Power Factor Correction Pfc Matlab

Bonn-Th-2025-17

Power corrections to the heavy electron form factor

Aniruddha Venkataa

aBethe Center for Theoretical Physics, Universit¨at Bonn, D-53115, Germany

Abstract

We study the first power correction to the heavy electron form factor in QED and show that it factorizes as a derivative operator. We discuss the result in QED with no light fermions, where the first power correction can be written explicitly in terms of one- loop integrals and the anomalous magnetic moment. In the presence of light fermions, the heavy electron form factor admits a representation as a sum over matrix elements, each of which receives corrections from higher orders in perturbation theory. From this analysis, we are able to extract the next-to-leading power soft photon theorem in the limit of heavy fermion-initiated dijet events. This is a first step towards studying the heavy quark form factor in the non-abelian theory.

Introduction

The investigation of power corrections to QCD observables is of importance in the era of precision collider physics ,,. Studying power corrections to jet observables at the cross-section level has been an area of active research and has attracted increasing interest ,,, , and . Such corrections to N-subjettiness , a jet shape ob- servable designed to identify hadronically decaying heavy particles, have been studied extensively ,, , . Therefore, the study of power corrections to observables involving the production and decay of heavy quarks is of interest to the collider physics community.

A complementary approach useful for resummation is studying power corrections at the level of amplitudes or form factors ,, . In these works, the form factor was investigated in QED in the light fermion limit. However, in this limit, the form factor consists of four independent components: two jet functions, a soft function, and

A Hard Function. Schematically, One May Write

Γ(p1, p2) = J1(p1, β2)H(p1, p2)J2(p2, β1)S(β1, β2),

P0

i . The functions J1, and J2 are the jet functions, the function S is the soft function, and H is the hard function. In studying power corrections to the form factor in the light fermion limit, all four functions contribute to the next-to-leading power

Arxiv:2504.20786V2 [Hep-Ph] 15 Aug 2025

(NLP) corrections. Further, as a result of pinch surfaces with physically polarized partons connecting hard and jet functions, which begin to contribute at NLP, the simple factorized form in Eq. (1.1) is modified into a more complicated convolution between hard and jet modes. This makes the study of factorization theorems formidable even in QED. Such an NLP study of the form factor in the light quark limit is useful to study the thrust cross section in the two-jet limit and is of importance phenomenologically.

At the level of a cross-section, there are multiple notions of a power correction since cross-section predictions are often at leading power in multiple dimensionless parameters. For example, the thrust cross-section in e+e−annihilation , depends

Q2

µ2 (here Q2 is the centre of mass energy and µ is the renormalization scale) as well as the thrust variable 1−T. Therefore, the thrust cross- section admits a power expansion in two parameters 1 −T, ΛQCD

. At Leading Power In

1 −T, the thrust cross-section develops a large logarithm of the form ∼log2(1 −T), which corresponds to a singularity in the two-jet limit. Leading power factorization of the Sudakov form factor in Eq. (1.1) allows us to resum all large logarithms in 1−T. In contrast, power corrections to the total cross-section in e+e−annihilation are

Q4

and arise from vacuum condensates . In what follows,



. Another simple class of questions involves power corrections to the form factor in the heavy quark limit. The heavy quark form factor which we denote Γρ(p1, p2, M) at

Leading Power Reads ,

Γρ(p1, p2, M) = Hρ(p1, p2, M)⟨Wv1(0, ∞)Wv2(0, ∞)⟩.

Here, Vµ

i are the 3-velocities in the pi direction and Wvi are Wilson lines in th vi direction. In particular, the velocities vi are four vectors defined to have unit energy.



.

(1.3)

The perturbative expansion of the Wilson lines in momentum space requires regular- ization to avoid eikonal singularities. This is completely determined by the limits of σ integration and demanding that integrals are well defined at ∞. It will be convenient to study the form factor with incoming momentum q = p1 + p2, in the centre of mass



.

2

In this frame, the velocity vectors can be made explicit

!

.

(1.5)

Further, we will use Feynman gauge, where the gluon propagator is

Ν(K′)⟩= −Iδabηµν

k2 + iϵ (2π)dδd k + k′
.

(1.6)

With these conventions in place, we can explain the formula in Eq. (1.2). The function H(p1, p2, M) is a IR finite, hard function which we may evaluate in perturbation theory, while the Wilson line expectation value is an IR divergent soft function which involves the exchange of soft modes. The absence of an IR divergent jet function is explained by the presence of massive quarks in the final state, with a mass scale M ≫ΛQCD.

The pinch singular surfaces of the form factor are shown in Fig. 1.

(B)

Figure 1: Two types of pinch singular surfaces that are present in the form factor. (a) Leading

Pinches (B) Non-Leading Pinches

The leading pinches in Fig. 1 have no soft gluons connecting the soft function and the hard function, while the sub-leading pinches in Fig. 1 do. Loop integration of the form factor in the vicinity of a soft pinch involves integration of modes that satisfy |⃗k|< ΛQCD and, by definition, are integrals over soft scales, yielding results of O(ΛQCD).

At leading power, integration of loop momentum modes satisfying |⃗k|< ΛQCD can be separated out, and the resulting expression is Eq. (1.2). Qualitatively, there are two sources of corrections to the formula in Eq. (1.2): integration around sub-leading soft

3

pinch surfaces and subleading terms near leading soft pinch surfaces. To arrive at a formula analogous to Eq. (1.2) at NLP, we must account for both types of corrections and separate out soft modes from hard modes. The presence of subleading pinches that connect the hard part and the soft subgraph suggests that at NLP, there is a convolution between the hard and soft functions.

We Will Show That In Qed, This

expectation is wrong, and conventional factorization extends to NLP, with new soft functions. Recently, it was shown that the soft photon theorem ,,, at leading power is modified in the presence of massless quarks . A detailed study at next- to-leading power in all two-loop QED graphs has been carried out in . A more comprehensive all-loop theorem is still missing in the massless quark limit. It is not likely that the radiative amplitude is related to the nonradiative form factor alone. It is therefore worthwhile to investigate the opposite, massive quark limit, and ask if the NLP soft photon theorem is modified in this limit as well. Developing such an NLP soft photon analysis requires NLP factorization, which is the subject of study here. We will also state the NLP soft photon theorem in the heavy electron limit in Sec. 4. In QCD, an explicit cutoff can be placed around each pinch surface, which is a UV cutoff for the soft modes and an IR cut off for the hard function as previously described.

Such a cutoff is physical when the form factor is embedded inside a cross-section (like a thrust cross-section). However, in carrying out explicit calculations, it is natural to consistently use dimensional regularization both in QCD and QED. In this setting, the leading power soft function is defined by its ultraviolet (UV) counter-terms, and the leading power soft function contains UV poles accompanied by a factorization scale, µ.

Power corrections to this soft function are therefore naturally organized by the ratio

Μ

pi·vi . This structure is observed in our explicit results in Eqs. (2.14),(3.28). In this work, we study the question of NLP corrections to the heavy quark form factor in the abelian theory. We will therefore study the heavy electron form factor in QED, with and without virtual light fermions in intermediate states. In Sec. 2, we study the one-loop form factor graphs in massive QED and arrive at an NLP factorization formula valid at one-loop. We will find that the one-loop factorization formula is gauge invariant and reproduces the one-loop triangle graph at NLP in a perturbative expansion of the matrix element. 3, we repeat this analysis at two loops and arrive at additional terms that begin at two loops. We find that the new terms are independently gauge invariant. 4 we argue that there are no new terms differing from those inferred in the two-loop analysis. 5, we summarize our results and list directions for generalization that we plan to study in future work.

P2, M

Figure 2: The one-loop contribution to the form factor

One-Loop Analysis

In this section, we study the one-loop soft function in QED. Let us first observe that in QED, the one-loop ladder graph is IR divergent in four dimensions at leading power. Further, it exponentiates and captures all IR divergences in QED . The soft function can be evaluated at all loops by evaluating a one-loop integral

(2.1)

The relevant graph at one-loop is the QED ladder graph shown in Fig. 2. The vertex

((P2 + K)2 −M2 + Iϵ)(K2 + Iϵ)

.

(2.2)

Here, we work in a normalization where ˜Γρ, (0) = γρ. This graph has only one pinch surface at kµ = 0, the soft pinch.

Such A Pinch Is A Leading Pinch Since The Tree

level hard function, ˜Γρ, (0), is independent of k. At one-loop, there are no non-leading pinches. In anticipation of the distinction from the two-loop case discussed in Sec. 3, we note that at two loops, both subleading and sub-subleading pinches can arise. Let us make an observation about the notation we use here. We use the symbol ˜Γρ to denote the ladder graph contribution to the full form factor Γρ. These differ by the

5

insertion of counter-terms as well as self-energy graphs, which we don’t analyze in this section. At leading power, in Eq. (2.2) we make the following approximations near the pinch • In the numerator, we drop /k in favour of /p1 + M. Clearly, the term with /k in the numerator has an additional vanishing scale, which is power suppressed.

• In the p1 denominator, we make the approximation (p1−k)2−M2+iϵ ≈−2p1·k+ iϵ, dropping the k2 relative to −2p1 · k. This follows from the soft approximation and is justified by the assumption of outgoing kinematics, as discussed in . In this case, since both p1, p2 are outgoing, we may make this approximation safely.

• In the p2 denominator, we make the approximation (p2+k)2−M2+iϵ ≈2p1·k+iϵ, dropping the k2 relative to 2p2 · k. Having made a list of approximations to the one-loop graph, we may use the Dirac equation on either spinor, ¯u(p1, s1)(/p1 −M) = (/p2 + M)v(p2, s2) = 0, to write the

(2.3)

which agrees with the leading term in the expansion of Eq. (2.1) multiplying the tree- level hard function γρ. We can now separate the next-to-leading power terms in Eq. (2.2). Of the two /k in the numerator, we may retain one and continue to make the eikonal approximation. Alternatively, we may expand the eikonal denominator and consider next-to-eikonal corrections. Next-to-eikonal corrections have been studied in non-abelian gauge theories in . Here, we specialize to the case of abelian gauge theory with fermionic matter.

To isolate the contributions, we analyze each fermion line separately and define

1 (P2)V(P2, S2)−Ie2Ηµν

k2 + iϵ . Here, the subscript indicates the number of gauge boson insertions on the fermion line. In what follows, we will find it convenient to use the Grammer-Yennie decomposition

1

p1 · k.

6

Let us note that there is a freedom in the choice of the K photon, with a numera- tor shifted by any vector proportional to kµ(O(1) + O(k) + . . .) and the denominator shifted by O(k2) + . .. Such shifts are higher order in k and do not affect the leading power argument. However, at NLP, such a shift begins to contribute to the integrand, and we make a choice which separates the leading power from all non-leading contri- butions. This choice differs from that made in . We now proceed to apply the K–G decomposition.

Ν(P2, K)F Ν

1 (p2).

(2.6)

Next, we observe that the G-photon satisfies the identity

(2.7)

which implies that the G-photon anti-commutes with /p1

Νγν( /P1 + M) = (−/P1 + M)Gµ

νγν.

(2.8)

As a result, the /p1 + M term in the first line of Eq. (2.4) and the /p2 −M term in the second line identically vanish when γµ is replaced by Gµ

Νγν. We Conclude That The

G-photon is power suppressed relative to the K-photon. The K-photon is longitudinal and satisfies the Ward identity

=

−1.

Ν(P2, K)F Ν

1 (p2, k) agrees with the leading power expression in Eq. (2.3). Therefore, all next-to-leading power terms at one loop must come from the insertion of a single G-photon. The Ward identity used here involves no approximation.

The leading power approximation is therefore neatly summarized by the assertion that all terms involving G-photons are to be dropped at leading power. We notice that the vertex correction with the F µ(p1, k) replaced by Gµ

1 (P2, K) Is Next-To-Next-To Leading Power And We

may absorb it into the hard part when working at next-to-leading power. We may now

K2 + Iϵ .(2.10)

Let us briefly comment on the qualitative features of G-K decomposition. The sum of the G, K photons is the identity, and the K is to be interpreted as the photon connected

7

to an eikonal (Wilson) line. Therefore, the G photon is to be identified as being the remainder after subtraction of the eikonal piece. It contains sub-eikonal corrections as well as numerator corrections to the eikonal approximation. At leading power, all G photon attachments are absorbed into the hard function. At NLP, only one G-photon escapes from the hard part (at one loop), while the rest remain within the definition of the hard function.

Finally, we would like to write an explicit expression for Gµ

(2.11)

where the approximation in the second line retains the leading term in the denominator, since a non-leading term has already appeared in the numerator. We can put the G-

1

2p1 · k.

(2.12)

The term inside the brackets on the right-hand side of the Eq. (2.12) , we may identify as the momentum space representation of the field strength tensor ˜Fµν(k). As a result, we may put one of the NLP terms into the matrix element form

Γµγνγρ, (2.13)

where we have replaced the K-photon on the p2 line by a Wilson line using the Ward identity in Eq. (2.9) as in Eq. (2.3). The single G-photon corresponds to the insertion of a single F in the matrix element. The Wilson lines capture all K-photons insertions, following the leading power result.

The insertion of the Wilson line Wv1(0, ∞) adds the leading power graph, and the denominator subtracts the connected, leading power ladder graph from the matrix element. This has been done so as to ensure that the NLP soft function generalizes to higher orders with more complicated connected graphs. In particular, disconnected graphs cancel between the numerator and denominator. The numerator in Eq. (2.13) is the position space analogue of Eq. (2.12).

A similar analysis applies to the second term in Eq. (2.10), and we write all next- to-leading power terms at one loop in the factorized matrix element form

⟨Wv1(0, ∞)Wv2(0, ∞)⟩

Γρ,(0)γνγµ.

8

Therefore, we have shown that at one-loop, the factorization is preserved at NLP and that the long-distance function involves an insertion of a gauge-invariant field strength.

Where We Have ˜Γρ,(1)

NLP is the function that appears in the right hand side of Eq. (2.14), and remainder ˜H defines a new hard function where both leading power and next-to- leading power long distance physics has been factored out. The full form factor has a vertex correction counter term insertion which involves no loop integrals, and therefore trivially shifts the hard function that appears on the right-hand side of Eq. (2.15) in a scheme-dependent way. The self-energy graphs that we have not studied here can be chosen to vanish on the mass shell by renormalizing on-shell .

In the next section, we show that the one-loop factorization formula in Eq. (2.15) does not hold at two loops. However, the two-loop NLP corrections factorize as a differential operator.

Two-Loop Analysis

In this section, we study next-to-leading power (NLP) factorization at two loops. As previously indicated, new terms arise at two loops that do not appear in the one-loop analysis.

These are of two types: (i) a soft photon emerging from the hard part, which factors as a derivative operator acting on the lower-order hard function, and (ii) a double G-photon insertion on a fermion line. These terms are absent at one loop because the tree-level hard function is trivial, so its derivative vanishes. Excluding self-energy corrections on the external lines, at most one gauge boson attaches to an outgoing fermion, so multi-G-photon terms do not contribute at one loop.

As before, instead of studying all graphs, we start by studying a representative subset. We will include other graphs that contribute to the form factor subsequently. We start our two-loop analysis by focusing on ladder-like graphs, denoted by ˜Γρ,(2).

The full form factor Γρ,(2) also includes additional diagrams, which we omit in this discussion. The relevant graphs at two loops are shown in Fig. 3. Additional contributions include self-energy diagrams, nontrivial counterterm insertions, and vertex correction ladders. Closed fermion loops at this order contain three-photon subgraphs, which vanish by Furry’s theorem. In the presence of other light fermions, the self-energy with a fermion loop is also pinched.

Let us list all pinch surfaces of the two graphs in question. • Both l, k are soft: this region contributes a new term involving a double G-photon insertion.

(B)

Figure 3: Graphs that contribute to the two-loop form factor. (a) Ladder graph (b) Crossed

Ladder Graph

• The loop momentum l is soft and k hard: this region is doubly power suppressed in the ladder diagram but contributes at NLP in the crossed-ladder case. We will obtain a p1 derivative operator acting on the one-loop hard function.

• The loop momentum l is hard and k soft: this region contributes at both lead- ing and next-to-leading power in the ladder diagram. These contributions are encapsulated by Eq. (2.15), albeit with a more intricate hard function. In the crossed-ladder case, this configuration again yields a p2 derivative acting on the one-loop hard function.

We begin by examining the double-soft limit. As in the one-loop case, we decompose both the ladder and crossed-ladder diagrams into contributions from two fermion lines.

V(P2, S2). (3.2)

From these expressions, we can reconstruct the two ladder-like graphs using

−Iηµ2Ν2

l2 + iϵ .

Here, The Factor Of 1

2 accounts for symmetrization over the loop momenta k, l. We can now carry out the K-G decomposition on each fermion line independently as in Eq. (2.6).

(P1) (3.4)

It is easy to check that each K-photon satisfies a Ward identity similar to Eq. (2.9).

1

p1 · l .

(3.5)

The structure of this Ward identity is K ⊗Fn(pi) = Fn−1(pi), where n is the number of attachments on the fermion line pi. Therefore, the Ward identity makes it possible to peel off each longitudinal photon one at a time, in a recursive fashion.

We now examine each term in Eq. (3.4) one at a time

(3.6)

which corresponds to the O(e2) expansion of the p1 Wilson line in Eq. (1.2). Next, we consider the KG terms in Eq. (3.4). We apply the Ward identity in Eq. (3.5) to obtain

Α1(P1, K)F Α1

1 (p1, k).

(3.7)

The action of G-photons on F1 has already been analyzed in Eqs. (2.11) and (2.12).

The Factor Of 1

2 in Eq. (3.3) cancels because the k, l symmetry gives two equivalent ways to assign G-polarization to the photons on the p1 line. Therefore, this corresponds to the expansion of the p1 Wilson line to O(e) and the momentum space analogue of ∂Fµν operator in Eq. (2.13).

As a result, any genuinely new contribution to the NLP factorization theorem must arise from the double G-photon term, which was absent at one loop. Let us analyze these terms carefully.

!

.

11

In picking out the relevant pieces at NLP, we first observe that in either term, choosing /p1 + M in the first numerator (from the left) yields zero identically since /p1 anti-

Commutes With Gµi

αiγαi, and the Dirac equation implies that this term vanishes when acting on the spinor. Further, choosing −/k −/l in the second numerator (from the left)next-to-next-to-leading power (N2LP). For the same reason, we apply the eikonal approximation to both denominators. After making the relevant approximations, we

!

.

(3.9)

We may anticommute the rightmost /p1 + M past the G-polarized γ matrices to give zero by use of the Dirac equation once again. The only nonvanishing term arises from the inner product of p1 with the soft momentum.

(3.10)

where the Dirac matrices reduce to the metric tensor ηα1α2. The G-photons satisfy an identity that manifests a gauge invariant coupling between G-photons and the fermion

=

kβϵα −kβϵα.



.

(3.12)

Finally, we write a position space matrix element as in Eq. (2.13) for the double G- photon insertion.

(3.13)

Once again, the two factors of F reproduce the double G photon insertion, while the Wilson lines capture all the K photons (of which there are infinite at all orders in perturbation theory).

12

Let us now turn to the remaining pinch surfaces. To derive the NLP factorization theorem for the form factor in the other regions, we must include additional diagrams that contribute at this order. First, consider the self energy ladders in Fig. 4.

(B)

Figure 4: Graphs that contribute to the two loop form factor but not at NLP when one loop is soft and the other is hard. (a) The p1 self-energy ladder graph (b) The p2 self-energy

Ladder Graph

Consider the graph with p1 self-energy ladder shown in Fig. 4(a). When k is soft and l is hard, the leading-power contribution from the soft photon attachment to the fermion lines can be replaced by their corresponding K-photon components. When combined with other attachments on the p1 line, a self-energy diagram remains, which vanishes under on-shell renormalization. When the soft photon, labelled by the loop momentum k, is G-polarized, the k integral factorizes from the self-energy, yielding a matrix element in the form of Eq. (2.13). This leaves behind an on-shell self-energy diagram, which again vanishes upon renormalization.

When l is soft and k is hard, both self-energy ladders are doubly power suppressed because they contain two fewer eikonal lines. When l and k are both soft, the G photon contributions are non-trivial, but as we will see in Sec. 4, such contributions are all captured by Eqs. (2.13) and (3.13). Next, we consider the graphs in Fig. 5.

The vertex-correction ladder diagrams possess a double-soft region, where both k and l are soft. In this region, at leading power, we may replace each attachment to the fermion lines by their K-photon components. At next-to-leading power, one or two attachments can be G photons instead, and the relevant graphs are generated O(e4) expansion of Eqs. (2.13) and (3.13). As we will argue in Sec. 4, the case of three or more G-polarized photons is a next-to-next-to-leading power (N2LP) effect.

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

Related Journal Articles & DOI Links

Selected peer-reviewed publications relevant to 12 Lead ECG Acquisition. Click the DOI to access the full paper (may require institutional access).

Why Choose Us?

Bangalore guidance for robotics, Spectre and autonomous systems projects.