2.3.3. Energy Absorption Method

In the side crash simulation, the various energies, such as elastic, plastic, kinetic, friction, and viscoelastic energy, were transformed by deformation during the collision. The energy absorbed by deformation was experimentally calculated using load-displacement data according to Equation (1) [22].

$$E\_{ab} = \int\_0^s F(\mathbf{x})d\mathbf{x} \tag{1}$$

where *Eab*, *s*, and *F* are absorbed energy, crash displacement, and impulsive force, respectively. The specific energy absorption (SEA) was obtained by Equation (2):

$$\text{SEA} = \frac{E\_a}{M} \tag{2}$$

where *M* and *Ea* are total mass and absorbed energy, respectively.

When the SEA value is high, the energy absorption capability is high. However, this is approximate (rather than exact) estimation method and cannot determine elastic and plastic deformation energy.

In this study, the integrated stress–strain curve data per unit element volume were used to calculate the accurate absorbed energy, such as elastic and plastic dissipated energies, and for analysis of the distribution of energy flow in the time domain.

Figure 7a shows the meshed shell with the S4 element type used to accurately calculate the energy variable data and avoid the hourglass effect. As shown in Figure 7a, the shell element has 4 points, which are inside the mid-surface, in contrast with a solid element type. Because the mid-surface was used for analysis, the change in shell thickness was not visually expressed, and the strain in the thickness direction of the shell element was calculated as Equation (3):

$$
\varepsilon\_{33} = \frac{\nu}{1-\nu} (\varepsilon\_{11} + \varepsilon\_{22}) \tag{3}
$$

Treating these as logarithmic strains,

$$\ln \frac{t}{t\_o} = -\frac{\nu}{1-\nu} \ln \left(\frac{l\_0}{l}\right) = -\frac{\nu}{1-\nu} \ln \left(\frac{A}{A^o}\right) \tag{4}$$

where *l*, *t*, *ν*, and *A* are the element length, thickness, Poisson's ratio, and area of the shell's reference surface, respectively.

The change in shell thickness is expressed by Equation (5).

$$\frac{t}{t\_o} = \left(\frac{A}{A\_o}\right)^{(-\frac{\nu}{1-\nu})}\tag{5}$$

According to the above equations, the element volume of each mesh can be measured by Equation (6) after deformation.

$$V\_{element} = A t\_o \left(\frac{A}{A\_o}\right)^{\left(-\frac{\nu}{1-\nu}\right)}\tag{6}$$

The stress–strain curve describing the calculation of absorbed energy per element volume during deformation is shown in Figure 7b.

As shown in Figure 7b, the specific elastic and plastic deformation energies are measured by Equations (7) and (8), respectively:

$$
\Delta E\_{\mathfrak{c}\mathfrak{s}} = \frac{1}{2} \sigma\_{\text{New}} \Delta \varepsilon\_{\mathfrak{c}\mathfrak{s}} \Delta V \tag{7}
$$

$$
\Delta E\_p = \frac{1}{2} (\sigma\_{Old} + \sigma\_{New}) \Delta \varepsilon\_p \Delta V \tag{8}
$$

where *σNew*, *σOld*, *σu*, and Δ*V* are the new stress, previous stress, user-defined equation stress, and specific element volume, respectively.

The internal energy can be expressed by integrating Equation (9):

$$E\_I = \int\_a^t \left( \int\_V \sigma^\mu : \dot{\varepsilon} \, dV \right) dt \tag{9}$$

Then, . *<sup>ε</sup>* <sup>=</sup> . *ε es* + . *ε <sup>p</sup>* + . *ε <sup>c</sup>* and *EI* can be separated as in Equation (10):

$$E\_l = \int\_{\boldsymbol{\theta}}^t \left( \int\_V \boldsymbol{\sigma}^{\mu} : \dot{\boldsymbol{\varepsilon}}dV \right) d\boldsymbol{t} = \int\_{\boldsymbol{\theta}}^t \left( \int\_V \boldsymbol{\sigma}^{\mu} : \dot{\boldsymbol{\varepsilon}}^{\boldsymbol{\varepsilon}}dV \right) d\boldsymbol{t} + \int\_{\boldsymbol{\theta}}^t \left( \int\_V \boldsymbol{\sigma}^{\mu} : \dot{\boldsymbol{\varepsilon}}^{\boldsymbol{\varepsilon}}dV \right) d\boldsymbol{t} + \int\_{\boldsymbol{\theta}}^t \left( \int\_V \boldsymbol{\sigma}^{\mu} : \dot{\boldsymbol{\varepsilon}}^{\boldsymbol{\varepsilon}}dV \right) d\boldsymbol{t} = E\_{\rm cr} + E\_p + E\_{\rm c} \tag{10}$$

where . *ε es*, . *ε p* , and . *ε <sup>c</sup>* are the elastic strain rate, plastic strain rate, and creep strain rate, respectively; and *Ees*, *Ep*, and *Ec* are the elastic energy, plastic energy, and creep strain energy, respectively.

The elastic strain energy (*Ees*) results from linear deformation, whereas energy is dissipated by plasticity (*Ep*) when permanent deformation of the meshed element begins. The elastic and plastic strain regions are linear and non-linear, respectively. The aforementioned energies are defined by Equations (11) and (12), respectively:

$$E\_{\varepsilon\mathbb{S}} = \int\_{o}^{t} \left( \int\_{V} \sigma^{\mu} : \dot{\varepsilon}^{\varepsilon} \,dV \right) dt = \sum\_{t} \sum\_{i=1}^{n} \frac{1}{2} \sigma\_{i} \varepsilon\_{i} \Delta V\_{i} \tag{11}$$

$$E\_p = \int\_o^t \left( \int\_V \sigma^\mu : \dot{\epsilon}^p dV \right) dt \tag{12}$$

In this study, the element set of PW, PS, and MDB was selected, as shown in Figure 8. The *Ees* and *Ep* of the four kinds of center pillars were compared during a side crash because the elastic strain and plastic deformation energies account for most of the internal energy, which is known as the absorbed energy during a collision. *Ees* and *Ep* of PS and PW were calculated using the above equation during the deformation of the element according to the four types of center pillars, as shown in Figure 1b.

**Figure 8.** Energy distribution of the center pillar and MDB.

2.3.4. Damage Initiation Criteria and Damage Evolution

Local necking in the shell element could not be realized because the sheet metal in the simulation was very thin. To predict the onset of necking instability, the forming limit diagram (FLD) curve was used in this study. Simulations were performed by applying the previously obtained FLD for 22MnB5 (HPF and PS) to Abaqus/explicit [23,24].
