A new empirical analytical approach is developed for predicting the stress concentration profile around an in-situ tow break in filament-wound composites. A shear-lag analysis is firstly performed to solve for the perturbational axial displacement of the broken tow and resultant debonding lengths. Solution of stress field caused by tangential load on the surface of a elastic half space is then utilized in combination with superposition concepts to obtain the overload magnitudes in the neighboring tows. Subsequently, high-fidelity finite element analysis on a representative uni-directional laminate model under different stress states is performed, and excellent overall agreement is observed between analytical and numerical predictions. The proposed method takes into account essential aspects such as transversely isotropic material properties, in-situ stress states and their effect on the interfacial frictional forces in the debonded interfaces, and thus provides a convenient way to evaluate the stress concentration factors in damaged filament-wound composites. In addition, this approach can be applied to yield auxiliary failure evaluation criteria for statistical strength prediction or finite element modeling of filament-wound composites or similar structures.

We investigate phenomena resulting from oblique impact of a right circular cylinder (RCC) and a steel sphere onto laminate targets consisting of multiple, cross-plied sheets of UHMWPE fiber in a polymer matrix at 10-20% mass fraction. Laminates were produced in both an un-pressed and hot-pressed condition using resin matrices differing widely in stiffness. Ballistic tests under both normal and 30°oblique impact, were conducted at multiple velocities to determine both V50 and the depth of penetration (DOP) below V50. Results were compared across all test conditions: RCC vs. sphere, oblique vs. normal, stiff vs. flexible matrix, and pressed vs. un-pressed. Pressed targets with 50% higher thickness and an intermediate stiffness matrix were also tested using an RCC. A numerical model was developed to simulate normal and oblique RCC impact into the various laminate targets. Simpler RCC oblique impact into a single yarn was also studied under varying sliding friction. The simulation model embodied a large array of coupled, mass-spring-damper yarn and matrix elements arranged to mimic the laminate geometric structure and mechanical properties, including both RCC and sheet on sheet, frictional slip. For a given target thickness, the impact velocity was the median experimental V50 for the set. Although not allowing yarn failure, the model still provided details on the evolution of layer-to-layer yarn displacements and strains, including their variability, allowing prediction of: (i) yarns most likely to fail on impact, (ii) target versions likely to have a higher V50 than others, and (iii) effects of obliquity on both V50 and DOP. Model predictions broadly agreed with experimental results, both demonstrating that a lower V50 under RCC oblique vs. normal impact strongly depends on target thickness where its increase reversed this trend while its reduction amplified it. Oblique RCC impact on a single yarn under sliding friction, reveals a strain concentration on the trailing side, however, this does not explain behavior seen in laminated targets as the full model uncovers far more intricate mechanisms.

Stress rupture (sometimes called creep-rupture) is a time-dependent failure mode occurring in unidirectional fiber composites under high tensile loads sustained over long times (e. g., many years), resulting in highly variable lifetimes and where failure has catastrophic consequences. Stress-rupture is of particular concern in such structures as composite overwrapped pressure vessels (COPVs), tension members in infrastructure applications (suspended roofs, post-tensioned bridge cables) and high angular velocity rotors (e.g., flywheels, centrifuges, and propellers). At the micromechanical level, stress rupture begins with the failure of some individual fibers at random flaws, followed by local load-transfer to neighboring intact fibers through shear stresses in the matrix. Over time, the matrix between the fibers creeps in shear, which causes lengthening of local fiber overload zones around previous fiber breaks, resulting in even more fiber breaks, and eventually, formation clusters of fiber breaks of various sizes, one of which eventually grows to a catastrophically unstable size. Most previous models are direct extension of classic stochastic breakdown models for a single fiber, and do not reflect the micromechanical detail, particularly in terms of the creep behavior of the matrix. These models may be adequate for interpreting experimental, composite stress rupture data under a constant load in service; however, they are of highly questionable accuracy under more complex loading profiles, especially ones that initially include a brief “proof test” at a “proof load” of up to 1.5 times the chosen service load. Such models typically predict an improved reliability for proof-test survivors that is higher than the reliability without such a proof test. In our previous work relevant to carbon fiber/epoxy composite structures we showed that damage occurs in the form of a large number of fiber breaks that would not otherwise occur, and in many important circumstances the net effect is reduced reliability over time, if the proof stress is too high. The current paper continues our previous work by revising the model for matrix creep to include non-linear creep whereby power-law creep behavior occurs not only in time but also in shear stress level and with differing exponents. This model, thus, admits two additional parameters, one determining the sensitivity of shear creep rate to shear stress level, and another that acts as a threshold shear stress level reminiscent of a yield stress in the plastic limit, which the model also admits. The new model predicts very similar behavior to that seen in the previous model under linear viscoelastic behavior of the matrix, except that it allows for a threshold shear stress. This threshold allows consideration of behavior under near plastic matrix yielding or even matrix shear failure, the consequence of which is a large increase in the length-scale of load transfer around fiber breaks, and thus, a significant reduction in composite strength and increase in variability. Derivations of length-scales resulting from non-linear matrix creep are provided as Appendices in the Supplementary Material.

An upper estimate for fiber/matrix modulus ratio in traditional fiber reinforced polymer (FRP) composites is 100. Matrices made from tough elastic gels can have modulus approaching kilopascals and increase this ratio to 107. We study how this extremely high modulus ratio affects the mechanical behavior of such fiber reinforced “soft” composites (FRSCs). We focus on unidirectional FRSCs with parallel fibers perfectly bonded to a soft elastic matrix. We show such composites exhibit the Mullins effect typically observed in rubbers and double network (DN) gels. We quantify size effect on mechanical properties by studying unidirectional composites consisting of finite length fibers. We determine the stress concentration factors (SCFs) for a cluster of fiber breaks in this geometry and show that there is a transition from equal load sharing (ELS) to local load sharing (LLS). We also determine the mean strength and work of extension assuming fibers obey Weibull statistics. We discuss the application of fracture mechanics to this emerging class of composites. We highlight similarities and differences between FRSCs and DN gels.

The projectiles have included RCC and spherical shapes, and viscoelastic interactions and slip between the both the projectiles and the target and between the target layers and fabric yarns have been modeled. Besides exploring new phenomena, these models have been useful for validating previous analytical models where various simplifying assumptions were necessary. Thus far, however, the numerical models have had the limitation (albeit a computational advantage) of assuming a symmetric framework whereby numerical solution of only one quarter of the projectile and target plane was necessary. Consequently, only normal (perpendicular) impact could be studied. To address this limitation, we have turned our attention to modeling projectile impact into the full target plane thus making it possible to treat oblique impact at an arbitrary angle of incidence. This would also make it possible to investigate boundary effects (clamped, free) where projectile impact occurs close to one boundary or near a corner. Making such generalizations has resulted in unanticipated challenges arising from modeling rapid projectile sliding on the target, and thus we have ‘backtracked’ to a simpler early version of the discretization model and begun with angled impact by a right circular cylinder (RCC) on a biaxial, plate-like structure. These challenges have led us to focus initially on scrutinizing features suspected of being possible numerical artifacts in the results, or, establishing through well-understood test cases that the features observed are indeed realistic and not artifacts. The most important findings come from a study varying the angle of projectile incidence as well as the coefficient of viscous sliding friction between the RCC projectile and fibrous target. It is shown that increasing the degree of viscous friction eventually leads to increasing fiber strain with angle of incidence up to some angle were a maximum is achieved. Generally, we find that oblique impact gives rise to important phenomena not captured by impact perpendicular to the target.

Composite Overwrapped Pressure Vessels (COPVs) are widely used in space launch vehicles and the automotive industry. Due to the potential for catastrophic loss caused by failure of COPVs, improved understanding of its mechanical and damage growth behavior is necessary to ensure its safe operation. In this paper, we described a developed mesoscopic model with the capability of modeling tows/yarns with inserted, in-ply interface elements and interface layers as resin rich zones. Furthermore, the model realizes 1) arbitrary winding angle build-up of overwrap plies, 2) controllable detailed dimensions, and 3) user defined material behavior for all regions. These features are critical for potential studies of in-situ and parametric overwrap failure in COPVs. In the present work, stress profiles and progressive failure around a single tow break are shown by case studies involving various interface and damage evolution properties. The investigation indicates that the strength and damage evolution law of interface elements both have direct effect on the stress concentration factors and damage evolution around a tow break/fracture. Resulting physical phenomena and stress overload profile are discussed as well.

We revisit the classic problem of determining stress concentrations on neighboring fibers to multiple, transversely-aligned fiber breaks in a planar, unidirectional fiber–matrix composite. Fibers are assumed to be perfectly bonded to the elastic matrix. Finite size effects on stress concentration are studied by varying the overall length of the composite relative to the characteristic load transfer length between broken and intact fibers. As an alternative to the discrete fiber and matrix framework in the classic analysis of Hedgepeth, and its extension by Hikami and Chou, the fiber stress distribution in the composite is obtained through continuum modeling of the composite as a highly anisotropic elastic plate, whereby the stresses and stress concentration factors at fiber locations in the discrete model are extracted in a closed form. For composites of finite length, the stress concentration factors determined using the continuum model compare favorably with numerical solution of the discrete shear-lag model. In the limit of a plate with infinitely long fibers, our stress concentration factors also agree well with the exact results of Hikami and Chou. For composites having a length less than the characteristic elastic load transfer length, and loaded under displacement boundary conditions, we show that local stress concentrations vanish irrespective of the size of the crack or the number of fiber breaks. This behavior becomes important when modeling and interpreting laboratory experiments on the mechanical behavior of recent soft composite specimens consisting of stiff fibers in an extremely compliant elastic matrix.

Stress rupture is an important failure phenomenon in composite overwrapped pressure vessels, which is highly unpredictable other than on a statistical basis. Even then, there are several statistical models, with varying bases in composite micromechanics and molecular failure mechanisms. Choosing among these models is not trivial, even when micromechanical details of the failure process are reasonably well appreciated, and one has available a reasonably large database of strength and lifetime data. As a result, there is little in the way of guidance to choose the most appropriate model. One important issue is that accurate predictions are desired at relatively low service loads compared to the strength, and low probabilities of failure that are far less, e.g., 10 −6 , than can be directly confirmed using the data itself. In essence, one needs a robust and accurate statistical model free of inconsistencies associated with such low stress levels and probabilities of failure. This paper performs an in-depth comparison of several current models, which have varying physical bases. The models compared differ in the number of parameters to be estimated from data. The results of this study, however, show that over a broad range of parameter values these models give surprisingly similar failure probability predictions. While practitioners may have a preference for one model over another, the basis for such a choice is not easily established, given the fidelity of typical data.

Abstract This research presents a novel approach using Natural Gas Clathrate Hydrates (NGH) for the production of natural gas from deep-water petroleum reserves. Liquefied Natural Gas (LNG) has limitations associated with its enormous capital cost (requiring significant production capacity), inherent hazards, and environmental impact. Storage and transport of natural gas must address the economic, safety and environmental requirements, as a whole, by designing the supply chain as an end-to-end process. The solution is the production of NGH at depths where the in situ pressure and temperature inherently favor the formation in situ, based on thermodynamics and kinetics. The remaining system is then created in order to manage these products. The engineering design, safety analysis and business case are assessed concurrently in order to optimize the system and meet all of the requirements. Containment technology, originating from space exploration, is adapted so that it will be strong, durable, flexible and lightweight. The processing rate and dimensional parameters are selected to mitigate system losses and energy dissipation during long distance ocean transport. The resulting proposed process is economically assessed to be significantly cheaper for the production of natural gas. It can be applied for small production rates and replace operations which flare gas byproducts. Theoretically, it could be scaled for the production of large natural gas fields. The proposed system presents an inherently safer system as the hazards inherent with compressed natural gas (CNG), LNG and pipeline are eliminated by design. There is no handling of compressed or cryogenically cooled fluids, and the transfer from platform to carrier is eliminated. Consistent with the industry direction to move crude oil processing to the bottom of the ocean, the new process can be managed autonomously and avoid the hazards associated with adverse weather conditions. A resulting by-product from this new process is significantly-desalinated water originating in the deep ocean away from man-made pollutants. It is proposed that this by-product can either be post-processed to yield potable water or used as-is, replacing consumption of local resources. The proposed system can be utilized to produce natural gas reserves which would otherwise be considered as stranded reserves. It can serve markets which are too small or ports are too shallow to handle the delivery of LNG from standard carriers. A Patent Cooperation Treaty application on this proposed method was filed and received a favorable preliminary review by the United States Patent and Trademark Office; claims were found to be novel and useful.

Stress rupture is a failure mode for unidirectional continuous fiber composites that is of increasing concern in composite overwrapped pressure vessels (COPVs). Stress rupture is a catastrophic failure mode with a large variance in failure times due to inherent randomness. Prediction of a composite structure's resistance to stress rupture is typically based on extensive testing at higher loads than used in service. The resulting datasets are then analyzed within the framework of a statistical model to determine an estimate of the probability that a composite structure will survive for a given lifetime under a particular load profile. For instance, in life safety applications the interest would be in ensuring a very small corresponding failure probability, such as one in a million. The statistical model parameters can be estimated in many ways, one of which is to use a maximum likelihood approach, but these estimates are of questionable value in the absence of a measure of their uncertainty. This paper presents a procedure for determining the uncertainty in such estimates, as well as removing any inherent bias. Details of the procedure are fully illustrated using data generated on model carbon/epoxy COPVs tested at the NASA White Sands Test Facility. This procedure is based in Monte-Carlo simulation of ‘typical’ datasets, which are then analyzed using the same method as the original dataset, thus giving a distribution of estimates. This distribution allows for quantification of uncertainty and bias. The Monte-Carlo procedure can also be used to evaluate experimental test design to determine the expected amount of uncertainty for a given test setup.

Century Fathom presents an innovative process to utilize clathrate hydrates for the production, storage and transportation of natural gas from off-shore energy reserves in deep ocean environments. The production scheme was developed by considering the preferred state of natural gas in the deep ocean and addressing the hazards associated with conventional techniques to transport natural gas. It also is designed to mitigate the significant shipping cost inherent with all methods. The resulting proposed scheme restrains transport in the hydrate form to the ocean and does not attempt to supply energy to the residential consumer. Instead; the target recipients are industrial operations. The resulting operational concept is intrinsically safer by design; environmentally sustainable and significantly cost-effective compared with currently proposed schemes for the use of natural gas hydrates and has the potential to be the optimal solution for new production of reserves; depending on the distance to shore and capacity of the petroleum reserve. A potential additional benefit is the byproduct of desalinated water.

Lightweight and ballistic resistance are significant parameters in the design of aircraft armors. Aircraft should not compromise the payload or its maneuverability due to the armors added to the system. In addition to this, the aircraft has to sustain high ballistic resistance under enemy fire. In the design of aircraft armors choosing light and high strength materials that respond to these demands ballistic impact resistant concepts are being developed. In this study numerical simulation of ballistic impact is carried out for the aircraft armors. The ballistic impact response of the fiber reinforced composite armor is computed using forward finite difference method. Cylindrical rigid projectile hitting the woven crimp composite fabric at an angle 90° is analyzed. The yarn segments between hinged joints at crossovers are modeled using discrete mass-spring-damper in pin-joint systems consisting of planar square lattices. After a certain time of impact; displacement of the fabric, change in the velocities and the failure in the material is computed and depicted graphically. The effect of crimp and slip viscosity on the ballistic performance of the fabric is examined and discussed.

Yarn shooting experiments were conducted to determine the ballistically-relevant, Young’s modulus and tensile strength of ultra-high molecular weight polyethylene (UHMWPE) fiber. Target specimens were Dyneema® SK76 yarns (1760 dtex), twisted to 40 turns/m, and initially tensioned to stresses ranging from 29 to 2200 MPa. Yarns were impacted, transversely, by two types of cylindrical steel projectiles at velocities ranging from 150 to 555 m/s: (i) a reverse-fired, fragment simulating projectile (FSP) where the flat rear face impacted the yarn rather than the beveled nose; and (ii) a ‘saddle-nosed projectile’ having a specially contoured nose imparting circular curvature in the region of impact, but opposite curvature transversely to prevent yarn slippage off the nose. Experimental data consisted of sequential photographic images of the progress of the triangular transverse wave, as well as tensile wave speed measured using spaced, piezo-electric sensors. Yarn Young’s modulus, calculated from the tensile wave-speed, varied from 133 GPa at minimal initial tension to 208 GPa at the highest initial tensions. However, varying projectile impact velocity, and thus, the strain jump on impact, had negligible effect on the modulus. Contrary to predictions from the classical Cole-Smith model for 1D yarn impact, the critical velocity for yarn failure differed significantly for the two projectile types, being 18% lower for the flat-faced, reversed FSP projectile compared to the saddle-nosed projectile, which converts to an apparent 25% difference in yarn strength. To explain this difference, a wave-propagation model was developed that incorporates tension wave collision under blunt impact by a flat-faced projectile, in contrast to outward wave propagation in the classical model. Agreement between experiment and model predictions was outstanding across a wide range of initial yarn tensions. However, plots of calculated failure stress versus yarn pre-tension stress resulted in apparent yarn strengths much lower than 3.4 GPa from quasi-static tension tests, although a plot of critical velocity versus initial tension did project to 3.4 GPa at zero velocity. This strength reduction (occurring also in aramid fibers) suggested that transverse fiber distortion and yarn compaction from a compressive shock wave under the projectile results in fiber-on-fiber interference in the emerging transverse wave front, causing a gradient in fiber tensile strains with depth, and strain concentration in fibers nearest the projectile face. A model was developed to illustrate the phenomenon.

A numerical code has been developed to model ballistic impact into a multilayered, nonwoven composite thin plate. The target consists of N unidirectional layers that are arranged in an alternating, cross-plied configuration. The projectile is either a hard sphere or a right circular cylinder (RCC). Motivation for such a model is the commercial development of soft and plate-like body armor whose building blocks are thin, nonwoven ultrahigh molecular weight (UHMWPE) fibrous sheets consisting of four bonded unidirectional layers with alternating (0o/90o/0o/90o) orientations. The matrix is typically a very flexible polyurethane just sufficient to fill the voids between the fibers, so about 17 percent by mass. Our interest is in modeling various features seen in ballistic impact into such structures. The code, which was written in MATLAB and runs on a PC, involves a forward, finite-difference (FD) scheme of numerical solution. Layers are sub-divided into multiple elastic ‘yarn’ elements with distributed masses. Elastic and viscous compression (normal) and shear (transverse) elements are used to model inter-layer yarn couplings. Sliding on the projectile follows a viscous friction law. Following impact, the evolution of tension and transverse waves in the layers are modeled, as well as projectile deceleration. Graphical results are presented for velocities, displacements, forces and strains versus time, which we interpret.

We present a numerical model that simulates ballistic impact of a rigid spherical projectile into a plain weave fabric target. The code, which runs in MATLAB on a highend PC or laptop, allows one to vary projectile radius, mass and velocity, as well as various fabric mechanical and geometric parameters. These include yarn spacing, stiffness and density, non-linear slip viscosity between fabric and projectile, non-linear viscoelastic contact stiffness (compressibility) at yarn crossovers together with viscoelastic shear stiffness resisting yarn sliding, and the angle of yarn crimp out-ofplane. The model accommodates a progressively spreading contact zone on the fabric, as occurs with an impacting spherical projectile. Fabric dimensions are chosen large enough to avoid any tension wave reflections at the boundary. After providing some details on the finite difference formulation of the model, we present a sampling of results obtained at various times after impact. The results include contour plots of out-of-plane displacements, out-of-plane and in-plane velocities and yarn strain distributions at a given time, as well as plots, over time, of maximum local strain level, and out-of-plane velocities and displacements in the contact region of the projectile. Some parameters will be varied to provide insight into the rich behavior possible.

Stress rupture is a sudden, stochastic failure mode that occurs in continuous unidirectional fiber composites and in particular composite overwrapped pressure vessels subject to long-term, steady loads. A common approach for modeling stress rupture is the probabilistic classic power-law model for material breakdown within a Weibull framework (CPL-W). This model includes a number of parameters, which need to be estimated from real data. These parameters may be estimated in a variety of different ways. This paper investigates how best to estimate the parameters of the CPL-W model given a set of experimental data for both composite strength and composite lifetime obtained at multiple stress levels. Eight different maximum likelihood estimation approaches are investigated regarding their differing errors of estimation. The accuracy of each method is estimated by repeated Monte Carlo simulations of specific instances of the CPL-W model typical of various carbon/epoxy and aramid/epoxy fiber composite systems; no actual experimental data are analyzed. One particular approach stands out as having the least estimation error while a commonly used approach does very poorly.

Fibers used in both soft and hard body armor have very high longitudinal tensile strength and stiffness, but differ drastically in their transverse mechanical properties. Glass and carbon fibers are stiff and brittle in the transverse direction and easily shatter upon projectile impact unless they are cushioned within a soft matrix to disperse the load. In contrast, aramid fibers (e.g., Kevlar 29 and Twaron) and ultra-high-molecular-weight polyethylene (UHMWPE) fibers (e.g., Dyneema and Spectra) have quasi-plastic transverse behavior, with a low yield strength, and thus tend to flatten upon projectile impact, yet retain much of their tensile load-carrying capability. Thus, these polymer fibers are especially suitable for ‘soft’ body armor consisting of stacked sheets or fabrics, whereas the former glass and carbon fibers are useful mainly when aligned in a strong polymer matrix to form a thick plate. In this work, we report on a study of the tensile mechanical properties of single UHMWPE fibers (i.e., single filaments) that have been transversely deformed from their original cylindrical shape to form thin flat micro-tapes with a width-to-thickness ratio of up to 60:1. The deformed, ribbon-like fibers show very high retention in fiber strength, though with increased variability resulting from locally induced defects. Because transverse deformation resulted in more than a factor of three increase in surface area per unit length, the stress transfer length necessary to fully load a fiber near a break was found also to decrease by the same factor, as the corresponding interfacial shear stress remained the same. A Weibull probability analysis revealed that the increase in variability in fiber strength was consistent with a more pronounced length effect. These changes in fiber strength properties were understood through an alteration of the crystalline domains within the fibers due to the extreme deformation.

This paper presents results from an in-house, finite-difference (FD) code to model ballistic impact into flexible composite body armor. The code, written in MATLAB, runs well on a high-end PC. The version we consider in this paper models the postimpact, pre-failure behavior of non-woven, cross-plied sheets made from UHMWPE yarns (e.g., DSM Dyneema®) in a polyurethane matrix at low volume fraction. This version can also model crimp imprinted in an up-down diamond pattern. Possible projectiles are right-circular cylinders (RCC) and spheres, though we focus on the former in this paper. We first review some assumptions regarding the element structure of the model where a single sheet is treated as a flat 2-D network of discrete, pin-jointed yarns and each node represents a non-sliding, yarn crossing point. We then adapt this model to apply also to a much thicker panel and then present various results, in graphical form, to illustrate the richness in behavior that the model captures.

In this study, interacting crack growth in an infinite plate is analyzed with new, fast and accurate Boundary Cracklet Method (BCM) developed by Phoenix and Yavuz. An interior crack is under consideration to watch its propagation because of cyclic loading which is very common for aerospace, naval and civil engineering structures. BCM is very useful to determine the overall stress field as well as stress intensity factors for crack tips and singular wedges at crack kinks. BCM uses integral equations expressed in terms of unknown edge dislocation distributions along crack lines. These distributions derive from an accurate representation of the crack opening displacements using power series basis terms obtained through wedge eigenvalue analysis, which leads to both polynomial and non-polynomial power series. The process is to choose terms of the series and their exponents such that the tractions on the crack faces are virtually zero compared to the far field loading. Applying the method leads to a set of linear algebraic equations to solve for the unknown weighting coefficients for the power series basis terms to make no use of numerical integrations unlike in other methods. Thats why, solution takes just a few seconds on a PC. A simple crack growth emanating from a triangular hole in an infinite plate is analyzed. The fatigue crack growth is assumed to follow Paris-Erdogan Law. The results are compared to those of other numerical methods. A parametric study is performed via graphs and tables to demonstrate the ability of BCM in analysis of fatigue crack growth.

In this work, numerical implementations of a new biaxial fabric of interlaced yarns model for ballistic impact into flexible body armor are presented. First the model is applied to a single-layer panel of polyethylene Dyneema® fibers in a matrix and impacted by various projectile geometries. We obtain the model results from a mass-spring, finite-difference model by modifying the code first developed at DSM. The results of the model are compared with experimental results and our previous biaxial model from firing 9 mm and FSP projectiles into Dyneema® panels. For both models, parametric studies are presented using 2-D and 3-D graphics. The new biaxial interlaced yarns model extends our previous biaxial and axisymmetric membrane model by addressing biaxial influences on material inflow to the impact cone. The new model is much more complete and produces velocity, strain and deformation histories for the full system up to perforation or projectile halting. It can also treat cases of mixing different fabric types, which can result in interference effects depending on the layer stacking order, and can treat multiple layers with air gaps in between, where we will show that such gaps can seriously degrade performance.