From 5c0054a185fa428c2fc988863ef576f6c9bb967c Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sun, 16 Aug 2026 19:08:03 +1000 Subject: [PATCH 1/4] Teaching examples: crossing fault zones, and what a gouge zone's width is for Two pages built on the finite-width (ribbon) representation, each testing a prediction rather than illustrating one. crossing-fault-zones.md answers the open question in #544: where two fused zones overlap, does the nearest-fault director reproduce an orthotropic treatment? The algebra comes first and reframes the question. In plane strain the deviatoric strain-rate space is two-dimensional and a rotation by theta rotates a state in it by 2 theta, so a weak plane occupies one LINE in that space and two planes separated by Delta in the rock are separated by 2 Delta there. Delta = 0 and Delta = 90 are therefore the same case, and the 90-degree X that "high angle" suggests is the one geometry where the rules provably coincide. Checked against the shipped Muehlhaus-Moresi tensor, which also confirms the eta -> 1/(4 eta) substitution of #539 and shows the orthotropic tensor is itself a TI tensor in 2-D, so it needs no new constitutive machinery. Measured on one mesh per angle, with the rules differing ONLY in the overlap cells and referenced to a no-fault control: the nearest-fault rule recovers 70% of the network's weakening at a 10-degree crossing, rising monotonically to exactly 100% at 90 degrees, where the two dissipations agree to seven significant figures. That null is required by the algebra and nothing was tuned to produce it. The error is largest where the issue expected none. The per-cell tensors differ most at 45 degrees, but the overlap AREA goes as 1/sin(Delta) and the area wins, so the case to worry about is the shallow crossing. gouge-zones.md takes the matched collapse eta_I = eta_1 / w and measures what it loses. It reproduces slip to 1.7% at w = 0.002 and 12% at w = 0.02, behaving as the w -> 0 limit it is. It preserves the ratio eta_1 / w and destroys the pair, so the gouge's own temperature excess, eta_1 V^2 / 8 kappa, is unrecoverable from a contact - matched to 1% at three widths with nothing fitted. Separately, a transport equation posed on a split-node mesh sees a perfect insulator: plain conduction with uniform diffusivity and no fault properties anywhere loses 19% of the flux and develops a temperature discontinuity of 0.47. Both pages record their resolution controls. The host mesh does not enter the crossing result at all (three host sizes give the same ratio to four figures); the zone mesh does, and refining it moves the nearest-fault rule closer to the orthotropic one, so the quoted percentages are bounds rather than estimates. Scripts and full working notes: ~/+Simulations/ribbon_network_2d/ Underworld development team with AI support from Claude Code --- docs/advanced/crossing-fault-zones.md | 284 +++++++++++++++++++ docs/advanced/figures/crossing_map.png | Bin 0 -> 189619 bytes docs/advanced/figures/crossing_weakening.png | Bin 0 -> 144877 bytes docs/advanced/figures/gouge_insulation.png | Bin 0 -> 788266 bytes docs/advanced/figures/gouge_mechanics.png | Bin 0 -> 124814 bytes docs/advanced/figures/gouge_thermal.png | Bin 0 -> 128493 bytes docs/advanced/gouge-zones.md | 165 +++++++++++ docs/advanced/index.md | 14 + 8 files changed, 463 insertions(+) create mode 100644 docs/advanced/crossing-fault-zones.md create mode 100644 docs/advanced/figures/crossing_map.png create mode 100644 docs/advanced/figures/crossing_weakening.png create mode 100644 docs/advanced/figures/gouge_insulation.png create mode 100644 docs/advanced/figures/gouge_mechanics.png create mode 100644 docs/advanced/figures/gouge_thermal.png create mode 100644 docs/advanced/gouge-zones.md diff --git a/docs/advanced/crossing-fault-zones.md b/docs/advanced/crossing-fault-zones.md new file mode 100644 index 00000000..8e2834ed --- /dev/null +++ b/docs/advanced/crossing-fault-zones.md @@ -0,0 +1,284 @@ +# Fault zones that cross + +Two fault zones that cross are the geometry the finite-width +representation exists for. Thickening each trace and letting the CAD +kernel fuse the results turns the junction into ordinary cells of a +single region: nothing downstream is told which branch a cell came from, +and no junction geometry is ever constructed. The zero-thickness +representation cannot take this route at all — the split refuses +overlapping cuts. + +That ignorance is deliberate, and it has a price. The rheology at the +overlap comes entirely from the `Surface` lookup, which gives every point +the **director of the nearest surface**. Where the two zones cross, the +material is weak in two directions and a single director represents one +of them. This page works out when that matters. + +Scripts: `~/+Simulations/ribbon_network_2d/`. + +## Building the crossing + +One call places the whole network. The two ribbons may overlap — that is +the point — and `assembly="fuse"` returns their union as one region with +no internal seam: + +```python +from underworld3.utilities.place_surface import place_thin_volume + +dm, info = place_thin_volume(mesh0.dm, [trace_a, trace_b], width=0.02, + label="Zone", label_value=31, + size=0.005, assembly="fuse") +``` + +Two details in that call decide whether the experiment can measure +anything. + +**`size` must be passed.** It defaults to `0.9 * width`, which puts about +two elements across the band at every width. Two elements cannot resolve +which half of an overlap a rule owns, and the ownership split is the +whole question. `width/4` is used throughout here. + +**`fragment` is the wrong assembly for a crossing.** It keeps each +overlap piece as its own region, so the mesh conforms to the boundaries +of the overlap; where two zones converge at a shallow angle the overlap +is a spike, and its fragmented tip meshes to arbitrarily bad angles. The +fused union has no such tip. + +## What the overlap should be + +Before solving anything, the question has a closed-form answer in 2-D, +and the closed form says where to look. + +In plane strain with an incompressible medium the deviatoric strain-rate +space is **two-dimensional**, spanned by + +$$ +\mathbf{e}_1 = \tfrac{1}{\sqrt 2}\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}, +\qquad +\mathbf{e}_2 = \tfrac{1}{\sqrt 2}\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}, +$$ + +and rotating the material by $\theta$ rotates a state in that space by +$2\theta$. A weak plane with normal at $\phi$ is compliant in exactly one +mode — on-plane shear — so it occupies a single **line** in that space, +at angle $2\phi + 90^\circ$. + +Everything follows from the doubling. Two planes separated by $\Delta$ in +the rock are separated by $2\Delta$ there, so: + +- $\Delta = 0^\circ$ and $\Delta = 90^\circ$ put both compliances on the + **same line**. Shear on a plane with normal $\hat{\mathbf x}$ *is* + shear on a plane with normal $\hat{\mathbf y}$ — the same $\tau_{xy}$. + A 90-degree X is degenerate in 2-D. +- $\Delta = 45^\circ$ makes the two compliances **orthogonal**. Together + they span the whole deviatoric space, and the overlap is isotropic. + +So the geometry the phrase "high angle" suggests — the square X — is the +one geometry where the question cannot be asked. The tensors to compare +differ most at 45 degrees. + +### The orthotropic tensor is a TI tensor + +Summing the two planes' compliances and inverting gives a tensor with two +eigenvalues and one principal direction in the deviatoric plane, which is +precisely the structure of +{py:class}`~underworld3.constitutive_models.TransverseIsotropicFlowModel`. +So the orthotropic treatment needs no new constitutive machinery in 2-D: +read off an effective director and effective $\eta_0$, $\eta_1$ and use +the shipped model. + +```python +S = np.eye(2) / (2 * eta_0) +s = 1 / (2 * eta_1) - 1 / (2 * eta_0) +for phi, lam in zip(normal_angles, weights): + u = weak_mode(phi) # unit vector at 2*phi + 90 + S = S + lam * s * np.outer(u, u) +C = np.linalg.inv(S) +eta_1_eff, eta_0_eff = sorted(np.linalg.eigvalsh(C) / 2) +``` + +### Two candidate rules, not one + +The weights are a modelling choice and the two ends of the family behave +differently: + +| $\Delta$ | compliance **sum** ($\lambda = 1$ each) | compliance **average** ($\lambda = \tfrac12$ each) | +|---|---|---| +| $0^\circ$ | anisotropy 199 — twice as weak as one plane | exactly the single plane | +| $45^\circ$ | isotropic, $\eta_1$ | isotropic, $2\eta_1$ | +| $90^\circ$ | anisotropy 199 | exactly the single plane | + +The sum is what "add the compliances" means literally: two independent +fabrics superposed. Read it at $\Delta = 0$ — two *coincident* weak +planes come out twice as weak as one — and note that the excess does not +switch off away from coincidence. The average treats the overlap as a +50/50 mixture of the two fabrics and degenerates to the single plane +exactly. Both are carried below; neither is obviously right. + +## The experiment + +Two straight ribbons crossing at the centre of a unit box, fused into one +zone, driven in horizontal extension with a free top. The crossing angle +$\Delta$ is swept, and five rheologies are solved **on the same mesh**: + +| case | the zone | the overlap | +|---|---|---| +| `none` | uniform $\eta_0$ | — (the no-fault control) | +| `isotropic` | weak in every mode | weak in every mode | +| `nearest` | TI, nearest trace's director | TI, nearest trace's director | +| `ortho_sum` | TI, nearest trace's director | the compliance-sum tensor | +| `ortho_avg` | TI, nearest trace's director | the compliance-average tensor | + +The last three are identical everywhere except the overlap cells, so any +difference between them is the overlap rheology and nothing else. +`isotropic` weakens the arms too, so it is not a candidate rule — only a +bracket on how compliant the network could be. + +The metric is the **dissipation weakening** + +$$ +W = 1 - \frac{D}{D_{\rm no\ fault}}, +$$ + +referenced to the `none` control on the same mesh at the same angle. It +is a single number with no probe geometry in it. (The control dissipates +exactly 4.000000 in this box, which is the analytic value, so the +integral is checked as well.) + +### What comes out + +| $\Delta$ | $W$ nearest | $W$ orthotropic | $W$ isotropic | nearest, as % of orthotropic | +|---|---|---|---|---| +| $10^\circ$ | 1.198% | 1.710% | 3.628% | **70.0%** | +| $20^\circ$ | 3.687% | 4.520% | 6.256% | 81.6% | +| $30^\circ$ | 6.618% | 7.679% | 9.240% | 86.2% | +| $45^\circ$ | 11.566% | 12.435% | 13.646% | 93.0% | +| $60^\circ$ | 15.789% | 16.253% | 17.110% | 97.1% | +| $75^\circ$ | 18.322% | 18.476% | 19.121% | 99.2% | +| $90^\circ$ | 18.918% | 18.918% | 19.465% | **100.0%** | + +```{figure} figures/crossing_weakening.png +:width: 100% + +Left: what the two faults are worth, against the no-fault control on the +same mesh. Right: the overlap rule on its own — the two runs share the +mesh and the arms exactly, so the ratio is the overlap and nothing else. +``` + +Read the last row first. At $\Delta = 90^\circ$ the two dissipations +agree to seven significant figures (3.243286 against 3.243286). That is +not a tuned result: the algebra says the orthotropic overlap tensor there +is *identically* the single plane's, and a 2-D director is only defined +mod $90^\circ$, so the nearest-fault rule assigns that same tensor on +both sides of the medial axis. The experiment reproduces it exactly, and +that is what licenses the other six rows. + +Now read up the column. The nearest-fault rule recovers only **70% of the +weakening** at a $10^\circ$ crossing, and the error closes monotonically +as the crossing squares up. + +```{figure} figures/crossing_map.png +:width: 100% + +The same three answers in physical space at $\Delta = 45^\circ$, on one +colour scale. They are indistinguishable by eye, and the region in +dispute is 34 cells out of 1480 (right). A difference of this kind is +only visible in an integral — which is why the metric above is a +dissipation and not a picture. +``` + +### The error is largest where the issue expected none + +The per-cell tensors differ most at $\Delta = 45^\circ$, and the measured +curve shows no feature there at all. Both facts are right, and the +resolution is geometric: the overlap **area** goes as $w^2/\sin\Delta$. +A shallow crossing has a long lens of shared material, a square one has +almost none — 141 overlap cells at $10^\circ$ against 24 at $90^\circ$. +The tensor error peaks at $45^\circ$, the area falls monotonically, and +the product is dominated by the area. + +So the geometry to worry about is the **shallow** crossing, not the +high-angle one — the opposite of what "high angle" suggests, and the +opposite of the case a designer would think to check. + +## Measuring the slip is harder than it looks + +Slip is read as the tangential velocity jump between probes either side +of the band. A **fixed** standoff — some fraction of the width — is wrong +at a crossing: near a shallow one, a probe placed at $0.75\,w$ from one +trace sits *inside the other ribbon*, in weak material, and reports +almost no jump. At $\Delta = 10^\circ$ that read 0.006 against a peak of +0.18 on the same fault. + +The probes have to be walked clear of **every** ribbon before they are +read. The jump then means the same thing at every angle: the velocity +difference across the whole weak region. + +Even walked out, the local probe **cannot** be trusted at the crossing +itself: that is exactly where the standoff changes discontinuously, and +the reading there dips sharply — at $\Delta = 90^\circ$ it goes negative +once the no-fault baseline is subtracted, which is not a physical slip. +This is the reason the metric above is a dissipation integral. Use the +slip probe for the arms and for pictures, not for the number that decides +the question. + +Two further cautions on reading absolute slip here: + +- The pair is symmetric about the loading axes, so changing $\Delta$ + necessarily changes how well each plane is oriented for slip. At + $\Delta = 10^\circ$ both planes are nearly vertical — nearly principal + planes — and barely slip at all. Compare ratios at fixed angle, not + slip across angles. +- The no-fault control read through the same probes gives the wall rock's + own contribution over the probe separation. Subtract it. + +## How much of this is the mesh + +The overlap is a few tens of cells, so the numbers above are worth +nothing without a resolution control. At $\Delta = 45^\circ$: + +| host cell | zone $h$ | zone cells | overlap | $W$ nearest | $W$ orthotropic | ratio | +|---|---|---|---|---|---|---| +| 0.030 | 0.0100 | 380 | 12 | 10.9522% | 12.4148% | 88.2% | +| 0.030 | 0.0050 | 1480 | 34 | 11.5662% | 12.4351% | 93.0% | +| 0.030 | 0.0025 | 5342 | 130 | 11.7717% | 12.4565% | 94.5% | +| 0.040 | 0.0050 | 1480 | 34 | 11.5593% | 12.4278% | 93.0% | +| 0.020 | 0.0050 | 1480 | 34 | 11.5714% | 12.4404% | 93.0% | + +The **host** mesh does not enter: 0.04, 0.03 and 0.02 give 93.0% three +times over, with $W$ agreeing to four significant figures. The embedded +zone mesh is identical in all three, which is the embedding working as +designed and worth having as a measurement rather than a claim. + +The **zone** mesh does. Halving it twice moves the ratio 88.2 → 93.0 → +94.5%, and the increments fall by about a factor of three, so the +converged value here is near 95%. Refinement always moves the +nearest-fault rule *closer* to the orthotropic one — so the table earlier +on this page, computed at $w/4$, **overstates the error**, and its +percentages should be read as bounds rather than estimates. + +## `clearance` is not monotone + +The carve deletes host vertices near the assembly's skin, and whether +what is left is one simple hole is a topological accident of the host +mesh against a plus-shaped cavity. It does not improve monotonically with +`clearance`: a 30-degree crossing builds only at `clearance=1.3`, a +45-degree crossing at 1.0 but *not* at 1.3. Walk a ladder and record +which value was used. + +```python +for clearance in (1.0, 0.9, 1.1, 0.8, 1.2, 0.7, 1.3): + try: + dm, info = place_thin_volume(..., clearance=clearance) + except RuntimeError: + continue + break +``` + +## See also + +- [Fault networks](fault-networks.md) — the zero-thickness toolkit, and + the offset-junction preparation the ribbon does not need. +- [Split-node faults](split-node-faults.md) — the contact representation. +- [Transverse isotropy](vep-transverse-isotropy-faults.md) — the weak-plane + rheology and its parameters. diff --git a/docs/advanced/figures/crossing_map.png b/docs/advanced/figures/crossing_map.png new file mode 100644 index 0000000000000000000000000000000000000000..7e5f411b46e21d674b64f14a0d4b8621c80cba5e GIT binary patch literal 189619 zcmZU*1z1&WyDh8&0#Yg<-3TJ7bc3{%NQty4-Q5T(NC*N-!xBVLLAp~wy1To(JJz|^ z_wWB)XYYMoye|sNHRpVuJH|c6cvkQ;6*;_H^x(Ns?dhc%xt-Hk*%9V}A%+mF(2%CN+IoHX+hV>N0x 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a/docs/advanced/index.md +++ b/docs/advanced/index.md @@ -47,6 +47,12 @@ throws away. **[→ Gouge Zones](gouge-zones.md)** +### Branching Faults and Junctions +What a crack model cannot do at a branch point, measured in a shear box +— and how to choose representations because of it. + +**[→ Branching Faults and Junctions](fault-branching-junctions.md)** + ### Custom Meshes Create complex geometries with gmsh for research problems. @@ -119,6 +125,7 @@ split-node-faults fault-networks crossing-fault-zones gouge-zones +fault-branching-junctions fault-mechanics-examples custom-meshes curved-boundary-conditions

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zj$b^pex{VqnOOwV+6xkiGhBP{Fwl>CuA-Q5!ByyRt&wVAvZ_}O{zWF5X%;7rU%LL+ z|4f>x*#-uT8o|jp;+Nz_2&8-0*1XVE63%bVoW)IGDU!@jPbEuZ6|K&>UQb-WazPA6 zA6<$j69uAgsh6uNe)EA9R=2_-iPM53eIm^}$l*SkgM^6%msXk{85r5?D`lL+yOuOE z$~UXz-2Akve)?+qd|KN3AHM|`z0RJ2T@>yDH8yi)aq$!aUJ5ubedmwroPpKIe^TA# zmJp9g9X%2mAn7L3fN8*(Uo&oX!t4*JH%}x!V?mVWDARkSAh1sN`>Tuppo>r{La#EN zFH?|_flK}NuWv+1cY?zujKFiE>83+grF7b` Date: Sun, 16 Aug 2026 19:28:20 +1000 Subject: [PATCH 2/4] Refine the crossing result: the quoted percentages were bounds, and here is where they converge The overlap is a few tens of cells at a zone mesh of w/4, so the sweep table needed a refinement study before its numbers could be read as measurements. Halving the zone mesh twice at the two angles that carry the argument: Delta = 10: 70.0 -> 73.8 -> 75.1% (overlap 141 -> 522 -> 2051 cells) Delta = 45: 88.2 -> 93.0 -> 94.5% The increments contract about threefold in both cases, extrapolating to roughly 76% and 95%. Refinement always moves the nearest-fault rule CLOSER to the orthotropic one, so the sweep overstates the error everywhere. Corrected headline: the rule loses about a quarter of the network's weakening at the worst crossing angle, not a third. The structure of the result is unchanged - monotone in the crossing angle, exactly zero at 90 degrees, and largest at shallow crossings rather than high-angle ones. The host mesh does not enter the answer at all: 0.04, 0.03 and 0.02 give the same ratio to four significant figures because the embedded zone mesh is identical in all three. That is the embedding contract holding, and it is worth having as a measurement. Underworld development team with AI support from Claude Code --- docs/advanced/crossing-fault-zones.md | 33 +++++++++++++++++++++------ 1 file changed, 26 insertions(+), 7 deletions(-) diff --git a/docs/advanced/crossing-fault-zones.md b/docs/advanced/crossing-fault-zones.md index 8e2834ed..12d37a40 100644 --- a/docs/advanced/crossing-fault-zones.md +++ b/docs/advanced/crossing-fault-zones.md @@ -173,9 +173,13 @@ mod $90^\circ$, so the nearest-fault rule assigns that same tensor on both sides of the medial axis. The experiment reproduces it exactly, and that is what licenses the other six rows. -Now read up the column. The nearest-fault rule recovers only **70% of the -weakening** at a $10^\circ$ crossing, and the error closes monotonically -as the crossing squares up. +Now read up the column. The nearest-fault rule loses a substantial part +of the network's weakening at a shallow crossing, and the error closes +monotonically as the crossing squares up. The percentages in that table +are computed at a zone mesh of $w/4$ and are **bounds**, not estimates — +refining the zone mesh moves the rule closer to the orthotropic one, and +the converged values are about 76% at $10^\circ$ and 95% at $45^\circ$ +(see the resolution control below). ```{figure} figures/crossing_map.png :width: 100% @@ -252,10 +256,25 @@ designed and worth having as a measurement rather than a claim. The **zone** mesh does. Halving it twice moves the ratio 88.2 → 93.0 → 94.5%, and the increments fall by about a factor of three, so the -converged value here is near 95%. Refinement always moves the -nearest-fault rule *closer* to the orthotropic one — so the table earlier -on this page, computed at $w/4$, **overstates the error**, and its -percentages should be read as bounds rather than estimates. +converged value here is near 95%. + +The same check at the worst angle, $\Delta = 10^\circ$, where the overlap +is largest and the resolution matters most: + +| host cell | zone $h$ | zone cells | overlap | $W$ nearest | $W$ orthotropic | ratio | +|---|---|---|---|---|---|---| +| 0.030 | 0.0050 | 1374 | 141 | 1.1980% | 1.7103% | 70.0% | +| 0.030 | 0.0025 | 4910 | 522 | 1.2731% | 1.7252% | 73.8% | +| 0.030 | 0.0013 | 19562 | 2051 | 1.3030% | 1.7361% | 75.1% | + +Increments of +3.8 then +1.3, the same threefold contraction, giving a +converged value near 76%. + +Refinement always moves the nearest-fault rule *closer* to the +orthotropic one, so the sweep table earlier on this page **overstates the +error** everywhere. The corrected statement is that the rule loses about +a quarter of the network's weakening at the worst crossing angle, not a +third — which changes the size of the effect and none of its structure. ## `clearance` is not monotone From 519743ec4f586b7cc26fdb10ea9a349b04e9ddc5 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sun, 16 Aug 2026 19:30:11 +1000 Subject: [PATCH 3/4] Answer the review: the 2-D degeneracy does not carry to 3-D, and there is no external reference Three additions from reviewing the diff wide. The 90-degree null is a property of the two-dimensional deviatoric space and nothing on the page said so. In 3-D that space has five dimensions, two weak planes at any angle occupy different subspaces, and there is no null at all. A reader carrying "a 90-degree crossing is safe" into a 3-D model would be wrong, so the page now refuses that reading explicitly. There is no external reference solution for what an overlap should be, and there cannot be one without modelling the fabric. Both orthotropic constructions are assumptions, so the measurement is a distance to a choice rather than to a truth. What makes it readable is that the two choices differ by a factor of two in overlap weakness and still bracket the answer tightly, which is now stated where the constructions are introduced rather than left for the reader to infer from the tables. At a shallow crossing the whole signal is a difference of about 0.03 between dissipations that agree to 1%. That is now said next to the metric, along with what makes it readable - the exact analytic control and the refinement study. Underworld development team with AI support from Claude Code --- docs/advanced/crossing-fault-zones.md | 28 +++++++++++++++++++++++++++ 1 file changed, 28 insertions(+) diff --git a/docs/advanced/crossing-fault-zones.md b/docs/advanced/crossing-fault-zones.md index 12d37a40..a6c3b8e3 100644 --- a/docs/advanced/crossing-fault-zones.md +++ b/docs/advanced/crossing-fault-zones.md @@ -77,6 +77,16 @@ So the geometry the phrase "high angle" suggests — the square X — is the one geometry where the question cannot be asked. The tensors to compare differ most at 45 degrees. +```{warning} +**This degeneracy is two-dimensional and does not carry to 3-D.** It +exists because the deviatoric space here has only two dimensions, so two +lines in it must either coincide or be distinct in one way. In 3-D that +space has five dimensions, two weak planes at any angle occupy different +subspaces, and there is no null. Do not carry "a 90-degree crossing is +safe" into a 3-D model — the 3-D question is open and nothing on this +page answers it. +``` + ### The orthotropic tensor is a TI tensor Summing the two planes' compliances and inverting gives a tensor with two @@ -115,6 +125,17 @@ switch off away from coincidence. The average treats the overlap as a 50/50 mixture of the two fabrics and degenerates to the single plane exactly. Both are carried below; neither is obviously right. +```{important} +There is **no external reference solution** here, and there cannot be one +without modelling the fabric itself. Both constructions are assumptions, +so what is measured below is the distance from the shipped rule to a +*choice*, not to a truth. What makes that worth reading is that the two +choices differ by a factor of two in how weak they make the overlap and +still bracket the answer tightly — 61.8% against 70.0% at the worst +angle, 92.3% against 93.0% at $45^\circ$. The conclusion is insensitive +to the part that had to be assumed. +``` + ## The experiment Two straight ribbons crossing at the centre of a unit box, fused into one @@ -145,6 +166,13 @@ is a single number with no probe geometry in it. (The control dissipates exactly 4.000000 in this box, which is the analytic value, so the integral is checked as well.) +Note what $W$ is made of at a shallow crossing: at $\Delta = 10^\circ$ +the two dissipations are 3.9521 and 3.9225 against a control of 4.0000, +so the whole signal is a difference of about 0.03 between numbers that +agree to 1%. The exact control and the resolution study below are what +make that difference readable; a single run at one resolution would not +be. + ### What comes out | $\Delta$ | $W$ nearest | $W$ orthotropic | $W$ isotropic | nearest, as % of orthotropic | From 9de6c7c84ec0c7876b39ea5c479cba383bdec333 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Mon, 17 Aug 2026 17:45:58 +1000 Subject: [PATCH 4/4] Branching faults and junctions: the shear-box study, and what a crack model cannot do The crossing-zones page measured overlap rheologies in pure extension and found the differences modest. That test was too kind, and the page now says so: the configuration is maximally symmetric about the load and the drive carries no rotation. This commit adds the test that removes both - a shear box (driven plates, open sides, spin in the kinematics) holding an asymmetric Y: a sub-horizontal main fault at 20/30/40 degrees with a horizontal branch fusing into it - and the verdict inverts. Measured, all from the P0 cell strain rate, isotropic bands as the reference: - The junction is a slip exchanger. The reference's main-fault slip at the junction RISES as the fault becomes misoriented: the well-oriented branch hands its offset into the main, the isotropic corner rotating the deformation. The TI variants collapse against it - 77% / 46% / 19% of the reference at theta = 20/30/40 - and the three overlap rules are indistinguishable from one another throughout. The overlap tensor was never the dominant term; the arms' constraint is. Mechanism: a slip gradient along a fault is a volumetric exchange, which is exactly the mode the crack constraint removes. - The peak is an exit throat with a law. It sits at r = 2.1 w up the main arm - self-similar across a fourfold width sweep - where the fused Y narrows back to one band, and its intensity is gamma = (s_main + s_branch) / w to 10% with nothing fitted. As w -> 0 the throat slides onto the corner and diverges: a branch point is a corner singularity of the continuum problem and the band width is its regularisation length. Junction width is a process-zone size, not a numerical parameter. - The country rock feels the difference as a pattern, not a magnitude. Matrix stress means agree to 2-4% across all cases; the locked TI junctions park a wall-rock stress concentration at the elbow that the reference does not have, and trade it for tip lobes as the junction is freed. Ranking by peak stress alone orders the models backwards. - Offered plasticity, the crack model builds a bypass. A threshold damage law nucleates at the TI junction's stress blob and marches a front along the shear plane, re-routing the network around the misoriented fault. The negative control (same law on the reference) grows faults at their free tips only - the physical pattern - with the junction silent. Both models grow tips; only the constrained model manufactures growth at junctions. The page closes with the frame that organises the whole comparison: the split surface and the TI band are crack models (hard and soft forms of the same kinematic constraint) and the isotropic band is not, so crack assumptions fight each other at branches. Choosing rules follow: surfaces for large-scale networks of distinct well-oriented segments; ribbons with TI where a zone should stay crack-like and isotropic where it must evolve internally; joins made kinematically compatible - bent in to meet tangentially, with damage zones sized as process zones - rather than patched after the fact. Scripts and caches: ~/+Simulations/ribbon_network_2d/ (branch_study.py, branch_width.py, branch_damage.py). Underworld development team with AI support from Claude Code --- docs/advanced/crossing-fault-zones.md | 15 ++ docs/advanced/fault-branching-junctions.md | 187 ++++++++++++++++++ docs/advanced/figures/branch_maps_z20.png | Bin 0 -> 947755 bytes docs/advanced/figures/branch_slip_z20.png | Bin 0 -> 194215 bytes .../advanced/figures/branch_stress_matrix.png | Bin 0 -> 1956078 bytes .../figures/damage_control_compare.png | Bin 0 -> 135360 bytes docs/advanced/figures/halo_recovery.png | Bin 0 -> 169348 bytes docs/advanced/figures/junction_peak_zoom.png | Bin 0 -> 487109 bytes docs/advanced/index.md | 7 + 9 files changed, 209 insertions(+) create mode 100644 docs/advanced/fault-branching-junctions.md create mode 100644 docs/advanced/figures/branch_maps_z20.png create mode 100644 docs/advanced/figures/branch_slip_z20.png create mode 100644 docs/advanced/figures/branch_stress_matrix.png create mode 100644 docs/advanced/figures/damage_control_compare.png create mode 100644 docs/advanced/figures/halo_recovery.png create mode 100644 docs/advanced/figures/junction_peak_zoom.png diff --git a/docs/advanced/crossing-fault-zones.md b/docs/advanced/crossing-fault-zones.md index a6c3b8e3..497cf0d5 100644 --- a/docs/advanced/crossing-fault-zones.md +++ b/docs/advanced/crossing-fault-zones.md @@ -322,6 +322,21 @@ for clearance in (1.0, 0.9, 1.1, 0.8, 1.2, 0.7, 1.3): break ``` +## Postscript: this test was too kind + +Everything above is measured in pure extension with the fault pair +symmetric about the load — a drive with **no rotation**. Repeating the +question in a shear box with an asymmetric branching geometry (spin in +the kinematics, nothing mirror-symmetric) inverts the verdict: the +difference between director rules stays immaterial, but the *entire +TI-arms family* loses factors — not percent — against the isotropic +reference wherever the network must hand slip between differently +oriented members. The overlap tensor was never the dominant term; the +arms' constraint is. See +[Branching Faults and Junctions](fault-branching-junctions.md), which +supersedes the recommendation implied here for any network that +redistributes slip. + ## See also - [Fault networks](fault-networks.md) — the zero-thickness toolkit, and diff --git a/docs/advanced/fault-branching-junctions.md b/docs/advanced/fault-branching-junctions.md new file mode 100644 index 00000000..6847acee --- /dev/null +++ b/docs/advanced/fault-branching-junctions.md @@ -0,0 +1,187 @@ +# Branching faults, junctions, and what a crack model cannot do + +The [crossing-zones page](crossing-fault-zones.md) compared overlap +rheologies for fused fault zones in pure extension and found the +differences modest at ordinary angles. That test had a hidden weakness: +the configuration is maximally symmetric about the load, and the drive +carries **no rotation**. This page repeats the question with both +removed — and the verdict inverts. It ends at the reason: two of our +fault representations are crack models, one is not, and the difference +is not a numerical detail but the mechanics of branching itself. + +Scripts and caches: `~/+Simulations/ribbon_network_2d/` +(`branch_study.py`, `branch_width.py`, `branch_damage.py`). + +## The shear box + +Simple shear supplies what extension cannot: spin, $\omega = +\dot\gamma/2$. The test frame is a driven box — top plate $v = (+0.5, +0)$, bottom $v = (-0.5, 0)$, sides open (traction-free) — holding an +asymmetric Y: a **sub-horizontal main fault** at $\theta = 20/30/40^\circ$ +and a **horizontal branch** running in from the side and fusing into it. +Nothing is mirror-symmetric; the branch lies in the shear plane +(optimally oriented) while the main degrades with $\theta$ (resolved +drive $\propto \cos 2\theta$). Bands at the union-jack default +resolution, structure verified per build; the isotropic weak band is the +reference throughout, with transversely isotropic (TI) variants against +it. Everything is measured from the P0 cell strain rate — local, no +recovery step, no probes. + +```{figure} figures/branch_maps_z20.png +:width: 100% + +Strain rate (P0 invariant, log scale, one range). Rows: $\theta$ = +20/30/40$^\circ$; columns: the isotropic reference and three TI +variants differing only at the junction. In the reference the junction +is the hottest point of the network and intensifies with $\theta$; in +every TI variant it cools, and the misoriented main fades until its +lower arm is a ghost. +``` + +## The junction is a slip exchanger — and the crack constraint cannot run it + +Slip on each fault, at the junction, TI variants as a fraction of the +reference: + +| $\theta$ | main fault | branch | +|---|---|---| +| $20^\circ$ | 77% | ~75% | +| $30^\circ$ | **46%** | ~78% | +| $40^\circ$ | **19%** | ~52% | + +Three facts organise this. First, the reference's main-fault slip at the +junction **rises** as the fault becomes misoriented (0.30 → 0.36): the +optimally-oriented branch delivers its offset *into* the main through +the junction, the isotropic corner rotating the deformation between the +two orientations. Second, the three TI overlap treatments (nearest +director, harmonic average, isotropic overlap cells) are +indistinguishable from each other while all collapsing against the +reference — the overlap rule was never the dominant term. Third, the +mechanism is general: **a gradient of slip along a fault is a +volumetric exchange between band and surroundings**, and that is +precisely the deformation mode the TI constraint (like the slip +surface) removes. Crack kinematics carry slip *along* a fault +faithfully and cannot hand it *between* faults. + +Imposed isotropic "damage halos" around the junction recover the +exchange roughly linearly in their radius (42% at $3w$, 73% at $5w$) +with no plateau: there is no compact junction patch, because the +constraint taxes the mechanics wherever slip varies. + +```{figure} figures/halo_recovery.png +:width: 100% + +Slip profiles and recovery against halo radius: freeing the constraint +where the slip gradient lives, by hand. +``` + +## The exit throat, and a law with nothing fitted + +Where is the reference's peak? Not at the junction point. It sits on +the main's up-dip arm at $r \approx 2.1\,w$ — a **self-similar** +position (2.05–2.11 $w$ across a fourfold width sweep) — where the +fused Y narrows back to a single band. The combined slip of both faults +must exit through that neck, and flux conservation predicts its +intensity outright: + +$$ +\gamma_{\rm peak} \;\approx\; \frac{s_{\rm main} + s_{\rm branch}}{w}, +$$ + +measured at 0.96–1.09 of the prediction over the three resolved widths. + +```{figure} figures/junction_peak_zoom.png +:width: 100% + +The peak (circled) at four band widths, one log scale: always at the +exit throat, always at $r \approx 2.1\,w$, in healthy cells. +``` + +The limit is the point: as $w \to 0$ the throat slides onto the corner +and its amplitude diverges. **A branch point is a corner singularity of +the continuum problem, and the band width is its regularisation +length.** Neither description converges there — the crack models +*exclude* the corner physics; the continuum model *concentrates* it +without bound. The junction's width is therefore not a numerical +parameter to refine away but a physical one: the process-zone size. (It +is the [gouge page's](gouge-zones.md) "width is physics", at the place +where it matters most.) + +## What the country rock feels + +Deviatoric stress in the matrix (where $\tau = 2\dot\varepsilon$ +exactly), zones masked: + +```{figure} figures/branch_stress_matrix.png +:width: 100% + +Matrix stress, one linear scale. Means agree to 2–4% across all six +cases — the load is conserved; its **location** is not. +``` + +The locked TI junctions park a wall-rock stress concentration at the +elbow (the tail rises 11%) that the reference does not possess; freeing +the junction trades it back for **tip lobes**, which is where the +reference — and a real fault — concentrates stress. Ranking these +models by peak stress alone would order them backwards; the location is +the diagnostic. A constrained-network model read at face value predicts +wall-rock damage and seismicity at junctions where the physics puts +none, and under-predicts tip process zones. + +## Offered plasticity, the crack model builds a bypass + +A deliberately primitive damage law — any matrix cell whose stress +exceeds $\tau_c$ becomes isotropic $\eta_1$, instantly and irreversibly; +re-solve; repeat — asks what a yielding background does with each +representation. With a negative control: + +```{figure} figures/damage_control_compare.png +:width: 100% + +Damaged cells coloured by the pass at which they yielded, same law and +threshold. Left, the isotropic reference: growth at the free **tips** +only — the physical pattern — with the junction silent throughout. +Right, the fully-TI network: the same tip growth **plus** a +junction-nucleated front marching along the shear plane — a +through-going bypass that retires the misoriented fault. +``` + +Both models grow faults at tips; that part is physics. Only the +constrained model manufactures growth at the junction — nucleated by +the stress its own kinematic restriction parks there — and its response +to that stress is not to heal the junction but to **re-route the +network around it**. The law is a cartoon (no softening, no length +scale, grid-width bands), so the nucleation sites and topology are the +trustworthy content, not rates or widths. One further consequence of +the corner singularity: at realistic (thinner) widths the *reference's* +corner stress also exceeds any finite yield threshold — so in a +yielding crust a branch point **must** develop a damage zone. The +damage halo at a junction is not a modelling patch; it is the +material's own regularisation of a corner that crack mechanics cannot +represent and continuum mechanics cannot bound. + +## Choosing, finally + +One of these is a crack model twice over — the split surface imposes +slip-only kinematics exactly; the TI band imposes the same as a +$10^{3}$ constitutive penalty — and one is not: the isotropic band is a +model of fault *rock*, not fault *kinematics*. The crack assumptions +fight each other at branches. Hence: + +- **Large-scale fault networks**: build from surfaces (contacts) — the + crack idealisation at its best, where segments are distinct, + well-oriented, and slip-dominated. +- **Zones with interior physics** (gouge, heating, damage evolution): + thin volumes/ribbons — with TI where the zone should stay + crack-like, isotropic where it must evolve internally. +- **Crossings and joins**: make them kinematically compatible by + construction — bend faults in to meet tangentially rather than + abutting at high angle, and surround the joins with isotropic damage + zones sized as process zones. Do not expect any director rule, patch, + or refinement to buy back a high-angle locked junction: the failure + is the crack assumption itself. +- **Evolution in time** — where a misoriented fault would dismember + into en-échelon segments and the network would rebuild itself through + its damage field — requires transporting faults as material chains + and re-deriving them with the evolving damage. That is a programme, + not a feature; this page records the mechanics that motivates it. diff --git a/docs/advanced/figures/branch_maps_z20.png b/docs/advanced/figures/branch_maps_z20.png new file mode 100644 index 0000000000000000000000000000000000000000..455a574d30a014f0a99e68a4bfcb3916c2168635 GIT binary patch literal 947755 zcmX`T3p~^B|32P9r3jTnjw2OCDCgrslFF$AIVa?7IUhDwh$19Ojte>EFo&EE3nAy^ zeA*o6Fbp%>?DT(mf4;x}t%#kBU&Klgi^XSMC z)?Y`Coczdo3^>CoE-`WB=#e7^ceI}bWo+1-`Z_aGKkD-GdunElXU)V(k8;E4zfiU} zN3x^1j%B|!T4iBX+~ak-=W@FF;ZkWKegSQ*7{VD<8`I_7HG}ua4W5uCHb-m(KFMounHqs@rjxQ(Wp~+vOc`xFJi0 zzI`RQt=UNZ@nqM`3P)+3U7U|@ru;LDY^g6xM1<^ zio8s1*}te7dGD*Ds51B7r+oj9Ixdpy&c~B^9)4t7z4E^5;DT5spGo2Uo?L6~zBt6FsDNq?HM;$)V|Ag@#Xzf z*I0T7i?0(VZ>Ms!-FAvuTX~=uY~P}lex~hNXaJKuC_6CDlbm{=L(I8hqaYKIDSIgOLACkF_P=#F7d9^lIrnfk2H6Qh+lGj1-z``G~&I` zmyk>6uf8q6thS>|$=BNL>RedomI#66FK4iD{678nch+=%{z z60MwTJKwaz4_Noq3KMQ4k2iDdsr|(j?x~H@PPEHyzIVASBp~yi-22F;%;3IK$IahT 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