Kirchhoff hypothesis

The implications of the Kirchhoff hypothesis on the laminate displacements u, V, and w in the x-, y-, and z-directions are derived by use of the laminate cross section in the x-z plane shown in Figure 4-4. The displacement in the x-direction of point B from the undeformed middle surface to the deformed middle surface is Uo (the symbol nought (°) is used to designate middle-surface values of a variable). Because line ABCD remains straight under deformation of the laminate, the displacement at point C is... 

The laminate strains have been reduced to e, Sy, and by virtue of the Kirchhoff hypothesis. That is, ez = Yw -r 0- small strains (linear elasticity), the remaining strains are ned in terms of displacements as... 

The Kirchhoff hypothesis of linear strain variation through the laminate thickness applies (prior to degradation, if any after degradation, linear only through the thickness of each lamina). 

The Kirchhoff hypothesis of negligible transverse shear strains, Yxz and tutes... 

Even though in classical lamination theory by virtue of the Kirchhoff hypothesis we assume the stresses and are zero, we can still obtain these stresses approximately by integration of the stress equilibrium equations... 

In plate theory, the problem is reduced from the deformation of a solid body to the deformation of a surface by use of the Kirchhoff hypothesis (normals to the undeformed middle surface remain straight and normal after deformation, etc., as discussed in Chapter 4). Then, we attempt to apply boundary conditions to that surface which is usually the middle surface of the plate. There should be no surprise that the boundary conditions for the unapproximated solid body are not the same as those for the solid approximated with a surface. The problem arises when these boundary conditions are applied to an approximate set of equilibrium equations that result when force-strain and moment-curvature... 

However, these transverse shearing stresses were neglected implicitly when we adopted the Kirchhoff hypothesis of lines that were normal to the undeformed middle surface remaining normal after deformation in Section 4.2.2 on classical lamination theory. That hypothesis is interpreted to mean that transverse shearing strains are zero, and, hence, by the stress-strain relations, the transverse shearing stresses are zero. The Kirchhoff hypothesis was also adopted as part of classical plate theory in Section 5.2.1. 

The first assumption allows the development of averaged stiffness values for each ply, which depend on the individual stiffnesses of the fibers and matrix. The second and third assumptions are necessary for the development of the weighted stiffnesses for each ply and for each laminate. They can be relaxed when dealing with some failure modes such as fiber pull-out and delamination. The last two assumptions form the Kirchhoff hypothesis and ensure small deflections and rotations. 

The assumption which underlies the derivation of (2.79), that the radial curvature is uniform everywhere in the film—substrate system, is an essential feature of the deformation in the linear range. In the nonlinear deformation range, on the other hand, there is no basis for expecting the curvature to be uniform. The finite element method of numerical analysis can be used to determine the deformed shape of the substrate midplane in the nonlinear range without a priori assumptions on the distribution of curvature. The deformation is constrained to be consistent with the Kirchhoff hypothesis but it is otherwise general. In particular, transverse deflections which are large compared to the substrate thickness hs are accommodated. 

The issue of bifurcation in equilibrium shape is pursued in two steps. First, a simple energy approach is taken. The deformed shape of the substrate midplane is assumed to be ellipsoidal so that the shape is characterized completely by two principal curvatures which need not be the same. Consistent in-plane displacements, which account for midplane stretching, are also assumed. The principle of stationary potential energy is then invoked to determine the relationship of the principal curvatures to system parameters to ensure that the system is in equilibrium. A more detailed examination of bifurcation on the basis of a finite element simulation, without a priori restrictions on deformation beyond the Kirchhoff hypothesis, is described... 

Let us derive a condition of nonpenetrating in general case (see Fig. 1.3). The Kirchhoff-Love hypothesis provides the linear dependence of the shell horizontal displacements on a distance from the mid-surface, namely... 

William Prout s composite atoms hypothesis. G. Kirchhoff and R. Bunsen discover spectral analysis and significance of Fraunhofer lines Kirchhoff s law. 

Kininogen, 4 86—87 Kirchhoff-Love hypothesis, 26 780 KirchhofPs law, 14 234 23 4 Kish, leaching, 12 783—784 Kiss coater, 7 12 Kitasamycin, 15 287... 

Further on, for sufficiently thin laminates, the following kinematic assumption shall be applicable, usually being denoted as the Kirchhoff-Love hypothesis in shell theory ... 

The following assumptions are made in order to simplify the problem. Displacements are small and a linear analysis is valid. The plate is isotropic and the material is uniform. No delamination opens and the frictional force is neglected. Kirchhoff s hypothesis is valid even in the neighborhoods of the delamination edges. The boundary conditions at r = are... 

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