Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity

: This paper concerns ﬁnite elasto-plasticity of crystalline materials with micro-structural defects. We revisit the basic concepts: plastic distortion and decomposition of the plastic connection. The body is endowed with a structure of differential manifold. The plastic distortion is an incompatible diffeomorphism. The metric induced by the plastic distortion on the intermediate conﬁguration (considered to be a differential manifold) is a key point in the theory, in deﬁning the defects related to point defects, or extra-matter. The so-called plastic connection is metric, with plastic metric tensor expressed in terms of the plastic distortion and its adjoint. We prove an appropriate decomposition of the plastic connection, without any supposition concerning the non-metricity of plastic connection. All types of the lattice defects, dislocations, disclinations, and point defects are described in terms of the densities related to the elements that characterize the decomposition theorem for plastic connection. As a novelty, the measure of the interplay of the possible lattice defects is introduced via the Cartan torsion tensor. To justify the given deﬁnitions, the proposed measures of defects are compared to their counterparts corresponding to a classical framework of continuum mechanics. Thus, their physical meanings can be emphasized at once.


Introduction
The study concerns finite elasto-plasticity of crystalline materials (such as metals) with micro-structural defects.
The main objective of the paper is to revisit the basic concepts, the definition of plastic distortion and the decomposition of the plastic connection, and to introduce the measures of micro-structural defects (and their interplay) in terms of the elements that characterize the plastic connection, emphasized through the decomposition theorem. As a basic assumption, we consider the multiplicative decomposition of the deformation gradient into its components, the so-called plastic and elastic distortions. These plastic and elastic distortions are constitutive concepts, which cannot be reduced to their geometrical aspects (see Mandel [1], Teodosiu [2], and Cleja-Ţigoiu and Soós [3]). Both components are incompatible (i.e., anholonomic) diffeomorphisms, which means that they are not differentials of certain appropriate vector fields. A non-local second-order theory allows us to define the measure of defects in terms of elements that characterize the so-called plastic connection. The defects existing inside elasto-plastic crystalline materials, such as dislocations, disclinations, and point-defects (involving micro-cracks and microvoids, which means that the material is damaged at the micro-level) are experimentally put into evidence (see Romanov [4]). The effects of defects on the material's behaviour are theoretically and numerically analyzed within different constitutive models (see, for instance Cleja-Ţigoiu et al. [5,6]). The proposed models are scale sensitive and can be successfully applied to very small length scale (nano-mechanic) and to processes that occur in micro-seconds. configuration of the body at time t, B t , is replaced by the so-called intermediate (plastically deformed) configuration, labeled B. Under the mentioned hypotheses, we proved the decomposition theorem, that form was assumed in different versions by Bilby [29] and Noll [20], as well as Minagawa [30], Le and Stumpf [31], and Clayton [26]. Minagawa [30] considered a Cosserat continuum viewed as an ensemble of trihedron, i.e., composed of a point and three vectors, which change their directions and lengths when "a material piece is torn apart from a body". Non-metric connections were introduced by Krö (see [32], Minagawa [30], and De Wit [10]), accounting for extra-matter defects, such as interstitial atoms, vacancies, or other defects (see [11]).
Yavari and Gorieli [27] show that "in multiplicative plasticity one can combine the reference and "intermediate" configurations into parallelizable material manifolds". F p is assumed to act on the local basis reference configuration and to give a local basis in the intermediate configuration. Using Cartan's moving frame, they prove that the intermediate configuration can be endowed with a so-called Weitzenböck connection, namely a connection compatible with a diffeomorphism, here with F p . In our decomposition of the plastic connection, we found again the presence of this type of connection (called crystal connection, following [31]), which is directly related to defects such as dislocations, and a part (a skew-symmetric tensor) that is related to disclinations, while the symmetric part concerns extra-matter defects. All these defects, physically described by Kleman and Fridel [33], correspond to the elements that enter the decomposition representation of the plastic connection.
The measures of the micro-structural defects are defined as third-order tensor fields, following [34], in terms of the elements that characterize the plastic connection, emphasized through the decomposition theorem. To justify the given definitions, we return to the classical framework of continuum mechanics. These densities can be rewritten in terms of second-order tensor fields, and thus comparison with the other definitions becomes possible, and their physical meanings can be emphasized at once. As a novelty, we mention the possible interplay between dislocations, disclinations, and extra-matter defects.

Metric Tensor and Deformation Gradient
We define the geometrical structure of the continuously deformed body as a part of the material geometric structure of the elasto-plastic body, in the sense adopted by Wang [21]: "the material geometric structure of a simple body is implied mathematically by these mechanical properties". In reverse context, the geometrical structure is described within a constitutive framework.
We introduce the definition of the continuous deformable body, the metric and the induced inner products by metrics, and the deformation gradient.

Definition 1.
A body B is a differential manifold with the structure defined by 1.
the set of n-dimensional differentiable the set of diffeomorphisms {χ(·, t) | t ∈ R, t ≥ 0} of the class C 2 , called the motion of the body, The velocity and acceleration of the particle p, at time t are defined by for k = 1 and k = 2, respectively. Here, the tangent spaces of the manifolds B and B t at the point p and q = χ(p, t), respectively, are denoted by T p B, and T q (B t ).
The differential of the diffeomorphism χ(·, t) at the point p ∈ B is denoted by F(p, t), which is linear. F(p, t) is called the deformation gradient. Notations At a point p ∈ B, the tangent space T p (B) is spanned by the basis ∂ ∂x µ , and the dual space or cotangent space T * p (B) is spanned by the basis d x µ , where x µ is the local coordinate on the chart (U i , ϕ i ) to which p belongs.
At a point q = χ(p, t) ∈ B t , the tangent space T q (B t ) is spanned by the basis ∂ ∂y ν , and the dual space or cotangent space T * q (B t ) is spanned by the basis d y ν where y µ is the local coordinate on the chart (V i , ψ i ) to which q belongs. F (B)-the set of all smooth functions from B to R, X (B)-the set of all smooth vector fields on B. Let us introduce a connection ∇ : X (B) × X (B) −→ X (B) on (B), with specific properties (see Nakahara [18] and Lee [19]). We use the notation ∇(U, V) = ∇ U V, ∀ U, V ∈ X (B), called the covariant derivative (or the directional derivative) of the vector field V along the vector field U; these values are vector fields from X (B).
Moreover, the dual bases {d x k } and {d y µ } of T * q (B) and T * q (B t ), respectively, satisfy the orthogonality conditions We denote by ∇ V the directional derivative along the vector V, for V ∈ T p (B), acting on the scalar field f ∈ F (B), as and on the tensor fields of arbitrary type, say T 1 and T 2 , as where T 1 and T 2 are tensor fields of arbitrary type.

Metric and Induced Inner Product
We now introduce the metric. Assumption 1. B is endowed with a metric expressed as with the matrix C ij (p, t) symmetric and positive definite.
where U = U j ∂ ∂x j and V = V j ∂ ∂x j ∈ T p (B).

Remark 2.
As χ(·, t) : B −→ B t is a diffeomorphism from a differentiable manifold with a metric, C(p, t), the induced natural metric, denoted by g(q, t), can be introduced on B t , namely in component representation. Here, q = χ(p, t).
We now introduce the inner product induced by the metric, between any two vectors or two dual vectors. Moreover, the inner product is extended to define an inner product between a vector and a dual vector, which is compatible with Nakahara's definition in [18].
As there exists a metric, the inner product can be extended to define the product between any two vectors. Nakahara, in [18], defined the inner product between a vector and a dual vector as as a consequence of (4). As the symmetry property < U, ω > is not yet defined, there is a so-called duality product and not an inner product.
For any U ∈ T p (B), there exists ω ∈ T * q (B), such that For any ω ∈ T * p (B), there exists U ∈ T p (B), such that where the matrix C ij (p, t) is the inverse matrix (C ij (p, t)) −1 .
Thus, the following inner product between a vector and a dual vector is well defined Obviously, the inner product defined by b. is compatible with (10).
The inner product is also defined between the elements of the same spaces in the case of differentiable manifolds with a metric, as follows: Remark 3. For the sake of simplicity, we do not introduce different notations (say, to mention the metric) for the above inner products if it is not necessary.

Proposition 2.
The metric C(p, t) gives rise to an isomorphism between T p B and T * p B.
Proof. As a consequence of the metric property of the differentiable manifold, via the relationships (8) and (13), we proved that, for all U ∈ T p (B), there exists ω ∈ T * p (B), and vice versa. Moreover, < ω, ω >=< U, U > .

Deformation Gradient and Its Adjoint
Proposition 3. The deformation gradient is given by the following representation: while the adjoint of the deformation tensor, F * (p, t), is given by Proof. We apply the definition of the adjoint of a tensor. Let A : M −→ N be a linear map, where M and N are differentiable manifolds, having dual manifolds denoted by M * and N * . The rule to define the adjoint A * : N * −→ M * is given by Following the rule (18), written for F(p, t) : T p B −→ T q B t , which is linear, we derive the equalities Formula (17) results.

Remark 4.
The rules ((15)c.) applied to the basis vectors in T q B t and in T p B, respectively, give rise to the scalars while the rule (14) acts on the basis vectors taken from the tangent space T q B t and its dual T * q B t as

Remark 5. Let us consider the tensors
Based on the inner product (see Formulas (21) and (22)), the following composition rules concerning the tensors are introduced: and so on. In the two last compositions, the presence of the components of the matrices associated with the metric tensor can be observed.

Assumption 2.
The metric tensor C(p, t) and the deformation gradient F(p, t), for any p ∈ B are related by The component representation Formula (25) follows as a consequence of the Formulas (16) and (17), together with the composition rule as in (24).
Traditionally, in continuum mechanics, the right Cauchy-Green tensor C is defined by C = F T F, being associated with a given deformation gradient F. Following Yavari and Goriely [25], we consider the right Cauchy-Green tensor, denoted here as C (G) to avoid confusion with the previously introduced formulae, to be defined by Here, the deformation gradient F is arbitrarily given as F : T p B −→ T q B t , when the differential manifolds B and B t with the metrics G and g, respectively, are considered. The transpose of the deformation gradient is introduced as The relations are written in our notations, while mentioning the appropriate metrics. In the component, the Formulas (28) and (27) become where {g ab } and {G AB } are components of the metric tensors. The last formula in (29) is derived by lowering the indices using the metric G.

Remark 6.
We give the arguments that justify our representation (25) for the metric tensor C in terms of F and its adjoint F * .

1.
Let us remark that (C (G) ) A B ≡ G A B , ∀ A, B = 1, 2, 3, (which means that the components of the Cauchy-Green tensor are the metric components) if and only if Thus, the metric components of the differential manifolds are related by the components of the deformation gradient, considered F.

2.
By definition of the Cauchy-Green tensor, (27), while the metric tensor, denoted here by G, applied to a vector field is a dual vector, namely G(p, t)U ∈ T * p (B), as it follows from (8). Thus, only if T * p (B) is identified with T p (B) can we say that a metric tensor can be viewed as a Cauchy-Green tensor. Moreover, from (30) the metric tensor with the components {G A B } induces the metric with the components {g ab (q, t)} by the deformation gradient, represented in terms of its components F a A .

3.
Formula (29) 1 expresses the components of the transpose tensor in terms of both metrics, while Formula (26) gives rise to the induced metric {g is } on B t by F.

4.
In classical continuum mechanics, all vector fields refer to the same vector space, V, which is associated with the Euclidean space E . The dual space is identified with the vector space. Consequently, the adjoint and the transpose of a tensor coincide. Moreover, the metric and non-metric properties are independent feature of the connection. 5.
All the formulas derived above remain valid if the deformation gradient is replaced by an anholonomic diffeomorphism.

Material Connection-Revisited Decomposition Theorem
Take a chart (U , ϕ), with the coordinate {x k } on B and define functions Γ s kj , s, k, j ∈ {1, 2, 3}, called connection coefficients by for the covariant (or directional) derivative along the vectors { ∂ ∂x j }, which forms a basis in T p (B). Here, {d x s } is the basis in T * p (B). Let us introduce the notation ∇ Γ for the connection ∇ with the connection coefficients Γ, given by Formula (31).

Remark 7.
Given the coefficient connection {Γ s kj }, we propose a tensor-type formulae to emphasize the coordinate basis in which the coefficients are computed.

Covariant Derivatives of the Tensor Fields
Definition 2. Let a tensor field on B at time t be given as an element of T q r (B), namely the tensor field p ∈ B −→ T(p, t) ∈ T q r (B), which is given in components by The covariant derivative of T(·, t) at p ∈ B is a tensor field, denoted by ∇ k T(p, t) ∈ T q r (B) and computed following the rule written in coordinate charts as The above tensor notations are similar to those used by Clayton [26].

Definition 3.
The connection ∇ Γ is metric if the covariant derivative of the metric tensor C(p, t) is vanishing, namely ∇ k C(p, t) = 0, ∀k Proposition 4. The covariant derivative of the field F is computed by The covariant derivative of the field F * is given by We compose Formulas (37) and (33) with (16) and (17), respectively, by applying the rules similar to those listed in (24) together with (21) and (22). We provide the formulas We now introduce a connection to the body manifold B, denoted by A, which is compatible with the tensor field F and allows a coordinate representation of By composing the metric tensor C(p, t), given by (7), with (39), we provide the equalities if we account for (26). Finally, Formula (41) becomes where the connection coefficients A m kj are written in (40). (38) and (39), respectively, can be written as

Proposition 5. Formulas
and if we take into account Formula (42).
We evaluate the following tensor and scalar fields, respectively, when the directional derivative along the vector basis defined by (31) is applied: as can be seen in the following proposition: Under the hypothesis that the connection ∇ Γ is metric, there exist scalar fields which are symmetric, i.e., Q k si = Q k is , ∀ k, i, s.

(46)
Moreover, the scalars Q k nj Q k nj = F s n F i j Q k si , have the same symmetry Q k nj = Q k jn , ∀ k, j, n.
Proof. By the map χ(·, t), the tangent field where V l s denotes the Jacobian matrix. We applied (31) to (48): The left hand side expression of (42) becomes The last equality in (50) is a direct consequence of the metric property of the connection ∇ Γ , characterized by Formula (35).
The symmetry of the appropriate fields, written in (46) as well as in (47), follows as a consequence of (50).

Decomposition Theorem of the Metric Connection
The abstract result concerning the representation of the so-called connection coefficient, specific to a continuous deformable body with microstructure, can now be formulated. Theorem 1. Let B, C, ∇ Γ be the set of the differential body manifold, B, endowed with the metric C, and a connection, ∇ Γ , with the coefficient connection denoted by Γ.
Let B t , g, ∇ γ be the set of the differential body manifold at the moment t, B t , endowed with the metric g and the connection ∇ γ , with the coefficient connection denoted by γ.
a connection A compatible with the anholonomic diffeomorphism F, i.e., in the gradient , such that Qu is a second-order symmetric tensor ∀u ∈ T (B); c. Ω ∈ T 0 3 (B) such that Ωu ∈ Ω 2 (B), i.e., Ωu, is 2-form ∀u ∈ T (B), such that The tensor fields Q and Ω are expressed as In the relation (52), the tensor type representation (32) associated with the connection coefficients is introduced.
Proof. By this hypothesis, the connection ∇ Γ is metric. Thus, the covariant derivative of the metric tensor is vanishing, namely ∇ k C(p, t) = 0 (see formula (35)). Taking into account the relationship (25), the covariant derivative of C(p, t) allows the expression: From (51) together with the formula (54), the following representation is derived: due to the symmetry of Q(k ·). From (54), we read that the second-order tensor field that is written inside the bracket is a skew-symmetric second-order field. Consequently, there exists a two-form on B, i.e., which characterizes the skew-symmetry mentioned above, namely Remark 8. The result proved in Theorem 1 is in agreement with the statement: if Γ and Γ are two connection coefficients on B, then Γ − Γ is a tensor of type (1,2), i.e., in T 1 2 (B).
In order to emphasize the specificity of the representation for the connection coefficients given by (52), we recall the basic results (see, for instance, Schouten [35]). Theorem 2. Let ∇ Γ be a metric connection, with connection coefficients Γ and ∇ γ as the Levi-Civita connection of the metric tensor C. Both connections are defined on the differential manifold B.
There exists a so-called contortion tensor W ∈ T 1 2 (B), such that 1.
The contortion W and the torsion S are tensors of the type T 1 2 (B), which determine each other by taking into account the definition of the torsion S, 2.
The following skew-symmetries hold: To see the difference between the Schouten result and the decomposition derived in Theorem 1, we write Theorem 2, which is related to the decomposition of a metric connection when the metric is given. In Theorem 1, there exists the hypothesis that the metric tensor is associated with an anholonomic diffeomorphism.

Plastic Connection for Material with Micro-Structure
In the elasto-plastic description of the material body, we assume that -B, and B t , are differentiable manifolds, at any time t; -the existence of a differentiable manifold (time dependent, but we omit the t in the description for the sake of simplicity), B, called intermediate configuration, such that B and B are C 2 − diffeomorphic; i.e., there exists a diffeomorphism, say f : B −→ B, of the class C 2 , for all X ∈ B, and at every time t; -the deformation gradient F is induced by the motion function, i.e., F(·, t) = d χ(·, t) : the existence of the plastic and elastic C 1 − non-induced diffeomorphisms, F p and F e , (called distortions), (62) Remark 9. Consequently, F p and F e are involved in the multiplicative decomposition of the deformation gradient The following representations for the plastic distortion, F p , and its adjoint, F p * , hold, namely The plastic metric tensor on B is defined as and C p (X, t) induces the metric g p (q, t) on T q ( B) by F p , which is given by Here, C ij are the coordinate components of the metric tensor C p . We apply the abstract result, Theorem 1, concerning the representation of the so-called connection coefficient, specific to an elasto-plastic continuous deformable body with microstructure. Theorem 3. Let B, C p , ∇ Γ p be the set of the differential body manifold, B, endowed with the metric C p , and a connection, ∇ Γ p , with the coefficient connection denoted by Γ p .
Let B, g p , ∇ γ be the set of B t -the differential manifold representing the intermediate configuration, at the moment t endowed with the metric g, and the connection, ∇ γ , with the coefficient connection denoted by γ.
Under the hypotheses: There exists a C 2 diffeomorphism F p (·, t) : T (B) −→ T ( B), such that the metric tensor is given by C p (·, t) = F p * (·, t) • F p (·, t); • The metric g p is induced from C p by F p via the Formula (66). The following representation for the plastic connection coefficients is provided as Here,

a.
A p is a connection compatible with the diffeomorphism F p , i.e., in the gradient notation A similar decomposition to that given in (67) can be seen in Minagawa [30]. Despite working with the intermediate configuration B viewed as a smooth differential manifold (motivated by the simplification in the notations), we have in mind a material neighbourhood of a given material point, N , cut off from the deformed body configuration, and which is free of the loading, by a certain relaxation process. As we mentioned, it is not justified from a mathematical point of view to associate a differentiable manifold structure to the assembly of these locally relaxed configurations. Hence, when we refer to the assembly of locally relaxed configurations, the hypothesis of the vector bundle, whose basis is the differential manifold of the body in reference configuration, seems to be more adequate (from the differential geometry point of view) than the previous assumption. In a previous paper, we adopt a new definition for the plastic distortion to be a diffeomorphism acting from the tangent vectors on the differential manifold B on a vector bundle, E .

Measure of Microstructural Defects
We characterize the lattice defects in terms of the elements that enter the decomposition of the so-called plastic connection. The defects are treated as continuously and locally distributed in the body. Geometrical properties of the non-Riemannian connection, in their different approaches, are considered to characterize dislocations (Bilby [29] and Noll [20]), disclinations (De Wit [10,26]), and point defects (Minagawa [30] and Kröner [11].

Density of Dislocations
In contrast with the deformation gradient, F, the plastic distortion F p is not integrable. Only the non-integrability of F p is associated with defects such as dislocations. The nonintegrability condition for F p means that, on a certain simple connected set, such as a part of the body, the skew-symmetric part of the third-order field A p , denoted here as SkwA p , is not vanishing. Starting from the definition of SkwA given for any third-order tensor field, we found that Remark 10. The tensor field SkwA p can be a measure of dislocations, following Bilby [29] and Noll [20].
The third-order tensor field Skw (p) A is related to the second-order tensor α = (F p ) −1 curl(F p ) (which is defined in [34], the geometrically necessary dislocation, GND-tensor) by which are written for all vectors u and v. Here, we use the definition for curlF p . By definition, curlA is the second-order tensor field, which allows the given representation in a local basis, associated with a curvilinear coordinate system {x j }.
Concerning the plastic connection, we account for two cases.
In the first case, case I, in the expression of the plastic connection (see Formula (67)), the so-called symmetric part of the third-order tensor field,Q p is vanishing, i.e., Q p = 0.
In the second case, case II, the full representation for the plastic connection, derived in Formula (67), is considered.

Density of Disclinations
Case I. We look at the body B with the geometric structure (B, ∇ Γ p , C p ), in which Γ p , written in (67), is replaced by with a metric C p = F p * • F p , but under the hypothesis that the induced metric g (p) (see (66)), is such that with the crystal connection ∇ p A with the coefficient connection described by (40).
In reference to (73), Clayton [26] notices that this implies either B is a Euclidean space, or the basis vectors in B are not tangent to any coordinate system, i.e., they correspond to some external reference frame.
The torsion of the connection considered in (72) is given by written for all vectors u, v ∈ T (B). The tensor field Ω p , which enters (74), is expressed as in (53) 2 and has the property The expression of the Cartan torsion, S p , written in (74), is given by Remark 11. The tensor field Skw(Ω p ) can be a measure of disclinations.
Using the tensorial representation given by Cleja-Ţigoiu in [17], the following representation for Ω p can be derived: in terms of the second order field Λ, I being the identity second-order tensor. Thus, the Cartan torsion (76) is given by and it can be expressed in terms of the second-order torsion, denoted by N p , The following result is proved in [17,36].

Lattice Defects and Their Interplay
In case II, the plastic connection is given by (67), with Q (p) written as in (53), and such that the second-order tensor, Q (p) u, is symmetric, namely If we compute the Cartan torsion attached to the plastic connection (67), the contribution due to Q p has to be considered. As Q p u is a second-order symmetric tensor ∀ u ∈ T (B), we evaluate the appropriate contribution as follows: The skew-symmetry with respect to the indices k and j is mentioned through the notation Q [k j]n . Finally, we obtain A , given by (68); -the disclination density Skw(Ω p ); -the defect of non-metricity type (such as point defects, extra-matter, vacancy)), SkwQ (p) , written in (82) asS = Skw Several particular cases can be emphasized in reference to the field Q (p) , for instance: (i) The point defect characterized by Q (p) =q ⊗ C, withq a 1-form, i.e.,q = q n d x n , or in component representation as In this case SkwQ (p) = 0; thus, no contribution to the torsion follows.
(ii) The gradient-type defect occurs if there exists a symmetric, second-order field,Q, such that In this case, by direct calculus, we obtain Remark 12. The measures of the lattice defects are defined in Proposition 8 as third-order tensor fields. To justify the given definitions, we return to the classical framework of continuum mechanics. These densities can be rewritten in terms of second-order tensor fields, and thus comparison to the other definitions becomes possible, and their physical meanings can be emphasized at once.

Remark 13.
If the crystal connection A p is symmetric, i.e., SkwA p = 0, then the so-called geometrically necessary dislocations are absent, i.e., GND-tensor=0. The connection A p , called the crystal connection, following Le and Stumpf [31], is an integrable connection. Thus, the Riemannian curvature tensor is vanishing.
The connections associated with non-zero curvature are able to describe the presence of rotational defects in the lattice, defects that were interpreted as disclinations by De Wit [10].
Contrarily, if we compute the curvature tensor, R Λ , formed with the third-order tensor, Λ × I, we proved in [17] that there exists a second-order tensor r Λ , such that for all vectors u, v, w, z. Here, (AdjΛ) T (u × v) = Λu × Λv.

The Cartan torsion,S, defined by (83) is written under the form
We conclude that Skw

(p)
A can be related to defects such as dislocations, while Skw(Ω p ) can be interpreted to be responsible for defects such as disclination. Concerning the last term in the expression of Cartan torsion written in (88), let us introduce a second-order tensor, Ω Q , such that ((SkwQ (p) )u)v = Ω Q (u × v), following the procedure proposed in [34]. Consequently, the second-order torsion tensor, N , associated with Cartan torsion, written in (88), via Formula (79) 1 , reads Let us remark that, for the special case considered in (81), Remark 14. The physical characteristics can be attached to the defect variables, following Kröner, in the interpretation performed by de Wit [7] in linear approximation, as follows:Q is introduced via Formula (85), and it is a symmetric second-order tensor. As it is related to a measure of nonmetricity,Q is called the quasi-plastic strain (see also Kröner [32]). The second-order field Ω Q introduced via (89) can be called quasi-dislocation. To justify the attributed name, let us remark that the special case considered in (86) leads us to the Cartan torsion (see (90)), which contains curlF p and curlQ in a linear approximation. To paraphrase (for large deformation) De Wit [7], we concluded that each of these tensor fields "plays a valuable role in the understanding of the theory of extra-matter".
In order to take into account the interplay between defects, the following densities can be useful: A + (C p ) −1 Skw(Λ × I) A + (C p ) −1 SkwQ (p) + Skw(Λ × I) . (92) The interplay between dislocations and disclinations is measured by D 1 , while the interplay between dislocations and defect variable Q (p) is described by D 2 . Finally, the possible interplay between dislocations, disclinations, and extra-matter defects can be characterized by D 3 .

Conclusions
Based on the theorem concerning the decomposition of the plastic connection, measures of lattice defect densities are introduced as third-order tensor fields. The definition of plastic distortion as an incompatible diffeomorphism is based on the existence of the intermediate configuration considered to be a differential manifold. The metric induced by the plastic distortion on the intermediate configuration is a key point in the theory. No assumptions concerning the non-metricity of the plastic connection is done. As a direct consequence of the metric induced by the plastic distortion, defects such as extra-mater are infered in model. If the metric on the intermediate configuration is Euclidean, then the non-metric property of the connection does not result. All types of lattice defects, dislocations, disclinations, and point defects can be described in terms of the densities related to the elements that characterize the decomposition theorem for plastic connection. As a novelty, the measure of the interplay of the possible lattice defects is introduced via the Cartan torsion tensor. We justify the given definitions by comparing the proposed measure of defects with their counterparts corresponding to classical framework of continuum mechanics. Thus, their physical meanings can be emphasized at once.