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#### Energy Research, Vol. 2, Issue 1, Mar 2018, Pages 24-40; DOI: 10.31058/j.er.2018.21003 10.31058/j.er.2018.21003

### Any Seal under Pressure Shall Have a Poisson's Deformability of 0.5 Ratios—also as a Statement of Local Revisions to The Special Issue of Xu's Sealing and Flowing Theory of Fluids

#### , Vol. 2, Issue 1, Mar 2018, Pages 24-40.

#### DOI: 10.31058/j.er.2018.21003

####
Xu Chang-Xiang ^{1*} ,
Zhang Xiao-Zhong ^{1}

Baoyi Group Co. Ltd., Wenzhou, China

#### Received: 2 January 2018; Accepted: 23 February 2018; Published: 8 March 2018

### Abstract

Any sealing ring shall be able to effectively instantly convert a loading pressure on it into its sealing stress orthogonal to the pressure. For example, any self-sealing ring for opposing faces shall be able to exactly instantly convert a fluid pressure on its cylindrical surface into its sealing stress on its end faces. Firstly, what can effectively instantly performs the conversion is either a rigid wedge or a closed liquid; secondly, a common substance, with a Poissons ratio ranging from 0 to 0.5, has both some solid behavior and some liquid behavior, and the substance whose Poissons ratio value more approaches 0.5 more has full liquid behavior and volumes incompressibility and is abler to effectively instantly finish the conversion; thirdly, Poissons ratio is the substances orthogonal strain ratio of its non-loading direction to its loading direction, and compensable; fourthly, a different substance has a different Poissons ratio lagging behind its stress under a different temperature and a different stress; and hence any cavity for sealing rings, no matter how great their Poissons ratio at room temperature is, shall have such a compensating ability to result in the ring therein having a Poisons deformability of 0.5 ratio on being under pressure as to effectively instantly finish the orthogonal conversion of a loading pressure into a sealing stress.

### Keywords

Sealing ring, Poissons ratio, Poissons deformability

### 1. Introduction

Any sealing ring shall be able to effectively instantly orthogonally convert a loading pressure on it into its sealing stress. For example, a pressure-tight ring for stem cylinders shall effectively instantly convert its live coaxial pressure from fasteners into its live radial sealing stress, and any self-sealing ring for opposing faces shall exactly instantly convert a live fluid pressure on its cylindrical surface into its live sealing stress on its end faces. The reason why a liquid can perform the orthogonal conversion of pressure is its softness and incompressibility;its softness results in it having a deforming and flowing power on being under pressure, and its incompressibility results in it having the same deformation and flowing power in all directions. A common substance with a Poisson's ratio ranging from 0 to 0.5 has both some solid behavior and some liquid behavior, and the substance whose Poisson's ratio value more approaches 0.5 more has full liquid behavior and volume's incompressibility [1], and is abler to effectively instantly finish the conversion. Therefore, the designing of a sealing ring is to ensure that it always has a Poisson's deformability of 0.5 ratio. However, all prior sealing arts have not yet known any dependence of sealing on Poisson's deformation.

Rubber is very soft and its Poisson's ratio value approaches 0.5 very much, and thus can be regarded as a full liquid that is so thick as to have a fixed shape at atmospheric temperature and pressure. Of course, a ring of infinitely thin wall tubes fully filled with liquid can be used to simulate a coreless rubber O-ring, the infinitely thin wall tube being used to represent the fixed shape of the liquid (see Fig.1a). Thus, first, from that the simulating metallic tubing ring shall have the same circumferential tensile strength as the simulated coreless rubber O-ring shall, it can be seen that:2πr_{u}δ_{k}R_{mk} = πr_{u}^{2}R_{mr} → 2πr_{u}kr_{u}R_{mk} = πr_{u}2R_{mr} → kR_{mk} = 0.5R_{mr};then, from the pressure withstanding equation pm =Rmδr of a common metallic thin wall tubing, it can be seen that the maximum pressure (pme) that a simulating metallic tubing ring for the same coreless rubber O-ring can withstand in its cavity not only changes with its maximum extruded arc radius (re), but also the product of the pressurepme and the arc radius re is also constant, or pmere ≡ Rmkδk = a constant → pmere = (Rmkk)ru = (0.5Rmr)ru; i.e. the maximum pressure withstandable by the simulating metallic tubingring after installed into its cavity:

(1a)

and the maximum pressure withstandable by the simulating metallic tubing ring before installed into its cavity:

or the compressed stress or the inner pressure of a coreless rubber O-ring

(1b)

where ru = cross-sectional radius or free extrusion arc radius of a coreless rubber O-ring whose entire outside surface is not under pressure,

re = extrusion arc radius of a coreless rubber O-ring through the maximum gap of its closed cavity = 0.5x the maximum gap of its closed cavity,

*r _{x}* = extrusion arc radius of the uncompressed surface of a coreless rubber O-ring whose partial outside surface is under pressure,

*δ _{k}* = kr

_{u}= wall thickness of simulating metallic tubing rings,

*R _{mk}* = material's maximum strength of simulating metallic tubing rings,

*R _{mr}* = material's maximum strength of rubber O-rings

*R _{m}* = material's maximum strength of a common metallic thin wall tubing,

*δ* = wall thickness of a common metallic thin wall tubing,

* r* = cross-sectional radius of a common metallic thin wall tubing;

and finally, it can be seen from equation (1b),

● that a coreless O-ring will have an inner pressure *p _{x}0.5* or an assembled stress

*S*equal to its material's maximum strength Rmr when its being assembled causes its extrusion arc radius

_{a}0.5*r*= 0.5

_{x}*r*, or that the compressed stress

_{u}*p*of a coreless O-ring at 50% squeeze just equals its material's maximum strength

_{x}*R*, and

_{mr}● that any sealing micro-stretch deformation that makes sealing surfaces of a coreless O-ring squeeze into leaking microdepressions and microchannels can happen only when they have a compressed stress *p _{x}* equal to their maximum strength Rmr and have no tensile resistance, because their sealing micro-stretch deformation making its

*r*≈ 0 will need a compressed stress

_{x}*p*≈ ∞ before they does not yet have a

_{x}*R*becoming null and still have some tensile resistance.

_{mr}There is a statement that rubber at room temperature is a liquid from a perspective of its molecular chain segments in local motion, and a solid from a perspective of its molecular chains in general stillness [2]. There is another statementthat rubber's molecular chains tend to line up straight in its stretch direction when stretched, which is like a crystallizing process causing it to have more solid behavior, and tend to become more wound and tangled when compressed, whichis like a crystallization-reversing process causing it to have more liquid behavior [3]. There is also another statement that material's Poisson's ratio before yielding will reduce to zero with its increase of stretching load, and, after yielding andbefore any strengthening, will gain to 0.5 with its increase of load [4,5]. Therefore, a rubber O-ring unassembled into its cavity, as shown in Fig.1b, can be regarded as a liquid ring with some concentric circular elastic fibers on its cross-section; a rubber O-ring assembled into a square cavity, as shown in Fig.1c, can be regarded as a liquid ring with some concentric square elastic fibers on its cross-section, but its four corners have been stretched to become stronger andable to withstand a higher pressure; and a rubber O-ring fully compressed in its cavity, as shown in Fig.1d, can be regarded as a liquid ring with some concentrated arc elastic fibers only in its extrusion corners situated outside its cavityincircle (Φ2a), or its incircle region has been so fully compressed as to have more liquid behavior that can exactly equally transmit pressure to each directions and its extrusion corner regions have been so fully stretched as to have moresolid behavior that can withstand a higher extrusion pressure.

Because a rubber O-ring is restrained from producing any Poisson's deformation when compressed by fluid in its cavity, it can be seen from constrained modulus:

(2)

= ∞ (when Poisson's ratio ν = 0.5), or

= E (when Poisson's ratio ν = 0.0)

that the ring material stretched in the extrusion corner and having a Poisson's ratio closer to 0 [3, 4] has such a tensile modulus close to Young's modulus E as to have some possible occurrence of stretched breakage, and the ring materialcompressed in the cavity incircle and having a Poisson's ratio closer to 0.5 [3, 4] has such a compressive modulus close to infinity as to completely have no possible occurrence of compressed breakage, or the ring material constrainedly compressedup to having full liquid behavior and no tensile resistance can gradually restore its tensile strength with its being decompressed or being re-stretched.

Figure 1. Deformation of coreless rubber O-rings in square cavities.

Parker Corporation used a butyl O-ring of 4.850 inch I.D. as face seals (see Fig. 2 left) to do their leak tests at 15%, 30% and 50% squeezes under a pressure of 0.41 MPa at a temperature of 25 °C, and achieved a squeeze-leak ratecurve shown in Fig. 2 (right) (note: the leak rate shall be scientifically called “fluid current” according to “electric current”). Parker's leak rate curve shows that grease coatings can reduce the leakage of O-rings at squeezes less than 50%,but cannot reduce the leakage of an O-ring at 50% squeeze [6].

Since the compressed stress of a coreless O-ring at 50% squeeze just makes its sealing surfaces become a surface with no tensile resistance, of course,

● a sealing surface from a squeeze less than 50% and still with tensile resistance cannot directly squeeze into and block up all leaking microdepressions and microchannels by any high inner pressure, and can only squeeze some grease coatings partially into and partially block up those leaking microdepressions and microchannels, because a lower and higher inner pressure respectively created by a smaller and greater squeeze can only respectively produce a smaller and greater squeezing pressure on the grease coatings by sealing surfaces and respectively less and more partially reduce the leakage,

● a sealing surface from a squeeze equal to 50% and with no tensile resistance also needs an inner additional pressure increment not less than atmospheric pressure (0.1 MPa) to be able to squeeze in a micro-stretching way into allleaking microdepressions and microchannels and fully extrude air therein, whereas, at the moment, the Parker's test pressure of 0.41 MPa is far less than its inner pressure equal to its material's maximum strength and cannot at allcause its inner pressure to increase and make it produce any sealing micro-stretch deformation;

i.e. Parker have inadvertently in advance proved equation (1b) correct by experiments.

Figure 2. Parker's tests:leak rate and squeeze percent of rubber O-rings [6].

A fully leak-free connection results from loading a sealing contact layer first up to its fully deformed contact and then up to its fully tight contact, or needs first to create a deformed contact that can fully seat a sealing surface into irregularities on a sealed surface and then to create a tight contact that can fully resist a strongest seepage through the tight contact. The effective elastic modulus Ec of a sealing contact layer or surface of a constrained sealing ring under compression is a ring's capability index to resist a sealing deformation and a tight contact, while the effective elastic modulus *E _{s}* of the substrate of a sealing contact layer or surface is a ring's power index to achieve and maintain the sealing deformation and the tight contact. Hence, the difficulty for a sealing ring to achieve or maintain its fully leak-free contact is determined by material's effective elastic modulus ratio m1, m1 = Ec/Es, of its contact layer or surface to its substrate, and a rubber O-ring whose m1max = 1 because Ec =

*E*

_{s}_{ }is the most difficult to achieve or maintain a fully leak-free contact [12].

*Fig. 2 Parker's tests: leak rate and squeeze percent of rubber O-rings [6].*

Therefore, theories and experiments both have proved that the initial sealing stress Si = (Rm + 0.2) MPa required by a seal at atmospheric temperature and pressure, where Rm = the material's maximum tensile strength of sealing surfaces =the stress needed to compress an initial sealing surface into a semi-finished sealing surface with no tensile resistance in position, and 0.2 = the maximum additional stress needed to make a semi-finished sealing surface achieve and maintain itsfully tight contact at atmospheric temperature and pressure = the maximum value of seal's minimum necessary sealing stress y, because it needs an additional stress of 0.1 MPa to make each semi-finished sealing surface made of elastomers(rubber) and materials without any tensile resistance achieve its full sealing deformation at atmospheric temperature and pressure, but the maintaining of former sealing deformation needs another additional stress of 0.1 MPa, and, of the latter,does not need any, whereas both need an additional stress of 0.1 MPa to achieve each full tight contact that can resist the atmospheric seepage, i.e. the minimum necessary sealing stress y for elastomers (rubber) equals 0.2 MPa, and formaterials without any tensile strength, 0.1 MPa, or, as a whole, seal's minimum necessary sealing stress y ≤ 0.2 MPa.

However, ASME have mistaken the stress needed to assemble or work a seal in position up to no visually detectable leakage, which is equivalent to the aforesaid initial sealing stress Si, for the minimum necessary sealing stress required by the seal, and thus have mistaken a rubber O-ring seal that is the most difficult to achieve or maintain its tight connection for the seal that is the easiest to achieve or maintain that connection [7].

From the above-mentioned, it can be seen that the designing of seals is to ensure that a sealing element under pressure achieves a full deformation depending on Poisson's effect, and different at all from the designing of common machineries that is to ensure that the deformation or stress of a structural element under loads shall not exceed its material's permissible maximum. Hence, the prior sealing art whose seal-designing concept has been away from a Poisson's effect cannot provide any scientific seal for the present world. In fact, except for this, prior arts have not known the physical quantity of leak resistance (tightness) and any scientific means for measuring, detecting and controlling it. It is imaginable that the prior technical standards of seals cannot be scientific at all.

### 2. Compensation for Poisson's Deformability

*2.1. Material's Poisson's Ratio and Its Change*

Material's Poisson's ratio ν is the orthogonal strain ratio of its non-loading direction to its loading direction, and hence, from the following three aspects [10]:

● creep is solid's permanent strains lagging behind its stress at a homologous temperature higher than 0.5 degree (note:material's homologous temperature is the ratio of its absolute temperature to its melting absolute temperature),

● creep strains include a short or transient deformation component and a long viscous flow component that lag behind their stress (see Figure 3 ), and

● creep strain values increase with stress σ and temperature T (see Figure 4 ),

or from the fact that the strain of sealing materials will not only change with temperature and stress but also lag behind their stress when their homologous temperature is higher than 0.5°, it can be seen that Poisson's ratio of sealing materials always not only changes with their temperature and loading pressure but also lags behind their loading pressure.

Total strain | Sudden strain | Transient strain | Viscous strain |

Figure 3. Components and superposition of creep strains [10].

(a) Strains increase with stress σ | (b) Strains increase with temperature T |

Figure 4. Creep strain values increase with stress σ and temperature T [10].

As shown in Figure 5 , Poisson's ratio of relaxing and creeping materials not only increases with and lags behind their loading stress almost the same as the relaxation and the creep but also approaches 0.5 with increase of their relaxation strength Δ [11]. Because the relaxation strength Δ = (E0 - E∞)/E∞ = the reduction in Younger's modulus ÷ the final Younger's modulus after the relaxation ends and the Younger's modulus will change with the loading stress only after materials yield, material's Poisson's ratio value after it yields increases with and lags behind loading pressure to 0.5.

From an expression of Poisson's ratio:

ν = (3K-2G)/(6K+2G)

= (3K/G–2)/(6K/G+2) → -1 (when K/G → 0)

= (3-2G/K)/(6+2G/K) → 0.5 (when G/K → 0 or when K → ∞ and G → 0)

it can be seen that Poisson's ratio of a substance at room temperature is the function of ratio of its shear modulus G to its bulk modulus K (see Figure 6 ) [1];or that the softer and also the more incompressible the substance, the closer to liquid Poisson's ratio value its Poisson's ratio;or that substance's Poisson's ratio is the index of its liquid behavior — the closer to 0.5 the substance's Poisson's ratio, the stronger both its liquid behavior and its incompressibility, and even liquids can be regarded as a substance with its Poisson's ratio equal to 0.5, and, solids, one with its Poisson's ratio less than to 0.5.

Δ = (E0 - E∞)/E∞ = the reduction in Younger's modulus ÷ the final Younger's modulus after the relaxation ends, and a greater Δ represents a greater load because the Younger's modulus E of a material after yield will change with its load. |

Figur* e 5*. Poisson's ratio of relaxing and creeping materials changes with relaxation strength Δ [11].

Figure 6. Substance's Poisson's ratio and its ratio of bulk modulus K to shear modulus G [1].

Because substance's homologous temperature is the ratio of its absolute temperature to its melting absolute temperature, any substance at homologous temperatures 1° and 0° is respectively at its softest liquid state and its hardest solid state, and, at homologous temperatures more and less than 0.5°, respectively has more liquid and solid behavior. Because the homologous temperature of different substances at room temperature is different from each other, a different substance at room temperature, as shown in Fig. 6, has a different Poisson's ratio value and thus a different liquid behavior. Because a substance at a higher temperature will become closer to its softest liquid state, thus, as shown in Fig. 4b, it has a more strain under a constant stress at a higher temperature T.

A Japanese test has proved that Poisson's ratio of isotactic-polypropylene decreases to 0 with its being stretched before its yielding (I-II), but first increases to 0.5 with its further being stretched after its yielding and before its stiffening (III), and then decreases to 0 with its furthest being stretched after its stiffening and before its fracturing (IV) (see Figure 7 ) [4]. A Chinese test has proved that Poisson's ratios of steel Q235 annealed respectively at 1000 °C and 200 °C both approach 0.5 after yield (see Figure 8 ) [5].

Figure 7. IPP's Change in Poisson's Ratio [4].

Q235: ReH ≥ 235 MPa, Rm = 370~400 MPa, almost no change in ReH after annealed at less than 400 °C, and a reduction by 60% or so in ReH after annealed at 1000 °C. | |

Figure 8. Q235 Steel's change in Poisson's Ratio [5]. | |

To sum up, Poisson's ratio value of any metallic and non-metallic material before/after yield not only changes with its temperature, load and loading time but also approaches 0.5 with its temperature rise to its melting point and with its load rise to its tensile limit. Therefore, all designers for a seal shall know if the material's change in Poisson's ratio value has any effect on leakage, and also, if having some effect, shall know how to overcome the effect.

*2.2. Angles Used to Compensate for Poisson's Deformability*

The essence of self-energized seals is causing a sealing ring to exactly orthogonally transmit a fluid pressure, for example, causing a face sealing ring 02 in Figure 9 to exactly convert a fluid pressure *p* on its inner cylinder into a sealing stress S on its end face, and hence any self-sealing ring shall have full liquid behavior.

Since substance's Poisson's ratio is the index of its liquid behavior or the closer to 0.5 the substance's Poisson's ratio, the stronger both its liquid behavior and its incompressibility and the greater its effective instant orthogonal pressure-transmitting capability (see clause 2.1), any pliable solid material, no matter how smaller than 0.5 its Poisson's ratio is, can be used for a self-sealing ring by compensating for its orthogonal strain ratio up to 0.5 by an angle *θ _{l} *compensating for material's liquid behavior (see Figure 10 ).

A Designed end B Fully flat end 02 Self-sealing ring | A Designed end B Fully flat end 02 Self-sealing ring | A Designed end B Fully flat end 02 Self-sealing ring |

Figure 9. Self-sealing essence. | Figure 10. Compensation for orthogonal deformation. | Figure 11. Offset toorthogonal deformation. |

As shown in Figure 11 , any self-sealing ring 02 with some radial clearance (C) between it and its bonding wall (Φd2′) will be away from its previous tight contact at a certain pressure because of a Poisson's deformation decrease in its height accompanied by a deformation increase in its circumference caused by the radial clearance under the pressure. Because some component-manufactured errors and some material's thermal expansion coefficient difference both will cause a sealing ring a possible radial contact clearance, any self-sealing ring with full liquid behavior also needs an angle *θc* used to in time offset its orthogonal deformation decrease proportional to its Poisson's ratio value and caused by its possible radial clearance at a pressure.

Since it is necessary for a self-sealing ring whose Poisson's ratio ranges from 0 to 0.5 to be compensated for its liquid behavior (see Fig. 10) and for its radial contact (see Fig. 11), and both are to increase its deformation in height under a fluid pressure, thus, if the compensation for the liquid behavior is done from 0 to 0.5 and the compensation for the contact is done from 0.5 to 0, any self-sealing ring will need or have one angle *θ _{l}* fully compensating for its liquid behavior and one angle

*θc*fully compensating for its contact whose magnitudes are both determined by the Poisson's ratio limit 0.5. The two full compensation angles can be unifiedly called an essential Poisson's deformation compensation or offset angle

*θe*= arctg(

*υh/r*) = arctg(h/d) [12], where υ = 0.5 (Poisson's ratio limit),

*h/d*= height/inner diameter of self-sealing rings.

If the essential Poisson's deformation compensation or offset angle *θe* of a self-sealing ring has some wedging function, the function can also cause the ring to more gain some useful sealing deformation. And hence, however great the angle *θe* is, what it changes is only the time for the ring material to reach its Poisson's ratio limit 0.5 or 0 but never the magnitude of the two limits, or at most compensates the ring material for its orthogonal deformation ratio from 0 to 0.5 or offsets its orthogonal deformation ratio from 0.5 to 0 as soon as possible, or at most eliminates the lagging of its orthogonal deformation ratio behind its final value. Therefore, any material of self-sealing rings, however great its Poisson's ratio is, can use one angle *θx* greater than angle θe as its Poisson's deformation compensation angle θl and offset angle *θc*.

Since the essential Poisson's deformation compensation angle *θe* for a self-sealing ring can compensates for its Poisson's ratio from 0 to 0.5, an actual Poisson's deformation compensation angle *θx* greater than angle θe, no matter how the Poisson's ratio changes with the material's temperature and pressure, can ensure that any sealing ring will have a Poisson's deformability of 0.5 ratio on moving under a pressure;i.e. the designing of a sealing ring only needs considering if it has a Poisson's deformation compensation angle greater than angle θe, not needing at all studying and investigating how its Poisson's ratio value changes with its materials and their temperature and pressure.

The essence of the Poisson's deformation compensation angle θx for a sealing ring is to ensure that its cavity can result in it moving from a great space to a small space on being under a loading pressure, enabling each of its annular layers with orwithout Poisson's deformability of 0.5 ratio in the great space to respectively have a Poisson's deformability of 0.5 ratio the moment it is compressed into the small space, and being equivalent to making it have a Poisson's deformability of 0.5 ratioat all times. Actually, the sealing ring compressed in that cavity by a loading pressure is so fully constrained by sealed surfaces that it cannot produce any Poisson's deformation or that it must have some Poisson's deformation that could occur. If thePoisson's deformation that could occur is regarded as a deformation compressed from the small space back into the great space, it is not hard to understand that the sealing ring is also so incompressible in the great space that its sealing surfaces,whether situated in the great space or in the small space, can produce a sealing stress equal to the loading pressure. Therefore, the first criterion for seal's design conformity is to see if its sealing ring has such a space whose angle is not less thanangle θe = arctg(hd) as to cause the ring to move from the great to the small on being under pressure.

### 3. Application of Sealing Grease or Sealant

A thick object compressed by two stiff surfaces, as shown in Figure 12 , has two thin contact layers and a thick middle. The contact layers are under fully constrained compression and cannot produce any Poisson's deformation. The middle is not under fully constrained compression and can produce some Poisson's bulge deformation. Hence, a film of sealing grease or sealant that is so compressed by two sealed stiff surfaces as to have no material that can be further extruded out is an object that has no bulge middle and is entirely under fully constrained compression [13].

Figure 12. Constrained and non-constrained compression [13].

Since the material constrainedly compressed up to having neither any tensile resistance nor any macro strain not only has no compressed breakage but also has a Poisson's ratio value close to 0.5 [4,5], it can be seen from equation (2) thatthe film of sealing grease or sealant so compressed by two sealed stiff surfaces as to have no material that can be further extruded out has such an elastic modulus close to infinity as to make its maximum permissible pressure only limited bythe strength of its compressing stiff surfaces.

Therefore, the tightness and the permissible working pressure of sealing grease or sealant entirely depend on its being fully circumferentially uniformly compressed that results in it forming an almost infinitely strong film. The means of ensuring two opposing faces of bolted joints a full circumferential uniform compression is adding a peripheral macrosawtooth ring whose sawtooth top is flush with the opposing face at the designed end [14], and the means of ensuring two opposing faces of threaded joints a full circumferential uniform compression, ensuring the perpendicularity of the opposing face to its thread axis. The means of ensuring that the two kinds of opposing faces a full strong compression is ensuring that their joints both can pass a burst pressure test at four times the permissible working pressure.

Sealing grease or sealant between engaged threads under a fluid pressure is also constrainedly compressed, and thus also can ensure that its joints can withstand a fluid pressure close to infinity according to equation (1b) as long as it isensured that it has no extrusion gap. The means of ensuring that it has no extrusion gap is using the mating of internal parallel threads Rp and external taper threads R with no incomplete thread in the last engagement thread in order toensure that the first engaged threads are multi-turns of fastening threads without any engagement at their crests and roots and that the last engaged threads is a turn of jointing threads with a full engagement at their crests and roots,respectively used to meet the powering and the deforming requirements for realizing and sustaining a threaded joint with no sealant's extrusion gap [14].

Because the various means of ensuring the sealing grease or sealant a circumferential uniform compression are very common and cheap, grease/sealant seals are the cheapest, simplest and most reliable seal, and should have been widely applied.

### 4. Application of O-Ring Seals

Because rubber O-rings in a rectangular cavity anytime has only a constant surface area exposed to pressure fluid, there is no method for using the same cavity to result both in their sealing force being greater than their unsealing forcefrom pressure fluid and in them being installed up to their initial tight state, or there is no method for using the same cavity to result both in them, as shown in Fig. 13a, having ra ≥ 0.75ru in service and in them, as shown in Fig. 13b, beinginstalled up to ra ≤ 0.5ru [12].

If the cavity of coreless rubber O-rings, as shown in Figure 14 , is a round wall cavity based on a cavity-polygon's incircle, they will have only one fluid compression corner and one free extrusion corner in the cavity. Hence, a weak fluid pressure can cause the O-ring to move from a small space to a great space with no resistance in its fluid compression corner and leave its round wall momentarily, and thus cause its surface area and sealing force actuated by fluid to spurt, making it move from a great space to a small space in its fluid extrusion corner and up to its fully deformed contact and its fully tight contact at the same time.

(a) As a working seal, ra ≥ 0.75ru | (b) As an initial seal, ra ≤ 0.5ru |

Figure 13. Pitiful rectangular cavities only have a constant ra.

Figure 14. A round wall cavity of O-ring seals.

The prior sealing art has not paid attention to the following basic aspects related to coreless rubber O-ring seals:

● rubber materials have a huge friction coefficient equal to 1 to 4, a rubber O-ring in standardized rectangular cavities has a huge frictional contact area equal to 3 to 4 times its surface area exposed to or actuated by pressure fluid, and it needs a very high fluid pressure to be able to cause it to start its sealing deformation and movement from a great space to a small space;

● on the one hand the maximum pressure withstandable by a thin wall tube is inversely proportional to the tube wall radius, and on the other hand the inner and outer pressure fields of rubber O-rings in cavities under a pressure cannot have any step change region, and thus any rubber O-ring in cavities can only adapt to its inner and outer pressures with an isodiametric arc surface tangent to its cavity wall;

● rubber O-rings under constrained compression in cavities is such an object that can flow along their cavity like a liquid but has an infinite compressive resistance and a limited tensile resistance that their withstandable pressure limit is determined according to equation (1b) by their stretch deformation in their extrusion corner;

and, hence, neither knows how a rubber O-ring deforms in its cavity, such as does not realize that its surface exposed to fluid shall be an isodiametric arc [15-17], nor knows how to design a rubber O-ring seal, such as does not know how tocalculate its withstandable pressure.

Since the coreless rubber O-ring in round wall cavities has only one fluid compression corner that can cause both its flow resistance to be up to zero and its flowing power to progressively increase, and one free extrusion corner that cancause its sealing stress to progressively increase, or has no other unnecessary flow that consumes power, any initial fluid pressure that can seep through its sealing contact surface and try to leak, undoubtedly, can momentarily cause it toleave its cavity round wall, form a structure shown in Fig. 14 right, and start its moving and sealing process from a great space to a small space. Because a coreless rubber O-ring under a weak fluid pressure still has definite solid behavior,the structure shown in Fig. 14 can be regarded as a great-small end piston that can amplify the weak fluid pressure on the great end face into the sealing stress not less than the material's maximum strength on the contact surface at thesmall end, thus causing the coreless rubber O-ring in round wall cavities not yet again to need achieving its initial tight contact by installation.

Therefore, it is undisputed that a circle-based system of O-ring seals based on a cavity polygon incircle is the most scientific. See references 12 and 14 for details.

### 5. Application of Rectangular Ring Seals

As shown in Figure 15 , the arc bonding wall (01) in designed ends (A) of opposing faces is so diametrally inward bulged as to be capable of providing both an elastic deformation rotation fulcrum with r as the rotating radius and two full Poisson's deformation compensation and offset angles (θx) for a rectangular self-sealing ring (02) that touches its cavity wall at the middle because of Poisson's deformation during installation, and hence any fluid pressure can cause the ring (02) to work as two rigid self-sealing wedges before it yields and as two pliable self-sealing lumps after it yields or can cause the ring to be up to its full tight contact as long as the ring can first deform in the joint system under a fluid pressure.

A Designed end B Fully flat end 01 Arc bonding wall 02 Self-sealing ring 04 Microsawtooth ring r = Elastic deformation rotation radius causing ring ends a wedging action h = Depth of ring cavity h1 = Height between ridge bottoms h2 = Height between ridge tops do = Nominal external diameter of connected pipes b = Nominal wall thickness of ring k >1 + bd1r to ensure Fs > Fu. |

Figure 15. Rectangular ring seals.

The self-sealing ring (02) is designed to outwards acuminate or dwindle its both end walls according to “height (h1) of ring between two weakening ridge bottoms < depth (h) of ring cavity < height (h2) of ring between two weakeningridge tops”, so that any fasteners that can cause the ring ends to be crushed cannot have any chance to cause the ring body to yield, thus absolutely ensuring that the ring's sealing difficulty factor (m1) is less than one; whereas any end partcompressed into a layer of thin films also has a compressive strength close to infinity, thus absolutely ensuring that the ring ends have no compressed breakage. Therefore, any bolted joint appropriately designed according to “total fastenertensile capacity/area > pipe tensile capacity/area > ring body section area > ring end contact area” can ensure that:

a. fasteners, being stronger than the connected pipe in tensile capacity, have a strength condition that can cause both the ring ends a full plastic deformation and the ring body a full elastic deformation, and

b. a deformation that first and more happens during assembly and service is the ring's sealing deformation but not any other deformation of the other components,

so that there will never be any failure caused by the seal in any bolted joint that selectively uses one kind, two kinds or the three kinds of sealants, microsawtooth rings (04) and O-rings or rectangular rings (02) as its sealing element(s)and also ensures it/them a full circumferential uniform compression when the ring is so designed in accordance with k > 1 + bd1r as to ensure “its sealing actuation force (Fs) > its unsealing actuation force (Fu)”.

In addition, rectangular rings can be made with a material similar to pressure vessel and not limited by any working temperature and pressure, any thermal expansion coefficient, any corrosion resistance and any manufacturing technology of materials, and hence the rectangular ring seal can be a most ideal face seal. See reference 12 for details.

### 6. Application of Triangular Ring Seals

Rules for designing an equilateral triangle ring seal used as indispensable pressure-tight seals between moving rods or shafts and their housing, as shown in Figs.16 and 17, are first, by means of wedging function of its hard gland 03a, to amplify its original axial clamping force 2f into a compression force 4f/√3 to its soft equilateral triangle ring 04 on stem 02 and ensure that its soft ring is so compressed from a great room to a small room as to be able to pass a pressure exactly to each different direction;then to cut off its off-stem corner to give its cavity an opening or an axial compressing allowance to its soft ring;and last, by means of an anti-extrusion ring, such as a metallic C-ring 05a, without axial resistance, to close the opening to provide a full support for the sealing deformation of its soft ring compressed in its cavity.

The maximum permissible working pressure (pmr) of soft triangular rings without an anti-extrusion C-ring is equal to the maximum pressure withstandable by their incircle-ring-simulating metallic O-ring thin wall tubing in the cavity, orpmr = 0.5Rmrru/rc, where *rc* = their extrusion arc radius at the off-stem corner, ru = the incircle radiusof their fundamental triangles and *R _{mr}* = their material's maximum tensile strength (see clause 1). The maximum permissible working pressure (pmrc) of a soft triangular ring with an anti-extrusion metallic C-ring 05a is the sum of the maximum pressure withstandable respectively by the soft ring without the anti-extrusion C-ring and by the anti-extrusion C-ring, or pmrc = 0.5Rmrru/rc + Rmcδc/rc, where

*δ*= the wall thickness of anti-extrusion metallic C-rings and

_{c}*R*= the tensile strength of anti-extrusion metallic C-rings.

_{mc}01 Housing 02 Stem 03a Hard gland 03b Bonnet 04 Soft sealing ring

05a Anti-extrusion C-ring 05b Single turn anti-extrusion ring

Figure 16. Small sizes of rising stem seals for valves.

Figure 17. Large sizes of rising stem seals for valves.

### 7. Conclusions

Achieving an initial tight joint needs loading its sealing surface to the extent of no tensile resistance or to the extent that its sealing stress is not less than the maximum strength of its sealing surface material, and hence it is necessary to cause the sealing element to be compressed either into a thin film or in a closed cavity, or ensure that the sealing element has no Poisson's deformation and an enough extrusion strength under compression, in order to ensure that sealing element has no possible occurrence of compressed and stretched breakage under its maximum working pressure. Maintaining the initial tight joint needs further loading its sealing surface or sealing element to the extent that its sealing stress is not less than the fluid pressure. Achieving and maintaining an initial tight joint often require that its sealing element can orthogonally pass a pressure or can effectively convert its loading pressure into its sealing stress perpendicular to the pressure.

Firstly, Poisson's ratio is the index of substance's liquid behavior, and the closer to 0.5 the substance's Poisson's ratio, it more has full liquid behavior and incompressibility and is abler to effectively instantly perform an orthogonal conversion of pressures. Secondly, material's Poisson's ratio not only changes with its temperature and loading pressure but also lags behind its loading pressure. Thirdly, both the fluid pressure and the sealing stress needed by moving stems constantly change, or are live. Therefore, achieving and maintaining a fully tight joint must ensure that its sealing element has a Poisson's deformability of 0.5 ratio at all times in order to be able to effectively instantly orthogonally convert a loading pressure into a sealing stress.

Because Poisson's ratio is substance's orthogonal strain ratio of its non-loading direction to its loading direction, if cavities have such a space whose angle is not less than angle θe = arctg(hd), where hd = height/inner diameter of thesealing ring, as to cause a sealing ring therein to move from the great space to the small space under pressure, then each annular layer, with or without Poisson's deformability of 0.5 ratio, of the sealing ring in its great space will respectivelyhave a Poisson's deformability of 0.5 ratio the moment the ring is compressed into its small space, being equivalent to making the ring have a Poisson's deformability of 0.5 ratio at all times.

Because any sealing element with a Poisson's deformability of 0.5 ratio in service at all times is constrainedly compressed and has no other macro tensile and compressive strain except for some macro tensile strains in its extrusion corner,its maximum working pressure is determined by its maximum tensile capacity in its extrusion corner, and it has fully been proved by the history of high pressure applications of rubber O-ring seals.

### Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this article.

### Copyright

© 2017 by the authors. Licensee International Technology and Science Press Limited. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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