Stage 1: Preliminary investigation

Four parameters of the finite element model were identified for attending to in some detail during a preliminary phase of the analytical study:

· Young’s modulus of steel · Yield stress of steel · The mesh size · Boundary conditions

In this preliminary investigation the static stress strain relation for mild steel is used (Fig. 5.4). Post yield tension hardening was ignored. The strain rate dependence features provided by ABACUS were also not used. The increase in strength was determined by a trial and error method by comparing the percentage errors in the

An initial value was chosen for each of the four parameters mentioned above: Young’s modules: 209 GPa

Yield stress: 800 MPa

Mesh size: An element thickness of 1 mm and length 5 mm. Boundary conditions: Pinned at both bottom and top surface.

The effect of changing each parameter on the percentage error of the computations was then individually investigated. While the effect of a particular parameter was being investigated the values of the remaining three parameters were fixed at their initial values.

Young’s modules of steel

The Young’s modulus of steel is expected to increase at high strain rates. The effect of changes in this parameter on the percentage error of the computations was investigated.

Four different values of young’s modulus of steel were used in this part of the exploratory study: 157 GPa, 209 GPa, 314 GPa and 418 GPa (representing 75, 100, 150 and 200 percent of the actual static elastic modulus of steel of 209 GPa).

The percentage errors between the computations and the experimental results for both the peak strain and the recovery for each value of the Young’s modulus was calculated and is listed in Table 5.2.

As can be seen from Table 5.2, the percentage error in the computations of the peak strains decreased as the Young’s modulus value was increased. The Young’s modulus value of 314 GPa recorded the least error of 11% in the computations. Thus at

Chapter5: Analytical Modeling of Impact Response

However the percentage errors in the computations of the recovery did not coincide with the above trend. As the Young’s modulus value was increased the percentage error in the computation of the recovery value decreased. The computations indicated very low recoveries at higher values of Young’s modulus.

Thus based on the contradicting trends, it cannot be convincingly concluded that a higher Young’s modulus value decreases the percentage error of the computations at higher strain rates. Judging from the errors in computations of both the peak strain and the recovery it was decided to retain the actual Young’s modulus of mid steel of 209 GPa.

The Yield Stress of Steel

Yield stress of steel varies with the strain rate of loading. Due to the relatively high strain rate loading in the experiments being investigated, it would be expected that steel would behave as if it had a strength which is greater than its static yield strength. Thus five yield stress values were investigated here in. Starting from the theoretical static yield stress for mild steel of 275 MPa, the yield stress value was increased to values of 450 MPa, 500 MPa, 600 MPa and 800 MPa

The first trial was run using the static yield stress of grade 43 mild steel of 275 MPa. The peak strains were significantly higher than the actual values indicating an increase in yield stress under dynamic loading as expected. As the yield stress value increased the peak strains decreased as expected. The percentage errors for both the peak strain computation and the recovery computation for each case of yield stress value investigated is summarized in Table 5.3.

The average peak strain percentage errors were 2% for a yield stress value of 500 MPa. The average percentage error in the recovery compuations were 65% for this case but this was still the least percentage error for the values of yield stress tried.

Based on the above results from the test runs the value of 500 MPa was chosen. However thicker specimens experienced lower peak strains and therefore lower strain rates during impact. Since the increase in yield strength of steel is related to the strain rate, it was required to have a proportional adjustment of the increment in yield strength corresponding to the strain rates experienced by each specimen. A fixed yield stress value of 500 MPa was therefore only an initial approximation indicating the range of the increased yield stress. Further rigorous calibration by incorporating the strain rate effect within the material model was employed in the second stage of the model development.

The mesh size

In general the smaller the size of the mesh the greater is the accuracy of the results in finite element modeling. However a very small mesh size leads to an increase in the number of calculations and the error term gets magnified and achieves relatively significant proportions. Thus an optimal mesh size would exist that minimizes deviation of results from the experimental results. This optimal value was explored using trial and error.

For the wire elements used the mesh size is defined by both the thickness and the length of the element. Four different mesh sizes were investigated (Fig. 5.5):

Chapter5: Analytical Modeling of Impact Response

· Element thickness of 1 mm

o Element length of 5 mm (Fig. 5.5c) o Element length of 2.5 mm (Fig. 5.5d)

The percentage errors in the computed results for the peak strain and the recovery when compared to the experimental results for each mesh size mentioned above are listed in Table 5.1.

It can be seen from the table that in both the cases of peak strain and recovery, the percentage error in the computations seemed to decrease as the mesh size was decreased. However for the last extremely fine mesh the percentage errors seemed to have increased over the previous relatively coarser mesh.

Based on the above results the optimal mesh refinement was chosen as an element with a 1 mm thickness and 5 mm length.

Boundary Conditions

Finally the effect of the boundary conditions on the accuracy of the computations was also explored. Chapter 3 explains the exact laboratory conditions for the experiments. The edge conditions around the 100 mm diameter impact portion of the specimens were neither fixed nor pinned. Different combinations of boundary conditions were tested here to see which set of conditions record the minimum percentage error in the computations of the peak strains and the recovery.

Initially only the 100 mm portion of the specimen that was exposed directly to the impact head was modeled. The boundary conditions were assigned to the edge elements.

A pinned condition represents restraint against translation alone while a fixed condition represents restrain against rotation as well. Five different boundary conditions were tried (Fig. 5.6):

· Single pin: Only the bottommost element at the far end is pinned. (Fig. 5.6a)

· Pinned: The bottommost and the topmost elements at the far end of the specimen were pinned. (Fig. 5.6b)

· Fixed: The bottommost and the topmost element at the far end of the specimen were fixed. (Fig. 5.6c)

· Fully pinned: All the elements at the far end of the specimen were pinned. (Fig. 5.6d)

· Fully fixed: All the elements at the far end of the specimen were fixed. (Fig. 5.6e) The percentage errors in the computations of the peak strain and the recovery using the above boundary conditions were calculated and are recorded in Table 5.4a. It is seen that using pinned conditions recorded much lower percentage errors in the computations.

Attempts were also made to model the entire experimental setup instead of only the 100 mm exposed portion of the specimen. The 10 mm thick cover plate on top of the specimen and the 20 mm bottom plate below the specimen were also modeled using the same 1 mm thick and 5 mm long axisymmetric wire elements. The full specimen was now modeled. Again two end conditions were tried with this setup, pinned and fixed. However in this case the conditions were assigned to the ends of the top and bottom plates holding the specimen in between them rather than to the specimen (Fig. 5.7).

Chapter5: Analytical Modeling of Impact Response

Thus extending the modeling beyond the exposed portion of the specimen by incorporating the cover plates and the entire specimen did not improve the accuracy of the computations.

Based on the above results the pinned condition using only the 100 mm exposed portion of the specimen in the model was chosen.