Assignment 4

Objective: The objective of the assignment to get familiarised with the procedure to define different material modelling law and compare their performance in terms of the total number of cycles, energy error, mass error and simulation time using a generic example of the rigid ball impacting on a deformable wall.

Case Setup and Execution:

The keyword file FAILURE_JOHNSON_0000.rad was imported in the HyperWorks browser. The overview of the model under study is illustrated in Figure-1, describing the rigid ball and the deformable wall. The element properties were updated to the recommended values as described in Table-1.

Figure-1: Problem overview.

Table-1: Recommended element formulation properties.

Parameters	Comment
Ishell =24	QEPH 4 nodes shells are the best combination of cost and accuracy.
Ismstr=2	Full geometric nonlinearities (default)with possible small strain formulation activation in RADIOSS Engine.
Ish3n=2	Standard 3 noded shell (C0)with modification for large rotation(default)
N=5	The number of integration points set to 5 for accuracy bending.
Ithick=1	Thickness change is taken into account for accuracy.
Iplas=1	Iterative plasticity for good accuracy.

Firstly, the imported model was solved with a plastic strain at failure (eps_p_max) was set as 0.151 and was saved as Law2_eps_max_failure. For the second run, the parameters defined in Table-2 were used and the simulation was saved as Law2_eps_max_crack.

Table-2: Element failure parameters for Law2_eps_max_crack

Parameters	Comment
Ifail_sh =1	Shell is deleted or cracked when D=∑Δεpεf≥1">D=∑Δεpεf≥1D=∑Δεpεf≥1 for one integration point or layer.
Dadv = 1	Criterion for the crack advancement (Only active if with Ixfem =1)
Ixfem = 1	Element crack formulation using extended finite element method

Next, in the same model, the /FAIL/JOHNSON failure card was deleted and the simulation was saved as Law2_epsmax_nofail. For the next model, the value of the Eps_p_max was deleted and the simulation was saved as Law2. This was followed by importing the LAW27_0000.rad keyword file. The material and failure criteria defined in the model has been summarised in Table-3. The results were saved under the name of LAW27.

Table-3: Material and failure definitions of LAW27.

Parameters	Comment
ε_t1 = 0.14	Tensile failure strain at which stress starts to reduce in the principal strain direction 1.
ε_m1 = 0.15	Maximum tensile failure strain in principal strain direction 1 at which the stress in the element is set to a value dependent on dmax1.
d_max1 = 0.999	Maximum damage factor in principal strain direction 1.
ε_f1 = 0.151	The number of integration points set to 5 for accuracy bending.
ε_t2 = 0.14	Maximum tensile strain for element deletion in principal strain direction 1
ε_m2 = 0.14	Maximum tensile strain in principal strain direction 2 at which the stress in the element is set to a value dependent on dmax2.
d_max2 = 0.999	Maximum damage factor in principal strain direction 2
ε_f2 = 0.151	Maximum tensile strain for element deletion in principal strain direction 2.

Finally, the FAILURE_JOHNSON_000.rad keyword file was imported again and the material card was changed to M36_PLAS_TAB. The material and failure definition along with the tabular plasticity data for aluminium are defined in Table 4 and Table 5, respectively.

Table-4: Material and failure definitions for LAW36.

Parameters	Value	Comments
Rho_initial	0.0027	Initial density.
E	71000	Young's modulus.
Nu	0.33	Poisson's ratio.
Eps_p_max	0.16	Failure plastic strain.
EPS_t	0.1	Tensile failure strain at which stress starts to reduce
EPS_m	0.11	Maximum tensile failure strain at which the stress in the element is set to zero.

Table-5: Aluminium plasticity data.

Strain	Stress [Mpa]
0	300.0
0.000484405	360.13
0.00111878	390.56
0.0032105	442.82
0.00478134	465.98

Observations:

-Total Number of cycles

Figure-2 illustrates a bar graph comparison of the total number of cycles required by each material model. It is observed that for all the material models except LAW 36 the number of cycles remain fairly similar.

Figure-2: Comparison of the total number of cycles.

-Energy Error

Figure-3 illustrates a bar graph comparison of the energy error of the material models. It is observed that for every material model except Law 36, energy is created which is characteristic of the feature of the QEPH shell formulation. For material model following LAW36, energy is lost in the process. As the acceptable energy error for QEPH shell formulation is 1-2%, material models LAW2 (eps_max_crack), Law2 and Law36 do not portray an accurate simulation.

Figure-3: Comparison of energy error.

-Mass Error

For all the cases, a mass error of 0% was recorded.

-Simulation Time

Figure-4 illustrates a bar graph comparison of the total time required for a complete solution. The values below were obtained using four threads of Intel-i5 8250U processor and are subjected to change depending on the processor.

Figure-4: Comparison of the simulation time.

Table-6: Summary of deformation mechanism for different material models.

Case	Animation	Comments
Law 2 (ε_max failure)		The results represent an isotropic elastoplastic material model defined using the Johnson-Cook material model coupled with a Jhonson-Cook ductile damage model without the XFEM formulation. Here, the elements get deleted as damage parameter reaches or exceeds unity.
Law 2 (ε_max crack)		The results represent an isotropic elastoplastic material model defined using the Johnson-Cook material model coupled with a Jhonson-Cook ductile damage model with the XFEM formulation. Here, the elements undergo crack invitation and propagation as damage parameter reaches or exceeds unity. This is followed by element deletion which is governed by the Failure Plastic Strain (εp_max) defined in the Johnson-Cook material model.
Law 2 (ε_max no-fail)		The results represent an isotropic elastoplastic material model defined using the Johnson-Cook material model. Here, the element gets deleted (without crack initiation) as the element approached the Failure Plastic Strain (εp_max).
Law 2		The results represent an isotropic elastoplastic material model defined using the Johnson-Cook material model without damage.
Law 1		The results represent the isotropic linear elastic material model using Hooke's law and are one of the simplest material models available. This material model is best suited for small deformation in the linear-elastic range.
Law 27		The results represent an isotropic elastoplastic Johnson-Cook material model coupled with an orthotropic brittle failure model. Here, the material damage is accounted for prior to failure and Failure and damage occurs only in tension.
Law 36		The results represent an isotropic elastoplastic material model using user-defined plastic stress-strain data and can accommodate values for different strain rates.

Table-7: Variation of internal energy, kinetic energy and normal reaction force obtained for different material models.

Case	Energy Plots	Comments
Law 2 (ε_max failure)		The plot describes energy absorption up to 4ms and failure of deformable wall beyond that. Here, the element deletion does not accurately represent the real-world performance but may be considered as a computationally efficient way to obtain approximate behaviour.
Law 2 (ε_max crack)		Here, as the elements do not fail suddenly and instills crack initiation and propagation, hence the energy absorption capacity of the elements is higher. The failure initiates at approximately 4.5 ms. The crack initiation and progression I the most detailed model that captures the real-world behaviour.
Law 2 (ε_max no-fail)		Here, the performance is similar to the material model following LAW2 (ε_max failure). The failure initiates at 4 ms.
Law 2		Here, as there are no failure criteria defined the elements tend to absorb superficial energy without getting damaged. Although, the plasticity of the elements is still considered. The material model is not able to capture the energy absorption of the characteristics accurately as the model does not take damage into consideration. But for cases where damage is not a parameter understudy, LAW2 provides a computationally efficient way to model just the energy absorption characteristic.
Law 1		Here, the energy absorption capacity of the elements tends to be higher as they follow linear-elastic behaviour with geometric non-linearity and plasticity of the material is completely ignored. The model is only valid for small strain linear-elastic applications and hence is not able to capture the real work performance of the model under study.
Law 27		Here, a linear energy absorption characteristic is observed before failure. Failure initiates just after 4ms. Law 27 is used to stimulate anisotropic brittle failure and hence is not able to capture the ductile failure of the deformable aluminium wall accurately.
Law 36		Here again, the energy absorption characteristic of the deformable wall is linear and failure initiates early at 3 ms. The model is not an accurate representation of the model under study as it required plastic stress-strain data which are obtained experimentally for multiple strain rates.

Conclusion

From the observations above it was concluded that the material model LAW2 (ε_max crack) represents the real-world performance of the model under study most accurately.