Residual stress and deformation that develop during heat treatment of steel parts can be determined quantitatively using numerical simulation.

Crankshaft with numerical analysis grid overlay

The ability of a component to handle higher load strains can be accomplished by heat treating surface layers to improve mechanical properties. Compressive residual stress in the surface layer contributes to increased durability (higher fatigue limit) by reducing crack growth velocity and by its influence on the mean stress. In the case of a crankshaft, induction surface layer hardening produces high hardness at the bearing surface for improved wear resistance, high toughness in the center and compressive residual stress in the surface layer for a higher fatigue limit. Methods are needed to describe this stress state.

The relationship of all the parameters that influence the residual stress state is highly complex, thus the influencing effects of the hardening process can only be covered and controlled numerically. Numerical simulation of the induction hardening process of crankshafts by means of finite-element calculations using ESI's SYSWELD software provides a good understanding of the time history of temperature, stress and deformation [1]. A simulation concept to calculate the residual stress state obtained by induction hardening of a crankshaft is discussed in this paper.

Fig 1 FEM model for the calculation of a crankshaft

Description of the calculation

The induction heating process for a crankshaft is simulated using moving surface heat sources. The temperature cycle for austenitization is determined empirically with a time- and location-dependent distribution of the heat sources to obtain hardness penetration depths for the hardening process.

The structure of the component after heat treatment, as well as resulting residual stress and deformation are determined by the time history of temperature, phase transformations, stresses, strains and the interactions between those partial processes.

The calculation consists of two different partial steps that are coupled based on existing interactions. Simulation of the hardening process (heating and quenching) is based on a three-dimensional (3-D) model of a section of a crankshaft consisting of the main bearing, cheek and connecting rod bearing.

Fig 2 Measurement points and nodes on the rod bearing; Fig 3 Temperature cycles for measurement point 1

Thermometallurgical calculation

Temperature distribution in the part is determined using a thermometallurgical calculation, which considers initial thermal and boundary conditions and temperature- and structure-dependent material parameters, depending on location and time. Time-temperature dependent diffusion-controlled phase transformations are calculated using the Leblond model [2]; the mathematical model gives the metallurgical behavior of steel along continuous cooling curves. It reproduces quantitatively all CCT-diagram characteristics, as well as experimental results, and distinguishes the different phases of austenite, ferrite-pearlite, bainite and martensite. The effects of short-time austenitization (austenite grain size and carbon distribution) can be considered, but are neglected in this case. The martensite transformation is simulated using the Koistinen-Marburger Law [3].

Material properties needed for the calculation include thermal conductivity, specific heat capacity and density. Thermal properties, especially high-temperature data, can be obtained from the steel producer or software manufacturer, or must be determined experimentally. Thermal boundary conditions to simulate induction heating and quenching are defined by the heat sources and heat-transfer coefficients, respectively.

Fig 4 Martensite distribution at time = 24 s; Fig 5 Transient evolution of temperature and phases in the radius for measurement points 1 and 5

Mechanical calculation

Strain, deformation and stress are calculated using a mechanical calculation based on the results of the thermometallurgical calculation, which takes into consideration elastoplastic material behavior, isotropic hardening and transformation-induced plasticity. Young's modulus, Poisson's ratio, thermal expansion and yield stress (temperature- and phase-dependent mechanical properties) also need to be defined for the calculation. Mechanical properties are rather complex because of their dependence on temperature and phases, and usually are determined experimentally, or can be provided by the software company. Mechanical boundary conditions describe the symmetry planes of the crankshaft and the way the crankshaft is fixed during the hardening process.


Figure 1 shows the model for calculating the connecting rod bearing. Due to limited computer resources and to minimize calculation time for simulating the hardening process, the 3-D model of the crankshaft section has a mesh that is sufficiently fine and regular around the bearing surface and the fillet, while the rest of the model is meshed free and rather coarse. Thermometallurgical and mechanical calculations are performed on the side of the connecting rod bearing using this model.

The results of this calculation are projected onto a second model (main bearing) having a fine mesh around the main bearing surface and the fillet, while the rest of the mesh becomes coarse again. With the new model and the projected results of the calculation of the connecting rod bearing, a second thermometallurgical and a mechanical calculation are performed, in which the main bearing is hardened.

The results of the calculation include temperature distribution, phase proportions, stresses, strains and deformations of the whole part of the crankshaft. Consequently, the sequence of the hardening process of the single bearings, which has a big influence on the total deformation of the crankshaft, can also be simulated.

Fig 6 Comparison of the transient evolution of hydrostatic stresses and temperature for measurement point 3; Fig 7 Comparison of the evolution in time of the accumulated plastic strains in the austenite with the hydrostatic stresses for measurement point 2


There is an overlap of thermal stresses and stresses due to phase transformation in the component. Thermal stresses are the result of a temperature difference (and therefore a difference in volume) between the center and the edge of the part. Transformation stresses are caused by a change in volume due to phase transformation (for example, the specific volume of martensite is larger than that of austenite). The creation of residual stresses depends on many parameters, such as dimensions, transformation behavior (CCT diagram), quenching velocity, high temperature strength of the material, etc.

Points and nodes of the connecting rod bearing used in the evaluation are defined in Fig. 2, and the resulting time-temperature plot during hardening of the connecting rod bearing at measurement point 1 is shown in Fig. 3. The bearing is heated during the first 12 seconds and quenched via a shower for 12 seconds, followed by cooling the crankshaft to room temperature.

The time-temperature plot around the bearing surface is shown for three nodes distributed over the cross section, on the surface (# 2964), 4 mm (0.16 in.) below the surface (#4241) and 10 mm (0.4 in.) below the surface (#4460). At 4 mm deep, the temperature is just high enough for austenitization.

Figure 4 shows the development of depth of hardness penetration. The martensitic zone is about 4 mm deep around the bearing surface and about 2 mm (0.08 in.) deep around the fillet. All austenite has transformed into martensite after 24 s around measurement point 1 (Fig. 5), but the temperature around point 5 at 24 s has not reached the martensite-end temperature, and the martensite transformation at this point is not complete after quenching 12 s, lasting a few seconds longer. Quenching around point 5 is a little slower due to part shape.

Fig 8 Profile of the normal principal stresses through the thickness at the quenched surface measurement line 11 (rod bearing); Fig 9 Profile of the normal principal stresses through the thickness at the quenched surface (main bearing)

Deformations, strains and stresses

The calculation gives a better understanding of stress as a function of time during heating and quenching. Figure 6 shows tensile and compressive stress versus time during austenitization and quenching for point 3. The surface layer expands more during heating than the center due to higher temperatures at the surface layer. The cooler center prevents the surface layer from expanding freely, which creates tensile stresses in the center and compressive stresses in the surface layer during heating.

Compressive stresses in the surface layer are reduced with progressive austenitization due to a volume reduction that occurs with austenitization. Furthermore, stresses are balanced; i.e., the difference between tensile and compressive stresses becomes smaller due to decreasing differences in temperature over the cross section. Stresses in the surface layer are very low when the material becomes austenitic due to the low mechanical properties of austenite at high temperatures.

Further heating of the center causes thermal expansion, which decreases tensile stresses in the center. Now, compressive stresses change to tensile stresses due to thermal contraction when the center starts to cool. Quenching starts at 12 s. Austenite contraction in the surface layer during cooling results in tensile stresses in the surface layer and compressive stresses in the center. At 15 s, austenite in the surface layer starts to transform into martensite with a corresponding volume increase, and, therefore, decreases tensile stresses in this layer. The reduction of tensile stress is sufficiently severe to result in a change in the sign of the stress. This results in compressive stresses in the surface layer and tensile stresses in the center. The volume increase from the austenite to martensite transformation overlaps with the volume contraction caused by further cooling. Compressive stresses in the surface layer decrease again. As the center keeps cooling, compressive stresses in the surface layer increase again.

Figure 7 shows a plot of accumulated plastic strains in the austenite compared with hydrostatic stress over time.

Plastic strain develops in the austenite during heating and continues to increase during cooling caused by tensile stresses in the cooled austenite layer and by the austenite to martensite transformation. The accumulated plastic strain remains constant upon complete transformation of austenite to martensite. The largest accumulated plastic strains occur in the fillet at the surface (as expected) around points 1 and 2, where higher temperatures are necessary for austenitization because of more adjacent material.

Figures 8 and 9 show the resulting depth profiles of the principal stresses for the main and connecting rod bearing. Hardness penetration of the main bearing is substantially lower compared with that in the connecting rod bearing. Hardness penetration is only about 2.5 mm (0.1 in.) in the middle of the bearing surface and 1.5 mm (0.06 in.) around the fillet for the main bearing. This lower penetration has a direct influence on stresses; there is a smaller area of compressive stresses and the transition from the area of compressive to tensile stresses in the main bearing moves closer to the surface.

Fig 10 Distribution of residual stresses for the Me Blinie 23

Compressive stresses are about the same for both bearings because they are directly influenced by transformation stresses. However, higher tensile stresses exist in the center of the main bearing. This is mainly due to thermal strains, which increase with increasing diameter because there is more material available for thermal conduction, leading to steeper temperature gradients. Therefore, the center cools considerably faster in the main bearing as in the connecting rod bearing, resulting in higher tensile stresses.

Two connecting rod bearings were hardened using a lower power to determine the influence of inductor power on depth of hardness penetration. The heat sources were reduced by 10% for test 1 and 20% for test 2 (corresponding to a voltage step-down). Heating the bearings to a maximum temperature of 1040 and 920 C (1905 and 1690 F) for test 1 and 2, respectively, compared with heating to about 1170 C (2140 F) correspondingly decreases the depth of hardness penetration, resulting in lower stresses and causing the area of tensile stresses to move toward the surface and stresses.

Verification of results

The team at the Hahn-Meitner-Institut performed residual-stress analysis using neutron diffraction to verify calculated residual stresses. The biggest differences in compressive residual stresses [s(axial) - s(radial)] appear in the area of the fillet (Fig. 10) and decreases continuously from the surface to the inside. The depth of compressive stress changes to tensile stress at a depth between 4 and 5 mm in the area of the fillet. Compressive stress is larger in the area of the bearing surface than in the fillet. (Note: Radial stresses should be zero on the surface because there is a two-axial state of stress. The small deviation of calculated values from zero is caused by numerical inaccuracy; for example, extrapolation of stress values from the integration points to the nodes.)

Generally, there is quite good agreement between calculated values and diffraction analysis. A considerably larger amount of measured values at smaller radial increments are required to more precisely verify the calculated residual stresses over the part cross section. This is especially true in the transition zone from compressive to tensile stresses, where there are very steep stress gradients requiring more dense measurement points. IH