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Description and Model of the Bi-directionally Corrugated Sandwich Structure

The great motivation of this thesis is the understanding of the behavior of a new type of formable all-metal sandwich sheet: the bi-directionally corrugated sandwich structure. However, it was not possible to carry on all the physical experiments required to have an overview of the sandwich behavior. Analytical analyses are also restricted by the complex shape of the core structure. As an alternative, and thanks to the periodic and symmetric nature of the core geometry, numerical simulations of a representative unit cell of the sandwich material are performed to investigate the effective behavior of the material under different loading conditions. These simulations are referred as “virtual” experiments.
Predicting the effective behavior of cellular materials based on FE analysis of the underlying unit cell (for periodic media) has been successfully used by several research groups. For example, Mohr and Doyoyo (2004) investigated the crushing response of aluminum honeycomb using a detailed shell element model of the hexagonal cell structure; Youseff et al. (2005) built a finite element model of a PU foam based on X-ray tomography images; while Caty et al. (2008) developed a micrsotructural FE model of a sintered stainless steel sphere assembly (similar to closed-cell foam) based on X-ray tomography images. It is worth mentioning that the unit cell computations are only representative for the material behavior of the microstructures remain mechanically stable. In the case of instabilities (which are frequent in cellular materials of low relative density), a careful analysis of the type of instabilities is needed to check the validity of the homogenized material description (Triantafyllidis and Schraad, 1998).
This chapter starts by a detailed description of the physical core structure. Then, the finite-element model of a unit cell is presented. Since the exact geometry of the dimples of the bi-directionally corrugated core layers depends on the strain distribution after stamping, we perform numerical simulations of the all manufacturing process.

The bi-directionally corrugated core structure

Core architecture and stamping tool

The core architecture is made of two bi-directionally corrugated layers brazed together. Each layer is composed of a periodic array of domes (Fig. 1.1) which are introduced into an initially flat sheet through stamping.
Fig. 1.1 : (a) Side view of the four layer sandwich structure, (b) top view of a single core layer.
The material coordinate system ( , , ) as shown in Fig. 1.1 is introduced to describe the microstructure. The coordinate axis is aligned with the thickness direction of the sandwich sheet material (out-of-plane direction) whereas and denote the so-called in-plane directions. The L-direction is parallel to the connecting line of two neighboring domes while the W-direction is defined as . The core structure features seven symmetry planes; the bi-layer assembly is symmetric with respect to the central (W,L)– plane (Fig. 1.1). Furthermore, each layer is symmetric with respect to the (W,T)– and (L,T)– planes.
In the manufacturing process, we can control the geometry of the stamping tools, while the final geometry of a core layer after stamping depends also on the plastic properties of the basis sheet material. The stamping tool consists of a male and a female die. The male die comprises a periodic array of pins that are positioned on a triangular pattern at a spacing D receiving female die features the corresponding periodic array of holes of a diameter along with a corner radius (Fig.1.3).
The relative density of the core structure describes the ratio of the overall mass density of the core structure to the density of the basis sheet material. In the case of incompressible sheet materials, the relative density is given by 2 1.1 where denotes the initial sheet thickness and /2 is the effective height of a single core layer after stamping.

Basis material

The bi-layer core structure can be made of any sheet material that provides sufficient formability. Here, we focus on dimpled layers that are made from a tin mill product of the type “black plate” which is a light gage low-carbon , cold-reduced steel. According to ASTM A623-05, it features a maximum carbon and manganese content of 0.13% and 0.6% respectively. The material has been supplied by ArcelorMittal in the T4 temper.
Uniaxial tensile tests are performed under static loading conditions to characterize the anisotropic plastic properties of this sheet material. Dogbone shaped ASTM-E8 specimens are extracted from the sheets along the rolling direction, the cross-rolling direction and the 45° direction. We make use of a universal testing machine (MTS Model 318.10) with a 100kN load cell. The specimens are loaded at a constant cross-head velocity of 2mm/min. A random speckle pattern is applied to the specimen surface and monitored throughout testing using a digital camera (Allied Vision, PIKE) with 10mm Nikon Nikkor lenses. The axial and width strains are determined based on a series of 500-1000 pictures using digital image correlation; the virtual extensometer lengths are 15mm and 9mm for the axial and width directions, respectively.
Fig. 1.2 : Engineering stress-strain curves for the 0.2mm thick low carbon steel sheet for loading along different in-plane directions.
The measured engineering stress-strain curves are presented in Fig. 1.2. Assuming a Young’s modulus of 210GPa and an elastic Poisson’s ratio of 0.3, we determined a yield stress of about 310MPa at 0.2% plastic strain for all loading directions. The Lankford ratios are determined from the average slope of a plot of the logarithmic plastic width strain versus logarithmic0.69plastic thickness1.15 strain (assuming0.76. plastic incompressibility). Upon evaluation, we find , and In our simulations, we make use of the Hill (1948) yield function along with an associated flow rule and isotropic hardening to model the sheet material behavior. The corresponding yield stress ratios are given in Table 1.1.
All specimens fractured at an engineering strain of about 0.25. Here, we extrapolate the measured true stress versus logarithmic plastic strain curve using the modified Swift law

Stamping experiments

Experimental set-up

A stamping experiment is performed to provide an experimental basis for the validation of the computational model that 2is.2 . used for the subsequent parametric study. We manufactured1.2 a set0.2 of dies with 1.8 The male0.2tool had the dimensions female die. The stamping tool features a matrix of 572 pins and dies over an area of 25 x 25 mm. A four-column low friction guidance system guaranteed the alignment of the die and punch matrices throughout stamping.
The tool is set up at the center of a universal testing machine. The force is applied via a cylinder on the upper clamping block (part #3 in Fig. 1.3). The oil lubricated sheet is positioned between the upper and lower clamping block of the tool (part #1 and #2). Throughout stamping, a 250kN load cell measures the total stamping force, while an LVDT inside the vertical actuator recorded the applied displacement. Each stamping experiment is performed under displacement control at an actuator velocity of 0.33mm/min.

Experimental results

The measured force-displacement curves throughout stamping are shown in Fig. 1.4. The superposition of the results from two experiments (red and black curves) demonstrates good repeatability.
After an initial linear response, the slope of the force-displacement curve decreases at a measured displacement of about 0.2mm. The stamping force continues to increase up to a displacement of about 1mm until the sheet fractures. The second experiment is stopped prior to fracture at a displacement of about 0.75mm. The slope of the corresponding loading branch of the force-displacement curve at the beginning of the experiments is smaller than for unloading. This is explained by the evolution of the specimen geometry throughout stamping. At the beginning of the experiment, the sheet material is predominantly subjected to bending, while the state of loading is more membrane-dominated once the sheet has formed its characteristic dimple shape.
A SEM picture of a dimpled layer after stamping is shown in Fig. 1.5. The corresponding cross-sectional cut through a single dimple elucidates pronounced necking in a region where the sheet material leaves the punch radius (near cross-section label 2 in Fig. 1.5). Optical thickness measurements indicate that the sheet thickness is reduced to 0.16mm right below the punch while its thickness remains more or less unchanged near the die contact areas. Within the neck region, we observe thicknesses as low as 0.11mm which corresponds to a thickness reduction of 45%.

Computational models for “virtual experiments”

A finite element model is built to simulate the making of a core layer using Abaqus 6.8.3. The unit cell model of the material microstructure is subsequently used for all the virtual experiments on the bi-layer core structure. Thus, the size of the periodic unit cell and the boundary conditions are chosen according to the symmetry of the experimental set-up. Note that the detailed modeling approach is only feasible with reasonable computational effort at the unit cell level, whereas a macroscopic model is required for the design of structures made from sandwich materials.

Important modeling assumptions

The mechanical behavior of sheet materials in forming and crash simulations can be predicted with remarkably high accuracy using state-of-the-art computational models. Since the proposed sandwich material corresponds to a sheet metal assembly, it is expected that the virtual experiments will provide representative estimates of its effective behavior. The key simplifications with respect to representing the real sandwich structure are:
(1) Assumption of perfect alignment of the two core layers; when manufacturing the sandwich material, it is very difficult to guarantee the perfect alignment of the core layers. It can be seen from the micrograph shown in Fig. 1.1 that small misalignment errors are present in the real material, while perfect alignment will be assumed in the virtual experiments;
(2) Negligence of property changes due to brazing; in reality, the four constituent layers of the sandwich material are brazed together; the temperature history throughout brazing may change the steel properties (preliminary experiments have shown this); this effect will be neglected in the present study;
(3) Assumption of rigid braze joints; it is assumed that the braze joints are very thin and strong, such that the deformation in these joints is negligibly small with respect to the deformation of the core layers.
An attempt was made to confront the results from virtual experiments with experiments on real prototypes. It was found that point #2 presents a first order effect which makes it almost impossible to achieve good agreement.

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Manufacturing simulations

A finite element model of a unit cell of the core structure is obtained after simulating three manufacturing steps:
(1) the stamping of flat sheets to create the dimpled shaped layers,
(2) the forming of flat bonding lands on each layer, and
(3) elastic springback.
The residual plastic strain fields are imported from one step to the next one. Once the core structure is created, flat face sheets are added to form the sandwich structure.


The dashed rectangles in Fig. 1.6 indicate the size of the unit cell models which are used to perform the virtual experiments. The green lines define the smallest model; the model defined by the red rectangle is twice as long as the green model, while the blue model is twice as wide. The different dimensions are needed to facilitate the definition of periodic boundary conditions (which depend on the specific loading case to be studied). In the case of the small green model, two punches (male die) along with their receiving female dies are needed for the stamping of this unit cell. All forming tools are modeled as analytical rigid surfaces. A mesh with five first-order solid elements (type C3D8R from the Abaqus element library) in the thickness direction is chosen to account for high through–thickness stresses as well as through-thickness necking. The receiving dies are fixed in space while the punches move along the T–direction (Fig. 1.7). To guarantee quas i-static conditions throughout the stamping simulations, the punch velocity increases linearly from 0 to 1m/s over a time interval of 40 µs.
Subsequently, it is kept constant until the maximum stamping depth is reached. A kinematic contact formulation with a friction coefficient of 0.1 is employed to model the contact between the tools and the sheet surfaces. Throughout stamping, the in-plane displacement 0 component is settozerof orall nodes on the boundary surface of normal,while on all boundaries of normal vector .
Fig. 1.7 : Sequence of computed geometries during the stamping of a unit cell of a single core layer.
The results from the stamping experiments are compared with the numerical predictions to validate the model assumptions. We define the stamping pressure as the total stamping force2363/per unit area of the stamped sheet. The initial slope of the simulation65/ curve ( ) is much higher than that of the experiment ( ). This is attributed to the finite stiffness of of the universal testing machine and the forming tool.
Considering the 67/forming tool as a spring in series with the work piece, we find a machine stiffness of . Figure 1.4 shows the comparison of the simulation and the experimentally-measured stamping pressure versus displacement curves. Note that both curves are initially identical since we added the displacement associated with finite machine stiffness, ∆ / , to the simulation result (stiffness correction). The agreement between the two responses is remarkably good. Note that the simulations are performed without any fracture criterion. Consequently, the maximum load in the simulation is determined by the post-necking behavior of the simulation model, whereas the fracture initiates at an earlier point in the experiment.

Forming of the bonding land, springback and joining of the core layers and face sheets

The four-layer sandwich material needs to be virtually joined together in order to estimate the effective shear properties. A tie contact model (Abaqus, 2008) is used to join the core layers to each other as well as to the respective top and bottom face sheets. After completing the previous stamping simulations (step #1, explicit time integration), an additional forming step is introduced (step #2, explicit time integration) where flat rigid plates are used to flatten the bonding lands (Fig. 1.6). A first rigid plate applies a pressure to the bottom surface of the corrugated sheet until the resulting material thickness below the centers of the punches equals about 80% of the initial sheet thickness. Similarly, a second rigid plate is used to apply a pressure to the top surface of the corrugated sheet. Here, the simulation is stopped as the initial sheet thickness above the dies equals about 90% of the initial sheet thickness. The flatness of the bonding lands (contact areas) is important to avoid an artificial mesh distortion when using the tie contact model. In reality, the flatness of the bonding lands is also important as it enhances braze joint strength. The flattened bonding lands of two opposing core layers are bonded to each other using the tie option with a position tolerance of 0.001mm.
After joining all layers together with the tie contact, a spring back analysis is performed (step #3). Since springback analyses are simply static simulations without external loading, Abaqus/Standard is preferred for that step whereas all the others were performed using Abaqus/Explicit. The final shape and dimensional changes associated with spring back are negligibly small for the present design, but it is still important to compute a macroscopically stress-free configuration before starting any virtual experiments on this unit cell model. Note that this last step is omitted for virtual experiments in the elastic range of the material as residual strains do not influence the elastic behavior.
Two 0.2mm thick face sheets are created and bonded onto the core layers to form the sandwich structure. A tie contact with a position tolerance of 0.002mm is used to create the virtual bond between the face sheets and the core structure.

Table of contents :

Existing core structures
Forming of sandwich panels
Modeling the behavior of cellular solids
Overview of the thesis
1.1. The bi-directionally corrugated core structure
1.1.1 Core architecture and stamping tool
1.1.2 Basis material
1.2. Stamping experiments
1.2.1 Experimental set-up
1.2.2 Experimental results
1.3. Computational models for “virtual experiments”
1.3.1 Important modeling assumptions
1.3.2 Manufacturing simulations Stamping Forming of the bonding land, springback and joining of the core layers and face sheets
2.1. Estimation of the effective shear stiffness
2.2. Parametric study of the effective shear stiffness
2.2.1 Input parameters
2.2.2 Results
2.2.2 Comment on the optimal design of formable sandwich sheets
2.3. Experimental validation
2.3.1 Four-point bending experiment
2.3.2 Model for the bending of the sandwich structure
2.4. Conclusions
3.1. Models for virtual experiments
3.1.1 Out-of-plane compression
3.1.2 Out-of-plane shear
3.1.3 Uniaxial in-plane loading
3.1.4 Combined in-plane loading
3.2. Results from virtual experiments
3.2.1 Uniaxial out-of plane compression
3.2.2 Out-of-plane shear
3.2.3 Uniaxial in-plane tension
3.2.4 Uniaxial in-plane compression
3.2.5 Biaxial in-plane behavior
3.2.6 Volume change of core structure
3.3. Phenomenological macroscopic constitutive model
3.3.1 Modeling approach
3.3.2 Notation and kinematics
3.3.3 Elastic constitutive equation
3.3.4 Macroscopic yield surface
3.3.5 Distortional-isotropic hardening
3.3.6 Flow rule and volume change
3.3.7 Summary of material model parameters
3.4. Validation and discussion
3.4.1 Comparison: macroscopic model versus virtual experiments
3.4.2 Discussion
3.5. Conclusions
4.1. Calibration experiments
4.1.1 Calibration experiment #1: Uniaxial tension
4.1.2 Calibration experiment #2: Four-point bending
4.2. . Material model parameter identification
4.2.1 Summary of material model parameters
4.2.2 Elastic constants and thickness change parameter
4.2.3 Isotropic-distortional hardening functions
4.2.4 Composite shell element models Shell model for uniaxial tension Shell element model for four-point bending
4.2.5 Inverse model parameter identification
4.3. Structural validation: draw bending
4.3.1 Virtual experiment
4.3.2 Composite shell model predictions and discussion
4.4. Conclusions


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