Bree Interaction Diagrams have long been one of the major visual design guides for employing and evaluating shakedown in engineering applications. These diagrams provide representations of the realms in which elastoplastic behaviors, including shakedown, are found for a material and structure under variable loads. The creation of these diagrams often relies upon some combination of upper or lower bound shakedown theorems and numerical shakedown limit determination techniques. Part of the utility of these diagrams is that, for a given structure and loading conditions, inspecting them is sufficient to determine whether shakedown will occur or not. The diagrams cannot however, give the designer insight into how the conditions for shakedown are met. This chapter presents some graphical interpretations of one of the common methods for shakedown determination: the use of Melan’s Lower Bound Theorem. The intent is to provide additional insight for designers regarding how shakedown conditions are satisfied. In this way, additional directions for modifying designs to recover shakedown behavior may also be identified. Revisiting this well-established theorem from a graphical and pedagogical approach, also provides a foundation for interdisciplinary innovation. The particular focus is on simple examples that highlight ways in which Melan’s theorem may be applied to shakedown design problems.
Designing materials with exceptional combinations of properties at low weight is a continuous goal in many industries. Cellular (i.e., porous) materials with one or more phases topologically organized in a precisely designed configuration (often denoted as architected materials, or metamaterials) are excellent candidates to reach combinations of properties that are unattainable by existing monolithic materials. Additive manufacturing techniques are perfectly suited to implement the topological complexity that is often required for optimal performance. As beneficial size effects often arise in mechanical and functional properties as dimensions are shrunk to the nanoscale, 2PP becomes an ideal platform to investigate and ultimately fabricate topologically micro-architected and nano-architected materials with truly unique properties. The chapter reviews some notable features of architected materials, surveys commonly available manufacturing approaches, and presents challenges and opportunities for 2PP fabrication.
Plasticity theory is the mathematical formalism that describes the constitutive model of a material undergoing permanent deformation upon loading. For polycrystalline metals at low temperature and strain rate, the J2 theory is the simplest adequate model. Classic plasticity theory does not include any explicit length scale, and as a result, the constitutive behavior is independent of the sample dimensions. As the characteristic length of a sample is reduced to the micro (and nano) scale, careful experimental observations clearly reveal the presence of a size effect that is not accounted for by the classical theory. Strain gradient plasticity is a formalism devised to extend plasticity theory to these smaller scales. For most metals, strain gradient plasticity is intended to apply to objects in the range from roughly 100 nm to 100 μm. Above 100 μm, the theory converges with the classical theory and below 100 nm surface and grain boundary effects not accounted for in the theory begin to dominate the behavior. By assuming that the plastic work (or in some theories, the yield strength) depends not only on strain but also on strain gradients (a hypothesis physically grounded in dislocation theory and, in particular, in the notion of geometrically necessary dislocations (GND) associated with incompatibility due to strain gradients), an intrinsic length scale is naturally introduced, allowing the theory to capture size effects. According to most theories, the intrinsic length scale is of the order of the distance between dislocation-clipping obstacles or cellular dislocation structures (typically, submicron to tens of microns). This continuum theory is appropriate for length scales that remain large relative to the distance between dislocations. As the sample length scale is dropped below this level, dislocations must be modeled individually, and discrete dislocations simulations (DSS) are the preferred approach. At even smaller scales, molecular dynamics (MD) becomes the applicable tool. This article presents a brief overview of one of the simplest continuum strain gradient plasticity theories that reduces to the classical J2 theory when the scale of the deformation becomes large compared to the material length scale. This simple theory captures the essence of the experimental trends observed to date regarding size effects in submicron to micron scale plasticity.