# NONLINEAR DYNAMIC EQUATIONS FOR ELASTIC MICROMORPHIC SOLIDS AND SHELLS. PART I

## Abstract

The present paper develops a general approach to deriving nonlinear equations of motion for solids whose material points possess additional degrees of freedom. The essential characteristic of this approach is the
account of incompatible deformations that may occur in the body due to distributed defects or in the result of the some kind of process like growth or remodelling. The mathematical formalism is based on least action principle and Noether symmetries. The peculiarity of such formalism is in formal description of reference shape of the body, which in the case of incompatible deformations has to be regarded either as a continual family of shapes or some shape embedded into non-Euclidean space. Although the general approach yields equations for Cosserat-type solids, micromorphic bodies and shells, the latter differ significantly in the formal description of enhanced geometric structures upon which the action integral has to be defined. Detailed discussion of this disparity is given.

## Introduction

There is a widespread tendency at present to derive dynamic equations for shell-like solids by direct approach. In doing so, shells are regarded as two-dimensional Cosserat continuum, which additional degrees of freedom are associated with speciﬁc shell kinematics. From this point of view, models are similar to three-dimensional micromorphic solids and can be considered within a single mathematical formalism. However the geometric structure of the manifold representing the body has fundamentally diﬀerent properties, depending on which model is considered. In the language of modern diﬀerential geometry, these diﬀerences are characterised by a particular bundle structure deﬁned over a manifold that formalizes the shape of the body or its material counterpart. In this paper we have sought to answer on the question: how does the formal structure of bundle relate to the commonly accepted hypotheses in micropolar, micromorphic and shell-like bodies.

Mathematical shell theory is conventional ﬁeld of continuum mechanics so it is unlikely that a complete literary list can be given here. We will give only some references [1–4]. The mathematical formalization of shell theory in Cosserat framework is also the subject of quite a lot of literature [5–8]. Non-linear models of shells whose deformations do not satisfy the compatibility conditions are much less developed. The works [9; 10] are the ﬁrst such that should be mentioned here. In them, non-linear models are introduced within a general theory of materially inhomogeneous bodies, primarily developed by Truesdell and Noll in the framework of the material connection theory [11]. This approach has been developed in [12–14]. A slightly diﬀerent approach based on the modeling of maps between manifolds with dimensions greater than 3 is represented in [15; 16]. Formally, this approach retains the idea of a geometric description for measures of deformation incompatibility in terms of material metric and connection, but deﬁnes them in high-dimensional spaces, which correspond to a set of material points, each of which has a continuous set of orientations. Models of this type have speciﬁc physical meaning, but it is very diﬀerent from what is usually found in classical shell models. The latter can be obtained by considering the sections of bundles that form high-dimensional spaces. This issue is discussed in detail in present work.

1. ## Physical Space and Time

#### 1◦. Euclidean physical space and time. Generally, the geometrical approach developed in the paper, regards the body and the physical space as smooth manifolds of the common structure. However, with the application of the developed theory, we will conﬁne ourselves to the speciﬁc case of a physical space with Euclidean structure and absolute time, as it is required by classical (non-relativistic) approach. Meanwhile, for the body, as the set of material points, we leave the possibility of being a smooth manifold of a general kind, because the diﬀerence between its geometry and Euclidean geometry is precisely that which characterizes the incompatible deformations.

We suppose that physical space can be formalized as three-dimensional Euclidean point space, which can be formulated as the structure2

E = (E, V, vec, ·, or). (1.1)

The ﬁrst element of the structure represents the underlying set E. Second element of (1.1) is three-dimensional real vector space V = (V, +V , ·V ), in which V is the underlying set of translation vectors, +V : V × V V is operation of addition, and ·V : R × V V is operation of multiplication on scalars. Points, i. e., elements

of E, and translation vectors, i. e., elements of V , are related via the map

vec : E × E V, (a, b) 1→ ab := vec(a, b),

which belongs to the structure (1.1). The following Weyl axioms are satisﬁed [17]:

1. for all three points a, b, c E the Chasles’ relation holds:

ab + bc = ac.

2One and the same set X can be endowed with various structures, commonly represented by ordered lists of sets and mappings, that turn it into certain mathematical object. Algebraic, topological, smooth and geometric structures may serve

as example. If Struct1 and Struct2 are structures on the set X then to distinguish between objects, deﬁned by them, we denote these objects as tuples (X, Struct1) and (X, Struct2). For example, topology T and binary operation on X deﬁne two structures: topological, (X, T ), and algebraic, (X, ).

Вестник Самарского университета. Естественнонаучная серия. 2021. Том 27, № 1. С. 81–103

Vestnik of Samara University. Natural Science Series. 2021, vol. 27, no. 1, pp. 81–103 83

2. for arbitrary point a E and arbitrary vector v V there exists a unique point b E, such that

ab = v.

The symbol (·) from (1.1) stands for inner product3 · : V × V R on V, which is bilinear, symmetric and

i=1

positive-deﬁnite functional. Finally, or is the chosen orientation of vector space V, i. e., class [(ei)3

] of

bases under the following equivalence relation: bases (ei)3

and (e )3

, e = Ωj ei, are similarly oriented if

i=1

i i=1 i i

det[Ωij ] > 0. Here [Ωi ] is the transition matrix from basis (ei)3

to basis (e )3

. By choosing an orientation,

j i=1

i i=1

we restrict the set of acceptable bases to the corresponding class.

k=1

Having reference standards (ik )3

, one can construct reference standards for area and volume, which

are represented by 2-vectors ikil, i. e., oriented unit squares, and 3-vector i1i2i3, i. e., oriented unit cube. Moreover, the choice of inner product and orientation imply isomorphism : V V V, the

Hodge star [17, 18], from vector space of 2-vectors to vector space V. This allows to introduce cross-product operation

[·, ·] : V × V V, [u, v] := (u v).

The absolute time is formalized as one-dimensional manifold T, called chronometric. Its elements are instants of time. Appealing to daily observations, we suppose that T can be ordered, from “past” to “future”. Besides spatial reference standards one has another, temporal reference standard. It deﬁnes coordinatization

T R of chronometric manifold T. Assuming that reference standard is ﬁxed, we identify chronometric

manifold with real line R.

#### 2◦. Enhanced physical space. The above mathematical formalization of the physical space E contains

points which have conventional degrees of freedom only. To describe solids with extra degrees of freedom one needs to enhance the structure of E. Possible way is to associate some m-dimensional manifold F with each point of E. To formalize this structure in terms of smooth manifolds, further simple reasoning is needed.

One can deﬁne (3 + m)-dimensional product manifold E × F. The submanifold {xF, which is topologically equivalent to F, serves as manifold of possible extra spatial degrees of freedom associated with place x E. It is preimage of canonical projection prE : E × F E, prE : (x, f ) x, and by this reason, instead of pointing out the collection of all manifolds of the form {x} × F, it is suﬃcient to specify the projection prE. Thus, one arrives at the structure of a bundle (E × F, E, prE, F). Here E × F is enhanced physical space over

the manifold of places E, and the manifold F is collection of extra degrees of freedom. Enhanced physical space and conventional physical space are related via the map prE. Note, that here we take extra degrees

of freedom from one manifold F, although one may associate with each point x E some manifold Fx, and

manifolds Fx and Fy for distinct points x and y may diﬀer. Such the case is beyond of this study.

The structure of manifold F may be rather general. One can consider the following particular cases. In the ﬁrst case F is m-dimensional vector space. In the second case we deal with Lie group (G, ) with group operation . Elements of G are associated with orientations of points. The structure of Lie group allows to

introduce the following map:

E: (E × G) × G E × G, (x, g) E h := (x, gh),

the action of Lie group G on enhanced physical space E × G. It deﬁnes translations along each ﬁber {xG,

i. e., one can obtain other orientations of point x from some given one.

We went into detail on the description of these cases because their diﬀerences are signiﬁcant in the derivation of various models for oriented bodies, namely, micropolar, micromorphic and shell-like solids.

2. ## Material Manifolds

1. ### General Description for Material Manifold

#### 3◦. Body manifold. As it was mentioned at the beginning of the study, we leave the possibility for a material manifold to have the most common non-Euclidean geometry. Its formalization therefore calls for more general considerations, presented below. In what follows we use the concepts of a body, its shapes, and mappings between them. All these notions are introduced within the theory of connections on smooth manifolds [19, 20]. In particular, the body, i. e., a set of material points, is formalized as a smooth manifold

B of dimension n 3 (we refer on it as body manifold) [11, 21]. This manifold does not carry any metric

3Let us dwell into the issue, how to choose inner product (·) and orientation or. Within framework of classical physics it is assumed that one has three rigid reference standards (one for each dimension) and rigid protractor. The reference standards are formalized by ﬁxed three non-coplanar vectors (i1, i2, i3) from V. Upon measurements it is deﬁned that vectors ik have unit length, mutually orthogonal, and form right-hand triad. From formal viewpoint this means that basis

k=1

(ik )3

deﬁnes an inner product, with respect to which it is orthogonal: u · v := δmnumvn for u = uk ik and v = vk ik ; it

k=1

deﬁnes orientation as well: or = [(ik )3 ].

Lychev S.A., Koifman K.G., Digilov A.V. Nonlinear Dynamic Equations for Elastic Micromorphic Solids and Shells

84 Лычев С.А., Койфман К.Г., Дигилов А.В. Нелинейные динамические уравнения для упругих микроморфных тел...

or connection and deﬁnes only topological features of the body. In this regard the body can be viewed as the following structure:

B = (B, TB, DB ), (2.1)

where B is the underlying set, TB is a Hausdorﬀ topology on B, that satisﬁes the second countability axiom, and DB is a smooth structure on the topological manifold (B, TB ). We denote the points of B by uppercase

Fraktur symbols like X, Y, Z. It should be noted that often a structure for B can be derived from the

structure for some shape clearing it of Euclidean geometry. This issue will be discussed in detail further, in section 5.2.

#### 4◦. Geometry on body manifold. Vividly speaking, the body manifold is a trunk of a tree which

branches represent particular spaces. The only question is which branch should be used in modelling. One can obtain the structure of geometric space by adding Riemannian metric, or connection, or volume form, or some of their combination, to the basic structure (2.1). Table 1 contains examples of geometric spaces4, commonly used in geometric continuum mechanics [22–24].

#### Geometric spaces over B

Table 1

 Space Structure Primary ﬁelds Secondary ﬁelds Riemann space (B, TB, DB, gB, ∇B, dVB ) gB ∇B , dVB Weitzenb¨ock space (B, TB, DB, gB, ∇B, µB ) H, µB gB , ∇B Weyl space (B, TB, DB, gB, ∇B, µB ) gB , ν, µB ∇B

On Table 1 the ﬁeld gB corresponds to Riemann metric, and the ﬁeld B corresponds to aﬃne connection. The body manifold B is supposed to be oriented [25] and its volume form is denoted by µB . The column

“Primary ﬁelds” contains ﬁelds that can be prescribed from some physical reasons that don’t depend on the structure of geometry. The latter column, “Secondary ﬁelds”, contains ﬁelds, which can be derived from the primary ones and geometric properties of smooth manifolds. Note that if the space is Riemannian, then the

volume form µB = dVB is determined by metric as dVB = det gB dX1 ∧ · · · ∧ dXn. The connection B is

Levi-Civita connection, being also determined by metric. The aﬃne connection and metric of Weitzenb¨ock space are generated by prescribed ﬁeld H of linear transformations. Finally, the aﬃne connection of Weyl space can be completely determined by 1-form ν [26].

2. ### Enhanced Material Manifold

5. Body bundle. In this study we consider bodies, which particles are, in turn, perform themselves as continual sets, consisting of microparticles. With a view to introduce more detailed labelling for constituents of the body, one can endow the body manifold B with ﬁber bundle structure [27, 28] (F B, B, πB, F). Here

F B is a smooth manifold, the total space of the bundle, to which we refer as enhanced body, B is the original manifold of labels of particles, the base of the bundle, πB : F B B is a smooth surjective map, called

B

B

projection, and F is a smooth manifold, called model ﬁber. Its elements serve as labels for microparticles. Submanifold F BX := π1({X}) is the ﬁber over point X. To stay in conventional framework of diﬀerential geometry, the following condition is required to be hold: for each point X B there exist a neighborhood U in B and a diﬀeomorphism Φ : π1(U ) U × F, called local trivialization, such that

prU Φ = πB, (2.2)

π1

where prU : U × F U is the canonical projection of the Cartesian product onto the ﬁrst factor. Thus, diﬀeomorphism Φ has the following representation: Φ(p) = (πB(p), φ(p)), for some smooth mapping φ :

B (U ) F.

The local trivialization property means that enhanced body F B is locally arranged as Cartesian product

B

of some part of B and the model ﬁber F. This allows one to introduce special local coordinates on F B. Indeed, let m = dim F, and let Φ : π1(U ) U × F be local trivialization, where U is also coordinate domain

of some chart (U, σB) on B; here σB : U Rn

is coordinate map. Moreover, let (V, σF) be a chart on F;

B

here σF : V Rm is also coordinate map. Then put O := Φ1(U × V ) π1(U ). The restriction

is homeomorphism and by this reason5

ΦO = Φ|O : O U × V

Σ = (σB × σF) ΦO : O Rn × Rm, Σ(p) = (X1, ..., Xn; f1, ..., fm),

4More detailed description of what metric and connection give to body manifold can be found in paragraph 23of the Appendix.

5Here f1 × f2 : X1 × X1 Y1 × Y2 is designation for product of mappings fi : Xi Yi, i = 1, 2, deﬁned as f1 ×

× f2(x1, x2) := (f1(x1), f2(x2)).

Вестник Самарского университета. Естественнонаучная серия. 2021. Том 27, № 1. С. 81–103

Vestnik of Samara University. Natural Science Series. 2021, vol. 27, no. 1, pp. 81–103 85

is coordinate homeomorphism. Thus, (O, Σ) is a chart on enhanced body F B. We refer to coordinates, induced by this chart, as natural coordinates.

The chart (O, Σ) has the property, that any point p from O can be completely determined by coordinates of the base manifold B and coordinates of the model ﬁber F. Moreover, the coordinate representation

πB = σB πB Σ1 : Rn × Rm Rn

of the projection πB coincides with the canonical projection of Rn × Rm onto Rn.

The structure of general ﬁber bundle is very ﬂexible. One can obtain particular cases by assuming that

model ﬁber F is either vector space or Lie group and imposing the corresponding restrictions on local trivializations. Due to importance of these cases in our study, we discuss them individually. Examples of bundles, listed below, would be used in various theories of micropolar ﬁelds considered in the study.

6. The case of vector bundle. Material vector bundle of rank m [19] is the ﬁber bundle (V B, B, πB, F)

B

B

over B, where the substructure F is m-dimensional real vector space, which elements are orientations of the medium. It is supposed that for every X B the preimage π1({X}) =: V BX is m-dimensional real vector space. Moreover, local trivialization Φ : π1(U ) U × F satisﬁes, beyond the property (2.2), the following requirement: for all Y U , the mapping Φ|V BY : V BY F is isomorphism between vector spaces V BY and {Y} × F = F.

Thus, like in the case of general ﬁber bundle, the whole enhanced body V B is split into disjoint union V B = ⨿xB V BX of ﬁbers V BX, which contain all possible material orientations of particle X. Since ﬁbers V BX are linearly isomorphic to typical ﬁber F, i. e., V BX = F, in reasonings one can replace them by

elements of the model ﬁber.

i=1

The bundle structure allows one to introduce natural coordinates on V B. Choose some coordinates (Xi)n

i=1

on the base manifold B and choose some basis (ei)m

for F. Then every point of V B can be completely

characterized by (m + n)-tuple (X1, ..., Xn; v1, ..., vm) Rm+n.

7. The case of principal bundle. Material principal bundle with structure group G [29] is a structure (P B, B, πB, G, , B), in which P B, B, and πB : P B B are, as earlier, enhanced body, body manifold, and projection. The substructure (G, ) is Lie group with binary operation , and B: P B × G P B is

a smooth right action of the group G on manifold P B. The following properties are assumed:

1. The action preserves ﬁbers of πB:

p P B g G : πB(p B g) = πB(p).

B

B

2. For every point X B there exists a neighborhood U in B and a diﬀeomorphism Φ : π1(U ) U × G, such that Φ(p) = (πB(p), φ(p)), where the mapping φ : π1(U ) G is equivariant, that is,

B

p π1(U ) g G : φ(p B g) = φ(p)g.

Thus, material principal bundle (P B, B, πB, G, , B) is particular case of material ﬁber bundle

(F B, B, πB, F). The typical ﬁber is “linear” in the sense that it is represented by Lie group (G, ), and

Lie group structure is aligned with ﬁber bundle structure through group action. In particular, manifold P B

can be endowed with natural coordinates (X1, ..., Xn; g1, ..., gm). Here m is dimension of G. With respect to

natural coordinates projection, as well as group action, can be represented in laconic form. Like for general case of ﬁber bundle, the coordinate representation of projection πB : P B B is canonical projection:

πB : Rn × Rm Rn,

B

π (X1, ..., Xn; g1, ..., gm) = (X1, ..., Xn).

The coordinate representation of action B is the map

(X1, ..., Xn; g1, ..., gm)B(h1, ..., hm) = (X1, ..., Xn; (g1, ..., gm)(h1, ..., hm)),

where : Rm × RmRm is coordinate representation of group operation .

8. Frame bundle. If general linear group GL(n; R) is chosen as structure group, then one arrives at

material frame bundle (LB, B, πB, GL(n; R), ., B), in which total space is manifold over the set

i=1

LB = ⨿ LXB, LXB = {(ei)n

i=1

| (ei)n

is a basis for TXB},

XB

the projection πB : LB B is deﬁned as

and action B: LB × GL(n; R) LB is

i=1

πB(X, (ei)n

) := X,

(X, (ei)n

) B [Ωi ] := (X, (Ωj ej )n ).

i=1 j

i i=1

Lychev S.A., Koifman K.G., Digilov A.V. Nonlinear Dynamic Equations for Elastic Micromorphic Solids and Shells

86 Лычев С.А., Койфман К.Г., Дигилов А.В. Нелинейные динамические уравнения для упругих микроморфных тел...

Here (.) denotes matrix product. Local trivialization Φ can be deﬁned as follows. Choose some chart (U, σ)

i=1

on body manifold B and let (i)n

be coordinate frame with respect to this chart. Then for any basis

(ei)n

LXB, X B, one has decomposition ei = σj j |X, where [σi ] GL(n; R). Introduce the

i=1 i j

B

mapping Φ : π1(U ) U × GL(n; R) by equality

i=1

Φ(X, (ei)n

) := (X, [σij ]).

In particular, φ : π1(U ) GL(n; R) acts as φ(X, (ei)n

) = [σi ], which implies the equivariance property.

B i=1 j

The dimension of enhanced body LB is dim LB = n + n2, and the representation on natural coordinates is

2

of the form

LB p 1→ (X1, ..., Xn; g11, ..., gnn) Rn × Rn .

3. ## Kinematic Description

1. ### Conventional Kinematics

#### 9◦. Conventional conﬁgurations and deformations. Body manifold is not directly observable to us, inhabitants of three-dimensional Euclidean space E. One can observe shapes, that are the images of

conﬁgurations. By a conﬁguration [11] we mean a smooth embedding6 κ : B E. Thus, we formalize

shapes as images Sκ = κ(B). Note, that the image Sκ may not coincide with the whole physical space E,

even if B has three dimensions. By this reason, κ is not invertible. To remedy the issue, one can deﬁne a new map κ : B Sκ, such that κ (X) = κ(X). Then the new map is invertible, as desired.

Let κR, κ : B E be conﬁgurations which images are shapes SR and S respectively. The change of

1

R

shapes is characterized by mapping γ := κ κ

: SR S, to which we refer as deformation. Relations

between the conﬁgurations and deformations are illustrated on Fig. 1.

Fig. 1. Relations between the conﬁgurations and deformations

On Fig. 1, SR = κR(B) and S = κ(B). The assignments ιSR : SR 'E and ιS : S 'E are inclusion maps7.

10. Motion. Suppose that we have ﬁxed some reference shape SR. We deﬁne motion of the body as a family {γt}tT of deformations γt : SR St, which index set T R is an interval. This is conventional

viewpoint, presented in such monographs as [31]. It is also convenient to represent the motion in terms of one smooth mapping γ : SR × T E deﬁned as γ(X, t) := γt(X).

Another viewpoint on motion uses family of shapes instead of one global shape. The need of such generalization arises in the case, when body doesn’t have reference shape with desired properties (in particular, stress-free), but there is a family of shapes, locally satisfying the required properties. This viewpoint is discussed in detail in section 5.1.

It is appropriate to note about the following possibility for position deﬁnition in physical space. Aﬃne structure of physical space E allows to describe points through translation vectors (“vectorization”). To do

this, one chooses some origin o E and deﬁnes invertible ﬁeld p : E V of radius-vectors p(x) := ox. If we

denote radius-vectors of points X from reference shape SR by X, i. e., X = p(X), then we can represent mapping γ in vectorial form:

χ(X, t) := p[γ(p1(X), t)].

6A smooth embedding from one smooth manifold M to another smooth manifold N is a mapping κ : M N , which is

1) smooth, i. e., its coordinate representation is smooth in the sense of conventional Calculus; 2) at every point of M the rank of the Jacobi matrix formed upon coordinate representation of κ is equal to dim M ; 3) κ is a homeomorphism onto its image [25]. By requiring from conﬁguration to be an embedding we exclude undesired situations like self-intersections.

7Apart formal mathematical need to use inclusion maps, there is a physical need as well. Suppose that body manifold is two-dimensional. Then one can describe its stress-strain state by means of inner geometry, i. e., using two-dimensional vectors, tensors, etc. [30]. Meanwhile, to relate ﬁelds of strains and stresses with their three-dimensional counterparts, one needs to use inclusion map. Contact of three-dimensional body with two-dimensional membrane may serve as example of such the problem.

Вестник Самарского университета. Естественнонаучная серия. 2021. Том 27, № 1. С. 81–103

Vestnik of Samara University. Natural Science Series. 2021, vol. 27, no. 1, pp. 81–103 87

The mapping χ : VR × T V, where VR := p(SR), is used in further considerations as well. Note the distinction between mappings γ and χ. The former mapping transforms points to points and its deﬁnition doesn’t require Euclidean structure of the ambient manifold, i. e., it can be generalized to arbitrary manifolds. Meanwhile, the latter mapping transforms radius-vectors to radius-vectors an by this reason exists in Euclidean manifold only.

2. ### Enhanced Kinematics

11. Enhanced conﬁgurations and deformations. In a manner similar to body manifold we suppose that physical manifold E is endowed with ﬁber bundle structure (F E, E, πE, F) with the same model ﬁber F

as for the body (in 2 we have considered trivial bundle; here we suppose that this bundle may be of general

form). Total space of the bundle F E plays role of enhanced physical space. Its k-dimensional submanifolds (here k = dim F B) generalize conventional shapes of the body; thus, shape of the body is not only region of places, but also a collection of extra degrees of freedom.

Following the general methodology of continuum mechanics, we consider conﬁgurations and deformations

as embeddings of one smooth manifold into another smooth manifold. By this reason, we deﬁne enhanced conﬁguration as smooth embedding [15, 16] F κ : F B F E of enhanced body F B to enhanced physical space F E. We suppose that this mapping preserves ﬁbers, i. e., there exists an embedding κ : B E of

the body manifold to the physical space, the conventional conﬁguration, such that

κ πB = πE F κ. (3.1)

If one introduces natural coordinates (X1, ..., Xn; f1, ..., fm) on enhanced body and natural coordinates (x1, x2, x3; f 1, ..., fm) on enhanced physical space, then coordinate representation of F κ would have the form

where κ

Fκ(X1, ..., Xn; f1, ..., fm) = (κ(X1, ..., Xn); τ (X1, ..., Xn; f1, ..., fm)),

: (X1, ..., Xn) 1→ (x1, x2, x3) is a coordinate representation of conﬁguration κ, and τ :

(X1, ..., Xn; f1, ..., fm) 1→ (f 1, ..., fm) characterizes those part of embedding (in coordinate representation),

which with material degrees of freedom associates their spatial counterpart.

Let F κR, F κ : F B F E be enhanced conﬁgurations, to which correspond conventional conﬁgurations

κR, κ : B E via (3.1). Then their images F SR = F κR(F B) and F S = F κ(F B) correspond to shapes

of enhanced material manifold in enhanced physical space. Each of them is smooth submanifold of F E and

one can restrict ﬁber bundle structure from F E to these shapes, i. e., to consider ﬁber bundles

(F SR, SR, πE|F SR , F), (F S, S, πE|F S, F),

where SR = κR(B), S = κ(B) are n-dimensional submanifolds of physical space E, that represent conventional

shapes of the body. Of course, they are, in some sense, projections of enhanced shapes F SR and F S, and by this reason the information about extra variables is lost. Mappings πE|F SR : F SR F SR and πE|F S : F S F S

are restrictions of the projection πE to the corresponding shapes.

Bearing in mind the intuitive meaning of deformation as change of shapes we deﬁne composition8

R R

= F κ F κ1 : F S F S,

to which we refer as enhanced deformation. If κR, κ : B E are conventional conﬁgurations that correspond

1

R

to enhanced ones, then the composition γ = κ κ

: SR S is the conventional deformation, i. e., change

of places. This is exactly what one observes in physical space if not using experimental setup that allows

to identify extra degrees of freedom. Enhanced deformation and conventional deformation γ are related by formula analogous to (3.1):

γ πE|F SR = πE|F S Fγ. (3.2)

Relations between shapes, conﬁgurations, deformations and their enhanced counterparts are shown on Fig. 2.

For coordinate representation of enhanced deformation choose natural coordinates9 (X1, ..., Xn; F 1, ..., Fm) on reference shape F SR and natural coordinates (x1, ..., xn; f 1, ..., fm) on actual shape F S. Then coordinate representation of deformation has the form

Fγ(X1, ..., Xn; F 1, ..., Fm) = (γ(X1, ..., Xn); ε(X1, ..., Xn; F 1, ..., Fm)).

8Like in the case of conventional conﬁgurations (see 9), the mappings F κR and F κ are not invertible in general. To deﬁne deformation as change of shape, we restrict the codomains of these mappings to the images. So obtained maps are denoted with hat signs.

9In these tuples, (X1, ..., Xn) are coordinates on the shape SR and (x1, ..., xn) are coordinates on the shape S; both

shapes are considered as manifolds on its own right.

Lychev S.A., Koifman K.G., Digilov A.V. Nonlinear Dynamic Equations for Elastic Micromorphic Solids and Shells

88 Лычев С.А., Койфман К.Г., Дигилов А.В. Нелинейные динамические уравнения для упругих микроморфных тел...

Fig. 2. Enhanced conﬁgurations and deformations

1→

Here γ : (X1, ..., Xn) (x1, ..., xn) is coordinate representation of deformation γ, and

...,

ε(X1, Xn; F 1, ..., Fm) 1→ (f 1, ..., fm) is the mapping, which assigns with reference values of extra degrees

of freedom their actual values.

R

n

Note, that in general, mapping ε : m+n R

doesn’t have counterpart among “point to point” mappings

F SR F. Meanwhile, there is a particular case, when it has such the counterpart. Suppose that the physical

E

bundle (F E, E, πE, F) is trivial, i. e., local trivialization Φ : π1(U ) U × F is global: U = E. Then one can replace F E by E × F and πE by canonical projection prE : E × F E. By this reason, one can consider enhanced deformation as mapping : SR × F S × F, and the condition (3.2) implies, that there exists a mapping ε : SR × F F, such that

(X, F ) = (γ(X), ε(X, F )). (3.3)

One sees, that in this case deformation of solids with extra degrees of freedom is characterized by two

deformations

γ and ε. Deﬁned above enhanced conﬁgurations and deformations are mathematically close to the generalized introduced in [15].

#### 12◦. Sections. Mathematical structure based on the idea of embedding of an enhanced material manifold

into an enhanced physical space is rather general. In particular, it allows for continual set of possible orientations, associated with a material point. To derive conventional model of oriented solid, one has to limit this set in a certain way. One possible approach is to deﬁne a section over the bundle, thereby choose a single reference orientation for each material point.

Let (F B, B, πB, F) be material body bundle. A smooth section of this bundle is a smooth mapping

o : B F B, such that πB σ = IdB. The latter equality means that σ(X) F BX for each X B. Choosing section σ, we arrive at the structure (F B, B, πB, F, σ), to which we refer as oriented solid. In what follows

we suppose that σ is an embedding. Then conﬁguration of oriented solid is formalized as an embedding

F κσ := F κ σ : B F E,

where F κ : F B F E is some enhanced conﬁguration. With regard to deformations, we obtain that the equality (3.3) takes the form

U (X) = (γ(X), ε(X, UX )),

where γ is conventional deformation, while SR X 1→ UX F is section of ﬁber bundle (F SR, SR, πE|F SR , F), induced by section σ. Thus, we arrive at the following representation for deformation of micromorphic solid:

U

(X) = (γ(X), ε(X)),

and it means that deformation is completely characterized by mappings γ : SR S and ε : SR F.

endix),

Similarly, when one deals with material vector bundle (see paragraph 24 of the App the formula

(5.10) for enhanced deformation reduces to

V γU (X) = (γ(X), M(X, UX )),

or, denoting MU (X) := M(X, UX ), to

V γU (X) = (γ(X), MU (X)). (3.4)

Here X 1→ MU (X) is ﬁeld SR F of actual directors.

Вестник Самарского университета. Естественнонаучная серия. 2021. Том 27, № 1. С. 81–103

Vestnik of Samara University. Natural Science Series. 2021, vol. 27, no. 1, pp. 81–103 89

There is an issue related with the above considerations. As it is assumed, with each point X of reference

X

shape SR one associates some vector dR

from F. A deformation causes change of shape and then particles

X

of body occupy some shape S. Orientations dR , in turn, deform to orientations dx, associated with points x from S. Meanwhile, this picture depends on observer: diﬀerent observers may choose diﬀerent reference orientations, which can transform to diﬀerent actual orientations as well. To make considerations objective, one

needs to consider relations between orientations instead of the orientations themselves. But these relations are hidden in (5.10). Indeed, for each X SR one has linear partial mapping10 MX Lin(F; F). This mapping transforms reference orientation U to actual orientation u = MX [U ]. If we deal with sections, then

formula (3.4) should be replaced by

V γU (X) = (γ(X), MX [UX ]). (3.5)

Here U : X 1→ UX is ﬁxed section on reference shape, i. e., ﬁeld of reference directors. Thus, in fact, deformation is completely characterized by mapping γ : SR S, the conventional deformation, and ﬁeld M : SR Lin(F; F) of linear maps, which relate reference and actual orientations.

Note, that since shapes SR and S are equitable, the enhanced deformation V γ is a diﬀeomorphism. This implies, in particular, that linear map MX from formula (3.5) is invertible, as well as conventional

deformation γ.

4. ## Action Integral

1. ### Action Integral of Conventional Elasticity

13. Classical Lagrangian. The main focus of this study is the action integral, which will be denoted with the symbol J . We believe that J can be deﬁned as the integral of a smooth scalar function of a certain set of independent and dependent variables (ﬁelds). All follow-up analysis depends upon the choice of these variables and their geometric structure (scalar, vectorial or tensorial).

For classical (non-polar) continuous medium action integral can be written in the following form:

t2

∫ ∫

J = L dV dt, L = L (X, t, χ, χ, χ˙ ), (4.1)

t1 SR

where the integration is performed over the arbitrary time interval [t1, t2] and arbitrary reference domain

SR E. In this expression L is a Lagrangian density per unit of the volume of reference shape, X is the

position of material particle in reference shape identiﬁed with radius-vector, t is a time variable, χ = χ(X, t)

represents positions of material particles in actual shape, and χ is position gradient11.

It should be noted that, generally, the dimension of SR may be less than for E. Then instead of using three-dimensional ﬁeld χ one needs to use two-dimensional surface gradient sχ introduced in [30]. This

will play a signiﬁcant role in the derivations for shell-like bodies. Above all, let us make no assumption about the property of reference shape to be free from stresses.

2. ### Solids with Orientations

14. Preliminary reasonings. Even in the framework of conventional elasticity one has to take into account that every deformation within inﬁnitesimal neighborhood of any point X from reference shape has “rotation” part. During deformation each inﬁnitesimal ﬁber emanating from X stretches and rotates according

to Cauchy’s polar decomposition theorem [18]: deformation gradient χ can be represented as the product

of an orthogonal tensor O and symmetric tensor V , i. e.,

χ = V ·O, OT = O1, V = V T = (χ·(χ)T)

1/2

. (4.2)

The ﬁrst equality of (4.2) implies that the tensor ﬁeld O is related with actual positions χ of particles:

1/2

O = V 1 ·χ = (χ·(χ)T)

·χ. (4.3)

Formula (4.3) shows that rotations of inﬁnitesimal ﬁbers are embodied in the deformation gradient and are not independent from χ: elementary volume, associated with X, stretches and rotates as the whole; after

10Hereafter the symbol Lin(U ; V ) denotes vector space of linear maps U V .

∂χk

11Here χ is considered as indivisible symbol; χ =

∂Xl

ik il.

Lychev S.A., Koifman K.G., Digilov A.V. Nonlinear Dynamic Equations for Elastic Micromorphic Solids and Shells

90 Лычев С.А., Койфман К.Г., Дигилов А.В. Нелинейные динамические уравнения для упругих микроморфных тел...

deformation these volumes again constitute a connected region. The natural step is to introduce another rotations, independent from χ, and to associate them with “microparticles”.

Note, that rotation of inﬁnitesimal ﬁber, represented by translation vector dX, can be determined within Rodrigues’ rotation formula12 [33]:

dX = cos θdX + sin θ(k × dX) + (k · dX)(1 cos θ)k. (4.4) Here k is a unit vector directed along the rotation axis and θ is rotation angle. Thus, instead of dealing with second-rank tensor ﬁeld O one can deal with pair (θ, k), which consists of rotation angle and axis

direction. To this end, we introduce vector ﬁeld φO := θk, which we would use as a basis for the future generalization.

Let us discuss relation between rotation tensor O and pair (θ, k) in more detail. The following formula allows to obtain O upon pair13 (θ, k):

O = cos θ 1 + (1 cos θ)k k + sin θK,

where K is second-rank tensor such that14 Kv = k × v. Conversely, suppose that O is known. To determine pair (θ, k), one needs to do the following steps [33]:

1. Solve eigenvalue problem Ok = k under condition k= 1. This gives k.

tr(KO)

2. The angle θ is obtained from the solution of equation sin θ = 2 .

#### 15◦. Micropolar model. Within the micropolar theory of elasticity, in contrast with conventional

(symmetric) theory, it is assumed that the orientation of the elementary volume is determined independently. One introduces independent vector ﬁeld φ = φ(X, t) and considers it as an additional generalized coordinate. In this case the density of Lagrangian L is a function of material coordinates X, time t, ﬁelds χ(X, t), φ(X, t) and their ﬁrst gradients, i. e.

L = L (X, t, χ, χ, χ˙ , φ, φ, φ˙ ). (4.5)

Note, that vector ﬁeld φO , associated with orthogonal tensor O from (4.2) through Rodrigues’ formula, is independent from φ. The independent ﬁeld φ can be regarded as a section of vector bundle (see

paragraph 12).

#### 16◦. Generalized micropolar model. One can give the following geometric interpretation of the

kinematics of the micropolar continuum. The non-deformable orthonormal trihedrons (triples of non-coplanar vectors) are associated with points of the medium, that are oriented along the base vectors in the reference state, and when the medium is deformed, change its spatial orientation. A more general interpretation may be given, when one allows these trihedrons to distort.

k=1

Suppose that one has ﬁxed some orthonormal basis (ik )3

k=1

. Then for every trihedron (dk )3 of

non-coplanar vectors there exists invertible linear map L : V V, such that L[ik ] = dk . In dyadic representation, L = dk ik . Thus, instead of deformable trihedrons one can consider invertible linear maps,

which are elements of general linear group. Meanwhile, the following issue arise here. The physical space E is too narrow to contain simultaneously places of particles and linear operators, associated with them. If we think of shapes, as of sets, which elements are pairs (X, L), then these sets cannot be deﬁned as

2

12One can obtain another version of formula (4.4). If put τ = tan θ , then from (4.4) directly follows that

dX =

1

1 + τ 2

[(1 τ 2)dX + 2τ (k × dX) + 2τ 2(k · dX)k] .

After some algebraic transformations one gets [32]

dX = dX +

1

4

1 + 1 ϑ2 ϑ ×

(

dX +

1 )

2 ϑ × dX ,

where ϑ = 2τ k, ϑ = ϑ= 2τ . The vector ϑ is usually called as ﬁnite rotation vector, although its magnitude have no common with real rotation angle.

s=1

13If some orthonormal basis (is)3

in V has chosen, then, according to (4.4), operator O is represented by matrix

1

(1cos θ)k2 +cos θ k1k2(1cos θ)k3 sin θ k1k3(1cos θ)+k2 sin θ

2

3

[O]= k1k2(1cos θ)+k3 sin θ (1cos θ)k2 +cos θ k2k3(1cos θ)k1 sin θ , k1k3(1cos θ)k2 sin θ k2k3(1cos θ)+k1 sin θ (1cos θ)k2 +cos θ

where k1, k2, k3 are components of k.

14In matrix form K is given by

[K] =

0 k3 k2

k3 0 k1 .

k2 k1 0

Вестник Самарского университета. Естественнонаучная серия. 2021. Том 27, № 1. С. 81–103

Vestnik of Samara University. Natural Science Series. 2021, vol. 27, no. 1, pp. 81–103 91

images of embeddings to physical space. Usually one employs the artiﬁcial construction, when “shape” is a region of physical space, while linear operators are considered separately, being associated with each point of this region. This reasoning is suitable for conventional problems, but hardly suitable to situations when one considers incompatible deformations. To use methodology of non-Euclidean shape one needs to introduce somehow shapes, that contain places and linear operators simultaneously.

To ﬁx the issue it seems appropriate to consider enhanced physical space LE, which is the total space of frame bundle (LE, E, πE, GL(3; R), ., E) discussed in paragraph 8. In this case conﬁguration can be deﬁned in usual manner as embedding κ : B LE and images of conﬁgurations are shapes of the body.

This viewpoint is systematically developed in [34].

Any conﬁguration κ : B LE deﬁnes an embedding κ : B E, conventional conﬁguration, as follows: κ := πE κ. Thus, one indeed deals with enhanced picture, which contains all information about places of points. In the particular case, when LE = E × GL(3; R), one has the following representation for κ: κ(X) =

= (κ(X); L(X)). Here κ : B E is conventional conﬁguration and L : B GL(3; R) corresponds to ﬁeld

of trihedrons.

If one chooses natural coordinates on LE, generated by coordinate map φLE : LE R3 × R9, and some coordinates on the body manifold B, generated by coordinate map φB : B Rn, then coordinate representation of conﬁguration κ : B LE is composition

κ = φLE κ φ1,

κ(X1, ..., Xn) = (κi(X1, ..., Xn); [L]ij (X1, ..., Xn)),

B

where i, j = 1, 2, 3 and κ : Rn R3, [L] : Rn R9 are some mappings, uniquely deﬁned by κ. Moreover,

κ is coordinate representation for κ:

B

κ = φE κ φ1, where φE : E R3

is coordinate map for physical

space.

With a view to describe micropolar body with incompatible deformations in the framework of geometric continuum mechanics one needs to enhance the body manifold as well, since origin of incompatibility is more general here, as it involves particles and trihedrons, while body manifold contains information of particles only.

Thus, one arrives at the frame bundle (LB, B, πB, GL(n; R), ., B). In this case embeddings Lκ : LB LE

represent enhanced conﬁgurations15 (see paragraph 25of the Appendix). Within suitable coordinates, Lκ

is represented as mapping

LB

Lκ = φLE Lκ φ1 , where φLB : LB Rn × R9 is coordinate map on total

space LB. All above reasonings are illustrated on Fig. 3. The dashed lines from LB illustrate that this part

Fig. 3. Embedding of generalized micropolar body into enhanced physical space

of the ﬁgure corresponds to the case of incompatible deformations. One can obtain the conventional case of micropolar continuum if replaces general linear group GL(3; R) by special orthogonal group SO(3).

#### 17◦. Micromorphic model. In reasonings of paragraph 16◦we have associated with each particle some invertible linear transformation L : V → V, which is a counterpart for deformable trihedron of directors.

15It would be too restrictive to assume that embedding Lκ is related with conﬁguration κ : B LE as Lκ = κ πB. Indeed, in this case the image Lκ(LB) would be a surface in LE, not an open set.

Lychev S.A., Koifman K.G., Digilov A.V. Nonlinear Dynamic Equations for Elastic Micromorphic Solids and Shells

92 Лычев С.А., Койфман К.Г., Дигилов А.В. Нелинейные динамические уравнения для упругих микроморфных тел...

Within this formalism, a motion of such the solid is represented by pair (κ(X, t), L(X, t)), in which κ : B×

×]t1, t2[E is family of conventional conﬁgurations and L : B×]t1, t2[GL(3; R) is family of linear

maps. Meanwhile, one can adopt another point of view. First, one can choose reference directors from some

1. imensional Euclidean vector space F instead of choosing them from translation vector space V. In actual picture one still observes trihedrons of vectors from V. Second, one can suppose that relation between abstract directors, i. e., vectors from F, and real directors, i. e., vectors from V, is linear. The corresponding linear

transformation is deﬁned by a second-rank tensor X = X (X, t) Lin(F; V), operating in a m-dimensional

Euclidean vector space F. This tensor is referred to as the tensor of microdeformations16.

In view of the above, the density of Lagrangian can be described as:

X

L = L (X, t, χ, χ, χ˙ , X , ∇X , ˙ ). (4.6)

It seems appropriate to mention about relation between ﬁeld X : SR Lin(F; V), the argument of Lagrangian density (4.6), and ﬁeld M : SR Lin(F; F) from (3.5). Values of both ﬁelds are linear maps

that transform reference orientations to actual orientations. It is assumed here that orientations are vectors from some abstract vector space F. Meanwhile, in conventional theories of elasticity orientations are considered to be visible deformable trihedrons of translation vectors, i. e., elements from V. By this reason, we suppose

the following relation between ﬁelds X and M: X = In M. Here In : SR Lin(F; V) is ﬁeld of linear

mappings, that perform inclusion of abstract orientations from F into visible elements from V. The form of

this ﬁeld depends on physical nature of the model.

There is another way of justifying microdeformations, when one considers particles as aggregates, that consist of microparticles [35]. The body manifold B contains labels of particles and its structure is too narrow to ﬁt microparticles also. By this reason, suppose that each particle can be formalized as 3-dimensional Euclidean vector space F, which, in turn, contains labels of microparticles, and introduce material vector

bundle (V B, B, πB, F) over B. Here each ﬁber V BX over X B represents single cell, associated with

label X. The distinction between particles and microparticles requires to introduce two types of conﬁgurations. The ﬁrst type is conventional conﬁguration represented by embedding κ : B E, X 1→ κ(X), associated

X

with particles. The second type, for X B, is an embedding κF

κF

X

: F V, U 1→ κF (U ). We suppose

X X can be interpreted as assignment, that

returns radius-vectors of microparticles within center of mass of the single cell X.

X

Note, that one cannot think of κF

as of embedding into physical manifold E, since in this case there would

not be clear distinction between particles and microparticles. Instead of this, one can treat microparticles as inﬁnitesimal translation vectors emanating from center of mass of the cell κ(X).

As it was mentioned, one cannot consider positions of particles and microparticles in physical space E simultaneously. Meanwhile, using vector bundles again, one can introduce vector bundle V E over E with model ﬁber V. Then place and “microplace” can be uniﬁed into a single point from V E. Both mapping κ

X

and the family (κF )XB of mappings induce global map V κ : V B V E, i. e., enhanced conﬁguration, from manifold V B to manifold V E. In what follows we suppose that V B = B × F and V E = E × V, i. e., vector

bundles over B and E are trivial.

X

To get conventional picture, described by Mindlin [35], one can provide the following modeling. As enhanced conﬁguration is completely deﬁned by mappings κ and κF , we can deal only with them. Introduce

the mapping κ : B × F E by the relation

X

κ(X, U ) := κ(X) + κF (U ).

X

This mapping returns ﬁctive position of microparticle, equal to the sum of position of the corresponding cell and relative position of microparticle within the cell. Consider ﬁrst order Taylor expansion of κF (U ):

κF F

X(U ) = κX(0) + X X(U ) + o(U ). Here X X : F V is linear map. Taking into account the calibration

condition, one gets

κ(X, U ) := κ(X) + X X(U ) + o(U ).

Thus, with up to o(U ), the ﬁctive position of microparticle is deﬁned by pair (κ, X ), where κ : B E

is conventional conﬁguration and X : B × F V is tensor ﬁeld of microdeformations. Therefore, the density

of Lagrangian has the form (we identify body manifold with some of its shapes):

X

L = L (X, U , t, χ, χ, χ˙ , X , ∇X , ˙ ).

16Let us make more clear the distinction between models discussed in paragraphs 16and 17. Linear maps in them play diﬀerent roles:

• In 16linear map is a counterpart for directors triad;

I=1

• In 17 linear map relates family (f I )m

i=1

of abstract reference directors with family (di)3

of their spatial counterpart.

Вестник Самарского университета. Естественнонаучная серия. 2021. Том 27, № 1. С. 81–103

Vestnik of Samara University. Natural Science Series. 2021, vol. 27, no. 1, pp. 81–103 93

Believing that Lagrangian density is independent of U , i. e., cells are homogeneous and isotropic, we arrive at (4.6). If the cell, above that, is absolutely rigid, the equations are reduced to the equations of the micropolar continuum (4.5).

Reasonings, provided so far, are illustrated on Fig. 4. On the ﬁgure: φF, φV B, φB, φV and φE are

Fig. 4. Embedding of micromorphic body into physical space

coordinate mappings of manifolds indicated in lower indices. Mappings

X

κ F , κ

and

Vκ are coordinate

X

representations of the corresponding mappings κF , κ and V κ. The mapping ι is inclusion map. The word “model” on the right upper part of the ﬁgure indicates the transition from enhanced conﬁguration V κ to ﬁctive conﬁguration κ.

#### 18◦. Shell models. Although there are many distinct models of shells, the ﬁber bundle formalism allows

to think of these models as one. To this end a shell can be formalized as material ﬁber bundle (F B, B, πB, F) where total space F B is smooth manifold of dimension 3 and base B has dimension 2. The manifold F B contains all labels of shell particles, while manifold B contains labels of points from reduction surface. Every ﬁber F BX, one-dimensional manifold, consists of material transversal elements of the shell over point X.

In physical space E one observes shape S of the shell, the image of embedding κ : F B E, and shape S of the body B, the image of embedding κ : B E. In general, κ and κ are independent and shape S may not be contained in S. We refer to the shape S as reduction surface. Let φF B : F B R3 be coordinate map, that returns natural coordinates of points from F B. That is, for p F B one has triple (X1, X2, ξ) R3,

where (X1, X2) are coordinates on the manifold B and ξ is “transversal” coordinate, i. e., coordinate on model ﬁber F. Let κ : F B S be mapping, that obtained from κ by restriction of codomain (see 9). Then

deﬁne coordinate map φ

: S R3 as φ = φ

• 1. Thus, points from shape S are represented by tuples

(X1, X2, ξ). S

S F B κ

One arrives at the following general vectorial relation between positions of points from S and natural coordinates on F B:

r = f (X1, X2; ξ), (4.7)

where f depends on the model chosen for description of conﬁguration. The semicolon separates coordinates on reduction surface from transversal coordinate. If one assumes that transversal elements are associated with normal elements, then one gets the following approximation for conﬁguration κ: r = ρ(X1, X2) + ξ n(X1, X2), where ρ(X1, X2) is position vector of reduction surface and n is unit normal vector ﬁeld.

In general, the choice of particular approximation for (4.7) is provided in two stages. One pulls out the reduction surface S from S and considers it as base manifold for some vector bundle (V S, S, πS, F1) with

model ﬁber F1, an m-dimensional vector space. Elements of the space F1 serve as objects, which are used to approximate partial ﬁeld f (X1, X2; ·) : ξ 1→ f (X1, X2; ξ). In particular, one may consider some functional

Hilbert space. Choosing some basis, e.g., Legendre polynomials, one then restricts its attention in linear

Lychev S.A., Koifman K.G., Digilov A.V. Nonlinear Dynamic Equations for Elastic Micromorphic Solids and Shells

94 Лычев С.А., Койфман К.Г., Дигилов А.В. Нелинейные динамические уравнения для упругих микроморфных тел...

span of dimension m. This is what exactly F1 is. In the next step one combines ﬁelds f (·; ξ) : (X1, X2) 1→

f (X1, X2; ξ) and f (X1, X2; ·) : ξ 1→ f (X1, X2; ξ) into one global ﬁeld from R3 to E. The image of this

ﬁeld is denoted by S. The obtained ﬁeld is the result of approximation. It is expected that after modeling

S = S, but it is worth to note, that the shape S is not the image of embedding from F B.

k=1

Suppose that (ek )

is some Schauder basis of Hilbert space L2[1, 1] (not necessary orthonormal).

Choosing m N, we replace f from (4.7) by additive decomposition r = ρ(X1, X2) + ρ1(X1, X2; ξ), where

1 k=1

ρ (X1, X2; ξ) = m ck (X1, X2)ek (ξ). Thus,

m

r = ρ(X1, X2) + ck (X1, X2)ek (ξ).

k=1

Here ck (X1, X2) are vectorial coeﬃcients of the expansion, which components deﬁne elements from vector space F1.

All reasonings provided above are illustrated on Fig. 5. On the ﬁgure, symbols φ with lower indices

Fig. 5. Shell modeling scheme

denote the corresponding coordinate maps. Mappings ι are inclusion maps and “model” denotes the process of transition from manifold V S to the manifold S.

5. ## Incompatible Deformations

1. ### Local Discharging

19. Family of shapes. In conventional elasticity one assumes that body B has stress-free shape SR in Euclidean physical space E. Like all other shapes, this shape is considered as a subspace of physical manifold, with metric and connection induced from it:

SR = (SR, TE |SR , DE |SR , gE |SR , E |SR ).

Here SR is the underlying set of the shape, while other elements of the tuple are, respectively, topology, smooth structure, metric and connection of Euclidean space E induced to SR. In this case deformation is a mapping

γ : (SR, TE |SR , DE |SR , gE |SR , E |SR ) (S, TE |S, DE |S, gE |S, E |S ),

between two open (hereafter we consider the case dim B = 3) submanifolds of E with Euclidean geometry. If solid has defects, e.g., dislocations, disclinations, etc., then the assumption of existence of global Euclidean stress-free shape fails. Before stating weaker assumption, let us provide auxiliary consideration.

Вестник Самарского университета. Естественнонаучная серия. 2021. Том 27, № 1. С. 81–103

Vestnik of Samara University. Natural Science Series. 2021, vol. 27, no. 1, pp. 81–103 95

Suppose that one have chosen some shape SI in Euclidean physical space E, the image of conﬁguration κI : B E. We refer to it as intermediate shape. Moreover, let SR and S be, respectively, reference and actual shapes in E, which are images of conﬁgurations κR, κ : B E respectively. The shape SR is assumed

to be stress-free, while SI and S are self-stressed in general. Deﬁne maps

I

1

χ = κ κ

1

I

: SI S, ξ = κR κ

: SI SR,

which are deformations from the intermediate shape to actual and reference shapes respectively. Then ξ

corresponds to relaxation from the shape SI to the shape SR, and the composition γ = χ ξ1 : SR S is

the conventional deformation from stress-free reference shape to the actual shape.

Instead of considering motion γ : SR × [t0, t1] E as a ﬁeld variable, one can choose mappings χ : SI ×

× [t0, t1] E and ξ : SI × [t0, t1] E, deﬁned on SI , to be ﬁeld variables. Thus, the Lagrangian density

in (4.1) is replaced by the following function:

L = L (X, t, χ, χ, χ˙ , ξ, ξ, ξ˙). (5.1) Assuming independence on rigid translations, one can replace (5.1) by

L = L (X, t, χ, χ˙ , ξ, ξ˙). (5.2)

Independence on rigid rotations, in turn, allows to replace (5.2) by

L = L (X, t, B, χ˙ , β, ξ˙). (5.3)

In (5.3) B and β are, respectively, B = χχT and β = ξξT.

Now, bearing in mind the auxiliary consideration, consider the case, when all Euclidean shapes of the

body are self-stressed. We suppose the weaker assumption, to which we refer as principle of local discharging :

there exist a shape SI in Euclidean space and a family (γ(X))X

SI

of deformations γ(X) : SI

S(X) from the

shape SI to some shapes S(X), such that the inﬁnitesimal neighborhood of point γ(X)(Y )|Y =X is stress-free for any X SI .

F (X)

For each deformation γ(X) one can obtain deformation gradient F (X) = γ : SI Lin(V; V), F (X) : Y 1→

Y . Due to the principle of local discharging, tensor ﬁeld F

(X)

satisﬁes the following property: the linear

Y

map F (X)|Y =X : V V transforms inﬁnitesimal neighborhood of point X SI to inﬁnitesimal neighborhood

of point γ(X)(Y )|Y =X , which is stress-free. Synthesize upon the family (F (X))X

mappings as follows:

SI

the new ﬁeld H of linear

Y

H : SI Lin(V; V), X 1→ HX := F (X)|Y =X . (5.4)

According to deﬁnition, for each X SI the linear map HX transforms inﬁnitesimal neighborhood of X to

stress-free state. We refer to the ﬁeld H as ﬁeld of local deformations. Since it is synthesized upon distinct deformation gradients, it is not the gradient of some map ξ : SI E that globally relaxes shape SI to

stress-free state.

Within our considerations, the Lagrangian density (5.1) of conventional elasticity should be modiﬁed to the form

L = L (X, t, γ, γ, γ˙ , H, H˙ ).

Here γ : SI × [t0, t1] E and H : SI × [t0, t1] Lin(V; V) are, respectively, the motion of the shape SI

and time-dependent ﬁeld of local deformations. Both are considered as independent ﬁeld quantities.

#### 20◦. Principle of local discharging: micromorphic kinematics. For deﬁniteness, suppose that we deal

with enhanced physical space and shapes which are modeled as vector bundles. Then enhanced deformation between two shapes is represented by formula (5.10) from Appendix, while micromorphic deformation is represented by formula (3.5):

V γ(X) = (γ(X), M(X)[UX ]),

where γ is conventional deformation and M(X) Lin(F; F) is linear map, which relates reference and actual directors. In this case the principle of local discharging is assumed to be of the form: there exist a shape

SI in physical space E, a family (γ(X))X

SI

of deformations γ(X) : SI

S(X) from the shape SI

E to

some shapes S(X) and a family (M(X))X

SI

of directors M(X) : SI

Lin(F; F), such that the inﬁnitesimal

neighborhood of point V γ(X)(Y )|Y =X = (γ(X)(Y )|Y =X , M(X)(Y )|Y =X ) is stress-free for any X SI .

We utilize the procedure of synthesizing to deformation gradients F (X) = γ : SI Lin(V; V), and to director gradients ∇M(X) : SI Lin(V; Lin(F; F)). This results in ﬁeld (5.4) of local deformations (like in

the case of simple solid), and in ﬁeld

D : SI Lin(V; Lin(F; F)), X 1→ DX = Y M(X)|Y =X ,

synthesized upon gradients of microdeformation tensors.

Lychev S.A., Koifman K.G., Digilov A.V. Nonlinear Dynamic Equations for Elastic Micromorphic Solids and Shells

96 Лычев С.А., Койфман К.Г., Дигилов А.В. Нелинейные динамические уравнения для упругих микроморфных тел...

In the case of micromorphic kinematics the Lagrangian density (4.6) of micromorphic elasticity is modiﬁed to the form

M

L = L (X, t, γ, γ, γ˙ , M, ∇M, ˙

, H, H˙ , D, D, D˙ ).

Here γ : SI × [t0, t1] E, M : SI × [t0, t1] Lin(F; F) and H : SI × [t0, t1] Lin(V; V), D : SI ×

× [t0, t1] Lin(V; Lin(F; F)) are, respectively, the motion of the shape SI and time-dependent ﬁelds of local deformations. Maps γ, M, H, and D are considered as independent ﬁeld quantities.

2. ### Geometrical Viewpoint

#### 21◦. Simple body. Considerations of section 5.1. were based on the rejection of using the global stress-free shape. Instead of this, one deals with family of locally stress-free shapes. Meanwhile, the concept of global stress-free shape is attractive enough, since one can deal with material description of processes, which is commonly convenient in non-linear elasticity. To allow global stress-free shapes, one just needs to go beyond Euclidean structure and consider manifolds equipped with non-Euclidean geometry. Choosing appropriate geometry, one obtains global stress-free shape and then deformations become embeddings of non-Euclidean stress-free shape to Euclidean space.

The stress-free shape is modeled on the intermediate shape SI = (SI , TE |SI , DE |SI , gE |SI , E |SI ). To do this, one needs to erase Euclidean structure from it. Thus results in manifold MR = (SI , TE |SI , DE |SI ), pure from any geometry. Let H : MI Lin(V; V) be ﬁeld of local deformations. One can use it to introduce

geometry on MI .

Riemannian metric G : MI T MI T MI is deﬁned on MI upon ﬁeld of local deformations as follows:

GX (u, v) := HX [u] · HX [u], (5.5)

for all X MI and u, v TX MI . Let us explain the action of “Euclidean” tensor HX Lin(V; V) on abstract tangent vector from TX MI . Since we consider the case dim B = 3, the shape MI is open submanifold of E.

Then there is natural vector space isomorphism TX MI = V [25]. By this reason, we tacitly identify element

of Lin(V; V) with element of Lin(TX MI ; V).

The value GX of ﬁeld G is non-degenerate symmetric positive-deﬁnite bilinear form on TX MI , that deﬁnes metric structure on TX MI . Upon tangent vectors u, v TX MI , which are inﬁnitesimal material ﬁbers, it returns the inner product u · v of their relaxed images u = HX [u] and v = HX [v]. Thus, one obtains lengths

of material ﬁbers and angles between them as if they where in relaxed state.

There are several possibilities to introduce aﬃne connection on MI [26, 36]. One of them, the

Weitzenb¨ock connection, is chosen in the study. Its connection functions in coordinate frame are given by

relations

Γijk = HckXj [H1]ic = [H1]icXj Hck, (5.6)

X

where H1 : X 1→ H1 is ﬁeld of inverse local deformations. The deviation of this connection from Euclidean one is completely characterized by torsion tensor ﬁeld T. Thus, T may serve as the measure of inhomogeneity. Thus, one arrives at non-Euclidean space S = (MI , G, ), to which we refer as non-Euclidean reference

shape. Any deformation is then an embedding λ : S → E of non-Euclidean manifold into Euclidean one. It

can be decomposed into two deformations,

λ = γ ψ,

where ψ : S → SI is a smooth map that “imprints” non-Euclidean shape into Euclidean space. More formally, it replaces the structure (MI , G, ) with underlying manifold MI by the structure17 (MI , gE |S , E |S ) with

I I

the same underlying manifold MI , such that X 1→ X. Other map γ : SI E is conventional deformation

between Euclidean shapes.

Since S is manifold with non-Euclidean parallel transport rule, one cannot use Euclidean diﬀerentiation operator . Instead of it, one can use tangent map operation T [25]. Being applied to deformation λ, it results in the map : T S → T E, which can be considered as the collection of linear maps TX λ : TX S → Tλ(X)E, X ∈ S, with matrices [] in coordinate frame deﬁned as follows:

[] = D(φ λ σ1),

where σ : S → R3 is coordinate map on S and φ : E R3 is coordinate map on E. The symbol D stands for total derivative in R3.

Assuming all considered ﬁelds as time-dependent, one can introduce the following Lagrangian density:

L = L (X, t, λ, Tλ, λ˙ , G, G˙ , T, T˙ ). (5.7)

17Here gE is metric tensor on E.

Вестник Самарского университета. Естественнонаучная серия. 2021. Том 27, № 1. С. 81–103

Vestnik of Samara University. Natural Science Series. 2021, vol. 27, no. 1, pp. 81–103 97

Here X is point on the shape S, λ : S × [t0, t1] E is a motion, and G, T are, respectively, time-dependent metric and torsion.

22. Micromorphic body. The case of micromorphic continuum can be considered in a similar manner. One starts with some submanifold V MI V E, the substructure of intermediate shape V SI , pure from

any geometry. Meanwhile, in contrast to the previous considerations, one has two tensor ﬁelds of local deformations: H : MI Lin(V; V) and D : MI Lin(V; Lin(F; F)). Here MI is the base manifold.

Moreover, we suppose that model ﬁber F has inner product structure, which, in turn, induces inner product

(:) on Lin(F; F).

Using natural isomorphism TX MI = V, we deﬁne two Riemannian metrics on MI . One is the Riemannian metric G, introduced by formula (5.5). The second ﬁeld GF : MI T MI T MI is deﬁned by the following

equality:

GF

X [u, v] = DX [u] : DX [v], (5.8)

for all X MI and u, v TX MI . The metric GF is a metric over orientations, i. e., with each point X MI

it associates a machine, that measures lengths of orientations associated with point X and angles between

them.

Similarly to metrics, we introduce two connections on MI . One is aﬃne connection on MI with connection functions deﬁned as (5.6). Its torsion tensor T characterizes the inhomogeneity within particles.

Using metric ﬁeld GF deﬁned by (5.8), one can induce Levi-Civita connection F on MI . Its curvature tensor

RF characterizes the inhomogeneity within micro-particles. Thus, instead of (5.7), one arrives at the following

Lagrangian density:

L = L (X, t, λ, Tλ, λ˙ , G, G˙ , T, T˙ , GF, G˙ F, RF, R˙ F), (5.9)

where X MI and mapping λ : MI × [t0, t1] E corresponds to motion.

## Conclusions

We point out the following positions of the classical method, which are speciﬁc for oriented solids and shell-like solids with incompatible deformations.

1. The action is formulated with respect to intermediate shape, which is related with the continuous family of local stress free shapes. This brings additional ﬁeld arguments, namely implants (with translation and microdeformational character), to the Lagrangian density.

2. It is possible to formulate the action with respect to single stress-free shape, but, generally, this shape will be non-Euclidean.

3. The principal diﬀerence between models for micropolar, micromorphic and shell-like solids is encoded in the structure of bundles, that are used for enhanced manifold deﬁnition.

4. Most general structure that characterize embedding of smooth material bundle into smooth physical bundle is too wide for all conventional models, because it takes into account a continuum of possible orientations for each material point. Meanwhile one can derive from it conventional models for oriented solids and shells by constructing speciﬁc sections over the bundles.

## Appendix

23. Metric and connection on body manifold. Let us consider in more detail what gives the use of metric and connection in the structure of body manifold. Riemannian metric gB : B T B T B satisﬁes the following deﬁning properties at each point X B [25]:

• for all tangent vectors u, v TXB one has that gB |X(u, v) = gB |X(v, u);

• for each vector u TXB, gB |X(u, u) 0;

• for any vector u TXB, gB |X(u, u) = 0 iﬀ u = 0.

Thus, the bilinear form (u, v) 1→ gB |X(u, v) is an inner product on tangent space TXB. One can measure lengths of tangent vectors and angles between them. After embedding κ : B E to physical manifold one

can compare metric information on B with metric information on the image κ(B) of the embedding. In the

framework of conventional elasticity this description gives nothing new, since one deals not with body, but with some ﬁxed privileged subspace of E, shape of the body. Usually it is assumed that this shape is stress-free.

Lychev S.A., Koifman K.G., Digilov A.V. Nonlinear Dynamic Equations for Elastic Micromorphic Solids and Shells

98 Лычев С.А., Койфман К.Г., Дигилов А.В. Нелинейные динамические уравнения для упругих микроморфных тел...

Meanwhile, for solid with defects one cannot ﬁnd such the shape in Euclidean physical space. Geometric methodology, developed in the papers [22; 23; 36–38] suggests to ﬁx the issue by using non-Euclidean shapes, modeled on the body manifold. Then metric gB acquires the certain sense: it gives reference (in stress-free state) lengths and angles of inﬁnitesimal material ﬁbers, formalized as tangent vectors. At the same time, one needs an extra information: to formulate balance equations on the body manifold one needs certain parallel

transport rule. This rule can be established be means of the aﬃne connection B , which to any pair (u, v)

of vector ﬁelds on B assigns vector ﬁeld (B )uv on the same body manifold. The following requirements

are satisﬁed [39]:

(B )u+vw = (B )uw + (B )vw,

• (B )f uv = f (B )uv,

(B )u(v + w) = (B )uv + (B )uw,

(B )u(f v) = f (B )uv + (uf )v,

for vector ﬁelds u, v, w : B T B and scalar function f : B R. Aﬃne connection endows body with some geometry. Each geometry is characterized by tensor ﬁelds of torsion T, curvature R and nonmetricity

Q [24]. The correspondence to each space is shown on Table 2, where the symbol illustrates that the corresponding ﬁeld vanishes, while designates that ﬁeld takes nonzero values.

Table 2

#### Correspondence between geometries and tensor ﬁelds of torsion, curvature and nonmetricity

 Geometry Torsion (T) Curvature (R) Nonmetricity (Q) RiemannWeitzenb¨ock Weyl ◦•◦ •◦◦ ◦◦•

#### 24◦. Enhanced kinematics for vector bundles. The mapping ε from (3.3) has general form in

the case of arbitrary ﬁber bundles. Meanwhile, vector bundle and principal bundle structures induce some particular properties of ε. Consider the case of vector bundle ﬁrst. Physical bundle is represented by structure (V E, E, πE, F), which is vector bundle of rank m, like material bundle. The enhanced conﬁguration V κ :

V B V E and enhanced deformation V γ : V SR V S are related to conventional conﬁguration κ : B E

and conventional deformation γ : SR S according to the general formulae (3.1) and (3.2):

κ πB = πE V κ, γ πE = πE V γ.

Moreover, it is required that mappings V κ and V γ are compatible with vectorial structure of the

X

bundle18 [25]: for any X B the restriction V κ|V B

: V BX V Eκ(X) to ﬁbers of V B is linear map,

and for any X SR the restriction19 V γ|V S

R; X :

V SR; X V S

γ(X)

to ﬁbers of V SR

is also a linear map.

Introduce natural coordinates

(X1

, ..., X

n; U

1, ..., U

m) on the enhanced reference shape V SR and natural

coordinates (x1, ..., xn; u1, ..., um) on the enhanced actual shape V S. Let γ˜ : (X1, ..., Xn) 1→ (x1, ..., xn) be

coordinate representation of conventional deformation γ. Then bundle structure gives the following equality for coordinate representation Vγ : Rm+n Rm+n of V γ:

M

Vγ(X1, ..., Xn; U 1, ..., Um)=(γ˜(X1, ..., Xn); ˜ (X1, ..., X3; U 1, ..., Um)),

M

where ˜

: Rm+n Rm gives coordinate representation of actual orientations. The linearity condition imposed

M

on enhanced deformation imply that partial map ˜ (X1, ..., Xn, ·) : Rm Rm is linear.

Consider particular case, when physical vector bundle is trivial, i. e., V E is diﬀeomorphic to product space E × F via local trivialization Φ. Then, since vector bundle structures are trivial, the enhanced deformation is the mapping V γ : SR × F S × F, which values are of the form

(x, u) = V γ(X, U ) = (γ(X), M(X, U )), (5.10)

where assignment M : SR × F F returns actual orientations and it is such that its partial map M(X, ·

·) : F F, M(X, ·) : U 1→ M(X, U ) is linear. Thus, in the case of trivial vector bundle we obtain that deformation of solid with extra degrees of freedom is characterized by conventional deformation γ : SR S and so-called microdeformation tensor M : SR × F F.

18If not so, we won’t obtain “nice” properties of the map ε.

R

19Here, as in general case of ﬁber bundle, we induce vector bundle structures on shapes: (V SR, SR, πE|V S

, F), and

(V S

, S, πE|V S, F).

Вестник Самарского университета. Естественнонаучная серия. 2021. Том 27, № 1. С. 81–103

Vestnik of Samara University. Natural Science Series. 2021, vol. 27, no. 1, pp. 81–103 99

25. Enhanced kinematics for principal bundles. Finally, we equip physical space E with principal bundle structure (P E, E, πE, G, , E), which structure group G is similar to the structure group of material principal bundle (P B, B, πB, G, , B). For further reasonings it would be convenient to denote coordinate

tuples on base and structure group by putting tilde above. In particular, X = (X1, ..., Xn) and g = (g1, ..., gm).

By enhanced conﬁguration of enhanced body P B to enhanced physical space P E we mean a smooth mapping P κ : P B P E, that satisﬁes the equivariance condition [29]:

p P B g G : P κ(p B g) = P κ(p) E g. (5.11)

In other words, P κ preserves principal bundle structure. In particular, it preserves layers and by this reason it induces embedding κ : B E, conventional conﬁguration, such that the equality (3.1) holds.

Relation (3.1) induces representation of enhanced conﬁguration P κ in natural coordinates:

Pκ(X ; g) = (κ (X ); τ (X ; g)),

x is coordinate representation of conventional conﬁguration κ, while τ : (X ; g) 1→ g corresponds to conﬁguration of orientations. Moreover, equivariance condition (5.11) gives

p p)h.

τ ( B h) = τ (

Let P κR, P κ : P B P E be enhanced conﬁgurations of enhanced body P B, and let κR, κ : B E be corresponding conventional conﬁgurations of the body manifold B. Then images P SR = P κR(P B) and P S = P κ(P B) correspond to enhanced shapes of P B in enhanced physical space P E. The composition

R R

= P κ P κ1 : P S P S of enhanced conﬁgurations correspond to enhanced deformation in enhanced

1

R

physical space P E, while γ = κ κ

: SR S is conventional deformation of reference set of places SR to

the actual one, S.

In natural coordinates on enhanced shapes the coordinate representation of enhanced deformation has the form

γ(X ); ε(X ; G )).

g returns actual orientations.

ε : (X ; X ) 1→

It can be shown that is equivariant, i. e., it satisﬁes the property similar to (3.1). Then the mapping

o has the “homogeneity” property with respect to elements of G:

ε(X ; G )H .

ε(X ; 1), where 1 is coordinate

Suppose that G can be covered by one chart. Then, if set Ψ(X ) :=

representation of identity of G, one obtains ε(X ; G ) = Ψ(X )G , that gives

Pγ(X ; G ) = (γ(X ); Ψ(X )G ).

Supposing that physical principal bundle is trivial20, i. e., P E = E × G, one obtains that enhanced deformation : SR × G S × G can be expressed in the form

(X, g) = (γ(X), ε(X, g)),

where γ : SR S is conventional deformation, and ε : SR × G G returns actual orientations. If we put

Ψ(X) := ε(X, 1), where 1 is the identity of G, then, since ε(X, g) = Ψ(X)g, one gets

(X, g) = (γ(X), Ψ(X)g).

Thus, deformation of oriented solid is characterized by mappings γ : SR S and Ψ : SR G.

×

### S. A. Lychev

IPMech RAS

Author for correspondence.
Email: lychevsa@mail.ru
ORCID iD: 0000-0001-7590-1389

Doctor of Physical and Mathematical Sciences, associate professor, Laboratory of Mechanics of Technological Processes

Russian Federation, 101-1, Prospect Vernadskogo, Moscow, 119526, Russian Federation.

### K. G. Koifman

Bauman Moscow State Technical University

Email: koifman.konstantin@gmail.com
ORCID iD: 0000-0002-7891-9995
Russian Federation, 5, 2-nd Baumanskaya Street, Moscow, 105005, Russian Federation

### A. V. Digilov

Ishlinsky Institute for Problems in Mechanics,

Email: avdigilov@gmail.com
ORCID iD: 0000-0001-6892-7740

Russian Federation, 101, Vernadsky Avenue, bldg. 1, Moscow, 119526, Russian Federation

## References

1. Lebedev L.P., Cloud M.J., Eremeyev V.A. Tensor Analysis with Applications in Mechanics. World Scientific, 2010. DOI: http://doi.org/10.1142/7826.
2. Ciarlet P.G. Mathematical elasticity. Volume II, Theory of Plates. Elsevier, 1997. DOI: http://doi.org/10.1016/s0168-2024%2897%29x8001-4.
3. Ciarlet P.G.: Mathematical elasticity. Volume III, Theory of Shells. North Holland, 2000, 659 p. Available at: https://readli.net/theory-of-shells/.
4. Libai A., Simmonds J.G. The Nonlinear Theory of Elastic Shells. Cambridge: Cambridge University Press, 1998. DOI: http://doi.org/10.1017/CBO9780511574511.
5. Then action of Lie group is defined as (x; g) ▹E h := (x; g⊤h).
6. Ambartsumian S.A. Micropolar Theory of Shells and Plates. Erevan: Nats. akad. nauk Resp. Armeniya. In-t problem informatiki i avtomatizatsii. Lab. komp’yuter. poligrafii, 1999. (In Russ.)
7. Eremeyev V.A., Altenbach H. Basics of Mechanics of Micropolar Shells. In: Altenbach H., Eremeyev V.A. (eds.) Shell-like Structures. CISM International Centre for Mechanical Sciences (Courses and Lectures), vol. 572. Heidelberg: Springer, Cham, 2017, pp. 63–111. DOI: http://doi.org/10.1007/978-3-319-42277-0_2.
8. Zubov L.M., Eremeyev V.A. Mechanics of elastic micropolar shells. Far Eastern Mathematical Journal, 2003, vol. 4, no. 2, pp. 182–225. Available at: http://mi.mathnet.ru/dvmg158. (In Russ.)
9. Zhilin P.A. Basic equations of non-classical theory of elastic shells. Dinamika i prochnost’ mashin: Trudy Leningradskogo politekhnicheskogo instituta, 1982, vol. 386, pp. 29–46. (In Russ.)
10. Wang C.C. Universal Solutions for Incompressible Laminated Bodies. In: Noll W., Toupin R.A., Wang C.C. Continuum Theory of Inhomogeneities in Simple Bodies. Heidelberg: Springer, Cham, 1968, pp. 149–180. DOI: http://doi.org/10.1007/978-3-642-85992-2_6.
11. Cohen H., Wang C.C. Some equilibrium problems for fibrillar bodies. Archive for Rational Mechanics and Analysis, 1993, vol. 123, no. 4, pp 337–375. DOI: http://doi.org/10.1007/BF00375584.
12. Noll W. Materially uniform simple bodies with inhomogeneities. Archive for Rational Mechanics and Analysis, 1967, vol. 27, no. 1, pp. 1–32. DOI: http://doi.org/10.1007/BF00276433.
13. Lewicka M., Mahadevan L., Pakzad M.R. Models for elastic shells with incompatible strains. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2014, vol. 470, issue 2165. DOI: http://doi.org/10.1098/rspa.2013.0604.
14. Lychev S.A. Equilibrium equations for transversely accreted shells. ZAMM — Journal of Applied Mathematics and Mechanics. Zeitschrift fur Angewandte Mathematik und Mechanik, 2014, vol. 94, no. 1-2, pp. 118–129. DOI: https://doi.org/10.1002/zamm.201200231.
15. Lychev S.A., Altenbach H. Growing Solids and Thin-Walled Structures. Procedia IUTAM, 2017, vol. 23, pp. 13–32. DOI: http://doi.org/10.1016/j.piutam.2017.06.002.
16. Epstein M., Leon M. Geometrical theory of uniform Cosserat media. Journal of Geometry and Physics, 1998, vol. 26, issue 1–2, pp. 127–170. DOI: https://doi.org/10.1016/S0393-0440(97)00042-9.
17. Bucataru I., Epstein M. Geometrical theory of dislocations in bodies with microstructure. Journal of Geometry and Physics, 2004, vol. 52, no. 1, pp. 57–73. DOI: http://doi.org/10.1016/j.geomphys.2004.01.006.
18. Postnikov M.M. Lectures in Geometry: Analytic Geometry. Moscow: URSS, 1994. Available at: http://alexandr4784.narod.ru/pmmgeo1.html. (In Russ.)
19. Postnikov M.M.: Lectures in Geometry: Linear Algebra and Differential Geometry. URSS, 1994. Available at: http://alexandr4784.narod.ru/pmmgeo2.html. (In Russ.)
20. Rudolph G., Schmidt M. Differential Geometry and Mathematical Physics. Part I. Manifolds, Lie Groups and Hamiltonian Systems. Springer Science+ Business Media Dordrecht, 2013. DOI: http://dx.doi.org/10.1007/978-94-007-5345-7.
21. Rudolph G., Schmidt M. Differential Geometry and Mathematical Physics. Part II. Fibre Bundles, Topology and Gauge Fields. Springer Science+ Business Media Dordrecht, 2017. DOI: http://doi.org/10.1007/978-94-024-0959-8.
22. Wang C.C. On the geometric structure of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. In: Noll W., Toupin R.A., Wang C.C. Continuum Theory of Inhomogeneities in Simple Bodies. Springer, Heidelberg, 1968, pp. 87–148. DOI: http://doi.org/10.1007/978-3-642-85992-2_5.
23. Yavari A., Goriely A. Riemann — Cartan geometry of nonlinear dislocation mechanics. Archive for Rational Mechanics and Analysis, 2012, vol. 205, no. 1, pp. 59–118. DOI: http://doi.org/10.1007/s00205-012-0500-0.
24. Yavari A., Goriely A. Weyl geometry and the nonlinear mechanics of distributed point defects. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012, vol. 468, no. 2148, pp. 3902–3922. DOI: http://doi.org/10.1098/rspa.2012.0342.
25. Lychev S.A., Koifman K.G. Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics. De Gruyter, 2018. DOI: http://dx.doi.org/10.1515/9783110563214.
26. Lee J.M. Introduction to Smooth Manifolds. New York: Springer, 2012. 486 p. Available at: http://inis.jinr.ru/sl/M_Mathematics/MD_Geometry%20and%20topology/MDdg_Differential%20geometry/Lee%20J.M.%20Introduction%20to%20smooth%20manifolds%20(draft,%202000)(486s).pdf.
27. Lychev S.A., Koifman K.G. Material Affine Connections for Growing Solids. Lobachevskii Journal of Mathematics, 2020, vol. 41, no. 10, pp. 2034–2052. DOI: http://doi.org/10.1134/S1995080220100121.
28. Husemoller D. Fibre Bundles. New York: Springer, 1994. Available at: https://www.maths.ed.ac.uk/ v1ranick/papers/husemoller.
29. Epstein M. Differential Geometry: Basic Notions and Physical Examples. Springer International Publishing, 2014. DOI: http://doi.org/10.1007/978-3-319-06920-3.
30. Naber G.L. Topology, Geometry and Gauge fields: Foundations. New York: Springer-Verlag, 2011. DOI: http://doi.org/10.1007/978-1-4419-7254-5.
31. Gurtin M.E., Murdoch A.I. A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis, 1975, vol. 57, no. 4, pp. 291–323. DOI: http://doi.org/10.1007/BF00261375.
32. Marsden J.E., Hughes T.J. Mathematical foundations of elasticity. New York: Courier Corporation, 1994. Available at: https://resolver.caltech.edu/CaltechBOOK:1983.002.
33. Lurie A.I. Analytical Mechanics. Moscow: GIFML, 1961. Available at: https://bookree.org/reader?file=450294&pg=1.
34. Liang K.K. Efficient conversion from rotating matrix to rotation axis and angle by extending Rodrigues’ formula. arXiv, 2018. Available at: https://arxiv.org/pdf/1810.02999.pdf.
35. Delphenich D.H.: Mechanics of Cosserat media: I. non-relativistic statics. arXiv, 2012. Available at: https://arxiv.org/ftp/arxiv/papers/1212/1212.0399.pdf.
36. Mindlin R.D. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 1964, vol. 16, pp. 51–78. DOI: http://doi.org/10.1007/BF00248490.
37. Lychev S.A., Koifman K.G. Contorsion of Material Connection in Growing Solids. Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 8, pp. 1852–1875. DOI: http://doi.org/10.1134/S1995080221080187.
38. Kupferman R., Olami E., Segev R. Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies. Journal of Elasticity, 2017, vol. 128, no. 1, pp. 61–84. DOI: http://doi.org/10.1007/s10659-016-9617-y.
39. Lychev S.A, Kostin G.V., Koifman K.G., Lycheva T.N. Non-Euclidean Geometry and Defected Structure for Bodies with Variable Material Composition. Journal of Physics: Conference Series, 2019, vol. 1250, no. 1, p. 012035. DOI: 10.1088/1742-6596/1250/1/012035' target='_blank'>http://doi: 10.1088/1742-6596/1250/1/012035.
40. Chern S.S., Chen W.H., Lam K.S. Lectures on Differential Geometry. World Scientific, 1999. DOI: http://doi.org/10.1142/3812.

Copyright (c) 1970 Lychev S.A., Koifman K.G., Digilov A.V.