Dynamic Mechanical Simulation of Miniature Silicon Membrane during Air Blast for Pressure Measurement †

: The development of new ultra-fast sensors for pressure air blast monitoring requires taking into account the very short rise time of pressure occurring during explosion. Simulations show here that the dynamic mechanical behavior of membrane-based sensors depends significantly on this rise time when the fundamental mechanical resonant frequency of the membrane is higher than 10 MHz


Introduction
The real time and dynamic measurements of pressure during air blasts is very challenging due to the abrupt variation of pressure from the atmospheric pressure to the so-called overpressure peak Pmax (between few bars and several ten of bars) with very short rise time tm (< 100ns) (Figure 1).For accurate measurement of Pmax, sensors with high fundamental mechanical resonant frequency Fo are then required [1].When simulating a pressure sensor for which Fo << 1/tm, the rise-time is assumed to be close to zero.However, we show here that this assumption is no more valid when performing the mechanical simulation of a sensor for which Fo > 10 MHz.

Sensor description and simulation conditions
As shown in Figure 2, the proposed sensor is based on a miniature rectangular silicon membrane (5µm*30µm*90µm) with four piezoresistive gauges located at its center [1].Simulations were performed using Abaqus software [2] for real membrane clamping conditions.Considering the gauges dimensions, the stresses in the gauge areas are close to ones calculated at the membrane center (error lower than 7% on the sensor response).Dynamic behavior of the sensor is then modelled by the differential stress ∆σ given by equation 1 at the center of the membrane, and normalized by the static pressure : where   (resp.,   ) is the stress applied to the gauge parallel (resp., to the gauge perpendicular) to the current in the gauge.The fundamental mechanical resonant frequency Fo, obtained from harmonic module of Abaqus software, is of 32.7 MHz and consequently 1/Fo is close to 30ns.Due to the short reaction time tr of the sensor (< few µs, see Figure 5), we assume here that the pressure profile (shown on Figure 1) can be modelled by the Heaviside step function (the decrease of pressure is lower than 2% after 1µs).
Moreover, assuming that the acoustic damping is predominant [3], the quality factor Q of the membrane inversely related to the equivalent pressure Pe applied on both sides of the membrane as shown in equation 2.

𝑄𝑄 ≅ 95 P 𝑒𝑒 (𝑏𝑏𝑏𝑏𝑏𝑏)
(2) Abaqus simulations indicates that the pressure Pe is the average pressure applied between the two membrane sides.Table 1 reports the Q factor for typical applied pressure on one side of the membrane while the other side is in vacuum.We observe that the Q factor decreases rapidly and is lower than 20 for the applied pressure greater than 10 bars.For a shock wavefront normally incident upon the membrane surface, the dynamic mechanical response to a linear variation of pressure was obtained from the Second Order Transfer Function (SOTF) where the rise time differs from zero (Equation 3).This model is consistent with Abaqus simulations results, as it can be noticed from Figure 3.We can observe that the difference is lower than 4% after the rise time.Consequently, the effects on sensor reaction time can be neglected.

Results
Figure 4 displays an example of the dynamic mechanical response of the membrane for rise time tm up to 100ns and for a Q factor of 19.The response is normalized by the static value.The reaction time tr of the sensor (±5% of the static response) is extracted from these figures for Q factor between 5 and 100 (Figure 5).The following observations can be made: i. for tm < 1/ Fo, we retrieve the classical damped oscillation for which tr is mainly driven by the Q factor (tr ≅ Q / Fo); ii. as tm increases, the reaction time tr decreases and, due to the decreasing of the fundamental mode amplitude (Figure 6), we obtain tr ≅ tm when tm = 1/ Fo; iii.when tm = n/ Fo where n is a natural number, no oscillation occurs and tr is close to tm; iv.when tm > 1/ Fo, small oscillations appear during the rise time.
The reduction of the reaction time related to the increase of the rise time is given in Figure 7 (reaction time for tm=0 is taken as reference).We can observe that for pressure rise time tm greater than 20ns, the sensor reaction time may be underestimated almost by 20% compared with the zero pressure risetime (tm=0) assumption.This error may reach 100% when tm is close to n/ Fo.

Conclusions
The dynamic mechanical behavior of membrane-based piezoresitive sensors was modelled by a second order transfer function with non-zero pressure rise time.For miniaturized silicon membrane (5µm x 30µm x 90µm) the fundamental resonant frequency Fo is of 33 MHz.The results show that pressure rise time tm larger than 1/(3 Fo) plays a crucial role for the accurate estimation of the sensor reaction time.Consequently, when designing ultra-fast sensors for pressure air blast monitoring, the increase of frequency Fo for a given rise time is expected to provide a significant reduction of the sensor reaction time.

Figure 1 .
Figure 1.Typical dynamic pressure variation during an air blast experiment : (a) Illustration of the different phases and (b) example of pressure measurement at 1m from 1kgTNT by using a Tourmaline piezoelectric sensor.

Figure 2 .
Figure 2. a) Cross sectional view of the sensor; (b) Wheatstone bridge on the rectangular membrane.

Figure 3 .
Figure 3.Comparison of SOTF model and Abaqus simulation results for the dynamic response

Table 1 .
Q factor versus absolute pressure from equation 2.