Project 4
The Breathing Sphere: Dilatant Compression as Impulsed Expansion at T = 0
Reynolds BEng
Abstract
Project 2 wrote an extra hoop term. Project 3 put that term on Reynolds’ misfit surface and trapped it as water-stretch stitched into a shell. This paper writes the time of the pulse that created the stretch.
A constrained pack cannot take compression without opening volume. The opening is dilatancy. Dilatancy is the expansion response to the impulse, delayed by the rebuild of the contact patch. From the residual
0i2 = r2 m unit package k.g.s2
the return after each pulse is
m4 / k.g.s2
The delay is the s2. The closed body therefore breathes.
Two unit columns are kept distinct until a converter exists. Strength of water is written N m s2. Wall stress is written N/m2. Their balance is a three-way square product: impulse, the area the impulse makes, and the stress on that area. The area is the contact patch. Under stretch the patch thickens. That thickening is auxetic. It is measured as a negative Poisson ratio. Water is the ordinary material in which this already occurs.
The piTensor holds both readings of one breath at 0i2. T = 0 is the instant before either reading is discarded.
1. What is already on the table
Project 2:
σ_θ = pr / t + σ_R
Project 3: σ_R is trapped water-stretch on a singular surface of misfit, frozen by hydration into a stiff, brittle shell.
Neither paper wrote the duration between clamp and return.
2. Two columns of units
Rest Time column
residual geometric twist, Rest Time mass-energy reading
0i2 = r2 m, unit package k.g.s2
strength of water, residual F
N m s2
return after the pulse
m4 / k.g.s2
delay of the pulse
the s2 inside k.g.s2
celerity c
m/s
moment duration h
N m s2
Engineering column
σ_θ and σ_R
N/m2
shear path
m
judder
m/s
converter
Z_R = m4 s2
σ_R = F / Z_R
F in N m s2
σ_R in N/m2
These columns are not yet the same SI chain. Z_R is the converter that would take N m s2 on a path of length m and write N/m2. Until Z_R is written, the columns stand side by side.
The balance already visible is this. N m s2 and N/m2 differ by a four-power of length:
(N m s2) / (N/m2) = m4 s2
That is why the residual r2 m, once given duration s2, returns as an m4 spring. The impulse does not stress a ready-made area. It makes the area it stresses. The area is the contact patch.
3. Dilatant compression is impulsed expansion
Slip is instant. Clamp is instant. Rebuild of the two-dimensional patch is not.
Sequence:
compressive pulse
→ six-kite ring cannot reseat in the old plan area
→ volume opens
→ stretch is stored
→ return arrives as m4 / k.g.s2
The return is after the pulse. That delay is the extra time element. The sphere is seen to breathe: clamp, open, push back, clamp.
4. The contact patch makes its own area
The three-way product is impulse, area, stress.
The impulse is the residual force package, N m s2.
The area is not imported. Stretch thickens the patch. The patch becomes the area that carries the stress.
The stress is the engineering reading, N/m2.
Thickening under stretch is auxetic. Lateral dimension increases as the patch is pulled. That is a negative Poisson ratio, already the language used for living dilatancy. Water does this at the contact. Cement later occupies the extra length and stops the return. Project 3 was that stop. Project 4 is the breath before the stop.
5. Two readings of one breath
State A
disc
live load that relies on bending
k.g.s2 / m4
E_A = 2c / h
disc perimeter / sphere diameter = π
adds to the sum
State B
sphere
bending that relies on live load
k.g.s2 / m4
E_B = hbar / c
polar height / sphere circumference = 1/π
spends potential
Product of the geometric ratios = 1.
Product of the conjugate unit writings = 1.
Same engine.
6. PiTensor
A coefficient keeps one reading. The piTensor keeps both at 0i2.
It is two-dimensional because the generating figure is the disc. Polar height is already the third direction inside the State B ratio. Ordinary xyz strain can still be written. What it cannot show is that the strain is the disc answering the sphere after a pulse.
Lewe’s rule stands: after the cut, both actions remain visible. The piTensor is that rule applied to the breath.
7. T = 0
T = 0 is the instant at which the patch has opened and the return has not yet been spent. The cut and the whole are both present. Shear path and hoop are both present. N m s2 and N/m2 are both present. State A and State B are both present.
If that instant is discarded, only a coefficient remains. If it is kept, the sphere is allowed to breathe, and σ_R has a time as well as a surface.
8. What remains
Z_R, the converter from N m s2 into N/m2.
Whether the locked residual in the wall still breathes, or only remembers one breath.
One measurement that would show the delay: a hoop excess, a thickening of the contact, or a negative Poisson response in the paste before set.
Working sentence:
Dilatant compression is impulsed expansion, delayed by s2, returned as k.g.s2 / m4. The contact patch makes the area it stresses. The sphere breathes. The piTensor holds the breath at T = 0.
9 Conclusion — visibility of every term, and the 2008 question answered
Project 2 wrote an extra hoop term. Project 3 named its source as water trapped on Reynolds’ misfit surface. This paper gave that term a time: dilatant compression is impulsed expansion, delayed by s^2, returned as m^4 / k.g.s^2. The converter that lets the two unit-columns speak to one another is
Z_R = A_* t_* L_* τ² = m⁴ s²
σ_R = F / Z_R
A_* t_* is m3. L_* is the fourth metre
A_* is the contact patch after the pulse has opened it.
t_* is the auxetic thickening of that patch.
L_* is the in-plane opening that supplies the fourth metre.
τ2 is the rebuild delay.
F is the strength of water, N m s2.
No coefficient is required to move from F to σ_R. The fourth metre is not a decoration. It is the same fourth-order length that a beam element already carries when d4w / dx4 is written for a wall strip. The 2008 thesis already noted that shell starting-point as m4. Z_R is that m4 given duration.
The hoop equation is then fully visible:
σ_θ = pr / t + F / Z_R
The first term is the stored liquid. The second is the wall’s remaining share of the misfit-stretch. After hydration the second term is locked and the duration is spent: high hoop, brittle fracture. Before the stitch both readings of the breath are present at T = 0. The pi-tensor keeps them. After the stitch only sigma_R remains. That is the engineering practice: write both actions after the cut, do not hide the stitch inside K1, K2, K3.
The 2008 question
The BEng thesis asked whether tank loadings could be calculated without coefficients. In 2008 the honest answer was no. The answer now is yes.
T(x) = (pr / t + sigma_R) tM_base = cantilever share of the same water load after base restraint.
σ_R = F / Z_R is the term the coefficients were carrying without naming. Once Z_R is on the page, the distribution between ring tension and bending is a partition of one water, not a lookup. Maximum tension, height of maximum tension, and base moment remain the three design values of 2008. They are no longer read from K1, K2, K3.
Map onto §6.1 of the 2008 thesis
| 2008 task | Where it now sits |
|---|---|
| 1. Understand derivation of the PCA tables | Project 1 |
| 2. Translation of Lewe | Completed |
| 3. Follow up further references | Project 1 |
| 4. Create a design method from scratch using one basic shell theory | Projects 2–4. Z_R = m4 s2 closes Task 4 |
| 5. Effect of base-slab deflection | Remains open. Experiment still required |
| 6. Effect of prestress on deflection | Remains open. Prestress is an imposed F on the same surface |
Completing Project 4, with Z_R written, completes Task 4. That is the conclusion of the 2008 thesis, eighteen years on.
Practice rule
Keep every term on the calculation sheet:
Liquid hoop pr/t.
Water-stretch hoop σ_R = F / Z_R.
Joint restraint that moves part of those two into M and shear.
Duration: live Z_R before set; locked Z_R after set.
If a number cannot be pointed at as one of those four, it is a coefficient and should not replace the geometry.
Working close:
The sphere breathes. Z_R is the volume-time of one breath. Concrete is the breath trapped. A tank wall can now be calculated from those sentences. The PCA tables remain as a check. They are no longer the method.
ace-consultancy.uk 