Analogy Methods to Address Warping and Plasticity in Torsion

This project is primarily an experimental study on the membrane analogy and the sand-heap analogy associated with plasticity (along with warping) when a prismatic bar (circular or noncircular) is twisted. This is a supplemental lab activity for the junior level Mechanics of Materials course. When a prismatic bar of arbitrary cross-section is twisted, it gets deformed into helical curves as the cross sections rotate during torsion. The analogy between the elastic torsion of a bar and the small deflection of a laterally loaded membrane is what is known as the membrane analogy. The membrane is stretched by a uniform tension and is attached to a die whose edge plane is the same as the cross section of the twisted bar. A uniform lateral pressure is applied to the membrane to produce a deflection. Except for a circular cylindrical bar, the crosssections become warped and it plays a significant role in structural design. As the torque is increased, the material passes from the elastic to plastic state. There is a limit to the ability of the cross section to withstand increasing torque. This feature is not typically addressed in the junior level Mechanics of Materials course, but we felt the need for it because this deformation can be readily shown in a torsion experiment with a ductile material such as aluminum. When the torque is increased to a critical value, the resultant shear stress attains the yield value ‘k’ at one or more points, somewhere along the boundary of the cross section. The extension of the membrane analogy to elastic plastic torsion leads to the sand-heap analogy. A sand heap is formed on a base having a cross-section similar to the cross-section under investigation. In the case of membrane analogy, the volume bounded by the deflected membrane is proportional to the applied torque. For the sand heap analogy, the volume of the sand heap is proportional to the limiting torque applied to the specimen. Experimental estimates for the elastic and inelastic torques in twisting circular and non-circular shafts are in reasonable agreement with the values predicted by the membrane analogy and the sand heap analogy. INTRODUCTION In this work, prismatic bars of circular and square sections have been loaded in torsion to study their responses in elastic and elastic-plastic domains. The purpose was to experimentally demonstrate the plastic deformation that takes place when the torque exceeds the amount that causes the bar to yield. This is an extension of the torsion test where the torques are such that yielding does not take place, and the torque twist characteristic is linear. In such tests typically bars of circular cross are used. Bars of square cross section were used in our work to experimentally demonstrate the phenomenon of warping that takes place for bars of non-circular sections under torsion. When a bar is twisted beyond the elastic range (yielding) there is a non-linear torque-twist behavior. Specifically, for bars of non-circular cross-section it is hard to separate the warping deformation from the plastic deformation. Simple analytical solutions for the deformation and stresses in a bar subject to axial torsion exist only for circular cross-sections. However, it is sometimes necessary to design shafts of noncircular cross-sections. As an example, the design of a connecting rod would require determination of torsional stresses in non-circular sections (typically I-sections). Such situations also exist in the design of various machine parts such as brackets and supports that are sometimes loaded in torsion. The determination of the stresses and deflections for the torsion of non-circular shafts involve equations that are quite complicated. The assumptions that are valid for circular cross-sections do not apply here. For non-circular shafts under torsion, the plane cross-sections perpendicular to the shaft axis do not remain plane after twisting, and deformation takes place in the axial direction which is called warping. One of the simplest ways to study elastic as well as elastic-plastic behavior analytically is to assume an elastic-perfectly plastic representation of the stress-strain curve. For the case of torsion this would be the shear stress vs. shear strain curve as shown in Figure 1. The shear stress varies linearly with shear strain in the elastic region, and reaches a constant value and stays constant in the plastic region. Sometimes this representation is termed as one of “zero hardening.” However, most ductile materials exhibit strain hardening where in the plastic region the shear stress monotonically increases with shear strain. The focus of this work is on analogy methods to study elastic and elastic-plastic torsion. The two analogies associated with torsion are described below. The Membrane Analogy for Elastic Torsion An analogy between the elastic torsion of a bar and the small deflection of a laterally loaded membrane has been pointed out by Prandtl The membrane is stretched by a uniform tension F per unit length of its boundary, and is attached to a die whose edge plane is of the same shape as Figure 1 the cross section of the twisted bar. A uniform lateral pressure is then applied to the membrane to produce a deflection w at a generic point). The boundary condition is w = 0 along the edge of the die. It turns out that the contours of constant deflection correspond to the lines of shearing of the twisted bar. Also the applied torque is proportional to the volume bounded by the deflected membrane and the xy plane. Since the membrane is everywhere concave to the applied pressure, the greatest value of the shear stress must occur somewhere on the boundary. Consider a cylindrical or prismatic bar of constant cross section which is twisted and held in equilibrium by twisting moments applied at its ends. The bar is considered to be composed of an isotropic material possessing the idealized stress-strain relationship for an elastic, perfectly plastic material shown in Figure 1. Increasing torque causes the material to pass from the elastic region (line AB, Figure 1) into the perfectly plastic range (line BC, Figure 1). After a point in the cross section reaches the yield stress in shear (point B), this maximum shearing stress remains a constant value k as increasing torque causes an increase in the plastic region of the bar. Before examining the plastic behavior of the prismatic cross section, we will consider the stress characteristics in the elastic range The Sand-Heap Analogy for Plastic Torsion An extension of the membrane analogy to elastic/plastic torsion has also been suggested by Prandtl It is necessary to erect a roof of constant slope having its base similar to the boundary of the cross section. If the bar of certain cross-section is twisted beyond the yield point, certain parts of the bar will be deformed plastically. Similar to the case of the elastic torsion, the shearing stresses are directed tangentially to the contour lines of the plastic stress surface. The plastic stress function may be considered as a "roof" under which the membrane, geometrically the same as the cross section, expands. When the membrane touches the "roof," the condition of plasticity is satisfied and plastic yielding begins at that point in the cross section. As the torque is increased, as represented by increasing air pressure on the membrane, the membrane expands and touches more of the "roof." At the limit the membrane fills the entire volume under the roof. The cross section is considered to have attained a fully plastic state. Following the membrane analogy, the torque required to achieve the fully plastic state is proportional to the volume under the "roof." The mathematical and physical interpretation of the plastic response for the case of complete yielding, of the entire prismatic bar can be demonstrated by sand heaps covering a plate similar in cross section to the twisted bar. This analogy was first presented by Nadai [2] at a meeting of the German Society of Applied Mathematics and Mechanics in Marburg, Germany in 1923. ."A plate whose shape is geometrically similar to the cross section of the twisted bar series as a horizontal tray to hold a heap of dry uniform sand. The heap is to be as big as it is possible to pile by pouring a gentle stream of sand on top of the model, the excessive sand rolling freely down the slopes of the heap and falling off the elevated tray. .” In a solid bar made of non-hardening material, the fully plastic stress distribution represents a limiting state which is approached in an asymptotic manner as the angle of twist increases. The fully plastic value of the torque has a physical significance, since it is very closely attained while the deformation is still of the elastic order of magnitude. The stress surface for a fully plastic cross section can be obtained experimentally by piling dry sand on a horizontal base whose shape is geometrically similar to that of the cross section of the bar. The limiting torque is proportional to the volume of sand forming the hill. Once the limiting torque has been reached, the bar is free to twist in an unrestricted manner. Some results [3] of the computation for a square cross section of side 2a, based are shown in Figure 2. Choosing n=9, which very nearly represents a perfectly plastic material, the positions of the elastic/plastic boundary for increasing angles of twist are obtained as contours of τ =k, and plotted in (a). The influence of work-hardening on the torque-twist relationship is indicated in (b). Figure 2: Numerical results for torsion of a square cross-section