PSI - Issue 34

Sigfrid-Laurin Sindinger et al. / Procedia Structural Integrity 34 (2021) 78–86 S.-L. Sindinger et al. / Structural Integrity Procedia 00 (2019) 000–000

81

4

The linear material behavior for the rollers was defined using standard isotropic steel properties (MAT1). Definition of orthotropy, as required for the additively manufactured structure, is in Optistruct ™ available for 3D (MAT9ORT) and 2D elements (MAT8) alike. Thereby, one essential di ff erence exists between element types. For 3D elements, a user-defined coordinate system can be assigned that dictates the 123-axes of the material independently of element orientation. In 2D elements, however, only an in-element-plane rotation of the material system is possible. While this is adequate for laminated composites, in which the continuous fibers lie within a ply, it is insu ffi cient for the present material, wherein material orientation does not depend on the geometry but is fixed corresponding to the build coordinate system (Chen et al., 2021). If for instance a shell element is rotated about the global y -axis by some angle θ , as depicted in Fig. 2b, the material parameters in local 1-direction do not coincide with any principal material axis and thus must be transformed prior to assignment. The thickness dependency of each material parameter was modeled by fitting a curve (Curve Fitting Toolbox ™ The MathWorks, Inc., Natick, USA) through the corresponding mean values of each coupon orientation. The utilized function was a two-terms power law of the form given in Eq. 2 (2) with the element specific shell thickness t and the fitted model coe ffi cients a , b and c . This yields the global thickness dependent compliance matrix S x , y , z ( t ) that using transformation matrices T (Schu¨rmann, 2007) can be transformed to the local compliance matrix of the respective shell element S 1 , 2 , 3 ( t ) via Eqs. 3 f ( t ) = a t b + c for t min ≤ t ≤ t max whereby per 2D element formulation only the in-plane entries of S 1 , 2 , 3 ( t ) are considered for property definition. Based on these relations, a script was developed that automatically extracts the shell thickness and local coordinate sys tem orientation, computes the elastic parameters and creates material cards for each unique element configuration. Thereby, the inhomogeneous elastic material behavior is automatically mapped throughout the shell mesh. Failure prediction is conducted in a post-processing step once the element stresses are computed. For assessment, the latter need to be transformed back to the global coordinates from the local element system using the transformation relation T T · σ 1 , 2 , 3 = σ x , y , z , wherein the stress tensor components are represented in Voigt notation. Due to shell formulation, the out-of-plane components in σ 1 , 2 , 3 are zero. Consequently, the transformed stresses σ x , y , z can be compared to corresponding ultimate strength values in global coordinates. Therefore, the three-dimensional criterion for orthotropic materials proposed by Hill (1948) was adapted by inserting thickness dependent instead of conventional homogeneous denominators. This yields Eq. 4 H ( t ) =  σ x σ ∗ x ( t )  2 +  σ y σ ∗ y ( t )  2 +  σ z σ ∗ z ( t )  2 +  τ xy τ ∗ xy ( t )  2 +  τ yz τ ∗ yz ( t )  2 +  τ zx τ ∗ zx ( t )  2 −   1 σ ∗ 2 x ( t ) + 1 σ ∗ 2 y ( t ) − 1 σ ∗ 2 z ( t )   σ x σ y −   1 σ ∗ 2 x ( t ) + 1 σ ∗ 2 z ( t ) − 1 σ ∗ 2 y ( t )   σ x σ z −   1 σ ∗ 2 y ( t ) + 1 σ ∗ 2 z ( t ) − 1 σ ∗ 2 x ( t )   σ y σ z (4) 2.4. Material Modeling      1 E x ( t ) − ν xy ( t ) E x ( t ) − ν zx ( t ) E z ( t ) − ν yz ( t ) E y ( t ) 0 0 0 0 0 0 0 0 0 1 0 0 1 E y ( t ) 1 E z ( t ) G yz ( t ) sym. 1 G xz ( t ) 0 1 G xy ( t )         S x , y , z ( t ) −→ T T · S x , y , z ( t ) · T = S 1 , 2 , 3 ( t ) (3)

in which element failure occurs if the index H ( t ) > 1. This criterion accounts for interactions between di ff erent stress components but does not distinguish between tension and compression. Other well established criteria exist that in-