Prediction of machining quality due to the initial residual stress redistribution of aerospace structural parts made of low-density aluminium alloy rolled plates

Abstract

During the machining of thick, large and complex aluminium parts, the redistribution of initial residual stresses is the main reason for machining errors such as dimensional variations and the post-machining distortions. These errors can lead to the rejection of the parts or to additional conforming operations increasing production costs. It is therefore a requirement to predict potential geometrical and dimensional errors resulting from a given machining process plan and in taking into consideration the redistribution of the residual stresses. A specific finite element tool which allows to predict the behaviour of the workpiece during machining due to its changing geometry and to fixture-workpiece contacts has been developed. This numerical tool uses a material removal approach which enables to simulate the machining of parts with complex geometries. In order to deal with industrial problems this numerical tool has been developed for parallel computing, allowing the study of parts with large dimensions. In this paper, the approach developed to predict the machining quality is presented. First, the layer removal method used to determine the initial residual stress profiles of an AIRWARE^Ⓡ 2050-T84 alloy rolled plate is introduced. Experimental results obtained are analysed and the same layer removal method is simulated to validate the residual stress profiles and to test the accuracy of the developed numerical tool. The machining of a part taken from this rolled plate is then performed (experimentally and numerically). The machining quality obtained is compared, showing a good agreement, thus validating the numerical tool and the developed approach. This study also demonstrates the importance of taking into account the mechanical behaviour of the workpiece due to the redistribution of the initial residual stresses during machining when defining a machining process plan.

Appendix: The redistribution of the residual stresses and the principle of the layer removal method

Let h and b denote respectively the height and width of a panel sampled from a rolled plate. A strain gauge has been bonded to it. Assuming plane stress conditions (σ _{z z} =0), the strains ε _{x x} and ε _{y y} in the x- and y-directions can be expressed as

$$ \varepsilon_{xx} = \frac{\sigma_{xx} - \nu \sigma_{yy}}{E} \quad\quad \varepsilon_{yy} = \frac{\sigma_{yy} - \nu \sigma_{xx}}{E} $$

(5)

where σ _{x x} and σ _{y y} are the stresses in the x- and y-directions.

In order to simplify the problem presented in this introduction, the assumption that σ _{x x} =σ _{y y} =σ(z) is made. σ(z) therefore represents the residual stress profiles which evolve only through the thickness of the plate. Using this assumption the strains in both the x- and y-directions are given by

$$ \varepsilon(z) = \frac{(1-\nu)\sigma(z)}{E} $$

(6)

The forces and bending moments associated with the initial residual stress profiles being balanced, the following conditions can be written

$$ \begin{array}{ll} F_{x} = F_{y} = b{{\int}_{0}^{h}}\sigma(z) dz = 0\\ M_{x} = M_{y} = b{{\int}_{0}^{h}}\sigma(z)z dz = 0 \end{array} $$

(7)

with F _x and F _y being the forces and M _x and M _y the bending moments in the x- and y-directions.

When the material is removed layer by layer, a new state of equilibrium is reached and therefore a new residual stress profile is obtained. If one layer with a thickness of a is removed, the new residual stress σ ^′ can be expressed as

$$ \sigma^{\prime}(z) = \sigma(z) + \sigma_{l1}(a) $$

(8)

with σ _l1(a) being a uniform balancing stress due to the removal of the first layer.

Considering balanced forces in the resulting specimen, the following equation can be written

$$ b{\int}_{0}^{h-a}\sigma^{\prime}(z) dz = 0 $$

(9)

Whereas the forces will be balanced, the bending moments will not be, due to the new asymmetric residual stress profile. A moment has to be created to counterbalance the moment $M(a) = b{\int }_{0}^{h-a}\sigma ^{\prime }(z)z dz \ne 0$ [22]. The new residual stress profile that will satisfy both the force and moment equilibrium can be written as

$$ \sigma_{new}(z) = \sigma^{\prime}(z) + \frac{M(a)\left( \frac{h-a}{2}-z\right)}{I} $$

(10)

with I the moment of inertia expressed as

$$I = \frac{b\times(h-a)^{3}}{12} $$

The sample will deform due to the redistribution of the residual stresses and due to the re-equilibrium of the forces and moments. Figure 21 illustrates the evolution of the residual stress profile, starting from the initial residual stress profile σ(z) to the residual stress profile allowing to obtain the force equilibrium σ ^′(z) and then the residual stress profile allowing to fulfil both the force and moment equilibrium σ _{n e w} (z). It is the re-equilibrium of the moment which will then provoke distortions as also illustrated in Fig. 21.

Fig. 21

Illustration of the residual stress redistribution after the removal of a layer with a simplified residual stress distribution: (a) the initial residual stress profile (b) the residual stress profile after the layer removal with only balanced forces (c) the new residual stress profile after the layer removal respecting both force and moment equilibrium

The strain gauge is used to measure the strain linked to the redistribution of the residual stresses. It is bonded to the center of the lower surface of the sample and measures the evolution of the strain after each removal of a layer. Using this method, authors in [10, 18, 22] have demonstrated that it is possible to link the initial residual stress profiles to the measured strain. Equation 11 represents this relation and gives the measured strain depending on the thickness of the layer removed. For more details on this technique and the theoretical analysis, readers can refer to [18, 22].

$$ \varepsilon(a) = \frac{1-\nu}{E(h-a)^{2}}{\int}^{h-a}_{0}[6z-4(h-a)]\,\sigma(z)\,dz $$

(11)

with ε(a) being the strain detected by the strain gauge due to the removal of a layer of thickness a and σ(z) the initial residual stress profile.