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What if your phone could fit to your ear when you make a call? Could turn up at the edges to hide what you’re looking at from others? Could mold to fit your hand depending on whether you’re texting, playing a game or just need something solid to squeeze?

Just days after LG's mobile division announced plans to release a phone with a flexible OLED screen by the end of 2013, researchers at the University of Bristol in the U.K. have published a paper outlining six different prototypes for "self-actuated flexible mobile devices."

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Dubbed Morphees, these flexible phones would be able to change their shape to best suit the task in which the user is engaging. If you were typing in a password, for example, most of the prototypes would automatically curl up at the edges to hide the screen from prying eyes.

That doesn’t mean you could grab these phones and twist, though — Morphees work by automatically re-forming themselves into one of several learned positions depending on what the user is asking of it.

One of these Morphees is made entirely of wood.  Called Morphee-couture, the device is made up of interlocking wooden pieces that are moved via wires that connect them. These wires are made of a material called shape memory alloys and can be "taught" to move themselves into remembered positions, thus operating as the joints of the Morphee-couture. This serves as the smartphone body; the screen is created via a projector and camera, mounted on the user’s shoulder, that can track the position of the user's finger relative to the device surface.

Another prototype, called Morphee-forged, is a little more like what you might imagine a flexible smartphone to be. The casing, circuits and E Ink screen are all made of flexible materials, and built around a net of shape memory alloy wires that bend in various remembered positions.

Aside from the prototypes, the team has also developed a measurement for flexible devices that they’ve dubbed "shape resolution." Just as screen resolution refers to the sharpness of an image, shape resolution refers to the degree of detail to which a device can form a relevant, ergonomic shape.

In their paper "Toward High 'Shape Resolution' in Self-Actuated Flexible Mobile Devices," the team outlines 10 criteria for measuring shape resolution, including the device’s curvature, stretchability and the speed at which it can change its shape.

There are still plenty of problems to overcome before we start seeing Morphees in stores, though. For one, the added moving parts mean a significant additional drain on the phone's battery life.  The constant bending also takes a physical toll on the devices, causing them to wear out and break relatively quickly.

Still, University of Bristol postdoctorate researcher Anne Rodaut said Morphees are "just a step behind" the flexible screens expected to become commercially available in the next one to two years.

"The reason behind this research is much more than just … 'because it’s cool,'" Rodaut told TechNewsDaily. "When I see my phone with all the possible functions that it can achieve but with the same static rectangular shape, I feel that we are missing something," she added. "I think shapes should fit functionalities to help us better interact and manipulate our devices."