Further Development of the Theory of Internal Buckling of Multilayers

The approximate theory of internal buckling of multilayers is obtained by a direct method and generalized to nonsinusoidal deformations. The numerical discussion includes the case of unequal thickness of competent and incompetent layers. Simple expressions are derived for the dominant wave length for rigid confinement or selfZntroduction For geological applications it is important to supplement exact theories of buckling of multilayered structures by simplified approximate treatments which bring out the essential mechanics and lead to simple formulas. For internal buckling of confined multilayers, such an approach was initiated in two previous papers (Biot, 1963; 1964a). In this paper the equations that govern internal buckling are derived in a different and more direct way. They are also expressed in more general form, as a set of field equations with two unknowns: the vertical deflection and the vertical stress. These field equations are not restricted to sinusoidal deflections. Values of the dominant wave length were given earlier (Biot, 1964a) for competent and incompetent layers of equal thickness. In this paper we have extended the numerical discusssion to layers of unequal thickness and obtained results quite similar to those of the previous analysis, showing that the thickness of the competent layer, as well as the confinement thickness, is a significant parameter. Internal buckling may be caused by rigid confinement or by self-confinement when occurring in a medium of infinite extent. Either case follows from the same analysis. The theory is presented in the context of viscous media and is derived in two different ways: from solid and fluid mechanics. Both methods lead to equations which for all practical purpose are identiconfinement showing the influence of interstitial flow. The theory is presented in the context of viscous media using both solid and fluid mechanics and is valid for large compressive strain and moderate slopes. Its applicability to elastic and viscoelastic media is indicated. cal. The results are valid for large compressive strain with variable thickness of the layers. As will be pointed out further on, equations applicable to elastic and viscoelastic media are immediately obtained by viscoelastic correspondence. Equilibrium Equations For Plate Buc&ng Consider a plate of thickness 2/r under an average compressive stress P along its axis (Fig. 1). We will assume plane strain in the x,y plane of the figure. When the plate is deformed with moderate slopes, a compressive stress approximately equal to P continues to act along the deformed axis (Fig. 1). At the same time the deformation generates a bending moment x and a total shear z acting over a cross section. This cross section is assumed to be of unit width in the direction perpendicular to the x,y plane. Hence X and X have the dimensions of a moment and force per unit width. Also per unit width and per unit distance along the axis we apply a vertical load q and a clockwise moment m. The equilibrium conditions of these forces for the deformed plate with moderate slopes are approximately The vertical deflection is denoted by v. Except for the additional external moment m, these Geological Society of America Bulletin, v. 76, p. 833-840, 5 figs., July 1965 833 834 M. A. BIOT-THEORY OF INTERNAL BUCKLING OF MULTILAYERS Figure 1. Forces and stresses on a composite plate containing one competent layer equations are the same as those considered in the previous theory of similar folding (Biot, 1965a). Elimination of X yields a2m -=a.2 q--f&+2&‘/@! ax2 (2) A derivation of this equation is also given in an earlier paper dealing with the effect of interfacial adherence on viscous and viscoelastic folding (Biot, 1959). It can be shown that equation (2) is a consequence of the general mechanics of initially stressed media (Biot, 1965b, p. 127). Y 4X Buck$ing Equation for a Competent Layer in a Multilayered Structure Consider a multilayered structure of viscous, incompressible material. Competent layers of thickness hr alternate with incompetent layers of thickness hs (Fig. 2). The system may be regarded as a stacking of plates, each plate being composed of a competent layer sandwiched between two incompetent layers of thickness h2/2. The total thickness of the composite plate is denoted by 2h = hr + hs. We shall apply equation (2) to this composite plate. We Figure 2. Structure of composite plate containing one competent layer