The material model requires the user to specify 9 material parameters and 7 physical constants. This work is limited to the identification ofA, Qi=1,2 andCi=1,2 for the aluminium material, while the remaining parameters and physical constants were taken from the literature [162, 167, 195]. Remember that the material data and parameters provided for the steel material in [162]
also apply in this thesis (see Section 3.2.2). The parameters A, Qi=1,2 and Ci=1,2 were obtained by inverse modelling using a finite element (FE) model of the material tests presented in Section 3.2.2. It was decided to perform the inverse modelling using the optimization package LS-OPT. This provides a simulation environment where the objective is to minimize the mean-squared-error between the experiment and simulation for a user-defined curve. LS-OPT reads input and result files from the FE software and optimizes the parameters of a constitutive relation through sequential analyses on the same FE model by varying the input parameters. It was considered convenient to establish the FE model in LS-DYNA [196] due to its tailored interface with LS-OPT and the fact that a similar material model exists in LS-DYNA (*MAT_107). The target curve was chosen as the force-displacement curve from a typical tensile test in the rolling (0◦) direction, and the FE model consisted of Belytschko-Tsay shell elements with an initial element size in the gauge area equal to the thickness of the specimens in an attempt to capture the local necking. Material constants for both materials are listed in Table 4.1, while physical constants taken from Holmen et al. [162,195] are provided in Table 4.2. The strain-rate sensitivity constantcfor the 1050A-H14 aluminium alloy was taken from [167].
However, since the optimization simulations were carried out using a different FE software it was considered necessary to verify that the material parameters
4.5. Material parameter identification 95
Table 4.1: Material parameters for the modified Johnson-Cook constitutive relation.
Material A Q1 C1 Q2 C2 c m p˙0 Wc
[MPa] [MPa] [-] [MPa] [-] [-] [-] [s−1] [MPa]
Docol 600DL [162] 370.0 236.4 39.3 408.1 4.5 0.001 1.0 5×10−4 473.0 1050A-H14 80.0 49.3 1457.1 5.2 121.5 0.014 1.0 5×10−4 65.0
Table 4.2: Physical constants for the materials taken from the literature.
Material E ν ρ cp χ Tr Tm
[GPa] [-] [kg/m3] [J/kgK] [-] [K] [K]
Docol 600DL [162] 210.0 0.33 7850 452 0.9 293 1800 1050A-H14 [195] 80.0 0.30 2700 910 0.9 293 893
were applicable also in EPX. This also served as a validation of the implemen-tation of the VPJC material model. The uniaxial tension tests presented in Section 3.2.2 were therefore modelled in EPX by the same shell elements to be used in the airblast simulations, and by prescribing the same elongation history as in the material tests. The prescribed velocity was ramped up over the first 0.5 % - 1.0 % of the total computational time using a smooth transition curve. It was used exactly the same mesh as in LS-DYNA, resulting in a spatial discretization of 3116 4-node quadrilaterals. The elements were 4-node quadri-laterals (calledQ4GS) with 6 dofs per node and 20 Gauss integration points (5 through the thickness). Mass-scaling by a factor 109and 108was used to speed up the computational time for the steel and aluminium specimens, respectively.
Larger scaling factors resulted in non-physical inertia effects during the necking and a non-negligible kinetic energy in the simulations.
Comparisons between FE analyses and tensile tests are shown in Figure 4.7.
Since necking occurred at very small strains for the aluminium alloy, results from the numerical simulations were compared to the experimental data in terms of nominal stress-strain curves. The trend was that the numerical models were able to describe the overall response for both materials, and that the material parameters in Table 4.1 were valid also in EPX. Since necking occurred already at strains of approximately 0.7 % in the aluminium tests, it was necessary to include the strain-rate sensitivity term in Eq. (4.32) to capture the post-necking behaviour (see Figure 4.7b). After the initiation of necking, the strain rate increased by an order of magnitude and delayed the evolution of the neck by increasing the load-carrying capacity of the material.
This was also observed in the experiments since barely any diffuse necking occurred before localized necking and failure, which may be explained by the high rate sensitivity in these types of alloys (see e.g. [164–167]).
The CL parameterWc in Eq. (4.36) was determined based on the numerical simulations by inspecting the element exposed to the largest plastic work. This
0 0.05 0.1 0.15 0.2 0.25 0.3
Figure 4.7: Nominal stress-strain curves from uniaxial tensile tests along three different loading directions for (a) the Docol 600 DL steel and (b) the aluminium alloy 1050A-H14. Numerical results from EPX (FEA) with material data from Table 4.1 and Table 4.2 are included for comparison. The red dots denote the point of failure in the calculation ofWc.
element is always located inside the neck, since localization is the first sign of material failure. The accuracy ofWc is therefore highly dependent on a proper representation of the localized necking. That is, when necking localizes in the tension test, damage will evolve rapidly in the critical element. The parameter, as obtained in this work, is therefore mesh dependent because the mesh size influences the representation of the localized necking. Only the tension tests in the rolling direction of the plate were used in the calibration, although the failure strain for the aluminium alloy was somewhat lower in the 45◦ and 90◦ directions (see Figure 4.7b). This also implies a spread inWc between each material direction, which (at least to some extent) may affect the numerical results. However, modelling of anisotropic failure was beyond the scope of this thesis. The points used to extractWc from the numerical results are indicated by red dots in Figure 4.7 and the values are given in Table 4.1. Figure 4.7 shows that the numerical simulation of the steel captured the localized necking very well, while the simulation of the aluminium did not manage to predict localized necking at the same strain as in the material test. TheWc parameter for the aluminium alloy was therefore determined at the value of the major principal strain where failure occurred in the test.
4.6 Concluding remarks
A material model was implemented and calledVPJC in EPX. This is an elastic-thermoviscoplastic model which is formulated in a corotational framework allowing for finite strains and finite rotations. Ductile failure is also included by using an energy-based criterion which is uncoupled from the constitutive
4.6. Concluding remarks 97 equations. TheVPJC material model is applicable for a wide range of elements and coupled with the element deletion options available in EPX.
The respective material parameters were identified for the materials considered in Part II of this thesis and the VPJC model was used in the numerical simulations of the tension tests in Section 3.2.2. It was found that the numerical results were in good agreement with the experimental data, indicating that the VPJC model is properly implemented in EPX and can be used in the simulations of the blast-loaded plates in Chapter 5.
5
Numerical simulations
The experimental observations of the counter-intuitive behaviour (CIB) and reversed snap buckling (RSB) at relatively small scaled distances in Chapter 3 attracted special attention as it occurred both during and after the elastic rebound. However, since it was challenging to conclude on the effects producing this abnormal response based on the experimental data, the influence of the negative phase and the elastic effects on the dynamic response is investigated numerically in this chapter. The numerical work presented in this chapter is also presented in the third paper published in International Journal of Impact Engineering [197].
5.1 Introduction
The experimental study presented in Chapter 3 investigated the effect of stand-off distance on the dynamic response of thin aluminium and steel plates subjected to airblast loading. The tests covered the entire range of structural response from complete tearing at the supports to a more CIB where the final configuration of the plate was in the opposite direction of the incident blast wave due to RSB. RSB attracted special attention as it occurred at relatively small scaled distances and both during and after the elastic rebound.
However, since it was challenging to conclude on the governing parameters for this abnormal response based on the experimental data in Chapter 3, the influence of the negative phase and the elastic effects on the dynamic response is investigated numerically in the present chapter. The numerical simulations are performed by the FE code EUROPLEXUS (EPX) [130] using a Lagrangian formulation. Pressure-time histories are prescribed to the plates based on the mass and position of the charge. This is often called an uncoupled approach and makes the inherent assumption that the blast properties are unaltered by the structural motion and the surroundings [69]. The uncoupled approach is usually the preferred procedure in blast-resistant design [16], due to the increased complexity and computational costs when using fully coupled fluid-structure interaction simulations.
A well-established reference for the properties of the positive phase from airblast experiments is the work by Kingery and Bulmash [4]. The most commonly used negative phase parameters seem to be those given in the traditional diagrams in [16]. However, there still seems to be some uncertainty regarding the representation and treatment of the negative phase of the pressure-time history. As already mentioned in Section 1.2.1, the literature contains three basic representations of the pressure-time history when modelling this phase, i.e., a bilinear approximation, an extended Friedlander equation based on the waveform of the positive phase and a cubic representation. Rigby et al. [30]
reviewed these methods and found that bilinear and cubic approximations resulted in the best agreement with experimental data. Before simulating all the 0.8-mm-thick plates in Table 3.1 it was therefore decided to perform a parametric study on the effect of bilinear and cubic representations on the dynamic response. Based on the findings in this parametric study it is carried out numerical simulations of the blast-loaded plates. The numerical model is first validated against the experimental data in Chapter 3, before performing a numerical study to determine the governing parameters for the observed RSB.
Special focus is placed on the influence of elastic effects and negative phase on the structural response. The capabilities of the Cockcroft-Latham (CL) failure criterion and element erosion in predicting the crack patterns observed in the experiments are also evaluated. Due to trigger problems and flaking of the paint at the centre part of the plate in some of the tests, 3D-DIC analyses were only possible in 13 out of the 21 experiments conducted. The displacement histories reported in this section are therefore limited to the tests where 3D-DIC analyses were possible. However, the tests showed good repeatability and the reported results are considered to be a good representation of the experimental observations. All deformation profiles presented herein were corrected for the slight movement of the mounting frame during the tests.