The microstructure algorithm used here was motivated by the work of Kirkaldy , and originally developed for welding by Watt et al., Henwood et al. and enhanced by Khoral .
The various microstructural transformation temperatures are a function of carbon and alloy content of the low alloy steel.
For low alloy steel, the liquidus and solidus temperature can be
determined as a function of carbon content .
TL =1530.0-80.581C (4-1)
Ts =1527.0-181.356C (4-2)
С = carbon content of the steel
TL = liquidus temperature of the steel (°С)
7’. = solidus temperature of the steel (°С)
In modem micro-alloyed steels, austensite grains are pinned by carbide/nitride precipitates. Ashby, Easterling and Ion [6, 9 and 10] presented a modified relationship to calculate the precipitate dissolution temperature. This relationship takes into account the effect of these grain growth retarding precipitates and is given as:
A&B = constants that depend on the precipitate species. A=3.11 and B=7520.0 from [4 and 36]
Cm &Cc = concentration of metal (Nb, Ті, V etc.) and non metal (carbide, nitride or boride) in precipitate respectively a & b = stoichiometry constants. a=l,0 and b=l,0 for VC from [4 and 36]
The concentrations are in wt % and temperature is in degree Kelvin.
The upper critical temperature Аъ depends on the carbon content of the steel as shown in the iron-carbon phase diagram Figure 2-30. The most accurate representation for the Аъ line is given by
Kirkaldy and Baganis . But this is not convenient in the computer model because of the difficulty involved in inverting this relation for use in the lever law. To avoid this difficulty an alternative relation proposed by Leslie  is used in this algorithm.
A3 (°С)=912- 200jc -15.2Ni+ 44.75'/+31 Шо ґл ^
+U.W-Q0Mn+\Cr+2OCu-100P-A00Al-20As-A0m) where the composition are in wt %.
The lower critical temperature (Д) or pearlite start temperature is also dependent on the composition of the low alloy steel and is given as :
Д (°С) = 723-10.1 Mn— 16.9M+29Si+6.9Cr+290As+6.4W (4-5)
As the weld cools and the temperature drops below the bainite start temperature (BS), the formation of pearlite and ferrite stops and the austenite decomposition to bainite starts. The bainite start temperature is given as :
BS(°C) = 656 - 58С - ЪЪМп - 15Si -15 Ni - 34 Cr -4Mo (4-6)
On rapid cooling of the weld, austenite in the heat affected zone starts decomposition into martensite. This transformation starts below the martensite start temperature, which is given as :
MS (°С) = 561- 474C - 3 5Mn -1 INi -1 ICr - 21 Mo
The carbon content of eutectoid alloy is also a constant for a particular low alloy steel and is given by the following relations :
фх =910-15.2M + 44.7S/+104F + 315Afo + 13.WT (4-9)
ф2 = ЗОМп +1 ІСг + 20Cu -70QP - 400Al -120 As - 400Л (4-10)
All these parameters given by equations (4-1) to (4-10) are used to divide the HAZ thermal cycle into eight distinct regions as shown in Figure 2-30.
Phase transformations taking place in the HAZ during the weld thermal cycle can be divided into two sections a) transformations that take place during heating and b) transformations that take place during cooling.
Initialization of ferrite and pearlite spans from room or pre-heat temperature to the lower critical temperature Ax. In this region, in the absence of knowledge of the real microstructure, the microstructure is assumed to be in equilibrium and is a mixture of ferrite and pearlite. The volume fraction of ferrite and pearlite is given by the lever law :
XF =------- (4-і 1)
С - С
= fraction of ferrite
= fraction of pearlite
= carbon content of steel
= carbon content of the eutectoid
= carbon content of ferrite
The carbon content of ferrite below Д is given by an empirical equation assuming a linear decrease from eutectoid value to a value of zero at room temperature (20 °С) [4 and 20]:
T — 20 0
C= ' (0.105-115.3x10~6xA) (4-13)
“ A,-20.0 1
As the temperature in the HAZ exceeds the A] line during heating, pearlite colonies transform to austenite. The intercritical region or partially transformed region of the HAZ is the region where temperature varies between Д and A3. As the peak
temperature goes towards A3, the austenite content increases under
near equilibrium conditions. Because of the very high heating rate in the welds а-phase is not likely to get superheated substantially before the a —* у phase transformation takes place. Under
equilibrium conditions the amount of austenite and ferrite is given as:
Ca-cr y }
XA = fraction of austenite
С = carbon content of austenite
Ca = carbon content of ferrite
The carbon content of austenite can be obtained by rearranging equation (4-4):
c 1 (4.16)
г 2032 v 7
ф = 910-15.2M +44.75/+ 104F + 315Мо + 13.1Ж (4-17)
ф2 =30Ми + 11Сг + 20См-700Р-400^/-120^-4007ї (4-18)
Above Ах the carbon content of ferrite is given by the following relation:
Ca =0.105-115.0х10“6хГ (4-19)
As the peak temperature goes above A} the remaining ferrite matrix also transforms to austenite. Thermal cycles with peak temperature between A} and precipitate dissolution temperature represent the recrystalized zone.
The growth in the grain size of the austenite that forms can be described as a function of temperature and time. However, growth will not begin until carbon-nitride precipitates such as VC and NbC (niobium) dissolve.
It is assumed that grain growth is diffusion controlled and the driving force is the surface energy and it does not require any nucleation. The grain growth equation is given as :
g = grain size (jum)
к = grain growth constant (jumVs)
Q = activation energy for grain growth (kJ/mol)
R = universal gas constant (kJ/mol K)
T = temperature ( °K)
t = time (s)
Q and к are dependent on the type of precipitate [4 and 10]. The austenite grain growth which begins at precipitate dissolution continues up to the peak temperature of the thermal cycle.
On cooling as the temperature drops below the A3 line, austenite starts decomposing into its daughter products. The kinetics of austenite decomposition into its daughter products are described by ordinary differential equations (ODE), [4, 21 and 22], of the form: dX
= B(G, T) Xra(l - X)p 4-21)
X = volume fraction of the daughter product
В - effective rate coefficient
G = austenite grain size
m, p = semi-empirical coefficients, were determined by experiment
by Kirkaldy .
The term B(G, T) is a function of grain size, under-cooling, the dependence of the carbon diffusivity on the alloy and temperature and the phase fractions present.
This is essentially the model presented by Watt et al . The microstructure algorithm has been coupled with the three dimensional finite element heat transfer program developed by Goldak et al [26 and 27] to predict the transient microstructure of the HAZ.
The mechanical properties of metals are sensitive to their microstructure. By intentionally changing the microstructure, one can control the properties to provide the best service. However, not all the changes are beneficial. Many manufacturing processes can produce unfavorable microstructures, which can reduce the reliability and performance of the product. In welding, undesirable microstructures can cause failures and are a serious concern. Whether the microstructural changes bring fortune or catastrophe will depend on one’s capability to predict why and how these changes happen. The prediction of the microstructural changes caused by thermo-mechanical processing continues to be an area of great practical benefit. •
The major contribution of this chapter to the simulation of microstructures is to create a data structure for the coupling of a steady state thermal field to the formulation of microstructures, Gu et al. . Some changes have been made in order to satisfy conditions in the weld pool and fusion region.
Methods to predict the steady state temperature field around a weld pool were first reported by Rosenthal  and Rykalin .
In the previous chapter, a FEM model of a steady state temperature field around the weld pool was described. Since this is a true three-dimensional model, it can investigate different welds in detail for their unique temperature distributions. Such distributions are sometimes critical to the simulation of microstructures. Compared to a time marching Lagrangian formulation for a transient thermal analysis, the cost of computing a steady state temperature field has been reduced from hours or days to seconds. The difficulty of coupling the steady state temperature field with a microstructure prediction is to extract the thermal cycle from the steady state temperature field, especially when the mesh is complex, i. e. unstructured.
Microstructure evolution is associated with a material point. In a Lagrangian formulation, nodes and Gauss points are material points. In this case, one simply solves the equations for microstructure evolution as a function of temperature, time and possibly stress for any or all nodal or Gauss points. However, in an Eulerian formulation, e. g., a steady state formulation for a moving weld, the nodal and Gauss points are spatial points and not material points. In this case, one must first compute the curves traced out in space for each material point for which one wishes to compute the microstructure evolution. In the general case, where the velocity field is a function of time, such curves are called flow lines. At each instant of time, the tangent of the curve is parallel to the velocity field. In the steady state case, the velocity field is not a function of time. In this case, the tangent of each point on a curve is parallel to the constant velocity field. Such curves are called streamlines.
In an arbitrary Eulerian Lagrangian (A EL) formulation, the mesh is allowed to move with the material points or with spatial points or allowed any other convenient mesh motion or velocity. In this case, the task of computing flow lines or streamlines is not fundamentally different from the Eulerian case. One simply deals with both the velocity of material points and the velocity of the mesh.
To capture the thermal cycle in a steady state temperature field, the applied data structure should meet several requirements. The data structure must support efficient ordered traversals of the spatial points on a streamline (starting at the head and proceeding downstream). It must also allow the element that contains the head of a streamline to be found. Lohner used a heap of faces , supporting insertion and deletion in 0( log2 n) operations for n faces. However, given different sized faces this may require a complete search to find the face containing a point. A data structure has been developed by Gu et al.  to represent properly the streamlines in a steady state field and to meet all the above requirements.
In a steady state temperature field described in a Eulerian frame, flow lines and streamlines are equivalent. Integration along a flow line always gives the thermal history of a material point. In a general FEM mesh with refinement around the weld pool, the creation of flow lines can consume more time than the microstructure calculation itself. Data structures to organize elements, points and flow lines are necessary. The functionality of the data structure is critical to the success of the model.