**2. The principle of critical energy**

The principle of critical energy is stated as follows [2, 4]:

'The critical state in a process or phenomenon is reached when the sum of the specific energy amounts involved, considering the sense of their action, becomes equal to the value of the critical specific energy characterizing that particular process or phenomenon'.

The principle of critical energy allows the calculation of the effect produced upon a physical body by the simultaneous or successive action of several external actions or loads *Yj* (where *j* =1; 2; 3...). The mathematical expression of the principle of critical energy was stated as follows (1):

$$\sum\_{j} \left( \frac{E\_j}{E\_{j,\alpha}} \right) \cdot \mathbf{S}\_j = \mathbf{1},\tag{1}$$

where *Ej* is the specific energy (expressed in J/kg, J/m3 or J/m2 ) introduced in the material by loading *Yj* , while *Ecr*, *<sup>j</sup>* is the critical value of the specific energy *Ej* . If *Ej* =*E <sup>j</sup>*,*cr* the critical state is reached, namely, fracture, excessive deformation, buckling, and so on:

The Principle of Critical Energy as a Transdisciplinary Principle with Interdisciplinary Applications http://dx.doi.org/10.5772/64914 3

$$
\delta\_{\rangle} = \begin{cases}
1, \text{if the specific energy acts in the direction of the process;} \\
0, \text{if it has no effect upon the respective process;} \\
\tag{2}
$$

The expression under the sum in relation (1) represents the participation of the specific energy introduced by the action or load *Yj* and is written as:

$$P\_j = \left(\frac{E\_j}{E\_{j,cr}}\right) \cdot \mathfrak{G}\_{j,r} \tag{3}$$

thus, the sum in the left part of relation (10) is called the total participation of specific energy:

$$P\_T = \sum\_{j} P\_{j\prime} \tag{4}$$

where *PT* is a sum of dimensionless variables calculated with respect to the critical state; this particularity gives a high degree generality of the Eq. (4).

If the loading is caused by normal stress, one writes *Pj* =*Pj* (σ). But in the case of shear stress, τ, loading *Pi* =*Pi* (*τ*). For multiaxial loading:

$$P\_T = \sum\_{j} P\_j \left( \sigma \right) + \sum\_{i} P\_i \left( \tau \right)$$

For real materials whose mechanical characteristic values range inside a dispersion interval, the critical specific energy also ranges inside a dispersion interval. Consequently, the right part of relation (1) can be replaced by the condition *Pcr* (*t*)≤1. Relation (1), by taking into account the relations (3) and (4), becomes:

$$P\_T = P\_{cr}\left(t\right),\tag{5}$$
