Introduction
In this book, we utilize established symbols and conventions, where Greek and Latin subscripts represent the values 1, 2, and 1, 2, 3, respectively Summation over repeated indices is implied, and points in orthogonal Cartesian coordinates are denoted as x = (x1, x2) for R² and x = (x1, x2, x3) for R³ A superscript T signifies matrix transposition, while α = ∂( )/∂xα represents partial derivatives The Laplacian is denoted by Δ, and the Kronecker delta is indicated by δij Additional notation will be defined as it appears throughout the text.
The elastostatic behavior of a three-dimensional homogeneous and isotropic body is described by the equilibrium equations t i j , j + f i =0 (1.1) and the constitutive relations t i j =λu k , k δ i j +μ(u i , j +u j , i ) (1.2)
In the context of internal stresses, the notation t i j = t j i represents the stress components, while u i denotes displacements and f i indicates body forces The Lamé constants, λ and μ, characterize the material properties The resultant stress vector t in a specified direction n = (n 1, n 2, n 3) T is expressed as t i = t i j n j Additionally, the internal energy density, which refers to the internal energy per unit volume, plays a crucial role in understanding material behavior.
C Constanda, Mathematical Methods for Elastic Plates, 1 Springer Monographs in Mathematics, DOI: 10.1007/978-1-4471-6434-0_1, © Springer-Verlag London 2014 www.MathSchoolinternational.com
A thin plate is defined as an elastic body occupying a region in three-dimensional space, represented as S¯× [−h 0 /2,h 0 /2], where S is a two-dimensional domain and h 0 is a constant thickness The unique geometry of thin plates allows for the application of simplifying assumptions in the analysis of small deformations, enabling the development of two-dimensional theories that effectively capture the essential characteristics of the deformation state This article focuses specifically on the bending process of thin plates.
The first truly systematic theory of bending of thin elastic plates was proposed by Kirchhoff (1850) Under his assumptions the displacement field becomes u α = −x 3 u 3 ,α , u 3 =u 3 (x γ ), (1.5) and from (1.1) and (1.2) it follows that ΔΔu 3= p
D, where p is the resultant load on the faces x 3= ±h 0 /2 of the plate and
The rigidity modulus is represented by 3(λ+2μ), and while this theory provides reasonable approximations in various practical scenarios, it entirely overlooks the impact of transverse shear forces, as indicated by equations (1.2) and (1.5), which yield t 3 α =0 across the plate This oversight leads to mathematical inconsistencies, where certain stress components are omitted in some equations but included in others Furthermore, the unknown deflection u 3 is only able to meet two boundary conditions, falling short of the three that are physically anticipated.
Reissner (see Reissner1944;1976) takes transverse shear into account by assum- ing that t αβ = h 2 0
The study presents a sixth-order theory based on the principle of least work, which accommodates three boundary conditions and offers a more comprehensive model than Kirchhoff’s, although it only provides average displacement values Notable contributions from Hencky (1947), Bollé (1947), Uflyand (1948), and Mindlin (1951) incorporate the effects of transverse shear deformation, beginning with the displacement assumptions u α = x 3 v α (x γ ) and u 3 = v 3 (x γ ).
This article introduces an approximate sixth order theory by averaging equations (1.1) and (1.2) over the thickness of the plate, similar to Reissner's approach, which permits the specification of three boundary conditions However, these equations exhibit a lack of rigor, primarily because the parameter t33 is disregarded in the constitutive relations, which also include correction factors.
The various theories on bending have been refined to provide valid information on bending characteristics while simplifying the problem to two dimensions, as noted by Reissner (1985) This analysis focuses on the mathematical treatment of these theories rather than their physical advantages We adopt an approximation based on the kinematic assumption (1.6) to avoid inconsistencies from oversimplification Our method can be easily adapted to apply to all existing sixth-order theories with elliptic equilibrium equations, requiring minimal adjustments to the coefficients.
Geometry of the Boundary Curve
In this article, we simplify notation by using the same symbol to represent both a point and its position vector in R² Additionally, we do not differentiate between vector functions and scalar functions through special markings, as their nature is clear from the context.
The boundary ∂S of a set S is defined as a simple closed curve with a length denoted by l This curve has a natural parametrization, represented as x = x(s), where s ranges from 0 to l and satisfies the condition x(0) = x(l) This parametrization is a bijection, meaning it has a unique inverse s = s(x) for points x along the boundary ∂S.
Throughout what follows,∂S is a C 2 -curve; in other words, x is twice continuously differentiable on[0,l]and d x ds (0+)=d x ds (l−), d 2 x ds 2 (0+)=d 2 x ds 2 (l−).
As is well known, d x ds =τ(s)=τ(x) www.MathSchoolinternational.com
The orientation of the local frame axes at a point x on the boundary ∂S is defined by the unit tangent vector, which indicates the direction of increasing parameter s The unit outward normal vector ν(x) at x is oriented away from the surface S, and the local frame formed by τ(x) and ν(x) is configured to be left-handed This relationship is mathematically expressed as τ α = ε βα ν β, where ε αβ represents the two-dimensional Ricci tensor, also known as the alternating symbol.
Figure1.1shows the orientation of the local frame axes.
The Frenet–Serret formulas d dsτ(x)= −κ(x)ν(x), d dsν(x)=κ(x)τ(x)
(1.8) connectτ(x),ν(x), and the algebraic valueκ(x)of the curvature of∂S at x.
The selection of the normal vector's direction aligns with established practices in the literature, ensuring that the subsequent analytic arguments involving ν are formulated coherently.
In a domain with holes, the orientation conventions for τ and ν are applicable not only to the outer boundary but also to the boundaries of each individual hole.
(iii) Since∂S is a C 2 -curve, we can define κ 0= sup x ∈ ∂ S
It is obvious thatκ 0 >0, forκ 0 =0 would imply that∂S were a straight line and, therefore, not a closed curve.
|x| 2 =x 1 2 +x 2 2 be, respectively, the standard inner product and the Euclidean norm onR 2
While the estimates provided may not be the most precise, more accurate ones are available However, for the sake of simplicity in manipulation, we choose acceptable numerical coefficients that facilitate the inequalities.
1.2 Geometry of the Boundary Curve 5
Proof Let s and t be the arc length coordinates of x and y We have
The Taylor series expansion now yields
1−κ(x )ν(x ),x −y⇔(s−t) 2 , where s is the value of the arc length coordinate of a point x lying between x and y on∂S.
Following the same procedure, we have
∂s 2 ν(y),x−y⇔ s = s (s−t) 2 , where s is the arc length coordinate of a point x lying between x and y on∂S; hence, by (1.12),
On the other hand, if|x−y|>1/(2κ 0 )(or, what is the same, 2κ 0|x−y|>1), then
Combining the two cases, we conclude that for any x and y on∂S,
=κ(x )τ(x )(s−t), where s is the arc length coordinate of a point x lying between x and y on∂S. Hence, in view of (1.12), for|x−y| ≤1/(2κ 0 )we have
At the same time, for|x−y|>1/(2κ 0 ),
|ν(x)−ν(y)| ≤ |ν(x)| + |ν(y)| =2r ,
|s−t| ≤l≤ l r|x−y|, www.MathSchoolinternational.com we conclude that for all x,y∈∂S, c|s−t| ≤ |x−y| ≤ |s−t|, where c=min
Properties of the Boundary Strip
This book presents various results that are established by analyzing the behavior of specific two-point functions near the boundary To enhance the clarity of these proofs, we will first explore some commonly utilized properties The normal displacements of the boundary ∂S are defined as follows.
, σ =const, 00; hence, d x ds = dξ ds +σdν(ξ) ds =τ(ξ)+σ κ(ξ)τ(ξ) 1+σ κ(ξ)τ(ξ) (1.21) Since, in terms of the arc parameter t on∂S σ , d x=d x dt dt =τ(x)dt, www.MathSchoolinternational.com
1.3 Properties of the Boundary Strip 11
Fig 1.6 Arcs of ∂ S and ∂ S σ it follows that, by (1.21), dt= |d x| d x ds ds d x ds ds
1+σκ(ξ) ds dt dt, so ds dt 1+σκ(ξ) − 1 ; therefore, τ(x)=d x dt =d x ds ds dt
Suppose that there areξ, ξ ∈∂S,ξ =ξ , such that the support lines ofν(ξ)and ν(ξ )intersect at some point x located at a distance less than 1/κ 0from∂S; that is, x=ξ+σ ν(ξ)=ξ +σ ν(ξ ), |σ|,|σ |< 1 κ 0
Then x ∈∂S σ ∩∂S σ , so, by (1.22), τ(x)=τ(ξ)=τ(ξ ), which implies thatν(ξ)=ν(ξ ) Since this contradicts our assumption, we conclude that∂S σ is well defined.
Figure1.6illustrates an arc of∂S and the arc of a typical curve∂S σ 1.8 Definition Letσ 0be a fixed number such that 0< σ 0 8|x−x |}, (1.29) www.MathSchoolinternational.com
1.3 Properties of the Boundary Strip 15
Fig 1.9 A portion of Σ ξ, r and Σ 1 (heavier arc) thenΣ 1 lies strictly withinΣ ξ, r ,ξ ∈Σ 1, and for all y∈Σ 2,
In the diagram illustrated in Fig 1.9, the heavier arc denotes Σ 1, with boundary points a, b for Σ ξ and p, q for Σ 1 positioned on opposite sides of ξ The arc length coordinates for points p and q are represented as t p and t q, where t p < s < t q It is established that the distances |ξ−a| and |ξ−b| are both equal to r, as derived from equations (1.17) and (1.29).
0 is independent of x and y If S is unbounded, then the above definition must hold on every bounded subdomain of S.
We denote by C 0 ,α (S)¯ the vector space of (real) Hửlder continuous (with index α∈(0,1]) functions onS, and by C¯ 1 ,α (S)¯ the subspace of C 1 (S)¯ of functions whose first-order derivatives belong to C 0 ,α (S¯).
The proof consists in the verification of (1.52).
The spaces C 0 ,α (∂S)and C 1 ,α (∂S)are introduced similarly, with (1.52) required to hold for all x, y ∈ ∂S In view of Lemma 1.4, we will not distinguish between
C 0 ,α (∂S)and C 0 ,α [0,l], which is defined by means of the inequality
Obviously, Lemma 1.23 also holds for functions on∂S. www.MathSchoolinternational.com
If a function \( f \) is bounded within the set \( S \), meaning that \( |f(x)| \leq M \) for a constant \( M \) across all \( x \) in \( S \), and if the condition (1.52) is satisfied for all \( x, y \) in \( S \) where the distance \( |x - y| \) is less than or equal to a constant \( \delta > 0 \), then this condition will also hold for all \( x, y \) in \( S \), potentially with a different constant \( c \) This can be easily demonstrated.
1.25 Remark Ifϕ ∈ C 0 ,α (∂S)as a function of x and x =x(s), then, by Lemma 1.4,ϕ∈C 0 ,α (∂S)also as a function of s, and vice versa.
1.26 Definition A two-point function k(x, y)defined and continuous for all x ∈S 0
(x ∈∂S ) and y∈∂S, x =y, is called aγ-singular kernel in S 0(on∂S ),γ ∈ [0,1], if there is p=const>0, which may depend on∂S, such that for all x ∈S 0(x ∈∂S ) and y∈∂S, x=y,
If the above inequality holds and, in addition, for all x,x ∈S 0(x,x ∈∂S ) and y∈∂S satisfying 00 depends only onγ.
Proof It is obvious that K(x)is an improper integral for x ∈∂S.
LetΣ x , r ,Σ 1, andΣ 2be defined by (1.16) and (1.29) In view of Remark 1.24, we may consider x,x ∈S 0satisfying (1.43).
Setting, as before, x=ξ+σ ν(ξ), x =ξ +σ ν(ξ ), ξ, ξ ∈∂S, we can write
K(x)−K(x )=I 1+I 2+I 3 , where, by Definition 1.22, Remark 1.14, and Lemmas 1.10–1.13,
|ϕ(x)|, where l is the length of the boundary curve∂S The assertion now follows from the fact that the constants c 1 , ,c 8 > 0 are independent of x and x (although they depend onγ).
The result is established for x, x ∈∂S as a particular case of the above, by setting x=ξand x =ξ
1.34 Remark It is obvious that Theorem 1.33 holds if the kernel k(x, y)is contin- uous on S 0×∂S (∂S×∂S ).
1.35 Theorem If k(x, y)is a proper 1-singular kernel in S 0(on∂S),ϕ∈C 0 ,α (∂S), α∈(0,1], and Φ(x)
∂ S k(x, y) ϕ(y)−ϕ(ξ) ds(y), (1.54) where x =ξ +σ ν(ξ)∈S 0 (x=ξ ∈∂S), thenΦ∈C 0 ,β (S 0 ) (Φ ∈C 0 ,β (∂S))for anyβ ∈(0, α) If, in addition,α∈(0,1)and
≤c=const>0 (1.55) for all x ∈S 0 (x∈∂S)and all 0< δ