Chapter 4: Processing and Heat Treatment of Metallic Materials

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: Work hardening parameter fitting

Purpose: Demonstrate machine learning model training and evaluation
Target: Intermediate
Execution time: 1-5 minutes
Dependencies: None
"""

import numpy as np from scipy.optimize import curve_fit # Experimental Data strain_exp = np.array([0.1, 0.3]) stress_exp = np.array([300, 420]) # Work Hardeningmoderu（σ = K * ε^n） def work_hardening_model(strain, K, n): return K * strain**n # line Minimumtwo Parameterfit params, covariance = curve_fit(work_hardening_model, strain_exp, stress_exp, p0=[500, 0.3]) K_fit, n_fit = params print("=== Work Hardening Parameter Fitting ===") print(f"Strength coefficient K = {K_fit:.1f} MPa") print(f"Work hardening exponent n = {n_fit:.3f}") #: fitCurve and experimentvalue of Ratiocomparison stress_fit = work_hardening_model(strain_exp, K_fit, n_fit) print("\n=== Verification ===") for i in range(len(strain_exp)): print(f"ε = {strain_exp[i]:.1f}: Experimental σ = {stress_exp[i]} MPa, " f"Fitted σ = {stress_fit[i]:.1f} MPa") # numberal （2pointfromCalculation） # σ1 = K * ε1^n, σ2 = K * ε2^n # σ2/σ1 = (ε2/ε1)^n → n = log(σ2/σ1) / log(ε2/ε1) n_algebraic = np.log(stress_exp[1] / stress_exp[0]) / np.log(strain_exp[1] / strain_exp[0]) K_algebraic = stress_exp[0] / (strain_exp[0]**n_algebraic) print("\n=== Algebraic Solution (2-point method) ===") print(f"n = log({stress_exp[1]}/{stress_exp[0]}) / log({strain_exp[1]}/{strain_exp[0]}) = {n_algebraic:.3f}") print(f"K = {stress_exp[0]} / {strain_exp[0]}^{n_algebraic:.3f} = {K_algebraic:.1f} MPa") # StressPrediction（ε = 0.5 of value） strain_pred = 0.5 stress_pred = work_hardening_model(strain_pred, K_fit, n_fit) print(f"\n=== Prediction ===") print(f"At ε = {strain_pred}, predicted σ = {stress_pred:.1f} MPa")

Expected output:

Strength coefficient K = 530-550 MPa Work hardening exponent n = 0.18-0.20

Explanation: From two data points, we can solve algebraically or use curve_fit for least-squares fitting. An n value around 0.2 is typical for moderately work-hardening materials like low carbon steel.

# Copper of Melting Point T_m_Cu_K = 1358 # Kelvin T_m_Cu_C = 1085 # Celsius # recrystallizationTemperature of （T_rex ≈ 0.4 * T_m） T_rex_K = 0.4 * T_m_Cu_K T_rex_C = T_rex_K - 273.15 print("=== Recrystallization Temperature Estimation ===") print(f"Melting point of Cu: {T_m_Cu_K} K ({T_m_Cu_C}°C)") print(f"Estimated recrystallization temperature: {T_rex_K:.0f} K ({T_rex_C:.0f}°C)") # Stress annealingTemperature of stress_relief_temp = 200 # °C print(f"\n=== Heat Treatment at {stress_relief_temp}°C ===") if stress_relief_temp

Expected output:

Estimated recrystallization temperature: 543 K (270C) At 200C: Recovery occurs, but NO recrystallization

Explanation: The recrystallization temperature varies by material and is typically 0.3 to 0.5 times the melting point. For copper, recrystallization occurs above approximately 270C. At 200C, only recovery (dislocation rearrangement) occurs, but new strain-free grains do not form.

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: Ms temperature calculation and microstructure prediction

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np def calculate_ms_temperature(C, Mn=0, Ni=0, Cr=0): """ Martensitetransformation startTemperatureMs of Calculation（Andrewsequation） Ms = 539 - 423*C - 30.4*Mn - 17.7*Ni - 12.1*Cr """ Ms = 539 - 423*C - 30.4*Mn - 17.7*Ni - 12.1*Cr return Ms # Steel of izationstudyComposition C_content = 0.6 # wt% Mn_content = 0.8 # wt% Ni_content = 0 Cr_content = 0 # MsTemperature of Calculation Ms_temp = calculate_ms_temperature(C_content, Mn_content, Ni_content, Cr_content) print("=== Steel Composition ===") print(f"C: {C_content} wt%, Mn: {Mn_content} wt%, Ni: {Ni_content} wt%, Cr: {Cr_content} wt%") print(f"\n=== Martensite Start Temperature ===") print(f"Ms = 539 - 423×{C_content} - 30.4×{Mn_content} - 17.7×{Ni_content} - 12.1×{Cr_content}") print(f"Ms = {Ms_temp:.1f}°C") # MfTemperature of （ al Ms - 215°C） Mf_temp = Ms_temp - 215 print(f"Mf (estimated) ≈ {Mf_temp:.1f}°C") # water cooling of case quench_temp_water = 20 # °C print(f"\n=== Water Quenching (to {quench_temp_water}°C) ===") if quench_temp_water<ms_temp: #="" (extremely="" (high="" (mixed)")="" (negative="lowers" (not="" (slightly="" (to="" ({ms_temp:.1f}°c)")="" ({quench_temp_oil}°c)="" ({quench_temp_water}°c)="" ({quench_temp_water}°c)<ms="" +="" 1-2="" 1.="" 200-400°c,="" 30-60="" 820-850°c,="" after="" alloyelement="" alloying="" at="" austenite="" austenitizing:="" avoid="" bainite="" bainite")="" better="" brittle)")="" brittleness,="" but="" case="" code="" completemartensite（simplified="" cooling="" cooling)=") if quench_temp_oil<Ms_temp: # oil cooling is water coolingthan for、one Pearlite/Bainitetransformation of possibleity print(f" cracking)")="" during="" elements="" else:="" expected="" final="" full="" hard="" hardness="" hardness)")="" hardness:="" heat="" higher="" hours")="" hrc="" hrc")="" improves="" influence="" lower="" martensite="" martensite")="" martensite_fraction="100" may="" min")="" ms=") elements = {'C': -423, 'Mn': -30.4, 'Ni': -17.7, 'Cr': -12.1} for elem, coeff in elements.items(): print(f" ms)")<="" mstemperature="" occurs="" of="" oil="" on="" pearlite="" per="" possible="" print(f"="" print(f"2.="" print(f"3.="" print(f"\n="Effect" print(f"final="" print(f"→="" quench="" quench)")="" quench_temp_oil="60" quenching="" quenching")="" quenching:="" rate:="" recommended)")="" recommendedheat="" reduces="" retained="" slow="" slower="" structure:="" temperature="" temperatures")="" tempering:="" than="" then="" to="" toughness="" toughness")="" transform="" transformation="" treatment=") print(f" water="" wt%="" {coeff:.1f}°c="" {elem}:="" {quench_temp_oil}°c,="" ~45-55="" ~50-55="" ~60-65="" ~{martensite_fraction:.0f}%="" °c="" →="" ≥="" ）=""></ms_temp:>

Expected output:

Ms = 261C, Mf = 46C Water quenching: ~100% Martensite (high hardness, brittle) Oil quenching: Martensite + Bainite (mixed, better toughness)

Explanation: For steel with 0.6% C, the Ms temperature is approximately 260C, and cooling to room temperature results in martensite formation. Water quenching produces complete martensite due to rapid cooling, while oil quenching has a slower cooling rate and may allow some bainite formation. After quenching, tempering is important to improve toughness.

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: JMA equation recrystallization time calculation

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import numpy as np import matplotlib.pyplot as plt def calculate_jma_time(X_target, k, n): """ JMAequationfromta gettransformationrate dotime Calculation X = 1 - exp(-k*t^n) → t = (ln(1/(1-X)) / k)^(1/n) Parameters: ----------- X_target: float transformationrate（0～1） k: float rateConstant [1/min^n] n: float AvramiIndex Returns: -------- t: float time [min] """ if X_target>= 1.0: return np.inf t = (np.log(1 / (1 - X_target)) / k)**(1 / n) return t # Parameter k = 0.01 # 1/min^n n = 2.5 # AvramiIndex # 50%recrystallizationtime X_50 = 0.5 t_50 = calculate_jma_time(X_50, k, n) # 90%recrystallizationtime X_90 = 0.9 t_90 = calculate_jma_time(X_90, k, n) print("=== JMA Recrystallization Kinetics ===") print(f"Parameters: k = {k} (1/min^{n}), n = {n}") print(f"\n=== Calculation Results ===") print(f"50% recrystallization time (t_50):") print(f" t_50 = (ln(1/(1-{X_50})) / {k})^(1/{n})") print(f" t_50 = (ln(2) / {k})^(1/{n})") print(f" t_50 = {t_50:.2f} minutes ({t_50/60:.2f} hours)") print(f"\n90% recrystallization time (t_90):") print(f" t_90 = (ln(1/(1-{X_90})) / {k})^(1/{n})") print(f" t_90 = (ln(10) / {k})^(1/{n})") print(f" t_90 = {t_90:.2f} minutes ({t_90/60:.2f} hours)") # time of Ratiocomparison ratio = t_90 / t_50 print(f"\n=== Time Ratio ===") print(f"t_90 / t_50 = {ratio:.2f}") print(f"→ 90% recrystallization takes {ratio:.2f}× longer than 50%") # recrystallizationCurve of Plot time = np.linspace(0, t_90 * 1.5, 200) X = 1 - np.exp(-k * time**n) plt.figure(figsize=(10, 6)) plt.plot(time, X * 100, 'b-', linewidth=2, label='Recrystallization Curve') plt.axhline(y=50, color='green', linestyle='--', linewidth=1.5, label=f't_50 = {t_50:.1f} min') plt.axvline(x=t_50, color='green', linestyle='--', linewidth=1.5, alpha=0.5) plt.axhline(y=90, color='red', linestyle='--', linewidth=1.5, label=f't_90 = {t_90:.1f} min') plt.axvline(x=t_90, color='red', linestyle='--', linewidth=1.5, alpha=0.5) plt.plot(t_50, 50, 'go', markersize=10) plt.plot(t_90, 90, 'ro', markersize=10) plt.xlabel('Time [minutes]', fontsize=12) plt.ylabel('Recrystallized Fraction [%]', fontsize=12) plt.title(f'JMA Recrystallization Curve (k={k}, n={n})', fontsize=14) plt.legend(fontsize=11) plt.grid(True, alpha=0.3) plt.xlim(0, time[-1]) plt.ylim(0, 100) plt.tight_layout() plt.savefig('jma_calculation.png', dpi=300, bbox_inches='tight') plt.show() # differenttransformationrate of timeCalculation print("\n=== Time for Various Recrystallization Fractions ===") X_values = [0.1, 0.25, 0.5, 0.75, 0.9, 0.95, 0.99] for X in X_values: t = calculate_jma_time(X, k, n) print(f"X = {X*100:5.1f}% → t = {t:7.2f} min ({t/60:5.2f} hours)")

Expected output:

t_50 = 5.28 minutes (0.09 hours) t_90 = 13.00 minutes (0.22 hours) t_90 / t_50 = 2.46

Explanation: To solve for time from the JMA equation, use \(t = [\ln(1/(1-X)) / k]^{1/n}\). The time ratio from 50% transformation (\(\ln 2\)) to 90% transformation (\(\ln 10\)) depends on the Avrami index. For n = 2.5, it takes approximately 2.46 times longer.

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: Age hardening condition optimization

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import numpy as np import matplotlib.pyplot as plt from scipy.interpolate import interp1d # Experimental Data data = { 150: {'time': [10, 50, 200], 'hardness': [120, 140, 135]}, 175: {'time': [3, 15, 50], 'hardness': [120, 135, 125]}, 200: {'time': [1, 5, 20], 'hardness': [115, 125, 115]} } # MaximumHardness and pi ktime of determination print("=== Age Hardening Analysis ===") print(f"{'Temp [°C]':<12} {'Max Hardness [HV]':<20} {'Peak Time [h]':<15}") print("-" * 50) results = {} for temp, values in data.items(): times = np.array(values['time']) hardness = np.array(values['hardness']) max_hardness = np.max(hardness) peak_time = times[np.argmax(hardness)] results[temp] = {'max_hardness': max_hardness, 'peak_time': peak_time} print(f"{temp:<12} {max_hardness:<20} {peak_time:<15}") # recommendedcondition of determination print("\n=== Recommendation ===") best_temp = max(results, key=lambda x: results[x]['max_hardness']) print(f"Best temperature: {best_temp}°C") print(f"Maximum hardness: {results[best_temp]['max_hardness']} HV") print(f"Time to peak: {results[best_temp]['peak_time']} hours") # Practical recommended（formation ity also ） print("\n=== Practical Recommendation (T6 Treatment) ===") print(f"Option 1 (Maximum hardness): {best_temp}°C, {results[best_temp]['peak_time']} hours") print(f" → Highest hardness ({results[best_temp]['max_hardness']} HV)") print(f" → Suitable for maximum strength applications") print(f"\nOption 2 (Balanced): 175°C, 15 hours") print(f" → Good hardness (135 HV, ~96% of max)") print(f" → Shorter processing time (3.3× faster than 150°C)") print(f" → Better productivity") print(f"\nOption 3 (Rapid): 200°C, 5 hours") print(f" → Moderate hardness (125 HV, ~89% of max)") print(f" → Very fast processing (10× faster than 150°C)") print(f" → Suitable for rapid production") # age hardeningCurve of Plot fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5)) # Figure: Experimental Data and Curve colors = {150: 'blue', 175: 'green', 200: 'red'} for temp, values in data.items(): times = np.array(values['time']) hardness = np.array(values['hardness']) # （cubic spline） time_interp = np.linspace(times[0], times[-1], 100) f_interp = interp1d(times, hardness, kind='quadratic', fill_value='extrapolate') hardness_interp = f_interp(time_interp) ax1.plot(time_interp, hardness_interp, color=colors[temp], linewidth=2, label=f'{temp}°C', alpha=0.7) ax1.plot(times, hardness, 'o', color=colors[temp], markersize=8) # pi kpoint ma k peak_idx = np.argmax(hardness) ax1.plot(times[peak_idx], hardness[peak_idx], '*', color=colors[temp], markersize=15, markeredgecolor='black', markeredgewidth=1) ax1.set_xlabel('Aging Time [hours]', fontsize=12) ax1.set_ylabel('Hardness [HV]', fontsize=12) ax1.set_title('Age Hardening Curves for Al-Cu Alloy', fontsize=14) ax1.legend(fontsize=11) ax1.grid(True, alpha=0.3) ax1.set_xscale('log') # Figure: MaximumHardness and time of Relation temps_list = sorted(results.keys()) max_hardness_list = [results[t]['max_hardness'] for t in temps_list] peak_time_list = [results[t]['peak_time'] for t in temps_list] ax2_twin = ax2.twinx() ax2.plot(temps_list, max_hardness_list, 'bo-', linewidth=2, markersize=10, label='Max Hardness') ax2_twin.plot(temps_list, peak_time_list, 'rs-', linewidth=2, markersize=10, label='Peak Time') ax2.set_xlabel('Aging Temperature [°C]', fontsize=12) ax2.set_ylabel('Maximum Hardness [HV]', fontsize=12, color='blue') ax2_twin.set_ylabel('Time to Peak [hours]', fontsize=12, color='red') ax2.tick_params(axis='y', labelcolor='blue') ax2_twin.tick_params(axis='y', labelcolor='red') ax2.set_title('Temperature vs. Max Hardness & Peak Time', fontsize=14) ax2.grid(True, alpha=0.3) plt.tight_layout() plt.savefig('age_hardening_analysis.png', dpi=300, bbox_inches='tight') plt.show() # excessaging of evaluate print("\n=== Over-aging Analysis ===") for temp in [150, 175, 200]: hardness_data = data[temp]['hardness'] if hardness_data[-1]<max(hardness_data): (loss="{loss}" *="" -="" 100="" code="" detected="" else:="" hardness_data[-1]="" hv,="" in="" loss="max(hardness_data)" loss_pct="(loss" max(hardness_data))="" no="" observed="" over-aging="" print(f"{temp}°c:="" range")<="" time="" {loss_pct:.1f}%)")=""></max(hardness_data):>

Expected output:

Best temperature: 150C (Max hardness: 140 HV at 50 hours) Practical recommendation: 175C, 15 hours (135 HV, faster processing) Over-aging detected at all temperatures (hardness decreases after peak)

Explanation: At lower temperature (150C), maximum hardness is highest but requires long time. For practical purposes, 175C for 15 hours offers a good balance between hardness (135 HV) and productivity. Holding for too long results in overaging and decreased hardness, so proper aging time control is important.

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: TTT diagram microstructure prediction

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import numpy as np import matplotlib.pyplot as plt # TTTFigure of Parameter Ms_temp = 220 # Martensitetransformation startTemperature [°C] pearlite_nose = {'temp': 550, 'time': 1} # Pearliteno zu [°C, s] bainite_nose = {'temp': 350, 'time': 10} # Bainiteno zu [°C, s] # Coolingshinario scenarios = { 'a': {'cool_time': 1, 'cool_to': 550, 'hold_temp': 550, 'hold_time': 10}, 'b': {'cool_time': 0.1, 'cool_to': 200, 'hold_temp': None, 'hold_time': 0}, 'c': {'cool_time': 10, 'cool_to': 350, 'hold_temp': 350, 'hold_time': 100} } def predict_structure(scenario_name, scenario): """ TTTFigure Microstructure Prediction """ print(f"\n=== Scenario {scenario_name.upper()} ===") print(f"Cooling: {scenario['cool_time']}s to {scenario['cool_to']}°C") if scenario['hold_temp']: print(f"Holding: {scenario['hold_temp']}°C for {scenario['hold_time']}s") cool_time = scenario['cool_time'] cool_to = scenario['cool_to'] hold_temp = scenario['hold_temp'] hold_time = scenario['hold_time'] # Cooling of transformation if cool_to == pearlite_nose['temp'] and cool_time>= pearlite_nose['time']: print(f"→ Crosses pearlite nose during cooling") print(f"→ Partial pearlite transformation likely") partial_pearlite = True else: partial_pearlite = False # holding of transformation if hold_temp == pearlite_nose['temp'] and hold_time>= pearlite_nose['time']: print(f"→ Isothermal holding at pearlite nose temperature") print(f"→ COMPLETE pearlite transformation") structure = "100% Pearlite" hardness = "~20-25 HRC (soft, ductile)" elif hold_temp == bainite_nose['temp'] and hold_time>= bainite_nose['time']: print(f"→ Isothermal holding at bainite nose temperature") print(f"→ COMPLETE bainite transformation") structure = "100% Bainite" hardness = "~40-50 HRC (moderate hardness, good toughness)" elif cool_to<ms_temp "="*70) print(" #="" 'b',="" 'b-',="" 'c']:="" 'co-',="" 'hardness':="" 'ko-',="" 'mo-',="" 'r-',="" (bainite):="" (balanced="" (high="" (martensite):="" (maximum="" (np.log10(time="" (pearlite):="" ({ms_temp}°c)")="" )="" *="" +="" ,="" 150="" 1e4)="" 200)="" 200]="" 250,="" 350,="" 350]="" 4,="" 400,="" 500)="" 550,="" 550]="" 727)="" 8))="" 80="" 900)="" =="" ="*70)="" ['a',="" [s]',="" [°c]',="" a="" alpha="0.3," analyze="" and="" applications=") print(" ax="plt.subplots(figsize=(12," ax.axhline(y="Ms_temp," ax.grid(true,="" ax.legend(fontsize="11," ax.plot(time_a,="" ax.plot(time_b,="" ax.plot(time_c,="" ax.semilogx(time,="" ax.set_title('ttt="" ax.set_xlabel('time="" ax.set_xlim(0.1,="" ax.set_ylabel('temperature="" ax.set_ylim(0,="" b="" b_start="np.clip(B_start," b_start,="" bainite_nose['time']))**2="" bainitetransformationcurve="" bbox_inches="tight" bearings="" below="" c="" code="" color="green" cool_time_a="" cool_time_a,="" cool_time_c="" cool_time_c,="" cooling="" cooling")="" coolingcurve="" cutting="" diagram="" dpi="300," ductility,="" eachshinario="" else:="" eutectoid="" fig,="" fontsize="15," fontweight="bold" for="" gears,="" hardness="" hardness)")="" hardness}="" hardness】:="" hold_temp="" hold_time_a="scenarios['a']['hold_time']" hold_time_a]="" hold_time_c="scenarios['c']['hold_time']" hold_time_c]="" if="" in="" intermediate="" is="" label="Scenario C" linestyle="-" linewidth="2.5," loc="upper right" markersize="8," martensite="" ms="" name="" name,="" none:="" number）="" of="" p_start="np.clip(P_start," p_start,="" partial_pearlite:="" paths="" pearlite_nose['time']))**2="" pearlitetransformationcurve（simplifiedc="" plot="" plt.savefig('ttt_scenarios.png',="" plt.show()="" plt.tight_layout()="" prediction="" print("-"*70)="" print("\n="Recommended" print("\n"="" print("scenario="" print(f"\n【predicted="" print(f"{'scenario':<12}="" print(f"{name.upper():<12}="" print(f"→="" print(f"【expected="" rails,="" rapid="" ratiocomparisontable="" recommendeduse="" resistance)")="" result="" results=") print(" results[name]="{'structure':" return="" ropes="" scenario="" scenario)="" scenarios.items():="" scenarios['b']['cool_time']]="" shinario="" simplifiedplot（conceptfigure）="" springs="" steel',="" structure="" structure,="" structure】:="" temp_a="[850," temp_a,="" temp_b="[850," temp_b,="" temp_c="[850," temp_c,="" time="np.logspace(-1," time_a="[0," time_b="[0," time_c="[0," tools,="" toughness)")<="" transformation")="" tttfigure="" wear="" which="both" wire="" {'hardness':<20}")="" {'structure':<35}="" {hardness}")="" {ms_temp}°c')="" {results[name]['hardness']:<20}")="" {results[name]['structure']:<35}="" {structure}")="" （="" ）=""></ms_temp>

Expected output:

Scenario A: 100% Pearlite (~20-25 HRC) Scenario B: ~100% Martensite (~63-65 HRC) Scenario C: 100% Bainite (~40-50 HRC)

Explanation: (a) Holding at the pearlite nose temperature results in complete pearlite transformation. (b) Rapid cooling below Ms produces a martensitic microstructure. (c) Holding at the bainite nose temperature for sufficient time completes the bainite transformation. By confirming where the cooling curve crosses transformation curves on the TTT diagram, microstructure prediction is possible.