a certain sense, e.g., point or line singularities. The discussion in [4] of the stationary edge dislocation also shows that in the multiply connected or punctured region \(\Omega \) the symmetric strain tensor \(\pmb {\textit{E}}^{(E)}\) satisfies the Saint-Venant relation (2.17) and is derivable from an associated vector field \(\pmb {\textit{u}}^{(E)}\). Suitable cuts convert \(\Omega \) into a singly connected region and enable the corresponding Burgers vector to be calculated from the Volterra–Cesaro formula (3.13) in which \(\pmb {\textit{e}}^{(\gamma )}\) is replaced by \(\pmb {\textit{E}}^{(E)}\). The reader is referred to [4] for further details.

Previous sections concern the derivation of certain dislocation solutions from linear elastic solutions whose strains are compatible. This section studies the inverse process, namely, the recovery of the elastic solutions from a uniquely specified dislocation solution. As already mentioned, body- forces are often essential to achieve complete generality. Non-uniqueness is inherent since one constituent initial boundary problem can be arbitrarily selected for the elastic field \((\pmb {\textit{u}}^{(2)},\pmb {\textit{e}}^{(2)},\pmb {\sigma }^{(2)})\) provided the strain \(\pmb {\textit{e}}^{(2)}\) is compatible. The solution is not required to be unique. In consequence, however, the second constituent elastic solution \(( \pmb {\textit{u}}^{(1)},\pmb {\textit{e}}^{(1)},\pmb {\sigma }^{(1)})\), which must also be compatible, is unambiguously determined such that the difference between it and the first constituent solution corresponds to the given dislocation solution.

We examine further implications of the inverse procedure. Assume as before, that the mass density is unaltered throughout the discussion. The compatible strain \(\pmb {\textit{e}}^{(2)}(x_{i},t)\) is derived from the displacement \(\pmb {\textit{u}}^{(2)}\) and belongs to the solution of the elastodynamic problem subject to body-force \(\pmb {\textit{f}}^{(2)}(x_{i},t)\), boundary conditions \(\pmb {\textit{h}}^{(2)}(x_{i},t),\,\pmb {\textit{g}}^{(2)}(x_{i},t)\), and initial conditions \(\pmb {\textit{u}}^{(2)}_{0}(x_{i}),\,\pmb {\textit{u}}^{(2)}_{1}(x_{i})\).

The Filon relation (5.2) formally leads toBut in general for given \(\pmb {\textit{E}}^{(P)}\), we haveand consequently (6.1) is not necessarily consistent with the assumed compatibility of the strain \(\pmb {\textit{e}}^{(2)}\) and satisfaction of the Saint–Venant relation (2.17). The difficulty is again resolved by introduction of an auxiliary second-order tensor field \(\pmb {\textit{E}}^{**}\) and replacement of (6.1) by the relationwhere \(\pmb {\textit{E}}^{**}(x_{i},t)\) is chosen to satisfyAn obvious choice isAlthough the last relation is formally identical to (5.7), and the tensors \(\pmb {\textit{E}}^{*}\) and \(\pmb {\textit{E}}^{**}\) are superficially the same, we prefer to retain a separate notation for purposes of conceptual clarity.

Another method of selection is to arbitrarily choose the vector \(\pmb {\textit{u}}^{(2)}(x_{i},t)\) to ensure compatibility of the strain \(\pmb {\textit{e}}^{(2)}\), and also to arbitrarily select the elastic modulus tensor \(\pmb {\textit{C}}^{(2)}\). The corresponding body-force is then determined from the equations of motionBoundary and initial conditions are derived by a similar semi-inverse method. Provided the chosen elastic modulus tensor \(\pmb {\textit{C}}^{(2)}\) satisfies the major symmetry in (2.9), the corresponding elastodynamic solution is unique. (c.p., [26]).

Accordingly, \(\pmb {\textit{u}}^{(2)}(x_{i},t),\, \pmb {\textit{e}}^{(2)}(x_{i},t)\), body-force   \(\pmb {\textit{f}}^{(2)}(x_{i},t)\), boundary conditions \(\pmb {\textit{h}}^{(2)}(x_{i},t),\,\pmb {\textit{g}}^{(2)}(x_{i},t)\), and initial conditions \(\pmb {\textit{u}}^{(2)}_{0}(x_{i}),\,\pmb {\textit{u}}^{(2)}_{1}(x_{i})\) can be calculated by any of the procedures just described. In order to determine the second constituent displacement \(\pmb {\textit{u}}^{(1)}\), body-force \(\pmb {\textit{f}}^{(1)}\), boundary conditions \(\pmb {\textit{h}}^{(1)},\,\pmb {\textit{g}}^{(1)}\), and initial conditions \(\pmb {\textit{u}}^{(1)}_{0},\,\pmb {\textit{u}}^{(1)}_{1}\), we select the elastic tensor \(\pmb {\textit{C}}^{(1)}\) to be the same as for the dislocation problem, i.e., \(\pmb {\textit{C}}^{(1)}=\pmb {\textit{C}}\). The strain \(\pmb {\textit{e}}\) is compatible and derived from the dislocation total displacement \(\pmb {\textit{u}}\) which is assumed known along with the dislocation body-force \(\pmb {\textit{f}}\), boundary conditions \(\pmb {\textit{h}},\,\pmb {\textit{g}}\), and initial conditions \(\pmb {\textit{u}}_{0},\,\pmb {\textit{u}}_{1}\). In consequence, define \(\pmb {\textit{u}}^{(1)}\) byand use this expression to derive the compatible strain \(\pmb {\textit{e}}^{(1)}(x_{i},t)\) and corresponding stress \(\pmb {\sigma }^{(1)}=\pmb {\textit{C}}^{(1)}\pmb {\textit{e}}^{(1)}=\pmb {\textit{C}}\pmb {\textit{e}}^{(1)}\). The body-force required to maintain these fields is obtained from the respective equations of motion (6.6) and (2.27) which, together with (6.5) and the identity \(\pmb {\textit{e}}=\pmb {\textit{e}}^{(1)}-\pmb {\textit{e}}^{(2)}\), gives whereBoundary and initial conditions are obtained by semi-inverse methods.

Notwithstanding Remark 6.2, in Sect. 8.1, we present an example in which body-forces are zero.

It is convenient to list explicit expressions for the Cartesian components of the fourth-order tensors \(\pmb {\textit{D}},(\pmb {\textit{I}}-\pmb {\textit{D}})\) and \((\pmb {\textit{I}}-\pmb {\textit{D}})^{-1}\) for isotropic linear elasticity. Corresponding expressions for the elastic modulus and its inverse, or compliance, are given by (3.4)–(3.9).

From Definition (4.10), we obtainwhere components \(I_{ijkl}\) of the identity matrix are given by (4.9), and Consequently, we havewhereNow supposewhere the scalar quantities \(A_{1},\,A_{2}\) are chosen to ensure thatWe conclude that When \(\mu ^{(1)}\ne \mu ^{(2)}\) but \(\nu ^{(1)}=\nu ^{(2)}\), expressions (7.2) and (7.8) give \(D_{1}=A_{1}=0\). On the other hand, when \(\mu ^{(1)}=\mu ^{(2)}\) but \(\nu ^{(1)}\ne \nu ^{(2)}\) both \(A_{1}\) and \(A_{2}\) become singular and the respective inverses require alternative derivation. In this case, \(D_{3}=0\), andwhereandIt follows from (5.7) thatwhich gives \(e^{(2)}_{kk}=(E^{*}_{kk}+E^{(P)}_{kk})/3D^{*}_{1}\). Only the polar parts of the respective tensors are immediately involved so that an appropriate representation is where the deviatoric part \(\pmb {\varepsilon }^{(2)}\) of \(\pmb {{ e}}^{(2)}\) satisfies \(\varepsilon ^{(2)}_{kk}=0\). Both \(\pmb {\textit{E}}^{*}\) and \(\pmb {\epsilon }^{(2)}\) are chosen such thatOn using (3.7) and (3.8), we may derive analogous tensors for plane strain isotropic elasticity. When \(\nu ^{(1)}\ne \nu ^{(2)},\,\mu ^{(1)}\ne \mu ^{(2)}\), we havewherewhile the inverse matrix has componentswhere When \(\nu ^{(1)}=\nu ^{(2)},\,\mu ^{(1)}\ne \mu ^{(2)}\), we easily conclude that \(D_{1}=A_{1}=0\). When \(\nu ^{(1)}\ne \nu ^{(2)},\,\mu ^{(1)}=\mu ^{(2)}\), we have \(D_{3}=0\), and \(D_{1}=D^{**}_{1}\) wherebut as before \(A_{1},\,A_{2}\) become singular. Similar arguments, however, to those previously employed lead to the inverse relationswhere \(\varepsilon _{\kappa \kappa }=0\), and \(\varepsilon _{\alpha \beta }\) and \(E^{**}_{\alpha \beta }\) are selected to ensure that the strain tensor \(\pmb {\textit{e}}^{(2)}\) is compatible.

Both the inclusion problem and the Filon construct for stationary edge and screw dislocations are discussed in [4, 5] for isotropic linear elasticity. Therefore they are not considered here.

Instead, we apply the inverse Filon construct to the stationary edge dislocation and to the uniformly moving screw dislocation both in a homogeneous isotropic linear elastic whole space from which the positive Cartesian \(x_{3}\)-axis is deleted. The aim is to determine a pair of elastic states from the respective dislocation fields.

The original Filon construct for plane isotropic linear elastostatics has been generalised to three-dimensional elastodynamics. Body-forces and especially an auxiliary plastic distortion tensor, crucial for a complete extension, are absent in the original formulation. The construct, however, does not generate the total plastic distortion tensor \(\pmb {{ U}}^{(P)}\) but only its symmetric part \(\pmb {{ E}}^{(P)}\) which is related to the dislocation density \(\pmb {\alpha }\) through (2.23). Expression (2.23), and consequently Filon’s construct, is insufficient to uniquely determine \(\pmb {\alpha }\) despite probably it being more appropriate as data. In this respect, it is established in [11] that the dislocation density does not uniquely determine the dislocation solution. The full implications of this last remark await investigation not only for the construct itself, but just as importantly for the inverse procedure.

Several other aspects of the generalised Filon construct and its inverse provide scope for future detailed examination. These include the following topics.

The approach adopted in this study has utilised the linearity of the theories involved. Inspection reveals, however, that a fundamental concept is that of internal stress which in turn may be explained by heuristic cut-and-weld operations. Neither notion is intrinsically linear. In consequence, the technique might be extendable to certain non-linear theories.

Other developments that perhaps warrant examination include the application of a generalised Filon construct to elastic inclusions contained in a vibrating body, or to wave motion in laminates. Interrelationships with other dynamic coupled theories, including thermoelasticity, present further open problems.