**3. Practical use of PCE**

The criterion of truth in scientific research is the experiment. For this reason, down below there have been presented the predictions resulting from the application of the principle of critical energy in comparison to experimental results. In this way, the critical energy principle has been validated by the experimental data reported in the literature by various authors.

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The principle of critical energy has been used, for example, to solve the problems of superpo‐ sition and/or cumulation of the effects of actions definitory for such disciplines/chapters of engineering sciences as mechanical engineering, electrical and electromagnetic engineering, chemical engineering, etc. The question is whether we are dealing in this case with an inter‐

Interdisciplinarity means bringing together elements of two or more academic disciplines in order to solve a specific theoretical or practical problem. The result would not be possible without the 'cooperation' of different academic disciplines, out of which use is made of elements that have been time proven. Creation, in this case, refers to the combination of

Transdisciplinarity essentially means concerns that go beyond any discipline ('trans' = beyond) or away from a particular discipline, concerns based on the existing academic disciplines and finally capable of generating new areas of knowledge. The new concepts in the field can be

Consequently, the PCE features transdisciplinarity but in conjunction with just one more

'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

The principle of critical energy allows the calculation of the effect produced upon a physical

*j* =1; 2; 3...). The mathematical expression of the principle of critical energy was stated as

δ 1, *<sup>j</sup> j*

å (1)

) introduced in the material by

. If *Ej* =*E <sup>j</sup>*,*cr* the critical state

or J/m2

(where

body by the simultaneous or successive action of several external actions or loads *Yj*

,

is the critical value of the specific energy *Ej*

*E E* æ ö ç ÷× = ç ÷ è ø

*j j cr*

is reached, namely, fracture, excessive deformation, buckling, and so on:

critical specific energy characterizing that particular process or phenomenon'.

disciplinary or a transdisciplinary issue?

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knowledge already extant in the academic fields under scrutiny.

retrieved and applied to other areas or academic disciplines.

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

where *Ej* is the specific energy (expressed in J/kg, J/m3

academic discipline it becomes interdisciplinary.

**2. The principle of critical energy**

follows (1):

loading *Yj*

, while *Ecr*, *<sup>j</sup>*

The total participation of specific energies is a dimensionless value that expresses the loading level of any physical body by considering its behaviour.

For example, if a load featuring stress *Y* produces effect *X* upon a physical body, the interde‐ pendence between the two expresses the body behaviour. In general, one resorts to the law of non-linear behaviour, function of power

$$\mathbf{Y} = \mathbf{C} \cdot \mathbf{X}^{k} \,, \tag{6}$$

where *C* and *k* are constants of the physical body. The specific energy in this case is:

$$E = \bigcap\_{0}^{\chi} Y \cdot \mathbf{dX} = \frac{\mathbb{C} \cdot \mathbf{X}^{k+1}}{k+1}. \tag{7}$$

Expressed as a result of loading *Y*, relation (8) becomes:

$$E = \frac{\mathbf{Y}^{(1/k)\cdot\mathbf{1}}}{\left(k+1\right)\cdot\mathbf{C}^{1/k}}.\tag{8}$$

The specific critical energy corresponds to *Y* =*Ycr*, so that:

$$E\_{cr} = \frac{Y\_{cr}^{(1/k)\*1}}{\left(k+1\right)\cdot \mathbb{C}^{1/k}}.\tag{9}$$

From relations (4), (9) and (10) one acquires the expression of the specific energy participation:

$$P\left(Y\right) = \left(\frac{Y}{Y\_{\alpha}}\right)^{\mathcal{Y}k \ast 1} \cdot \delta\_{Y'} \tag{10}$$

where δ*Y* means the same thing as δ *<sup>j</sup>* .

When several loads *n* act, written as *Yi* , where *i* = 1; 2; … *n* (Figure 1), the total participation of the specific energies of action is:

$$P\_T = \sum\_{i=1}^{n} P\left(Y\_i\right). \tag{11}$$

The Principle of Critical Energy as a Transdisciplinary Principle with Interdisciplinary Applications http://dx.doi.org/10.5772/64914 5

**Figure 1.** Loading with stresses *Yi* (*Ei* ), carrying specific energies *Ei* , produces upon the physical body effects *Xj* .

If for all loads *Yi* , the behaviour of the material is given by a relation of the form (7), where the values of constants *Ci* and *ki* are different, Eq. (12) becomes:

$$P\_T = \sum\_{l=1}^{n} \left(\frac{Y\_i}{Y\_{i,cr}}\right)^{a\_l+1} \cdot \delta\_{Y\_{l1'}} \tag{12}$$

where α*<sup>i</sup>* =1 / *ki* , and δ*Yi* stands for δ *<sup>j</sup>* .

For example, if a load featuring stress *Y* produces effect *X* upon a physical body, the interde‐ pendence between the two expresses the body behaviour. In general, one resorts to the law of

Proceedings of the International Conference on Interdisciplinary Studies (ICIS 2016) - Interdisciplinarity and Creativity

,

d . 1

( )

*k C*

1/ 1 1/ . <sup>1</sup> *k*

( )

*k C*

1/ 1 1/ . <sup>1</sup> *k cr cr k <sup>Y</sup> <sup>E</sup>*

From relations (4), (9) and (10) one acquires the expression of the specific energy participation:

1 1

( )

.

1

*n T i i P PY* =

+

*k*

*cr*

æ ö = × ç ÷ ç ÷ è ø

,

*Y*

d

+

( )

( )

*<sup>Y</sup> P Y Y*

.

+

*k*

( )

*<sup>Y</sup> <sup>E</sup>*

*k*

1

where *C* and *k* are constants of the physical body. The specific energy in this case is:

*<sup>X</sup> <sup>k</sup> C X E YX*

0

Expressed as a result of loading *Y*, relation (8) becomes:

The specific critical energy corresponds to *Y* =*Ycr*, so that:

where δ*Y* means the same thing as δ *<sup>j</sup>*

When several loads *n* act, written as *Yi*

the specific energies of action is:

*<sup>k</sup> Y CX* = × (6)

<sup>+</sup> <sup>×</sup> =×= <sup>+</sup> <sup>ò</sup> (7)

<sup>=</sup> + × (8)

<sup>=</sup> + × (9)

, where *i* = 1; 2; … *n* (Figure 1), the total participation of

<sup>=</sup> å (11)

(10)

non-linear behaviour, function of power

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The critical participation of specific energies, a time-dependent dimensionless variable, has the general expression [2, 12]:

$$P\_{cr}\left(t\right) = P\_{cr}\left(0\right) - D\_T\left(t\right) - P\_{res}\tag{13}$$

where *Pcr* (0) is the initial value of the critical participation at *t* =0.

The value of *Pcr* (0) depends on the probability of achieving the critical condition at *t* =0. Generally, *Pcr* (0)∈ *Pcr*,min(0);*Pcr*,max(0) , where *Pcr*,min(0)>0 and *Pcr*,max(0)≤1. The critical partic‐ ipation *Pcr* (0) expresses the value distribution of physical characteristics (e.g. tensile strength, σ*u*, yield stress, σ*y*, etc.). If one considers as deterministic (fixed; statistical averages) values of the critical physical characteristics *Yi*,*cr*, then *Pcr* (0)=1.

For the *deterministic values* of the physical characteristics *Yi*,*cr* one replaces *Pcr* (0)=1, so that relation (14) becomes:

$$P\_{cr}\left(t\right) = 1 - D\_T\left(t\right) - P\_{ns}.\tag{14}$$

Relation (14) is used to interpret the experimental data, and relation (15) is used to calculate structures with deterministic calculation methods [12].

The total damage *DT* (*t*), a dimensionless value, depends on the duration of exposure, *t*, and one calculates it by using the general relation [3, 7, 12]:

$$D\_T\left(t\right) = \sum\_k D\_k\left(t\right). \tag{15}$$

where *Dk* (*t*) is the deterioration produced by loading or by cause *k*: crack *D*(*a*; *c*), pre-loading *<sup>D</sup>*(−*t*), corrosion *D*(*tcs*), creep *D*(*tc*), hydrogen action *D*(*<sup>H</sup>* +), neutron action *D*(*n*), magnetic action *D*(*B*), vibration action *D*(ω), radiation flow action *D*(Φ), pollutant action *D*(*c*), etc.

In the manufacturing of engineering components (by plastic deformation, welding, moulding, forging, etc.) there are induced residual stresses, σ*res*, that map out the participation [2],

$$P\_{res} = \left(\frac{\sigma\_{res}}{\sigma\_u}\right)^2 \cdot \delta\_{res} \tag{16}$$

where δ*res* =1 if the residual stresses act in the direction of the process taking place in the body under load and δ*res* = −1 if not.

The practical use of the results obtained lie in comparing, for a certain given moment *t* (Figure 2), the values of *PT* (*t*) and *Pcr* (*t*). If

$$\begin{aligned} P\_T\left(t\right) &< P\_{cr}\left(t\right)-\text{subcritical loading};\\ P\_T\left(t\right) &= P\_{cr}\left(t\right)-\text{critical loading};\\ P\_T\left(t\right) &> P\_{cr}\left(t\right)-\text{overcritical loading}.\end{aligned} \tag{17}$$

After equalizing the expressions of participations (13) and (14),

$$\sum\_{i=1}^{n} \left(\frac{Y\_i}{Y\_{i,cr}}\right)^{\alpha\_i+1} \cdot \delta\_{\mathbf{y}\_i} = P\_{cr}\left(\mathbf{0}\right) - D\_T\left(t\right) - P\_{\text{res}}\,\tag{18}$$

one obtains the time life *tl* , or the duration down to the moment when the body under load is destroyed (Figure 2).

In the general case of statistical value distribution of critical characteristics *Yi*,*cr* between a minimum (*Yi*,*cr*)min value and a maximum (*Yi*,*cr*)max one, the initial critical participation is itself a statistical distribution and consequently, it lies probabilistically between the corresponding curves written as *Pcr*,min(*t*) and *Pcr*,max(*t*).

The Principle of Critical Energy as a Transdisciplinary Principle with Interdisciplinary Applications http://dx.doi.org/10.5772/64914 7

The total damage *DT* (*t*), a dimensionless value, depends on the duration of exposure, *t*, and

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where *Dk* (*t*) is the deterioration produced by loading or by cause *k*: crack *D*(*a*; *c*), pre-loading *<sup>D</sup>*(−*t*), corrosion *D*(*tcs*), creep *D*(*tc*), hydrogen action *D*(*<sup>H</sup>* +), neutron action *D*(*n*), magnetic action *D*(*B*), vibration action *D*(ω), radiation flow action *D*(Φ), pollutant action *D*(*c*), etc.

In the manufacturing of engineering components (by plastic deformation, welding, moulding, forging, etc.) there are induced residual stresses, σ*res*, that map out the participation [2],

> 2 , *res res res u*

where δ*res* =1 if the residual stresses act in the direction of the process taking place in the body

The practical use of the results obtained lie in comparing, for a certain given moment *t* (Figure

– subcriticalloading; – criticalloading; – overcriticalloading.

( ) ( )

*Yi cr T res*

*<sup>Y</sup> P Dt P*

In the general case of statistical value distribution of critical characteristics *Yi*,*cr* between a minimum (*Yi*,*cr*)min value and a maximum (*Yi*,*cr*)max one, the initial critical participation is itself a statistical distribution and consequently, it lies probabilistically between the corresponding

ç ÷ ×= - - ç ÷

0 ,

è ø <sup>å</sup> (18)

, or the duration down to the moment when the body under load is

d

s

æ ö = × ç ÷ ç ÷ è ø

s

*P*

(*t*). If

*T cr T cr T cr*

*Pt Pt Pt Pt Pt Pt*

After equalizing the expressions of participations (13) and (14),

1 ,

*Y*

*i i cr*

=

curves written as *Pcr*,min(*t*) and *Pcr*,max(*t*).

*<sup>i</sup> n i*

æ ö

α 1

+

d

< = >

( ) ( ) ( ) ( ) ( ) ( )

under load and δ*res* = −1 if not.

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2), the values of *PT* (*t*) and *Pcr*

one obtains the time life *tl*

destroyed (Figure 2).

*Dt Dt* <sup>=</sup> å (15)

(16)

(17)

( ) ( ), *T k k*

one calculates it by using the general relation [3, 7, 12]:

**Figure 2.** Lifetime with critical characteristics featuring probabilistic variables. *Pcr*,min(*t*) corresponds to maximum prob‐ ability and *Pcr*,max(*t*) corresponds to the minimum probability of recovery of a value of the critical characteristic (e.g. the maximum probability of survival and the minimum probability of survival, respectively).

If *PT* (*t*)= constant, the lifetime lies between values *tl*,min and *tl*,max, if *PT* (*t*)= *<sup>P</sup>* ′ (*t*) rises in time, then the lifetime decreases and it lies between *t* ′ *<sup>l</sup>* ,min and *t* ′ *<sup>l</sup>* ,max. Similarly, if the total participation *PT* (*t*)= *<sup>P</sup>* ″(*t*) decreases in time, the lifetime rises and it lies between *<sup>t</sup>* ″ *<sup>l</sup>* ,min and *t* ″ *<sup>l</sup>* ,max.

**Figure 3.** Lifetime when critical characteristics are deterministic variables.

In case critical characteristics *Yi*,*cr* feature deterministic values, critical participation *Pcr* (0)=1, which yields unique, precise values in the three cases analogous to Figure 2. One obtains lifetimes *t* ′ *<sup>l</sup>* <sup>&</sup>lt;*tl* <sup>&</sup>lt;*<sup>t</sup>* ″ *<sup>l</sup>* (Figure 3).
