Mechanical apparatus for mechanical axial and lateral gradation of elastomeric foams

Abstract

The present disclosure generally relates to a device and method for generating a foam sheet with nonuniform density. The device or method may include or use a nonlinear timing mechanism with at least one nonlinear damper and at least one nonlinear spring.

Description

SDSUF.021WO PCT APPLICATION MECHANICAL APPARATUS FOR MECHANICAL AXIAL AND LATERAL GRADATION OF ELASTOMERIC FOAMSSTATEMENT REGARDING FEDERALLY SPONSORED R& D

[0001] This invention was made with government support under grant number 2035663 awarded by the National Science Foundation. The government has certain rights in the invention,RELATED APPLICATIONS AND INCORPORATION BY REFERENCE

[0002] This application claims the benefit of U.S. Provisional Ser. No. 63 / 751740, filed January 30, 2025, which is hereby incorporated by reference in its entirety.FIELD

[0003] Aspects of the present disclosure relate to devices and methods for generating foams. In some embodiments of the present disclosure, the foam is a semi-closed cell hybrid polyurea foam with pronounced viscoelastic properties and impact mitigation attributes.BACKGROUND

[0004] Foams have been integrated into numerous impact mitigation mechanisms, such as protective pads, football helmets, walking and running shoes, and biking helmets, to name a few. However, the functionalities of polymer-based foams are well understood for impact mitigation; the underlying polymer's mechanical and physical properties, as well as its ability to be foamed, play a more prominent role in the effectiveness of the foam. For polymer-based foam to be effective in mitigating impact, the foam layer should be able to reduce the amplitude of the impact load while increasing the duration of the impact. In other words, the foam shields lessen the severity of the incoming impact forces by reducing the transmitted energy through energy-absorbing mechanisms that include elastic and plastic deformation of the foam materials.

[0005] Polyurea has been heavily investigated in the past decade and has effectively mitigated impact in different applications (i.e., civilian and military) when used in bulk form. Existing methods are cumbersome, expensive, and not scalable for processing polyurea foams.SUMMARY

[0006] Disclosed herein is a device for generating a foam sheet. In some embodiments, the foam has varying pore sizes, and / or has axial and lateral density gradation. In some embodiments, the device comprises a mold; a movable lid; and a nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring, wherein the nonlinear timing mechanism is capable of controlling the expansion rate of the foam sheet in the axial and / or lateral direction. In some embodiments, the device further comprises at least one holding bracket, holding lid, mounting bracket, or any combination thereof. In some embodiments, the device further comprises at least one spacer. In some embodiments, the at least one spacer is rigid. In some embodiments, the device comprises at least one spacer for each nonlinear timing mechanism. In some embodiments, the device comprises the same number of spacers and nonlinear timing mechanisms. In some embodiments, the device comprises a spacer that functions as a secondary cover. In some embodiments, the nonlinear timing mechanism is tuned. In some embodiments, the at least one nonlinear damper and the at least one nonlinear spring are connected. In some embodiments, the at least one nonlinear damper and / or the at least one nonlinear spring is attached to the movable lid, and / or to the mold. In some embodiments, the device further comprises at least 2, 3, or 4 nonlinear timing mechanisms. In some embodiments, the resulting foam sheet has varying or nonuniform density. In some embodiments, the nonlinear timing mechanism comprises a coil-over-shock. In some embodiments, the nonlinear timing mechanism comprises a nonlinear material, optionally wherein the nonlinear material is a nonlinear cellular material and / or a foam with nonlinear mechanical behavior. In some embodiments, the nonlinear cellular material and / or a foam comprises at least one pore, wherein the diameter of the pore is between about 10, 20, 30, 40, 50, 75, 100, 200, 300, 400, 500, and 600 gm. In some embodiments, the nonlinear timing mechanism comprises a pneumatic and / or hydraulic timing element. In some embodiments, the foam sheet comprises a polyurea foam. In some embodiments, the foam doesnot require applied heat and / or an exothermic reaction to form. In some embodiments, the mold comprises a base, wherein the base has straight or serrated edges. In some embodiments, the combined spring and damper length is about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, or 30 mm, optionally wherein the combined spring and damper weight is between about 3 and 25 mm. In some embodiments, the diameter of the nonlinear timing mechanism is about 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, or 60 mm.

[0007] Also disclosed herein is a method of generating a foam sheet with nonuniform density gradation. In some embodiments, the method comprises expanding a foam mixture within a mold, wherein the expansion rate is controlled by a nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring,

[0008] Also disclosed herein is a method of generating a foam with auxetic structure. In some embodiments, the method comprises expanding a foam mixture within a mold, wherein the expansion rate is controlled by a nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring,

[0009] Also disclosed herein is a nonlinear timing mechanism. In some embodiments, the nonlinear timing mechanism comprises at least one nonlinear damper and at least one nonlinear spring. In some embodiments, the nonlinear timing mechanism has use in regulating the expansion of a foam surface. In some embodiments, the nonlinear timing mechanism comprises at least one nonlinear spring and at least one nonlinear damper in the configuration as shown in FIG. 1.

[0010] Also disclosed herein is a foam produced by the device of any one of the embodiments of the present disclosure. In some embodiments, the foam comprises a top, a middle, and a bottom section, wherein foam pores in the middle section vary in size, wherein the average diameter of the foam pores in the top and / or bottom section are smaller than the average diameter of the foam pores in the middle section. In some embodiments, the average diameter of the foam pores in the middle section is at least 50, 100, or 150 μm. In some embodiments, the average diameter of the foam pores in the top and / or bottom section is at most 1, 10, or 50 μm. In some embodiments, the density of the foam pores in the middle section is at most 50, 100, 150, 160, 180, or 200 kg / m3. In some embodiments, the density of the foam pores in the top and / or bottom section is at least 200, 300, 350, 400, 450, or 500 kg / m3.BRIEF DESCRIPTION OF THE DRAWINGS

[0011] The foregoing is a summary and, thus, necessarily limited in detail. The above-mentioned aspects, as well as other aspects, features, and advantages of the present technology, are described below in connection with various embodiments, with reference to the accompanying drawings.

[0012] FIG. 1 show’s a non- limiting example schematic of a nonlinear timing mechanism comprising a nonlinear spring and nonlinear damper.

[0013] FIG. 2 shows a non-limiting example quantification for the rise displacement of foam over time through the convention rise rate (black line), using a nonlinear timing mechanism (grey line), and using a nonlinear timing mechanism along with optimizing the spring constant and dampening coefficient (dotted line).

[0014] FIG. 3 shows a non-limiting example representation of the device capable of generating a foam with axial and lateral density gradation. In this example, the device comprises a foam mold with a nonlinear timing mechanism mounted on top of a movable lid / cover. Also depicted are controlling spacers, which independently adjust the preload on the timing mechanism to tune the rising and expansion rates, as well as top holding brackets.

[0015] FIG. 4 shows a second non-limiting example representation of the device capable of generating a foam with axial and lateral density gradation. In this example, the device comprises a nonlinear timing mechanism mounted between a movable lid and a top, holding lid.

[0016] FIG. 5 shows a third non-limiting example representation of the device capable of generating a foam with axial and lateral density gradation. In this example, the device comprises a foam mold with a nonlinear timing mechanism mounted on top, below a movable lid / cover. Also depicted are mounting brackets, and a base with serrated edges.

[0017] FIG. 6 shows a non-limiting example foam that is produced using the device of any one of the present embodiments. The foam comprises three sections (p1, pv, and p3), wherein at least the pvsection has pores of varying sizes.DETAILED DESCRIPTION

[0018] The foregoing is a summary and, thus, necessarily limited in detail. The above-mentioned aspects, as well as other aspects, features, and advantages of the presenttechnology, will now be described in connection with various embodiments. The inclusion of the following embodiments is not intended to limit the disclosure of these embodiments but rather to enable any person skilled in the art to make and use the claimed subject matter. Other embodiments may be utilized, and modifications may be made without departing from the spirit or scope of the subject matter presented herein. Aspects of the disclosure, as described and illustrated herein, can be arranged, combined, modified, and designed in a variety of different formulations, all of which are explicitly contemplated and form part of this disclosure.

[0019] Density-graded foams, composed of layers or adjacent regions with varying densities, significantly advance impact protection technology. Each layer, or region, within these structures, may be engineered to absorb specific impact energy and velocity levels. By strategically layering these foams, the research team can achieve optimal impact resistance w'hile minimizing weight. Traditional methods for creating density gradients often involve complex and costly processes, such as mechanical, thermal, or chemical treatments. These techniques frequently require post-processing steps, adhesives, or controlled environmental conditions.

[0020] Some aspects of the present disclosure relate to a nonlinear spring-damper timing mechanism. The mechanism represents a simplified manufacturing process since the gradation is achieved concurrently during fabrication. This mechanism progressively slows the rise of foam layers during manufacturing, resulting in a density gradient without the need for vacuum, heat, or advanced electronics. By precisely tuning the stiffness and damping coefficients of the nonlinear springs and dampers, the density gradient can be tailored to meet specific impact requirements. The method used in some aspects of the present disclosure enables the creation of vertical and lateral density gradients. In some embodiments, vertical gradation is achieved by unifying the stiffness and dampening of all the plugs between the movable top and spaces. In some embodiments, lateral gradation is produced by varying the properties of the non-linear timers such that each spring / damper combination is distinctly chosen. Vertical gradation optimizes the foam's response to repeated impacts at a single location, while lateral gradation enhances its performance against random impact sites. The dual-gradient approach provides superior protection against a wide range of impact conditions based on the frequency and intensity of the impact events.

[0021] The benefits of density-graded foams are numerous. They offer lightweight, high-performance solutions for various applications, including aerospace, automotive, sports equipment, and military technologies. By reducing the weight of protective structures, these foams can improve fuel efficiency and enhance the overall performance of vehicles and equipment. Additionally, their ability to absorb impact energy effectively can help to protect delicate components and reduce the risk of injury.

[0022] Current gradation methods also penalize the overall thickness of protective gear, whereas achieving a higher gradient is limited to relatively tall samples or extremely laborious manufacturing and assembly processes. Pursuing continuous density gradation is limited to the constant infiltration of gases or chemicals during the curing process, resulting in unidirectional gradation based on complex steps.

[0023] Current density gradation in the field relies on the nonuniform distribution of foaming or blowing agents, heat, or diffusion of gas species throughout the curing process. In some embodiments, the methodology of the present disclosure comprises (1) the progressive, reactive force of the rising foam slurry during the curing process within the mold and (2) a purely mechanical, nonlinear timing mechanism. Elastomeric polyurea foam is fabricated with a self-foaming process that leverages violent emulsion of the forming chemicals in water, where localized reaction with water results in carbon dioxide bubbles responsible for the resulting microcellular structure comprising of partially perforated cells surrounded by a chain of closed cells. When cast in a rigid mold, the curing foam slurry induces a reactive force on the sides based on the available mold cavity's size, altering the foam's final microstructure. This progressive reactive force is also associated with the nonlinear rise rate of the foam within the mold, such that a fast rise rate corresponds to a relatively empty mold cavity. In contrast, a slower rate is associated with a nearly filled counterpart. On the other hand, a nonlinear timing mechanism comprising a nonlinear spring and damper is attached to the mold cover or sides to control the axial rising rate and lateral expansion rate, respectively, resulting in axial and lateral density gradations during the curing process. Therefore, the methodology of the present disclosure combined the progressive reactive forces induced by the foam slurry curing on the sides of the mold and mechanical timing mechanisms to control the density gradation of polyurea foams by tuning the rise and expansion rates.

[0024] The present disclosure constitutes the first purely mechanical gradation apparatus capable of concurrently achieving nonuniform density gradation throughout the foam sheet thickness and lateral gradation in width and breadth. In some embodiments, the methodology can also be used to induce auxetic structure to enhance the overall impact efficacy as a byproduct of tuning the parameters of the timing mechanisms with respect to the rise and expansion rates of the foam slurry.High-performance elastomeric foams

[0025] High-performance elastomeric foams are emerging as superior impact mitigation materials for civilian and military applications under various loading, operating, and environmental conditions. Specifically, polyurea foams produced using green and simple manufacturing processes outperform the state-of-the-art in impact efficacy, weight, and environmental stability. Polyurea foams may have superior mechanical properties under impact loading scenarios with a projectile velocity ranging from 2 m / s to >25 m / s and beyond. Remarkably, polyurea foams recovered >98% of their original height within milliseconds of the impact event, evidencing efficacy and recoverability. Furthermore, they studied discretely- graded foam structures with adhered or seamless interfaces by assembling multiple layers with different densities (properties) using strong, thin adhesive layers or sequentially casting several polyurea foam layers. The assembly of density-graded foams, irrespective of the interfacing strategy (adhered vs. seamless), complicates the manufacturing process via additional steps and limits the density gradient to a few discrete layers.

[0026] Mold cast foam, irrespective of the foaming mechanism (self-foaming, heated-activated chemical blowing agent, vacuum-controlled rising, etc.), is confined by the rigid sides of the mold cavity, resulting in nearly uniform cell size throughout the thickness and lateral dimensions of the mold. The foaming process induces progressive reactive forces on all sides of the mold due to axial rise and lateral expansion that often require relief vents or escape routes to ensure the properties' consistency and reduce the mold stresses. The relief mechanisms can be achieved by strategically metering the pour weight and volume or including channels or vents throughout the molds. This resulting reactive force within the rigid mold cavity is a non-obvious aspect of this disclosure, which is leveraged to dynamicallycontrol the size of the mold cavity throughout the curing process to tune the cell size (and the densities and properties).Timing Mechanism

[0027] In some embodiments, the device or methodology of the present disclosure further comprises a timing mechanism. In some embodiments, the timing mechanism is a nonlinear timing mechanism. In some embodiments, the timing mechanism comprises at least one nonlinear spring(s) connected in parallel with at least one nonlinear damper(s) that is attached to the cover or sides of the mold to control the rising rate. Analogous timing mechanisms can be connected to the mold sides to control the lateral expansion rate and achieve in-plane density and functional gradation. The timing mechanism can be constructed using coil-over-shocks (e.g., like those used in automobile suspension), nonlinear materials (e.g., foams with nonlinear mechanical behavior that resembles the response of parallelized nonlinear springs and dampers), or pneumatic or hydraulic timing elements without the need for any advanced electronics or control algorithms. The rise time (and, by induction, the lateral expansion rate) is based on the overall timing mechanism's effective spring constant and damping coefficient.

[0028] FIG. 1 illustrates a nonlimiting example timing mechanism that can be attached to a typical mold used to manufacture polyurea foams. FIG. 2 and illustrates how the timing mechanism affects the rising rate of the foam. Naturally, polyurea foam possesses a linear free rise rate of about 1mm / min, plateauing after 15 minutes. Changing the spring constant and damping coefficient in some embodiments alters the time towards the steady-state condition. This allows for the foam rise rate and force to be adjusted as desired. In some embodiments, the nonlinear timing mechanism is preloaded by clamping the spring and damper between the movable lid and foam surface.

[0029] The embodiments of the present disclosure are also as described in any of the following non-limiting numerated alternatives:1. A device for generating a foam sheet, wherein the device comprises:a mold;a movable lid; anda nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring,wherein the nonlinear timing mechanism is capable of controlling the expansion rate of the foam sheet in the axial and / or lateral direction.2. The device of alternative 1, further comprising at least one holding bracket, holding lid, mounting bracket, or any combination thereof.3. The device of any one of the previous alternatives, further comprising at least one spacer.4. The device of alternative 3, wherein the at least one spacer is rigid.5. The device of alternative 3 or 4, wherein the device comprises at least one spacer for each nonlinear timing mechanism.6. The device of alternative 5, wherein the device comprises the same number of spacers and nonlinear timing mechanisms.7. The device of any one of alternatives 3-6, wherein the device comprises a spacer that functions as a secondary cover.8. The device of any one of the previous alternatives, wherein the nonlinear timing mechanism is tuned.9. The device of any one of the previous alternatives, wherein the at least one nonlinear damper and the at least one nonlinear spring are connected.10. The device of any one of the previous alternatives, wherein the at least one nonlinear damper and / or the at least one nonlinear spring is attached to the movable lid, and / or to the mold.11. The device of any one of the previous alternatives, further comprising at least 2, 3, or 4 nonlinear timing mechanisms.12. The device of any one of the previous alternatives, wherein the resulting foam sheet has varying or nonuniform density.13. The device of any one of the previous alternatives, wherein the nonlinear timing mechanism comprises a coil-over-shock.14. The device of any one of the previous alternatives, wherein the nonlinear timing mechanism comprises a nonlinear material, optionally wherein the nonlinear material is a nonlinear cellular material and / or a foam with nonlinear mechanical behavior.15. The device of alternative 14, wherein the nonlinear cellular material and / or a foam comprises at least one pore, wherein the diameter of the pore is between about 10, 20, 30, 40, 50, 75, 100, 200, 300, 400, 500, and 600 pm.16. The device of any one of the previous alternatives, wherein the nonlinear timing mechanism comprises a pneumatic and / or hydraulic timing element.17. The device of any one of the previous alternatives, wherein the foam sheet comprises a polyurea foam.18. The device of any one of the previous alternatives, wherein the foam does not require applied heat and / or an exothermic reaction to form.19. The device of any one of the previous alternatives, wherein the mold comprises a base, wherein the base has straight or serrated edges.20. The device of any one of the previous alternatives, wherein the combined spring and damper length is about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, or 30 mm, optionally wherein the combined spring and damper weight is between about 3 and 25 mm.21. The device of any one of the previous alternatives, wherein the diameter of the nonlinear timing mechanism is about 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, or 60 mm.22. A method of generating a foam sheet with nonuniform density gradation, the method comprising expanding a foam mixture within a mold, wherein the expansion rate is controlled by a nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring.23. A method of generating a foam with auxetic structure, the method comprising expanding a foam mixture within a mold, wherein the expansion rate is controlled by a nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring.24. A nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring.25. A nonlinear timing mechanism for use in regulating the expansion of a foam surface, the nonlinear timing mechanism comprising at least one nonlinear spring and at least one nonlinear damper in the configuration as shown in FIG. 1.26. A foam produced by the device of any one of alternatives 1-21, wherein the foam comprises a top, a middle, and a bottom section, wherein foam pores in the middle section vary in size, wherein the average diameter of the foam pores in the top and / or bottom section are smaller than the average diameter of the foam pores in the middle section.27. The foam of alternative 26, wherein the average diameter of the foam pores in the middle section is at least 50, 100, or 150 μm.28. The foam of alternative 26 or 27, wherein the average diameter of the foam pores in the top and / or bottom section is at most 1, 10, or 50 μm.29. The foam of any one of alternatives 26-28, wherein the density of the foam pores in the middle section is at most 50, 100, 150, 160, 180, or 200 kg / m3,30. The foam of any one of alternatives 26-29, wherein the density of the foam pores in the top and / or bottom section is at least 200, 300, 350, 400, 450, or 500 kg / m3,EXAMPLE

[0030] The non-limiting example methodology and devices as disclosed herein are used in the following working Example. An apparatus, as shown in FIGs. 3-5, was employed to fabricate a series of density-graded foam sheets. In some embodiments, the timing mechanism comprises nonlinear material, integrating nonlinear spring and damper in one material plug. A range of density gradations was achieved by adjusting the configuration of nonlinear springs and dampers and applying appropriate preloads to the timing mechanism. Fig. 6 shows the resulting axial gradation within a polyurea foam cross-section, exemplified in different pore sizes, gradually ranging from 10s μm at the top and bottom surfaces to 100s μm at the center of the sample. Small pore size is attributed to densities of 350-450 kg / m3, while cells with mean diameters within 100 pm correlate to 100-160 kg / m3densities. The stiffness of springs was used to experiment with gradation effects and combat the forces provided by the foam. Microscopic analysis of the fabricated foam sheets confirmed the presence of distinct density gradients characterized by varying average cell sizes across the thickness. These density gradients directly translate into functional gradations, enabling the foam structures to withstand impacts at diverse loading rates and energy levels effectively.

[0031] The amount of time required to generate the foam is related to stiffness and damping parameter, such that the time constant is defined as the reciprocal of the product ofthese mechanical properties (i.e., 1 / (ζωn), where ζ is the dampening ratio and ωnis the natural frequency (√k / m). Hence, adjusting the size of the pores in the nonlinear material plug as used herein results in changes in the damping ratio and stiffness values,

[0032] References in the specification to “one embodiment,” “an embodiment,” “an illustrative embodiment,” “some embodiments,” etc,, indicate that the embodiment described may include a particular feature, structure, or characteristic, but every embodiment may or may not necessarily include that particular feature, structure, or characteristic. Moreover, such phrases are not necessarily referring to the same embodiment. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is submitted that it is within the knowledge of one skilled in the art to effect such feature, structure, or characteristic in connection with other embodiments, whether or not explicitly described.

[0033] As used in the description and claims, the singular form “a”, “an” and “the” include both singular and plural references unless the context clearly dictates otherwise. For example, the term “scaffold” may include, and is contemplated to include a plurality of scaffolds. At times, the claims and disclosure may include terms such as “a plurality,” “one or more,” or “at least one;” however, the absence of such terms is not intended to mean, and should not be interpreted to mean, that a plurality is not conceived.

[0034] The term “about” or “approximately,” when used before a numerical designation or range (e.g., to define a length or pressure), indicates approximations, which may vary by ( + ) or ( - ) 5%, 1% or 0.1%. All numerical ranges provided herein are inclusive of the stated start and end numbers. The term “substantially” indicates mostly (i.e., greater than 50%) or essentially all of a device, substance, or composition.

[0035] As used herein, the term “comprising” or “comprises” is intended to mean that the devices, systems, and methods include the recited elements, and may additionally include any other elements. “Consisting essentially of’ shall mean that the devices, systems, and methods include the recited elements and exclude other elements of essential significance to the combination for the stated purpose. Thus, a system or method consisting essentially of the elements as defined herein would not exclude other materials, features, or steps that do not materially affect the basic and novel character istic(s) of the claimed disclosure. “Consisting of” shall mean that the devices, systems, and methods include the recited elements and excludeanything more than a trivial or inconsequential element or step. Embodiments defined by each of these transitional terms are within the scope of this disclosure.

[0036] The examples and illustrations included herein show, by way of illustration and not of limitation, specific embodiments in which the subject matter may be practiced. Other embodiments may be utilized and derived therefrom, such that structural and logical substitutions and changes may be made without departing from the scope of this disclosure. Such embodiments of the inventive subject matter may be referred to herein individually or collectively by the term “invention” merely for convenience and without intending to voluntarily limit the scope of this application to any single invention or inventive concept, if more than one is in fact disclosed. Thus, although specific embodiments have been illustrated and described herein, any arrangement calculated to achieve the same purpose may be substituted for the specific embodiments shown. This disclosure is intended to cover any and all adaptations or variations of various embodiments. Combinations of the above embodiments, and other embodiments not specifically described herein, will be apparent to those of skill in the art upon reviewing the above description.

Claims

WHAT IS CLAIMED IS:

1. A device for generating a foam sheet, wherein the device comprises:a mold;a movable lid; anda nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring,wherein the nonlinear timing mechanism is capable of controlling the expansion rate of the foam sheet in an axial or a lateral direction.

2. The device of claim 1, further comprising at least one holding bracket, holding lid, mounting bracket, or any combination thereof.

3. The device of claim 1, further comprising at least one spacer.

4. The device of claim 3, wherein the at least one spacer is rigid.

5. The device of claim 4, wherein the device comprises at least one spacer for each nonlinear timing mechanism.

6. The device of claim 5, wherein the device comprises the same number of spacers and nonlinear timing mechanisms.

7. The device of claim 3, wherein the device comprises a spacer configured to function as a secondary cover.

8. The device of claim 1, wherein the nonlinear timing mechanism is tuned.

9. The device of claim 1, wherein the at least one nonlinear damper and the at least one nonlinear spring are mechanically connected.

10. The device claim 1, wherein the at least one nonlinear damper or the at least one nonlinear spring is attached to the movable lid, or to the mold.

11. The device of claim 1, further comprising at least 2 nonlinear timing mechanisms.

12. The device of claim 1, wherein the resulting foam sheet has varying or nonuniform density.

13. The device of claim 1, wherein the nonlinear timing mechanism comprises a coil-over-shock.

14. The device of claim 1, wherein the nonlinear timing mechanism comprises a nonlinear material, wherein the nonlinear material is a nonlinear cellular material or a foam with nonlinear mechanical behavior.

15. The device of claim 14, wherein the nonlinear cellular material or the foam comprises a pore, wherein the diameter of the pore is between about 10 and about 600 μm.

16. The device of claim 1, wherein the nonlinear timing mechanism comprises a pneumatic or a hydraulic timing element.

17. The device of claim 1, wherein the foam sheet comprises a polyurea foam.

18. The device of claim 1, wherein the foam does not require applied heat or an exothermic reaction to form.

19. The device of claim 1, wherein the mold comprises a base, wherein the base has straight or serrated edges,20. The device of claim 1, wherein the combined spring and damper length is between about 3 and about 25 mm.

21. The device claim 1, wherein the diameter of the nonlinear timing mechanism is between about 5 and about 60 mm.

22. A method of generating a foam sheet with nonuniform density gradation, the method comprising expanding a foam mixture within a mold, wherein the expansion rate is controlled by a nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring.

23. A method of generating a foam with auxetic structure, the method comprising expanding a foam mixture within a mold, wherein the expansion rate is controlled by a nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring.

24. A nonlinear timing mechanism comprising at least one nonlinear damper and at least one nonlinear spring.

25. A nonlinear timing mechanism for use in regulating the expansion of a foam surface, the nonlinear timing mechanism comprising at least one nonlinear spring and at least one nonlinear damper in the configuration as shown in FIG. 1.

26. A foam produced by the device of any one of claims 1-21, wherein the foam comprises a top, a middle, and a bottom section, wherein foam pores in the middle section varyin size, wherein the average diameter of the foam pores in the top and / or bottom section are smaller than the average diameter of the foam pores in the middle section.

27. The foam of claim 26, wherein the average diameter of the foam pores in the middle section is at least 50 pm.

28. The foam of claim 26, wherein the average diameter of the foam pores in the top or bottom section is at most 50 pm.

29. The foam of claim 26, wherein the density of the foam pores in the middle section is at most 200 kg / m3,30. The foam of claim 26, wherein the density of the foam pores in the top and / or bottom section is at least 200 kg / m3.

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