**2.2 Coulomb-Mohr fracture criterion**

The Coulomb-Mohr fracture criterion gives reasonably accurate predictions of fracture in brittle materials for which the compressive strength far exceeds the tensile strength, e.g., concrete or cast iron [45]. It is presumed that fracture occurs in a certain stress plane of material when a critical combination of normal stress and shear stress acts on the plane. The linear relation of the combination of critical stresses is given by:

*Ultimate Compressive Strength of Steel Stiffened-Plate Structures Triggered by Brittle… DOI: http://dx.doi.org/10.5772/intechopen.97155*

**Figure 5.** *Brittle fracture of a full-scale steel stiffened plate structure under axial-compressive loads at -160°C [7].*

$$|\mathfrak{r}| + \mu \sigma = \mathfrak{r}\_i \tag{2}$$

where *τ* is the shear stress, *σ* is the normal stress, *μ* and *τ<sup>i</sup>* are constants for a given material.

### **2.3 Johnson-Holmquist fracture criterion**

The Johnson-Holmquist fracture criterion [46] is useful for modelling brittle materials, e.g., ceramic and glass, over a range of strain rates. It is one of the most widely used models in dealing with the ballistic impact on ceramics, which is expressible as follows:

$$
\sigma^\* = \sigma\_i^\* - D\left(\sigma\_i^\* - \sigma\_f^\*\right) \tag{3}
$$

where *σ* <sup>∗</sup> *<sup>i</sup>* is the uniaxial failure strength of intact material, see Eq. (4), *σ* <sup>∗</sup> *<sup>f</sup>* is the uniaxial failure strength of completely fractured material, see Eq. (5), and *D* is a damage accumulation variable, see Eq. (6).

$$
\sigma\_i^\* = A \left( p^\* + T^\* \right)^\eta \left[ \mathbf{1} + C \ln \left( \frac{d\epsilon\_p}{dt} \right) \right] \tag{4}
$$

$$
\sigma\_f^\* = B(p^\*)^m \left[ \mathbf{1} + C \ln \left( \frac{d\varepsilon\_p}{dt} \right) \right] \tag{5}
$$

$$\frac{dD}{dt} = \frac{1}{\varepsilon\_f} \frac{d\varepsilon\_p}{dt} \tag{6}$$

where *A*, *B*, *C*, *m*, *n* are material constants, *t* is time, *ε<sup>p</sup>* is the inelastic strain, and *ε <sup>f</sup>* is the plastic strain to fracture. The asterisk indicates a normalised quantity, where the quantities of each variable are defined as follows:

$$\sigma^\* = \frac{\sigma}{\sigma\_{hel}}, p^\* = \frac{p}{\sigma\_{hel}}, T^\* = \frac{T}{p\_{hel}} \tag{7}$$

where *σ* <sup>∗</sup> and *p*<sup>∗</sup> are the stresses normalised by the stress at the Hugoniot elastic limit, and *T* <sup>∗</sup> is the tensile hydrostatic pressure normalised by the pressure at the Hugoniot elastic limit.

The Johnson-Holmquist damage model was modified by Deshpande and Evans [47] and Bhat et al. [48], where it is considered that the propagation of an initial crack is a function of the stress state, the fracture toughness and the flaw characteristics.
