Electronic systems that offer elastic mechanical responses to highstrain deformations are of growing interest because of their ability to enable new biomedical devices and other applications whose requirements are impossible to satisfy with conventional wafer-based technologies or even with those that offer simple bendability. This article introduces materials and mechanical design strategies for classes of electronic circuits that offer extremely high stretchability, enabling them to accommodate even demanding configurations such as corkscrew twists with tight pitch (e.g., 90°in Ϸ1 cm) and linear stretching to ''rubber-band'' levels of strain (e.g., up to Ϸ140%). The use of single crystalline silicon nanomaterials for the semiconductor provides performance in stretchable complementary metal-oxide-semiconductor (CMOS) integrated circuits approaching that of conventional devices with comparable feature sizes formed on silicon wafers. Comprehensive theoretical studies of the mechanics reveal the way in which the structural designs enable these extreme mechanical properties without fracturing the intrinsically brittle active materials or even inducing significant changes in their electrical properties. The results, as demonstrated through electrical measurements of arrays of transistors, CMOS inverters, ring oscillators, and differential amplifiers, suggest a valuable route to high-performance stretchable electronics.flexible electronics ͉ stretchable electronics ͉ semiconductor nanomaterials ͉ plastic electronics ͉ buckling mechanics I ncreasingly important classes of application exist for electronic systems that cannot be formed in the usual way, on semiconductor wafers. The most prominent example is in large-area electronics (e.g., back planes for liquid crystal displays), where overall system size rather than operating speed or integration density, is the most important metric. Similar systems that use flexible substrates are presently the subject of widespread research and commercialization efforts because of advantages that they offer in durability, weight, and ease of transport/use (1, 2). Stretchable electronics represents a fundamentally different and even more challenging technology, of interest for its unique ability to flex and conform to complex curvilinear surfaces such as those of the human body. Several promising approaches exist, ranging from the use of stretchable interconnects between rigid amorphous silicon devices (3) to ''wavy'' layouts in singlecrystalline silicon CMOS circuits (4), both on elastomeric substrates, to net-shaped structures in organic electronics on plastic sheets (5). None offers, however, the combination of electrical performance (high electron and hole mobility), scalability (with relatively modest modifications to conventional microelectronic technologies), integrated circuit applicability in complementary designs and mechanical properties required of some of the most demanding, and most interesting, systems. Here, we introduce design concepts for stretchable electronics th...

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The flow theory of mechanism-based strain gradient (MSG) plasticity is established in this paper following the same multiscale, hierarchical framework for the deformation theory of MSG plasticity in order to connect with the Taylor model in dislocation mechanics. We have used the flow theory of MSG plasticity to study micro-indentation hardness experiments. The difference between deformation and flow theories is vanishingly small, and both agree well with experimental hardness data. We have also used the flow theory of MSG plasticity to investigate stress fields around a stationary mode-I crack tip as well as around a steady state, quasi-statically growing crack tip. At a distance to crack tip much larger than dislocation spacings such that continuum plasticity still applies, the stress level around a stationary crack tip in MSG plasticity is significantly higher than that in classical plasticity. The same conclusion is also established for a steady state, quasi-statically growing crack tip, though only the flow theory can be used because of unloading during crack propagation. This significant stress increase due to strain gradient effect provides a means to explain the experimentally observed cleavage fracture in ductile materials

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