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How to use
Pick a preset or change material, rake angle, friction, feed, width and cutting speed. The picture is the tool frame: the workpiece streams in from the right, the thin layer is sheared on the dashed yellow plane and leaves as a thicker, curling chip. The grid shows how material is stretched. Colour of the chip is its temperature (straw, then blue, then red for steel); False colour switches to a heat map.
Try: raise the rake angle and watch the shear angle and the chip ratio change while the force falls; set friction to 1.1 and watch the chip thicken; switch the shear-angle rule and compare; turn coolant on; move the speed slider and watch tool life collapse in graph 4 and the two economic speeds in graph 6; start with a worn tool (90 %) and press play to watch flank wear reach 0.3 mm, then press Change edge. Space pauses, R resets.
The science
Orthogonal cutting is the cleanest model of machining: the cutting edge is a straight line perpendicular to the cutting speed V, and a layer of thickness t0 (the feed per revolution when you turn the end of a thin tube) is sheared into a chip of thickness tc. All the deformation happens on one thin shear plane at angle φ to the cut surface; the chip then slides up the rake face, tilted by the rake angle α, against friction.
1. Geometry and velocities. Continuity of volume gives the chip thickness ratio, and the velocity triangle gives the shear and chip speeds:
r = t0/tc = sinφ / cos(φ − α); γ = cotφ + tan(φ − α)
Vs = V cosα / cos(φ − α); Vc = V r
The chip is always thicker than the layer it came from (r < 1), and the shear strain γ in a steel chip is typically 1.5 to 3, far beyond a tensile test.
2. Forces: the Merchant circle. If the shear plane carries a constant shear stress τ (with the size-effect factor 1 + 0.5 √(0.04 mm / t0) switched on), the shear force and the resultant R of the tool on the chip are
Fs = τ b t0 / sinφ; R = Fs / cos(φ + β − α); tanβ = μ
Fc = R cos(β − α); Ft = R sin(β − α); F = R sinβ; N = R cosβ
All the components lie on a circle whose diameter is R: that is the Merchant force circle drawn in the corner of the canvas. The same resultant is resolved three ways (cutting axes, rake face, shear plane), and the page checks that they agree.
3. Which shear angle? Mechanics alone does not fix φ. Three classical rules are offered: Ernst-Merchant (the φ that minimises Fc) φ = 45° + (α − β)/2; Lee-Shaffer (slip-line field for a rigid-plastic chip) φ = 45° + α − β; and Merchant's second solution 2φ + β − α = C with C = 75°. They disagree by several degrees, which is the honest state of the theory: real φ has to be measured.
4. Power and heat. The cutting power splits between the shear plane and the rake face, and all of it becomes heat:
Fc V = Fs Vs + F Vc; u = Fc / (b t0) (specific cutting energy, J/mm3)
Temperatures follow a Boothroyd/Weiner-style partition. With RT = ρ c V t0 / k, the fraction of shear-plane heat that goes into the workpiece is βf = 0.5 − 0.35 log10(RT tanφ) (0.3 − 0.15 log10 above 10). Then
ΔTs = (1 − βf) Fs Vs / (ρ c V t0 b); ΔTc = F Vc / (ρ c V t0 b); ΔTinterface ≈ ΔTs + 1.5 ΔTc
The factor 1.5 for the peak-to-mean ratio on the rake face is a representative constant, not a derivation. Low-conductivity metals (titanium, Inconel) have small RT, so the heat stays in the chip and tool.
5. Tool life, wear and money. Taylor's law V Tn = C (extended here with weak feed and width exponents: C′ = C (0.2/t0)0.15 (2.5/b)0.1) gives the edge life T. Flank wear VB grows fast at first, nearly linearly in the middle, then accelerates; the ISO 3685 criterion VB = 0.3 mm is the end of life. A worn flank rubs the new surface and adds a ploughing force 0.6 τ b VB to Fc and Ft. Turning a part of diameter D and length L at feed f takes tm = π D L / (1000 V f) minutes, so the cost per part is
Cpart = Cm (th + tm) + (tm/T)(Cm tc + Ct); Tmin cost = (1/n − 1)(tc + Ct/Cm); Tmax rate = (1/n − 1) tc
The vertical lines in the cost graph mark those two economic speeds; the page confirms them by a numerical search.
Honest simplifications. One shear plane (no thick shear zone), constant τ and μ (real friction is sticking-sliding, with a built-up edge at low speed), no strain-rate or temperature dependence of τ (no Johnson-Cook), no chip breaking, no tool-edge radius, no chatter. Material constants are indicative handbook-range values for coated carbide, not a cutting-data table; use them to learn the trends, not to set a real machine. Flank wear and the Taylor constants are empirical. No brand or product is modelled.