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.

A swinging theory

DGDGA∧D, California Dreamin', Chris Proctor

About once every four years, the sea-surface temperature in the Eastern Equatorial Pacific is a few degrees higher than normal (Philander, 1990). Near the South American coast, this warming of the ocean water is usually at its maximum around Christmas. Long ago, Peruvian fishermen called it El Niño, the Spanish phrase for the Christ Child.

Phenomena

During the past several decades, El Niño has been observed in unprecedented detail thanks to the implementation of the TAO/TRITON array and the launch of satellite-borne instruments (McPhaden et al., 1998). The relevant quantities to characterise the state in the equatorial ocean and atmosphere are sea level pressure, sea-surface temperature (SST), sea level height, surface wind and ocean subsurface temperature.

The annual mean state of the equatorial Pacific sea-surface temperature is characterised by the zonal contrast between the western Pacific “warm pool” and the “cold tongue” in the eastern Pacific. The mean temperature in the eastern Pacific is approximately 23°C, with seasonal excursions of about 3°C. What makes El Niño unique among other interesting phenomena of natural climate variability is that it has both a well-defined spatial pattern and a relatively well-defined time scale. The pattern of the sea-surface temperature anomaly for December 1997 is plotted in Fig. 8.1 and shows a large area where the SST is larger than average.

Dynamical systems theory is an extremely powerful framework for understanding the behavior of complex systems. Its concepts apply to many scientific fields, and hence its language provides a multidisciplinary and unifying communication tool. The theory provides a systematic approach for assessing the sensitivity of a mathematical model of a particular phenomenon to changes in parameters and initial conditions. As such, it finds application in stability problems, transition behavior and predictability studies. In addition, techniques and concepts from dynamical systems theory have led to the development of a diverse set of nonlinear methods of time series analysis.

For many phenomena, existing models cannot resolve all relevant spatial and temporal scales, and hence small-scale features are often represented as ‘noise’. As a result of the increase in computational power, solutions of the resulting stochastic partial differential equations are now within reach. Although stochastic dynamical systems are difficult to deal with, in recent years, the theory of stochastic dynamical systems has matured and is ready to be applied to many scientific areas.

This book developed from a course on climate dynamics that I taught at Colorado State University in 2005 and a course on stochastic climate models that I taught at Utrecht University in 2008. My main motivation in writing this book was to provide both an introduction into stochastic dynamical systems theory and to show the application of these methods to problems in climate dynamics.