The effects of pressure on the yield locus can be confused with the effects of the sign of the stress. For example, twinning is sensitive to the sign of the applied stresses and causes the yield behavior under compression to be different from that under tension.

S-D EFFECT

With a so-called strength differential (SD) effect in high strength steels [1], yield strengths under tension are lower than under compression. The fractional magnitude of the effect, 2(|σc| – σT)∕[(|σc| + σT] is between 0.10 and 0.20. Figure 14.1 shows the effect in an AISI 4330 steel and Figure 14.2 indicates that it is a pressure effect.

Although the flow rules, equation 4.18, predict a volume increase with yielding, none has been observed.

POLYMERS

For polymers, the stress-strain curves in compression and tension can be quite different. Figures 14.3 and 14.4 are stress strain curves for epoxy and PMMA in tension and compression. Figure 14.5 compares the yield strengths of polycarbonate as tested in tension, shear and compression. The effect of pressure on the yield strength of PMMA is plotted in Figure 14.6.

In 1950, R. Hill wrote an authoritative book, Mathematical Theory of Plasticity that presented a comprehensive treatment of continuum plasticity theory as known at that time. Much of the treatment in this book covers some of the same ground but there is no attempt to treat the same topics treated by Hill. This book, however, includes more recent developments in continuum theory, including a treatment of anisotropy that has resulted from calculations of yielding based on crystallography, analysis of the role of defects and forming limit diagrams. There is a much greater emphasis on deformation mechanisms, including chapters on slip and dislocation theory and twinning.

This book should provide a useful resource to those involved with designing processes for sheet metal forming. Knowledge of plasticity is essential to those involved in computer simulation of metal forming processes. Knowledge of the advances in plasticity theory are essential in formulating sound analyses.

INTRODUCTION

For pencil glide, the five independent slip variable necessary to produce an arbitrary shape change can be the amount of slip in a given direction and the orientation of the plane (angle of rotation about the direction). There are two possibilities for five systems: Either three or four active slip directions can be active. Chin and Mammel [1] used a Taylor type analysis for combined slip on {110}, {123}, and {112} planes, finding that Mav for axially symmetric flow = 2.748 (Figure 10.1). Hutchinson [2] approximated pencil glide by assuming slip on a large, but finite number of slip planes. Both of these analyses used the least work approach of Taylor. Penning [3] described a least-work solution considering the possibility of both three and four active slip directions. Parniere and Sauzay [4] described a least work solution.

METHOD OF CALCULATION

Piehler et al [5, 7, 8] used a Bishop and Hill-type approach, by considering the stress states capable of activating enough slip systems. Explicit expressions were derived for the stress states in the case of four active slip directions. Instead of explicit solutions for the case of three active slip directions, a limited number of specific cases were considered. The stress states are:

Calculation of exact forces to cause plastic deformation in metal forming processes is often difficult. Exact solutions must be both statically and kinematically admissible. This means they must be geometrically self-consistent as well as satisfying stress equilibrium everywhere in the deforming body. Slip-line field analysis for plane strain deformation satisfies both and are therefore exact solutions. This topic is treated in Chapter 15. Upper and lower bounds are based on well-established principles [1, 2].

Frequently, it is difficult to make exact solutions and it is simpler to use limit theorems, which allows one to make analyses that result in calculated forces that are known to be either correct or too high or too low than the exact solution.

UPPERBOUNDS

The upper bound theorem states that any estimate of the forces to deform a body made by equating the rate of internal energy dissipation to the external forces will equal or be greater than the correct force. The analysis involves:

1. Assuming an internal flow field that will produce the shape change.

2. Calculating the rate at which energy is consumed by this flow field.

3. Calculating the external force by equating the rate of external work with the rate of internal energy consumption.

Of concern in plasticity theory is the yield strength, which is the level of stress that causes appreciable plastic deformation. It is tempting to define yielding as occurring at an elastic limit (the stress that causes the first plastic deformation) or at a proportional limit (the first departure from linearity). However, neither definition is very useful because they both depend on accuracy of strain measurement. The more accurately the strain is measured, the lower is the stress at which plastic deformation and non-linearity can be detected.

To avoid this problem, the onset of plasticity is usually described by an offset yield strength that can be measured with more reproducibility. It is found by constructing a straight line parallel to the initial linear portion of the stress strain curve, but offset from it by a strain of Δe = 0.002 (0.2%). The yield strength is taken as the stress level at which this straight line intersects the stress strain curve (Figure 2.1). The rationale is that if the material had been loaded to this stress and then unloaded, the unloading path would have been along this offset line resulting in a plastic strain of e = 0.002 (0.2%). This method of defining yielding is easily reproduced.

INTRODUCTION

Slip-line field analysis involves plane-strain deformation fields that are both geometrically self-consistent and statically admissible. Therefore, the results are exact solutions. Slip lines are really planes of maximum shear stress and are oriented at 45 degrees to the axes of principal stress. The basic assumptions are that the material is isotropic and homogeneous and rigid-ideally plastic (that is, no strain hardening and that shear stresses at interfaces are constant). Effects of temperature and strain rate are ignored.

Figure 6.1 shows a very simple slip-line field for indentation. In this case, the thickness, t, equals the width of the indenter, b and both are very much smaller than w. The maximum shear stress occurs on lines DEB and CEA. The material in triangles DEA and CEB is rigid. Although the field must change as the indenters move closer together, the force can be calculated for the geometry as shown. The stress, σy, must be zero because there is no restrain to lateral movement. The stress, σz, must be intermediate between σx and σy. Figure 6.2 shows the Mohr's circle for this condition. The compressive stress necessary for this indentation, σx = −2k. Few slip-line fields are composed of only straight lines. More complicated fields are considered throughout this chapter.

Introduction

In the previous chapters, we have investigated lumped parameter electromechanical conversion, linear multiport representations for actuators and sensors, and the effects of external constraints on transducer response. We used a powerful electromechanical analog to synthesize circuit models for transducers, examined the signal conditioning and amplification stages needed to turn MEMS devices into practical systems, and finally developed a methodology to represent mechanical continua, viz., beams and plates, as lumped parameter systems. We are now ready to put these modeling tools to use in the analysis of some practical MEMS devices.

This chapter provides concise system-level technical presentations of four important MEMS applications: pressure sensors, accelerometers, gyroscopes, and energy harvesters. These applications have been selected according to diverse criteria. For example, pressure sensors, relatively uncomplicated as MEMS go, were among the first types of sensors and actuators to be miniaturized. MEMS accelerometers, more complicated than pressure sensors, are very widely used in automotive airbag systems. Micro-mechanical gyroscopes, now available commercially, are considerably more complex than accelerometers, with respect to both the dynamics and the system-level electromechanical drive–sense scheme needed to make them work. Finally, the MEMS energy harvester is included because it is a concept that shows great promise but needs further development before real market penetration can occur.

This appendix reviews the essentials of lumped parameter mechanical systems relevant to microelectromechanical sensors and actuators. Our focus here is mechanical resonators. This choice is based on the reality that, in the MEMS world, stictive phenomena such as van der Waals forces or capillarity are always lurking. Like a tree snagging a kite from the air, these forces can fasten upon a moving element and affix it permanently to an adjacent surface. For this reason, resonance is usually the best way to exploit large motions reliably in a MEMS device. We start with a simple spring-mass system with one degree of freedom and Newton's second law in scalar form. All the main concepts are derived from the single-degree-of-freedom linear model, and then generalized to multiple-degree-of-freedom models and continuous systems using modal analysis.

The approach taken in this appendix is tailored to students having some familiarity with Laplace transforms, from the perspective of either resonant electric circuits or mechanical resonators. The starting point is the differential equation of motion; simple Laplace techniques are introduced so that solutions to the equation can be obtained and the important properties of this solution can be studied.

Preliminaries

For the first century of its existence, the engineering discipline of electromechanics focused almost exclusively on high-current relays, magnetic actuators, and, of course, rotating machines, the latter ranging from fractional horsepower induction motors all the way to gigawatt-rated synchronous alternators [1, 2]. Capacitive transducers never really figured in the discipline because of the focus maintained by the electric power industry on electrical ↔ mechanical power conversion and control. It might seem ironic then that the first eight chapters of this text exclusively concern electrostatic transducers. This emphasis, warranted by the dominance of such devices in the technology of MEMS, testifies to the stern rule of physical scaling in the engineering of useful devices. On the scale of millimeters and below, electrostatic forces hold a considerable advantage.

It would be a mistake, however, to foster the impression that magnetic MEMS will have no role in the future of microsystems technology. They already fill some significant niches in micromechanics and many believe their full potential remains to be tapped. The area-normalized scaling analysis presented in Section A.6 of Appendix A reveals that, based on the present state of materials science and process technology for magnetic materials, the dimensional “break-even” point between capacitive and magnetic MEMS devices is near one millimeter, that is, ~103 microns. This result might seem to suggest that magnetic devices face insurmountable challenges in overtaking capacitive devices in the size range of ~100 microns; however, scaling analyses are prone to the biases built into the assumptions used to formulate them, and as a result, may misrepresent the situation.