Creep Age Forming Constitutive Eqations

Experimental Studies and Constitutive Modelling of the HardeningofAluminium Alloy 7055 under Creep Age Forming Conditions

Lihua Zhana, Jianguo Linb*，TrevorA. Deanc and Minghui Huanga

a School of Mechanical & Electrical Engineering, Central South University, Changsha, China

b Department of Mechanical Engineering, Imperial College London, UK

c School of Mechanical Engineering, University of Birmingham, UK

Abstract

A test programme has been designed to characterise the creep-ageing behaviour of Aluminium Alloy 7055, commonly termed AA7055,under creep age forming (CAF) conditions. Creep ageing tests have been carried out for a range of stress levels at 120oC for 20 hours, which is the typical period for a CAF process. Interrupted creep tests have also beencarried out to rationalise the effect of stress levels on age hardening. Based on experimental observations, a set of mechanism-based unified creep ageing constitutive equations has been formulated, which models creep induced evolution of precipitates, dislocation hardening, solid solution hardening and age-precipitation hardening.A multiple-step reverse process has been introduced to determine, from creep ageing test data, the values of constants arising in the constitutive equations.Close agreement between experimental data and computed results are obtained for creep and age hardening data for the stress range tested. The determined equation set has been integrated with the commercial FE code MSC.MARC via the user defined subroutine, CRPLAW, for CAF process modelling. In addition to springback, the evolution of precipitate size and creep induced precipitation hardening can be predicted.

Keywords: Creep age forming, precipitation hardening, constitutive modelling, FE process modelling, springback, AA7055.

Corresponding author: J Lin,

1. Introduction

As a relatively new metal forming method, creep age forming (CAF)is advantageous for manufacturing large integrally stiffened lightweight structures in light alloys for the aircraft and aerospace industry, compared to traditional mechanical metal forming methods. Two phenomena (age hardening and creep/stress relaxation) are combined into one in CAF. Ageing is a process that can increase the strength of a metal, while creep during ageing is the mechanism to promote forming and retention of the formed shape of the part. Although some studies have been carried out on creep and ageing as individual processes[1-4], when the two processes are combined, as in CAF, many new problems arise. One of the greatest challenges to improvement of the efficiency of the CAF technique is to predict exactly the amount of springback that will arise, in order that a tool shape may be defined to compensate for it. High levels of springback often result in the formed component being out of tolerance, extra after-work problems during final assembly and also deficient aerodynamic behaviour.Another important feature in CAF is the generation of high mechanical properties in a work-piece.Since usually, CAF is used to manufacture extra large panel components, a trial and error method is costly, laborious and time consuming.Materials and process modelling are efficient ways to understand such problems arising in the new forming method, to accurately predict the ultimate mechanical properties and shape of thetool required for an accurate part to be formed.

To gain an understanding of the mechanism of springbackfor conventional sheet forming processes,many modelling activities[5-8] on springback prediction for componentshave been carried out. These are not suitable for CAF, as they contain no consideration of the influence of precipitation hardening on mechanical properties. Less research has been done on the prediction of the ultimate mechanical properties of formed parts in CAF and only Ho et al. [9-10]have published results of a preliminary study on predicting the evolution of mechanical properties of AA7050.A comprehensive material model capable of predicting relationships between forming conditions, microstructure, mechanical properties and springback should be developed to establish a sound scientific basis for CAF processes. In addition to shape production, the most important requirement is to obtain maximum possible values of work-piece mechanical properties through CAF.

Thus to model precipitation hardening and mechanical property evolution of alloys in CAF, it is necessary to derive fully determined physically based unified creep ageing constitutive equations.

Some research has been carried out to investigate and establish new constitutive equations for light alloys under CAF conditions and carry out relative process simulation since the invention of CAF.For instance; Sallah et al [11] used a conventionalinelastic constitutive equation for autoclave age forming simulation, in which two stress relaxation models (linear Maxwell model and Walker/Wilson model) were used for predicting stress relaxation curves under CAF conditions.Beam specimens were used to experimentally validate the modelling process. Their work provides closed form solutions for age forming simulation of beams using a cylindrical tool and an easy way to determine springback. Guines et al [12] used a traditional power-law creep model to predict the creep deformation, stress relaxation phenomena, which take place during the thermal exposure (creep-ageing stage) in CAF. A single curved tool was used in the process modelling. The influence of different mechanical clamping conditions on the final shape of a single curvature integrally stiffened structure was carried out. Both these investigations considered only conventional high temperature stress relaxation and/or creep in CAF conditions, while individual physical mechanisms of hardening, such as dislocation hardening, precipitation hardening, during CAF were not modelled.

Ho, et al. [9] and Huang et al.[13] used a set of creep damage constitutive equations, developed by Kowalewski et al.[1] to describe creep damage. While two state variables; , which is related to the over aged condition, and ,which models failure at the tertiary creep stage,were not used in the CAF simulation. These equations have been used mainly to model the basic creep behaviour of metals. However, CAF is a combination of creep deformation and age-hardening, the interaction between them is not considered in their equations, although the equations can be used to predict springback in CAF.

To consider precipitate nucleation, growth and their effects on mechanical property evolution and creep deformation under CAF conditions, Ho et al. [10] developed a set of constitutive equations, which models primary and secondary creep and precipitate nucleation and growth for AA7010 aged at 150 oC. These material models were then introduced into the commercial FE solver ABAQUS through the user defined subroutine CREEP. The increase in yield strength of formed parts was predicted, in addition to the creep deformation, stress relaxation and springback. However, the equation set is only rudimentary as some important process factorssuch as, the influence of the variation of volume fraction of precipitates, the shape of the precipitates and so on, are not properly considered in them. For example, only one parameter, the radii of precipitates, is used in the equation set to consider the effects of precipitate hardening on creep deformation, thus it is only suitable for spherical precipitates. Further efforts are required to develop mechanism-based creep ageing constitutive equations for aluminium alloys with different forms of precipitates under different CAF conditions.

The work presented in this paper consists of experimental studies and constitutive modelling of creep age hardening of Aluminium Alloy 7055 (AA7055)under CAF conditions.First, the overall experimental programme is introduced and relevant experimental results are analysed. Second a set of mechanism-based unified creep ageing constitutive equations are established based on the experimental observations. Then, a multiple-step reverse process is introduced for determining the values of constants arising in the constitutive equations from creep ageing test data. In the end, the equation set is integrated with the commercial FE solver, MSC.MARC via the user defined subroutine, CRPLAW, for CAF process modelling for a double curvature panel part. In addition to springback, the evolution of creep induced precipitation hardening is predicted.

1. Experimental programme
2. Test materials

The material, AA7055, of composition shown in Table 1, used in the experiments was provided in the hot rolled condition. Specimens were machined from sheet of 2.85 mm thickness to a length of 150 mm with a gauge length of 50 mm and thensolution heat-treated and water quenched. Subsequently, the samples were kept in a refrigerated condition to reduce natural ageing.The geometry and dimensions of the specimen are shown in Figure 1.

Table 1. Main CompositionalElements of Aluminium Alloy 7055, Weight %.

Cu / Mg / Zn / Al
2.0~2.6 / 1.8~2.3 / 7.6~8.4 / Remainder

2.2The overall test programme

Tests were designed to investigate both creep and ageing behaviour of AA7055 under constant stressfor a controlled amount of time (e.g. 20 hours) at 120C. The testsare very similar to conventional creep tests, apart from the fact that the material used was not artificially aged, but was solution heat-treated andquenched. Therefore, the material was expected to be less strong initially but to exhibit a lot of hardening during the test. The hardening can be attributed to ageing due to thermal exposure and to creep deformation. At the end of the test, the material’s yield strength (measured as 0.2% proof stress) was expected to increase due to precipitation and dislocation hardening mechanisms.

Constant stress creep-ageing tests were carried out at 120 C for the stress range of 190 to 357.8 MPa. The total test duration was 20 hours, which is approximately similar to the duration for a complete industrial creep age-forming process. The experimental procedure, as shown in Fig.2,can besummarised as follows:

• First, the specimen was fitted and aligned in the middle of the furnace and one additional thermocouple was wired in the middle of the specimen gauge length.
• The furnace was closed and the heating was switched on. Thermal cotton was used to cover the top and bottom of the furnace to reduce heat loss. The closed furnace took up to 1.5 hours to rise from room temperature to reach a steady 120C.
• When the temperature became steady at 120 C, a load was applied and the elongation of the specimen was measured.
• The extension was measured every 10 seconds initially for the first 30 minutes. The time interval was then increased to 60 seconds for the rest of the experimental period.
• The data logger was stopped when the time reached 20 hours. The heating was switched off, the furnace was opened and the load was removed.

In order to determine the influence of ageing time (5 hours and 8 hours) onmechanical properties,interruptedcreep ageing tests were also carried out. Room temperature tensile tests werethen conductedto obtain the 0.2% proof strengthof the CAFed test-pieces.

1. Constitutive modelling

3.1Mechanisms of CAF and age-hardening

To model precipitation hardening, springback and mechanical property evolution of materials in CAF, it is important to have fully determined physically based unified creep ageing constitutive equations. Thus an understanding of relevant forming and age-hardening mechanisms is important.

Stress-strain-time relations of the basic CAF mechanism are shown schematically in Fig. 3. Fig. 3(a) shows the stress redistribution between initial loading and after ageing for a simple single plane bending condition. Figs. 3(b)-(d) show the corresponding stress-relaxation, creep curve and the stress-strain relationship during CAF. The stress relaxation (as shown in Fig. 3(d)) may be due to the thermally-activated diffusion and/or dislocation recovery and/or creep. The stress level (below yield stress) in the work-piece reduces from to , even though the total strain, , remains constant throughout the period of ageing. The amount of the inelastic strain, is responsible for shaping the part. It is believed that both thermally-activated stress relaxation and creep take place during the ageing.

The ageing behaviour of 7000 series aluminium alloys under isothermal heat treatment has been examined in detail by several investigators [14-17]. The generally accepted ageing sequence is

where is the aluminium matrix, GP zones are Guinier-Preston zones, is a transition phase and is the equilibrium phase, MgZn2. Fig.4 (a) shows a schematic representation of the microstructure evolution of a 7000 series aluminium alloy undergoing ageing (precipitation hardening) process. Fig. 4(b) shows experimentally determined data for time related growth of precipitate radius for AA7010 and Fig.4 (c) is a schematic ofthe individualcontributions of solute hardening, age hardening and dislocation hardening to yield strength.

The age-hardening mechanism basically is divided into several stages, as shown in Fig. 4. Upon ageing, the quenching process retains the supersaturated solid solution (SSSS) within the aluminium matrix (stage I) as there has been insufficient time for precipitate to nucleate in the water quenched aluminium alloy. The initial yield strength reflects contributions from the intrinsic strength and solute hardening.

At the early stage of ageing, ordered and solute rich clusters, so called GP zones, form (stage II). The yield strength starts to increase as the precipitate nucleates and coarsens. As the coarsening process proceeds, the SSSS in the matrix phase decreases. This decrease in the concentration of solute atoms in the matrix results in a decrease in the solute hardening. However, the decrease in solute hardening is less than offset by the increase owing to precipitate (age) and dislocation hardening. Therefore, the overall strength of the material continues to increase with time as the precipitates continue to increase in size.

Eventually the GP zone itself is replaced by the more stable phase, i.e. stage III in Fig. 4. There is no further decrease in strength owing to a decrease in the concentration of solute in the matrix, as it has reached its equilibrium value. Precipitation hardening will reach a maximum value at this stage as further nucleation and coarsening of precipitates to its peak ageing size (the best match of precipitates radius and spacing), thus the peak strength can be obtained.

During stage IV, the number of precipitates decreases and precipitate spacing increases further, where the equilibrium phase forms as large incoherent particles. The strength of the material begins to decrease. The material is now in the over-aged condition.

Another mechanism contributing to hardening is due to dislocations which are generated due to creep deformation. At the ageing temperature, annealing takes place, which reduces the dislocation density. Thus the dislocation density reaches a saturated or a balanced condition at the end of primary creep. This indicates that dislocation hardening increases quickly at the initial stage of the creep ageing process and reach to a fixed value when secondary creep starts. Dislocation also accelerates the precipitate formation and growth process. At high stress levels creep is higher, thus the dislocation density is higher, which results in a shorter ageing time necessary to reach peak strength.This interaction between age hardening and dislocation motion hardening is very important and should be included in the model comprising a set of constitutive equations.

3.2Development of unified uni-axial creep aging constitutive equations

Based on the understanding of forming and hardening mechanisms of CAF, a new set of unified physically-based constitutive equations, for uni-axial deformation,has been formulated. For ease of description, the equation set is listed first.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Where A1, B1, k0, , CA, m1, CSS, m2, A2, n, Cr, Q, m3, , m4, A3, Cp, m5are material constants.

Equation (1) describes the evolution of creep strain. Creep rate is not only a function of stress, , and dislocation density, , , but also the function of ageing (precipitation) hardening, , the solute hardening, , and dislocation hardening, , which altogether contribute to the material’s yield strength , which varies during a CAF process.

In the present model, the yield strength, derives from three sources, , and (also refer to Figure 4(c)). Deschamps et al. [18] used the precipitate radius to simplify the modelling of the ageing mechanism as the precipitate evolves or grows monotonically during isothermal ageing. However two problems arise due to this simplification: First, the initial precipitate radius is difficult to determine and varies with different aluminium alloys. Second, according to classic ageing mechanisms [19], in addition to precipitate radius, the spacebetween precipitates is another important factor to influence precipitate hardening behaviour and this should also be considered in the hardening equations. Thus a more general parameter, normalized precipitate size, is introduced here:

(8)

Where is the precipitate size at peak ageing state, which considers the best match of precipitate size and spacing for the alloy.When , under-ageing exists; , represents peak-ageing and , over-ageing. This approach simplifies the modelling process significantly. The evolution of the normalized precipitate size is given in equation (6), which will be detailed later.

The term in Equation (1) is used to model the primary creep, which is the contribution of dislocation density to creep rate. This is the same as strain hardening effect on creep. But the dislocation density variation is related to dynamic and static recovery, in addition to creep deformation. A detailed description of the normalized dislocation density will be given later.

Equations (2) and (3) represent the evolution of age hardening and solute hardening, which are described in terms of normalized precipitate size,and its evolutional rate (Equation (6)). The strengthening contribution from shearable precipitates can arise from a variety of mechanisms, such as chemical hardening, coherency strain hardening, etc. However, the overall strengthening contribution from various mechanisms is summarized in Equation (2), where describes the interaction between dislocations and shearable precipitates. Equation (3) approximates the contribution from solid solution strengthening (solute hardening), where resistance is caused by solute atoms to obstruct dislocation motion. With reference to Equation (3), is a constant related to the size, modulus and electronic mismatch of the solute; describes the depletion of solute into precipitate. As the concentration of the solute atom decreases, the solid solution strengthening decreases, acting more or less like a ‘softening’ mechanism. Equation (4) describes the evolution of dislocation hardening, which is a function of normalized dislocation density, , which is defined by [20]: