Soft Switching Inverter Matlab

How soft repulsion enhances the depletion mechanism Lorenzo Rovigatti,1, 2 Nicoletta Gnan,3 Alberto Parola,4 and Emanuela Zaccarelli3, 2

Boltzmanngasse 5, A-1090 Vienna, Austria

2Dipartimento di Fisica, Sapienza Universit`a di Roma,

Piazzale A. Moro 2, 00185 Roma, Italy

3CNR-ISC, Uos Sapienza, Piazzale A. Moro 2, 00185 Roma, Italy

4Dipartimento Di Scienza E Alta Tecnologia,

Universit`a dell’Insubria, Via Valleggio 11, 22100 Como, Italy

Abstract

We investigate binary mixtures of large colloids interacting through soft potentials with small, ideal depletants. We show that softness has a dramatic effect on the resulting colloid-colloid effective potential when the depletant-to-colloid size ratio q is small, with significant consequences on the colloidal phase behaviour. We also provide an exact relation that allows us to obtain the effective pair potential for any type of colloid-depletant interactions in the case of ideal depletants, without having to rely on complicated and expensive full-mixture simulations.

We Also Show

that soft repulsion among depletants further enhances the tendency of colloids to aggregate. Our theoretical and numerical results demonstrate that — in the limit of small q — soft mixtures cannot be mapped onto hard systems and hence soft depletion is not a mere extension of the widely used Asakura-Oosawa potential.

Arxiv:1409.0903V2 [Cond-Mat.Soft] 12 Nov 2014

Colloidal particles can be considered as super-atoms , moving in a background fluid, and are usually described in terms of effective interactions . The latter are not fixed by chemistry but can be tuned almost arbitrarily by a careful design of the suspension , making soft matter systems display states and phases that have no counterpart in atomic and molecular systems . As an example, the polymeric nature of some colloidal systems allows for a fine tuning of their molecular architecture as well as their softness.

Indeed, while hard-sphere (HS) colloids such as PMMA particles have become a favourite model system to study phase transitions and dynamics , more recently soft colloids have gained increasing attention. Among these, microgel PNIPAM particles have emerged as a prototype for soft repulsive colloids . While a consensus has not been reached yet about microgel effective interactions, comparisons with experiments have shown that they can be described by soft potentials .

In addition to the possibility of changing the nature of colloidal particles themselves, a well-established way to tune colloidal interactions is to add a co-solute to the suspension, often in the form of non-adsorbing polymers or surfactants . The resulting depletion forces, controlled by the size and the concentration of the additives (also called depletants), give rise to an effective colloid-colloid attraction. Depletion interactions have been known for about sixty years since the pioneering works of Asakura and Oosawa (AO) and Vrij . In the case of a mixture of HS colloids and ideal polymers, with polymer-colloid HS interactions, the effective colloid-colloid potential can be derived analytically. This AO formulation has become the reference model system for depletion interactions and its use is widespread . For small depletant-to-colloid size ratio q the most unusual features of colloidal behaviour arise. Among these, we mention the metastability of the gas-liquid spinodal , which enhances nucleation and gelation , and the appearance of two different glasses separated by a reentrant liquid at packing fractions larger than a simple HS glass . On top of this, depletion plays an important role also in biological systems, from the folding of single biopolymers to systems where macromolecular crowding can significantly impact both structure and kinetics [33, 36–38].

In the last decade, considerable efforts have been devoted to the study of non-ideal colloid-polymer mixtures, building on the simple AO model with the aim of improving the description of polymer-polymer and polymer-colloid interactions.

Obtaining Reliable

effective potentials under these conditions requires sophisticated theoretical and numerical

2

calculations . Recent work has also been devoted to the effects induced by interactions between depletant molecules, such as the investigation of the phase behaviour of non-ideal mixtures of hard spheres with very short-range Yukawa tails and a few studies on the effect of attractive interactions on depletion forces . Moreover, when colloid- depletants interactions are not hard, as in all these systems, it has been shown that the depletion mechanism, which in the AO model is controlled solely by entropy, can become dominated by enthalpy [38, 50, 51].

Even though a qualitative difference between the AO model and depletion effects in non- hard systems has already been noted [38, 42, 46, 52], the generalisation of the AO mixture to the case of soft colloid-depletant interactions has not been thoroughly tested, except for the work of Zausch and coworkers , who have shown that for q = 0.8 the system can be assimilated to a HS mixture with an effective particle diameter. In this work we take a step forward and investigate the effects of softness in colloid-depletant interactions. Treating the depletant molecules as ideal, we show that even a small amount of softness produces a dramatic effect on the resulting depletion attraction at small q-values. By focusing on representative model systems for soft potentials, such as inverse power-law and hertzian potential, we calculate numerically and theoretically the effective interaction between soft colloids immersed in ideal depletants, in the case of colloid-depletant soft repulsion. We find that the resulting depletion attraction is strongly enhanced with respect to the AO case both in range and in depth, a feature that is generic for any kind of soft interaction. By comparing soft depletion with the corresponding AO case in terms of second virial coefficients, we show that, unlike one-component systems [53, 54], soft mixtures cannot be mapped onto hard ones. These results hold true also when interactions between depletant particles in the form of soft repulsion are considered. Our results have, thus, profound consequences on the phase behaviour of depletion-interacting soft colloids, which have not been really appreciated so far, except for few sporadic studies [42, 45].

Theory

Without loss of generality, the solvent-mediated effective pair potential can be formally expressed in terms of grand canonical averages in a pure depletant reservoir at fixed temper- ature T, volume V and activity zd . In the special case of ideal depletants (but arbitrary colloid-depletant interaction), this expression can be written in closed form as

(1)

where ρd is the reservoir depletant number density, vcd(r) is the colloid-depletant potential, β = 1/kBT and kB is the Boltzmann constant. Equation 1 reduces to the well-known AO potential between two colloids whose centres are at distance R when vcd is a pure hard-core interaction. By use of the convolution theorem in Eq. (1), Vdepl can be easily evaluated numerically by Fourier transform and added to the direct colloid-colloid interaction Vcc, to yield the total colloid-colloid potential Vtot = Vcc + Vdepl.

As representative models of soft spheres, we focus on interactions (both for colloid-colloid and for colloid-depletant) given by (i) an inverse power-law potential with exponent n, Vn(r) = ϵ (σµ/r)n, where ϵ = 1 is the energy strength in units of kBT and (ii) a hertzian potential, VHZ = λ (1 −r/σµ)5/2 Θ(σ −r), where Θ is the Heaviside step function and λ = 500kBT is the strength of the interaction (fixed by recent comparison with experiments).

Here the subscript µ refers to the three length-scales of the problem σc, σd and σcd, where

2

. A key role is played by the depletant-to-colloid size ratio, formally defined as q = σd/σc. As shown in the Appendix, for a steep power-law colloid-depletant interaction (n ≫1), the resulting effective interaction between two colloids can be approximated as

N

σcd and Qi are fourth degree polynomials in R −2σcd. This analytical result shows that an approximately exponential tail, absent in the celebrated AO expression, is generated by the softness of the direct interaction.

Soft-To-Hard Mapping

In order to compare Vdepl with the AO expression for hard colloid-depletant interactions, it is convenient to first map the colloid-colloid interaction into an effective hard-sphere potential. As usual, we define an effective colloidal hard-core diameter σc

Eff= Hσc By Imposing

the equality of second virial coefficients [53, 54]. Such an equivalence is known to faithfully reproduce the properties of the pure colloidal particle suspension at low density. We now ask whether an analogous mapping can be carried out for the colloid-depletant interaction, thereby reducing the system to the well studied AO model.

[Exp(−Βv (R)) −1]R2Dr

allows to compare different potentials V by means of a single parameter, which provides a measure of the two-body potential strength. It normally depends on temperature or, in the case of depletion interactions, on depletant density. It is now well established, thanks to the work of Noro and Frenkel , that the thermodynamic and static properties of a wide class of different potentials, including a hard-core repulsion plus a short-range attraction, are identical when different systems are compared at the same normalized second virial

(Σ) = 2Πσ3/3 And Σ The Hs Diameter. Thus,

we map the soft repulsion onto the HS system through the definition of an effective hard

2

(σeff). The use of this formula yields σeff= hσ, where h = 1.01818 for V36 and h = 0.9272 for VHZ with λ = 500kBT. In order to avoid the introduction of too many length scales, we adopt throughout the manuscript the convention that, σc, σd, σcd identify the characteristic lengths defining the soft particles. These are indeed the quantities directly accessible experimentally, e.g. for hertzian particles they correspond to the experimentally determined diameters (by means, for example, of dynamic light scattering ). When we compare to the AO case we then use the corresponding rescaled effective diameters, multiplying them by the factor h, which is potential-dependent. In the case of hard particles (and thus for the AO case) h = 1, by definition. Note that mapping Vcc onto an effective hard-core potential requires a rescaling of σc, while to map the depletant density or the AO range we need to rescale σd.

Simulations

To evaluate effective potentials we perform parallel runs of Monte Carlo (MC) simulations in the canonical ensemble of two large colloids of diameter σc in solution with small particles of size σd. We use umbrella sampling to constrain the distance between the two colloids : in each run, the two large particles can explore only a limited range ∆i of reciprocal distances, and the probability P(x, ∆i) to find the colloids at a given surface-to-surface distance x within such interval is computed. Since different runs are allowed to have a small overlap in the probed ∆i, we obtain the total P(x) by merging together all the P(x, ∆i) by means of a least-squares-based algorithm. Finally, we extract the effective potential from the relation βVdepl = −ln(P(x)) + C, where C is a constant that is set by imposing Vdepl(∞) = 0. We explore different cases in which the colloids interact between them and with the co-solute particles with different interaction potentials.

To test the validity of the calculated effective two-body potentials, we also perform MC simulations of a monodisperse system of colloids interacting with Vtot (thereby neglecting many body interactions) and compare them with the corresponding full binary mixture.

Due to the large number of depletant particles, especially at the small q-values studied here, we use brownian dynamics on GPUs to simulate the full mixture. For the one-component system we simulate Nc = 10000 colloids at a density ρσ3

Ch3 = 0.1,

interacting through Vtot = V36 + Vdepl, with Vdepl obtained from Eq. 1. For comparison, we also simulate the same system interacting with the AO potential given by

(3)

in the range hσc < r ≤2hσcd and 0 for r > 2hσcd.

Simulations At Ρdσ3

dh3 = 0.27. We recall that ρd is the depletant number density of a reservoir at fixed temperature T, volume V and activity zd.

In the full mixture case, the system is in thermal equilibrium with the depletant reservoir, and thus the two share the same activity zd. However, the presence of the colloids makes it so that the resulting depletant density of the system is not ρd but ρr

D = Αρd, Where The Factor

α = α(q, ρc) depends solely on the size ratio q and the colloid density ρc. We compute α by means of the free-volume theory , obtaining the value α = 0.93. We confirm this value by explicitly computing the excess chemical potential µex of the full mixture with the Widom

6

insertion method, since α = exp(−µex/kBT). Therefore, we simulate a system composed by Nc = 100 colloids and Nd = 250000 depletants with size ratio q = 0.1. The colloid number

Dh3 = Αρdσ3

dh3 = 0.25. FIG. 1: (a)-(c): Cartoon of the system. (a) When two colloids (in red) are far apart, depletants (in blue) do not induce any interaction. (b) In the case of hard interactions, if the depletion layers (in black) do not overlap, the resulting effective interaction is null. In the case of soft interactions there is still a depletion area (in green) inducing a non-zero effective attraction. (c) Depletion is enhanced even when the two colloids are very close. (d): Numerical data for Vdepl between two large colloids interacting via V36 both between themselves and with the depletants, as a function of the surface-to-surface colloid distance x and for various values of the size ratio q (dashed lines with points). Full lines show the AO interaction. Inset: Numerical data for q = 0.1 at two different values of n. (e): q-dependence of Vdepl for n = 36 calculated with Eq. 1 (full lines), Eq. 2 (dashed lines) and numerical data (points). Inset: Vdepl from Eq. 2 rescaled onto the exact one of Eq. 1 using the ratio between contact energies.

Effective Potentials From Soft Depletion

Fig. 1(d) reports Vdepl between two soft colloids for several values of q and fixed depletant

Density Ρdσ3

dh3 = 0.158. The two colloids interact through Vn, both between themselves and with the depletants. The latter behave as ideal among themselves. As clearly shown, particularly for small q, the potential is much more attractive than its AO counterpart.

Not only the contact energy decreases, e.g. by more than 30% in the case q = 0.1, but, most importantly, it develops a long-distance contribution which grows dramatically upon decreasing q.

Indeed, while for an AO mixture the range of the interaction is exactly σd = qσc, the corresponding soft version has as exponential tail with a range roughly twice as large in the low-q limit. The AO behaviour is recovered only at large q . Finally, decreasing n leads to an even larger discrepancy with AO (inset).

To provide a physical interpretation of these findings, we refer to the cartoons shown in Fig. 1(a)-(c). Within the standard AO picture, small particles are excluded from the large particles volume plus the volume of a corona of size σd/2 around them. When the surface- to-surface distance x between large particles is smaller than σd (and up to contact), the two coronas overlap, giving rise to a larger available volume for the small particles. In the case of soft interactions particles can (i) partially interpenetrate paying a small energy penalty and (ii) feel a residual repulsion from a colloid even when the latter is relatively distant, due to the potential tail. It is precisely this tail that causes the effect of increasing the effective attraction among larger soft spheres. Indeed, when large particles are found at x = hσd as in Fig. 1(b), the depletants that are in the region between the two colloids feel roughly twice the long-tail soft repulsion than those closer to one colloid only. As a result, depletants prefer to stay outside the region between the two colloids, thus producing a residual imbalance in the osmotic pressure that generates an attractive force even when colloids are located at a relative distance x > hσd. The range and strength of the tail are controlled by the functional form of the potential, making this effect more pronounced for softer particles, while naturally recovering AO for hard particles, i.e. for n →∞(see Appendix). Since this enhanced depletion depends entirely on the behaviour of the soft tail of the interaction potential, it is expected to be generic for any soft system. In addition, this effect is important

8

only for small q-values, because the range of the tail must effectively compete with the range of the effective potential. Thus, for larger q the potential is typically negligible even at a surface-to-surface distance σd, so that one still recovers the AO behaviour even for small n. This explains the findings of Zausch et al , who showed the equivalence between a soft and the AO mixture for q = 0.8. For small q values, on the other hand, our results show that a mapping between AO and soft depletion is not possible. We believe that this large effect of softness on the interaction potential between colloids could be experimentally observed through different experimental approaches. For instance, a direct way to obtain the potential mean-force can be achieved through the use of holographic microscopy , which allows to compute the histogram of distances (or, equivalently, the radial distribution function), of a diluted colloidal system in the presence of soft depletants. Also, a possible way to establish the effective depletion potential is by using confocal microscopy even in more dense colloidal systems, to compare the measured radial distribution function with theoretical/numerical predictions .

We now show that Eq. 1 quantitatively describes the numerical results. The comparison between the theoretical and simulation data is reported in Fig. 1(e), showing that the two sets fall on top of each other. In addition, we also plot the predictions obtained by the asymptotic formula in Eq. 2, which applies to the case of inverse power-law pair potentials.

Despite a systematic overestimation of the contact energy, the description of the data is qualitatively correct. Moreover, scaling the predictions to the contact energy of the exact results provides a very good agreement (inset of Fig. 1(e)), thus proving that soft depletion in this case has a ∼exp(−n x/σcd) tail for x/σd > 1 which is essentially controlled by n (and to a smaller extent by q). Therefore, this contribution is always present for finite n, and one can never recover the AO result, e.g. by a simple rescaling of ρd.

Consequences On Colloidal Phase Behaviour

As we mentioned before, potentials of different shape behave in an identical way when

Compared Using B∗

2 as a control parameter . Indeed, it was shown that such systems

Exhibit A Gas-Liquid Phase Separation At B∗

2 ∼−1.2 [22, 27, 56, 60–63].

We Calculate B∗

2 for the total potential among colloids, Vtot = Vn + Vdepl, for different q values as a function of ρd. The results, reported in Fig. 2, clearly show the dramatic

Dh3 For Vtot = V36 + Vdepl

(full lines) and for the AO case (dashed lines). The dashed-dotted horizontal line indicates where colloids should phase separate according to the extended law of corresponding states. Filled squares are data for Vtot at q = 0.1 as a function of the depletant density, rescaled by a factor ≈1.72 that best fits the AO case, to highlight the different ρd dependence between the two. For n = 36, h = 1.01818.

consequences of softness, which leads to a significant decrease of the critical ρd with respect to the AO case. For small q, this decrease is of the order of a factor 2. Only for large n the AO behaviour is recovered (see Appendix). However, for small and intermediate n, a simple rescaling of the density or of the size ratio is not sufficient to reproduce the AO behaviour in the whole ρd range. Indeed, leaving ρd as a free parameter is not sufficient to rescale the soft and AO curves on top of each other, as shown by the purple squares in Fig. 2.

A perfect curve collapse is unattainable even for different soft potential, i.e. inverse power law interactions with distinct n exhibit different B∗

2 Ρd-Dependences Owning To The Peculiar

functional forms of the resulting Vtot.

2 Becomes Crucial In Order To Simulate The Full

mixture, a practice that is becoming more and more common thanks to the increasing usage of GPU computing . While the thermodynamics and structure of the depletants can be investigated via MC simulations by employing non-local update algorithms [68, 69], brownian dynamics simulations also allow to assess the dynamics of depletants and colloids.

To this aim, we consider an effective one-component colloidal system interacting with Vtot and compare it with the corresponding soft binary mixture. We find perfect agreement between the two, as shown in Fig. 3(a), where the two colloidal structure factors fall on top

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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