**Dislocation Density Tensor of Thin Elastic Shells at**

**Finite Deformation**

**Bohua Sun**

April 4, 2018

**Abstract** The dislocation density tensors of thin elastic shells have been
for-mulated explicitly in terms of the Riemann curvature tensor. The formulation
reveals that the dislocation density of the shells is proportional to *KA*3*/*2,
where*K* is the Gauss curvature and*A*is the determinant of metric tensor of
the middle surface.

**Keywords** dislocation density tensor, *·* thin shells *·* Riemann curvature
tensor

**1 Introduction**

Fracture and elasto-plasticity of plates and shells is of great practical as well as theoretical interest[1, 9–12]. For example, a large number of engineering structures, such as pressurized aircraft fuselages, ship hulls, storage tanks and pipelines are constructed of shells and plates. Concern over the safety of such structures has led to tremendous amounts of productive research in the ﬁeld. The dislocation density tensor has drawn a great deal of attention [15, 2, 5–8] due to the fact that the dislocation is being considered as the physi-cal mechanism of both fracture and plasticity. Recently Birsan and Neﬀ [3] proposed a dislocation density tensor for the Cosserat theory of shells [4]. From literature survey, we noticed that the dislocation density tensors of the Mindlin-Reissner (M-R) shells [14] and Kirchhoﬀ-Love (K-L) shells [13], have not been formulated yet. The aim of this short article is that we will apply the 3-D dislocation density tensor formulation obtained by Sun [18] to the shell models and derived their dislocation density tensor, respectively.

Bohua Sun

Department of Mechanical Engineering, Cape Peninsula University of Technology, Bellville 7535, Cape Town, South Africa

Tel.: +27 21 959 6766 E-mail: sunb@cput.ac.za

The paper is organized as that following an introduction, section 2 in-troduces the dislocation density tensor obtained by Sun [18]. Section 3 gives displacement ﬁeld of thin shells. Section 4 formulates deformation gradient. Section 5 derives the dislocation density tensor for the Mindlin-Reissner and K-L theory of shells. Finally, Section 6 concludes the paper.

**2 Dislocation density tensor in terms of the Riemann curvature**
**tensor**

In 2016 Sun [18] proved that the dislocation density tensor can be explicitly expressed by the Riemann curvature tensor, and shown that the dislocation density is proportional to the Riemann curvature tensor.

**Definition 1** Let* F* be the deformation gradient, the deﬁnition of the
dislo-cation tensor is given by

* T* =

*−*

**F***× ∇*(1)

**X**,where, the gradient operator* ∇X* =

*=*

_{∂}∂_{X}*(see appendix).*

_{∂X}∂i**G**i**Lemma 1** *Let* **u**be displacement field,**R**the Riemann curvature tensor, the*dislocation density tensor T*

*can be presented explicitly in terms of the Riemann*

*curvature tensor as follows*

* T* =

*−*1

2* R*(

*)*

**X**,**Y***(2)*

**u**.*or in conventional form*

* T* =

*−*1 2

*uiR*

*i*

*.jkl Gj⊗Gk×Gl*=

*−*

1
2*u*

*i _{R}*

*ijkl Gj⊗Gk×Gl.* (3)

The equation (2) and (3) were formulated by Sun [18].

*Proof* Let B be undeformed conﬁguration and *d u* =

**F***·d*=

**Y***, (*

**∇****Y****u**·d**Y***)*

**X**,**Y***∈*B. According to the Stokes integration theorem, we have the dis-placement vector change along an arbitrary close loop as follows

* u*= H

_{∂Ψ}d**u**= ∫* _{Ψ}d*(

*d*) =∫

**u***(*

_{Ψ}d*)*

**∇****Y****u**·d**Y***,*

(4)

in which,*∂Ψ* is close boundary of a surface*Ψ* *∈*B, and *d u*=

*is a vector-valued 1-form, diﬀerentiating the above equation once more, hence we obtain the vector-valued 2-form*

**∇****Y****u**·d**Y***d*(* ∇Yu·dY*) =

*d*(

*)*

**∇****Y****u***∧d*+ (

**Y***−*1)0

*2*

**∇****Y****u**d*=*

**Y***(*

**∇****Y***)*

**∇****X****u***·d*=

**X**∧d**Y**

**∇****Y****∇****X****u**·d**X**∧d**Y**,(5)

Since the antisymmetric nature of exterior algebra,*d X∧dY* =

*−d*

**Y***∧d*, thus

**X***d*(* ∇Yu·dY*) =

*−*1

_{2}[

*]*

**∇****X****∇****Y****u**−**∇****Y****∇****X****u***·d*(6)

**X**∧d**Y**.According to the deﬁnition of the Riemann operator

* R*(

*)*

**X**,**Y***=*

**u***]*

**∇****X****∇****Y****u**−**∇****Y****∇****X****u**−**∇[****X**,**Y***(7)*

**u**.and in the coordinate frame, the torsion curvature* ∇[X,Y*]

*= 0, we have*

**u*** ∇X∇Yu−∇Y∇Xu*=

*(*

**R***)*

**X**,**Y***(8)*

**u**.If we expand the vector ﬁelds in terms of the coordinate basis *∂I*, the

Rie-mann tensor * R*(

*)*

**X**,**Y***= [*

**u***RI*

*J KLX*

*K _{Y}L_{u}J*

_{]}

_{∂}*I* and its components*RIJ KL*:=
*∂KΓLJI* *−∂LΓKJI* +*Γ*

*I*
*KMΓ*

*M*
*LJ−Γ*

*I*
*LMΓ*

*M*

*KJ*. The symbols*Γ*
*I*

*J K* are called the

co-eﬃcients of the aﬃne connections, or the Christoﬀel symbols, with respect to
the frame* GJ*, that is,

*=*

**∇****G**J**G**K

**G**IΓ*I*
*J K*.

Therefore, we have

*d*(* ∇Yu·dY*) =

*−*1

_{2}[

*]*

**∇****X****∇****Y****u**−**∇****Y****∇****X****u***·d*

**X**∧d**Y**= *−*1

2* R*(

*)*

**X**,**Y**

**u**·d**X**∧d**Y**.(9)

Finally, we have the displacement change in diﬀerential forms

* u*=

*−*∫

*(*

_{Ψ}d*)*

**∇****Y****u**·d**Y**=*−*1_{2}∫* _{Ψ}*[

*]*

**∇****X****∇****Y****u**−**∇****Y****∇****X****u***·d*=

**X**∧d**Y***−*1

_{2}∫

*(*

_{Ψ}**R***)*

**X**,**Y**

**u**·d**X**∧d**Y**.(10)

Note that the area element *d A* =

*d*

**X***∧d*, hence the dislocation density tensor

**Y***and the incompatibility operator*

**T***Inc*(

*)*

**F*** T* =

*Inc*(

*) =*

**F***d*

**u***d A* =

*−*

1

2* R*(

*)*

**X**,**Y***(11)*

**u**.The proof is complete. *⊓⊔*

Equations (2) and (3) provide a general framework on formulation of
dis-location density tensor for a given displacement ﬁeld* u*, especially in the case
of beams, plates and shells, where the displacement ﬁeld are always assumed
with constraints.

**3 Displacement field of thin shells**

Let* X* =

*+*

**P***ξ*be position vector in the undeformed state, and

**A3***=*

**x***+*

**p***ξ*

**η**be position vector in the deformed state, where the unit normal vector* A3* is
orthogonal to the undeforemd middle surface, however, in general

*is neither unit or normal to the deformed middle surface.*

**η**The base vectors in the undeformed state * G3* =

*=*

**X**,ξ*, and*

**A3***=*

**G**α*=*

**X**,α*+*

**P**,α*ξ*=

**A3**,α*+*

**A**α*ξ*=

**A3**,α*=*

**A**α−ξBαβ**A**β*µαβ*=

**A**β*µαβ*, where

**A**βthe shitter tensor *µαβ* = *Aαβ* *−ξBαβ*, *µβα* = *δβα−ξBβα*. The metric tensor
*Gαβ*=*µλαµλβ*=*Aλρµλαµ*

*ρ*

*β*,*Gα*3= 0*, G*33= 1.

The base vectors in the deformed state* gα*=

*=*

**x**,α*+*

**p**,α*ξ*=

**η**,α*+*

**a**,α*ξ*

**η**,αand* g3*=

*=*

**x**,ξ*.*

**η**3.1 The displacement ﬁeld of the Kirchhoﬀ-Love theory of shells

The Kirchhoﬀ-Love (K-L) theory of shells is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love [13] using assumptions pro-posed by Kirchhoﬀ.

The K-L kinematic assumptions are: 1. straight lines normal to the surface remain straight after deformation; 2. straight lines normal to the mid-surface remain normal to the mid-mid-surface after deformation; and 3. the thick-ness of the plate does not change during a deformation.

For the Kirchhoﬀ-Love theory of shells [13], the displacement of thin shells
can be represented as* u*=

*vσ*+

**A**σ*v*3A3

*−ξ*(

*v*3

*,σ*+

*vγBσγ*)

*= [*

**A**σ*vσ−ξ*(

*v*3

*,σ*+

*vγBσγ*)

*] +*

**A**σ*v*3A3and/or

*=*

**u***vσ*+

**A**σ*v*3

*(*

**A3**−ξ*v,σ*3 +

*vγBγσ*)

*= [*

**A**σ*vσ−ξ*(

*v,σ*3 +

*vγB*)]

_{γ}σ*+*

**A**σ*v*3

*.*

**A3**The K-L theory of shells has three unknown variables, namely, *v*1* _{, v}*2

*3*

_{, v}_{.}

3.2 The displacement ﬁeld of the Mindlin-Reissner theory of shells

The Mindlin-Reissner theory of plates is an extension of Kirchhoﬀ-Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1951 by R. Mindlin. [14] A similar, but not identical, theory had been proposed earlier by Eric Reissner in 1945.[13] Both theories are intended for thick plates in which the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. The Mindlin-Reissner theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of one tenth the planar dimensions while the Kirchhoﬀ-Love theory is applicable to thinner plates.

Thus* u*=

*+*

**v***ξ*=

**d***vβ*+

**A**β*v*3

*+*

**A3***ξdβ*= (

**A**β*vβ*+

*ξdβ*)

*+*

**A**β*v*3

*, so we*

**A3**have the components of the arbitrary displacement vector* u*=

*uβ*+

**A**β*u*3

**A3**as follows*uβ*=*vβ*(*ξ*1*, ξ*2) +*ξdβ*(*ξ*1*, ξ*2) and*u*3=*v*3(*ξ*1*, ξ*2).

The M-R theory of shells has ﬁve unknown variables, namely,*v*1* _{, v}*2

*3*

_{, v}*1*

_{, d}*2*

_{, d}_{.}

**4 Deformation gradient of thin shells**

There are four base vectors in the analysis of shell deformation, namely,

* Ai,ai,Gi,gi*. The relationship of base vectors in both underformed and

**G**i

* F* //

**g**i**A**i

**µ**

O

O

* f* //

**a**i

**ψ**O

O

Fig. 1. Relationship of base vectors.

The deformation gradient of the thin shells is deﬁned as

* F* =

**g**i⊗**G**i=* gα⊗Gα*+

*3*

**g3**⊗**G**= (* p,α*+

*ξ*)

**η**,α*⊗*+

**G**α*3*

**η**⊗**G***.*

(12)

In the diagram 1, there are four tensors, namely,* F* =

*is*

**g**i⊗**G**ideforma-tion gradient of arbitrary surface,* f* =

*is middle surface deformation*

**a**i⊗**A**igradient,* µ*=

*is shifter tensor from*

**G**i⊗**A**i*to*

**A**i*in undeformed state,*

**G**iand* ψ*=

*is shifter tensor from*

**g**i⊗**a**i*to*

**a**i*in deformed state.*

**g**iWith the help of* f,µ,ψ*, the deformation gradient

*can be expressed in a multiplication format:*

**F*** F* =

*= (*

**g**i⊗**G**i*)*

**g**i⊗**a**i*·*(

*)*

**a**i⊗**A**i*·*(

*) =*

**A**i⊗**G**i

**ψ**·**f***·*1

**µ**−*.*(13)

The equation (13) indicates that the deformation gradient of shells can be
decomposed into the multiplication of middle surface deformation gradient
with two shifter tensors* µ* and

*. In order words, the study of general shell deformation can be considered as the subject of deformation of the middle surface. This relation can be mathematically interpreted as follows: the middle surface deformation gradient*

**ψ***is push-forwarded to the deformation gradient*

**f*** F* by two shifter tensor

*and*

**µ***.*

**ψ**The inverse of shifter tensor * µ*is

*1*

**µ**−_{=}

_{A}*i⊗ Gi*, namely,

* µ−*1=

**A**i⊗**G**i=* Aα⊗Gα*+

*3*

**A3**⊗**G**=* Aα⊗*(

**A**α−ξBαβ**A***β*

_{) +}

_{A}3*⊗ G*3

=* Aα⊗Aα*+

*3*

**A3**⊗**A***−ξ*

**B**=**A**−ξ**B**,

(14)

where the tensor * A*=

*=*

**A**i⊗**A**i*is an identity tensor, and the 2nd*

**I**funda-mental tensor* B*=

*Bα*

*β Aα⊗Aβ*.

The deformation gradient of the middle surface* f* =

*=0 is given by*

**F**|ξ* f* =

*=*

**a**i⊗**A**i*+*

**a**α⊗**A**α*3*

**a3**⊗**A**= (* Aα*+

*)*

**v**,α*⊗*+

**A**α*3*

**a3**⊗**A**=* Aα⊗Aα*+

*+*

**v**,α⊗**A**α*3*

**a***⊗*3

**A***.*

**5 The dislocation density tensor of thin shells**

5.1 The dislocation density tensor on the middle surface

The dislocation density tensor of middle surface is deﬁned as ¯* T* =

**f***×*¯, where the gradient operator ¯

**∇***=*

**∇***∇i*=

**A**i

_{∂ξ}∂λ**A***λ* _{+} *∂*
*∂ξ A*

3_{, namely, ¯}_{T}_{=}

*−*1
2*v*

*α _{R}*

_{¯}

*αβγω Aβ⊗Aγ×Aω*.

For the two dimensional middle surface, the Riemann tensor has 16
compo-nents, in which four are no-zero, namely, ¯*R*1212= ¯*R*2121=*−R*¯2112=*−R*¯1221.

If we denote ¯*K* as a the Gauss curvature of the middle surface, then we have
¯

*R*1221= ¯*KA*, where*A*= det(*Aαβ*).

Hence, the dislocation density tensor of the middle surface of the thin shells can be expressed as follows:

¯

* T* =

*−*1 2

*v*

*α _{R}*

_{¯}

*αβγω Aβ⊗Aγ×Aω*

=*−*1
2[*v*

1_{R}_{¯}_{1212A}2* _{⊗}_{A}*1

*2*

_{×}_{A}_{+}

*1*

_{v}

_{R}_{¯}

_{1221A}2

*2*

_{⊗}_{A}*1*

_{×}_{A}+*v*2*R*¯2112A1*⊗ A*1

*×*2+

**A***v*2

*R*¯2121A1

*⊗*2

**A***×*1]

**A**=*−*1
2

¯

*KA*[*v*1* A*2

*⊗*1

**A***×*2

**A***−v*1

*2*

**A***⊗*2

**A***×*1

**A***−v*2* A*1

*⊗*1

**A***×*2+

**A***v*2

*1*

**A***⊗*2

**A***×*1] =

**A***−KA*¯ [

*v*1

*2*

**A***⊗*1

**A***×*2

**A***−v*2

*1*

**A***⊗*1

**A***×*2]

**A**=*−KA*¯ [*v*1* A*2

*⊗*(

*1*

**A***×*2)

**A***−v*2

*1*

**A***⊗*(

*1*

**A***×*2)]

**A***.*

(16)

Notice* A*1

*×*2=

**A***√A*3, hence

**A**¯

* T* =

*−KA*¯ [

*v*1

*2*

**A***⊗*(

*√A*3)

**A***−v*2

*1*

**A***⊗*(

*√A*3)]

**A**=*−KA*¯ *√A*[*v*1* A*2

*⊗*3

**A***−v*2

*1*

**A***⊗*3]

**A**= ¯*KA*3*/*2[*v*2* A*1

*⊗*3

**A***−v*1

*2*

**A***⊗*3]

**A***.*

(17)

or in component format ¯*T*13= ¯*KA*3*/*2*v*2and ¯*T*23=*−KA*¯ 3*/*2*v*1.

The relation (17) is obtained for the ﬁrst time by this paper. The (17)
clearly shows that the dislocation density tensor ¯* T* of the middle surface is
proportional to the Gauss curvature and

*A*3

*/*2

_{, and has nothing to do with}

the displacement component*v*3, in other words, the component*v*3has no any

5.2 The dislocation density tensor on the Mindlin-Reissner theory of shells

Now let us attack the dislocation density tensor * T* =

*−*

**F***×*, where

**∇***=*

**∇***∇i Gi*=

_{∂ξ}∂λ**G***λ*

_{+}

*∂*

*∂ξ G*

3_{. Hence}

* T*(

*M−R*)= (

*1)*

**ψf µ**−*×*(

**G**λ*∂*
*∂ξλ* +**G**

3 *∂*

*∂ξ*)

= *∂*(**ψf µ**

*−*1_{)}

*∂ξλ* *× G*

*λ*_{+}*∂*(* ψf µ−*
1

_{)}

*∂ξ* *× G*

3_{.}

(18)

If you carry out this calculation, it will generate quite complicated expressions,
we are not going to continue this painful calculation. We will ﬁnd the
disloca-tion density tensor* T* by the above formula

*=*

**T***−*1

_{2}

*ukRkβγω*,

**G**β⊗**G**γ×**G**ωwhere the Greek indices range over the values 1,2 while the Latin *k* indices
range over the values 1,2,3.

From previous study of deformation geometry of thin shells, we have the
shell displacement vector* u*=

*+*

**v***ξ*.

**d**Hence the dislocation density tensor* T*(

*M−R*) is given as follows

* T(M_{−}R*)=

*−*

1
2*u*

*k _{R}*

*kβγω Gβ⊗Gγ×Gω*

=*−*1
2*u*

*α _{R}*

*αβγω Gβ⊗Gγ×Gω−*

1
2*u*

3_{R}

3*βγω Gβ⊗Gγ×Gω.*

(19)

Notice*R*3*βγω*= 0, hence

* T*(

*M*)=

_{−}R*−*

1
2*u*

*α _{R}*

*αM CD GM*

*⊗*

**G**C×**G**D=*−*1
2[*u*

1_{R}

1212* G*2

*⊗*1

**G***×*2+

**G***u*1

*R*1221

*2*

**G***⊗*2

**G***×*1

**G**+*u*2*R*2112G1*⊗ G*1

*×*2+

**G***v*2

*R*2121G1

*⊗*2

**G***×*1]

**G***.*

(20)

For the two dimensional parallel surface, the Riemann tensor has 16
compo-nents, in which four are no-zero, namely, *R*1212=*R*2121=*−R*2112=*−R*1221.

If we denote ¯*K* as a the Gauss curvature of the middle surface, then we
have *R*1221 = *KG*, where *G* = det(*Gαβ*) = *A*(1*−*2*ξH*+*ξ*2*K*)2 and *K* =

(*R*1+*ξ*)*−*1(*R*2+*ξ*)*−*1= ¯*K*(1 +*ξ/R*1)*−*1(1 +*ξ/R*2)*−*1, where both*R*1*, R*2are

the principal radius of curvature of the surface, and the Gauss curvature of
the middle surface ¯*K*= * _{R}*1

1*R*2.

The dislocation density tensor of thin shells can be expressed as follows:

* T*(

*M−R*)=

*KG*3

*/*2[(

*v*2+

*ξd*2)

*1*

**G***⊗*3

**G***−*(

*v*1+

*ξd*1)

*2*

**G***⊗*3]

**G**= *KG*¯

3*/*2_{[(}* _{v}*2

_{+}

*2*

_{ξd}_{)}

*1*

_{G}*3*

_{⊗}_{G}

_{−}_{(}

*1*

_{v}_{+}

*1*

_{ξd}_{)}

*2*

_{G}*3*

_{⊗}_{G}_{]}

(1 +*ξ/R*1)(1 +*ξ/R*2)

*≃ _{KG}*¯ 3

*/*2

_{Q}_{[(}

*2*

_{v}_{+}

*2*

_{ξd}_{)}

*1*

_{G}*3*

_{⊗}_{G}

_{−}_{(}

*1*

_{v}_{+}

*1*

_{ξd}_{)}

*2*

_{G}*3*

_{⊗}_{G}_{]}

_{,}= ¯*KA*3*/*2*Q*4[(*v*2+*ξd*2)* G*1

*⊗*3

**G***−*(

*v*1+

*ξd*1)

*2*

**G***⊗*3]

**G***,*

in which*Q*(*ξ*) = 1*−*2*ξH*+*ξ*2*K*,* G*3=

*3=*

**A***, and*

**A3***=*

**G**β*λβ*, where the shifter

_{α}**A**α*λβα*is deﬁned as

*λβαµαδ*=

*δ*

*α*
*δ*,*λ*

*β*

*α*=*δβα*+*ξBαβ*+*ξ*2*BγβBγα*+*...*, hence the

metric of the parallel surface is*Gαβ*_{=}** _{G}**α

_{·}

**β**

_{G}_{=}

_{λ}α*γλ*

*β*
*δA*

*γδ*_{.}

* T*(

*M−R*)= ¯

*KA*3

*/*2

*Q*4[(

*v*2+

*ξd*2)

*1*

**G***⊗*3

**G***−*(

*v*1+

*ξd*1)

*2*

**G***⊗*3]

**G***.*

= ¯*KA*3*/*2*Q*4[(*v*2+*ξd*2)*λ*1*α Aα⊗A*3

*−*(

*v*1+

*ξd*1)

*λ*1

*β*3]

**A**β⊗**A***.*

(*α, β*= 1*,*2)*.*

(22)

The equation (22) is derived for the ﬁrst time by this paper. The formula
shows that the dislocation density tensor of arbitrary surface of thin shells
is proportional to ¯*KA*3*/*2_{, and is an order 9 polynomial function of thickness}

coordinate*ξ*, which can be simpliﬁed to a lower order of*ξ*. The equation (22)
also indicates that the component*v*3 has no any contribution to the* T*(

*M−R*).

5.3 The dislocation density tensor of the Kirchhoﬀ-Love theory of thin shells

For Kirchhoﬀ-Love (K-L) theory of thin shells, the rotation vector * d* =

**η**−*3=*

**A***3*

**a***−*3, which can be represented by the middle surface displacements

**A***vα*_{and thickness-through displacement}* _{v}*3

_{, namely,}

_{d}σ_{=}

_{−}_{(}

*3*

_{v}*,σ*+*vγBσγ*).

Substitute the *dσ* _{into the equation (22), therefore the dislocation density}

tensor* T*(

*K−L*) of the K-L theory of thin shells can be obtained as follows

* T*(

*K−L*)= ¯

*KA*3

*/*2(1

*−*2

*ξH*+

*ξ*2

*K*)4

*{*[

*v*2

*−ξ*(

*v*3

*,*2+

*v*

*γ*2

_{B}*γ*)]*λ*
1
*α A*

*α _{⊗}_{A}*3

*−*[*v*1*−ξ*(*v*3_{,}_{1}+*vγB _{γ}*1)]

*λ*1

*3*

_{β}**A**β⊗**A***},*(

*α, β*= 1

*,*2)

*.*(23)

The equation (23) is obtained for the ﬁrst time by this paper, which shows that
the dislocation density tensor of arbitrary surface of thin shells is proportional
to ¯*KA*3*/*2_{. The equation (23) reveals that the displacement component}_{v}

3will

have contribution to the * T*(

*K−L*) because of the Kirchhoﬀ-Love kinematical

constraints.

**6 Conclusions**

This paper studied the dislocation density tensor of thin shells. The dislocation
density tensor for both Mindlin-Reissner and K-L theory of shells have been
explicitly expressed in terms of the Riemann curvature tensor. The formulation
reveals that both dislocation density of the shells are proportional to*KA*3*/*2_{,}

where *K* is the Gauss curvature and *A* = det(*Aαβ*) is the determinant of

metric tensor of the middle surface. The study ﬁnds that the displacement
component*v*3_{has no any direct contribution to the dislocation density of the}

**Table 1** Dislocation density tensor of thin shells

Thin shells Dislocation density tensor

¯

**T***KA*¯ 3*/*2_{[}* _{v}*2

*1*

_{A}*3*

_{⊗}_{A}*1*

_{−}_{v}*2*

_{A}*3*

_{⊗}_{A}_{]}

* T*(

*M−R*)

*KA*¯ 3

*/*2(1

*−*2

*ξH*+

*ξ*2

*K*)4[(

*v*2+

*ξd*2)

*λ*1

*α*3

**A**α⊗**A***−*(

*v*1+

*ξd*1)

*λ*1

*β*3]

**A**β⊗**A*** T*(

*K−L*)

*KA*¯ 3

*/*2(1

*−*2

*ξH*+

*ξ*2

*K*)4

*{*[

*v*2

*−ξ*(

*v*3

*,*2+

*vγBγ*2)]

*λ*1

*α*3

**A**α⊗**A***−*[*v*1*−ξ*(*v,*31+*vγBγ*1)]*λ*1*β A*

*β _{⊗}_{A}*3

_{}}**Acknowledgements**

This paper is dedicated to the memory of my mentor, Prof. Kaiyuan Ye. Supports by the South Africa National Research Foundation (NRF) (Grant no. 93918) is gratefully acknowledged.

**Author contributions statement**

Bohua Sun initiated, completed the project and prepared the manuscript.

**The conflict of interest statement**

The author declare no conﬂict interests with any parties.

**References**

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(1987).

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application of the methods of non-Riemannian geometry,*Proceedings of the Royal Society*
*of London*A231(1185): 263-273 (1955).

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4. Cosserat, E. and Cosserat, F.:*Th´eorie des corps d´eformables*. Hermann et Fils (reprint
2009), Paris (1909).

5. Kondo, K.: Geometry of elastic deformation and incompatibility, Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, (K. Kondo, ed.), vol. 1, Division C, Gakujutsu Bunken Fukyo-Kai, 1955, pp. 5-17 (1955).

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7. Kr¨oner, E: Das Fundamentalintegral der anisotropen elastischen Versetzungsdichte und Spannungsfunktionen, Z. Phys., 142, 463-475 (1955).

8. Kr¨oner, E.: Allgemeine kontinuumstheorie der versetzungen und eigenspannungen.*Arch.*
*Rational Mech. Anal.*, 4(4):273-334 (1960).

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*Engi-neering Science*34: 339-358 (1995).

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*Plasticity*, 53, 164C178 (2014).

12. Le, K.C.: Three-dimensional continuum dislocation theory,*Int. J. of Plasticity*76,
213-230 (2016).

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(1927).

14. Mindlin, R.D.: Inﬂuence of rotatory inertia and shear on ﬂexural motions of isotropic,
elastic plates,*ASME Journal of Applied Mechanics*, Vol. 18 pp. 31C38, 1851.

15. Nye, J. F.: Some geometrical relations in dislocated crystals,*Acta Metallurgica* 1(2):
153 - 162(1953).

16. Poincar´e, H.: Analysis Situs.*J. ´Ecole Polytech*. ser 2 1,1-123 (1895).

17. Reissner, E.: The eﬀect of transverse shear deformation on the bending of elastic plates,
*ASME Journal of Applied Mechanics*, Vol. 12, pp. A68-77 (1945).

18. Sun, B.: Compatibility conditions and dislocation density tensor in termas of the
Rie-mann curvature tensor, to*Applied Mathemathics and Mechanics*(2016).

**Appendix 1: Notation, preliminaries on finite deformations**

**1.1 Notations of this paper**

To distinguish the undeformed and deformed conﬁguration, the quantities with the
undeformed body (conﬁguration)Bwill be denoted by upper case (majuscules), and those
associated with the deformed body (conﬁguration) b by lower case (minuscules). When
these quantities are referred to Lagrange coordinates*XK*_{, their indices will be upper case}
(majuscules); and when they are referred to Euler*xk*_{, their indices will be by lower case}
(minuscules). For example, a displacement vector* u*referred to

*XK*will have components

*uK*, and referred to*xk* will have the components*uk*. If denoting* GK* and

*as covariant base vector in undeformed body and deformed body, respectively, then we can write the displacement vector in a total form*

**g**k*=*

**u***uK*

_{G}*K*=*uk gK*.

**1.2 Preliminaries on ﬁnite deformations**

In undeformed conﬁguration, let* X*=

*(*

**X***XA*

_{) as position vector of a partcle,}

_{X}A_{is }La-grange coordinates of the particle, then its diﬀerential is

*d*=

**X**

_{∂X}∂**X**AdX*A*_{=}_{X}_{∇}

*XAdXA*=

* X∇AdXA* =

*, where*

**G**AdXA*=*

**∇**A*and*

**G**A_{∂X}∂A*are covariant derivative and tan-gent vector in the underformed conﬁguration, respectively. After time*

**G**A*t*, the position moves to deformed conﬁguration, its position is

*=*

**x***(*

**x***), its diﬀerential is*

**X**, t*d*=

**x***∂*

**x***∂xidxi* =

* x∇_{x}idxi*=

*=*

**x****∇**idxi*, where*

**g**idxi*=*

**∇**i*and*

**g**i_{∂x}∂i*are gradient operator and tangent vector in the deformed conﬁguration, respectively.*

**g**iLet*d X* be line element between two particles

*XA*and

*XA*+

*dXA*, after deformation, the line element becomes

*d*between the corresponding particles

**x***dxi*

_{and}

_{x}i_{+}

_{dx}i_{, then}

*d x*=

*(*

**x***+*

**X***d*)

**X**, t*−*(

**x***) =*

**X**, t

_{∂}∂_{X}**x***·d*=

**X***=*

**x****∇**A·d**X**

_{∂X}∂**x**A*∂XA*

*∂ X*

*·d*=

**X***∂ x*

*∂XA GA·dX*=

*∂*

**x***∂xi*
*∂xi*

*∂XA GA·dX*=

**g**i*∂x*

*i*

*∂XA GA·dX*=

*=*

**F**·d**X***d*, where the deformation gradient tensor

**X**·**F**T*=*

**F***=*

_{∂}∂_{X}**x***=*

**x****∇**A

_{∂X}∂**x**A**G***A*_{=}_{g}

*iF _{A}iGA*=

*F*, its components

_{A}i**g**i**G**A*F*=

_{A}i*∂x*

*i*

*∂XA*=

*xi*_{;}* _{A}*=

*∇Axi*, transpose

*=*

**F**T

_{∂X}∂**x**A**G***A*

_{=}

_{F}i*A G*

*A*

_{g}*i*, and inverse* F−*1=

_{∂x}∂**X**i**g***i*

_{=}

_{F}A*i* * GAgi*.
Let

*be displacement vector, then*

**u***=*

**x***+*

**u***, then deformation gradient tensor*

**X***=*

**F*** I*+

*=*

**u****∇****X***+*

**I***, transpose*

**g**i**G**A∇Aui*=*

**F**T*+*

**I***=*

**∇****u***+*

**I***and inverse*

**G**A**g**i∇Aui* F−*1

_{=}

_{I}_{−}_{u}_{∇}* x*=

*.*

**I**−**g**j**g**i∇iujThe materials time derivative of*d x*leads to

*d*˙= ˙

**x***=*

**F**·d**X***, in which the velocity gradient tensor is deﬁned as*

**lF**d**x***= ˙*

**l***=*

**u****∇****x***∂*˙ = ˙

_{∂}_{x}**u***ui*

_{;}

*=*

_{j}**g**i**g**j*∇jui*= ˙

**g**i**g**j*1, which can be decomposed as sum of*

**F F**−*and*

**d***,*

**w***=*

**l***+*

**d***, where the rate of deformation tensor*

**w***=1*

**d**2(* l*+

**l***T*_{)}

and spin tensor* w*=1

_{2}(

**l**−**l**T_{).}

If denote*d A*as area element in undeformed conﬁguration, then the area in the deformed
one

*d*=

**a***J*and

**F**−T·d**A***d*=

**A***J−*1

*, where the Jacobean*

**F**T·d**a***J*= det(

*).*

**F**The tangent vectors between the undeformed and deformed conﬁguration can be easily
transferred as* gi*=

*δiA*

**G**A,*=*

**g**i*δiA*

**G**A,*=*

**G**A*δAi*

**g**i,*=*

**G**A*δAi*, where the shifters

**g**i*δiA*=* gi·GA*=

*δAi,*

*δiA*=

*=*

**g**i·**G**A*δAi*.

**Appendix 2: Introduction of shell geometry**

**2.1 Concepts of shells**

A thin shell/plate is a body that is bounded primarily by two closely spaced curved surfaces with boundary. The geometric feature of the shells is that the thickness length scale is much smaller compare with other two dimensions. This feature make the shells can easily be bended, which make one surface in stretching another in compression. This anti-symmetrical deformation respect to the middle surface will have a possibility to generate cracks on the tension surface.

The distance in between the curved surfaces is called the thickness of the shell*h*(*ξ*1* _{, ξ}*2

_{).}The surface in the middle of the curved surface is called the middle surface of the shell. The middle surface in undeformed conﬁguration can be expressed by the parametric represen-tation

*=*

**P***(*

**P***ξ*1

*2*

_{, ξ}_{)}

_{,}_{(}

*1*

_{ξ}*2*

_{, ξ}_{)}

_{∈}_{Ω}_{, where}

_{P}_{is the position vector from the origin}

_{O}_{to}points on the mid-surface, and the domain of the deﬁnition of the parameters is a closed region

*Ω*in the (

*ξ*1

*2*

_{, ξ}_{)-plane. We assume that there is one-to-one mapping between the}pairs of numbers (

*ξ*1

*2*

_{, ξ}_{)}

_{∈}_{Ω}_{and the points of the mid-surface.}

Let* A*3=

*3(*

**A***ξ*1

*, ξ*2) the unit normal vector on the middle surface, then the points out of the mid-surface in the shell are given by the position vectors

*(*

**X***ξ*1

*2*

_{, ξ}*3*

_{, ξ}_{) =}

_{P}_{(}

*1*

_{ξ}*2*

_{, ξ}_{) +}

*ξ A*3(

*ξ*1

*, ξ*2), where

*ξ*3 is thick-through-coordinate, so that

*ξ*3= 0 is the middle surface. We assume that 1

_{2}

*h <|Rmin| ̸*= 0, where

*Rmin*is the numerically smallest radius of curvature of the middle surface. A shell is called thin when the thickness is small compact with the other dimensions of the shell.

**2.2 Diﬀerential geometry of shell surfaces in the undeformed state**

It is known that the parameters (*ξ*1* _{, ξ}*2

_{) are called curvilinear coordinates of the surfaces,}and the

*ξ*1

_{and}

*2*

_{ξ}_{are called the coordinate curves.}

The covariant base vectors* Aα* of the surface are deﬁned by

*=*

**A**α*=*

**P**,α*, where the Greek subscripts represent the numbers 1, 2. The inﬁnitesimal vector connecting two points on the surface with coordinates*

_{∂ξ}∂**P**α*ξα*

_{and}

_{ξ}α_{+}

_{dξ}α_{is given by}

_{d}_{P}_{=}

_{P}*,αdξα*=* Aαdξα*.
The length

*dL*of this vector is therefore determined by (

*dL*)2

_{=}

_{d}_{P}_{·}_{d}_{P}_{=}

_{A}*α· Aβdξαdξβ*=

*Aαβdξαdξβ,*which is the ﬁrst fundamental form of the surface and*Aαβ*=* Aα·Aβ*is metric
tensor of the surface. The surface area

*dΣ*is given by

*dΣ*=

*√Adξ*1

*2*

_{dξ}

_{,}

_{A}_{=}

_{det}_{(}

_{A}*αβ*)*.*
The contra-variant base vectors of the middle surface are given by**A**α_{=}_{A}αβ_{A}

*β,*where

*Aαβ*_{=}** _{A}**α

_{·}

**β**

_{A}_{and}

**α**_{A}_{·}_{A}*β*=

*δβα*.

The unit normal vector * A*3 is deﬁned as

*3 =*

**A***√*1

*1*

_{A}**A***×*2, and

**A***3*

**A***·*= 0 and

**A**α* A*3

*·*= 0. If we introduce two dimensional permutation tensor

**A**α*eαβ*and

*eαβ*, then we have following useful relations

*=*

**A**α×**A**β*eαβ*3

**A***,*

*=*

**A**α×**A**β*eαβ*3

**A***,*3

**A***×*=

**A**α*eαβ Aβ,*

*3*

**A***×*=

**A**α*eαβ*where the permutation tensors are given by

**A**β,*eαβ*:

*e*12=

*−e*21=

*√*

*A,* *e*11=*e*22= 0; *eαβ* : *e*12=*−e*21 = 1*/*

*√*

*A,* *e*11 _{=}* _{e}*22

_{= 0}

_{,}_{and}

*identity*

_{e}_{−}_{δ}*eαλeβµ*=

*δαβδ*

*λ*
*µ−δαµδλβ*.

Diﬀerentiating* A*3

*·*= 0 with respect to

**A**β*ξα*we get

*3*

**A***,α·*+

**A**β*3*

**A***·*= 0

**A**β,α*.*The second fundamental tensor

*Bαβ*is deﬁned by

*Bαβ*=

*−A*3

*,α·*=

**A**β*3*

**A***·*=

**A**β,α*3*

**A***·*=

**A**α,β*Bβα.*If we introduce the Christofell connection

*Γλ*

*αβ*, the we have derivatives of the base vectors
as follows* Aα,β*=

*=*

**A**β,α*Γαβλ*

*+*

**A**λ*bαβ*3

**A***,*

*3*

**A***,α*=

*−Bαβ*and derivatives of the base vectors

**A**β,

**A**α*,β*=*−Γλβα* * Aλ*+

*bαβ*3

**A***.*If introduce the covariant derivatives of the base vectors

*=*

**A**α|β

**A**α,β−Γ_{αβ}λ*, then we have*

**A**λ*=*

**A**α|β*Bαβ*3

**A***,*

*A*3

*|α*=

*−B*

*β*
*α Aβ.*

of application of the theory. In order to be able required and the determination of these curves for a given surface is, in general, a fairly complicated problem. However, for many of the types of shells used in practice, the geometry of the middle surface is of a simple nature (e.g. surfaces of revolution or cylindrical surfaces), so that the lines of curvature are already known. In such cases the following formulae can be used directly.

As the coordinate curves are lines of curvature, the *ξ*1 _{and} * _{ξ}*2

_{curves are mutually}orthogonal families of curve. It follows that

*1*

**A***·*2= 0. The length of the base vectors ar denoted by

**A***A*(1)and

*A*(2), and we have

*A*(

*α*)=

*|*(

**A***α*)

*|*=

*√*

* Aα·Aα*, which is called Lam´e
parameters of the surface. By thie deﬁnition, we fnd for the length of

*dL*(

*α*)of line elements along the coordinate curves (

*dL*(

*α*))2=

*(*

**A**α·**A**α*dξα*)2=

*A*2

*αdξα*2, namely,

*dL*(

*α*)=

*Aαdξα*. The area

*dΣ*of a small surface bounded by coordinate curves with side lengths

*dL*(1) and

*dL*(2)is given by*dΣ*=*dL*(1)*dL*(2)=*A*1*A*2*dξ*1*dξ*2.

Arbitrary parallel surface is the surface with distance *ξ* to the middle surface. The
position in this surface will be* X*=

*+*

**P***ξ*3, accordingly, the corresponding quantities can be deﬁned and listed in the table 2.

**A****Table 2** Geometry of shell surface in undeformed state

middle surface parallel surface of middle surface

*ξ*= 0 *ξ̸*= 0

Position vector **P****X**

Line element *d P*

*d*

**X**Base vector * Aα,Aα,A*3=

*3*

**A***3=*

**G**α,**G**α,**G***3*

**G***=*

**A**α*=*

_{∂ξ}∂**P**α

**P**,α*=*

**G**α*=*

_{∂ξ}∂**X**α*Unit normal vector*

**X**,α*3=*

**A***√*1

*1*

_{A}**A***×*2

**A***3=*

**G***√*1

*1*

_{G}**G***×*2 Lam´e parameters

**G***A*(

*α*)= (

*)1*

**A**α·**A**α*/*2

*G*(

*α*)= (

*)1*

**G**α·**G**α*/*2

1st fund. form *d P·dP*=

*Aαβdξαdξβ*

*d*=

**X**·d**X***Gαβdξαdξβ*

*Aαβ*=**A**α·**A**β*Gαβ*=* Gα·Gβ*
2nd fund. form

*d*3=

**P**·d**A***Bαβdξαdξβ*

*d*3=

**X**·d**G***Cαβdξαdξβ*

*Bαβ*=*− A*3

*,α·*=

**A**β*3*

**A***·*

**A**α,β*Cαβ*=

*−*3

**G***,α·*=

**G**β*3*

**G***·*

**G**α,βThe base vectors in the undeformed state* G*3 =

*=*

**X**,ξ*3, and*

**A***=*

**G**α*=*

**X**,α*+*

**P**,α*ξ A*3

*,α*=

*+*

**A**α*ξ*3

**A***,α*=

*=*

**A**α−ξBαβ**A**β*µβα*=

**A**β*µαβ*, where the shiter tensor

**A**β*µαβ*=

*Aαβ−ξBαβ*,*µβα*=*δβα−ξBβα*, and metric tensor*Gαβ*=*µλαµλβ*=*Aλρµλαµρβ*,*Gα*3= 0*, G*33=

1.√*G/A*= 1_{2}*δαβρµµραµµβ*= 1*−*2*ξH*+ (*ξ*)2*K*, mean curvature*H*=

1
2*B*

*α*
*α*=12(*B*

1

1+*B*22), the
Gauss curvature*K*=1_{2}*δαβρµBαρBµβ*=*B*

1

1*B*22*−B*12*B*12.

**2.3 Diﬀerential geometry of shell surfaces in the deformed state**

**Table 3** Geometry of shell surface in deformed state

middle surface parallel surface of middle surface

*ξ*= 0 *ξ̸*= 0

Position vector **p****x**

Line element *d p*

*d*

**x**Base vector * aα,aα,a*3=

*3*

**a***3=*

**g**α,**g**α,**g***3*

**g***=*

**a**α*=*

_{∂ξ}∂**p**α

**p**,α*=*

**g**α*=*

_{∂ξ}∂**x**α*Unit normal vector*

**x**,α*3=*

**a***√*1

*1*

_{a}**a***×*2

**a***3=*

**g***√*1

*1*

_{g}**g***×*2

**g**Lam´e parameters *a*(*α*)= (* aα·aα*)1

*/*2

*g*(

*α*)= (

*)1*

**g**α·**g**α*/*2 1st fund. form

*d*=

**p**·d**p***aαβdξαdξβ*

*d*=

**x**·d**x***gαβdξαdξβ*

*aαβ*=**a**α·**a**β*gαβ*=* gα·gβ*
2nd fund. form

*d*3=

**p**·d**a***bαβdξαdξβ*

*d*3=

**x**·d**g***cαβdξαdξβ*