thermal upheaval buckling of heavy sheets

If a thin heavy sheet of very large in-plane dimensions lying on a flat surface, is subject to an increase in temperature, it will if restrained at its outer edges experience the development of an in-plane compressive force. Indeed, if the friction between the sheet and the surface upon which it lies is great enough, these compressive forces could develop even if the outer edges are not restrained. Under certain circumstances these compressive forces can induce an upward buckle against the weight of the sheet tending to prevent the buckle. This phenomenon of thermally induced uplift buckling of thin heavy heets has been considered by Hobbs (1989, 1990) and more recently by Croll (2004a, b). It is closely related to the more extensively studied thermal buckling of one dimensional components such as railway tracks, subsea oil and gas pipelines, concrete road pavements (see for example Martinet (1934), Kerr (1974,1984), Hobbs (1984), Taylor and Gan (1986), Croll (1997, 1998)). A full description of the mechanics is beyond the scope of this short posting, but a summary of the behaviour required to appreciate its applicability to the development of pingos is as follows.

Before considering the buckling of a heavy sheet it might be worth briefly discussing the buckling of a circular disk for which the effects of its own weight are neglected.

Perfect Behaviour: If a circular disk of finite radius, a, is restrained at its outer edges against both out-of-plane deformation and rotation and is subject to an in-plane stress that is equal in all directions (sometimes referred to as a hydrostatic or isostatic stress state), it will, at a certain critical level of this stress, develop an out-of-plane buckling mode taking the shape depicted in Figure 1(a). When the plate is initially perfectly flat the behaviour, exhibited in the form of a stress resultant n against out-of-plane displacement, w, is as shown in Figure 1(b). The stress resultant, n, is simply the average stress through the thickness of the plate times the plate thickness – it can be thought of as the amount of force per unit length. The plot of Figure 1(b) shows that there are two possible equilibrium paths. A fundamental path for which there is no deformation for ever increasing in-plane stress resultant, n, and a post-bucking path that shows increasing deformation, w, for little change in stress resultant, n. The two paths intersect at a stress level that is usually referred to as the critical stress (or sometimes the bifurcation stress although strictly it is an intesection rather than a bifurcation). In reality, for such a restrained plate there will be an increase in stress resultant for increasing out-of-plane displacement on the post-bucking path. For present purposes however, this complication will be ignored. The situation summarised in Figure 1 is often referred to idealised or “perfect” behaviour; it is not realistic in the very practical sense that it cannot actually be observed.

Imperfect Behaviour: Even the smallest out of flatness, sometimes described as an imperfection, will cause the above bifurcation behaviour of Figure 1 to be modified as shown in Figure 2. For example, the weight of the plate will cause a downward deformation (represented here as a negative deformation) of say magnitude, w0, even before any in-plane load is applied. As the in-plane load, n, is increased the non-linear equilibrium path will take the form shown as a-b in Figure 2. That is, the in-plane load will amplify the imperfection and tend to push the plate down at a rate which increases as the stress approaches that of the critical or bifurcation stress – or strictly as the imperfect path a-b asymptotically approaches the perfect post-buckling path. However, if the plate is sitting on a rigid foundation this downward displacement is not possible. Under these circumstances the only option open to the plate is for it to buckle upwards. These deformations will be controlled by the upper path d-e, which is sometimes referred to as the complementary equilibrium path for this same initial imperfection. While for most buckling problems these complementary paths are of little consequence, for the case of uplift buckling from a rigid or semi-rigid base support they are of controlling significance.

Perfect Uplift Buckling: If the plate of very large in-plane dimensions is sitting on the rigid foundation then the buckling behaviour will be close to that described above except for one important complicating factor. As the amplitude of the uplift buckle, w, increases the radius, a, of the uplifted buckle will also increase. The result is a propagation path that takes the form shown by path a-b in Figure 3. As demonstrated by Croll (1997, 2004a, 2004b) the approximate form for this propagation curve may be expressed in terms of the projections, shown by the heavy dashed lines in Figure 3, for which the equations take the forms shown. What these equations show is that the in-plane stress resultant required to propagate the uplift buckle, np, is inversely proportional to the inverse square root of the amplitude, w, of the uplift buckle. The radius, a, of the uplifted disk increases in direct proportion to the quarter power of the deflection, w - or it is proportional to the square root of the square root of w. In these equations q is the net downward weight of the plate per unit area and D is the bending stiffness of the plate defined in Figure 1. It is apparent from Figure 3  that if the plate starts out as perfectly flat, it would take an infinite load np before the system could get onto the falling propagation path a-b. How in reality the system will develop a propagating uplift buckle is all to do with initial imperfections.

Imperfect Uplift Behaviour: Probably of most direct relevance to our subsequent consideration of how the thermal uplift buckling model could help to explain the mechanics of pingo development, is that of an unstressed continuous geometric imperfection. This has been outlined by Croll (2004a) but considered in greater detail and possibly greater clarity in Croll (2004b). Imagine that the foundation were not perfectly flat but contained an out-of-flatness of the form shown in Figure 4(a). Studies have shown that for a given amplitude of this out-of-flatness, w0, the worst possible choice of wavelength would be that which has an affinity to the equivalent propagation shape given by eqn (3). This means the wavelength, a0, giving the lowest stress, and therefore temperature since we seem to have forgottren that the stress is being produced by a temperature increase, required to induce an uplift buckle would be related to the amplitude of imperfection, w0, given by eqn(5) shown in Figure 4(a).

With this amplitude and shape of continuous geometric out-of-flatness the stress required to cause the plate to initiate a lift-off condition, nL, would be exactly one quarter the equivalent propagation load that would apply for a displacement equal to the initial imperfection amplitude. The load required to reach the maximum or buckling load, nb, would be exactly twice this value and would occur at a total displacement twice the amplitude of the initial out-of-flatness, as depicted in Figure 4(b).

Lift-off From a Geometrically Imperfect Base: Railway track and subsea pipelines are in certain situations prone to a form of possibly sudden upheaval or restrained lateral buckling when the temperature becomes high enough that the associated in-plane stress reaches the magnitude nb. In these circumstances there will be a sudden jump in deformation with the railtrack or pipeline becoming highly distorted when it finally stops buckling. Just as for the asphalt lift-off discussed in an earlier posting, the evidence for pingos seems to point to their undergoing incremental increases over each annual thermal cycle. In these circumstances the temperature would reach a value in excess of that needed to induce a stress resultant, nL, leading to an incremental lift-off, but probably not great enough to reach nb and thereby cause a dynamic jump in deformation. Here the stress n is related to the temperature T through the relationship n = alfa.T.E/(1 - nu) where alfa is the coefficient of linear thermal expansion, E is the elastic modulus for the sheet, and nu is the Poisson's ratio for the material of the sheet.


I will return to some of the these considerations in the next posting where I hope to discuss how such a thermal upheaval of a heavy sheet might help to explain the emergence of pingos.

This form of 2-dimensional thermal buckling is briefly covered at pavement-blisters