Kirchhoff-Love Plate Theory

5 Kirchhoff-Love Plate Theory

In this chapter we will discuss the Kirchhoff-Love plate model, where the current director vector a3 is forced to be of unit length and orthogonal to the deformed surface. The advantage of this model is that the unknowns (both kinematic and force unknowns) are fewer than in the Cosserat and Reissner-Mindlin models. From the kinematic point of view, for instance, there are no more than 3 unknown kinematic fields, namely the displacement vector of the mid-surface. The director vector being completely determined from the displacement field, it is not an additional unknown. However, to set against this we have the fact that establishing equations for the Kirchhoff-Love model, as well as the final equations, is more complicated. 5.1. Current position of the plate mid-surface The description of the initial position of the plate is common to all plate models and has been discussed in Chapter 2. We will now describe the deformed position of the mid-surface in a Kirchhoff-Love plate. The current position, at a given time t, of a particle P0 located on the initial mid-surface is defined by the mapping φ( . , t) : P0 → P = φ(P0 , t), or, again, by the composite of φ( . , t) with the parametrization [2.1] of S 0 (see Fig. 3.1) : Bounded domain e ⊂ R2 → plane Oe1 e2 → (ξ1 , ξ2 )

→

P0

E

→ P(ξ1 , ξ2 , t) = φ(P0 (ξ1 , ξ2 ), t)

[5.1]

Definition. The displacement vector of the mid-surface, denoted by U, is, by definition U(ξ1 , ξ2 , t) ≡ P(ξ1 , ξ2 , t) − P0 (ξ1 , ξ 2 )

[5.2]

Definition. At every point P ∈ S , we define two vectors tangent at P to the deformed midsurface, S : aα ≡

∂P ∂ξα

, α ∈ {1, 2}

From relationship P = P0 + U we immediately derive:

[5.3]

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Nonlinear Theory of Elastic Plates

Theoreom. ∀α ∈ {1, 2}, aα = Aα + U,α

[5.4]

The following theorem is another way of writing the above relationship between aα and Aα and it is proven exactly as in [3.5] : Theorem. ∀α ∈ {1, 2}, aα = F.Aα

where F = I + gradU

[5.5]

Hypothesis. All the points on the current mid-surface S are regular. Thus, the vectors a1 , a2 are linearly independent at all the points (ξ1 , ξ2 ). The vectors a1 , a2 thus define the plane tangent to S at the point P. Definition. At every point P ∈ S , the unit normal to the deformed mid-surface is, by definition, n≡

a1 × a2 a1 × a2 

⊥a1 , a2

[5.6]

Unlike the Cosserat and Reissner-Mindlin plate models, the Kirchhoff-Love model requires the introduction of several new variables on the current mid-surface S . However, this being said, all the new variables are analogous in every respect to those introduced on the initial surface S 0 in section 2.1, and it is, therefore, not necessary to present them in detail. Only the most important results are listed below. 5.1.1. First fundamental form of S - Primal basis and dual basis Definition. The first fundamental form of S is defined by the components ∀α, β ∈ {1, 2},

aαβ ≡ aα .aβ

[5.7]

The component aαβ has the same dimension as Aαβ . In tensor theory, when we work in 3D space we have a 3-vector basis at each point and we can build up its dual, also made up of three vectors. Here, because we stay on the surface S (embedded in the 3D space) we have only two independent vectors a1 , a2 instead of three. Nevertheless, we can follow the same steps as in 3D (ref. tensor algebra in section 1.1) to define the dual vectors of a1 , a2 and to build a dual basis made up of only 2 vectors. In order to do this, let us introduce the following notations: Notations. – Knowing the coefficients aαβ in [5.7], we denote the following matrix by [a. . ](2×2) :  [a. . ](2×2) ≡

a11 a12 a21 a22

 [5.8]

(the 2 × 2 index is a reminder that this is a matrix of dimensions 2 × 2, not 3 × 3). It is a (symmetric and) invertible matrix, as the vectors a1 , a2 are linearly independent.

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85

– The inverse of the above matrix is denoted by [a. . ](2×2) ≡ [a. . ]−1 (2×2) ≡





a11 a12 a21 a22

[5.9]

(symmetric square matrix of dimension 2 × 2). It can be verified that, ∀α, β ∈ {1, 2}, the dimension of the component aαβ is the inverse of that of aαβ , i.e. of Aαβ . Knowing the coefficients aαβ , we can define the dual vectors of a1 , a2 as follows: Definition. The dual vectors of a1 , a2 , denoted by a1 , a2 , are defined by ∀α, β ∈ {1, 2},

aα = aαβ aβ

[5.10]

We can easily verify that the vectors a1 , a2 are linearly independent and that they satisfy the following relationships ∀α, β ∈ {1, 2},

aα .aβ = δβα

Definition. The basis (a1 , a2 ) is said to be the dual basis of the basis (a1 , a2 ). In contrast, the basis (a1 , a2 ) is called the primal basis. Even though the pairs (a1 , a2 ) and (a1 , a2 ) cannot be bases in 3D space, the term ’basis’ is
out of habit and is justified if we restrict ourselves to the plane that is tangent to S at P, still used which is a 2D space. The basis (a1 , a2 ) and the dual basis (a1 , a2 ) are represented in Fig. 5.1. 








 










Figure 5.1: Local basis (a1 , a2 ) and its dual basis 5.1.2. Second fundamental form of S - Curvature tensor of S Physically, the curvature of a surface is related to the fact that the direction of the normal vector varies from one point to the other when travelling along coordinate lines. The concept here is to measure the curvature of S using the rate of variation of the normal n when a single curvilinear coordinate, ξ1 or ξ2 , varies. Definition. The second fundamental form of S is defined by the components ∀α, β ∈ {1, 2},

bαβ ≡ −n,α .aβ

[5.11]

It can be verified that, ∀α, β ∈ {1, 2}, the dimension of the component bαβ is that of aαβ (that is, of Aαβ ) divided by a length.

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Nonlinear Theory of Elastic Plates

Note that n.n = 1 as n is of unit length. By differentiating this equality with respect to a given curvilinear coordinate ξα , we obtain 0 = (n.n),α = 2n.n,α , that is n,α ⊥n : the vector n,α belongs to the plane tangent to S . Theorem. Other expressions for coefficients bαβ are bαβ = −n,α .aβ = n.aβ,α  bβα = −n,β .aα = n.aα,β

[5.12]

which show, in particular, that bαβ = bβα . Proof. By differentiating the equality n.aβ = 0 with respect to ξα , we obtain n.aβ,α +n,α .aβ = 0. Hence the first line of [5.12]. ∂ ∂P ∂ ∂P Moreover, aβ,α = α β = β α = aα,β . This entails the symmetry bαβ = bβα and the ∂ξ ∂ξ ∂ξ ∂ξ second line of [5.12].  Fig. 5.2 depicts the variation in the unit normal n between two close points P and P , all along a coordinate line. It is because of the curvature of S that the variation dn of vector n has a non-zero component in the plane tangent to S at P and, also, that the variation daα of the (non-unit) vector
aα has a non-zero component along the normal. The negative sign in the equality −n,α .aβ = n.aβ,α is also geometrically justified in Fig. 5.2.

















Figure 5.2: Effect of the curvature of S Definition. The curvature tensor of S is, by definition: b ≡ bαβ aα ⊗ aβ

[5.13]

(the components aα .b.aβ in the tangent plane are equal to bαβ and the out-of-plane components are taken to be equal to zero). The curvature tensor b is symmetric. Theorem. Below is an intrinsic expression for b : b = −n,α ⊗ aα = −aα ⊗ n,α

[5.14]

where again the symmetry of b can be seen. Proof. Using [5.12], we can write the tensor b in two forms: b ≡ bαβ aα ⊗ aβ = −(n,β .aα )aα ⊗ aβ = −[(aα ⊗ aα )n,β ] ⊗ aβ  =I−n⊗n β α

=

n.n,β =0

QED

b ≡ bαβ aα ⊗ aβ = −aα ⊗ (n,α .aβ )aβ = −a ⊗ [(a ⊗ aβ )n,α ] = QED n.n,α =0  =I−n⊗n



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Theorem. The mixed components of the curvature tensor b have the following expressions: bα β = bβ α

=

denoted thus by

bβα = −n,α .aβ = n.aβ,α

[5.15]

As bα β and bβ α are equal, we can simply denote them by bβα , without distinguishing the order of the indices α, β. In practice, we calculate the mixed components bβα by bβα = bαλ aλβ = aβλ bλα

[b.. ] = [a. . ](2×2) [b. . ] (product of 2 × 2 matrices)

or

[5.16] Moving from the component relationship to the matrix relationship we respect uprow-lowcol convention, mentioned earlier in [1.37]. Thus, here β is the row number and α the column number! We also deduce from [5.16] that, regardless of the dimension of the curvilinear coordinates ξ1 , ξ2 , the dimension of the components bβα is always the inverse of a length. Proof. As relation [5.16] is straightforward, we only need to prove [5.15]. (i) Let us first prove the first equalities of [5.15]. We have  ⇒ bα β = aα .b.aβ = −n,α .aβ [5.14] : b = −aλ ⊗ n,λ λ [5.14] : b = −n,λ ⊗ a ⇒ bβ α = aβ .b.aα = −n,α .aβ hence the desired result.



bα β = bαλ aλβ . As aλβ = aβλ and bλα = bαλ , we can deduce bβ α = aβλ bλα = −n,α aλ aλβ . Hence the first equalities of [5.15]. 

Second proof. Let us write that bα β = bβ α = bαλ aλβ

aβ

(ii) By differentiating relationship n.aβ = 0 with respect to ξα , we obtain n,α .aβ + n.aβ,α = 0. Hence the last equality in [5.15].  The Weingarten equation below gives the decomposition of vector n,α in the local basis (a1 , a2 ) or (a1 , a2 ) : Theorem. Weingarten equation on the current mid-surface S : n,α = −bαβ aβ = −bβα aβ

[5.17]

Proof. It has been seen that the vector n,α belongs to the tangent plane and we can, thus, write n,α = cβ aβ or = cβ aβ , where the coefficients cβ and cβ are given by  β c = n,α .aβ = −bβα according to [5.15]  cβ = n,α .aβ = −bαβ according to [5.11] Calculating the curvature components bαβ , bαβ The curvature components bαβ , bβα must be calculated precisely as they are important quantities in Kirchhoff-Love plate theory. It will be seen that they come into play in the governing equations and boundary conditions [5.84]-[5.89], either explicitly or via the internal forces Rαβ , Lαβ .

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Nonlinear Theory of Elastic Plates

In order to understand the dependency of Rαβ , Lαβ on curvatures, we shall anticipate some results proven subsequently. Relation [5.60] shows that: – the forces Rαβ , Lαβ explicitly depend on bαβ , – they also depend implicitly on bαβ , as the bending moments M αβ are functions of changes in curvature κγδ = −bγδ (relation [5.40]) via the constitutive law [6.71].

There are two ways of calculating the curvature components: 1. First method : (a) Knowing the parametrization [5.1] of the current mid-surface of the plate, we start by calculating the vectors a1 , a2 using [5.4]. From this we deduce the normal n as a function of ξ1 , ξ2 using [5.6]. (b) We then calculate the 4 vectors aα,β by differentiating aα with respect to ξβ . (c) We deduce the 4 components bαβ using [5.12], bαβ = n.aα,β . We deduce the 4 components bβα using [5.16], bβα = bαλ aλβ . 2. Second method : (a) The first step is the same as in the previous method: we begin by calculating the vectors a1 , a2 using [5.4] and from this we then deduce the normal n as a function of ξ1 , ξ2 , using [5.6]. (b) We then calculate the 2 vectors n,α by differentiating n with respect to ξα . (c) Knowing n,α , we deduce the 4 components bαβ using [5.11], bαβ ≡ −n,α .aβ . The 4 components bβα are deduced using [5.16], bβα = bαλ aλβ , or are directly obtained using [5.15], bβα = −n,α .aβ . The second method is effective when the analytical expression of the derivatives n,α is easy to obtain. When this is not the case, the first method is preferable. Thus, in the case of finite elements, for example, the parametrization [5.1] is known piecewise only, or to be more precise, element by element. And the analytical expression for the normal n in terms of ξ1 , ξ2 – a fortiori the derivative n,α – are complicated. In this case, we use the first method to calculate n,α . Remarks on the calculation of the derivatives n,α : – Let us point out that in finite elements, there is also a special formula adapted for the numerical computation of n,α . – If we follow the first method, the Weingarten equation [5.17] can be used to derive n,α , knowing the curvatures bαβ , bβα . If we follow the second method, we calculate n,α by differentiating n with respect to ξα and, in this case, the Weingarten equation is not useful.

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89

5.1.3. Third fundamental form of S Definition and theorem. The third fundamental form of S is the symmetric form defined by ∀α, β ∈ {1, 2},

cαβ ≡ n,α .n,β = bαλ bλβ = bλα bλβ

[5.18]

The matrix of the components Cαβ is obtained by [c. . ] = [b. . ][b.. ] = [b. . ][a. . ](2×2) [b. . ] (products of 2 × 2 matrices)

[5.19]

Proof. We have     cαβ ≡ n,α .n,β = bαλ aλ . bβμ aμ from the Weingarten equation [5.17] this is the 2nd equality in [5.18] = bαλ bβμ aλμ  =bλβ

By permuting α and β in the 2nd equality in [5.18] obtained above, and by taking into account cαβ = cβα , we arrive at the final equality in [5.18]. The matrix relation [5.19] is obtained by combining [5.18] and [5.16].  ∀α, β ∈ {1, 2}, the dimension of the component cαβ is that of aαβ (that is, of Aαβ ) divided by a squared length. 5.2. Current position of the plate - Displacement field The current position of a Kirchhoff-Love plate is defined as for a Reissner-Mindlin plate but with the additional condition that the current director vector a3 in the Reissner-Mindlin model is constrained, here, to be equal to the unit normal to the current surface n: Kirchhoff-Love kinematics hypothesis ∀P0 ∈ S 0 , ∀Q0 ∈ the normal fiber passing through P0 , the vector P0 Q0 = Ze3 transforms into PQ = Zn

[5.20]

see Fig. 5.3. The normal vector n, image of the initial director vector e3 under a certain (with no explicit expression) rotation tensor R is called the current director vector, Thus, every normal fiber in the initial position transforms itself during the motion into a rectilinear fiber of the same length and which is orthogonal to the deformed mid-surface. The theory discussed in this chapter is called the Kirchhoff-Love plate theory, and it leads to the Kirchhoff-Love plate model. This theory is constructed based only on the kinematics assumption [5.20] and is not subject to any other restriction. It is valid for finite displacements, finite rotations and finite strains. It is the working framework we will use for this chapter. Having said this, in practice the kinematic hypothesis [5.20] is better verified if the plate is ’thinner’ and it turns out that in a ’thin’ plate, the strains can remain small even for large rotations.

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Nonlinear Theory of Elastic Plates 


















































































Figure 5.3: Kinematics of the Kirchhoff-Love plate Displacement field Noting that U(Q0 , t) = Q − Q0 = (Q − P) + (P − P0 ) +(P0 − Q0 ) = U + PQ − P0 Q0  ≡U

recalling from definition [5.2] that U = U(ξ1 , ξ2 , t) is the displacement vector of the mid-surface, we have the following equivalence: Theorem. The Kirchhoff-Love kinematics hypothesis [5.20] is tantamount to assuming that the displacement of an arbitrary particle Q0 is U(Q0 , t) = U + Z(n − e3 )

[5.21]

On account of [5.4] and [5.6], the displacement field in the plate, or equivalently, the current position of the plate, is determined uniquely by the three scalar fields U(ξ1 , ξ2 , t) . Natural covariant bases and dual bases in the current configuration The natural covariant basis and its dual basis are defined on the deformed position, as done for the initial position (section 2.2.2) and for the deformed position of Cosserat plates (section 3.2). Definition. At a point Q in the current volume of the plate, we define the vectors of the natural basis associated with the coordinates (ξ1 , ξ2 , ξ3 ≡ Z) by: gi ≡

∂Q ∂ξi

, i ∈ {1, 2, 3}

(g1 , g2 , g3 ) is called the natural covariant basis at point Q.

[5.22]

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91

Theorem. ∀α ∈ {1, 2}, gα = aα + Zn,α = aα − Zbβα aβ = (δβα − Zbβα )aβ g3 = n

[5.23]

Relations [5.23] show that the vectors g1 , g2 are in the plane tangent to the mid-surface at the point P(ξ1 , ξ2 ), while the vector g3 is orthogonal to this plane. Proof. We have ∂Q ∂P = + Zn,α where n,α = −bβα aβ in virtue of the Weingarten equation [5.17] ∂ξα ∂ξα ∂Q =n  g3 ≡ ∂Z [5.24] gα ≡

We can now introduce the following definition: Definition and property. We define a3 ≡ n , such that g3 = a3 and such that we can deduce from [5.23] that ∀i ∈ {1, 2, 3}, lim gi = ai : when the point Q Z→0

tends towards P ∈ S , the basis (g1 , g2 , g3 ) tends towards (a1 , a2 , a3 ). This enables us to call (a1 , a2 , a3 ) the natural covariant basis at the point P ∈ S .

[5.25]

Fig. 5.4 shows that the relative positions of the local bases (a1 , a2 , a3 ) and (g1 , g2 , g3 ) in the Kirchhoff-Love model are much simpler than those of the Cosserat and Reissner-Mindlin models, cf. Fig. 3.3.

   





  





















  


Figure 5.4: Local bases (a1 , a2 , a3 ) and (g1 , g2 , g3 ) with Kirchhoff-Love plate model As in [3.17] for the Cosserat plates: Definition. ∀i, j ∈ {1, 2, 3},

gi j ≡ gi .g j

[5.26]

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Nonlinear Theory of Elastic Plates

Here, with the Kirchhoff-Love model, the matrix [g. . ] has a particular form that resembles that of [2.22] : ⎤ ⎡ ⎢⎢⎢ g11 g12 ⎥⎥⎥ ⎥ ⎢⎢ ⎢ [g. . ] = ⎢⎢ g21 g22 ⎥⎥⎥⎥ [5.27] ⎦ ⎣ 1 The dimension of a component gαβ , α, β ∈ {1, 2}, is the same as that of aαβ (that is, the same as Aαβ ). As ξ3 ≡ Z has the dimension of a length, g33 = 1 is dimensionless. Theorem. The coefficients gαβ are 2nd-degree polynomials in Z, whose coefficients are the first, second and third fundamental forms of S : ∀α, β ∈ {1, 2},

gαβ = aαβ − 2bαβ Z + cαβ Z 2

[5.28]

Proof.

 "  ! gαβ ≡ gα .gβ = aα + Zn,α . aβ + Zn,β = aα .aβ + Z(n,α aβ + n,β .aα ) + Z 2 n,α .n,β    ≡−bαβ

≡−bβα



≡cαβ

According to the form [5.27] of matrix [g. . ] and the fact that this matrix is invertible, we deduce the form of its inverse:   ⎤ ⎡ 11 12 ⎤ ⎡ 1 g22 −g12 ⎢⎢⎢ g g ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ ⎢ ⎥⎥⎥ ⎥ ⎢ [g. . ] = ⎢⎢⎢⎢ g21 g22 ⎥⎥⎥⎥ = ⎢⎢⎢⎢ det[gαβ ] −g21 g11 [5.29] ⎥⎥⎦ ⎣ ⎦ ⎣ 1 1 The dimension of a component gαβ , α, β ∈ {1, 2}, is that of aαβ , that is, the inverse of that of aαβ or of Aαβ . The component g33 = 1 is dimensionless. Definition. The dual basis (g1 , g2 , g3 ) at the point Q is defined based on the natural covariant basis (g1 , g2 , g3 ) by ⎧ ⎪ α αβ ⎪ ⎪ ⎪ g = g gβ , α ∈ {1, 2} ⎨ ∀i ∈ {1, 2, 3}, gi = gi j g j that is ⎪ [5.30] ⎪ ⎪ ⎪ ⎩ g3 = g 3

• We proceed in the same manner with the basis (a1 , a2 , a3 ) instead of the basis (g1 , g2 , g3 ) : 1. We define the coefficients ai j ≡ ai .a j . ⎡ ⎢⎢⎢ a11 a12 ⎢ 2. We construct the matrix [a. . ] which is of the form [a. . ] = ⎢⎢⎢⎢ a21 a22 ⎣ 3. As the matrix [a. . ] is invertible, ⎡ 11 12 ⎤ ⎢⎢⎢ a a ⎥⎥⎥ ⎢ ⎥ .. [a ] = ⎢⎢⎢⎢ a21 a22 ⎥⎥⎥⎥ . ⎣ ⎦ 1

1

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎦ .

we can deduce the form of its inverse:

4. Finally, we define the vectors of the dual basis as follows:

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93

Definition. The dual basis (a1 , a2 , a3 ) at the point P is defined based on the natural covariant basis (a1 , a2 , a3 ) by ⎧ ⎪ α αβ ⎪ ⎪ ⎨ a = a aβ , α ∈ {1, 2} ∀i ∈ {1, 2, 3}, ai = ai j a j that is ⎪ [5.31] ⎪ ⎪ a3 = a ⎩ 3

The dual bases (a1 , a2 , a3 ) and (g1 , g2 , g3 ) are related through the same relation as [5.25] : Theorem. ∀i ∈ {1, 2, 3}, lim gi = ai : when the point Q tends towards P ∈ S , the dual basis at Z→0

Q tends towards that at P.

[5.32]

Proof. We have the following successive implications: The continuity property [5.25] ⇒ lim gi j = ai j Z→0

⇒ lim [g. . ] = [a. . ] Z→0

⇒ the continuity on the inverse matrices: lim [g. . ] = [a. . ] Z→0

Hence we deduce, taking the limit for gi in definition [5.30] : lim gi = lim gi j lim g j

Z→0

Z→0

Z→0

= ai j a j = ai from definition [5.31]



Note the equalities a3 = g3 = a3 = g3 = the unit normal n to the current mid-surface S . From this point onwards, we will write a3 or n interchangeably. Gauss formula The results below are related to the mid-surface S , but we can only prove them now, after having exited the surface S to define ξ3 , a3 , a3 , g3 . The Christoffel symbols γikj have been defined in [1.52]. In the sequel, we will need these symbols evaluated on the mid-surface S , that is at ξ3 = 0, and we will denote them by γ¯ ikj with a bar over it: Notation. ∀i, j, k ∈ {1, 2, 3}, γ¯ ikj ≡ γikj (ξ1 , ξ2 , 0) The following theorem gives the values for some specific γ¯ ikj symbols: Theorem. ∀α, β ∈ {1, 2}, 3 = bαβ γ¯ αβ α γ¯ β3 = −bαβ 3 γ¯ 3α = 0

[5.33]

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Nonlinear Theory of Elastic Plates

Proof. We have ∂aα ∂gα |ξ3 =0 ∂gα λ 3 3 = = β  = γ¯ k ak = γ¯ αβ aλ + γ¯ αβ a3 ⇒ bαβ ≡ aα,β .a3 = γ¯ αβ ∂ξβ ∂ξα ∂ξ ξ3 =0 [1.52] αβ α ∂aα ∂g|ξ3 =0 ∂gα α λ α 3 α = = β  = −¯γα ak = −¯γβλ a − γ¯ β3 a ⇒ bαβ ≡ a3 .aα,β = −¯γβ3 a3 .a3  ∂ξβ ∂ξα ∂ξ ξ3 =0 [1.54] βk

[5.34]

=1

Similarly, we have ∂g ∂a3 β 3 = 3 = γ¯ k ak = γ¯ 3α aβ + γ¯ 3α a3 ∂ξα ∂ξα [1.52] 3α 3 Upon comparing this relation with the Weingarten equation [5.17], we see that γ¯ 3α must be equal to 0. 

The following Gauss formula is homologous to [2.29] on the initial mid-surface and complementary to the Weingarten equation [5.17] : Theorem. Gauss equation on the current mid-surface S : derivatives of the basis vectors on S . ∀α, β ∈ {1, 2},

λ γ¯ αβ

λ aα,β = γ¯ αβ aλ + bαβ a3 α α λ a,β = −¯γβλ a + bαβ a3

[5.35]

Thus, the components in the tangent plane of aα,β or aα,β are given by the Christoffel symbols α or γ¯ βλ , while the components normal to the plane are given by bαβ or bαβ .

Proof. Just return to the expressions for aα,β and aα,β in [5.34], taking into account the abovefound Eqs. [5.33].  5.3. Strain tensor The Green-Lagrange strain tensor E is given by E = Ei j Ai ⊗ A j

with Ei j =

1 1 (gi j − Gi j ) = (gi j − Ai j ) 2 2

We will express the components Ei j as functions of the kinematic fields. While these expressions are not essential in this chapter, where we use the PVP to obtain the governing equations, they will be required in Chapter 6 where we will express the stresses as functions of the kinematic fields via the constitutive law. Theorem. ∀α, β ∈ {1, 2},

Eαβ =

  1 aαβ − Aαβ + Z aα .a3,β + a3,α .aβ + Z 2 a3,α .a3,β 2

[5.36]

Thus, Eαβ is a 2nd-degree polynomial in Z of the form (0) (1) (2) ∀α, β ∈ {1, 2}, Eαβ = Eαβ + ZEαβ + Z 2 Eαβ

(0) where Eαβ ≡

surface.

[5.37]

1 (aαβ − Aαβ ) , equal to Eαβ taken at Z = 0, is the strain component of the mid2

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95

The other components of strain are zero: ∀α ∈ {1, 2}, Eα3 = 0

and

E33 = 0

[5.38]

Proof. The reasoning is the same as for the Reissner-Mindlin plate. The only new feature here is that aα3 = 0 as a3 ⊥a1 , a2 . Using [5.23], we successively obtain ∀α, β ∈ {1, 2}, gαβ ≡ gα .gβ = (aα + Za3,α ).(aβ + Za3,β ) = aαβ + Z(aα .a3,β + a3,α .aβ ) + Z 2 a3,α .a3,β gα3 ≡ gα .g3 = (aα + Za3,α ).a3 = aα3 + Za3,α .a3 = 0 g33 ≡ g3 .g3 = a3 .a3 = 1  Expression [5.36] for the components Eαβ is the same [3.24] as in Cosserat theory, except that here the vector a3 is of unit length and ⊥a1 , a2 . Relations [5.38] result from the kinematics hypothesis, according to which the normal fiber is transformed during motion as a rigid inextensible body, orthogonal to the deformed mid-surface. It should be noted that in any curvilinear coordinates the strain components are not necessarily dimensionless, as has already been seen after [3.28]. The following theorem gives us another expression for Eαβ : Theorem. ∀α, β ∈ {1, 2}, Eαβ =

1 aαβ − Aαβ − 2Zbαβ + Z 2 cαβ 2

[5.39]

Proof. We just need to use the Weingarten equation [5.17] to replace the derivatives a3,α in [5.36] by a3,α = −bαβ aβ = −bβα aβ .  1 Second proof. Calculate Eαβ using Eαβ = (gαβ − Aαβ ) and using expression [5.28] for gαβ .  2 Relation [5.39] shows explicitly that the curvature tensor terms come into play in the strain tensor. This is why we need to carefully calculate curvatures. (1) Definition. The bending strains Eαβ in [5.37] are usually renamed καβ for brevity: (1) = −bαβ καβ ≡ Eαβ

[5.40]

These strains – the negatives of curvatures bαβ – are also called the changes of curvature of the midsurface . These strains can be expressed as functions of the displacement U of the mid-surface. To do a1 × a2 (relation [5.6]) and aα by Aα + U,α (relation a1 × a2  [5.4]). However, the expressions obtained are quite lengthy and cannot be used except when we consider the linearized strain tensor, as will be seen in section 7.6. this, one just has to replace a3 by a3 =

Example. Consider again the example of the bending of a metal sheet discussed in section 1 (0) ≡ (aαβ − Aαβ ) of the mid-surface is zero. Let us 3.4. It has been seen therein that the strain Eαβ 2 calculate the strain in the volume of the sheet by assuming that the sheet follows Kirchhoff-Love kinematics. From deformation [3.34] we can deduce the vectors of the current natural basis a1 =

∂P = e1 ∂X

a2 =

∂P = −eθ ∂Y

⇒

a3 =

a1 × a2 = er a1 × a2 

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Nonlinear Theory of Elastic Plates

Hence a3,1 = 0, a3,2 = − R1 eθ . Then, on applying [5.36], we find E22 =

Z Z2 , + R 2R2

the other Eαβ = 0



5.4. Velocity field The tool used to establish the plate governing equations is the principle of virtual power (PVP) whose expression in terms of Lagrangian variables is given in [3.36]. In preparation of the PVP, we will calculate the field of real velocities in this section: 1. the real velocity makes it possible to calculate the acceleration in the PVP, 2. the virtual velocity field involved in the PVP will be chosen analogous to the field of real velocities. The real velocity is obtained by differentiating [5.21] with respect to time: ˙ 0 , t) = U ˙ + Z a˙ 3 U(Q

[5.41]

˙ is, in fact, the velocity V ≡ V(P0 , t) of the particle of initial position P0 . The derivative U ∂U ∂P Indeed, knowing that P = P0 + U, we have V = = . As concerns a˙ 3 , it is given by the ∂t ∂t following theorem: Theorem. The derivative of vector a3 with respect to time is: " ! a˙ 3 = − V,α .a3 aα

⊥a3

[5.42]

˙ Proof. Differentiating the equation n.n = 1 with respect to time gives n.n˙ = 0, or n⊥n. We can, thus, decompose the vector n˙ in the tangent plane as n˙ = cα aα , where the coefficients ˙ α are to be determined. cα = n.a ˙ α + n.˙aα = 0. Hence By differentiating equality n.aα = 0 with respect to time, we obtain n.a cα = −n.˙aα . ∂P By differentiating definition [5.3] with respect to time, aα ≡ α , we have ∂ξ a˙ α =

∂ ∂P ∂ ∂P ∂V = = α ∂t ∂ξα ∂ξα ∂t ∂ξ



[5.43]

5.5. Virtual velocity field We decide to give the virtual velocities the same form as the real velocities [5.41]: U∗ (Q0 ) = U∗ + Za∗3 In this expression : – U∗ ≡ U∗ (P0 ) is the virtual velocity vector of the mid-surface and is an arbitrary function of (ξ1 , ξ2 ). – We decide to choose a∗3 in the same form as a˙ 3 in relation [5.42], namely:   a∗3 = − U∗,α .a3 aα ⊥a3 Thus, a∗3 is completely defined by the virtual velocity field U∗ .

[5.44]

Kirchhoff-Love Plate Theory

97

This gives us the virtual velocity field when we work with the Kirchhoff-Love model:   U∗ (Q0 ) = U∗ − Z U∗,α .a3 aα

[5.45]

Thus, the virtual velocity field is determined uniquely by the virtual velocity field U∗ (ξ1 , ξ2 ) of the mid-surface (3 scalar functions). 5.6. Virtual powers of inertia forces Further down, we will calculate the different terms of the PVP [3.36], namely the virtual power of inertia forces, internal and external forces.. The same calculation as used in Cosserat theory leads to expression [3.41] for the virtual power of internal forces:    ¨ + ρ0 H (1) a¨ 3 .U∗ + (ρ0 H (1) U ¨ + ρ0 H (2) a¨ 3 ).a∗ dS 0 P∗accel = ρ0 H (0) U 3 S0

Here, in Kirchhoff-Love theory, the vector a∗3 is of the form [5.44]. Hence P∗accel =

      ¨ + ρ0 H (1) a¨ 3 .U∗ − ρ0 H (1) U.a ¨ α + ρ0 H (2) a¨ 3 .aα a3 .U∗,α dS 0 ρ0 H (0) U S0

Using [2.35] to carry out the integration by parts, we obtain   √    ¨ + ρ0 H (1) a¨ 3 .U∗ + √1 ¨ α + ρ0 H (2) a¨ 3 .aα a3 A ρ0 H (1) U.a ρ0 H (0) U P∗accel = S0 A  ¨ α + ρ0 H (2) a¨ 3 .aα a3 .U∗ ν0α ds0 − ρ0 H (1) U.a

,α

 .U∗ dS 0

∂S 0

where ν0β ≡ ν0 .eβ , ν0 is the outward unit vector normal to ∂S 0 and belonging to the plane tangent to S 0 , Fig. 2.7. Note that the Weingarten equation [5.17], a3,α = bλα aλ , enables one to write for any scalar function f :  1 √ 1 √  3 A f a3 = √ A f a − bλα f aλ √ ,α ,α A A

[5.46]

¨ α + ρ0 H (2) a¨ 3 .aα and after some changes to the By applying this relation to f = ρ0 H (1) U.a indices, we finally arrive at the virtual power for inertia forces  

 ¨ + ρ0 H (1) a¨ 3 .U∗ dS 0 ρ0 H (0) U S0   1 √  ¨ α + ρ0 H (2) a¨ 3 .aα + A ρ0 H (1) U.a a3 √ ,α S 0 A   ¨ β + ρ0 H (2) a¨ 3 .aβ aα .U∗ dS 0 −bαβ ρ0 H (1) U.a   ¨ α + ρ0 H (2) a¨ 3 .aα a3 .U∗ ν0α ds0 ρ0 H (1) U.a −

P∗accel =

[5.47]

∂S 0

Remark. In practice, the Kirchhoff-Love theory is made for thin plates and we can ignore the terms in ρ0 H (1) and ρ0 H (2) . However, having said this, we will not make these approximations in this chapter and it is up to the reader to make approximations as per their convenience. 

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Nonlinear Theory of Elastic Plates

5.7. Virtual power of internal forces The same calculation as used in section 3.10 leads to expression [3.45] for the virtual power of internal forces: P∗int = −

   Rα .U∗,α + Lα .a∗3,α + S.a∗3 dS 0

[5.48]

S0

where the internal forces Rα , Lα and S are defined in [3.44]. Using the divergence theorem [2.35] to carry out integrations by parts, we arrive at the following relation, similar to [3.74] : −P∗int = − +

  

S0

   1  √ α 1  √ α AR .U∗ + √ AL − S .a∗3 dS 0 √ ,α ,α A A   Rα .U∗ ν0α + Lα .a∗3 ν0α ds0

∂S 0 \corners

[5.49]

It will be seen that unlike the Cosserat and Reissner-Mindlin plate theories, it is essential in the Kirchhoff-Love theory to take into account the possible presence of corners along the edge ∂S 0 of the plate, at the spot where the tangent vector is not unique. A circular edge has no corners, while a rectangular edge has 4. According to the  divergence theorem [2.35], we know that when there are corners on ∂S 0 , Fig. 5.5, the integral

∂S 0

is interpreted as the sum of the line integrals

along the smooth curves   that make up ∂S 0 . It is to remember this, that we have written in [5.49] rather than . ∂S 0 \corner

∂S 0















Figure 5.5: The edge ∂S 0 of the plate with the corners Kp , p = 1, 2, . . . The rest of the calculations is specific to the Kirchhoff-Love plate theory. Taking into account the fact that here a∗3 must be in the form [5.44], we have −P∗int =

 

   1  √ α 1  √ β α −√ AR .U∗ + √ AL .a − S.aα a3 .U∗,α dS 0 ,α ,β S0 A A   α ∗ β α 3 ∗ + R .U ν0α − (L .a )a .U,α ν0β ds0 ∂S 0 \corners

[5.50]

Kirchhoff-Love Plate Theory

99

Once again using the divergence theorem [2.35], let us carry out a second integration by parts over [5.50] : 

    1  √ β α A Rα + √ AL .a − S.aα a3 .U∗ dS 0 ,β S0 A ,α      1  √ β α α ∗ α 3 ∗ R .U ν0α + √ + AL .a − S.a a .U ν0α −(Lβ .aα )a3 .U∗,α ν0β ds0 ,β  ∂S 0 \corners A  1
[5.51]

− P∗int = −



1 √ A

√

While the integration by parts in [5.49] is common to all the plate models, the second integration by parts that we just carried out, as well as others below, are unique to the Kirchhoff-Love model.  • In order to process the term 1
in [5.51], let us introduce some preliminary definitions. Definition. [5.52] At a regular point along the edge ∂S 0 , we define the local orthonormal basis (ν0 , s0 , e3 ) as follows (Fig. 5.5) : – the vector e3 is, let us recall, normal to the mid-surface S 0 , – the vector ν0 is the outward unit normal to the edge ∂S 0 , belonging to the plane tangent to S 0 at the point considered, – the vector s0 is the unit tangent vector s0 = e3 × ν0 ⇔ ν0 = s0 × e3 ⇔ e3 = ν0 × s0 . The vectors ν0 and s0 are decomposed in the dual natural basis (A1 , A2 ) as follows ν0 ≡ ν0α Aα ≡ ν0α Aα

s0 ≡ sα0 Aα ≡ s0α Aα

[5.53]

Thus, ν0α = ν0 .Aα , s0α = s0 .Aα , ν0α = ν0 .Aα , sα0 = s0 .Aα . Let us consider the vector field U∗ defined over S 0 , whose gradient tensor is g r a dP0 U∗ = ∂U∗ ⊗ Aα . ∂ξα Definition. At a regular point along the curve ∂S 0 , we define – the normal derivative

∂U∗ ≡ g r a dP0 U∗ .ν0 ∂ν0

– the tangential derivative

∂U∗ ≡ g r a dP0 U∗ .s0 ∂s0

Upon decomposing the vector Aα in the local orthonormal frame (s0 , ν0 ) as Aα = ν0α ν0 +s0α s0 , we can express U∗,α at a regular point on the curve ∂S 0 as a function of the normal and tangential derivatives: U∗,α = g r a dP0 U∗ .Aα = g r a dP0 U∗ .(ν0α ν0 + s0α s0 ) =

∂U∗ ∂U∗ ν0α + s0α ∂ν0 ∂s0

[5.54]

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Nonlinear Theory of Elastic Plates

 The derivatives U∗,α contained in the term 1
of [5.51] are not operable insofar as, at a point ∗ on edge ∂S 0 , U,1 is not generally independent of U∗,2 . The interest of working with the normal and tangential derivatives, rather than with the derivatives U∗,α , is that we know that one derivative can vary independent of the other virtual quantities while the other cannot. Indeed: – The tangential derivative ∂U∗ /∂s0 is not independent of the function U∗ defined on the edge ∂S 0 , because knowing U∗ on ∂S 0 implies that we know ∂U∗ /∂s0 . – At a regular point on the edge ∂S 0 , the normal derivative ∂U∗ /∂ν0 is independent of U∗ along ∂S 0 . Note that, on the contrary, at a corner of the edge, ∂U∗ /∂ν0 is not independent of U∗ over ∂S 0 (we can see this when staying in the initial mid-surface, at a 90◦ corner). We will revisit this dependency after we obtain the boundary conditions [5.88]. Further on, we will preserve the normal derivative ∂U∗ /∂ν0 , on the other hand we will dispose of tangential derivative ∂U∗ /∂s0 . Finally, we introduce the useful following definition for an angular point on the edge ∂S 0 : Definition. Let f be a function defined on the edge ∂S 0 . We assume that ∂S 0 is oriented by the tangent vector s0 . The discontinuity or the jump of f at a point K p on the edge ∂S 0 , denoted  f K p , is defined by  f K p ≡ f |K+p − f |K−p

[5.55]

where f |K+p (resp. f |K−p ) designates the value of f calculated just after (resp. just before) the point K p , the ‘after’ or ‘before’ is understood in accordance with the pre-orientation of ∂S 0 .  • Having introduced the previous definitions, we can now return to the term 1
in [5.51] and ∗ transform it by replacing U,α with the right-hand side of [5.54] :  1
≡ −



(Lβ .aα )(a3 .U∗,α )ν0β ds0   ∂U∗ ∂U∗ (Lβ .aα )(a3 . =− )ν0β ν0α + (Lβ .aα )(a3 . )ν0β s0α ds0 ∂ν ∂s 0 0 ∂S 0 \corners ∂S 0 \corners

A final integration by parts of the term in ∂U∗ /∂s0 leads to   ∂  β α 3 ∗  (L .a )(a .U )ν0β s0α ds0 1
= −  ∂S 0 \corners ∂s0   ∂ ∂U∗ + )ν0β ν0α ds0 (Lβ .aα )a3 ν0β s0α .U∗ − (Lβ .aα )(a3 . ∂ν0 ∂S 0 \corners ∂s0 where

 ∂S 0 \corners

[5.56]

'  ∂  β α 3 ∗ (Lβ .aα )a3 ν0β s0α K p .U∗ (K p ) [5.57] (L .a )(a .U )ν0β s0α ds0 = − ∂s0 corners K p

  Remark. A term like ∂s∂0 (Lβ .aα )(a3 .U∗ )ν0β s0α is the derivative with respect to the curvilinear abscissa s0 defined along the edge ∂S 0 , i.e. the directional derivative along the tangent vector s0 . It must not be calculated by means of the formula ∂(•) ∂s0 ≡ g r a dP0 (•).s0 , since ν0β and s0α are defined on the edge ∂S 0 only, not over the whole surface S 0 . 

Kirchhoff-Love Plate Theory

101

The discontinuity (Lβ .aα )a3 ν0β s0α K p ≡ ((Lβ .aα )a3 ν0β s0α )|K+p − ((Lβ .aα )a3 ν0β s0α )|K−p of (L .aα )a3 ν0β s0α at any corner K p on the edge ∂S 0 is due to the discontinuities of the tangent vector s0 and of the normal vector ν0 at this corner. By grouping [5.51], [5.56]-[5.57], we arrive at the expression for the opposite of the virtual power of internal forces β

     √ 1 1  √ β α A Rα + √ AL .a − S.aα a3 .U∗ dS 0 √ ,β S0 A A ,α   1  √ β α Rα ν0α + √ AL .a − S.aα a3 ν0α ,β ∂S 0 \corners A   ∂  β α 3 + (L .a )a ν0β s0α .U∗ ds0 ∂s 0 ' ∂U∗ − (Lβ .aα )ν0β ν0α (a3 . )ds0 + (Lβ .aα )ν0β s0α a3 K p .U∗ (K p ) ∂ν 0 ∂S 0 \corners corners K

−P∗int = −  +



p

[5.58] recalling that the discontinuities  .  are due to the presence of possible corners K p on the edge ∂S 0 . In the previous expression for the virtual power of internal forces, we have only the following quantities, which can vary independently of one another: – U∗ on the mid-surface S 0 , – U∗ (s0 ) and the normal derivative curvilinear abscissa on the edge).

∂U∗ (s0 ), along the edge ∂S 0 (remember that s0 is the ∂ν0

The components Rαβ and Lαβ Before transforming expression [5.58] for the virtual power of internal forces we will provide some intermediary results. Let us recall expressions [3.46], [3.47] and [3.48] established in Cosserat plate theory: Rβ = N αβ aα + Qβ a3 + M αβ a3,α Lβ = M αβ aα + Q(1)β a3 + M (2)αβ a3,α  H/2 S = Qα aα + Q(1)α a3,α + Σ33 dZa3 −H/2

αβ

where the stress resultants N , M αβ , M (2)αβ , Qα , Q(1)α are defined in [3.49]-[3.50]. Here, in Kirchhoff-Love theory, a3 is orthogonal to a1 , a2 and we can apply the Weingarten equation [5.17] : a3,α = −bλα aλ . Hence   Rβ = N αβ − bαλ M λβ aα + Qβ a3   Lβ = M αβ − bαλ M (2)λβ aα + Q(1)β a3 [5.59]  H/2   S = Qα − bαλ Q(1)λ aα + Σ33 dZa3 −H/2

Definitions. We denote Rαβ ≡ N αβ − bαλ M λβ = aα .Rβ Lαβ ≡ M αβ − bαλ M (2)λβ = aα .Lβ

[5.60]

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Nonlinear Theory of Elastic Plates

Using [3.49]-[3.50], we can get explicit expressions for the above components in terms of components Σi j of the 2nd Piola-Kirchhoff stress tensor Σ = Σi j Gi ⊗ G j . The components N αβ , M αβ and M (2)αβ are symmetric with respect α, β. On the contrary, expressions [5.60] show that in general:Rαβ  Rβα and Lαβ  Lβα . With the notations [5.60], the internal forces Rβ , Lβ in [5.59] write Rβ = Rαβ aα + Qβ a3 Lβ = Lαβ aα + Q(1)β a3

[5.61]

Remark. In practice, Kirchhoff-Love theory is made for thin plates and we can neglect the Z 2 -terms as M (2)αβ as compared to the term M αβ in Z to get Lαβ  M αβ . This being said, we do not make these approximations in this chapter; it is up to the reader to do this according to their convenience.  We do not need to know whether N αβ , M αβ , M (2)αβ , Rαβ , Lαβ are or are not the 2-contravariant components of 2nd-order tensors. One has just to take these as the functions defined by [3.49] and [5.60]. In fact, it is pointless to define the tensors R, L, M(2) whose 2-contravariant components are Rαβ , Lαβ , M (2)αβ , even if it were possible to do so. On the other hand, as will be seen in Chapter 6 when studying of the Kirchhoff-Love constitutive law, we define the membrane force tensor N as N αβ Aα ⊗ Aβ and the bending moment tensor M as M αβ Aα ⊗ Aβ , see [6.67]. This implies that N αβ , M αβ are the 2-contravariant components in the basis Aα ⊗ Aβ of the tensors N, M. However for the present, it is not necessary to know this. The derivative ∦ If we formally consider that Rαβ is the 2-contravariant component of a certain 2nd-order tensor denoted by R, we can apply to it the covariant derivative restricted to S 0 , defined by [2.33] : αλ ¯ β λβ ¯ α Rαβ = Rαβ ,β + R Γλβ + R Γλβ . Or, by changing some indices: β ¯ λ αβ ¯ α λβ = Rαβ Rαβ ,β + Γλβ R + Γβλ R β In the sequel, we will require a similar notation: Notation. The operator denoted by ∦, applied to a function Rαβ of the curvilinear coordinates ξ1 , ξ2 , is defined by: αβ α λβ ¯ λ αβ ¯ βλ R Rαβ ∦β ≡ R,β + Γλβ R + γ

[5.62]

The operator ∦ differs a little from the covariant derivative  : it involves both the Christoffel α symbols Γ¯ αβλ related to the initial surface S 0 and those γ¯ βλ related to the current surface S ! Transforming expression [5.58] After having introduced the components Rαβ , Lαβ and the derivativ ∦, we can now transform expression [5.58] for the virtual power of internal forces. • The following lemma makes explicit the expression which appears twice in [5.58] :

Kirchhoff-Love Plate Theory

103

Lemma.   1  √ β α αβ 3 Rα + √ AL .a − S.aα a3 = Rβα aβ + L∦β a ,β A

[5.63]

Proof. The proof for the lemmas [5.63] and [5.64] makes use of relation [2.31] : Γ¯ λλα . We have

√ ( A),α = √ A

1  √ β AL = Lβ,β + Γ¯ λλβ Lβ where Lβ = Lλβ aλ + Q(1)β a3 according to [5.61] √ ,β A λβ (1)β = L,β aλ + Lλβ aλ,β + Q(1)β a3,β + Γ¯ λλβ Lβ ,β a3 + Q Hence: 1  √ β α αβ β α AL .a = L,β + Lλβ aλ,β .aα + Q(1)β a3,β .aα + Γ¯ λλβ L .a √  ,β A αβ L

α Using the Weingarten equations [5.17] and Gauss equations [5.35], we have aλ,β .aα = γ¯ λβ and a3,β .aα = −bαβ . Hence, with expression [5.59] for S :

1  √ β α  ¯ λ αβ αβ  ) = Lαβ − Qα α λβ  AL .a − S.aα = L,β + γ¯ λβ L − bαβ Q(1)β + Γλβ L − (Qα −  bαλ Q(1)λ √ ∦β ,β  A Finally, with the expression for Rα given by [5.61], we get   1  √ β α αβ 3   Qα Qα Rα + √ AL .a − S.aα a3 = Rβα aβ +  a3 + L∦β a − a3 ,β A



The previous lemma shows that the vector S as well as the components # H/2 Qα , Q(1)α , −H/2 Σ33 dZ will now vanish from expression [5.58] for the virtual power of internal forces and, therefore, from Kirchhoff-Love plate theory. All that will remain now are the components Rαβ , Lαβ , that is, the stress resultants N αβ , M αβ , M (2)αβ . • From the above lemma, the expression that features in the surface integral of [5.58] is written as 1 √ A



√

     1  √ β α 1  √  βα αβ 3 A Rα + √ AL .a − S.aα a3 = √ A R aβ + L∦β a ,β ,α A A ,α

Let us continue to make explicit the previous expression using the following lemma: Lemma. ) (     1  √  βα αβ 3 αβ α βλ αβ a3 A R aβ + L∦β a = Rαβ − b L + L + b R a √ α αβ β ∦λ ∦β ∦β α ,α A

[5.64]

where we denote 

αβ L∦β

 α

1  √ αβ  αβ αβ ≡ √ AL∦β = (L∦β ),α + Γ¯ λλα L∦β ,α A

[5.65]

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Nonlinear Theory of Elastic Plates

Proof. We have   1  √ βα  ¯ λ βα aβ + Rβα aβ,α AR aβ = Rβα √ ,α + Γλα R ,α A where aβ,α is given by the  Gauss formula [5.35]  ¯ λ βα ¯ β Rλα aβ + bβα Rβα a3 = Rβα ,α + Γλα R + γ λα  Rβα ∦α

Similarly  αβ  1  √ αβ 3  αβ αβ 3 AL∦β a = (L∦β ),α + Γ¯ λλα L∦β a,α a3 + L∦β √ ,α A 3 λ a,α = −bα aλ using Weingarten Eq. [5.17] where αβ αβ αβ = (L∦β ),α + Γ¯ λλα L∦β a3 − bλα L∦β aλ  

αβ

L∦β



α

By grouping together the two previous relations and after changing some indices, we obtain [5.64].  • The following lemma gives the explicit expression for a term that appears in the line integral of [5.58]: Lemma.   ∂  αβ ∂  αβ L ν0β s0α a3 = L ν0β s0α a3 − bαβ sβ0 Lμλ ν0λ s0μ aα ∂s0 ∂s0

[5.66]

Proof. We have  ∂ αβ ∂a3 ∂  αβ L ν0β s0α a3 = (L ν0β s0α )a3 + Lαβ ν0β s0α ∂s0 ∂s0 ∂s0 where ∂a3 = g r a dP0 a3 .s0 = (a3,β ⊗ Aβ ).s0 = (Aβ .s0 )a3,β ∂s0 = −sβ0 bαβ aα using [5.17] and [5.53] Hence the desired result, after changing some indices.  Caution: we must not write Lαβ ν0β s0α = s0 .L.ν0 because, whereas s0α , ν0β are components of vectors s0 , ν0 in the initial dual basis (A1 , A2 ), the Lαβ defined by [5.60] are not the 2-contravariant components (in the basis (A1 , A2 )) of a 2nd-order tensor L (no such tensor L has ever been defined). • Finally, the lemmas [5.63]-[5.66] make it possible to rewrite the opposite of the virtual

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105

power of internal forces [5.58] as follows  *

) + (   αβ α βλ αβ a Rαβ a3 .U∗ dS 0 − b L + + b R L α αβ β ∦β ∦λ ∦β α S 0    αβ + ν0α Rαλ ν0λ − bαβ sβ0 s0μ Lμλ ν0λ aα + L∦β ∂S 0 \corners   ∂ αβ + (L ν0β s0α ) a3 .U∗ ds0 ∂s0 ' ∂U∗ − Lαβ ν0β ν0α a3 . ds0 + Lαβ ν0β s0α a3 K p .U∗ (K p ) ∂ν0 ∂S 0 \corners corners K

−P∗int = −

[5.67]

p

5.8. Virtual power of external forces The external forces are calculated in the same manner as in section 3.11 for a Cosserat plate. The virtual powers of the body forces and the forces on the upper and lower faces of the plate is   P∗volume ∪S sup ∪S inf = p.U∗ dS 0 + c.a∗3 dS 0 [5.68] S0

S0

where the force p and the couple c are defined over S 0 by [3.53], [3.55] and [3.69]. As concerns forces along the plate edge, there are several small new features that are peculiar to Kirchhoff-Love theory and that are due to the possible presence of corners along the plate edge ∂S 0 . As a matter of fact, it turns out that the PVP treatment involves several discontinuities at the corners, and it proves useful to explicitly introduce concentrated forces at the corners among the external forces. Thus, we consider two types of forces along the plate edge: (i) The forces applied along the regular portions of the plate edge, whose virtual power is analogous to [3.63]:    P∗regular portions of the edge = [5.69] q.U∗ + C.a∗3 ds0 ∂S 0 \corners

In this expression, the line force q and the line couple C are defined as in [3.61]-[3.62] but this time, they are restricted to the regular portions of edge ∂S 0 . (ii) The concentrated forces, denoted by ZK p a3 at the possible corners K p of the edge; these corner forces are new – they do not occur in either Cosserat or Reissner-Mindlin plate theory. Their virtual power is straightforward: '   P∗corners = [5.70] ZK p a3 .U∗  Kp corners K p

We can think of a more general expression for the concentrated forces at the corners, namely XK p a1 + YK p a2 + ZK p a3 . However, the calculations show that in Kirchhoff-Love plate theory, the components XK p and YK p are necessarily zero and that only the component ZK p along a3 appears in the final equations. This is why one just has to consider the forces of the form ZK p a3 . To summarize: the external load applied to the Kirchhoff-Love plate is made up of the following forces:

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Nonlinear Theory of Elastic Plates

Notations.

[5.71]

1. The surface force p defined on the mid-surface S 0 . 2. The surface couple c defined on the mid-surface S 0 . 3. The line force q defined at the regular points of the plate edge ∂S 0 . 4. The line couple C – or, equivalently, the line couple Γ (see [5.76] below) – defined at the regular points of the edge∂S 0 . 5. Finally, the concentrated forces ZK p a3 at the possible edge corners K p .

Except for corner forces ZK p a3 , the above forces are applied to the current configuration but defined on the initial surface S 0 . The forces considered may be dead or follower (except the corner forces ZK p a3 which are, by definition, follower). The forces p, q and the couples c, C are summarized in Fig. 3.7. Fig. 5.6 represents a force concentrated at an
edge corner. This force is drawn on the current surface S , contrary to the other forces which are drawn on the initial surface S 0 .

 







Figure 5.6: Concentrated force at an edge corner in the Kirchhoff-Love plate model Relations [5.68]-[5.70] lead to the following expression for the virtual power of external forces: P∗ext =



p.U∗ dS 0 + S0

 S0

c.a∗3 dS 0 +

 ∂S 0 \corners

(q.U∗ + C.a∗3 )ds0 +

' corners K p



 ZK p a3 .U∗  Kp [5.72]

Compared to [3.68], the line integral here is restricted to the regular portions and there is, in addition, the last term which is due to the corner forces. Upon replacing a∗3 in [5.72] with [5.44], we obtain P∗ext =



      p.U∗ dS 0 − (c.aα ) U∗,α .a3 dS 0 + (q.U∗ − (C.aα ) U∗,α .a3 )ds0 S0 S0 ∂S 0 \corners '   ∗ + ZK p a3 .U  corners K p

Kp

Kirchhoff-Love Plate Theory

107

Using [2.35] to carry out integration by parts, we obtain P∗ext =





 1 √ A(c.aα )a3 .U∗ dS 0 √ ,α S S0 A 0 '       + ( q − (c.aα )a3 ν0α .U∗ −(C.aα ) U∗,α .a3 )ds0 + ZK p a3 .U∗  Kp  ∂S 0 \corners corners K p  1 
[5.73] p.U∗ dS 0 +

 As in the calculation of the virtual power of external forces, let us transform the term 1
by ∗ ∗ ∂U ∂U and the tangential derivative on the edge: bringing in the normal derivative ∂ν0 ∂s0     1 ≡ − (C.aα ) U∗,α .a3 ds0 
∂S 0 \corners   ∂U∗ ∂U∗ (C.aα )(a3 . = − )ν0α + (C.aα )(a3 . )s0α ds0 ∂ν0 ∂s0 ∂S 0 \corners  ∂  = − (C.aα )(a3 .U∗ )s0α ds0  ∂S 0 \corners ∂s0  ∗  ∂ α 3 ∗ α 3 ∂U + )ν0α ds0 (C.a )a s0α .U − (C.a )(a . ∂ν0 ∂S 0 \corners ∂s0 ' α 3 ∗ = (C.a )a s0α K p .U (K p ) 



corners K p

+

∂S 0 \corners

 ∂ ∂U∗ ((C.aα )a3 s0α ).U∗ − (C.aα )(a3 . )ν0α ds0 ∂s0 ∂ν0

  Remark. A term like ∂s∂0 (C.aα )(a3 .U∗ )s0α is the derivative with respect to the curvilinear abscissa s0 defined along the edge ∂S 0 , i.e. the directional derivative along the tangent vector s0 . It must not be calculated by means of the formula ∂(•) ∂s0 ≡ g r a dP0 (•).s0 , since ν0β and s0α are defined on the edge ∂S 0 only, not over the whole surface S 0 .  Inserting the last relation in [5.73] gives     1 √ P∗ext = p+ √ .U∗ dS 0 A(c.aα )a3 ,α S A 0     ∂  α 3 α 3 (C.a )a s0α − (c.a )a ν0α .U∗ ds0 q+ + ∂s0 ∂S 0 \corners ∂U∗ − (C.aα )(a3 . )ν0α ds0 ∂ν0 ∂S 0 \corners '   α 3 ZK p a3 + (C.a )a s0α K p .U∗ (K p ) +

[5.74]

corners K p

To further make explicit the previous expression let us use, on the one hand, relation [5.46] with f = c.aα : 1 √ A(c.aα ) a3 √ A

,α

 1 √ = √ Ac.aα a3 − bλα (c.aα )aλ ,α A

and on the other hand [5.66] with C.aα instead of Lαβ ν0β :  ∂  ∂ (C.aα )s0α a3 = (C.aα s0α )a3 − bαβ sβ0 (C.aμ )s0μ aα ∂s0 ∂s0

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Nonlinear Theory of Elastic Plates

By inserting the two relations above in [5.74] and after some index changes, we obtain the virtual power of external forces P∗ext =

 

  1 √ p − bαβ (c.aβ )aα + √ Ac.aα a3 .U∗ dS 0 ,α S0 A   ∂ q − bαβ sβ0 s0μ (C.aμ )aα + + (C.aα s0α )a3 − (c.aα )a3 ν0α .U∗ ds0 ∂s0 ∂S 0 \corners ∗ α 3 ∂U − (C.a )(a . )ν0α ds0 ∂ν0 ∂S 0 \corners '   α 3 ZK p a3 + (C.a )a s0α K p .U∗ (K p ) + corners K p

[5.75] • Expression [5.75] shows that, as in Reissner-Mindlin theory, the components c.a3 , C.a3 along a3 of the couples c, C do not come into play. We can, thus, restrict ourselves to considering the couples c, C tangent to S . The couples c, C, defined by [3.53], [3.55] and [3.62] are vectors which are not in the same direction as the couples physically applied. In some cases, it is preferable to replace them with μ, Γ defined in [4.48] : μ ≡ a3 × c and Γ ≡ a3 × C. The new feature here, with respect to the Reissner-Mindlin theory, is that since a3 is orthogonal to S , there are more precise results: − the couples μ and Γ are tangent to S , like the couples c and C − and

μ ≡ a3 × c ⇔ c = μ × a3 Γ ≡ a3 × C ⇔ C = Γ × a3

[5.76]




The couples μ and Γ represent the true couples physically applied, and they are represented in Fig. 5.7 in comparison with their homologues c and C.


















Figure 5.7: The couples c, C and their homologues μ and Γ, all tangent to S in the Kirchhoff-Love model (these are the couples defined on S 0 , but here we draw them on S in order to show that they are tangent to S ) We shall, as we go on, use the couples C or Γ interchangeably and the relations will be written with one and/or the other couple depending on the situation and requirements. Expression [5.75] contains the components C.aα of the couple C in the natural basis (a1 , a2 ). In practice, it is more convenient to work with the components of the couple in a local basis related to a point on the edge. For this, let us define the following orthonormal basis, (ν, s, a3 ), on ∂S , similar to the basis (ν0 , s0 , A3 ) seen in [5.52] :

Kirchhoff-Love Plate Theory

109

Definition. [5.77] At a regular point on the edge ∂S , we define the local orthonormal basis (ν, s, a3 ) as follows (Fig. 5.8): – the vector a3 is, let us recall, normal to the mid-surface S , – the vector ν = outward unit normal vector to the edge ∂S , belonging to the plane tangent
to S at the point under consideration, – the vector s = the tangent unit vector s = a3 × ν ⇔ ν = s × a3 ⇔ a3 = ν × s.












Figure 5.8: Local orthonormal basis (ν, s, a3 ) on the edge of S We decompose the couple Γ in the local basis (ν, s) as follows :  s Γ is the line bending couple, Γ ≡ Γ s s + Γν ν where Γν is the line twisting couple.

[5.78]

We thus have the following simple relation: C = Γ × a3 = Γ s ν − Γν s

[5.79]

which implies that C.aα = Γ s ν.aα − Γν s.aα

 ⇒

C.aα s0α = Γ s ν.aα s0α − Γν s.aα s0α C.aα ν0α = Γ s ν.aα ν0α − Γν s.aα ν0α

[5.80]

As s.aα , ν.aα are the components of s, ν in the current basis (a1 , a2 ), while s0α , ν0α are the components of s0 , ν0 in the initial basis (A1 , A2 ), the sums ν.aα s0α , s.aα s0α , ν.aα ν0α and s.aα ν0α are not necessarily 0 or 1. The expression for C.aα s0α may be left as they are. Further down, the expression for C.aα ν0α will be specified by first establishing the following preliminary results: Lemma. The unit vectors that are tangent and normal to the current deformed edge, s and ν, are related to their counterparts on the initial edge, s0 and ν0 , through s=

1 F.s0 λs

ν=

J −T F .ν0 λs

[5.81]

$ where F is the gradient deformation tensor, J ≡ det F, and λ s ≡ 1 + 2E ss is the stretch in the direction of the unit vector s0 orienting the edge ∂S 0 (Fig. 5.5), E ss being the strain in the same direction.

110

Nonlinear Theory of Elastic Plates

Proof. (i) We return to the proof [3.66] and extend it. Consider a small material segment dX on the edge ∂S 0 , originating from a point P0 ∈ ∂S 0 . This segment transforms into a segment dx on the edge ∂S$and originates from the point P, the transform of P0 . We have dx = F.dX, where dx = dX.FT .F.dX. From this we deduce the unit tangent vector s=

F.s0 dx F.dX F.s0 = $ = √ : that is [5.81]1 = $ T T dx 1 + 2E ss s .F .F.s dX.F .F.dX 0 0

(ii) To prove expression [5.81]2 for ν, let us write ν = s × a3 = λ1s F.s0 × F.A3 , taking into account [3.13] and [5.81]1 . Now, it can be verified, resuming the proof for the Nanson formula ndS = JF−T NdS 0 , that qT .(F.s0 × F.A3 ) = det F s0 × A3 = det F ν0 Hence F.s0 × F.A3 = det F F−T .ν0 .  The tangent vector s (resp. the normal a3 ) gives the direction of a small material segment over time, we say that it follows the direction of a small material segment: s follows the direction of a small material segment of the edge ∂S , a3 follows a normal fiber. This can also be seen in relation [5.81]1 , which shows that s is, to within a factor, the transform of s0 . On the contrary, the normal ν = s × a3 , constructed by the vector product of s and a3 , does not give the direction of a small material segment over time. In other words, a small material segment, originating from P0 ∈ ∂S 0 and parallel to ν0 , is not generally parallel to ν after deformation. Indeed, relation [5.81]2 shows that ν is not the transform of ν0 . Lemma. s.aα ν0α = 0

ν.aα ν0α =

λs 0 J

[5.82]

In the second equality, the value λ s /J is not interesting in itself; the only fact to remember is that the sum ν.aα ν0α is nonzero. Proof. (i) We have s.aα ν0α = (s.aα )(ν0 .Aα ), where s.aα is transformed using [3.21] and [5.81] successively: s0 s.aα = s.F−T .Aα = (F−1 .s).Aα = .Aα λs Hence s0 s0 = .(Aα ⊗ Aα ).ν0 s.aα ν0α = ( .Aα )(ν0 .Aα ) λs λs s0 s0 = .(I − A3 ⊗ A3 ).ν0 = .ν0 because A3 ⊥ν0 λs λs = 0 because s0 ⊥ν0 : this is [5.82]1 (ii) Similarly, we have  λs  T = (ν.aα ) (F .ν).Aα using [5.81] ν.aα ν0α = (ν.aα )(ν0 .Aα ) J λs λs = using [5.5] (ν.aα )(ν.F.Aα ) = (ν.aα )(ν.aα ) J J λs λs α 3 = ν.(a ⊗ aα ).ν = ν.(I − a ⊗ a3 ).ν Hence [5.82]2 J J



Kirchhoff-Love Plate Theory

111

Remark. If the strain level in the vicinity of the plate edge is small, then E ss  0, λ s  1, J  1, such that ν.aα ν0α  1. This approximation is not utilized in this book.  By inserting [5.82] in [5.80], we arrive at the expression for the sum C.aα ν0α appearing in the expression for the virtual power of external forces [5.75]: Theorem. C.aα ν0α = Γ s ν.aα ν0α 

[5.83]

0

5.9. Equations of motion and boundary conditions We now have all the ingredients to use the PVP and obtain the governing equations for KirchhoffLove plate dynamics together with the boundary conditions on forces. Theorem. (a) The 3 governing equations of motion are : 1. ∀t, ∀P0 ∈ S 0 , ∀α ∈ {1, 2},    ¨ + ρ0 H (1) a¨ 3 .aα − bα ρ0 H (1) U ¨ + ρ0 H (2) a¨ 3 .aβ ρ0 H (0) U β α βλ α α β −Rαβ ∦β + bβ L∦λ = p.a − bβ c.a



[5.84]

recalling that the operator ∦ is defined in [5.62]. This equation gives 2 scalar equations. 2. And ∀t, ∀P0 ∈ S 0 ,   √  ¨ + ρ0 H (1) a¨ 3 .a3 + √1 ¨ + ρ0 H (2) a¨ 3 .aα ρ0 H (0) U A ρ0 H (1) U ,α A  αβ  √  1 3 α − L∦β − bαβ Rαβ = p.a + √ Ac.a ,α α A  αβ  where L∦β is defined by [5.65]. relation [5.85] gives a scalar equation. 

[5.85]

α

(b) The boundary conditions are: 1. ∀t, ∀P0 ∈ ∂S 0 \{corners}, ∀α ∈ {1, 2},



 Rαλ − bαβ sβ0 s0μ Lμλ ν0λ = q.aα − bαβ sβ0 s0μ C.aμ

[5.86]

where C.aμ may be replaced by Γ s ν.aμ − Γν s.aμ . 2. ∀t, ∀P0 ∈ ∂S 0 \{corners}, ¨ + ρ0 H (2) a¨ 3 ).aα ν0α + Lαβ ν0α + −(ρ0 H (1) U ∦β q.a3 +

∂ αβ (L ν0β s0α ) = ∂s0

∂ (C.aα s0α ) − c.aα ν0α ∂s0

where C.aα s0α may be replaced by Γ s ν.aα s0α − Γν s.aα s0α .

[5.87]

112

Nonlinear Theory of Elastic Plates

3. ∀t, ∀P0 ∈ ∂S 0 \{corners}, Lαβ ν0β ν0α = C.aα ν0α = Γ s ν.aα ν0α

[5.88]

4. ∀t, ∀ angular point K p on the edge (if any), Lαβ ν0β s0α K p = ZK p + C.aα s0α K p

[5.89]

where C.aα s0α may be replaced by the same expression following [5.87]. Proof. By inserting the expressions for the virtual power of inertia forces [5.47], internal forces [5.67], and external forces [5.75] in the PVP [3.36], we arrive at an equality of the form: ∀t, ∀U∗ , 

(· · · ) .U∗ dS 0 + S0

 ∂S 0 \corners

(· · · ) .U∗ ds0 +

 ∂S 0 \corners

(· · · ).

' ∂U∗ ds0 + · · ·K p .U∗ (K p ) = 0 ∂ν0 corners K p

[5.90] where the (· · · ) are known vector expressions. The rest of the argument takes place over 4 steps: 1. First, we restrict ourselves to the functions U∗ which cancel themselves in the vicinity of ∗ the edge ∂S 0 (this implies that the normal derivative ∂U ∂ν0 is zero at every point on the edge) to reduce the equality [5.90] to  (· · · ) .U∗ dS 0 = 0 S0

As U∗ vary arbitrarily in S 0 , we deduce that the (· · · ) in the previous relation is zero. Hence the governing equation in vector form: ∀t, ∀P0 ∈ S 0 , ¨ + ρ0 H (1) a¨ 3 ρ0 H (0) U    1 √  ¨ α + ρ0 H (2) a¨ 3 .aα ¨ β + ρ0 H (2) a¨ 3 .aβ aα +√ A ρ0 H (1) U.a a3 − bαβ ρ0 H (1) U.a ,α A ) (     1 √ αβ α βλ αβ a3 = p − bαβ (c.aβ )aα + √ − Rαβ Ac.aα a3 ∦β − bβ L∦λ aα − L∦β α + bαβ R ,α A Projecting this relation on aα and a3 provides the governing Eqs. [5.84]-[5.85]. 2. Once the governing equations [5.84]-[5.85] are obtained, equality [5.90] reduces to  ∂S 0 \corners

(· · · ) .U∗ ds0 +

 ∂S 0 \corners

(· · · ).

' ∂U∗ ds0 + · · ·K p .U∗ (K p ) = 0 ∂ν0 corners K p

[5.91] We now restrict ourselves to the functions U∗ which cancel on the edge ∂S 0 to reduce the previous equality to  ∂U∗ (· · · ). ds0 = 0 ∂ν0 ∂S 0 \corners

Kirchhoff-Love Plate Theory

113

As ∂U∗ /∂ν0 can be given arbitrary values over any smooth arc of ∂S 0 , we deduce that the (· · · ) in the preceding relation is zero at every point of ∂S 0 \corners. There then results the following boundary condition: ∀t, ∀P0 ∈ ∂S 0 \{corners}, Lαβ ν0β ν0α a3 = (C.aα )ν0α a3 As both sides of the previous relation are vectors parallel to normal a3 , we arrive at the first equality of the boundary conditions [5.88]. The second equality comes from [5.83]. 3. Now that the governing equations [5.84]-[5.85] and the boundary conditions [5.88] are obtained, Eq. [5.91] is reduced to  ' (· · · ) .U∗ ds0 + · · ·K p .U∗ (K p ) = 0 [5.92] ∂S 0 \corners

corners K p

(a) If the edge ∂S 0 does not have any corners, the discontinuity terms in the preceding equation disappear. As the values of U∗ on the edge can be given arbitrary values, we deduce that the (· · · ) in the preceding relation is zero at any point on the edge. Hence the boundary condition: ∀t, ∀P0 ∈ ∂S 0 ,  ¨ α )a3 + ρ0 H (2) (¨a3 .aα )a3 .ν0α + − ρ0 H (1) (U.a     ∂ αβ αβ ν0α + (L ν0β s0α ) a3 Rαλ ν0λ − bαβ sβ0 s0μ Lμλ ν0λ aα + L∦β ∂s0 ∂ α β μ α 3 = q − bβ s0 s0μ (C.a )aα + (C.a s0α )a − (c.aα )ν0α a3 ∂s0 Projecting this relation on aα and a3 yields the boundary conditions [5.86]-[5.87]. As concerns the variants of these boundary conditions, they are provided by [5.80]. (b) If the edge ∂S 0 contains corners, we shall restrict ourselves to the functions U∗ that cancel themselves over the whole edge, except on a smooth arc. The argument in the above case remains valid, provided that the entire edge ∂S 0 is replaced by the smooth arc under consideration. 4. At this point, the equality [5.92] is reduced to ' · · ·K p .U∗ (K p ) = 0 corners K p

As the function U∗ can take arbitrary values, we obtain the following boundary condition at any angular points K p on edge ∂S 0 : Lαβ ν0β s0α a3 K p = ZK p a3 + (C.aα )s0α a3 K p As both sides of the previous relation are the vectors parallel to normal a3 , we arrive at boundary condition [5.89].  • It has been noted that if the plate edge has corners, then at these corners, ∂U∗ /∂ν0 is not independent of U∗ over ∂S 0 . The PVP terms at the 'respects this dependency: the discontinuity ' corners in the PVP are of the form · · ·U∗ (K p ) and not of the form · · ·U∗ (K p ) + ' corners K p

∂U∗ · · · . ∂ν0

corners K p

corners K p

114

Nonlinear Theory of Elastic Plates

On the contrary, at a regular point on the edge, ∂U∗ /∂ν0 is independent of U∗ on ∂S 0 , and this independence has been exploited to derive the boundary conditions [5.86] and [5.87]. • The governing equations [5.84]-[5.85] clearly show that there is extension-bending coupling, that is, coupling between the membrane forces N αβ on the one hand and and the bending moments M αβ , M (2)αβ on the other hand. The coupling takes place because of the curvatures bαβ or bαβ , this is why it is said that there is coupling through curvature. This is the same phenomenon that exists for a curved beam: a transverse force applied to the ends of a curved beam simultaneously introduces an axial force and a bending moment in the cross-sections. If we assume a priori that during the motion of the plate the curvature tensor does not change much, then the current curvature tensor is also zero: bαβ  0. Under these circumstances, it results from [5.60] that Rαβ  N αβ and Lαβ  M αβ . Then there remain only the membrane forces N αβ and the bending moments M αβ and these quantities appear decoupled in the equations: we obtain a governing equation in N αβ and another one in M αβ . We say we are studying the plate by neglecting geometric changes. The same phenomenon takes place in a straight beam where we ignore geometric changes: the axial force and the bending moment are decoupled. There are however situations where the approximation bαβ  0 is not licit in the plate, but where a more precise expression for bαβ must be used. We then say that we take into account the geometric changes of the system. This is what is done when studying pre-stressed plates or buckled plates (see Chapter 7). 5.10. Static problems In the framework of the statics, the inertia terms are zero and the following results can be derived straightforwardly from the theorem [5.84]-[5.88]: Corollary. (a) The 3 local equilibrium equations are: 1. ∀P0 ∈ S 0 , ∀α ∈ {1, 2}, α βλ α α β −Rαβ ∦β + bβ L∦λ = p.a − bβ c.a

[5.93]

(2 scalar equations). 2. And ∀P0 ∈ S 0 ,   αβ  1 √ − L∦β − bαβ Rαβ = p.a3 + √ Ac.aα ,α α A

[5.94]

(one scalar equation). (b) The boundary conditions are the same as [5.86]-[5.89], except [5.87] which simplifies to ∀P0 ∈ ∂S 0 \{corners}, αβ ν0α + L∦β

∂ αβ ∂ (L ν0β s0α ) = q.a3 + (C.aα s0α ) − c.aα ν0α ∂s0 ∂s0

where, as in dynamics, C.aα s0α may be replaced by Γ s ν.aα s0α − Γν s.aα s0α .

[5.95]

Kirchhoff-Love Plate Theory

115

5.11. Overview of equations and unknowns The field equations of the problem are the governing equations of motion [5.84]. As has been seen, they yield 3 scalar equations. The problem unknowns are of two types: 1. The 3 kinematic unknowns, namely the displacement field U of the mid-surface (the current director vector a3 , of unit length and orthogonal to the current mid-surface does not represent an additional unknown). 2. The force unknowns: the membrane forces N αβ and the bending moments M αβ , M (2)αβ . These stress resultants represent a total of 3 + 3 + 3 = 9 force unknowns. The internal forces Rαβ and Lαβ do not represent additional unknowns as they are expressed as functions of the stress resultants and the curvatures bαλ via Relations [5.60]. We thus have 3 scalar equations for 12 unknowns. That is, a deficit of 9 equations. The missing equations will be provided the so-called plate constitutive laws, which express the stress resultants in terms of kinematic unknowns. The results obtained in this chapter arise from the PVP and do not require any approximation or any assumption other than the Kirchhoff-Love kinematics hypothesis [5.20]. They are valid for plates made of any material. The plate constitutive laws, however, depend on the material and they will be established in Chapter 6 in the case of hyperelastic materials. 5.12. Example: Kirchhoff-Love plate in cylindrical bending 5.12.1. Statement of the problem We will study
the static deformation of a rectangular plate whose initial mid-surface belongs to the plane Oe1 e2 . The plate has the initial dimensions L along e1 , B along e2 , and initial thickness H, Fig. 5.9. 



  






  




  


  


  







  











 












Figure 5.9: Finite bending of a Kirchhoff-Love plate

116

Nonlinear Theory of Elastic Plates

The plate is homogeneous, made of a hyperelastic Saint Venant-Kirchhoff material (see Chapter 6), characterized by the Young modulus E and the Poisson ratio ν. In the reference state the stresses are zero. The Cartesian coordinates in the frame Oe1 e2 of a current particle initially located at P0 are denoted by X, Y. The edge X = 0 is clamped, while the edges Y = 0 and Y = B are in frictionless contact with rigid walls, such that it can be assumed that the displacement along e2 is identically zero throughout the plate and that the displacement vector U of the mid-surface has only two components U, W along e1 , e2 , independent of Y : U = U(X)e1 + W(X)e3

[5.96]

Thus, the deformed shape is cylindrical around the axis e2 and the plate is in plane strain state with respect to the plane e1 e3 . The plate is subject to the following given load, with notations [5.71] : (i) load on the midsurface: surface force p, surface couple μ2 e2 , and (ii) load on the edge X = L : line force q, line couple Γ s e2 , Fig. 5.10. All these loads are independent of Y.














 










Figure 5.10: Loading on the plate. The algebraic values for couples μ2 , Γ s are defined around vector e2 which is oriented inwards In the parametrization [2.1] of the plate, let us choose (ξ1 , ξ 2 ) = (X, Y). Hence  A1 = e1 ,

A2 = e2

⇒

[A. . ] =

  11 12    1 A A A11 A12 = = 1 A21 A22 A21 A22

[5.97]

and a1 = A1 + U,1 = (1 + U,1 )e1 + W,1 e3

 [a. . ] =

   a11 a12 a1 2 = a21 a22 1

a2 = A2 + U,2 = e2

⇒

[a. . ] =



  a11 a12 21 22 = a a

[5.98]



1 a1 2

1

[5.99]

On applying [5.12] and [5.16], bαβ = n.aβ,α and bβα = bαλ aλβ , the only nonzero curvatures are found to be b11 and b11 .

Kirchhoff-Love Plate Theory

117

5.12.2. Equations of the problem Governing equation • The governing equation [5.93] gives α βλ α α β −Rαβ ∦β +bβ L∦λ = p.a −bβ c.a

or according to [5.62]

⎧ αβ αβ α λβ ⎪ ⎪ ⎨ R∦β = R,β + γ¯ βλ R ⎪ βλ βλ β μλ ⎪ ⎩ L∦λ = L,λ + γ¯ λμ L

We deduce from this, for α = 1 :     1 11 11 1 11 ¯ 11 R + b11 L,1 + γ¯ 11 L = p.a1 − b11 c.a1 − R11 ,1 + γ

(Γ¯ λλβ = 0)

[5.100]

In this expression, c.a = (μ × a3 ).a = μ.(a3 × a ), where 1

a3 × a1 =

[1.12]

a2 √ , a

1

1

a ≡ det[a. . ] = a1 2

⇒

a3 × a1 =

e2 a1 

⇒

c.a1 =

μ2 a1  [5.101]

• The governing equation [5.94] gives   αβ  1 √ − bαβ Rαβ = p.a3 + √ Ac.aα [5.102] − L∦β ,α α A √ In this expression, A = 1 and  αβ   αβ  αβ αβ αβ αβ αβ – L∦β is defined by [5.65] : L∦β = (L∦β ),α + Γ¯ λλα L∦β = (L∦β ),α , where L∦β = L,β + α α  0

α λβ L . γ¯ βλ

    μ2 – (c.aα ),α = c.a1 = . ,1 a1  ,1 Thus, Eq. [5.102] becomes    αβ  μ2 α λβ − L,β + γ¯ βλ L − bαβ Rαβ = p.a3 + ,α a1  ,1 that is: −



11 L,1

+





1 11 γ¯ 11 L ,1

− b11 R

11

μ2 = p.a + a1 



3

[5.103] ,1

• By combining [5.100].a1 + [5.103].a3 , then by using the Gauss formula [5.35] and the Weingarten equation [5.17], we obtain       1 11 1 11 11 1 11 b11 a1 + L,1 − R11 ¯ 11 a1 + b11 a3 R11 + L,1 + γ¯ 11 L + γ¯ 11 L a3 ,1 a1 − γ ,1   −a3,1

a1,1

= p−

  μ2 μ2 1 b1 a1 + a3 a1   a1  ,1 −a3,1

that is:     11 1 11 + γ¯ 11 L a3 − R11 a1 − L,1 ,1

 ,1

=p+

μ2 a3 a1 

 [5.104] ,1

We can express R11 , L11 using their definitions [5.60] and knowing that b22 = 0 : R11 ≡ N 11 − b1λ M λ1 = N 11 − b11 M 11 L11 ≡ M 11 − b1λ M (2)λ1 = M 11 − b11 M (2)11

[5.105]

118

Nonlinear Theory of Elastic Plates

Constitutive laws We will anticipate the constitutive laws [6.70]-[6.71] for the Kirchhoff-Love plate, which will be established in Chapter 6. Because there are no pre-stresses in the plate, the pre-forces and the pre-moments defined by [6.66] are zero, N0αβ = 0, M0αβ = 0. The membrane force N 11 writes   EH (0) EH 1 1γ 1η 1η 1γ 11 γη (0) A + A A ) + νA A = E [5.106] N 11 = Eγη (1 − ν)(A [5.97] 1 − ν2 11 1 − ν2 2 (0) is derived from definition [5.37] and from [5.98] : where the membrane strain E11  1 1 (0) ≡ (a11 − A11 ) = U,1 + U,12 + W,12 E11 2 2 The bending moment M 11 is   1 M 11 = D (1 − ν)(A1γ A1η + A1η A1γ ) + νA11 Aγη κγη = Dκ11 = −Db11 [5.97] [5.40] 2

with D ≡

EH 3 . 12(1−ν2 )

[5.107]

[5.108]

The 2nd-order bending moment M (2)11 is given by the constitutive law [6.75] : ⇒

M (2)11 is negligible

L11 = M 11

[5.105]

[5.109]

Boundary conditions • The boundary conditions on the edge X = 0 are of kinematic type: ⎧ ⎪ W(0) = 0 ⎪ ⎨ U(0) = 0, ⎪ ⎪ and at X = 0, a ∥ e1 ⇔ W,1 (0) = 0 1 ⎩ [5.98]

[5.110]

• The boundary conditions on the edges X = 0, Y = 0 and Y = B are not of interest inasmuch as these conditions yield reaction forces. Let us write the boundary conditions [5.86] on the edge X = L:   ∀α ∈ {1, 2}, Rαλ − bαβ sβ0 s0μ Lμλ ν0λ = q.aα − bαβ sβ0 s0μ C.aμ On the edge X = L, we have ν0 = e1 , s0 = e2 , hence ν01 = 1, ν02 = 0, s01 = s10 = 0. Whence, for α = 1 : R11 − b11 s10 s0μ Lμ1 = q.a1 − b11 s10 s0μ C.aμ • Knowing that the derivatives [5.95] on the edge X = L gives

∂ ∂s0

⇒

R11 = q.a1

[5.111]

all along the boundary are zero, the boundary condition

αβ L∦β ν0α = q.a3 − c.aα ν0α

Taking α = 1 and using c.a1 =

μ2 a1 

found in [5.101], we arrive at

μ2 a1  • By combining [5.111].a1 + [5.112].a3 , we arrive at   μ2 11 1 11 + γ¯ 11 L a3 = q − a3 X = L, R11 a1 + L,1 a1  • Finally, let us write the boundary condition [5.88] on the edge X = L : 11 1 11 L,1 + γ¯ 11 L = q.a3 −

Lαβ ν0β ν0α = Γ s ν.aα ν0α

with

ν=

[5.112]

[5.113]

a1 1 ⇒ ν.a1 = a1  a1 

This gives us, for α = 1 : at X = L,

L11 =

Γs a1 

[5.114]

Kirchhoff-Love Plate Theory

119

Overview of the equations and unknowns The problem has 4 unknowns: two displacement unknowns U(X), W(X), and 2 force unknowns, which are the stress resultants N 11 , M 11 . The 4 equations that make it possible to determine these unknowns are – The local equilibrium equation [5.104]. In this equation, the vectors a1 and a3 of the current 1 , as well as the curvature b11 , are functions of the local basis, the Christoffel symbol γ¯ 11 displacements U, W. – The constitutive laws [5.106] and [5.108], in which the curvature b11 is a function of the displacements. These equations must be solved under the boundary conditions [5.110], [5.113] and [5.114]. 5.12.3. Comparison with Bernoulli’s beam in finite bending Before going further, let us compare the results obtained above with those for Bernoulli’s beam in finite bending. Let us consider a beam in finite bending, of length L, clamped at X = 0 as depicted in Fig. 5.11. In accordance with common usage, the problem is formulated in the plane Oe1 e2 , and not in the plane Oe1 e3 as previously done for the plate. The neutral axis of the beam is parametrized by ξ = X. The notations used for the beam are: – G : the current position for a particle on the neutral axis; a ≡ ∂G ∂X : the tangent vector to the a : the director vector indicating the position of the cross-section. the neutral axis; d = e3 × a The vectors a, d are homologous to the vectors a1 , a3 of the plate, resp. – χ : the curvature, equivalent to b11 of the plate. – N, M, M (2) : the stress resultants – axial force, bending moment, 2nd-order bending moment – defined like their homologues in plate theory. – R ≡ N + Mχ, L ≡ M + M (2) χ : the internal forces, homologous to R11 , L11 in plate theory. The symbol L has been used to avoid any confusion with the length L. – p, μ : the line force and line couple, distributed along the neutral axis; F, Γ : force and couple at the end X = L. The algebraic values for couples μ, Γ are defined around vector e3 which is oriented outwards to Fig. 5.11. In finite bending beam theory, we have the following results:     1 μ (aL),X d = p − Governing equation: − (Ra),X + d a a ,X ,X

Boundary conditions at X = L :

⎧ μ 1 ⎪ ⎪ ⎪ (aL),1 d = F + d ⎨ Ra − a a ⎪ ⎪ ⎪ ⎩ aL = Γ

[5.115]

[5.116]

In order to compare the equations for plates and those for beams, let us transform the governing equation [5.104] and the boundary condition [5.113] using the following lemma: Lemma. 11 1 11 L,1 + γ¯ 11 L =

 1  a1 L11 ,1 a1 

[5.117]

120

Nonlinear Theory of Elastic Plates














 












Figure 5.11: Comparison with a Bernoulli beam Compare this with Fig. 5.10 and note the differences in the choice of axes and the direction of the couples Proof. We have 

a1 L11

 ,1

11 = a1 L,1 + a1 ,1 L11

⇒

 a1 ,1 11 1  11 a1 L11 = L,1 + L ,1 a1  a1 

Moreover, a1 ,1 =

√

a1 .a1

 ,1

=

a1 .a1,1 1 = a1 ¯γ11 a1 

using the Gauss formula [5.35] 

Using this lemma, [5.104], [5.113] and [5.114] can be recast as follows        1  μ2 [5.118] a1 L11 a3 = p + Governing equation: − R11 a1 − a3 ,1 ,1 a1  a1  ,1 ,1

Boundary conditions at X = L :

⎧  1  μ2 ⎪ ⎪ ⎪ R11 a1 + a1 L11 a3 = q − a3 ⎪ ⎪ ,1 ⎨ a  a 1 1 s ⎪ ⎪ Γ ⎪ 11 ⎪ ⎪ (intact) ⎩L = a1  [5.119]

These relations are identical to those in [5.115]-[5.116] for the beam, using the evident # H/2 changes in notation and some changes in sign: definition M 11 ≡ −H/2 Z Σ11 dZ for a plate does # not have the negative sign as in definition M ≡ − S Y ΣXX dS 0 for the beam (S 0 is the initial 0 cross-section of the beam), see Fig. 5.12; on the other hand, the couples μ2 , Γ s for the plate are, respectively, opposite to μ, Γ for the beam, cf. Figs. 5.10 and 5.11. 5.12.4. Case of an inextensible plate Integrating [5.118] with respect to X, between current X and L, and taking in the boundary condition [5.119]1 leads to  L  1  11 μ2 pdX + q(L) − [5.120] L a1  a3 = R11 a1 + a3 ,1 a1  a 1 X

Kirchhoff-Love Plate Theory

121





 



 Figure 5.12: Moment M 11 and couple Γ s on the edge X = L of the plate #L Note that the sum X pdX + q(L) is equal to the force resultant applied on the portion at the right of the current section of abscissa X. To simplify the previous relation a little, we will adopt a hypothesis analogous to that in beam theory and verified in practice: Inextensibility hypothesis . The plate is inextensible in its mid-plane, more precisely: it is assumed that the extensional stiffness EH is very large, such that the mid-plane elongation is negligible: (0) E11

= U,1 +

[5.107]

 1 2 U,1 + W,12 = 0 2

⇔

a1  = 1

[5.121]

Remark. Under the inextensibility assumption, the constitutive law [5.106] takes the EH (0) indeterminate form N 11 = 1−ν 2 E 11 = ∞.0, which cannot be used. Using the hypothesis [5.121], we gain an additional equation a1  = 1, while at the same time losing an equation on N 11 . The membrane force N 11 is not determined by the constitutive law [5.106], but may be obtained using the governing equation [5.124] given below.  Theorem. a1  = constant over S

⇔

1 γ¯ 11 =0

[5.122]

1 Proof. Use just the Gauss formula [5.35] : a1 .a1,1 = γ¯ 11 .

In Bernoulli’s beam theory, a hypothesis similar to [5.121] is adopted, assuming that the neutral fiber of the beam is inextensible. The resulting theory is called elastica theory and yields the beam solutions with finite displacements and finite rotations. Eq. [5.120] becomes, under the inextensibility assumption [5.121] :  L 11 R11 a1 + L,1 a3 = pdX + q(L) − μ2 a3 [5.123] X

1 This equation can also be obtained by making a1  = 1 and γ¯ 11 = 0 in the governing equation [5.104]. Projecting relation [5.123] on a1 and on a3 , gives two equations equivalent to it:  L  N 11 − b11 M 11 = pdX + q(L) .a1 [5.124] X

 M,111 =

L X

 pdX + q(L) .a3 − μ2

[5.125]

122

Nonlinear Theory of Elastic Plates

As the boundary condition [5.119]1 has already been used to obtain [5.120], it remains to rewrite the boundary condition [5.119]2 under the inextensibility hypothesis [5.121] : L11 = Γ s

at X = L :

[5.126]

New overview of equations and unknowns The inextensibility of the mid-surface [5.121] implies that there is a relation between U and W, such that we can count only one of them as an unknown. At the same time, as has been seen earlier, the membrane force N 11 is no longer given by the constitutive law [5.106]. We find that there are 3 unknowns: a kinematic unknown W (or U), and 2 force unknowns N 11 , M 11 . The 3 equations that make it possible to determine these unknowns are – The equilibrium equations [5.124]-[5.125]. – The constitutive law [5.108] : M 11 = −Db11 . These equations must be solved under the boundary conditions [5.110] and [5.126]. Solution We will replace the current curvatures b11 , b11 with another variable that has a simpler meaning, namely, the rotation angle of the normal vector. Let us define the rotation angle θ as in Fig. 5.13; it is a kinematic unknown. For ease of reading, −e2 has been exceptionally chosen instead of e2 to define the positive direction of θ.









 















 


Figure 5.13: The rotation angle θ We have n ≡ a3 = − sin θe1 + cos θe3 ⇒ n,1 = −θ,1 (cos θe1 + sin θe3 ) a1 = a1  (cos θe1 + sin θe3 )

[5.127]

From which, by definition [5.12] and as per [5.16] b11 = −n,1 .a1 = a1 θ,1

b11 = b11 a11 =

θ,1 a1 

[5.128]

The constitutive law [5.108] thus becomes M 11 = −Db11 = −Dθ,1

[5.129]

Kirchhoff-Love Plate Theory

Inserting this in [5.125] gives the differential equation for θ :  L  −Dθ,11 = pdX + q(L) .a3 − μ2

123

[5.130]

X

knowing that vector a3 itself is a function of angle θ. This is the same equation as in elastica theory, except that here the plate bending stiffness D replaces the beam flexural rigidity EI. The differential equation [5.130] is associated with two boundary conditions in terms of θ: ⎧ ⎪ ⇔ θ(0) = 0 (this conditions replaces W,1 (0) = 0 in [5.110]) at X = 0, a1 ∥ e1 ⎪ ⎪ ⎨ [5.98] ⎪ s 11 ⎪ ⎪ and at X = L : Γ = 0 ⇒ M (L) =0 ⇒ θ,1 (L) = 0 ⎩ [5.126] [5.129] [5.131]


• For the sake of definiteness, assume that the external load is composed uniquely of a line force q = −qe1 on the edge X = L, where q > 0 is a given constant, Fig. 5.14. 



 












Figure 5.14: Plate compressed by a dead load q = −qe1 By then making p = 0, μ2 = 0 in [5.130], and by taking into account [5.127], we obtain θ,11 =

q q e1 .a3 = − sin θ D D

[5.132]

Multiplying [5.132] by 2θ,1 and then integrating it with respect to ξ1 = X provides 

dθ dX

2 =

2q cos θ + C1 D

[5.133]

The constant of integration C1 is determined by the boundary condition θ,1 (L) = 0 in [5.131] : 0=

2q cos θL + C1 D

with notation θL ≡ θ(L)

Consequently, [5.133] becomes 

dθ dX

2 =

2q (cos θ − cos θL ) D

⇒

dθ = dX

,

2q $ cos θ − cos θL D

[5.134]

dθ > 0. The sign + has been chosen assuming a priori that dX The integration of [5.134] between 0 and current X, by taking the boundary condition θ(0) = 0 in [5.131], leads to -  θ D dθ X−0= [5.135] √ 2q 0 cos θ − cos θL

124

Nonlinear Theory of Elastic Plates

• Determining the angle θL . By writing the last relation with X = L, we obtain a relation that makes it possible to determine the constant θL : -  θL D dθ [5.136] L= √ 2q 0 cos θ − cos θL This is a transcendental equation with unknown θL , which can be transformed using elliptic integrals. To do this, let us change the variables θ θL π = k sin ϕ where k ≡ sin , ϕ ∈ [0, ] 2 2 2 making it possible to recast the integral in [5.136] as  θL √ dθ = 2K(k) √ cos θ − cos θL 0 sin

where K(k) designates the complete elliptic integral of the first kind defined by  π 2 dϕ K(k) ≡ . 0 1 − k2 sin2 ϕ

[5.137]

[5.138]

Thus, [5.136] can be written in the form qL2 [5.139] = K 2 (k) D This equality establishes a relation between k – that is, the angle θL – and the compressive force q. The force q is given a value and Eq. [5.139] is then solved to obtain θL . Remark. If the rotation θ is small, [5.136] can be approximated by -  π D D θL dθ L = (independent of θL ) . q 0 2 q θ2L − θ2 Hence the compressive force q: π2 D 4L2 This value for q, corresponding to the onset of the out-of-plane deformation of the plate, is called the critical buckling force of the plate. Buckling will be studied in Chapter 7, where we will see how to directly obtain the critical buckling force by means of linearized equations, without needing to deal with the whole nonlinear problem.  q=

• Rotation angle θ(X). Once θL is known, relation [5.135] gives θ(X) as an implicit function of X. • Deformed shape of the plate. Let us denote x ≡ X + U and z ≡ 0 + W the current coordinates of the current particle of the plate, whose coordinates in the initial state are (X, Y, 0). Taking into account [5.134], we have ⎧ ⎪ ⎪ cos θdθ D ⎪ ⎪ ⎪ dx = cos θ dX = √ ⎪ ⎪ ⎪ 2q cos θ − cos θL ⎨ ⎪ ⎪ ⎪ ⎪ sin θdθ D ⎪ ⎪ ⎪ ⎪ ⎩ dz = sin θ dX = 2q √ cos θ − cos θL

Kirchhoff-Love Plate Theory

125

Integrating between 0 and current X, taking into account the boundary condition U(0) = W(0) = 0 in [5.110], that is, x(0) = z(0) = 0, and the boundary condition θ(0) = 0 in [5.131], leads to -  ⎧ θ ⎪ ⎪ D cos θ dθ ⎪ ⎪ ⎪ x−0= √ ⎪ ⎪ ⎪ 2q 0 cos θ − cos θL ⎨ [5.140] , ⎪  θ ⎪ ⎪  $ ⎪ 2D sin θ dθ 2q  $ ⎪ ⎪ ⎪ = 1 − cos θL − cos θ − cos θL √ ⎪ ⎩z−0= q 0 D cos θ − cos θL These are the parametric equations, with the parameter θ varying between 0 and θL , which enables us to determine the deformed shape of the plate. In particular, the current abscissa xL of the edge X = L is given by  θ xL D cos θ dθ D = [2E(k) − K(k)] = √ L 2qL2 0 qL2 cos θ − cos θL where E(k) designates the complete elliptic integral of the second kind, defined by:  π . 2 E(k) ≡ 1 − k2 sin2 ϕdϕ

[5.141]

0

The maximal deflection is 2D $ 1 − cos θL zL = q

⇒

zL = L

2D $ 1 − cos θL qL2

=

[5.137],[5.139]

2k K(k)

Fig. 5.15 shows the deformed shapes of the plate corresponding to 5 particular values for rotation angle θL of edge X = L : 30◦ , 60◦ , 90◦ , 120◦ , 150◦ . 




  


  




  



  
  

  
  





 



























Figure 5.15: Deformed shapes of the plate Table 5.1 gives some particular values for the force q, the rotation angle θL and the coordinates xL , zL of edge X = L.

126

Nonlinear Theory of Elastic Plates

qL2 /D π /4 = 0.7854 2.554 2.842 3.438 4.651 7.662 ∞ 2

θL 0◦ 30◦ 60◦ 90◦ 120◦ 150◦ 180◦

xL /L 1. 0.9324 0.7410 0.4569 0.1232 −0.2223 −1.

zL /L 0. 0.3239 0.5932 0.7628 0.8032 0.6979 0.

Table 5.1: Particular values of the deformed shapes • Internal force R11 . Relation [5.124], unused till now, gives the internal force R11 : R11 = N 11 − b11 M 11 = q(L).a1 This result is absolutely predictable. Indeed:  R11 ≡ N 11 − b11 M 11 = a1 .R1 [5.60]

where

R1 ≡

[3.44]

H/2 −H/2

 Π.A1 dZ =

[5.97]

H/2 −H/2

Π.e1 dZ [5.142]

As the vector Π.e1 is the nominal stress vector relative to direction e1 , the vector R1 is the through-thickness resultant force of the nominal stress vectors relative to e1 (cf. the remark after definition [3.44]). Here, with the load composed uniquely of a line force q(L) on the edge X = L, we have R1 = q(L). Finally, the equality R11 = a1 .R1 in relation [5.142] does indeed give R11 = q(L).a1 . • Membrane force N 11 . We can also calculate the membrane force N 11 . To do this, let us return to relation [5.124]: ⎧ θ,1 ⎪ ⎪ ⎪ = θ,1 b11 = ⎪ ⎪ ⎪ [5.128] a ⎪ 1  [5.121] ⎨ 11 N 11 − b11 M 11 = q(L).a1 where ⎪ M = −Dθ according to [5.129] ,1 ⎪ ⎪ ⎪ a1 ⎪ 1 ⎪ ⎪ = a1 ⇒ q(L).a1 = −q cos θ ⎩a = [5.99] a1 2 [5.121] Hence: 2 N 11 = −Dθ,1 − q cos θ = −q(3 cos θ − 2 cos θL ) based on [5.134]

In particular : at X = L, N 11 = −q cos θL at X = 0, N 11 = −q(3 − 2 cos θL ) It is not surprising that N 11 (0) is not equal to the external force −q applied at X = L! Indeed, relation [5.142] shows that in general N 11  R11 , except at those places where b11 M 11 = 0, that is, here, except for X = L. And we have seen that it is R11 that is equal q(L).a1 . To better understand the difference between N 11 (0) and R11 (0), let us directly obtain relation [5.142], R11 ≡ N 11 − b11 M 11 , with focus on X = 0. Use will be made of the relation σ = 1J F.Σ.FT

Kirchhoff-Love Plate Theory

127

(J = det F) relating the Cauchy stress tensor σ and the 2nd Piola-Kirchhoff stress tensor Σ. The deformation gradient tensor is F = I + H, where H(Q0 ) at X = 0 is given by [4.23]: H(Q0 )|X=0 = U,α ⊗ Aα + (a3 − e3 ) ⊗ e3 + Za3,α ⊗ Aα = U,1 ⊗ A1 + (a3 − e3 ) ⊗ e3 + Za3,1 ⊗ A1 ⎧ ⎪ a = e3 ⎪ ⎪ ⎨ 3 a3,1 = −b11 a1 = −b11 e1 Since, at X = 0: ⎪ , we obtain ⎪ ⎪ ⎩W ,1|X=0 = 0 ⇒ U,1 = U ,1 e1 + W,1 e3 = U ,1 e1 H(Q0 )|X=0 = (U,1 − Zb11 )e1 ⊗ e1 Hence F(Q0 )|X=0 = I + H(Q0 )|X=0 = (1 + U,1 − Zb11 )e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 Now 1 + U,1 = x,1 = cos θ = 1 at X = 0 = (1 − Zb11 )e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 Thus, the matrix of F(Q0 )|X=0 in the basis (e1 , e2 , e3 ) is ⎡ ⎤ ⎢⎢⎢ 1 − Zb11 ⎥⎥⎥ ⎢⎢⎢ ⎥ 1 ⎥⎥⎥⎥ [F]|X=0 = ⎢⎢ ⎣ ⎦ 1 We thus have: σ11 =

1 F11 Σ11 F11 = (1 − Zb11 )Σ11 J

Hence, by integrating in the thickness: 



H/2

σ11 dZ = −H/2  = a1 .R1 = R11

[3.44]

[5.60]

H/2

Σ11 dZ −H/2 

=N 11 according to [3.49]

 −b11

H/2

ZΣ11 dZ −H/2 

=M 11 according to [3.49]

We thus again arrive at relation [5.142], that is [5.60] written with α = β = 1 : R11 = N 11 − b11 M 11 , which confirms that N 11 (0)  R11 (0).