差排 Dislocation 3D 互動模型

⚠ 此模型已被修改，並非原始版本。原始著作：AEML, NTUST（國立臺灣科技大學 營建工程系）⚠ This model has been modified and is not the original. Original work: AEML, NTUST (National Taiwan University of Science and Technology)

差排 3D 互動模型Dislocation 3D Interactive Model

刃差排、螺旋差排、混合差排 ── 自行設定外力大小與方向，觀察差排如何在晶格中成核、滑移並留下滑移台階。Edge, screw and mixed dislocations: set the force and its direction, then watch a dislocation nucleate, glide through the lattice and leave a slip step.

b ξ ⊥ 刃型edge 螺旋型screw 混合型mixed S F S F 閉合缺口closure failure = b 閉合缺口closure failure = b 外力force F 反力reaction −F τ_R （沿 b）(along b) x y z

尚未開始：按「播放」開始施力。Not started: press Play to apply the force.

τ_R < τ_CRSS：只有彈性剪變形（已放大顯示），沒有差排產生。: elastic shear only (exaggerated), no dislocation.

τ_R ≥ τ_CRSS：差排沿滑移面前進中！: the dislocation is gliding!

反向剪應力把差排推回去Reversed shear pushes the dislocation back（τ_R < 0）

τ_R 不足：差排卡在原處不動。τ_R too low: the dislocation is stuck.

外力已移除：彈性變形恢復，但差排留在晶體內。Force removed: elastic strain recovers, the dislocation stays.

差排穿出晶體：上半部永久滑移一個 b，留下滑移台階。Dislocation exits: the top half is permanently shifted by b, leaving a slip step.

卸載：外力移除，彈性變形恢復。Unloading: force removed, elastic strain recovers.

本輪播放結束，0 秒後重播。Run finished, replaying in 0 s.

手動模式：拖曳「差排位置」滑桿推進差排。Manual mode: drag the dislocation-position slider.

已停止；滑移台階是永久的塑性變形。Stopped; the slip steps are permanent plastic deformation.

ψ = 90°：力垂直於滑移面，τ_R = 0，不會滑移。: force normal to the slip plane, τ_R = 0, no slip.

已播放完設定的次數。按「播放」可重新開始。All runs finished. Press Play to start again.

拖曳旋轉 · 雙指縮放Drag to rotate · pinch to zoom

此瀏覽器無法啟用 WebGL，3D 模型無法顯示。請改用最新版 Chrome、Edge、Firefox 或 Safari。WebGL is not available in this browser. Please use a recent Chrome, Edge, Firefox or Safari.

視角：View:

布格向量 b（含布格迴路）Burgers vector b (with circuit) 差排線 ξDislocation line ξ 剪力方向Shear force 晶格線Lattice lines 原子Atoms 滑移面Slip plane 僅顯示滑移面上下兩層原子Slip-plane atoms only

僅顯示滑移面正上方一層（藍色小球）與正下方一層（灰色大球）原子。顏色與大小只代表所在的層，不是不同元素。已自動切到俯視：未滑移區上下兩層重疊；已滑移區上層錯開一個 b；差排線附近是錯開一半的過渡區，也就是差排的「核心」。Only the layer just above the slip plane (small blue) and the layer just below it (large grey) are shown. Colour and size mark the layer, not a different element. The view switches to top: in the unslipped region the two layers overlap, in the slipped region the upper layer is offset by b, and near the line it is offset by half — the dislocation core.

晶格大小Lattice size （原子數 X × Y × Z，Z 為垂直方向）(atoms X × Y × Z, Z is vertical)

X × Y × Z 共total 512 顆原子atoms

快速選擇：Presets:

目前為內定最佳值 8 × 8 × 8：差排核心兩側都放得下 3 × 3 的布格迴路，512 顆原子也不會密到擋住差排線，手機上一樣順。Recommended 8 × 8 × 8: a 3 × 3 Burgers circuit fits on both sides of the core, and 512 atoms stay uncluttered and smooth on phones.

自訂尺寸。建議 X、Y 取 7～12，Z（垂直）取偶數 6～10（Z 為偶數時滑移面正好在正中間）。改尺寸會重設晶體。Custom size. Suggested X, Y = 7–12 and an even Z = 6–10 (the slip plane is then exactly in the middle). Changing the size resets the crystal.

尺寸偏小：差排離表面太近，布格迴路常放不下而不顯示。建議至少 7 × 7 × 6。Too small: the dislocation is close to the surfaces and the Burgers circuit often cannot be drawn. Use at least 7 × 7 × 6.

原子數超過 1600：畫面較擁擠，舊手機可能變慢。建議搭配「僅顯示滑移面上下兩層原子」，或關掉原子只看晶格線。More than 1600 atoms: crowded and possibly slow on older phones. Try “slip-plane atoms only”, or hide atoms and keep the lattice lines.

外力設定Applied force （作用在上表面，A = 1 mm²）(on the top face, A = 1 mm²)

力的大小Force F (N)自動auto

方位角Azimuth φ (°)
自 +x 轉向 +yfrom +x toward +y自動auto

仰角Tilt ψ (°)
向下壓入上表面pressing into the top自動auto

臨界分解剪應力CRSS τ_CRSS (MPa)自動auto

差排位置Dislocation position
可拖曳手動推進drag to move by hand

「自動」的意義：F 取 τ_R = 2 τ_CRSS 所需的力（差排明確滑移）；φ 每個循環轉 90°，依序展示四個滑移方向；ψ 每個循環依序取 0°、45°、70°，看 τ_R 隨 cosψ 下降（70° 時通常低於 τ_CRSS，不滑移）；τ_CRSS 取目前 τ_R 的一半（F 也自動時固定 30 MPa）。晶體會從 ±x、±y 中選出 τ_R 最大的方向當作 b。“Auto” means: F is the force giving τ_R = 2 τ_CRSS (clear glide); φ turns 90° every cycle to show all four slip directions; ψ steps through 0°, 45°, 70° so τ_R falls with cosψ (at 70° it is usually below τ_CRSS: no slip); τ_CRSS is half the current τ_R (30 MPa when F is also auto). The crystal picks the ±x/±y direction with the largest τ_R as b.

即時讀值Live readout

剪力分量Shear component τ = F cosψ / A	0 MPa
正向應力Normal stress σ = F sinψ / A	0 MPa
晶體選擇的滑移方向Slip direction chosen目前差排的布格向量Burgers vector of the dislocation b	[100][010][100][010]
水平分力與 b 夾角Angle between force and b φ − φb	0°
方位因子Orientation factor m = cosψ · cos(φ − φb)	0
分解剪應力Resolved shear stress τR = (F/A) · m	0 MPa
臨界分解剪應力CRSS τCRSS	0 MPa
差排位置 / 速率Position / velocity v	0% · 0.0 a/s
循環 / 播放Cycle / run	0 / 3 · 0 / ∞1
累計差排通過Dislocations passed	0
上半部累積淨滑移Net slip of top half Δ = (Δx, Δy)	(0, 0) b

0τ_CRSS2 τ_CRSS

表面台階最多畫到 3b；之後的滑移只累加在「累積淨滑移」讀值中。每滑移一個 b，晶體內部的原子排列都和原來相同。Surface steps are drawn up to 3b; further slip only adds to the net-slip readout. After each b of slip the interior arrangement is identical to the original.

|τ_R| < τ_CRSS → 不滑移（只有彈性變形）no slip (elastic only)|τ_R| ≥ τ_CRSS → 差排滑移（塑性變形）dislocation glide (plastic)

循環施力與動畫設定Loading cycles & animation

循環施力Loading cycles

循環施力次數Cycles per run 次× 自動auto

單向重複same direction 正反交替（每循環 φ 轉 180°）alternating (φ + 180° each cycle)

一次循環 = 加載（差排成核、掃過滑移面、穿出晶體）＋ 卸載。自動次數：φ 自動時 4 次（四個方向各一次）；正反交替 2 次（一來一回）；其他 3 次（台階畫到上限 3b）。φ 自動時「正反交替」不作用。One cycle = loading (the dislocation nucleates, sweeps and exits) + unloading. Auto count: 4 when φ is auto (one per direction), 2 when alternating (there and back), otherwise 3 (steps reach the 3b display limit). Alternating is ignored while φ is auto.

差排速率Dislocation velocity

v₀ (a/秒a/s)

速率隨超額應力變化velocity rises with excess stress

v = v₀ · (τ_R/τ_CRSS − 1)，下限 0.15 v₀、上限 3 v₀：力越大差排走得越快。v = v₀ · (τ_R/τ_CRSS − 1), limited to 0.15–3 v₀: the larger the force, the faster the dislocation.

固定速率：只要 |τ_R| ≥ τ_CRSS，差排就以 v₀ 等速前進。Constant velocity: whenever |τ_R| ≥ τ_CRSS the dislocation moves at v₀.

目前每循環動畫長度為固定值，差排速率改由長度決定，上面的 v₀ 暫不作用。The cycle length is fixed, so the velocity is set by that length and v₀ above is not used.

動畫Animation

每循環動畫長度Length per cycle 秒s 自動auto

動畫重複播放Repeat playback ∞ 無限次infinite 次× 自動auto

重播前停留時間Pause before replay 秒s 自動auto

每循環動畫長度包含加載與卸載，卸載約占 30 %（至少 0.8 秒）；長度固定時，差排會剛好在加載結束時穿出。自動長度：依差排速率算出差排掃過晶體所需時間，再加 1.5 秒卸載。
一輪播放 = 設定的循環次數；播完後停留，再重設晶體重播。自動重複：讓總播放時間約 1 分鐘。自動停留：取每循環長度的一半（至少 2 秒）。
模型第一次捲動到畫面中時會自動開始播放；捲出畫面時暫停運算以節省電力。The cycle length covers loading and unloading; unloading takes about 30 % (at least 0.8 s). With a fixed length the dislocation exits exactly when loading ends. Auto length: the time the dislocation needs to sweep the crystal at the set velocity, plus 1.5 s of unloading.
One run = the set number of cycles; after a pause the crystal is reset and the run replays. Auto repeat: about one minute of playback in total. Auto pause: half the cycle length (at least 2 s).
Playback starts automatically the first time the model scrolls into view, and the simulation pauses while it is off-screen.

觀察重點What to look for

刃差排：b ⟂ ξ。可以想成在滑移面上方多插入一片「額外半平面」（黃色原子），差排線 ξ 就是這片半平面的下緣，前表面以 ⊥ 標示。差排沿著 b 的方向移動，也就是與剪力同方向；上方原子受壓縮、下方受拉伸。按「沿差排線 ξ 觀察」最容易看出額外半平面。Edge dislocation: b ⟂ ξ. Think of an extra half-plane (yellow atoms) inserted above the slip plane; the line ξ is its lower edge, marked ⊥ on the front face. The dislocation moves along b, i.e. in the direction of the shear; atoms above are compressed and those below stretched. Use “Along ξ” to see the extra half-plane best.

螺旋差排：b ∥ ξ。沒有額外半平面；繞著差排線走一圈，原子面會像螺旋坡道一樣升降一格。差排移動方向垂直於 b，也垂直於剪力，但掃過之後的滑移量和刃差排一樣，都是一個 b。按「俯視滑移面」可以看到差排線一側的上層原子已錯開一格，另一側還沒有。Screw dislocation: b ∥ ξ. There is no extra half-plane; going once around the line takes you up or down one plane, like a spiral ramp. The dislocation moves perpendicular to b and to the shear, yet the slip it leaves is the same single b. In the top view the upper layer is already offset on one side of the line and not on the other.

混合差排：b 固定不變，差排線卻是彎的，所以 ξ 與 b 的夾角 θ 沿線連續改變。前表面 ξ ⟂ b 是刃型；左側面 ξ ∥ b 是螺旋型；中間兩者兼具：b_edge = b sinθ、b_screw = b cosθ。受剪時整條差排線向外擴張，每一段都沿自己的法線移動。兩個布格迴路（刃型端、螺旋型端）的閉合缺口都是同一個 b。Mixed dislocation: b is fixed but the line is curved, so the angle θ between ξ and b changes along it. At the front face ξ ⟂ b (edge); at the left face ξ ∥ b (screw); in between it is both: b_edge = b sinθ, b_screw = b cosθ. Under shear the whole line expands, each segment moving along its own normal. Both Burgers circuits (edge end and screw end) close with the same b.

布格迴路：綠色迴路沿原子走「右 3、下 3、左 3、上 3」。在完美晶體中會回到起點；繞著差排走一圈卻差一格，由終點 F 指向起點 S 的閉合缺口就是布格向量 b（FS/RH 約定）。Burgers circuit: the green loop steps “right 3, down 3, left 3, up 3” from atom to atom. In a perfect crystal it closes; around a dislocation it misses by one spacing, and the closure failure from finish F to start S is the Burgers vector b (FS/RH convention).

	刃差排Edge	螺旋差排Screw	混合差排Mixed
b 與vs ξ	垂直perpendicular ⟂	平行parallel ∥	夾角 0°～90°angle 0°–90°
差排移動方向Line motion	平行於 bparallel to b	垂直於 bperpendicular to b	沿差排線的法線along the line normal
額外半平面Extra half-plane	有yes	無no	刃型分量處才有where it has edge character
掃過後的滑移量Slip after passage	都是一個 balways one b

共同的核心觀念：差排線是滑移面上「已滑移區」與「未滑移區」的分界線。一次只需斷開、重接差排線旁的一排鍵結，所以金屬實際的降伏強度遠低於整面原子同時滑動所需的理論剪強度。Key idea: the dislocation line is the boundary between the slipped and unslipped parts of the slip plane. Only one row of bonds next to the line breaks and re-forms at a time, which is why real metals yield far below the theoretical shear strength of a whole plane sliding at once.

模型說明：簡單立方晶格（內定 8 × 8 × 8，可自訂），右手座標、z 軸朝上，滑移面為 (001) 面（z = 0）。原子位移採 Peierls–Nabarro 型平滑剖面，彈性剪變形已放大顯示；τ_R = (F/A) cosψ cos(φ − φ_b)，F 與 A 的數值僅供教學示意。Model: simple cubic lattice (8 × 8 × 8 by default, adjustable), right-handed axes with z up, slip plane (001) at z = 0. Atom displacements follow a smooth Peierls–Nabarro-type profile and the elastic shear is exaggerated; τ_R = (F/A) cosψ cos(φ − φ_b). Values of F and A are illustrative.

© 2026 AEML, NTUST · 國立臺灣科技大學 營建工程系Dept. of Civil and Construction Engineering, NTUST工程材料 教學模型Engineering Materials teaching model