Numerical Simulation of Thermal and Stress Fields in Medium-to-High Carbon Steel Weld Overlay

1. Definition and Fundamental Principles

1.1 Overview of the Subject

The numerical simulation of temperature field and stress field evolution during and after weld overlay on medium-to-high carbon steels (typically carbon content 0.25%–0.70% C) represents a critical computational engineering methodology. This technique employs finite element analysis (FEA) to predict the transient thermal history, residual stress distribution, distortion, and microstructural transformation zones that develop during multi-pass weld overlay operations. The simulation captures two distinct phases: the active deposition phase (heat source present) and the post-deposition cooling phase (heat source removed), each of which governs different metallurgical and mechanical outcomes.

1.2 Governing Physics

The simulation framework is built upon coupled thermo-mechanical finite element models governed by the following physical phenomena:

• Heat Transfer: Governed by the transient heat conduction equation with a moving heat source (Goldak double-ellipsoidal model or conical heat source), accounting for convective and radiative boundary conditions, latent heat of fusion, and temperature-dependent material properties (thermal conductivity, specific heat, density).
• Phase Transformation: Medium-to-high carbon steels are susceptible to martensitic transformation during rapid cooling, which introduces volumetric expansion (approximately 1–2% for fully martensitic transformation). This is modeled using Koistinen-Marburger or JMA (Johnson-Mehl-Avrami) kinetics equations.
• Residual Stress Development: Computed from elastic-plastic thermal strain, transformation strain, and mechanical constraint effects. The superposition of thermal contraction, plastic flow during deposition, and transformation-induced dilation determines the final residual stress state.
• Distortion Prediction: Accumulated plastic strain from successive weld passes leads to angular distortion, longitudinal warpage, and transverse shrinkage, which must be predicted to assess fixturing requirements and dimensional control.

1.3 Simulation Domain and Boundary Conditions

A representative simulation domain includes the base plate (medium-to-high carbon steel substrate), the weld overlay cladding layers, and a margin of unaffected base metal. Boundary conditions typically include:

• Convective heat transfer on exposed surfaces (heat transfer coefficient h = 5–25 W/m²·K depending on shielding gas and ambient conditions)
• Radiative boundary condition on top surfaces (emissivity ε = 0.6–0.9 for oxide-covered steel surfaces)
• Mechanical fixturing constraints representing clamps, welding tables, or backing plates
• Heat source parameters derived from WPS (Welding Procedure Specification) qualification data: welding current (I), arc voltage (V), travel speed (v), and heat input (q = VI/v)

2. Technical Purpose and Engineering Value

2.1 Rationale for Simulation on Medium-to-High Carbon Steels

Medium-to-high carbon steels present unique challenges in weld overlay applications that make numerical simulation indispensable:

• High Hardenability: Elevated carbon content (0.25%–0.70%) combined with alloying elements (Mn, Cr, Ni, Mo) results in high hardenability, making the Heat-Affected Zone (HAZ) and weld metal susceptible to hardening beyond 400 HV, sometimes exceeding 600 HV. This dramatically increases cold cracking susceptibility.
• Critical Cooling Rate Sensitivity: The critical cooling rate (t₈₀₀) for avoiding martensitic transformation in these steels is often below 20–40°C/s. Weld overlay processes frequently exceed this threshold, necessitating preheating, interpass temperature control, and post-weld heat treatment (PWHT).
• Residual Stress Magnitudes: Weld residual stresses in medium-to-high carbon steel cladding can approach or exceed the yield strength of the base material (σy = 350–620 MPa depending on grade and condition). These stresses can trigger delayed cracking, stress corrosion cracking (SCC), or fatigue failure.
• Complex Multi-Pass Thermal History: Thick cladding builds (commonly 6–25 mm for wear-resistant overlays) involve multiple passes with varying thermal histories, making analytical solutions intractable and empirical approaches unreliable.

2.2 Value to Qualification Building

Numerical simulation serves as a powerful tool in the qualification and optimization of weld overlay procedures:

1. WPS Optimization: Simulation enables virtual trials of heat input variations, preheat levels, interpass temperature windows, and backing plate configurations before physical qualification welds are performed, reducing qualification costs and cycle time.
2. HAZ Hardness Prediction: By correlating simulated cooling rates (t₈₀₀) with established hardness-cooling rate curves (e.g., from JIS Z 3146 or ASTM E 1079-based datasets), engineers can predict HAZ hardness profiles and identify risk zones requiring PWHT or procedure modification.
3. Residual Stress Assessment: Predicted residual stress fields inform the need for stress-relief heat treatment, mechanical stress relief (shot peening, hammer peening), or design allowances for residual stress in pressure vessel and piping applications governed by ASME Section VIII or NB/T 47014.
4. Distortion Control: Predicted angular and longitudinal distortion guides the design of welding fixtures, backing supports, and sequence planning to maintain dimensional tolerances specified in GB/T 19067 or relevant product specifications.

2.3 Value to Product Delivery and Customer Assurance

For Cladding Technology Shanxi Co., Ltd., simulation-derived data provides:

• Non-Destructive Verification: Computed residual stress distributions can be validated against experimental measurements (drill hole method per ASTM E837, neutron diffraction, or X-ray diffraction per ASTM E975), providing a non-destructive means of assuring stress levels in delivered products.
• Customer Technical Documentation: Simulation reports with temperature profiles, cooling rate maps, and residual stress contour plots serve as supplementary technical documentation in quality packages, demonstrating engineering rigor and compliance with customer specifications (e.g., API 579 FFS-1 assessment inputs).
• Risk Mitigation: Pre-qualification simulation identifies potential cracking zones, excessive hardness regions, and distortion hotspots, enabling proactive process adjustments that reduce field failure risk and warranty exposure.

3. Key Process and Implementation Points

3.1 Simulation Workflow

The numerical simulation workflow for weld overlay thermal-mechanical analysis follows a structured methodology:

1. Geometry Modeling: Create 3D or 2D axisymmetric geometry representing the base plate, weld passes, and boundary regions. For multi-pass overlay, a "dead reckoning" or "birth-death" element activation technique is used to sequentially activate weld elements as each pass is deposited.
2. Material Property Input: Define temperature-dependent properties for base metal, weld metal, and HAZ:
   • Base metal: thermal conductivity k(T), specific heat c(T), density ρ(T), elastic modulus E(T), yield strength σy(T), coefficient of thermal expansion α(T)
   • Weld metal: composition-specific properties based on filler metal selection (e.g., AWS A5.14 E71T-1, E80T-1, or E91T-1 cast iron for high-carbon base)
   • Phase transformation parameters: transformation start temperature (Ac1, Ac3), transformation kinetics coefficients, volumetric expansion upon martensite formation
3. Heat Source Definition: Implement a moving heat source model calibrated to the welding process:
   • Goldak double-ellipsoidal: suitable for TIG and MIG with asymmetric heat distribution (front and rear ellipsoids)
   • Conical heat source: alternative for deep-penetration processes
   • Point or line heat source: simplified model for preliminary analysis
4. Boundary and Initial Conditions: Apply preheat temperature, ambient conditions, and mechanical constraints reflecting actual welding setup.
5. Solver Configuration: Select coupled thermo-mechanical solver (e.g., ABAQUS, ANSYS, DEFORM, or specialized weld simulation software such as Sysweld or Q3D). Enable phase transformation module for medium-to-high carbon steels.
6. Post-Processing and Validation: Extract temperature histories at critical locations, compute cooling rates (t₈₀₀, t₆₀₀), map residual stress distributions, and compare with experimental data (thermocouple measurements, strain gauge readings, hardness surveys, X-ray stress measurements).

3.2 Critical Simulation Parameters for Medium-to-High Carbon Steel Overlay

Parameter	Typical Range	Engineering Significance
Base metal carbon content	0.25% – 0.70% C	Determines hardenability, HAZ cracking risk, and transformation behavior
Heat input (q = VI/v)	0.5 – 3.0 kJ/mm	Controls cooling rate; higher q reduces t₈₀₀ and hardening but increases distortion
Preheat temperature	100°C – 350°C	Reduces thermal gradient, slows cooling rate, mitigates cold cracking risk
Interpass temperature	150°C – 350°C	Controls successive pass thermal history and residual stress accumulation
Peak temperature (centerline)	1500°C – 2000°C	Defines fusion zone extent and dilution ratio
Critical cooling rate (t₈₀₀)	10°C/s – 100°C/s	Determines HAZ microstructure; below critical → martensite; above → ferrite-pearlite
Residual stress (longitudinal)	200 MPa – 600 MPa (tensile)	Must be assessed against base material yield strength and service stress
PWHT temperature	550°C – 680°C	Stress relief and microstructure softening; must be simulated for effectiveness

3.3 Post-Heat Source Removal Analysis

The "after heat source removal" phase of the simulation is particularly critical for medium-to-high carbon steels. Once the welding arc is extinguished, the following post-deposition phenomena dominate:

• Continued Thermal Contraction: The weld zone and HAZ cool from peak temperature to ambient, generating additional thermal compressive strain that converts to tensile residual stress upon unloading.
• Martensitic Transformation: As the HAZ cools below Ac1 (typically 720°C–780°C for medium carbon steels), martensitic transformation initiates. The associated volumetric expansion partially offsets thermal contraction but creates transformation-induced tensile stresses in constrained regions.
• Transformation Plasticity: In the two-phase region (austenite + martensite), transformation plasticity occurs under stress, redistributing local stress states. This must be modeled using anisotropic transformation plasticity models for accurate prediction.
• Hydrogen Diffusion and Embrittlement: While not directly simulated in thermal-mechanical models, the predicted cooling rate and hydrogen trapping potential in martensitic microstructures inform the need for post-weld bake-out (200°C–350°C for 2–4 hours) per NACE MR0175/ISO 15156 or relevant hydrogen embrittlement prevention guidelines.

3.4 Validation Approaches

Simulation credibility depends on rigorous validation against experimental data:

• Thermal Validation: Compare simulated temperature-time histories at thermocouple locations (embedded in base plate at distances of 5, 10, 15, 20 mm from weld centerline) with measured values. Acceptable deviation: ±50°C for peak temperature, ±5°C/s for cooling rate.
• Hardness Validation: Correlate simulated t₈₀₀ values with measured Vickers hardness profiles (per ASTM E92 or ISO 6507) across the weld, HAZ, and base metal. Use established hardness-cooling rate curves for the specific steel grade.
• Stress Validation: Compare simulated residual stress with drill hole method (ASTM E837) or neutron diffraction measurements. Acceptable deviation: ±30 MPa for longitudinal stress.
• Distortion Validation: Compare predicted angular distortion and transverse shrinkage with CMM or coordinate measurement data.

4. Applicable Standards and Acceptance Criteria

4.1 Welding Procedure and Qualification Standards

Standard	Scope	Relevance to Simulation
ASME Section IX, QW-250	Welding Procedure Qualification	Simulation supports PQR development by predicting variables affecting qualification
GB/T 985.1	Welding procedure qualification	Chinese standard for WPS/PQR; simulation data supplements qualification parameters
NB/T 47014	Qualification of welding procedures for pressure vessels	Simulation provides residual stress and distortion data for pressure vessel overlay procedures
ISO 15614-1	Qualification of welding procedures for metallic materials	International standard; simulation supports essential variables assessment
ASTM A388 / ASTM A516	Carbon and alloy steel plate specifications	Defines base material properties used in simulation material input
API 570 / API 579 (FFS-1)	Fitting and flange service / Fitness-for-service	Residual stress predictions feed into FFS assessment for in-service overlay repairs

4.2 Residual Stress and Post-Weld Treatment Standards

• ASME Section VIII, Division 1, UG-115: Specifies residual stress reduction requirements for pressure vessels. Simulation predicts whether PWHT is required to reduce residual stresses below 0.35σy (approximately 120–220 MPa for medium carbon steels).
• ASME Section VIII, Division 2, Part 5: Advanced pressure vessel design requires residual stress characterization; simulation provides the necessary stress field data.
• ISO 15156-3 (NACE MR0175): For sour service applications, simulation-informed PWHT procedures ensure hardness remains below 22 HRC (237 HV) in HAZ and weld metal.
• GB/T 150.4-2011: Chinese pressure vessel standard requiring residual stress assessment for critical welds; simulation provides quantitative support.
• EN 15614-1 / EN ISO 15614-1: European welding procedure qualification standard referencing residual stress limits.

4.3 Material and Testing Standards Referenced in Simulation

• ASTM E92 / ISO 6507: Vickers hardness test method for validating simulated hardness predictions.
• ASTM E837: Standard practice for determining residual stress by the incremental hole-drilling strain gauge method.
• ASTM E975: Standard practice for X-ray diffraction residual stress measurement.
• JIS Z 3146: Hardness-cooling rate correlation for carbon and low-alloy steels (widely used in simulation validation).
• ASTM A370 / GB/T 228.1: Tensile properties of base metal for simulation material input.
• ASTM A262 / ASTM G48: Intergranular corrosion testing, relevant for simulation-informed PWHT procedures on sensitized medium carbon steels.

5. Common Risks and Controls

5.1 Simulation-Specific Risks

Risk	Description	Control Measure
Material property uncertainty	Temperature-dependent properties vary between steel grades and heat numbers	Use measured properties from specific heats where available; apply sensitivity analysis with ±10% property variation
Heat source model inaccuracy	Goldak parameters may not accurately represent actual arc behavior for all processes	Calibrate heat source against thermocouple data from coupon welds; use energy balance verification
Phase transformation model oversimplification	JMA kinetics may not capture transformation plasticity or non-isothermal effects	Implement advanced transformation models (e.g., Koistinen-Marburger with transformation plasticity); validate against dilatometry data
Boundary condition mismatch	Actual fixturing, backing plate, and cooling conditions may differ from simulation assumptions	Document and model actual welding setup; perform parametric studies on boundary condition variations
Mesh sensitivity	Coarse mesh may miss peak thermal gradients; overly fine mesh increases computation without accuracy gain	Perform mesh convergence study; use adaptive mesh refinement near heat source; ensure element size ≤ 1/3 of weld bead width

5.2 Process Risks Identified Through Simulation

• Cold Cracking (Hydrogen-Induced Cracking): Simulation identifies HAZ zones where cooling rate exceeds critical threshold, leading to martensitic hardening above 400 HV. Controls: increase preheat temperature, reduce heat input, select low-hydrogen filler metal (Hd ≤ 5 mL/100g per AWS D1.1), apply post-weld bake-out.
• Hot Cracking (Solidification Cracking): High dilution with medium-to-high carbon base metal can create low-melting-point phases in the weld metal. Simulation predicts peak temperature and dilution ratio to identify risk zones. Controls: optimize filler metal selection, reduce base metal dilution, adjust travel speed.
• Excessive Residual Stress: Simulation may reveal longitudinal residual stresses exceeding 0.5σy in overlay welds. Controls: implement PWHT per ASME UG-115, use welding sequence optimization (symmetrical deposition), apply mechanical stress relief (shot peening, low-stress hammer peening).
• Excessive Distortion: Predicted angular distortion exceeding specification tolerances. Controls: implement backing plate support, use back-step welding sequence, apply mechanical clamping fixtures, reduce heat input per pass.
• Hardness Exceedance in Sour Service: Simulation predicts HAZ hardness exceeding 22 HRC (237 HV) in overlay on medium-to-high carbon steel for sour service applications. Controls: increase preheat, implement PWHT at 620°C–680°C for sufficient hold time, select appropriate filler metal with lower carbon equivalent.

6. Application Across the Company's Three Technology Routes

6.1 TIG/MIG Weld Overlay Route

Numerical simulation is most directly applicable to the TIG/MIG weld overlay route, which constitutes the primary technology domain for Cladding Technology Shanxi Co., Ltd. in terms of cladding plate and pipe fabrication:

• Multi-Pass Cladding Build: Simulation of sequential weld pass deposition on medium-to-high carbon steel substrates (e.g., 16Mn, 20CrMo, 15CrMo, C-12 steel) predicts the thermal history and residual stress state for each pass. This enables optimization of welding sequence (forward, back-step, or symmetrical) to minimize distortion and residual stress.
• Transition Layer Design: For overlaying austenitic stainless steel (309L, 310) or nickel-based alloys (625, 600) on medium-to-high carbon steel, simulation predicts the dilution profile and HAZ hardening. This informs the need for a transition layer (e.g., 309L on high-carbon base before 316L build-up) and validates the transition layer thickness and composition.
• Heat Input Optimization: For medium-to-high carbon steels, simulation demonstrates the trade-off between heat input and HAZ hardness. Higher heat input (1.5–3.0 kJ/mm) reduces cooling rate and HAZ hardening but increases distortion. Simulation identifies the optimal heat input window for each base material grade and overlay thickness.
• Preheat and Interpass Temperature: Simulation quantifies the effect of preheat temperature (100°C–350°C) and interpass temperature (150°C–350°C) on cooling rate and residual stress, providing data-driven recommendations for WPS development.
• PWHT Effectiveness: Simulation of post-weld heat treatment (550°C–680°C) predicts residual stress reduction and microstructural softening, validating PWHT procedures required by ASME UG-115 or GB/T 150.4.

6.2 Hydraulic Explosive Bonding Route

While hydraulic explosive bonding is a solid-state joining process without melting, numerical simulation principles apply to the surrounding weld overlay operations:

• Weld Overlay on Explosively Bonded Joints: When a cladding layer is subsequently weld overlayed onto a hydraulically explosively bonded laminate (e.g., carbon steel base + stainless steel cladding via hydraulic explosion), simulation predicts the thermal and mechanical response of the pre-existing bond interface. The bond interface, with its characteristic wave pattern and intermetallic compound layer, has different thermal and mechanical properties than homogeneous metal. Simulation accounts for this to predict potential bond degradation during subsequent welding.
• Residual Stress Interaction: Hydraulic explosive bonding generates its own residual stress field (typically compressive in the cladding layer, tensile in the base). When weld overlay is subsequently applied, the superposition of welding residual stress and bonding residual stress must be evaluated. Simulation provides the combined stress field for assessment against cracking and fatigue criteria.
• Heat-Affected Zone on Bonded Interface: Simulation predicts the thermal profile at the explosive bond interface during subsequent weld overlay, ensuring that the bond interface temperature remains below the threshold for intermetallic compound degradation (typically below 400°C for Fe-Cr intermetallics in stainless steel overlays).

6.3 Explosion Welding Route

Explosion welding (explosive cladding) is a high-energy solid-state joining process. Numerical simulation contributes in the following ways:

• Post-Explosion Weld Overlay: After explosive cladding of medium-to-high carbon steel with wear-resistant or corrosion-resistant alloys, subsequent weld overlay operations (e.g., adding a transition layer or additional build-up) require simulation to predict thermal and mechanical effects on the explosion-welded interface. The explosion bond interface has unique microstructural features (wave pattern, intermetallic layer, plastic deformation zone) that influence weldability and residual stress development.
• Residual Stress Assessment: Explosion welding generates significant residual stresses (tensile in flyer plate, compressive in base plate, with complex shear stresses at the interface). When weld overlay is subsequently applied, the combined residual stress state must be evaluated. Simulation provides the superimposed stress field for qualification and fitness-for-service assessment.
• Thermal History of Explosion Bond Interface: Simulation predicts whether subsequent welding operations raise the explosion bond interface temperature above critical thresholds that could degrade the bond (e.g., excessive intermetallic growth, bond softening). This is particularly relevant for Fe-Ni, Fe-Cu, and Fe-Stainless explosion welds on medium-to-high carbon steel substrates.
• Process Integration Optimization: For combined processes (explosion welding + weld overlay), simulation enables optimization of the overall process sequence, including preheat levels, welding parameters, and PWHT requirements, to ensure the final product meets specifications for both bond quality and overlay integrity.

7. Contribution to Qualification Building and Certification

7.1 WPS Qualification Support

Numerical simulation data directly supports the qualification of welding procedures for weld overlay on medium-to-high carbon steels:

• Essential Variables: Simulation predicts the effect of essential variables (heat input, preheat, filler metal, base metal PCM) on weldability outcomes, enabling rational selection of qualification ranges.
• PQR Data Supplementation: While physical qualification requires actual weld tests (tensile, bend, hardness, NDT), simulation provides supplementary data on residual stress, distortion, and thermal history that may not be directly measurable but are critical for process understanding.
• Procedure Transfer: Simulation enables prediction of process behavior when transferring a qualified procedure to a different base material grade, thickness, or geometry, reducing the need for full re-qualification.

7.2 Certification and Accreditation

The capability to perform validated numerical simulation of weld overlay thermal-mechanical fields contributes to the company's certification and accreditation status:

• ISO 9001 Quality Management: Simulation-based process optimization and validation demonstrates the systematic approach to quality planning required by ISO 9001:2015 (Clause 8.3 – Design and Development of Products and Services).
• ISO 3834-2 / EN ISO 3834-2: The quality requirements for welding of metallic materials standard references the use of simulation and analysis for welding procedure development. Validated simulation capability supports compliance with this standard.
• ASME "Q" Stamp / NB Pressure Vessel Certification: For pressure vessel and piping overlay applications, simulation provides the residual stress and distortion data required for design assessment and qualification documentation.
• API Q1 (Quality Management System): For oil and gas industry applications, simulation-based process control demonstrates the systematic approach to quality management required by API Q1.
• NACE/AMPP Certification: For sour service and corrosion-resistant overlay applications, simulation-informed PWHT procedures and hardness control demonstrate compliance with NACE MR0175/ISO 15156 requirements.

7.3 Customer Value Enhancement

The simulation capability provides direct value to customers across multiple dimensions:

• Technical Assurance: Customers receive simulation reports demonstrating that the overlay process has been engineered to control residual stress, distortion, and HAZ hardness within specified limits, providing confidence in product performance.
• Design Input: For customers specifying overlay requirements (e.g., wear-resistant cladding on medium-to-high carbon steel equipment), simulation provides design input on achievable overlay thickness, expected residual stress levels, and required PWHT conditions.
• Repair and Maintenance: For in-service repair applications, simulation predicts the thermal and mechanical effects of overlay repair on the existing structure, supporting fitness-for-service assessment per API 579 FFS-1.
• Cost Optimization: Simulation reduces the number of physical trials required for procedure development, reducing qualification costs and delivery time for customers.

8. Conclusion

Numerical simulation of temperature and stress fields during and after weld overlay on medium-to-high carbon steels is a critical engineering capability that bridges the gap between empirical welding practice and rigorous process engineering. For Cladding Technology Shanxi Co., Ltd., this capability enhances the company's position across all three technology routes—TIG/MIG weld overlay, hydraulic explosive bonding, and explosion welding—by providing quantitative predictions of thermal history, residual stress, distortion, and microstructural outcomes. The simulation data directly supports WPS qualification, certification compliance (ASME, NB, ISO, API, NACE), and customer technical assurance. By identifying and mitigating process risks (cold cracking, excessive hardness, distortion, residual stress) before physical production, the company delivers higher-quality products with reduced risk of field failure, strengthening its competitive position in the specialized cladding and overlay manufacturing market.