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Strain gauges are widely applied to measure mechanical deformation of structures and specimens. While metallic foil gauges usually have a gauge factor slightly over 2, single crystalline silicon demonstrates intrinsic gauge factors as high as 200. Although silicon is an intrinsically stiff and brittle material, flexible and even stretchable strain gauges have been achieved by integrating thin silicon strips on soft and deformable polymer substrates. To achieve a fundamental understanding of the large variance in gauge factor and stretchability of reported flexible/stretchable silicon-on-polymer strain gauges, finite element and analytically models are established to reveal the effects of the length of the silicon strip, and the thickness and modulus of the polymer substrate. Analytical results for two limiting cases,

Strain gauges are widely used across all engineering fields to measure mechanical deformation of a solid object. The most common type of strain gauges consists of a patterned metal foil on a stiff plastic backing sheet glued to the solid object. Deformation in the object leads to deformation in the foil, thereby causing its electrical resistance to change. The fractional change in resistance, Δ_{0}, is related to the mechanical strain

The

Strain measurement on curvilinear surfaces and/or soft, highly deformable objects calls for flexible or even stretchable strain gauges. Examples of applications include structural health monitoring on curvilinear surfaces [

Recent work shows that ultrathin sheets of single crystalline silicon,

In this paper, we describe finite element and analytical modeling of thin silicon strips bonded to polymer substrates of wide ranges of Young's modulus and thickness. Both gauge factor and stretchability can be predicted and effects of material and geometric variables are revealed. The tradeoff between

Silicon-based stretchable strain gauges are often composed of arrays of strips. The modeling work in this paper will just focus on a unit cell cut out of the periodic array. A schematic of a unit cell and its corresponding 2D plane strain model are depicted in _{app}, is applied to the substrate, the resistance of the silicon strip will change by Δ

Although strain distribution in silicon might not be uniform, it has been proven [_{Si} represents the intrinsic gauge factor of silicon. Depending on the crystal orientation as well as the doping type and concentration [_{Si} of p-type (110) silicon can reach as high as 200 [_{avg}, can be calculated by just averaging the strain of the neutral axis over the total length of silicon. This is because the strain of the neutral axis is the average strain across the thickness due to the linear strain distribution in the thickness direction. Using _{n}(_{avg} is defined as:

Plugging

The maximum strain in silicon, _{max}, will be related to the stretchability of the system. We define _{max} = _{cr}, in which _{max} represents the maximum tensile strain in silicon and _{cr} denotes the intrinsic critical strain-to-rupture of silicon that is to be measured experimentally. The failure criterion can be rewritten in the normalized form

When the polymer substrate is very thin or very soft, there will be slight concave bending in the silicon strip when the substrate is subject to tensile strain (_{b}(

As we have related the device performance indices, _{avg} and _{max} in silicon respectively, in the following we will calculate _{avg} and _{max} to ultimately determine ^{2}) represents the plane strain modulus with _{Si}/_{s}, _{avg}/_{app} and _{max}/_{app}, and then use analytical methods to derive the functional forms of

Over one hundred finite element models are built using commercial software package ABAQUS 6.11 to reveal the effect of the three variables in _{app} is fixed to be 10% for all models. Nonlinear geometry analysis has to be enabled in ABAQUS because large deformation can be found in the portion of the polymer substrate that is not covered by silicon. Silicon thickness

Representative contour plots of longitudinal strain in the substrate and in silicon are given in _{s} = 60 kPa and ^{−7} to 10^{−4}, depending on substrate thickness. It is because silicon is six orders of magnitude stiffer than Ecoflex substrate (_{s} = 60 kPa) so that it is highly resistant to elongation, whereas the portion of the substrate not covered by silicon has to accommodate the applied strain by tensile strains up to 25%. Due to huge elastic mismatch between silicon and Ecoflex, horizontally applied tensile strain on a thin substrate will cause bending deformation in the silicon strip, especially near the end (_{n}, is always smaller than the strain along the bottom of silicon, _{b}. Although _{n} is monotonic with _{b} might not be so due to localized bending near the end of the strip. When the substrate is thick enough (_{n} and _{b}. _{s} = 60 kPa, as the substrate becomes stiffer, the bending will be less significant.

_{n} and _{b} along _{s}, _{n} and _{b}. In all the plots, _{Si}_{n}(_{n} gradually builds up towards the center of silicon because interfacial shear force also builds up as we move away from the edge. The rate of strain growth decreases as _{b} shares some similarity with _{n}, as shown in _{b} is not always monotonic with _{s} = 60 kPa, edge bending can cause large variations in _{b}, as shown in

To study one effect at a time, we first vary _{s} with _{n} and _{b} for different _{s} (60 kPa, 2 MPa, and 2.5 GPa) are plotted in _{n} and _{b} increase as substrate modulus increases. It is simply because the stiffer substrate can apply higher shear stress to silicon, meaning that silicon can be stretched more by stiffer substrate. When substrate is very stiff, _{s} = 2.5 GPa, there is little difference between _{n} and _{b} curves because the assembly will stay almost flat. But when _{s} = 60 kPa, _{b} are larger than _{n} due to slight global concave bending.

To study the effect of _{s} = 60 kPa. The distribution of _{n} and _{b} for different _{n} and _{b} increase as _{n} monotonically decreases with increasing _{b} curves in

To study the effect of _{s} = 60 kPa and vary _{n} is higher when _{b} is more complicated than _{n} because when substrate is soft (e.g., _{s} = 60 kPa in this case), bending strain could become very significant. Depending on substrate thickness, bending strain distribution also varies a lot. When _{n}(_{b}(_{max}, in silicon, we need to find the maximum positive values of each _{b}(

With the insights from strain distribution in _{s}, on the average strain (_{avg}) and the maximum strain (_{max}) in silicon. Average strains are calculated from averaging the values of _{n} (_{b} (_{avg} as a function of _{s} and varying _{s}. All plots in _{avg} increases with increased film size (_{s}_{avg}. Due to experimental limitations on _{s} would be the most effective way to tune _{avg}, by orders of magnitude. According to _{avg} and _{avg} will always reach a plateau, _{avg} becomes independent of _{avg} in Section 4.

_{max} as a function of _{s} in a format similar to _{s} and varying _{s}.

The difference between _{max} and _{avg} is that _{max} has extra bending contribution. Therefore, similar to _{avg}, _{max} always increases with increased _{s}, and with increased H in most cases. However, when the substrate is very compliant and when _{max} could be higher in thinner substrate due to localized bending effect we discussed for _{s} increases to 2.5 GPa, as shown in _{max} and _{avg} become undistinguishable, _{max} will be derived in Section 4 as well.

In this section, we are going to develop analytical models for two limiting cases:

When _{avg} and _{max} only depend on _{s}/_{Si} and

To determine the _{s}. Applying Hooke's law in silicon, _{Si}_{s}, provided _{s} = 2.5 GPa and very small _{max}/_{app} and 0.279 for _{max}/_{app}. We then plot _{s} and _{avg}/_{app} in the upper frame and _{max}/_{app} in the lower frame. It is evident that _{s} when _{avg}/_{app} and _{max}/_{app}. When _{avg}/_{app} over a wide range of _{s} but is only able to capture _{max}/_{app} when _{s} = 2.5 GPa, due to the abnormal _{max} induced by large local bending.

In conclusion, when _{s} and _{s} can be easily changed by orders of magnitude as shown in

When _{Si}/_{s} and _{Si}/_{s},

Boundary conditions are decomposed to _{app}:

Following this practice, the simplest form of average strain in silicon can be written as:

Combining

Through curve fitting of FEM results, ^{−5} and _{avg}/_{app} and _{max}/_{app}, over wide ranges of _{max}/_{app} very well due to localized bending effects.

_{app} approaches 1. When the substrate becomes extremely soft or extremely thin, _{app} can be reduced by orders of magnitude, and should eventually die out.

With the two semi-analytical solutions given by

After finding out _{avg}/_{app} and _{max}/_{app}, the final step is to use _{Si} = 100 [_{cr} = 1%. Some representative results are shown in _{s}, as shown in _{s}. When the substrate is really soft and silicon is short, as the case for _{max} which is inversely proportional to
_{s} = 2.5 GPa as shown in _{s} increases, the larger strain in silicon suggests the higher _{s}, as shown in _{s} combinations. Similar effects are observed for _{s}, as shown in _{s} = 60 kPa, the critical applied strain to rupture silicon
_{s} should be 2 MPa and beyond, which still corresponds to stretchability beyond 100%.

The goal of plots in _{s} to make a strain gauge with a stretchability of 30% and maximized _{s} = 1.1 × 10^{7} Pa, which in turn predicts the

We have three final remarks on applying the analysis in this paper to real strain gauge samples [

The coupling between tension and bending is a generic problem associated with mechanical strain gauges. This study has also shed light on this issue. When _{s} ≪ _{Si}, the neutral axis of the bilayer almost overlaps with the neutral axis of silicon. In this case bending will induce minimum _{avg} in silicon and hence bending effect can be neglected. When _{s} is close to _{Si}, the neutral axis of the bilayer lies far away from the neutral axis of silicon. In this case the bending will induce large _{avg} in silicon. To minimize the bending induced signal when the substrate is a stiff polymer, one can add an identical polymer layer on top of silicon, forming a sandwiched structure to locate silicon along the neutral axis of the sandwich structure. This structure can be readily analyzed by replacing the old

Due to the mismatch in coefficients of thermal expansion (CTE) between silicon and polymer, temperature variation will induce stress and hence resistance change in silicon. The best known method to eliminate temperature effect is to use Wheatstone bridge instead of single resistors. The mechanical analysis on a single resistor offered in this paper is readily applicable to each linear arm of the Wheatstone bridge.

In conclusion, we performed strain analysis on polymer-bonded thin silicon strips using both FEM and analytical methods. The gauge factor and stretchability of a silicon-on-polymer strain gauge have been predicted as functions of the normalized length of silicon, and the normalized thickness and modulus of the polymer substrate. In general, we found that the longer strip, the thicker or the stiffer substrate will transfer a larger fraction of the applied strain to silicon. Silicon length and substrate thickness has only moderate effects on strain in silicon whereas varying the stiffness of the substrate could change the strain and hence the gauge factor and stretchability by orders of magnitude. A tradeoff between

This work is supported by the start-up fund provided by the Cockrell School of Engineering at the University of Texas at Austin.

The authors declare no conflict of interest.

Schematic and FEM contour plots of a thin silicon strip supported by polymer substrate subject to uniaxial tension. (_{n}(_{b}(_{s} in substrate and _{n} and _{b} in silicon when _{s} = 60 kPa and

Normalized _{n} and _{b} for different _{s} combinations. (_{n}(_{s} when _{n}(_{s} = 60 kPa are fixed. (_{n}(_{s} = 60 kPa are fixed. (_{b}(_{s} when _{b}(_{s} = 60 kPa are fixed. (_{b}(_{s} = 60 kPa are fixed.

Normalized average strains in silicon as a function of normalized silicon length for various combinations of _{s}. (_{s} = 60 kPa. (_{s} = 2 MPa. (_{s} = 2.5 GPa. (_{s}.

Normalized maximum strains in silicon as a function of normalized silicon length for various combinations of substrate modulus and thickness. (_{s} = 60 kPa. (_{s} = 2 MPa. (_{s} = 2.5 GPa. (_{s}, but not with _{s} are both small, as shown in (A).

Comparison of _{avg}/_{app} and _{max}/_{app} when _{avg}/_{app} and _{max}/_{app} for _{avg}/_{app} and _{max}/_{app} for _{avg}/_{app} and _{max}/_{app} for

Comparison of _{avg}/_{app} and _{max}/_{app} when _{avg}/_{app} and _{max}/_{app} as functions of _{Si}/_{s}. (_{avg}/_{app} and _{max}/_{app} as functions of _{s} = 2 MPa. Analytical solutions for two extreme conditions have found good agreement with FEM results.

Tradeoff between _{s} = 60 kPa are fixed. (_{s} = 2.5 GPa are fixed. (_{s} = 60 kPa are fixed. (_{s} when _{s} has the widest range of options and hence the most significant effect.

Elastic properties of materials used in FEM.

silicon | beam | (110) Silicon [ |
169 GPa | 0.27 |

Ecoflex [ |
60 kPa | 0.49 | ||

substrate | plane strain | 10:1 PDMS [ |
2 MPa | 0.49 |

Polyimide [ |
2.5 GPa | 0.34 |