Objectives

The motivation of this thesis is to contribute to the knowledge in the field of sheet metal plastic modelling. In particular, the contribution will advance two of the fundamental elements of classical plasticity, namely the yield locus description and plastic hardening curve, which are required to develop sound constitutive equations for anisotropic materials. First, this comprises a thorough and critical review of the anisotropic yielding approaches. It also includes the comprehensive details of the current mathematical models that are used to establish the plastic hardening curve when a continuous and in-line thickness measurement system, such as the Digital image correlation (DIC) system, is absent.

Two main goals are hoped to be reached within the scope of this manuscript:

● Accurate and cost-effective description of the yield locus

● Accurate determination of the biaxial flow curve when a continuous and in-line thickness measurement system, such as the DIC system, is absent

1.4.1 Accurate and cost-effective description of the yield locus 1.4.1.1 Yield locus for aluminium alloys

According to Mattiasson and Sigvant [18], the demands on the material models from the industrial viewpoint are the following:

● Accuracy and reliability. In other words, the models must describe accurately the plastic anisotropy or the yielding characteristics of the material. This requirement has been proved to be achieved by some of the recent advanced models such as Yld2000 [19] and BBC2008 [20].

● Simplicity. Some of the advanced models are complex and require advanced mathematical ability.

● Efficiency. A number of parameters are involved in the yield condition, and these parameters must be identified by conducting a certain number of experiments. The parameters have to be fitted to the experimental data. The greater the parameters are, the better the yield locus is determined. Therefore, a significant number of mechanical tests, which are time consuming and expensive, are required to identify the models’ parameters. As a result, most sheet forming industry analysts believe that this would not be suitable for industrial applications.

The central ingredient in the material characterizations is the yield function, which has a vital impact on the accuracy of numerical results such as thinning and splitting [17,21– 23].

The initial plastic anisotropy can be geometrically presented by a yield surface, while numerically it can be treated primarily through two approaches. The first approach is the phenomenological approach, which is a modelling of the macroscopic plasticity in which the average behaviour of all the grains is determined generally with experimental and mechanical tests. The second approach is the micro-macro or polycrystalline plasticity approach [24]; it is based on the crystal behaviour and averaging scheme used to determine the behaviour of the polycrystalline material. These two approaches will be described in detail in the literature review chapters (chapter 3 & chapter 4).

Today, the phenomenological approach is the only realistic approach to define the yield criteria [18,25,26]. However, some of the advanced models have a high number of parameters which must be computed by conducting more experimental work. Such mechanical tests that are involved in the identification process are costly and time consuming and must be conducted with care [7,27]; therefore, they are unrealistic for the sheet forming industry.

The main downside of the polycrystalline plasticity approach is the fact that the computational costs are still quite high. This hinders their applicability for industrial applications. Degree of complexity of the most accurate anisotropic yielding models- advanced polycrystalline plasticity models- increases analogously to the degree of their capabilities. However, the question remains whether models under this approach can replace experimental tests.

The trend in recent years is to combine the strengths of the physics-based models and phenomenological yield criteria. Consequently, the severity associated with the extensive and difficult tests that are required for calibrating advanced flexible yield functions can be reduced. These issues can be solved to some extent by combining the strengths of the polycrystalline and phenomenological approach (i.e. the possible accuracy of the polycrystal plasticity models and the computational efficiency of the phenomenological models) [25]. Crystal plasticity models can provide data points or “virtual points or experiments” that can be employed, in addition to experimental data, to calibrate advanced macroscopic yield functions. In other words, they can identify the anisotropy coefficients of the macroscopic yield functions [28].

The process of improving texture-based models and documenting knowledge about the performances of different polycrystalline models, as well as their performances in fitting advanced yield functions for different metallic sheets, is still in progress.

Yield locus derived from Taylor models (TP and TF) -CTFP

Pioneering previous work in the area of combining the strengths of the two approaches showed that the simple Taylor models, namely the full constraint model (TF) and its relaxed version (TP), were able to be useful tools to describe the behaviour of different considered steel alloys [7]. In their work, the correlation between the calculated texture- based yield loci and those measured from mechanical tests to define advanced yield functions efficiently and effectively for different steel grades was investigated. Based on the two texture-based yield loci calculated from the Taylor models, An et al. proposed a

new combined model referred to as CTFP [7]. The applicability of this model on the considered aluminium alloys is firstly examined.

Yield locus derived from Taylor model (TF) -CTF

Following a similar methodology to the CTFP model, a new yield loci description based on the simple (Full constraint) FC-Taylor model is proposed and compared with different macroscopic yield functions in chapter 7. One of the ultimate goals of this thesis is to develop and validate this new description using two aluminium alloys (Al-Mg-Si alloys). The new methodology focuses on calibrating the initial yield loci of the BBC2005 for two aluminium alloys. Simplicity and efficiency were put under consideration at the phase of proposing this new description for the anisotropic yielding behaviour.

1.4.1.2 Optimization of the phenomenological constitutive models parameters

Advanced yield criteria are used in academia and industry to describe the onset of plastic anisotropy for a material subject to given strain paths. Advanced yield criteria involve a certain number of plastic anisotropy parameters that have to be identified. The Newton– Raphson (NR) procedure has been widely used to identify these parameters. However, this procedure can fail to converge, especially for highly anisotropic materials such as AA2090-T3 or initial guesses far from the solution. Moreover, methods based on Newton iteration fail if the Jacobian of the system of nonlinear equations associated with yield function is singular. This research presents three solution methods for identifying material coefficients: the Levenberg (L), Levenberg–Marquardt (LM), and trust region dogleg (TRD) methods. These algorithms are hoped to overcome the problems encountered with the NR procedure. The constitutive models used to investigate the capability of these optimization methods are the BBC2003 and Yld2000-2d yield criteria, which are suitable for various materials. The applied algorithms are tested for various aluminium and steel alloys with different levels of anisotropy. The sensitivity of the yield functions to the initial guess is examined for the various applied solution methods and materials. The robustness and the effectiveness of the solution methods are investigated for the yield functions and materials. Finally, the performances of the yield functions are compared for the considered identification procedures for different materials.

1.4.2 Accurate and cost-effective description of the biaxial flow curve

In sheet metal forming simulation, flow curve and yield criterion are vital requirements for reliable numerical results. It is more appropriate to determine the flow curve using biaxial stress state tests such as the hydraulic bulge test instead of the uniaxial test because

hardening proceeds higher strains before necking occurs. With the uniaxial test, higher strains are extrapolated, which might lead to incorrect results. The bulge test, coupled with the digital image correlation (DIC) system, is utilized to determine the stress-strain data. In the absence of the DIC system, analytical methodologies are used to estimate hardening.

The hydraulic bulge test, in combination with a digital image correlation (DIC) system, is the state of the art to determine the biaxial stress-strain curves [29–31]. However, in the absence of continuous and in-line thickness measurement systems, such as the DIC system, researchers use simple analytical methods to determine the biaxial flow curves [30,32]. Furthermore, the usefulness of these analytical methodologies becomes apparent when deformation is reordered at high temperature [32,33]. It was reported by Koc et al. [33] that the results obtained with optical systems are inaccurate due to vapour and smoke occurring during deformation at high temperatures. These methods are concerned with identifying the bulge radius as well as the thickness at the dome apex.

Typically, such models incorporate a correction factor to achieve correlation to experimental data. An example is the Chakrabarty and Alexander method that uses a correction factor based on the n-value. Here, the Chakrabarty and Alexander approach is modified using a correction factor based on normal anisotropy.

Finally, because the biaxial flow curve is required to compute the biaxial yield stress, which is an essential input to define advanced yield functions, the effects of the different approaches of the biaxial stress-strain data determination on the shape of the BBC2005 yield loci is also investigated.