High-velocity impact of a ceramic target

Strength

The strength of the material is expressed in terms of the von Mises equivalent stress, $\sigma$, and is a function of the pressure, $P$, the dimensionless equivalent strain rate, ${\dot{ϵ}}^{*}={\dot{\overline{ϵ}}}^{pl}/{\dot{ϵ}}_0$ (where ${\dot{\overline{ϵ}}}^{pl}$ is the equivalent plastic strain rate and ${\dot{ϵ}}_0$ is the reference strain rate), and the damage variable, $D$ ($0\le D\le 1$). For the intact (undamaged) material, $D=0$, whereas $D=1$ for a fully damaged material.

For a dimensionless strain rate of ${\dot{ϵ}}^{*}=1.0$, the strength of the intact material ($D=0$) takes the form

where ${\alpha }_i={\sigma }_i/\left[\left({\sigma }_i^{\max }-{\sigma }_i\right)\left(P_i+T\right)\right]$ and $P_i$, $T$, ${\sigma }_i$, and ${\sigma }_i^{\max }$ are material parameters. The strength of the fractured material ($D=1$) is given by

where ${\alpha }_f={\sigma }_f/\left[\left({\sigma }_f^{\max }-{\sigma }_f\right)P_f\right]$ and $P_f$, ${\sigma }_f$, and ${\sigma }_f^{\max }$ are material parameters.

The intact and fractured strengths above are for a dimensionless strain rate of ${\dot{ϵ}}^{*}=1.0$. The effect from strain rates is incorporated by the Johnson-Cook strain rate dependence law as $\sigma ={\sigma }_0\left(1+C\ln {\dot{ϵ}}^{*}\right)$, where ${\sigma }_0$ is the strength corresponding to ${\dot{ϵ}}^{*}=1.0$. Plastic flow is volume preserving and is governed by a Mises flow potential.

Damage

The damage initiation parameter, $\omega$, accumulates with plastic strain according to

where $\mathrm{\Delta }{\overline{ϵ}}^{pl}$ is the increment in equivalent plastic strain and ${\overline{ϵ}}_f^{pl}\left(P\right)$ is the equivalent plastic strain to fracture under constant pressure, defined as

where $D_1$ and $D_2$ are material constants and $P^{*}=P/{\sigma }_i^{\max }$ and $T^{*}=T/{\sigma }_i^{\max }$. The optional parameters ${\overline{ϵ}}_{f,\min }^{pl}$ and ${\overline{ϵ}}_{f,\max }^{pl}$ are provided for additional flexibility to limit the minimum and maximum values of the fracture strain.

The JHB model assumes that the material fails immediately, $D=1$ when $\omega =1$. For other values of $\omega <1$, there is no damage ($D=0$) and the material preserves its intact strength.

Pressure

The equations for the pressure-density relationship without phase change are used in this study and are listed here.

where $\mu =\rho /{\rho }_0-1$. In the above, $K_1$, $K_2$, and $K_3$ are constants ($K_1$ is the initial bulk modulus); $\rho$ is the current density; and ${\rho }_0$ is the reference density. The model includes the effects of dilation or bulking that occur when brittle materials fail by including an additional pressure increment, $\mathrm{\Delta }P$, such that

The pressure increment is determined from energy considerations as

where ${\mu }_f$ is the current value of $\mu$ at the time of failure and $\beta$ is the fraction of the elastic energy loss converted to potential hydrostatic energy ($0\le \beta \le 1$ ). The bulking pressure is computed only for failure under compression (${\mu }_f>0$).

Strength

The strength of the material is expressed in terms of the normalized von Mises equivalent stress as

where ${\sigma }_i^{*}$ is the normalized intact equivalent stress, ${\sigma }_f^{*}$ is the normalized fractured equivalent stress, and $D$ is the damage variable. The normalized equivalent stresses (${\sigma }^{*}$, ${\sigma }_i^{*}$ and ${\sigma }_f^{*}$) have the general form ${\sigma }^{*}=\sigma /{\sigma }_{\mathrm{HEL}}$, where $\sigma$ is the actual von Mises equivalent stress and ${\sigma }_{\mathrm{HEL}}$ is the equivalent stress at the Hugoniot elastic limit (HEL). The model assumes that the normalized intact and fractured stresses can be expressed as functions of the pressure and strain rate as

The material parameters are $A,B,C,M,N,$ and the optional limits for the strengths ${\sigma }_i^{\max }$ and ${\sigma }_f^{\max }$. The normalized pressure is defined as $P^{*}=P/P_{\mathrm{HEL}}$, where $P$ is the actual pressure and $P_{\mathrm{HEL}}$ is the pressure at the $\mathrm{HEL}$. The normalized maximum tensile hydrostatic pressure is $T^{*}=T/P_{\mathrm{HEL}}$, where $T$ is the maximum tensile pressure that the material can withstand.

Damage

The damage initiation parameter, $\omega$, accumulates with plastic strain according to

where $\mathrm{\Delta }{\overline{ϵ}}^{pl}$ is the increment in equivalent plastic strain and ${\overline{ϵ}}_f^{pl}\left(P\right)$ is the equivalent plastic strain to fracture under constant pressure, defined as

The JH-2 model assumes that the damage variable increases gradually with plastic deformation by setting $D=\omega$.

Pressure

The equations for the pressure-density relationship are similar to the JHB model.

where $\mu =\rho /{\rho }_0-1$. The model includes the effects of dilation or bulking that occur when brittle materials fail by including an additional pressure increment, $\mathrm{\Delta }P$, such that

The pressure increment is determined from energy considerations as

where $\beta$ is the fraction of the elastic energy loss converted to potential hydrostatic energy ($0\le \beta \le 1$).

Strength

We use the general exponent form of the extended Drucker-Prager model (Extended Drucker-Prager Models) which, after some manipulations, can be written as follows:

We have replaced $q$ with $\sigma$ and $p$ with $P$ to be consistent with the symbols used in the JH-2 models. This expression is very similar to that of the intact strength of the JH-2 model

Comparing these two expressions, the equations to calibrate the material parameters in the Drucker-Prager model can be obtained as

After substituting the values for silicon carbide in Table 4, we find a = 3.920173 × 10⁻³ MPa ${}^{1-\frac{1}{N}}$ and b = 1.53846. After obtaining the material parameters $a$ and $b$, the uniaxial compression yield stress, ${\sigma }_c$, can be calibrated by solving the following equation.

For this silicon carbide, ${\sigma }_c$ = 6605.66 MPa. The Johnson-Cook type rate dependence can also be used together with the Drucker-Prager plasticity model.

Damage

The ductile damage initiation criterion in Abaqus (Damage Initiation for Ductile Metals) can be calibrated to reproduce the damage criterion used in the JH-2 damage model. The ductile criterion requires the specification of the equivalent plastic strain at the onset of damage as a function of the stress triaxiality. Along the intact strength curve of the JH-2 model, the stress triaxiality is given as

Given ${\overline{ϵ}}_f^{pl}$, the damage evolution relation for the JH-2 model gives the following expression for the pressure $P$:

Substituting this expression into the stress triaxiality expression, we finally obtain the functional relationship between ${\overline{ϵ}}_f^{pl}$ and the stress triaxiality needed for the ductile damage initiation criterion. An additional consideration when specifying the damage initiation criterion for the Drucker-Prager plasticity model is that the definition of the equivalent plastic strain is different from that used in the JH-2 model. The two are related by the following plastic work statement:

In summary, we specify some sampling points for ${\overline{ϵ}}^{pl}$ and calculate the corresponding pressure $P$ and, in turn, the intact strength ${\sigma }_i$ for the JH-2 model. We calculate the stress triaxiality using these $P$ and ${\sigma }_i$ values. We convert the ${\overline{ϵ}}^{pl}$ values into ${\overline{ϵ}}_c^{pl}$ through the above plastic work statement. In this manner, we finally obtain a table of data pairs for the equivalent plastic strain at damage initiation and the stress triaxiality for the Drucker-Prager model.

Pressure

The Mie-Grüneisen equation of state (Mie-Grüneisen Equations of State) is used to describe the hydrodynamic behavior of the silicon carbide material. The linear $U_s-U_p$ Hugoniot form is used. Without the energy contribution (${\mathrm{\Gamma }}_0$ = 0.0), the pressure is expressed as

where $\eta =1-{\rho }_0/\rho =\mu /\left(1+\mu \right)$. Using a Taylor expansion with respect to $\mu$, the linear and quadratic coefficients of the polynomial can be identified as $K_1$ and $K_2$ in the pressure density relation for the JH-2 model, which gives

We solve for the parameters $c_0$ and $s$. The values are $c_0$ = 8272.2 m/s and $s$ = 1.32.