2.2 Modelling of Hydraulic Autofrettage Process
2.2.6 Finite Element Method and Finite Difference Method Based
method (FEM) or finite difference method (FDM) for studying the stress behaviour induced in hydraulic autofrettage. At the earliest, Hill et al. (1947, 1951) developed a finite difference method of solution assuming plane strain condition for hydraulically autofrettaged closed-end cylinder. They considered Tresca yield criterion and Prandtl-Reuss plastic stress-strain law for the analysis. The model assumes compressibility of materials within the plastic region. Comparing the solution with the theories neglecting elastic strains in the plastic region, they observed a discontinuity in the axial stress at the elastic-plastic boundary radius. The difference between the two solutions increases as the elastic-plastic boundary radius increases. An error of greater than 60% in the axial stress was observed on the assumption of zero elastic strain in the plastic region for all ratios of outer to inner radii. A finite difference numerical solution of hydraulic autofrettage of open-ended thick cylinder of wall ratio 2:1 was carried out by Thomas (1953). The solution was
based on the Prandtl-Reuss incremental stress-strain relations, together with the Tresca yield criterion. The author compared his solutions with the previously available solutions of Allen and Sopwith (1951) and Steele (1952). A stiffness method involving finite difference solution for elastic-plastic problems using the incremental Prandtl-Reuss equations was suggested by Marcal (1965). The method assumes that the laws of plasticity are linear over an increment. Author tested the method for the closed-end thick-walled cylinder subjected to internal pressure. The results of Marcal (1965) agree with the results of Hill et al. (1951) for a cylinder with a diameter ratio of 2:1. Chen (1972) developed the incremental tangent- modulus approach of the finite element formulation together with the computer program for the generalized plane-strain case in the thick-walled cylinders subjected to internal pressure, external pressure and end force. The material model was assumed to be elastic-plastic strain-hardening, obeying the von Mises yield criterion and the Prandtl-Reuss flow theory. Later, the author proposed a finite difference approach for the problem (Chen, 1980). The numerical procedure uses an incremental approach and does not involve any iteration for each increment. The accuracy of the problem depends on the grid size and load increments.
In late 90’s and early 21st century, several researchers used FEM modeling of the hydraulic autofrettage process to study residual stress distribution, optimum autofrettage pressure, post machining effects on the distribution of residual stresses.
Feng et al. (1998) carried out the analysis of stresses and strains induced by low temperature hydraulic autofrettage of AISI 304 steel cylinder by using FEM method.
They investigated the influences of the autofrettage pressure, temperature, work hardening and reverse yielding on the residual stresses generated in the cylinder. The finite element calculation showed that more beneficial residual stress can be generated by autofrettage at low temperature than at room temperature. This may be due to the fact that at low temperature, the yield stress of the material is higher and the thermal strains accumulated during warm up to room temperature add to the autofrettage pre-compression. Authors found that the optimal autofrettage pressure and temperature of the cylinder were about 4000 bar and −90 °C, respectively. A 3- D FEM analysis of the autofrettaged cylinders with cross-bore was carried out by
Badr et al. (1999). They analyzed elastic-plastic and residual stress and strain fields including Bauschinger effect. The FEM results were compared with the analytically developed solution based on the strain energy density method. The strain energy density method provides relatively simple and reasonably accurate estimate of cross- bore intersection stresses and strains than the expensive and time consuming elastic- plastic FEM analysis. Hameed et al. (2004) also carried out a 3-D FEM analysis to evaluate the effect of radial cross-bore qualitatively in an autofrettaged cylinder using ANSYS finite element package. They observed that there is a severe localized change in the residual stress profile in the vicinity of the cross-bore. Authors carried out the analysis for variable cross-bore diameter and observed that by increasing the diameter of the radial hole, the residual hoop stress at the inner wall reduces, whilst it increases at the outer wall.
Gibson et al. (2006) presented a series of finite element models to predict the residual stress distribution in hydraulically autofrettaged cylinders for a range of end conditions— plane stress, plane strain and generalized plane strain. They assumed bilinear strain hardening material model including Bauschinger effect. Authors compared the results with an alternative numerical model (Parker, 2001) and Huang’s (2005) analytical model.
A multi-linear kinematic two-dimensional FEM model incorporating Bauschinger effect was developed by Hameed et al. (2003) using commercial ANSYS package to assess the post machining effects in a hydraulically autofrettaged gun barrel. They studied the effect of machining both at the inside and at the outside diameter on the depth of yield, maximum firing pressure and final residual stress distribution in the barrel. The FEM results were validated with experimental results. Authors observed that the reduction in the maximum compressive hoop stress is more sensitive to internal machining than external machining. They also observed that the depth of yield remains stable and there is no movement of the radius of elastic-plastic interface, relative to its location before material removal. The maximum firing pressure is not affected, if the internal machining removes material in which reverse yield has occurred. The post machining residual stress distribution in a hydraulically autofrettaged cylinder was
also studied by Bähre and Brünnet (2011) by using finite element analysis in ABAQUS and they made the similar observations.
Majzoobi et al. (2003) investigated the effect of autofrettage process on high pressure cylinders using finite element numerical simulation. They performed numerical simulation using finite element package NISA with an axisymmetric analysis. Authors incorporated kinematic hardening model for the analysis. The burst pressure of the cylinder was evaluated using von Mises yield criterion.
Experiments were also conducted to compare the numerical results. Authors observed that the best autofrettage pressure for strain hardened material is the pressure which brings the outer surface of the cylinder to the yield point. Any pressure higher than that will result in reduction of bursting pressure of the cylinder.
The value of autofrettage pressure obtained by finite element simulation and experiments closely agrees with the Xiaoying and Gangling (1984) analytical solutions.
A finite element analysis was carried out by Alegre et al. (2006) to obtain the residual stress distribution after the autofrettage of a 15–15 PH stainless steel incorporating Bauschinger effect. The finite element simulations were carried out considering an elastic-perfectly plastic behaviour of the material for the loading phase and a Ramberg–Osgood behaviour for the unloading phase, with its power coefficients depending on the previous equivalent plastic strain reached during the loading process.Authors observed a reduction of the residual stresses of about 25%
due to the Bauschinger effect. They also observed that if an autofrettage pressure value is exceeded, the residual stresses increase only slightly, whereas the plastic strains achieved during the autofrettage loading increase exponentially. Shim et al.
(2010) studied the Bauschinger effect in hydraulic autofrettage of SNCM8 high strength steel cylinder and observed that this effect is significant under some conditions. The tensile and uniaxial Bauschinger effect tests of SNCM8 were performed to evaluate BEF, and then this constant was used in calculating the residual stress distribution by analytical and finite element analysis.
Recently, Parker et al. (2012) carried out a finite element stress analysis of hydraulic autofrettage incorporating a material model whose unloading behaviour
varies with radius and Bauschinger effect. The finite element analysis was carried out by two methods— using a user programmable feature within a non-linear finite element analysis and using an elastic modulus and Poisson’s ratio adjustment procedure within a linear effective finite element analysis. Authors showed that both these two methods are in agreement with each other.
A finite difference method (FDM) was used by Perry and Aboudi (2003) to study hydraulic autofrettage of a thick-walled cylinder. The analysis was based on von Mises yield criterion, isotropic strain hardening in plastic region and Prandtl- Reuss theory. Authors assumed plane stress condition and accounted for Bauschinger effect during pressure release. The stresses were evaluated incrementally by FDM. The effect of machining of the cylinder after autofrettage on residual stresses was also studied. It was observed that strain hardening does not affect the residual stress distribution significantly, but Bauschinger effect does. A three-dimensional (3-D) finite difference numerical solution for hydraulic autofrettage incorporating the Bauschinger effect, using an accurate numerical representation of the experimentally measured material behaviour was given by Perl and Perry (2006). A 3-D computer code was developed for determining the stresses, strains and displacements throughout the autofrettage process. The analysis can also predict residual stress field in swage autofrettage process.