受力桿件拉伸試驗 3D 模擬器 Tension Test of an Axially Loaded Bar

工程材料Engineering Materials ｜ 機械性質Mechanical Properties of Materials

受力桿件拉伸試驗 3D 模擬器Tension Test Simulator of an Axially Loaded Bar

選擇延性或脆性材料、斷面形狀與加載條件後按「開始」，程式逐步施加應變直到破壞；3D 桿件同步顯示伸長、橫向收縮與頸縮，下方繪出工程與真實應力–應變圖。Choose a ductile or brittle material, a cross-section and the loading conditions, then press Start. Strain is applied step by step until fracture; the 3D bar shows elongation, lateral contraction and necking, and the engineering and true stress-strain curves are plotted below.

1材料類型與材料庫Material type & library

延性材料Ductile 脆性材料Brittle

材料Material Low-carbon steel (default) Rebar SD420W Aluminum alloy 6061-T6 Annealed copper Gray cast iron (default) Soda-lime glass Concrete (direct tension) Alumina ceramic Custom

材料庫數值為教科書典型值，僅供教學比較；選定後仍可逐項修改（自動切換為「自訂」）。Library values are typical textbook values for teaching comparisons. Any value can still be edited (the selection switches to "Custom").

延性材料：Ductile: 線彈性 → 降伏 → 應變硬化（均勻塑性變形）→ 最大荷重時開始頸縮 → 頸縮處斷裂。linear elastic → yielding → strain hardening (uniform plastic deformation) → necking begins at maximum load → fracture in the neck.

脆性材料：Brittle: 幾乎無塑性變形，不發生頸縮，達破壞強度即突然斷裂；斷面垂直於受力方向。almost no plastic deformation and no necking; sudden fracture at the fracture strength, on a plane normal to the load.

2材料參數Material properties

楊氏模數Young's modulus EGPa (= 10⁹ N/m²)

卜松比Poisson's ratio ν0 – 0.5

降伏強度Yield strength σ_yMPa (= 10⁶ N/m²)

抗拉強度Tensile strength σ_UTSMPa

破壞強度Fracture strength σ_fMPa

頸縮時應變Strain at necking ε_umm/mm

脆性材料不發生頸縮，ε_u 不適用。Brittle materials do not neck; ε_u does not apply.

破壞時應變Strain at fracture ε_fmm/mm, 工程應變engineering strain

3幾何形狀Geometry (SI: mm = 10⁻³ m)

圓形Circular 矩形Rectangular 橢圓形Elliptical

直徑Diameter d₀mm, 內定default 12.5

寬度Width b₀mm, 內定default 20

厚度Thickness t₀mm, 內定default 6

長軸Major axis 2a₀mm, 內定default 16

短軸Minor axis 2b₀mm, 內定default 10

標距長度Gauge length L₀mm, 內定default 50

內定尺寸參考 ASTM E8 標準試片（L₀ = 4d₀）。伸長百分率與 L₀/√A₀ 有關（Barba 定律），改變標距可觀察此效應。Default dimensions follow the ASTM E8 standard specimen (L₀ = 4d₀). Percent elongation depends on L₀/√A₀ (Barba's law); change the gauge length to see the effect.

4加載條件Loading

施加力上限Force limit F_maxN, 內定default 100 000

應變速率Strain rate ε̇s⁻¹, 內定default 0.001

動畫播放倍率Playback speed×, 最低 0.001；1 = 實際時間min 0.001; 1 = real time

自動（整個試驗約 12 秒播完）Auto (whole test in about 12 s)

加載循環次數Loading cycles內定無限次default: unlimited

無限次循環Unlimited cycles (∞)

3D 變形放大倍數3D deformation scale只影響 3D 顯示，數值輸出皆為真實值3D display only; all outputs are true values

自動（延性 ×1 真實比例；脆性自動放大）Auto (ductile ×1 true scale; brittle magnified)

請輸入有效的正數（卜松比須介於 0 與 0.5 之間）。Please enter valid positive numbers (Poisson's ratio must be between 0 and 0.5).

抗拉強度 σ_UTS 必須大於降伏強度 σ_y。Tensile strength σ_UTS must exceed yield strength σ_y.

頸縮時應變 ε_u 必須大於降伏應變Strain at necking ε_u must exceed the yield strain σ_y/E =

破壞時應變 ε_f 不可小於頸縮時應變 ε_u。Strain at fracture ε_f cannot be smaller than strain at necking ε_u.

破壞時應變小於 σ_f/E，已改以線彈性處理：Strain at fracture is below σ_f/E, so the material is treated as linear elastic: σ_f = E·ε_f = MPa

F_max = N < F_peak = N：桿件不會破壞。每次循環加載至上限後沿斜率 E 彈性卸載，再加載時沿同一直線回到上限。the bar will not fracture. Each cycle loads to the limit, unloads elastically along slope E, and reloads along the same line.

已完成設定的加載循環次數。按「重設」或「開始」重新試驗。The set number of loading cycles is complete. Press Reset or Start to test again.

待機Ready 彈性變形Elastic 均勻塑性變形Uniform plastic 頸縮Necking 斷裂Fracture 達施加力上限，卸載Force limit reached, unloading 再加載Reloading 完成Done F = 0 N ε = 0 試驗時間Test time t = 0 s 循環Cycle 1 / ∞ 材料最大荷重Peak load 0 N 播放倍率Speed ×1 單步Step Δε = 0

局部真實應力Local true stress σ_T (MPa)

00

3D 變形放大3D scale ×1

此瀏覽器無法啟用 WebGL，3D 模型無法顯示；應力–應變圖與數值輸出仍可正常使用。WebGL is not available in this browser, so the 3D model cannot be shown. The stress-strain chart and numerical outputs still work.

顯示原始外形（半透明）Show original shape (translucent) 尺寸標註Dimensions 應力色階Stress colours

拖曳可旋轉。藍色標註為目前標距長度 L，灰色為原始 L₀，紅色為最小斷面（頸縮處）的橫向尺寸；「斷面視」隱藏右半段，直接看中央斷面與原始輪廓（灰線）的差距。Drag to rotate. Blue marks the current gauge length L, grey the original L₀, and red the lateral size of the minimum (necked) section. "Section" hides the right half so the central section can be compared with the original outline (grey line).

應力Stress σ (MPa)

應變Strain ε (mm/mm)

目前點Current point: σ = 0 MPaσ_T = 0 MPaε_T = 0

工程應力–應變Engineering 真實應力–應變True 理論曲線（預覽）Theoretical curve (preview)

即時座標Live coordinates (ε, σ) & (ε_T, σ_T) UTS 與破壞點座標UTS & fracture coordinates 即時韌性Live toughness W = ∫σ dε 顯示面積（淡藍 = 韌性，淡綠 = 彈性能）Show areas (light blue = toughness, light green = elastic energy) 理論曲線Theoretical curve

縮放與平移：滾輪（或雙指）縮放，拖曳平移；游標放在 X 軸或 Y 軸刻度區滾動時只縮放該軸；雙擊或「重設視圖」回到全圖。Zoom and pan: scroll (or pinch) to zoom, drag to pan; scrolling over the X- or Y-axis labels zooms that axis only; double-click or "Reset view" to see the whole curve.
圖上標記：Y = 降伏點、UTS = 抗拉強度（頸縮開始）、深紅 X = 工程曲線斷裂點、淺紅 X = 真實曲線斷裂點；虛橫線為施加力上限對應的工程應力Markers: Y = yield point, UTS = tensile strength (onset of necking), dark red X = engineering fracture point, light red X = true fracture point; the dashed horizontal line is the engineering stress at the force limit F_max/A₀.

輸出結果Results (目前狀態current state斷裂瞬間；伸長率與斷面縮率為斷裂後彈性回復值at fracture; %EL and %RA after elastic recovery)

	項目Output	定義Definition	數值（SI 單位）Value (SI units)
1	工程斷面積Original (engineering) area	A0 (未變形斷面undeformed section)
2	實際斷面積（考慮頸縮）Instantaneous area (incl. necking)	A = A0·(d/d0)2, 取頸縮處最小斷面minimum section
3	縱向（受力方向）變形量及應變Axial deformation & strain	δ = L − L0; ε = δ/L0
4	橫向變形量及應變Lateral deformation & strain	最小斷面Minimum section: Δd = d − d0; εlat = Δd/d0 均勻段視卜松比Apparent Poisson's ratio (uniform part) −εlat/ε
5	工程應力Engineering stress	σ = F/A0
6	真實應力True stress	σT = F/A
7	工程應變Engineering strain	ε = (L − L0)/L0
8	真實應變True strain	εT = ln(1 + ε)頸縮後取頸縮處局部值after necking: local value in the neck εT = ln(l/l0)neck ≈ ln(A0/A)
9	伸長百分率Percent elongation %EL	目前now: (L − L0)/L0 × 100斷裂後after fracture: (Lf − L0)/L0 × 100 (扣除彈性回復elastic recovery removed)
10	面積縮小率Percent reduction in area %RA	目前now: (A0 − A)/A0 × 100斷裂後after fracture: (A0 − Af)/A0 × 100
11	彈性能（回彈模數）Modulus of resilience Ur	Ur = σy2/(2E)Ur = σf2/(2E) (脆性：以破壞強度計brittle: based on fracture strength) 整支標距段Whole gauge section: Ur·V0
12	韌性Toughness Ut	Ut = ∫0εf σ dε (工程曲線下面積area under the engineering curve) 目前已吸收Absorbed so far: W = ∮σ dε

模型假設Model assumptions

• 均佈載重：軸力 F 均勻分布於斷面，頸縮前整段標距均勻變形。彈性區依輸入的卜松比收縮；塑性變形體積不變（塑性卜松比 0.5）。Uniform loading: the axial force F is uniformly distributed over the section and the gauge length deforms uniformly before necking. Elastic contraction follows the input Poisson's ratio; plastic deformation conserves volume (plastic Poisson's ratio 0.5).
• 延性材料：σ_y 前為線彈性；σ_y 至 σ_UTS 為平滑的應變硬化曲線，於 ε_u 達最大荷重（dF = 0，Considère 準則）；之後真實應力以冪次律 σ_T ∝ ε_Tⁿ（n = ln(1+ε_u)）繼續硬化，但變形集中於中央頸縮區（寬度約 0.35 倍等效直徑的高斯分布），荷重因此下降。Ductile: linear elastic up to σ_y; smooth strain hardening from σ_y to σ_UTS, reaching maximum load at ε_u (dF = 0, Considère criterion). The true stress then keeps hardening as a power law σ_T ∝ ε_Tⁿ (n = ln(1+ε_u)), but deformation concentrates in a central Gaussian neck about 0.35 equivalent diameters wide, so the load falls.
• 脆性材料：以 Ramberg–Osgood 型曲線描述；若 ε_f = σ_f/E 即為完全線彈性。無頸縮。Brittle: a Ramberg-Osgood type curve; if ε_f = σ_f/E it is fully linear elastic. No necking. ε = σ/E + (ε_f − σ_f/E)(σ/σ_f)³
• 本模型為率無關：應變速率只決定試驗時間 t = ε/ε̇，不改變曲線。The model is rate-independent: the strain rate only sets the test time t = ε/ε̇ and does not change the curve.
• 矩形與橢圓斷面頸縮時假設兩向等比例收縮；夾持段與過渡段視為不變形。Rectangular and elliptical sections are assumed to contract proportionally in both directions when necking; grips and shoulders are treated as rigid.
• 加載循環：達破壞即為一次循環，換新試片重做；若施加力上限小於最大荷重，每次「加載至上限 → 彈性卸載至 F = 0」為一次循環。Loading cycles: reaching fracture ends one cycle and a new specimen is tested; if the force limit is below the peak load, each "load to the limit → unload elastically to F = 0" counts as one cycle.

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