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Modeling of Non-elastic Properties of Polymeric Foams Used in Sports Helmets
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 177 (2017) 314 – 317
XXI International Polish-Slovak Conference “Machine Modeling and Simulations 2016”
Modeling of non-elastic properties of polymeric foams used in sports helmets Leszek Radziszewskia*, Milan Sagab Kielce University of Technology, Al. Tysiąclecia Państwa Polskiego 7, 25314 Kielce Poland b University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia
a
Abstract Personal protection equipment for the head used in various sporting disciplines must meet at least three fundamental criteria: safety, comfort and aesthetics. Safety is understood as protecting as much head area as possible and, in case of a crash, spreading the resultant forces over the maximum area and absorbing as much impact energy as possible. After the occurrence of non-elastic deformations, this energy is absorbed by the helmet shell and protective liner, most commonly made of polymeric foams. Nonelastic properties of the foams are evaluated with the use of descriptors of energy absorption capability. 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license © 2017 © Published by Elsevier Ltd. This Peer-review under responsibility of the organizing committee of MMS 2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016 Keywords: helmets; protective foams; non-elastic properties;
1. Introduction Sporting helmets are composed of a shell, a protective liner, and a retention system. The retention system has a function of maintaining the helmet in a stable position. The main task of the shell is to distribute the impact-induced stress over the maximum area possible. The protective liner, typically made of polymer foams, e.g., expanded polyethylene (PE) or polystyrene (EPS), absorbs the impact energy so as to minimize the accelerations and forces acting on the head [1]. This paper aims at analysing the impact energy absorption capability of polymeric foams.
* Corresponding author. Tel.: +48-41-34-24-759; fax: +48-41-34-42-997. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016
doi:10.1016/j.proeng.2017.02.231
315
Leszek Radziszewski and Milan Saga / Procedia Engineering 177 (2017) 314 – 317
2. Analysis of mechanical properties of foam materials used in helmets Polymer foams have open- or closed-cell structures. Their mechanical properties are dependent on their structure and on the type of polymer of which the cells are made [2]. The mechanisms of cell edge and wall deformations are major factors in the mechanical properties of foams. Another factor is a ratio of the volume and amount of open cells to those of the closed cells. The cells can have a shape similar to a cube [3] or polyhedron with eight hexagonal and six square faces, as in the so-called Kelvin model. One of the best known analytical models of foam mechanical properties was described in [3], where geometric shapes of real cells were replaced with cubes, which allowed relating the mechanical characteristics to the relative density of the foams. Elastic properties of a closed-cell structure result from elastic deformations of the edges (beams) and walls (thin-walled plates) as well as from the pressure of gas present in the cell. Stresses in the elastic range cause 3% to 5% strains. Young’s modulus of the foam can be expressed as [3] ݂ܧ ݏܧ
ʹ
ߩ
ൎ ߔ ʹ ቀ ݂ ቁ ሺͳ െ ߔሻ ߩݏ
ߩ݂ ߩݏ
ܲͲ ൫ͳെʹߥ ݂ ൯ ߩ݂
ܵܧ൬ͳെ
ߩݏ
(1)
൰
where: ߔ - is the ratio of the material mass contained in the cell edges to the mass of the cell, ES- is Young’s modulus of the polymer, Ef- is Young’s modulus of the foam, νf- is Poisson’s constant, ρf- is the foam apparent density, ρS- is the density of the polymer, P0- is the initial pressure of the gas contained in the foam cell, and R=ρf/ρS – is the relative density of the foam. In the elastic strain region, the amount of absorbed impact energy is small. After exceeding the elastic limit, deformations of cell edges and walls occur in the plastic region and the gas pressure increases. These deformations make the cell walls collapse progressively. The walls of some of the cells meet and touch. A further, even slight, increase in stress produces considerable strains up to 70%. This is the main mechanism of impact energy absorption by polymer foams. The stress-strain behaviour can thus be written as [3] ߪ ൌ ߪܻ݂
ܲͲ ߝ
ͳ
ͳെߝെܴ
for ߝ ݂ݕ൏ ߝ ൏ ߝ ܦቀͳ െ ቁ ܦ
(2)
where: σYf – is the flow stress of the foam at compression, εyf – is the flow strain at compression, ߝ ܦ- is the strain at complete densification of the foam. Further loading causes the majority of cells to crush at a small increase in strain but with exponential rise in stress and loss of cohesion. ͳ
ߝܦ
ܦ
ߝ ܦെߝ
ߪ ൌ ߪܻ ቀ
݉
ቁ
ܲͲ ߝ ͳെߝെܴ
ͳ
for ߝ ߝ ܦቀͳ െ ቁ ܦ
(3)
where: D, m – coefficients (e.g., for EPS D=2.3 and m=1). Investigations carried out after impact tests showed that the layer of the material closer to the impact surface are subject to higher strains. This makes the thickness of the impact absorbing material an important structural parameter [4]. The compressive stress applied dynamically has a minor effect on Young’s modulus and quite a noticeable effect on the number of crushed cells. When the foam density increases, the values of Young’s modulus and flow stress also increase at decreasing strain value that starts the exponential stress rise. Foams with low density absorb impact energy in a large volume of the material adjacent to the impact site. Foams with high densities absorb this energy through brittle crushing of the cells that are near the impact site. These foams absorb larger amounts of energy than low-density foams (at the same strains) but transfer to the head higher maximum forces and accelerations that act near the impact site. To be able to absorb the same amount of energy, low-density foam has to be thicker. Elements made of foams of insufficient density may be destroyed under a high-energy impact and fail to protect the head. In the case of foams whose density is too high, a low-energy impact will fracture a certain number of cells but the effect sufficient to reduce the forces acting on the head will not occur. In this situation the foam acts as an obstacle and may itself pose a serious risk of head injury.
316
Leszek Radziszewski and Milan Saga / Procedia Engineering 177 (2017) 314 – 317
3. Descriptors of non-elastic properties of foams Let us assume that a body of mass of m falls from a height of ܿܪon the material with a thickness of ݄ܿ , an area of ܿܣ, Young’s modulus E, which results in the maximum strain ߝ݉ and stress ɐ . The impact energy absorbed by this material can be determined from the following dependence ߝ
݉݃ ܿܪൌ ߪ ݉ Ͳ ݄ܿ ܿܣሺߝሻ ݀ߝ
(4)
The force (weight) ݉݃ acting on area ܿܣdue to gravity generates static stress ɐͲ ൌ ݉݃Τ ܿܣ. Maximum stress in this material, normalized relative to the static stress, can be written as ݉ܩൌ
ߪ݉ ߪͲ
ൌ
ߪ݉ ݉݃
ܿܣൌ
ߪ݉ ܿܣ ߝ
ሺ ܿ ݄ ܿܣΤ ܿܪሻ ߪ ݉ Ͳሺߝሻ݀ߝ
ൌ
ߪ݉ ߝ
ሺ݄ ܿ Τ ܿܪሻ ߪ ݉ Ͳሺߝሻ݀ߝ
(5)
When we know the compressive stress-strain response (Fig. 1a), impact energy, type of cushioning material and acceptable force that can be acting on the object protected, the relationships above can be used to determine optimum foam density. The dependence of the absorbed energy on stress or strain occurring in the foam, as shown in Figs 1b and 1c, makes the task easier. Another descriptor of non-elastic properties is absorption effectiveness understood as a ratio of the energy absorbed by the material of unit volume subjected to strain ߝ to arising stress ɐሺߝሻ. ߝ
ܧሺߝሻ ൌ
ߪ Ͳሺߝሻ݀ߝ ߪ ሺߝሻ
(6)
Figure 1d shows that the highest effectiveness occurs for strains with values close to that of the foam densification strain, which facilitates comparison of absorption properties of various materials.
Fig.1 Mechanical characteristics of polyethylene foam with density of about 58 kg/m3 for 8 consecutive loadings, according to [5]: a) stress as a function of strain, b) absorbed energy level as a function of stress, c) absorbed energy level as a function of strain, d) energy absorption effectiveness as a function of stress.
317
Leszek Radziszewski and Milan Saga / Procedia Engineering 177 (2017) 314 – 317
One of the parameters used to assess the effectiveness of multiple impact absorption [5] is that of the ratio of the absorbed energy to the stress corresponding to strain ߝ݇ ܧሺߝ ǡ ߝ݇ ሻ ൌ
ߝ
ߪ ݇ ߝሺߝሻ݀ߝ ߪ ሺߝ ݇ ሻ
(7)
A high value of ܧሺߝ ǡ ߝ݇ ሻ means that the foam can absorb high amounts of impact energy. During each consecutive loading, Young’s modulus of the foam decreases, and the dependence of absorbed energy density on the strain is approaching a linear function. This causes the energy absorption capability of the foam to decrease progressively. The PE 58 foam has the maximum effectiveness for stresses of about 0.25 MPa in the range 0.3 to 0.2 [5]. Variations in ܧሺߝ ǡ ߝ݇ ሻ in other foams are similar. Energy absorption properties of a material can also be characterized by the ratio of the energy absorbed by the material investigated to that absorbed by the reference material at the same stress and strain levels maintained. This parameter can be written as ߝ
ܫൌ
ߪ Ͳሺߝሻ݀ߝ ߪߝ
(8)
Impact energy absorption capabilities can also be compared with the use of Janssen factor determined as a ratio of maximum acceleration, generated in the material to that generated in the reference material with the same energy of impact. The impact-induced acceleration can be determined from the equation of kinetic energy and work, thus ʹݒ
ܽ݅ ൌ
ʹ݄ ܿ
(9)
where v is the velocity of the impact. Following the definition, the Janssen factor can be written as ܬൌ
ܽ݉ ܽ݅
ൌ
ʹܽ ݉ ݄ ܿ ʹݒ
(10)
4. Summary The primary function of protective liners made of polymer foams for helmet applications is to cushion impacts and reduce the forces acting on the head. The foams of too low or too high density may be dangerous for the helmet user. The selection of a suitable foam type is facilitated by the analysis of the dependence of the absorbed impact energy contents on the stress or strain and energy absorption capability as a function of stress. These relationships indicate that each impact on the helmet may cause permanent damage to its structure. This means that another impact may induce stresses and forces which will be too high for the helmet to withstand. References [1] A. Sapietová, P. Novák, L. Gajdoš, M. Kowalski, "Analysis of Impact Force in Tensioning Mechanism in MSC.ADAMS with Consideration of Rigid and Flexible Bodies", Applied Mechanics and Materials, 816 (2015) 165-173. [2] S. Adamczak, J. Bochnia, B. Kaczmarska, An analysis of tensile test results to assess the innovation risk for an additive manufacturing technology, Metrology and Measurement Systems, XXII, 1 (2015) 127–138. [3] L. Gibson, M. Ashby, Cellular solids, Oxford, Pergamon Press, 1988. [4] L. Landro, G. Sala, D. Olivieri, Deformation mechanisms and energy absorption of polystyrene foams for protective helmets, Polymer Testing, 21 (2002) 217–228. [5] U. Ozturk, G. Anlas, Energy absorption calculations in multiple compressive loading of polymeric foams, Materials and Design, 30 (2009) 15–22.
ScienceDirect Procedia Engineering 177 (2017) 314 – 317
XXI International Polish-Slovak Conference “Machine Modeling and Simulations 2016”
Modeling of non-elastic properties of polymeric foams used in sports helmets Leszek Radziszewskia*, Milan Sagab Kielce University of Technology, Al. Tysiąclecia Państwa Polskiego 7, 25314 Kielce Poland b University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia
a
Abstract Personal protection equipment for the head used in various sporting disciplines must meet at least three fundamental criteria: safety, comfort and aesthetics. Safety is understood as protecting as much head area as possible and, in case of a crash, spreading the resultant forces over the maximum area and absorbing as much impact energy as possible. After the occurrence of non-elastic deformations, this energy is absorbed by the helmet shell and protective liner, most commonly made of polymeric foams. Nonelastic properties of the foams are evaluated with the use of descriptors of energy absorption capability. 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license © 2017 © Published by Elsevier Ltd. This Peer-review under responsibility of the organizing committee of MMS 2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016 Keywords: helmets; protective foams; non-elastic properties;
1. Introduction Sporting helmets are composed of a shell, a protective liner, and a retention system. The retention system has a function of maintaining the helmet in a stable position. The main task of the shell is to distribute the impact-induced stress over the maximum area possible. The protective liner, typically made of polymer foams, e.g., expanded polyethylene (PE) or polystyrene (EPS), absorbs the impact energy so as to minimize the accelerations and forces acting on the head [1]. This paper aims at analysing the impact energy absorption capability of polymeric foams.
* Corresponding author. Tel.: +48-41-34-24-759; fax: +48-41-34-42-997. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016
doi:10.1016/j.proeng.2017.02.231
315
Leszek Radziszewski and Milan Saga / Procedia Engineering 177 (2017) 314 – 317
2. Analysis of mechanical properties of foam materials used in helmets Polymer foams have open- or closed-cell structures. Their mechanical properties are dependent on their structure and on the type of polymer of which the cells are made [2]. The mechanisms of cell edge and wall deformations are major factors in the mechanical properties of foams. Another factor is a ratio of the volume and amount of open cells to those of the closed cells. The cells can have a shape similar to a cube [3] or polyhedron with eight hexagonal and six square faces, as in the so-called Kelvin model. One of the best known analytical models of foam mechanical properties was described in [3], where geometric shapes of real cells were replaced with cubes, which allowed relating the mechanical characteristics to the relative density of the foams. Elastic properties of a closed-cell structure result from elastic deformations of the edges (beams) and walls (thin-walled plates) as well as from the pressure of gas present in the cell. Stresses in the elastic range cause 3% to 5% strains. Young’s modulus of the foam can be expressed as [3] ݂ܧ ݏܧ
ʹ
ߩ
ൎ ߔ ʹ ቀ ݂ ቁ ሺͳ െ ߔሻ ߩݏ
ߩ݂ ߩݏ
ܲͲ ൫ͳെʹߥ ݂ ൯ ߩ݂
ܵܧ൬ͳെ
ߩݏ
(1)
൰
where: ߔ - is the ratio of the material mass contained in the cell edges to the mass of the cell, ES- is Young’s modulus of the polymer, Ef- is Young’s modulus of the foam, νf- is Poisson’s constant, ρf- is the foam apparent density, ρS- is the density of the polymer, P0- is the initial pressure of the gas contained in the foam cell, and R=ρf/ρS – is the relative density of the foam. In the elastic strain region, the amount of absorbed impact energy is small. After exceeding the elastic limit, deformations of cell edges and walls occur in the plastic region and the gas pressure increases. These deformations make the cell walls collapse progressively. The walls of some of the cells meet and touch. A further, even slight, increase in stress produces considerable strains up to 70%. This is the main mechanism of impact energy absorption by polymer foams. The stress-strain behaviour can thus be written as [3] ߪ ൌ ߪܻ݂
ܲͲ ߝ
ͳ
ͳെߝെܴ
for ߝ ݂ݕ൏ ߝ ൏ ߝ ܦቀͳ െ ቁ ܦ
(2)
where: σYf – is the flow stress of the foam at compression, εyf – is the flow strain at compression, ߝ ܦ- is the strain at complete densification of the foam. Further loading causes the majority of cells to crush at a small increase in strain but with exponential rise in stress and loss of cohesion. ͳ
ߝܦ
ܦ
ߝ ܦെߝ
ߪ ൌ ߪܻ ቀ
݉
ቁ
ܲͲ ߝ ͳെߝെܴ
ͳ
for ߝ ߝ ܦቀͳ െ ቁ ܦ
(3)
where: D, m – coefficients (e.g., for EPS D=2.3 and m=1). Investigations carried out after impact tests showed that the layer of the material closer to the impact surface are subject to higher strains. This makes the thickness of the impact absorbing material an important structural parameter [4]. The compressive stress applied dynamically has a minor effect on Young’s modulus and quite a noticeable effect on the number of crushed cells. When the foam density increases, the values of Young’s modulus and flow stress also increase at decreasing strain value that starts the exponential stress rise. Foams with low density absorb impact energy in a large volume of the material adjacent to the impact site. Foams with high densities absorb this energy through brittle crushing of the cells that are near the impact site. These foams absorb larger amounts of energy than low-density foams (at the same strains) but transfer to the head higher maximum forces and accelerations that act near the impact site. To be able to absorb the same amount of energy, low-density foam has to be thicker. Elements made of foams of insufficient density may be destroyed under a high-energy impact and fail to protect the head. In the case of foams whose density is too high, a low-energy impact will fracture a certain number of cells but the effect sufficient to reduce the forces acting on the head will not occur. In this situation the foam acts as an obstacle and may itself pose a serious risk of head injury.
316
Leszek Radziszewski and Milan Saga / Procedia Engineering 177 (2017) 314 – 317
3. Descriptors of non-elastic properties of foams Let us assume that a body of mass of m falls from a height of ܿܪon the material with a thickness of ݄ܿ , an area of ܿܣ, Young’s modulus E, which results in the maximum strain ߝ݉ and stress ɐ . The impact energy absorbed by this material can be determined from the following dependence ߝ
݉݃ ܿܪൌ ߪ ݉ Ͳ ݄ܿ ܿܣሺߝሻ ݀ߝ
(4)
The force (weight) ݉݃ acting on area ܿܣdue to gravity generates static stress ɐͲ ൌ ݉݃Τ ܿܣ. Maximum stress in this material, normalized relative to the static stress, can be written as ݉ܩൌ
ߪ݉ ߪͲ
ൌ
ߪ݉ ݉݃
ܿܣൌ
ߪ݉ ܿܣ ߝ
ሺ ܿ ݄ ܿܣΤ ܿܪሻ ߪ ݉ Ͳሺߝሻ݀ߝ
ൌ
ߪ݉ ߝ
ሺ݄ ܿ Τ ܿܪሻ ߪ ݉ Ͳሺߝሻ݀ߝ
(5)
When we know the compressive stress-strain response (Fig. 1a), impact energy, type of cushioning material and acceptable force that can be acting on the object protected, the relationships above can be used to determine optimum foam density. The dependence of the absorbed energy on stress or strain occurring in the foam, as shown in Figs 1b and 1c, makes the task easier. Another descriptor of non-elastic properties is absorption effectiveness understood as a ratio of the energy absorbed by the material of unit volume subjected to strain ߝ to arising stress ɐሺߝሻ. ߝ
ܧሺߝሻ ൌ
ߪ Ͳሺߝሻ݀ߝ ߪ ሺߝሻ
(6)
Figure 1d shows that the highest effectiveness occurs for strains with values close to that of the foam densification strain, which facilitates comparison of absorption properties of various materials.
Fig.1 Mechanical characteristics of polyethylene foam with density of about 58 kg/m3 for 8 consecutive loadings, according to [5]: a) stress as a function of strain, b) absorbed energy level as a function of stress, c) absorbed energy level as a function of strain, d) energy absorption effectiveness as a function of stress.
317
Leszek Radziszewski and Milan Saga / Procedia Engineering 177 (2017) 314 – 317
One of the parameters used to assess the effectiveness of multiple impact absorption [5] is that of the ratio of the absorbed energy to the stress corresponding to strain ߝ݇ ܧሺߝ ǡ ߝ݇ ሻ ൌ
ߝ
ߪ ݇ ߝሺߝሻ݀ߝ ߪ ሺߝ ݇ ሻ
(7)
A high value of ܧሺߝ ǡ ߝ݇ ሻ means that the foam can absorb high amounts of impact energy. During each consecutive loading, Young’s modulus of the foam decreases, and the dependence of absorbed energy density on the strain is approaching a linear function. This causes the energy absorption capability of the foam to decrease progressively. The PE 58 foam has the maximum effectiveness for stresses of about 0.25 MPa in the range 0.3 to 0.2 [5]. Variations in ܧሺߝ ǡ ߝ݇ ሻ in other foams are similar. Energy absorption properties of a material can also be characterized by the ratio of the energy absorbed by the material investigated to that absorbed by the reference material at the same stress and strain levels maintained. This parameter can be written as ߝ
ܫൌ
ߪ Ͳሺߝሻ݀ߝ ߪߝ
(8)
Impact energy absorption capabilities can also be compared with the use of Janssen factor determined as a ratio of maximum acceleration, generated in the material to that generated in the reference material with the same energy of impact. The impact-induced acceleration can be determined from the equation of kinetic energy and work, thus ʹݒ
ܽ݅ ൌ
ʹ݄ ܿ
(9)
where v is the velocity of the impact. Following the definition, the Janssen factor can be written as ܬൌ
ܽ݉ ܽ݅
ൌ
ʹܽ ݉ ݄ ܿ ʹݒ
(10)
4. Summary The primary function of protective liners made of polymer foams for helmet applications is to cushion impacts and reduce the forces acting on the head. The foams of too low or too high density may be dangerous for the helmet user. The selection of a suitable foam type is facilitated by the analysis of the dependence of the absorbed impact energy contents on the stress or strain and energy absorption capability as a function of stress. These relationships indicate that each impact on the helmet may cause permanent damage to its structure. This means that another impact may induce stresses and forces which will be too high for the helmet to withstand. References [1] A. Sapietová, P. Novák, L. Gajdoš, M. Kowalski, "Analysis of Impact Force in Tensioning Mechanism in MSC.ADAMS with Consideration of Rigid and Flexible Bodies", Applied Mechanics and Materials, 816 (2015) 165-173. [2] S. Adamczak, J. Bochnia, B. Kaczmarska, An analysis of tensile test results to assess the innovation risk for an additive manufacturing technology, Metrology and Measurement Systems, XXII, 1 (2015) 127–138. [3] L. Gibson, M. Ashby, Cellular solids, Oxford, Pergamon Press, 1988. [4] L. Landro, G. Sala, D. Olivieri, Deformation mechanisms and energy absorption of polystyrene foams for protective helmets, Polymer Testing, 21 (2002) 217–228. [5] U. Ozturk, G. Anlas, Energy absorption calculations in multiple compressive loading of polymeric foams, Materials and Design, 30 (2009) 15–22.