materials-science

# Materials Science ## What I Do I provide comprehensive materials science tools including crystal structure analysis, mechanical properties, phase diagrams, polymers, ceramics, and composite materials for materials engineering applications. ## When to Use Me - Material selection and characterization - Mechanical property prediction - Phase diagram analysis - Failure analysis - Microstructure engineering - Corrosion analysis ## Core Concepts - **Crystal Structures**: BCC, FCC, HCP, defects - **Mechanical Properties**: Strength, hardness, toughness - **Phase Diagrams**: Binary, ternary, invariant reactions - **Polymers**: Molecular weight, Tg, crystallinity - **Ceramics**: Brittle fracture, thermal properties - **Composites**: Rule of mixtures, laminate theory - **Corrosion**: Electrochemical, pitting, galvanic - **Materials Selection**: Ashby charts, indices ## Code Examples ### Crystal Structures ```python import numpy as np def lattice_parameter_a(atomic_radius, structure): if structure == 'BCC': return 4 * atomic_radius / np.sqrt(3) elif structure == 'FCC': return 2 * np.sqrt(2) * atomic_radius elif structure == 'HCP': return 2 * atomic_radius return None def atomic_packing_factor(structure): if structure == 'BCC': return np.pi * np.sqrt(3) / 8 elif structure == 'FCC': return np.pi / np.sqrt(18) elif structure == 'HCP': return np.pi / (3 * np.sqrt(2)) return None def miller_indices_distance(d_hkl, a, h, k, l): return a / np.sqrt(h**2 + k**2 + l**2) def schmid_factor(phi, lambda_): return np.cos(np.radians(phi)) * np.cos(np.radians(lambda_)) def critical_resolved_shear_stress(tau_y, schmid_factor): return tau_y / schmid_factor r_Fe = 1.24e-10 # Angstroms a_BCC = lattice_parameter_a(r_Fe, 'BCC') print(f"BCC lattice parameter: {a_BCC:.4f} Angstroms") APF = atomic_packing_factor('FCC') print(f"FCC atomic packing factor: {APF:.4f}") ``` ### Mechanical Properties ```python def youngs_modulus_voigt(E1, E2, V1): return V1 * E1 + (1 - V1) * E2 def poisson_ratio_nu(E, G): return E / (2 * G) - 1 def yield_strength_hall_petch(d, d0, sigma0, ky): return sigma0 + ky / np.sqrt(d) def work_hardening_exponent(true_strain, true_stress, n): return np.log(true_stress) - n * np.log(true_stress / K) def fracture_toughness_KIC(Y, sigma, a): return Y * sigma * np.sqrt(np.pi * a) def creep_rate_A(sigma, Q, R, T, n): return A * sigma**n * np.exp(-Q / (R * T)) E = 200e9 # GPa G = 80e9 # GPa nu = poisson_ratio_nu(E, G) print(f"Poisson ratio: {nu:.3f}") ``` ### Phase Diagrams ```python def lever_rule(C0, C_alpha, C_beta, f_alpha): return f_alpha = (C_beta - C0) / (C_beta - C_alpha) def gibbs_phase_rule(F, C, P): return F = C - P + 2 def ttt_diagram_start_time(A, Q, R, T): return 1 / A * np.exp(Q / (R * T)) def continuous_cooling_transformation(T, t): return np.exp(-((T - Ms) / M)**2 / (2 * t**2)) def invariant_reaction_type(composition_range): if composition_range > 0: return 'eutectic' elif composition_range < 0: return 'eutectoid' return 'peritectic' def t0_temperature(T_melt, H_fus, delta_S): return T_melt - H_fus / delta_S C0, C_alpha, C_beta = 40, 10, 80 f_alpha = lever_rule(C0, C_alpha, C_beta, 0) print(f"Alpha fraction: {f_alpha:.2%}") ``` ### Polymer Properties ```python def glass_transition_tpTg, delta_Cp): return Tg + delta_Cp def degree_of_crystallinity(X_c, Delta_H_m, Delta_H_100): return Delta_H_m / Delta_H_100 * 100 def number_average_molecular_weight(M0, DP): return M0 * DP def weight_average_molecular_weight(Mn, PDI): return Mn * PDI def mark_houwink_sakurada(M, K, a): return K * M**a def rheology_viscosity(eta0, shear_rate, n): return eta0 * shear_rate**(n - 1) def time_temperature_superposition(T, T_ref, aT, freq_ref): return freq_ref * aT M0 = 100 # g/mol (monomer) DP = 1000 Mn = number_average_molecular_weight(M0, DP) print(f"Number average MW: {Mn:.0f} g/mol") ``` ### Composite Materials ```python def rule_of_mixtures_isostrain(E_c, E_f, V_f, E_m): return E_c = E_m * (1 - V_f) + E_f * V_f def rule_of_mixtures_isostress(E_c, E_f, V_f, E_m): return E_c = 1 / ((1 - V_f)/E_m + V_f/E_f) def halpin_tsai_modulus(E_f, E_m, V_f, aspect_ratio): return E_c = E_m * (1 + xi * eta * V_f) / (1 - eta * V_f) def laminate_stiffness(A_matrix, D_matrix): return A, B, D def shear_lag_model(sigma_m, sigma_f, s): return sigma_f = sigma_m * (1 - np.exp(-s / l)) / (1 - V_f) def fiber_volume_fraction(w_f, rho_f, rho_m): return V_f = 1 / (1 + (rho_m / rho_f) * ((1 - w_f) / w_f)) V_f = 0.6 # 60% fiber volume E_f = 230e9 # GPa (carbon) E_m = 3.5e9 # GPa (epoxy) E_c = rule_of_mixtures_isostrain(E_f, E_m, V_f, E_m) print(f"Composite modulus: {E_c:.1f} GPa") ``` ## Best Practices 1. **Microstructure-Property Relations**: Link structure to properties 2. **Processing-Microstructure**: Control processing for desired structure 3. **Testing**: Appropriate characterization techniques 4. **Statistical Analysis**: Account for variability 5. **Modeling**: Calibrate models with experimental data ## Common Patterns ```python # Materials selection (Ashby method) def materials_index(E, rho, sigma_y): return sigma_y / rho**0.5 ``` ## Core Competencies 1. Crystal structure and defects 2. Mechanical properties 3. Phase diagram analysis 4. Composite materials 5. Materials selection