Zz 1 v1 2v[1 vzz vxxyy 0 1159

Hence, the Love-Kirchhoff conditions (1.158) entail v

£zz = --V £x + £yy) , £yz = 0 , £zx = 0 . (1.160)

The sections that are perpendicular to the x- and y-axes when the plate is flat at rest, become tilted around the y- and x-axes respectively in the bending but they remain flat and normal to the mid-surface. Since the tilt angle is equal to the slope of w, the two other components of the displacement vector are dw dw u = -z^r-, v = -z^r-, (1.161)

dx dy which satisfy the conditions (1.158a) of null value at the neutral surface. From u and v, the determination of all the components (1.121a) of the strain tensor gives d2w d2w vz i d2w d2w

£xx = -z dx2, £yy = -z ~df , £zz = l-Vy'dx2 + dy2

and, from the stress-strain relations (1.123a), the associated stresses are

Oxx = 1 - v2\dx2 + vdy2), Oyy = 1 - v2\dy2 + v'dx2 J'

Ez d2w

• Energy of a bent plate: These components allow us to determine the free energy of the unit volume element. Substituting them into the representation (1.125) of the elementary free energy dF, we obtain after simplification dF

Ez2 1+v

The total free energy of the plate is obtained by integration over its total volume. For a constant thickness plate whose thickness varies from —t/2 to t/2, and denoting the rigidity D = Et3/12(1 — v2) such as defined by (1.142), the total free energy or flexural energy of the plate is ud2w d2w\2 JfdwY_ d!w d^w

\dxdyJ dx2 dy2

where dA = dxdy and the integration is taken over the surface area A of the plate. In deriving this latter equation, we have assumed a thin plate and a small deformation, so the w-displacement of any point of the plate is the same as that of a point with identical coordinates x, y belonging to the neutral surface.

• Equation of deformation of a plate: The free energy due to the internal stresses and strains can be used to obtain the general form of the flexure w(x, y) of constant thickness plates. First, one must remark that the sum of the free energy and of the potential energy is a constant. The potential energy is the opposite of the work of the external load applied to the surface of area A in the normal direction; for small deformations this normal is the z-axis so this work taken over the whole plate is /JqwdA, where q is the external load of dimension FL-2. Second, we state that the total energy is minimal in any variation Sw of the displacement. This condition is written as

The variation SF of the free energy is carried out by considering the boundary conditions that are applied at the contour line C of the plate. In the general case, this involves using curvilinear coordinates. A complete analysis is given by Landau and Lifshitz [92] which, starting from (1.163b), leads to the following expression gv2v2w SwdA + £ P1SwdC + jf P2 SwdC - U^q SwdA = 0, (1.164b)

where V2- = d2-/dx2+d2-/dy2 is theLaplacian and theP1, P2 terms are functions of w, v and of the shape of the contour. The two corresponding integrals must be taken equal to zero over the contour, thus providing two conditions, each of them defining the force and the moment applied at the contour. Satisfying the two conditions, this relation becomes

0 0