PerspectiveMaterials Science

A figure of merit for flexibility

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Science  08 Nov 2019:
Vol. 366, Issue 6466, pp. 690-691
DOI: 10.1126/science.aaz5704

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Flexible indium gallium zinc oxide thin-film transistors could power flat-panel displays.


Flexible materials are widely used in health care, robotics, and other industries, but flexible electronic devices require that normally brittle electronic materials become flexible. Stiff materials can become flexible if they are sufficiently thin. Future applications, such as energy systems for the internet-of-things, will likely require new materials where trade-offs in performance and flexibility must be weighed with a metric of flexibility. The yield strain ϵy—that is, how much a material can stretch elastically (still recover its shape) before deforming plastically (stretching it out of shape)—for a given thickness of a material can serve as a figure of merit (FOM) for flexibility (fFoM).

Flexible displays (1), touch screens, and flexible solar cells (2) require electrical conductors that are both transparent and flexible and have spurred the development of transparent conducting oxides (3), conducting organic compounds and polymers (4), and exotic carbon-based or metallic nanowires and films (5, 6). Possible future applications for flexible electronics (7), sensors (8), supercapacitors (9), batteries (10), thermoelectrics (11), and other energy-harvesting concepts have even further broadened the interest in flexible materials. Qualitative demonstrations of flexibility, such as pictures of devices being bent or worn, do not allow for quantitative comparisons of different materials. The mechanical stresses of bending and its impact on materials are well studied, but a simple guideline for experts in other fields is lacking.

Brittle materials such as glass, silicon, and oxides can be flexible if made thin enough. The minimum bending radius rb before plastic deformation characterizes the flexibility of materials with thickness h.


The primary requirements for flexibility can be formulated into a simple FOM to facilitate comparisons and reporting. Flexible materials can be stretched reversibly—they undergo elastic strain—as opposed to a rigid material that would break or otherwise fail if repeatedly bent. The flexible materials must be strained in their elastic rather than plastic regimes (where the deformation persists) because of the repeated deformations expected for most applications. The material property that characterizes the elastic limit is the yield strength, or maximum stress (force per area) that a material can withstand before it breaks or deforms permanently. The yield strength σy is routinely measured by a tensile test, and its values are tabulated for many materials. During flexing, a material experiences compression and shear as well as tensile forces, but σy is considered a suitable descriptor for these other failure modes as well.

Flexibility is most simply demonstrated by bending a material along a radius of curvature without breakage. This working definition of flexibility can be easily quantified and related to materials parameters. The flexibility f is defined as 1/rb, where rb is the minimum bending radius of curvature (see the first figure), and thus the material is more flexible with a smaller rb. A material (sheet or wire) with a thickness h bent about a radius rb experiences the greatest stress as tensile stress on the outer surface and compressive stress on the inner surface. The maximum tensile (or compressive) strain, ϵ, which is the relative amount of elongation Δl/l on the outer (or inner) surface, is readily calculated from the geometry after recognizing that the middle (neutral layer) of the material is unstrained:

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The maximum bending strain before failure ϵb occurs when the tensile stress reaches the yield stress σy. Strain is related to stress (ϵ = σ/E) through the elastic (Young's) modulus E. Thus, the flexibility of a free-standing film or wire, defined as the inverse of the minimum bending radius before failure (1/rb), is related to the thickness and the material properties through the ratio σy /E:

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Thus, f depends on both the material properties through σy/E and the test geometry through h.

Several examples of normally brittle materials, such as glass, that can be flexible when they are made thin are shown in the first figure. Although glass is normally rigid and shatters easily if bent, thin glass fibers are actually quite flexible, and optical glass fibers for communications (which have a thickness of ∼250 µm) have become hugely successful, produced in massive quantity and, like metal wires, transported on spools. Silicon microelectronics, being made of crystalline silicon, are rigid and inflexible, but when they are thin enough (∼0.1 µm), they can be made flexible for numerous wearable and stretchable electronics (12). Another example is indium gallium zinc oxide (13), which can be made into flexible thin-film transistors. A drastic change in σy or E is not needed to make these devices; the challenge is simply to make them thin. Ultrathin two-dimensional materials such as graphene (14) are inherently flexible. Thus, simply demonstrating flexibility without reporting or recognizing the importance of thickness is of limited scientific value.

The intensive or material property that enters into f is the yield strain ϵy = σy/E, which makes it the materials fFoM. Comparing two materials of the same shape (thickness), the material with a higher σy/E will be more flexible. A higher fFoM can be achieved in a compliant material like a rubber with low E if it is not too weak (low σy). Alternatively, a strong material (high σy) can be flexible if it is not too stiff (high E). The trade-off between soft-stiff and weak-strong is precisely given by σy/E:

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A materials property (Ashby) chart (see the second figure) can be used to compare E and σy of various materials, including ceramics, polymers, metals, and carbon-based materials. The dashed lines of constant σy/E are materials with the same fFoM. Metals, although malleable (allow plastic deformation), do not necessarily have greater elastic flexibility than brittle ceramics.

Flexibility figure of merit for materials

Flexible materials need to be compliant (small elastic modulus) and strong (high yield strength), as opposed to stiff and weak. Dashed lines of constant-flexibility figure of merit σy/E divide materials into classes such as polymers and ceramics. Data sources are available in the supplementary materials.


For applications that use a different geometry, there may be a different geometric consideration for flexibility, but fFoM of the material will be the same. For example, the strain experienced by a film constrained on a substrate is determined more by the substrate than the film thickness when the substrate flexes. However, for films of equal thickness on the same substrate, σy/E will still determine which film can flex more. Thus, fFoM provides a metric for comparing and improving flexibilities of functional materials and setting up criteria for materials selection.

References and Notes

Acknowledgments: We acknowledge generous support from NASA-JPL Science Mission Directorate's RPS-TE Technology Development program.
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