TY - JOUR
AU - Middendorf,, Peter
AB - Abstract This paper investigates the possibility of producing an equivalent structural behaviour of two components each consisting of a different material. This is achieved through the implementation of structural optimizations. It is assumed that the initial structure is produced by conventional injection moulding and the structure to be optimized is 3D printed. For comparison, two material pairings currently used in both processes are considered. As a structural optimization method, thickness optimizations are performed in order to change the resulting cross-section of the prototype. At the beginning, the mechanical problem is formulated analytically and methods for structural optimization are evaluated. With finite element analysis, two methods are presented, which introduce the generation of a variable thickness distribution in rib structures. The first method represents a state-of-the art optimization. Ribs are directly optimized by approximating cross-section forces and moments of the prototype rib and the initial rib. The second method represents a new approach to the optimization of thin-walled structures. Local stress distributions and resulting triaxiality states, which are approximated in an intermediate step, are analysed. A newly developed finite element structure is presented, with which it is possible to generate discrete triaxiality fields and determine the necessary local thickening. This method can be used in order to produce functional prototypes in early design stage. The substituted plastic parts are usually produced by injection moulding, which initially requires a high expenditure of time and money for tool construction. Additive manufacturing represents a solution here to accelerate the development process. However, these 3D-printed prototypes are, regarding the material properties and resulting mechanical behaviour, different to the injection-moulded ones. Graphical Abstract Open in new tabDownload slide Graphical Abstract Open in new tabDownload slide size optimizations, 3D printing, equivalent structures, triaxiality stress states Highlights The potential of producing structures that are behaving mechanically equivalent is analysed. Conventional size optimizations as well as a new method to analyse and size optimise structures are presented. Local force–displacement curves and triaxiality states are examined. Material substitution of thermoplastic materials is evaluated. Method is used to produce fast functional prototypes with the help of additive manufacturing. 1. Introduction The conscious use of our resources and the continuous reduction of emissions is a major task for the current and future automotive industry. One possible way to achieve this goal is to reduce the amount of metal structures in the car’s body by integrating plastic components. However, the plastics used must meet the same high standards of durability and safety as the metal structures they replace. Virtual and real experiments are carried out to assess the plastic components and their influence on the overall vehicle behaviour in the test scenario. Functional prototypes are therefore required in the early development phase, especially for initial validation through real experiments. The semi-structural plastic components used are usually thermoplastics produced by injection moulding. In terms of economy in the series production process, this method has a major advantage in economic efficiency compared with other manufacturing processes. But with regard to early prototype production, the design and the production of the tooling take a lot of time. In order to produce early prototypes as quickly as possible, other manufacturing methods, such as generative manufacturing processes, can remedy this. In the last decade, there was a huge development of different 3D-printing processes and the used printing materials. As there was a demand for not only non-exclusive design prototypes but also functional prototypes, the processes and materials were further developed and secured in order to achieve the required quality standards. After process validation, the investigation and evaluation of the material properties started. With the background of attempts to substitute injection-moulded components in trials, significant differences in the mechanical properties of the prototypes were found. This can be explained by the fact that either the original material is not available in the printing process used or, if it is, the raw material and the subsequent printing process lead to different material and component behaviour (Lay & Thajudin, 2019). In order to properly use the prototypes in functional experiments, a method needs to be developed that copes with the difference in the component behaviour. 1.1. Shape and material substitution in structural components Considering the history of structural design, shape and material substitution is used from the beginning to influence the overall behaviour of structures. In early machine design, only steel as a high-performance material has been available at the beginning. Therefore, a needed increase of mechanical performance was only possible by changing the components’ shape. First evaluations of the resulting moment of inertia of profiles have been conducted by Thum and Petri (1941). Using the same thickness and cross-section area as fixed constraints, the differences of the regarded circular-, rectangular-, elliptical-, and H-shaped profiles were exposed. Over the years, various design rules and profile parameters have been derived in machine design, so that now, depending on the load case, suitable shape optimizations are available (Weck & Brecher, 2013). Related to lightweight applications, Wiedemann (2007) enlarged the studies on the evaluation of the performance of lightweight structures. With the definition of structural values, the comparison and the substitution of 1D and 2D structures were enabled. Furthermore, methods for the designing of structures using different materials were provided. The rules are based on constant elastic stiffness of the regarded materials and a maximum allowed stress. A similar approach to compare the performance, weight, and costs in the material and design process is given by Ashby (2000). Studies about the efficiency and the limits of material substitutions in complex vehicle assemblies were conducted by Patton, Li, and Edwards (2004). With the constraint of equivalent stiffness response in vehicle parts, Patton shows that the theoretical weight savings through material substitution is highly dependent on the local structure and joining techniques. An analytical and numerical evaluation of the crashworthiness of different profiles and materials in vehicles is provided by Han, Hou, and Hu (2011). Han uses the stiffness mass index, which relates the stiffness and density of materials to evaluate the performance of parts under bending and torsional loading conditions. In previous studies, profiles and structures with classical, homogeneous cross-sections were investigated. Pasini revaluated the performance of materials in the design process with a focus on shape modification. Pasini introduces the concept of shape transformers, which allows a geometrical independent shape optimization (Pasini, 2002). He published an adaption of this concept on lightweight structures, which allows a quick material and profile substitution with the determined material and shape graphs (Pasini, 2007). Singh extended Pasini’s studies for overlaid loading conditions and provided efficiency maps for scaled cross-section in open and closed profiles (Singh, Mirjalili, & Pasini, 2011). The described shape deformations are theoretically spoken applicable on every material. But in terms of practical use, the possible achievable geometry and shape is limited depending on the production process of each material. One exception of this production constraint is composite structures. Having same cross areas and shape of a profile, laminated structures can adjust the structural response by rearranging and orientating single plies in the ply stack. On base of Pasini’s shape transformers, An, Singh, and Pasini (2017) determined various material and shape graphs by the help of finite element analysis in order to enlarge the design space in lightweight structures. Besides the classical applications in the field of structural mechanics, current publications deal with hybrid structures, where the mechanical and physical properties are influenced by the shape of the structure under consideration (Hamdia et al., 2019). In contrast to the previous mentioned studies, where the aim was to find a material independent shape factor to calculate resulting properties, the recent studies use a new approach. In order to cope with the large amount of iterations in multifactor optimizations, machine learning techniques were used to find fast solutions. Samaniego uses in his article deep neuronal network to deal with partial differential equations (Samaniego et al., 2020). With his energy approach, a broad spectrum of physical and mechanical problems can be solved without using discretization techniques like the finite element method. Other studies present approaches to address major optimization problems using either new optimization algorithms (Li et al., 2020) or modified optimization models at different scale levels (Li, Luo, Xiao, Gao, & Gao, 2019). 1.2. Brief analytical overview on material substitution The mechanical behaviour of a structure is composed of its behaviour under static and dynamic loading conditions. In mechanical design, the static load cases outweigh the dynamic ones. The main reason for this is that there is often only a need to determine the stiffness and the allowed strength in case of elastic loading. This means that in the case of material substitutions almost exclusively the elastic behaviour is approximated. Expect taking into account the fact that in the past these structures generally consisted of metals or very brittle, fibre-reinforced materials, and almost ideally linear elastic mechanical behaviour can be assumed. From a mechanical point of view, using Hooke’s laws to calculate stresses and strains is legitimate. Under the further condition that small deformations are present, the stiffnesses, for a given modulus of elasticity, can be adapted by shaping the structure. In applied mechanics, there are four different types of stiffnesses like tensile, torsion, bending and shear stiffness. Their formulas show the influence of the cross area on the resulting parts’ stiffness. A direct proportionality of the cross area A on the resulting stiffness is present in tensile and shear load cases. In torsional and bending load cases, the cross-sectional area is considered in the moments of inertia. Here, the shape and distribution of the area along the local axes play a central role. In Table 1, the corresponding equations are listed up. Table 1: Overview of mechanical stiffness formulations. Structural stiffnesses . Moment of inertia rectangle . Tensile . Torsion . Bending . Shear . Bending . Torsion . |$E \cdot A$| |$G \cdot I_{T}$| |$E \cdot I_{yy}$| |$G \cdot A$| |$I_{yy} = \frac{b\cdot h}{12}$| |$I_{T} = chb^{3}$| Structural stiffnesses . Moment of inertia rectangle . Tensile . Torsion . Bending . Shear . Bending . Torsion . |$E \cdot A$| |$G \cdot I_{T}$| |$E \cdot I_{yy}$| |$G \cdot A$| |$I_{yy} = \frac{b\cdot h}{12}$| |$I_{T} = chb^{3}$| Open in new tab Table 1: Overview of mechanical stiffness formulations. Structural stiffnesses . Moment of inertia rectangle . Tensile . Torsion . Bending . Shear . Bending . Torsion . |$E \cdot A$| |$G \cdot I_{T}$| |$E \cdot I_{yy}$| |$G \cdot A$| |$I_{yy} = \frac{b\cdot h}{12}$| |$I_{T} = chb^{3}$| Structural stiffnesses . Moment of inertia rectangle . Tensile . Torsion . Bending . Shear . Bending . Torsion . |$E \cdot A$| |$G \cdot I_{T}$| |$E \cdot I_{yy}$| |$G \cdot A$| |$I_{yy} = \frac{b\cdot h}{12}$| |$I_{T} = chb^{3}$| Open in new tab In case of a tensile loading on a rectangular cross area, the same stiffness of a structure, where an initial material A is substituted with material B, is obtained by using Equation (1): $$\begin{eqnarray} E_{A} A_{A} = E_{B} A_{B}. \end{eqnarray}$$(1) As shown above, most of the structural components are built up with metals or fibre-reinforced materials, whose production process limits the freedom of shape. Conventional processes like forging, casting, or extrusion restrict the shape of the future structure to a preset of variants like I-, T-, O-, and H-beams. On base of the general formula (2) to calculate the second moment of inertia $$\begin{eqnarray} I_{y} = \underset{A}{\int }{z}^{2} dA \end{eqnarray}$$(2) for each section the influence of it’s parts area is summed up. 2. Problem Definition of Using Thermoplastic Structural Components When the material substitution consists of thermoplastics, the previous done assumptions to approximate the stiffness are not suitable any more. This material group has a variety of different mechanical properties and behaviours like non-linear elasticity, visco-elasticity, and plasticity. Taking into account that the regarded thermoplastic components are tested in static and dynamic loading conditions with large deformations, the previous method of calculation is, concerning the law of continuous mechanics, not possible any more (Altenbach, 2018; Erhard, 2008). The second aspect is that the structures have different loading conditions. While metal H-sections are often used to resist bending force, ribbed structures in automotive industries are subject to a variety of force boundary conditions. This means that the direct material substitution for an approximately rectangular rib cannot be fulfilled by a regular shape transformation like a homogeneous thickening. As shown in Table 1, this direct thickening is not able to achieve an equivalent behaviour in all stiffnesses at the same time. The possible approximation is therefore limited to a certain level. Finally, the injection moulding process also restricts the form and shape of the mostly rib-structured components. However, in terms of using 3D printing to build up the early prototypes, there is no need to stick to the present I-section of the ribs. A variable thickening over the height and length of the rib is possible, so that the locally needed cross-section area and resulting moment of inertia can be achieved. 2.1. Analytical problem formulation In the following section, an approach is presented that shows an analytical way to handle the material substitution and the needed thickness adjustment in order to obtain a similar mechanical behaviour. The brief study consists of a rectangular homogeneous plate that is exposed to overlaid loading conditions. As a reference behaviour, it is assumed that the plate was produced in an injection moulding process, which leads to a fixed homogeneous rectangular shape. Its mechanical behaviour should be approximated by using the 3D-printing process that allows shape modifications. The analytical derivation uses these basic assumptions: Handling non-linear material elasticity Using an elastic, ideal plastic modelling, the stiffness and the maximum force level are described. Handling large deformation It is assumed that at least in one region of the reference structure the given material yield stress is reached. The occurring strains are equal or larger to |$\epsilon _{ref} = \frac{\sigma _{yield,ref}}{E_{ref}}$|⁠. Shape modification The prototype structure is divided into sections. Each section can be modified by variable sizes (like length, width, and thickness). Besides that, the material properties of the reference structures have a higher elastic modulus as well as a higher yield stress than the prototype one. In Fig. 1, the elastic, ideal plastic materials and the regarded plate structure are illustrated. Figure 1: Open in new tabDownload slide Overview of the material substitution problem. (a) Stress–strain behaviour of material pair and (b) mounting and loading of a plate structure. Figure 1: Open in new tabDownload slide Overview of the material substitution problem. (a) Stress–strain behaviour of material pair and (b) mounting and loading of a plate structure. In general, due to the superimposed forces multiaxial planar stress states occur. This means that the stress tensor |$\sigma_{ij}$| provides values for |$\sigma_{x}$|⁠,|$\sigma_{y}$| and |$\tau_{xy}$|⁠. With the occurring local stresses, a determination of the normal and shear stress can be done by using an equivalent stress hypothesis. In this case, the Mises stress Equation (3) $$\begin{eqnarray} \begin{aligned} \sigma _{vM} &= \sqrt{\sigma _{x}^{2} + \sigma _{y}^{2} - \sigma _{x} \sigma _{y}+3\tau _{xy}^{2}}\\ \end{aligned} \end{eqnarray}$$(3) is used. With defining of the two ratios iand j, the Mises equation is defined by $$\begin{eqnarray} \begin{aligned} i &= \frac{\sigma _{y}}{\sigma _{x}} \quad\text{ and }\quad j = \frac{\tau _{xy}}{\sigma _{x}} \\ \sigma _{vM} &= \sqrt{\sigma _{x}^{2} (1+i^{2} -i +3j^{2})} . \end{aligned} \end{eqnarray}$$(4) Due to the fact that plastic strains occur and an ideal plastic behaviour is assumed, the |$\sigma_{vm}$| is equivalent to the |$\sigma_{yield}$| stress, so that |$\sigma_{x}$| is defined by $$\begin{eqnarray} \begin{aligned} \sigma _{x} &= \pm \sqrt{\frac{\sigma _{yield}^{2}}{1+i^{2}-i+3j^{2}}}. \end{aligned} \end{eqnarray}$$(5) The responding strains are calculated with Equation (6): $$\begin{eqnarray} \begin{aligned} \varepsilon _{x} &= \frac{1-\nu ^{2}}{E}\left(\sigma _{x}-\frac{\nu }{1-\nu }\sigma _{y}\right) ,\quad \varepsilon _{y} = \frac{1-\nu ^{2}}{E}\left(\sigma _{y}-\frac{\nu }{1-\nu }\sigma _{x}\right)\quad \text{ and }\quad \gamma = \frac{1-\nu ^{2}}{E}\frac{1}{1-\nu }\tau _{xy} . \end{aligned} \end{eqnarray}$$(6) Using the ratiosiand jfrom Equation (3), the Equation (6) is transformed by using the variables |$\alpha$| and |$\beta$| to $$\begin{eqnarray} \varepsilon _{x} &=& \frac{\sigma _{x}}{E}\left(1-\nu ^{2}\right)\left(1-\frac{\nu }{1-\nu }i\right) = \frac{\sigma _{x}}{E} \cdot \alpha \\ \varepsilon _{y} &=& \frac{\sigma _{x}}{E}\left(1-\nu ^{2}\right)\left(i-\frac{\nu }{1-\nu }\right) = \frac{\sigma _{x}}{E}\cdot \beta \\ \gamma &=& \frac{1-\nu ^{2}}{E}\frac{1}{1-\nu }\tau _{xy} . \end{eqnarray}$$(7) The determined stress and strains for both structures are defined by $$\begin{eqnarray} \sigma _{x,k} ,\, \sigma _{y,k},\, \tau _{xy,k},\, \epsilon _{x,k},\, \epsilon _{y,k}\, and\, \gamma _{k} \text{ with k = (ref;prot) for reference and prototype materials.} \end{eqnarray}$$(8) The size adjustment by using variable thickening of the structure will be performed and examined with two defined sectioning methods. In Fig. 2, the plate with the prototype material is divided into 4 and 9 sections. Figure 2: Open in new tabDownload slide Sectioning techniques of the plate. (a) Size-independent 2 × 2 fields and (b) fixed-size 3 × 3 fields. Figure 2: Open in new tabDownload slide Sectioning techniques of the plate. (a) Size-independent 2 × 2 fields and (b) fixed-size 3 × 3 fields. Starting with the 4-section structure, for the given reference material the resulting forces |$F_{m} $| (with |$m= x, y, \tau$|⁠) and the occurring strains are calculated with Equation (9) by $$\begin{eqnarray} F_{x} &=& \sigma _{xref} l_{y} t,\quad F_{y} = \sigma _{yref} l_{x} t \quad \text{ and }\quad F_{\tau } = \tau _{xyref}l_{x} t, \\ \Delta l_{x} &=& \varepsilon _{xref} l_{x} ,\quad\Delta l_{y} = \varepsilon _{yref} l_{y} \quad\text{ and }\quad \Delta x = tan(\gamma _{ref}) l_{x} . \end{eqnarray}$$(9) When replacing the reference material by the prototype material, the local forces |$F_{m}$| are the same. This boundary leads to the determination of the first two areas |$A_{mn} $| (with|$m=x,y$| and |$n=1,2$| ) of the 4-sectioned plate by $$\begin{eqnarray} \begin{aligned} A_{x2} &= \frac{F_{x}}{\sigma _{x,prot}} \text{ , }\quad A_{y1} = \frac{F_{y}}{\sigma _{y,prot}} \text{ .} \end{aligned} \end{eqnarray}$$(10) Regarding the deformations in x-direction, with Equation (11) $$\begin{eqnarray} \begin{aligned} \Delta l_{x} = \Delta l_{x1} + \Delta l_{x2} &= \varepsilon _{lx1}l_{x1} +\varepsilon _{x,prot} l_{x2} = \varepsilon _{lx1}l_{x1} +\frac{\sigma _{x,prot}}{E_{prot}} l_{x2} \\ \text{ with } \varepsilon _{lx1} &= \frac{\sigma _{Ax1}}{E_{prot}}\left(1-\nu ^{2}_{prot}\right)\left(1-\frac{\nu _{prot}}{1-\nu _{prot}}i\right) = \frac{\sigma _{A_{x1}}}{E_{prot}} \cdot \alpha _{prot} \text{,} \\ \text{ and with } \sigma _{A_{x1}} &= \frac{F_{x}}{A_{x1}} \text{ and } l_{x1} = l_{x} - l_{x2} \\ \end{aligned} \end{eqnarray}$$(11) the relative elongation |$\Delta l_{x}$| follows by Equation (12) $$\begin{eqnarray} \begin{aligned} \Delta l_{x} &= \varepsilon _{x,prot} l_{x2} + \frac{F_{x}}{A_{x1}E_{prot}}\cdot \alpha _{prot}\cdot (l_{x}-l_{x2}), \\ \Delta l_{x} &= l_{x2} \cdot \left(\varepsilon _{x,prot} - \frac{F_{x}\cdot \alpha _{prot}}{A_{x1}E_{prot}}\right) + \frac{F_{x} \cdot \alpha _{prot}}{A_{x1}E_{prot}}l_{x} .\\ \end{aligned} \end{eqnarray}$$(12) Using physical assumptions, the length |$\Delta l_{x2}$| is determined by Equation (13) $$\begin{eqnarray} \begin{aligned} l_{x2} \rightarrow 0 &\Rightarrow A_{x1} \rightarrow \frac{F_{x}\cdot \alpha _{prot}}{\Delta l_{x} E_{prot}} \cdot l_{x},\\ l_{x2} \rightarrow \frac{\Delta l_{x}}{\varepsilon _{x,prot}} &\Rightarrow A_{x1} \rightarrow \infty .\end{aligned} \end{eqnarray}$$(13) The extracted interval for the length |$\Delta l_{x2}$| is determined, which leads to the calculation of |$A_{x1}$|⁠. The same procedure can be done for the y-direction, so that the cross-sectional area |$A_{y2}$| is calculated. The shown derivation provides thickness and lengths of the four sections that are mathematically correct. But in terms of physical values, negative thicknesses or lengths can appear. In a second step, the 3 × 3 sectioned plate of Fig. 2 is evaluated. In contrast to the 2 × 2 sectioned plate, here the nine rectangles have a predefined length and width. The cross area |$A_{x1}$| is therefore calculated by $$\begin{eqnarray} A_{x1} = \frac{l_{y}}{3} \cdot (t_{1}+t_{4}+t_{7}). \end{eqnarray}$$(14) Using the previously defined equations from (3) to (9), the first two cross areas |$A_{x3}$| and |$A_{y1}$| are calculated by $$\begin{eqnarray} \begin{aligned} A_{x3} &= \frac{F_{x}}{\sigma _{x,prot}}\quad \text{ and }\quad A_{y1} &= \frac{F_{y}}{\sigma _{y,prot}} . \end{aligned} \end{eqnarray}$$(15) Regarding the deformation in x-direction, the relative elongation of the subdivided plates is determined by (16) $$\begin{eqnarray} \begin{aligned} \Delta l_{x} &=\varepsilon _{lx1}\frac{l_{x}}{3}+\varepsilon _{lx2}\frac{l_{x}}{3} + \varepsilon _{lx3}\frac{l_{x}}{3} .\\ \end{aligned} \end{eqnarray}$$(16) With the two strains |$\varepsilon_{lx1}$| and |$\varepsilon_{lx2}$| for the first two sections $$\begin{eqnarray} \begin{aligned} \varepsilon _{lx1} &= \frac{\sigma _{Ax1}}{E_{prot}}\left(1-\nu ^{2}_{prot}\right)\left(1-\frac{\nu _{prot}}{1-\nu _{prot}}i\right) = \frac{\sigma _{A_{x1}}}{E_{prot}} \cdot \alpha _{prot}\quad \text{ and }\quad \sigma _{A_{x1}} = \frac{F_{x}}{A_{x1}}, \\ \varepsilon _{lx2} &= \frac{\sigma _{Ax2}}{E_{prot}}\left(1-\nu ^{2}_{prot}\right)\left(1-\frac{\nu _{prot}}{1-\nu _{prot}}i\right) = \frac{\sigma _{A_{x2}}}{E_{prot}} \cdot \alpha _{prot}\quad \text{ and }\quad \sigma _{A_{x2}} = \frac{F_{x}}{A_{x2}}\\ \end{aligned} \end{eqnarray}$$(17) and |$\varepsilon_{lx3}$|⁠, which is directly determined to the general assumption 2, follows $$\begin{eqnarray} \begin{aligned} \varepsilon _{lx3} &= \varepsilon _{x,prot} = \frac{\sigma _{x,prot}}{E_{prot}} \left(1-\nu ^{2}_{prot}\right)\left(1-\frac{\nu _{prot}}{1-\nu _{prot}}i\right) = \frac{\sigma _{A_{x3}}}{E_{prot}} \cdot \alpha _{prot} \quad\text{ and }\quad \sigma _{A_{x3}} = \frac{F_{x}}{A_{x3}} ,\\ \end{aligned} \end{eqnarray}$$(18) so that final elongation is resulted in Equation (19) $$\begin{eqnarray} \begin{aligned} \Delta l_{x} &= \varepsilon _{x,prot} \frac{l_{x}}{3} + \frac{F_{x} \cdot \alpha _{prot}}{A_{x1}E_{prot}}\frac{l_{x}}{3}+\frac{F_{x} \cdot \alpha _{prot}}{A_{x2}E_{prot}}\frac{l_{x}}{3}, \\ \Delta l_{x} &= \frac{l_{x}}{3}\left( \varepsilon _{x,prot} + \frac{F_{x} \cdot \alpha _{prot}}{E_{prot}}\cdot \left(\frac{1}{A_{x1}}+\frac{1}{A_{x2}} \right) \right). \\ \end{aligned} \end{eqnarray}$$(19) The relative elongation in y-direction is determined similarly. Additional transformations of Equation (19) illustrate the dependency between the thicknesses and the given material properties and the occurring forces. Equations (20) and (21) describe this dependency for the x-direction $$\begin{eqnarray} \begin{aligned} \frac{1}{(t_{2}+t_{5}+t_{8})}+\frac{1}{(t_{1}+t_{4}+t_{7})} &= \frac{l_{y} E_{prot}}{3F_{x} \cdot \alpha _{prot}} \cdot \left(3\frac{\Delta l_{x}}{l_{x}}\right), \end{aligned} \end{eqnarray}$$(20) as well as for the y-direction $$\begin{eqnarray} \begin{aligned} \frac{1}{(t_{4}+t_{5}+t_{6})}+\frac{1}{(t_{7}+t_{8}+t_{9})} = \frac{l_{x} E_{prot}}{3F_{y} \cdot \beta _{prot}} \cdot \left(3\frac{\Delta l_{y}}{l_{y}}\right). \end{aligned} \end{eqnarray}$$(21) In contrast to the 2 × 2 plate sectioning in Equation (13), the solution provides no interval for the thicknesses. The given restrictions and boundaries can be used in an optimization process in order to find the thicknesses. A first general assumption, if the chosen material combination is suitable for a substitution process, is defined in the following section. In order to be able to make an estimation of the process limits, the Equation (19) can be transformed to $$\begin{eqnarray} \Delta l_{x} = \frac{l_{x}}{3}\left( \varepsilon _{x,prot} + \frac{F_{x} \cdot \alpha _{prot}}{E_{prot}}\cdot \left(\frac{1}{A_{x1}}+\frac{1}{A_{x2}} \right) \right) \rightarrow 3\cdot \frac{\Delta l_{x}}{l_{x}}-\varepsilon _{x,prot} = \frac{F_{x} \cdot \alpha _{prot}}{E_{prot}}\cdot \left(\frac{1}{A_{x1}}+\frac{1}{A_{x2}} \right). \end{eqnarray}$$(22) Further, the general deformation of the reference structure leads to the ratio |$\frac{\Delta l_{x}}{l_{x}}$|⁠, which is equal to |$\varepsilon_{xref}$|⁠. In addition, the resulting cross areas |$A_{x1}$| and |$A_{x2}$| have to be the same or larger in size than the |$A_{x3}$| to obtain the reference structure behaviour. Therefore, Equation (22) is transformed to $$\begin{eqnarray} \begin{aligned} 3\cdot \varepsilon _{xref}-\varepsilon _{x,prot} &\ge \frac{F_{x} \cdot \alpha _{prot}}{E_{prot}}\cdot \left(\frac{1}{A_{x3}}+\frac{1}{A_{x3}} \right) = \frac{F_{x}}{E_{prot}}\cdot \frac{1}{2A_{x3}}. \\ \end{aligned} \end{eqnarray}$$(23) With using |$A_{x3} = \frac{F_{x}}{\sigma _{x,prot}}$| and |$F_{x} = \sigma_{xref} \cdot A_{xref} $| follows $$\begin{eqnarray} \begin{aligned} 3\cdot \varepsilon _{xref}-\varepsilon _{x,prot} &\ge \frac{\sigma _{xref}A_{xref} \cdot \alpha _{prot}}{E_{prot}}\cdot \frac{\sigma _{x,prot}}{2\sigma _{xref}A_{xref}}, \\ \end{aligned} \end{eqnarray}$$(24) where |$A_{xref}$| is the initial cross area of the reference structure. Additionally, it follows that |$\sigma _{x,prot} = E_{prot} \varepsilon _{x,prot} \frac{1}{\alpha _{prot}}$|⁠. A further transformation leads to Equation (25), where |$\varepsilon_{ref}$| and |$\sigma_{x,prot}$| are in the assumption equal to |$\varepsilon_{ref}$| and |$\sigma_{prot}$|⁠: $$\begin{eqnarray} \begin{aligned} 3\cdot \varepsilon _{xref} E_{prot}-\frac{2\alpha _{prot}+1}{2\alpha _{prot}} \sigma _{x,prot} &\ge 0 \rightarrow 3\cdot \varepsilon _{ref} E_{prot} \alpha _{ref}-\frac{2\alpha _{prot}+1}{2\alpha _{prot}} \sigma _{prot} &\ge 0 . \end{aligned} \end{eqnarray}$$(25) With |$\varepsilon _{ref} = \frac{\sigma _{ref}}{E_{ref}}$|⁠, the last transformation leads to Equation (26): $$\begin{eqnarray} \begin{aligned} \frac{E_{prot}}{E_{ref}} \alpha _{ref} \ge \frac{2\alpha _{prot}+1}{6\alpha _{prot}}\cdot \frac{\sigma _{prot}}{\sigma _{ref}} . \\ \end{aligned} \end{eqnarray}$$(26) This equation allows to evaluate if the chosen material combination has a solution for the regarded subdivided plate. However, to perform the equation, the local state must already be known to get the necessary α. In daily work, often only the material properties and the loading condition of a structure are known. However, in order to be able to make a rapid assessment for some cases, the current method is manipulated by Assumption 1: Loading case induces uniaxial stress in x-direction. In this case, |$\sigma_{y}$| is zero or a small value. Then, the ratio i from Equation (4) will also become zero or very small, so that finally |$\alpha$| in Equation (12) is only dependent on the Poisson’s ratio ν. Assumption 2: Neglecting the Poisson’s ratio in deformation. With this major cut in the mechanical calculations, the proportion of the |$\sigma_{y}$| and the Poisson’s ratio to the resulting strain in x-direction |$\varepsilon_{x}$| are neglected. If the first assumption is true, α takes on a small constant value. If, in addition, the regarded material has a small Poisson's ratio, due to almost no necking behaviour in uniaxial deformation, |$\alpha$| is approaching 1. Using these assumptions in the previous derivation, equations for an estimation can be determined for both cutting techniques, using only the material characteristics. In addition, for the 2 × 2 sectioning similar to the 3 × 3 sectioning, fixed sizes of the division were defined. With Equation (27), an estimation with the defined assumption can be performed $$\begin{eqnarray} \begin{aligned} \text{for the 2 x 2: } 1 \le \frac{\sigma _{prot}}{\sigma _{ref}} \cdot \left(\frac{E_{prot}}{E_{ref}}\right)^{-1}\le 2 \text{ and } \text{for the 3 x 3: } \frac{E_{prot}}{E_{ref}} \ge \frac{1}{2} \frac{\sigma _{prot}}{\sigma _{ref}} .\\ \end{aligned} \end{eqnarray}$$(27) To illustrate the feasible and non-feasible material substitution pairs, in Fig. 3 for both sectioning techniques the results of the each related equation are plotted. A significant difference between the two sectioning methods in terms of suitable material pairs can be figured out. The 2 × 2 sectioning provides only a small region of the two property ratios, where an equivalent behaviour can be obtained. In contrast to that, the feasible solution space is larger by using the 3 × 3 sectioning method. Figure 3: Open in new tabDownload slide Feasible and non-feasible material combinations of the regarded sectioning techniques. (a) Fixed field size 2 × 2 and (b) fixed field size 3 × 3. Figure 3: Open in new tabDownload slide Feasible and non-feasible material combinations of the regarded sectioning techniques. (a) Fixed field size 2 × 2 and (b) fixed field size 3 × 3. The analytical problem formulation and the derivation of the sectioning methods show the complex interaction between the local stress states and the possible shape adjustment when approximating the mechanical behaviour. 2.2. Aim of the finite element study This paper investigates the possibility to achieve an equivalent mechanical behaviour in structures, which are built up with two different materials, by using finite element analysis. For four different load types, the resulting normal forces and moments are examined in defined cross-sections of the reference material. After that, thickness optimizations on a prototype structure are performed, in order to approximate the forces and moments of the reference one. The prototype structure is thereby built up with the second material. Besides the approximation of the stiffnesses, the main focus lies on the approximation of the resulting force levels, when the structures are subjected to large deformations. Therefore, material cards, which represent an elastic, ideal plastic mechanical behaviour, are used. 3. Introduction to the Finite Element Study The aim of this study is to evaluate if a thermoplastic material can be substituted by a thermoplastic 3D-printed one in early prototype testings. The resulting degree of accuracy of the approximation must also be determined. To start with, the study consists of two material combinations. The reference materials are conventional injection moulding thermoplastics, which are used in structural components in the car body or in semi-structural components in the interior. As prototype materials, two commonly used materials in the selective laser sintering (SLS) process are chosen. The SLS process enables an extensive geometric freedom, which is essential in terms of a later on transferability and testability of the theory in a testing scenario. In Fig. 4, the two material combinations are listed up. Material combination I consists of the reference material polyamide 6 (PA6) with 60% of glass fibres (PA6-GF60), which is especially used in structural crash components. As a substitution material, the PA12 short carbon fibre-reinforced SLS material is chosen. The elastic stiffness as well as the yield stress of the prototype material is lower than the reference material ones. The material pair II contains the high-density polyethylene, which is used in interior structures with high need of energy absorption. As substitution material, the polyamide 12 is defined. This SLS material is especially suitable for functional prototypes, because of its extraordinary isotropic elasticity and for small deformation isotropic plastic behaviour. Figure 4: Open in new tabDownload slide Overview of the two material combinations. (a) Defined stress strain behaviour and (b) defined material property values. Figure 4: Open in new tabDownload slide Overview of the two material combinations. (a) Defined stress strain behaviour and (b) defined material property values. Both materials have, either due to the reinforcement and/or due to the production process, more or less anisotropic local mechanical behaviour. However, regarding the later-on overall component behaviour, the materials often show rather isotropic mechanical behaviours. To start with, in this study for both material combinations isotropic properties are assumed. 3.1. Introduction to the FE model set-up The automotive structures under consideration consist of several reinforcing ribs, which are used to stiffen the structure and to avoid instabilities such as dents or warping. Common to all regarded ribs are their thin-walled structure, which means that their thickness dimension is significantly less than their length and width. Considering this, the resulting stresses over the thickness can be neglected and a modelling with finite shell elements is possible. From this, it follows that the manipulation of the targeted cross-section area can be performed by changing the thickness parameter of the shell elements. The thin-walled rib structures also have their several junctions in common. This allows to have a closer look on the resulting deformation and stress distributions of the ribs separately. In the modelling, a rib is cut out, mounted on the one side, and loaded on the other. After analysing the local behaviour of one rib, the four in Fig. 5 presented loading conditions are derived. Figure 5: Open in new tabDownload slide The four defined load cases of the FE studies. Figure 5: Open in new tabDownload slide The four defined load cases of the FE studies. The four loading cases induce deformations due to tension, rotation, bending, and compression forces. The chosen dimensions are 100 mm length, a width of 30 mm and an initial thickness of 2 mm. The discretization of the ribs uses fully integrated 4-node shell elements with a length of 1 mm. All loading conditions are performed with a predefined translation or rotation, to achieve the same deformation in the referenced and optimized prototype structures. The used material cards are not considering a failure or a damage behaviour. The finite element analysis is, due to no major occurring non-linearities, performed with an implicit solver. 3.2. Optimization variants and set-ups 3.2.1. Variant A–direct optimization The conducted size optimizations consist of two modelling methods, which illustrate different ways to handle and approximate the occurring local forces. Both variants divide the rib into several section groups to perform a thickening of smaller areas. Starting with variant A, the set-up shell model with the defined local properties regions is shown in Fig. 6. Figure 6: Open in new tabDownload slide Overview on the defined FE models (left: reference model with homogeneous thickness; right: prototype model with variable thickness regions). Figure 6: Open in new tabDownload slide Overview on the defined FE models (left: reference model with homogeneous thickness; right: prototype model with variable thickness regions). The rib is sectioned into 12 homogeneous rectangular regions by assigning different shell properties, which each consist of a single thickness parameter. The distribution and the arrangement resembles a conventional H-beam. If, for example, the optimized structure needs a higher bending stiffness to reach the reference stiffness and force level, the top and bottom property layers are relatively thickened up. In this case, the H-section is approximated. The distribution over the length makes it possible to increase or decrease the cutting forces and moments section by section. At the beginning of variant A, the loading and examination of the reference structure is conducted. The number and arrangement of the cross-sections that are needed to extract the resulting cutting forces and moments are defined. Figure 6 presents the sectioning of the reference FE model performed. In step three, the size optimization is set up. As input variables, the thickness parameters are defined continuously with thickness borders from 1 to 5 mm. The definition of borders is based on two aspects. First, the optimized thicknesses of the prototype structure should at least fit the stiffness level of the reference structure. With the regarded material combinations of Fig. 4, the ratios of the elastic modulus (Matcomb I = 1.92; Matcomb II = 0.80) multiplied by the initial thickness of 2 mm lead to upper and lower ranges of 1.6–3.8 mm. On the other hand, thinner and thicker regions in the structure are needed in order to realize H-shaped sections, which help to increase or decrease bending stiffnesses. With the chosen ranges a proper structural optimization was given. The boundary thickness remains constant in both cases. The optimization set-up defines a partial factorial sampling method to generate the parameter configurations. This method reduces the total number of conducted virtual tests. For each loop, 20 parameter configurations, based on the initial parameter values and the given upper and lower ranges, are calculated and examined. The gradient-based optimization iterates until the minimal deviation between the reference solution and the latest solution is found in the reduced range (Stander & Basudhar, 2019). The comparison takes places in form of a curve mapping algorithm, where the deviation between the reference curves and the calculated curves is evaluated. Figure 7 shows the process steps for both optimization methods. Figure 7: Open in new tabDownload slide Process steps of the methods. (a) Direct optimization variant A and (b) indirect optimization variant B. Figure 7: Open in new tabDownload slide Process steps of the methods. (a) Direct optimization variant A and (b) indirect optimization variant B. The determined thickness distribution of the section can then be used to build up the functional prototype rib by using 3D-printing processes. 3.2.2. Variant B – indirect, triaxiality state-based optimization The second optimization method is build up on a two-level optimization process. At first, on the basis of the analytical Section 2.1, the chosen materials are evaluated. It is checked whether for the 3 × 3 sectioning, as shown in Fig. 3, a solution is feasible. After that, the stress distribution in the rib is evaluated. Therefore, the local triaxialities η are computed with the following formula: $$\begin{eqnarray} \eta = \frac{\frac{1}{3} (\sigma _{x} + \sigma _{y} +\sigma _{z})}{\sqrt{\sigma _{x}^{2} + \sigma _{y}^{2} - \sigma _{x}\sigma _{y}+3\tau _{xy}^{2}}}. \end{eqnarray}$$(28) The stress triaxiality analysis illustrates the local stress state of the deformed structure. Stress triaxialities can change within the deformation process due to changing loading and mounting conditions or inhomogeneous material properties. Depending on the occurring type of stress state, which could result in a uniform or multiaxial stress state, the elements are grouped by discrete triaxiality values. This sectioning is affecting the following mapping process, in which the chosen degree of refinement will increase proportionally the optimization and mapping effort. After extracting the discrete triaxiality fields, a newly developed numerical model is used to generate homogeneous discrete triaxiality areas. In Fig. 8, the so called 'triaxiality dodecagon' is presented. Figure 8: Open in new tabDownload slide FE model of deformation dodecagon. (a) Applied deformation boundaries on FE-dodecagon model and (b) defined cross-sections and thickness fields on the inner region. Figure 8: Open in new tabDownload slide FE model of deformation dodecagon. (a) Applied deformation boundaries on FE-dodecagon model and (b) defined cross-sections and thickness fields on the inner region. The modelled dodecagon, which has an outer diameter of 160 mm and an initial thickness of 2 mm, contains two regions with different defined properties. The boundary region has a uniform thickness of 2 mm and a specific material, which acts like a regular spring. The defined material consists of a 25% higher elastic stiffness than the inner region and displays only linear-elastic behaviour. The inner region is, similar to the analytical study in Fig. 2, divided into nine squares. Each square consists of four 4-node shell elements with an element length of 5 mm and a defined thickness parameter. This internal configuration, based on the preliminary analytical study, allows the subsequent analysis of a large number of material combinations. Furthermore, 10 cross-sections, 5 horizontally and 5 vertically, are defined to examine the cutting forces and moment of the 9 sections. The assigned material corresponds to the reference materials of the regarded material pairs. The mounting of the dodecagon takes place by a fixed node in the centre with restricted translation and rotation degree of freedom. On the given 12 edges, the loading conditions are applied. The parametric conditions contain a scaling factor, which defines the maximum applied in-plane displacement from 0 to 100%. In addition, each displacement set has a variable continuous angle from 0 to 360°. The background of this set-up is to generate every possible configuration of the stress tensor. Equation (29) shows the general overlaid configuration of the normal and shear stresses as well as the configuration for an uniaxial tensile loading in a stress tensor S of the plane stress state $$\begin{eqnarray} S = \left( {\begin{array}{*{10}c}\sigma _{xx} & \quad \tau _{yx} \\ \tau _{xy} & \quad \sigma _{yy} \end{array}} \right); \text{ with uniaxial tension } \left(\eta =\frac{1}{3}\right) \rightarrow S_{\eta =\frac{1}{3} } = \left( {\begin{array}{*{10}c}\sigma _{xx} & \quad 0 \\ 0 & \quad 0 \end{array}} \right) . \end{eqnarray}$$(29) With the modelled dodecagon a parameter optimization is performed. As input variables, 24 parameters, which include the 12 loading angles and the 12 scaling factors, are used. As a point sampling method, the Latin Hyper Cube (Park, 1994) is used. A domain size reduction is used to decrease the sampling space and to find the local minimum of the optimization. The objective of the optimization is a defined discrete triaxiality value, which is compared with the average triaxiality value of the 36 regarded shell elements. The constraints of the optimization are the minimum and maximum existing triaxiality values in the nine squares. As lower and higher borders a 0.05 scatter band is defined. After determining the deformation configuration of the dodecagon, the resulting cutting forces and moments of the 10 cross-sections are extracted. These force and moment curves are used as reference curves for the second optimization. Another dodecagon, with the same deformation configuration but with the second material in the centre assigned, is set up. The input parameters for this optimizations are now the nine shell thicknesses. The objective of the optimization is, similar to variant A, to match the reference force and moment curves. As constraints, the thickness ratios between the nine squares are defined. In Fig. 9, the grouping of the constraints is illustrated. Figure 9: Open in new tabDownload slide Constraint configuration in size optimization. (a) Subgroups in the 3 × 3 property field and (b) thickness averaging and relative size constraint. Figure 9: Open in new tabDownload slide Constraint configuration in size optimization. (a) Subgroups in the 3 × 3 property field and (b) thickness averaging and relative size constraint. For each of the nine property fields, a local thickness group is created that averages the thickness of the fields in the adjacent fields. The ratio between the calculated thicknesses is restricted to a lower border of 80% and an upper border of 120%. These restriction rules ensure that no gross differences in thickness are iterated. As a result of the optimization, nine shell thicknesses are determined, which represent the equivalent force and moment curve for the discrete triaxiality distribution. In a final step, this thickness distribution needs to be mapped to the initial rib structure and the located triaxiality fields. Depending on the mesh sizes of the dodecagon fields and the reference model, a direct mapping is possible. If different discrete triaxiality fields are present, a separate adapted imaging technique must be used. 4. Results of Variant A 4.1. Presentation of the results 4.1.1. Material combination I The following section will separately show the results for the four defined load cases. Starting with load case one, where a tension load is applied, Fig. 10 illustrates the optimization result. Figure 10: Open in new tabDownload slide Optimization results of variant A – LC01. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). Figure 10: Open in new tabDownload slide Optimization results of variant A – LC01. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). The graph shows the reference force–displacement curve and the initial force displacement curve of the prototype material. Due to the significantly lower elastic modulus and the lower yield stress, the resulting curve of the prototype material with the same initial thickness of 2 mm is below the reference material. After the thickness optimization for the 12 regarded properties, thicknesses between 3.13 and 5.13 mm are determined. With this thickness distribution, the same force–displacement behaviour is achieved. Load case two induces a rotation on the structure, which means that in the outer regions tension stresses and in the middle shear and compression stresses occur. Figure 11 illustrates the resulting torque–rotation curve and the optimized thickness distribution. Figure 11: Open in new tabDownload slide Optimization results of variant A – LC02. (a) Torque–rotation curves, and thickness distribution at the front (b) and at the rear (c). Figure 11: Open in new tabDownload slide Optimization results of variant A – LC02. (a) Torque–rotation curves, and thickness distribution at the front (b) and at the rear (c). This case directly shows the non-uniaxial stress states in the structure. The curve is non-linear with an initial linear elastic stiffness slope and a later spline curve course. Again, the behaviour of the prototype material is non-proportional to the reference one. The deviation of both curves is increasing with the increased loading state. The optimization result determined a thickness distribution, with which the local reference torques can be approximated with the weaker prototype material. Only a small deviation of the curves occurs in the last sequence of the deformation. Concerning the thickness distribution, the optimization resulted in an inhomogeneous distribution with high thickness in the upper and middle regions as well as very thin parts at the bottom. The range includes thickness between 0.6 and 3.79 mm. In load case three, bending stresses occur due to the vertical loading. In Fig. 12, the resulting force–displacement curves are presented. Figure 12: Open in new tabDownload slide Optimization results of variant A – LC03. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). Figure 12: Open in new tabDownload slide Optimization results of variant A – LC03. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). The bending curve is non-linear and shows a high deviation between the reference curve and the prototype curve with the initial thickness results. In comparison with the results of the other load cases, the thickness distribution is quite homogeneous. The resulting thicknesses have values between 3.31 and 4.12 mm. A high approximation of the reference force–displacement curve is achieved by the prototype curve. The last load case applies a compression force on the structure. Similar to the load case one, the resulting force–displacement curves displays the linear elastic–ideal plastic behaviour of the material card. The optimized structure approximates the reference curve with no deviation. Also, similar to the tension case, the thickness distribution is homogeneous. In Fig. 13, the optimized structure and the force–displacement curves are presented. Figure 13: Open in new tabDownload slide Optimization results of variant A – LC04. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). Figure 13: Open in new tabDownload slide Optimization results of variant A – LC04. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). 4.1.2. Material combination II In the following section, the optimization results of the material combination II are presented. Beginning with load case one, the resulting curves and thicknesses are illustrated in Fig. 14. Figure 14: Open in new tabDownload slide Optimization results of variant A – LC01. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). Figure 14: Open in new tabDownload slide Optimization results of variant A – LC01. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). In contrast to the results of the material combination I, the prototype material has a higher stiffness and a higher yield stress. The resulting force–displacement curve is therefore higher than the reference one. With small deviations in the stiffness, the optimized curve is able to approximate the reference material behaviour. Compared with the material combination I, the derived thickness distribution has a wide range between 0.5 and 3.79 mm. The behaviour of the structures under rotational loading is presented in Fig. 15. Figure 15: Open in new tabDownload slide Optimization results of variant A – LC02. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). Figure 15: Open in new tabDownload slide Optimization results of variant A – LC02. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). Having a look at the structures with the same initial thickness, a rather small deviation can be noted. The optimized structure is able to approximate the behaviour under torsion loading. The thickness distribution varies between 0.5 and 2.69 mm. Regarding the bending load case, in Fig. 16 the calculated curves are displayed. The initial prototype structure behaves again stiffer than the reference one. After optimization, the bending behaviour was approximated with almost no deviation. The thicknesses were reduced significantly, in order to match the force–displacement curve. Especially in the middle section, the thicknesses reached the lower defined thickness threshold. In the outer areas, the upper limit of the allowed thickening is reached. The overall distribution consists of only three thicknesses, which are 0.5, 2.25, and 4 mm. Figure 16: Open in new tabDownload slide Optimization results of variant A – LC03. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). Figure 16: Open in new tabDownload slide Optimization results of variant A – LC03. (a) Force–displacement curves, and thickness distribution at the front (b) and at the rear (c). In the last regarded load case, higher deviations between the reference and the optimized prototype curve are observed. While the initial prototype structure is again stiffer with a higher force level, the optimized one can only approximate one behaviour of the reference curve. Either the prototype curve approximates the stiffness in small deviations or reference force level is reached. A total approximation could not be figured out. The results of the last load case of material combination II are presented in Fig. 17. Figure 17: Open in new tabDownload slide Optimization results of variant A – LC04. (a) Force–displacement curves, and thickness distribution for stiffness approximation at the front (b) and at the rear (c). Figure 17: Open in new tabDownload slide Optimization results of variant A – LC04. (a) Force–displacement curves, and thickness distribution for stiffness approximation at the front (b) and at the rear (c). Similar to the previous load case, the resulting thicknesses show a huge range in their distribution. The defined thickness limits are taken, in order to achieve the stiffness or the force level of the reference curve. 4.2. Discussion of the results – variant A The regarded four load cases could be approximated with the defined material combination I. With a 48.1% lower stiffness and a 44.7% lower yield stress, the prototype material needed to be strengthened in order to reach the reference material behaviour. To only reach tensile stiffness, a thickening from initially 2 to 3.85 mm is needed, by using the equations of Table 1. The resulting thickness distribution of the various structure varies around this thickness. Only in load case two, where torsional loading is applied, the structure has a larger range of the needed thicknesses, in order to approximate the stiffness and torque moments. Examining the results of material combination II, the prototype structure was thinned to achieve the same behaviour in the load cases. To obtain the same stiffness, the prototype rectangular cross-section needs to be reduced by 19.6%. The resulting thickness would then be 1.61 mm. The deviation of the thicknesses within one load case to this median thickness is in contrast to material combination I much larger. This shows the difficult management of the thickness reduction and the resulting mechanical behaviour. Overall, the mechanical behaviour of both material combinations could be achieved with this sectioning technique and arrangement of the shell properties. Except in load case four in material combination II, a decision on stiffness or force level is necessary. 5. Results of Variant B 5.1. Model preparation and intermediate results Optimization method B begins with a feasibility check of the material pairs. The results of the check, using the analytical derivation of Fig. 7, are shown in Table 2. Table 2: Feasibility check of the material combinations. Sectioning technique . 2 × 2 . 3 × 3 . . 1 ≤ feasible ≤ 2 . feasible ≥ 0 . Material combination I 1.066 0.485 Material combination II 1.071 1.156 Sectioning technique . 2 × 2 . 3 × 3 . . 1 ≤ feasible ≤ 2 . feasible ≥ 0 . Material combination I 1.066 0.485 Material combination II 1.071 1.156 Open in new tab Table 2: Feasibility check of the material combinations. Sectioning technique . 2 × 2 . 3 × 3 . . 1 ≤ feasible ≤ 2 . feasible ≥ 0 . Material combination I 1.066 0.485 Material combination II 1.071 1.156 Sectioning technique . 2 × 2 . 3 × 3 . . 1 ≤ feasible ≤ 2 . feasible ≥ 0 . Material combination I 1.066 0.485 Material combination II 1.071 1.156 Open in new tab For both material pairs, a material substitution is suitable on base of defined analytical derivations. The next step is to review the occurring local triaxiality states in the deformed plate. In Fig. 18, the extracted triaxiality states are displayed. Figure 18: Open in new tabDownload slide Triaxiality distribution η of the four load cases. (a) LC-01 tension, (b) LC-02 rotation, (c) LC-03 bending, and (d) LC-04 compression. Figure 18: Open in new tabDownload slide Triaxiality distribution η of the four load cases. (a) LC-01 tension, (b) LC-02 rotation, (c) LC-03 bending, and (d) LC-04 compression. The uniaxial load in cases one and four leads to almost homogeneous triaxialities of plus/minus one-third. Because of the differently defined boundary section, a not fully homogeneous distribution occurs. Load two provides nearly homogeneous stripes over the length with discrete triaxiality regions. In addition, the resulting fields present an axis symmetry. The values are mirrored at the middle axis, so that same values on the outer regions and the inner regions occur. The most complex load case concerning discrete regions is load case three. In general, the resulting triaxiality distribution shows two symmetry axis, one over the length of the middle axis and the second vertically in the middle of the structure. The regions are not, compared with the other three cases, set up with a homogeneous behaviour of the length or height. Therefore, a finer sectioning of the fields is needed. In Fig. 19, the defined discrete regions, which are relevant for the later mapping process, are illustrated. Figure 19: Open in new tabDownload slide Plate sectioning by discrete triaxiality fields η for the four load cases. Figure 19: Open in new tabDownload slide Plate sectioning by discrete triaxiality fields η for the four load cases. For the homogeneous stress state load cases one and four, there is in general only one sectioning needed. However, in order to see later mapping effects, two sections over the height are defined, where the later thickness distribution of the nine fields of the dodecagon is directly mapped. A second, more relevant evaluation of the influence of the sectioning on the resulting mechanical behaviour is performed with load case two. Two sectioning variants are defined, where the first one is a coarse division into three regions with a rough averaging of the local triaxiality fields. The second variant uses five regions, with a finer division over the height. Load case three uses the finest sectioning of the four load cases. Using the vertical symmetry, each half divides the structure into 30 fields. After analysing and extracting the average value for a region, some fields with the same value occur. In total, 20 different discrete triaxiality values are determined. In step 4 of the method chain in Fig. 7, the first optimization on the dodecagon with the found discrete triaxialities is performed. The results show a high accuracy of the method concerning the average resulted triaxialities in the inner 36 elements of the dodecagon. With the standardized examination of the calculated and preset triaxiality values, the small deviation is illustrated. In addition, over the regarded interval also the minimum and maximum values are in the preset boundary band. However, having a closer look at the shear stress states, the relative minimum and maximum values are higher compared with the uniaxial or bi-axial stress states. The determined triaxiality states and the achieved quality of the in Fig. 8 presented FE-dodecagon are shown in Fig. 20. Figure 20: Open in new tabDownload slide Approximation quality of the triaxiality η values with FE-dodecagon. Figure 20: Open in new tabDownload slide Approximation quality of the triaxiality η values with FE-dodecagon. Although the dodecagon is loaded with 12 sets of prescribed deformation conditions, the optimization achieved proper results for the required uniaxial stress states [|$\eta =-\frac{1}{3};\frac{1}{3}$|]. Especially in stress states [η = −0.1, 0.1] with high proportion of shear, the dodecagon uses its various deformation possibilities to generate the needed state of distortion. The achieved triaxiality states in the FE-dodecagon after the optimization are visualized in Fig. 21. Figure 21: Open in new tabDownload slide Triaxiality fields η values with FE-dodecagon. (a) |$\eta =-\frac{1}{3}$|⁠, (b) η = 0, (c) |$\eta =\frac{1}{3}$|⁠. Figure 21: Open in new tabDownload slide Triaxiality fields η values with FE-dodecagon. (a) |$\eta =-\frac{1}{3}$|⁠, (b) η = 0, (c) |$\eta =\frac{1}{3}$|⁠. After the triaxiality optimization, for each triaxiality-dependent deformations set, a calculation with the reference material in the inner part of the dodecagon is performed. From this, the necessary reference curves of the cross-sections can be extracted. Then, the second optimization, which finally provides the needed thickness distribution, is performed. The results of the curve matching optimization are shown in Fig. 22. Figure 22: Open in new tabDownload slide Results of size optimization for |$\eta =\frac{1}{3}$|⁠. Curve matching of (a) section forces and (b) section torques. Figure 22: Open in new tabDownload slide Results of size optimization for |$\eta =\frac{1}{3}$|⁠. Curve matching of (a) section forces and (b) section torques. The deviation within the horizontal and vertical section planes concerning the forces is small. Only in the resulting torque curves does the curve deviate over the defined 10 cutting planes. With the prototype material, the achieved accuracy is acceptable, as only small deviations occur. The resulting thickness distribution of the 3 × 3 square is illustrated in Fig. 23. Figure 23: Open in new tabDownload slide Thickness distribution in 9-element-shell structure of η = 0.33. Figure 23: Open in new tabDownload slide Thickness distribution in 9-element-shell structure of η = 0.33. The values of the nine thicknesses do not deviate much. The determined thicknesses are in a range between 3.16 and 5.00 mm. In general, the results of the other optimizations are similar, but show compared with |$\eta =\frac{1}{3}$| a smaller deviation between reference and prototype section curves. 5.2. Results of variant B – material combination I Starting with the uniaxial load cases (LC-01 and LC-04), the resulting force–displacement curves are illustrated in Fig. 24. Figure 24: Open in new tabDownload slide Results of variant B. (a) LC-01 tension and (b) LC-04 compression. Figure 24: Open in new tabDownload slide Results of variant B. (a) LC-01 tension and (b) LC-04 compression. The two defined mapping variants for load case one approximate the reference curve with small deviations. The main difference in the one- and two-layer mapping is that the two-layer mapping with a triaxiality |$\eta = \frac{1}{3}$| has a slightly higher force and stiffness level compared with the one-layer structure. Regarding the nearby discrete triaxiality values (η = [0.32; 0.40]), small and higher deviations are observed. In contrast to load case one, both mapping variants of load case four approximate the reference curves with almost no deviation. This is also shown in the thickness distribution of the mapped structure. The deviation of the thicknesses within one square is small, so that the finesse of the mapping does not have a significant influence on the structural behaviour itself. A substantial proportion of the chosen discretization and the resulting mechanical behaviour is observed in load case two. The torsional loading, which induces several continuous triaxiality distribution bands over the length, has a direct influence on the resulting torque–rotation curves. The greatest discretization results in an enormous deviation in stiffness and maximum torque. The finer mapping with five regions reaches a closer approximation of the reference curve. But also, the extracted discrete triaxiality values in the sectioning process influence the structural response. By changing the triaxiality value of the first and fifth regions, the resulting stiffness and the maximum torque was noticeably affected. The resulting torque–rotation curves as well as the two mapping variants are presented in Fig. 25. Figure 25: Open in new tabDownload slide Results of variant B LC02. (a) Torque–rotation curves and (b) mapped FE models. Figure 25: Open in new tabDownload slide Results of variant B LC02. (a) Torque–rotation curves and (b) mapped FE models. With the located discontinuous triaxiality fields in load case three, the defined amount of triaxiality regions was enough to approximate the reference force–displacement curve. Looking at the resulting thickness distribution over the plate, which is composed of the 20 defined regions with the nine local thickness configurations transferred in each case, the result is a wavy surface. Finally, the results of load case three are presented in Fig. 26. Figure 26: Open in new tabDownload slide Results of variant B – LC03. (a) Force–displacement curves and (b) mapped FE model with subsections. Figure 26: Open in new tabDownload slide Results of variant B – LC03. (a) Force–displacement curves and (b) mapped FE model with subsections. 5.3. Discussion of the results – variant B In variant B, similar to optimization variant A, all four load cases of material combination I are approximated with a comparatively less deviation. Within the several steps, the quality of the defined sectioning and the assigned values has a direct influence on the later optimization accuracy. Noticeable deviations are indicated by a change in the mapping technique or the use of slightly different average triaxiality values. The used FE-dodecagon provides the needed nearly homogeneous triaxiality field in the regarded 3 × 3 element inner core. The resulting mapped structures of load cases one, two, and four have a rather angular surface. Using a higher level of refinement like in load case three, where more smaller mapping squares are used, a smoother surface was obtained. 6. Conclusion Material substitutions have been successfully used for a long time and are essential in some cases. However, when considering thermoplastics, the material’s own non-linear behaviour presents challenges. In this paper, approaches to deal with non-linearity were identified. The desired goal of approximating the stiffness and force level in rib structures was achieved. With the analytical derivation, the limits of the rib sectioning were analysed on the basis of simple mechanical approaches and possibilities for determining the relevant thickness distribution were shown. For restricted load cases, material pairings can be evaluated before a mechanical calculation is performed. If first structural calculation results are available, a general evaluation of the substitutability of the material pairing can be carried out. The procedure based on numerical structural optimization enables an equivalent structural design with the two methods presented. The developed optimization methods can be used directly in larger components for the selected material pairings. An adaptation or extension of the methods with regard to the selected boundary conditions or selected material models can easily be realized. With the direct optimization variant A, a fast optimization of an existing structure is possible. The two times 6-fold sectioning of the used rib allowed the equivalent approximation to the given mechanical behaviour. Also, the possibility to adapt a stronger or weaker reference material with a prototype material was shown. Based on the analytical and numerical results, a division into 3 × 3 fields is recommended for future rib structures. This allows the user to substitute a larger group of materials, as can be seen in the analytical derivation. The indirect optimization in variant B provides a general solution to handle material substitutions on structural components. With the initially higher expenditure of the variant, further component or rib optimizations become obsolete. Once the triaxialities and the corresponding thickness distributions have been determined, larger components with complex rib strips can be designed equivalently. Therefore databases, which organize thickness distributions on base of triaxiality state and also, for example, on force levels, can be generated. With regard to the producibility of the results, additive manufacturing offers the possibility to produce the fine gradations and thickness differences with high accuracy. Processes such as SLS or PolyJet offer increased geometric freedom with high precision in the range of 100 μm. Since the structures under consideration are discretized by shell elements, the shape optimization was carried out along the longitudinal axis and within the plane. 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TI - Size optimization methods to approximate equivalent mechanical behaviour in thermoplastics
JO - Journal of Computational Design and Engineering
DO - 10.1093/jcde/qwaa069
DA - 2021-01-25
UR - https://www.deepdyve.com/lp/oxford-university-press/size-optimization-methods-to-approximate-equivalent-mechanical-BS9v1zsBT4
SP - 170
EP - 188
VL - 8
IS - 1
DP - DeepDyve
ER -