2 ωmax √ 1=ξ2−ξ (6.10) When considering an undamped material and applying lumped masses, this reduces to theCourant-Friedrichs-Lewy-criterion[128]:
∆tcr≤
l
c (6.11)
wherel = the length of the smallest element and cis the speed of sound in the material. This condition effectively limits the time step to be less than the time taken for a sound wave to traverse the smallest element in the model. The speed of sound in a material is defined as:
c=
√
E
ρ (6.12)
6.3
Material modelling
A comprehensive materials testing programme was performed to characterise the material properties of the Docol form 01 (DC01) mild steel test specimens at strain rates ranging from 3.3x10−4 s−1 (quasi static) to 534 s−1. The tensile tests showed that DC01 mild steel was highly ductile, tending to yield gradually and exhibited moderate strain rate sensitivity. Further details are available in Chapter 3. To accurately portray this behaviour the Modified Johnson-Cook constitutive relation and the Cockcroft-Latham fracture criterion have been implemented in LS-DYNA as MAT 107:MODIFIED JOHNSON COOK[123] and are used in this study. The formulation and implementation into LS-DYNA are discussed in the following sections.
6.3.1
Constitutive relation
The material was described using the Modified Johnson-Cook constitutive relation
[129]. The von Mises equivalent stress ¯σ in the Modified Johnson-Cook constitutive
relation, including the extended Voce hardening rule, is expressed as
¯ σ = A+ 2
∑
i=1 Qi(1−exp(−Ciε))¯ h 1+ ε˙¯ ˙¯ ε0 ich 1−T−Tr Tm−Tr mi (6.13)where, Adetermines the initial yield strength of the material at room tempera- ture. Qi andCi, i = 1,2, define the strain hardening. ¯ε is the equivalent plastic strain, ˙¯
ε is the equivalent plastic strain rate and ˙¯ε0is a user-defined reference strain rate. The
strain-rate sensitivity is represented by the constant c, while m models the thermal softening effect. The temperature dependence is given by the homologous temperature
T∗, defined asT∗= (T -Tr)/(Tm-Tr). T is the absolute temperature,Tr is the ambient temperature and Tm is the melting temperature. The temperature increment due to
adiabatic heating may be calculated as
∆T = ¯ ε Z 0 χσ¯d ¯ ε ρCp (6.14)
where ρ is the material density, Cp is the specific heat and χ is the Taylor- Quinney empirical coefficient that represents the proportion of plastic work converted to heat.
6.3.2
Identification of material parameters
The Modified Johnson-Cook (MJC) constitutive model, Equation 6.13 consists of three terms governing strain hardening, strain rate sensitivity and temperature softening. These parameters are usually calibrated separate to one another, and the processes are described below.
Strain hardening
The parameterAis the initial yield strength of the material at room temperature which is taken as the proof stress at 0.2% plastic strain, using corrected true-stress strain data at room temperature, following procedures outlined by LS-DYNA[122] . The strain hardening parametersQi andCi, i=1,2 are then fitted to the curve beyond the yield
6.3. MATERIAL MODELLING 119
Figure 6.4: Stress-strain (σ-εp) curve of DC01 and fit to the extended Voce hardening rule.
Strain rate sensitivity
The strain-rate sensitivity,cis determined by using the dynamic tensile test data shown for reference in Figure 6.5 and described in Chapter 3, whenT∗= 0. Figure 6.6 shows the flow stress as a function of the logarithmic strain rate at 5% plastic strain. The parameter c is then obtained by fitting a curve to the experimental data using least squares.
Figure 6.6: Flow stress as a function of strain rate. Experimental data points are shown as dots and the results obtained with the Modified Johnson-Cook model is shown as a line.
Temperature softening
To calibrate the thermal softening a series of quasi static tensile tests are conducted at elevated temperatures. Chung[130] and Langdon et al.[131] investigated the influence of incorporating temperature effects in numerical simulations for impulsively blast loaded steel plates. They concluded that it is beneficial to incorporate this parameter when predicting the onset of tearing, however incorporating temperature effects made very little difference when predicting Mode I response. Hsu[68] conducted a series of experimental tests on thin mild steel plates without holes at similar pressure levels and boundary configurations to the ones conducted in this study. She observed no thinning or tearing of the plates. With this taken into consideration it was expected that no thinning or tearing would occur in this present study either. Therefore no testing at elevated temperatures was deemed necessary. This assumption was confirmed when inspecting the plates within this experimental study (Chapter 5).
Borviket al.[132] states that it would be reasonable to assume a linear reduction in equivalent stress with temperature (i.e. m= 1) for steel when elevated temperature tests are not available. Therefore a value ofm= 1 was used in all future simulations.
6.3. MATERIAL MODELLING 121
Fracture criteria
A failure criteria is needed for a complete description of the material behaviour. Ductile failure is highly dependent on the stress triaxiality. One simple fracture criteria that includes stress triaxiality is the Cockcroft-Latham (CL) criteria.
Cockcroft-Latham model
The failure criteria proposed by Cockcroft and Latham[133]is based on the total plastic work per unit volumeW, defined as
εf
Z
0
hσ1idε¯ =W (6.15)
where ¯ε is the major principal stress,hσ1i=σ1whenσ1≥0 andhσ1i= 0 when
σ1<0. Damage accumulates during straining until it reaches a critical valueW =Wcr
at ¯ε = ¯εf. The material constantWcr is determined from a single uniaxial quasi-static
tensile test by calculating the plastic work (area) of the true stress-strain curve[18]. Researchers Rakvag[3] and Judge[134] have demonstrated that this simplified failure criteria is capable of describing the behaviour of a material when subjected to impact loading. Deyet al.[135] demonstrated that the single parameter derived using the (Wcr) Cockcroft-Latham fracture theory gave similar results to the Johnson-Cook fracture criterion which has a more involved process (five parameters). Therefore the Modified Johnson-Cook constitutive relationship and the Cockcroft-Latham fracture criterion have been used in the present study and are presented in Table 6.1.
Table 6.1: Material parameters for MJC constitutive relation and CL fracture criteria. Elastic constants Yield stress and Strain rate Cockcroft-Latham
and density strain-hardening sensitivity failure
A= 205 [GPa]
E = 205 [GPa] Q1= 263 [MPa] ε˙¯0= 0.341
υ = 0.33 [MPa] C1= 4.5 Wcr= 120 [MPa]
ρ= 7850 [kg/m3] Q2= 58.3 [MPa] C= 0.055