Quantifying the Scatter: Statistical Analysis and Stochastic Modelling of Microplasticity

When solid materials are loaded above a critical level, they may change their shape permanently: they undergo plastic deformation. Consider, for example, a cylinder which we compress by pushing from top to bottom. If the load is small, the cylinder first deforms elastically (it reverts to its original shape after the load is removed). Above a certain load, some permanent deformation remains. Now if we use a macroscopic cylinder, say, several centimetres in size, then the stress (the force per unit area) needed to obtain a given relative deformation will not depend on the size of the cylinder. It will increase gradually with increasing deformation, and this 'hardening behavior' will be identical for cylinders made of the same material and deformed under the same conditions. If the stress is everywhere the same in the cylinder, also the deformation will be homogeneous - the cylinder will get shorter and thicker but will retain its cylindrical shape.

But when the deforming body becomes very small - of the order of a few micrometers in diameter - then we observe quite different behavior: (1) The stress required to deform samples of material increases as the samples become smaller. (2) Even if the stress is increased slowly and steadily, the deformation does not increase gradually but in large jumps. These jumps occur randomly, and lead to large deformations in small parts of the sample. As a consequence, in our cylinder example the samples assume irregular accordeon-like shapes. If we bend very thin wires, they may not deform into smoothly curved but into random shapes resembling mis-shapen paperclips. (3) Even if the material properties are the same (for instance, if all our cylinders have been machined out of the same block) the stresses required to deform samples may scatter hugely. In two apparently identical micrometer sized samples, the stresses required to initiate or sustain plastic deformation may easily differ by a factor of two. Obviously this poses serious problems if we want to avoid or control irreversible deformation in very small components.

The first of these aspects have been studied in some detail, and some work has also been done on the second one. However, there is no systematic study which quantifies the scatter in deformation behaviour between different small samples and provides tools for assessing the risk of unwanted deformation behaviour. We have teamed up with German researchers who conduct micro-deformation experiments and with others who simulate such deformation processes by tracing the motion of material defects which produces the irreversible deformation. Together we will conduct and analyze large series of experiments and simulations to characterize the scatter in deformation behaviour and to understand how it depends on sample size, material preparation, and method of deformation. We will then use this database to develop simulation tools that allow engineers to assess the risk of undesirable outcomes.

Why is it important? Imagine you want to bend sheets of metal with a size of centimetres to meters, say for making them into cylinders for producing cans, or for making car doors. It is comparatively easy to get the desired shapes. However, if you try to do the same on a very small scale, the result might look quite different! Micro-scale scatter of deformation properties may affect our ability to form materials into very small shapes and to produce very small parts for microtechnologies. A striking example are the very thin wires that provide electrical connections for microchips. If the shape of these wires scatters too much, two of them may get into contact and produce a short-circuit that makes the device useless. As miniaturization of components and devices proceeds, we need to gain the knowledge and expertise needed to handle forming processes on the microscale. Our research wants to make a contribution to this purpose.

The proposed research aims at strengthening the knowledge base for advanced manufacturing industries in areas where plastic deformation properties are relevant to the manufacturing and performance of micro-scale components. This notably concerns the microelectronics and micro-electro-mechanical systems industries. Ultimately, the proposed research will provide these industries with knowledge that helps to improve the reliability of components and systems, to discover and mitigate problems already at the design stage, to optimize production processes, and thus to provide a competitive advantage to UK and European industries.
Specifically, plasticity can be of relevance (i) in manufacturing processes where desired microscopic shapes are to be produced by plastic deformation. A typical example is wire bonding, i.e. the use of metal wires for making interconnections between integrated circuits and circuit boards, and the production of bond wires. In these technologies, the shape of wires and interconnections needs to be controlled with sufficient accuracy to avoid undesired contacts. While control of the grain microstructure and alloy composition of bond wires has in the past been sufficient to ensure reproducible deformation, it has been predicted that intrinsic fluctuations of deformation behavior are likely to pose problems as soon as wire diameters - at present typically between 10 and 100 micrometers - reach the micrometer range. (ii) An even more important issue may be to avoid undesired plastic deformation. Most micro-mechanical and micro-electro-mechanical systems are fabricated using top-down approaches which do not rely on deformation processes to achieve desired shapes and configurations. However, the performance of such systems can be affected by stresses of thermal or mechanical origin, which may induce undesired deformation during manufacturing or service. In these cases, considerations of plastic deformation properties - and their scatter - are essential to ensure the reliability of such systems.
In a wider perspective, the proposed research will resolve fundamental issues regarding the construction of plasticity models that account for spatial and deformation-induced fluctuations of the microstructure, and that can be fit into a standard finite element framework. In particular, it will address the basic question how materials property fluctuations can be accounted for in meshes with variable resolution - in the language of physics, how fluctuation properties transform across scales (renormalization). This fundamental aspect of the work will, beyond the area of micro-manufacturing, be of interest in all applications where high-reliability predictions of plastic deformation behavior are needed.

How and on which time scale can it be achieved? The proposed research, in an area where little prior work has been done, will not immediately provide a standard software tool that could be used to optimize, say, a ball bonding cycle. Instead, it covers those first steps that, owing to their non-applied nature, cannot be easily made by researchers outside academia. Within three years we will conduct a detailed phenomenological study of statistical scatter in micro- deformation processes, use this for developing stochastic algorithms for deformation process simulation and resolve the associated conceptual questions, and validate the algorithms on experimental data including deformation of commercial micro-wires. The crucial next steps which constitute the logical follow-up of the project would be: to team up with partners from micro-manufacturing and software industries to (i) apply the developed method to benchmark problems taken from manufacturing applications, and (ii) to develop commercial software implementations that tie up with standard tools used in engineering design. This can in our estimate been accomplished within 2-3 years from the end of the project and will be detailed in the Pathways to Impact st