**4.1. Applications in mechanical engineering**

**a.** One examines the superposition of effects under loading combining tensile stress, σ, bending stress, σ*b* and torsion stress τ*t* (Figure 4): σ~ *F* ; σ*<sup>b</sup>* ~*Mb* and τ*<sup>t</sup>* ~*Mt*.

**Figure 4.** A tubular specimen loaded by axial force, *F*, bending moment, *Mb* and torsional moment, *Mt* .

The total participation is calculated with the general relation (12),

$$P\_T = \left(\frac{\sigma}{\sigma\_{cr}}\right)^{a\_\sigma + 1} \cdot \delta\_\sigma + \left(\frac{\sigma\_b}{\sigma\_b}\right)^{a\_b + 1} \cdot \delta\_{\sigma\_b} + \left(\frac{\tau\_t}{\tau\_{t,cr}}\right)^{a\_t + 1} \cdot \tag{19}$$

Since σ>0 (traction), δσ =1. In a stretched fibre, where σ*<sup>b</sup>* >0, δσ*<sup>b</sup>* =1, the denominators represent the critical values of the stresses corresponding in the numerator. Generally ασ =α*<sup>b</sup>* =α*<sup>t</sup>* =α, so that

The Principle of Critical Energy as a Transdisciplinary Principle with Interdisciplinary Applications http://dx.doi.org/10.5772/64914 9

$$P\_{T, \max} = \left(\frac{\sigma}{\sigma\_{cr}}\right)^{a+1} + \left(\frac{\sigma\_b}{\sigma\_{b,cr}}\right)^{a+1} + \left(\frac{\tau}{\sigma\_{t,cr}}\right)^{a+1} \,. \tag{20}$$

If the loading state is bound not to exceed the yield stress, then one has to accept that the material features a linear elastic behaviour, in which case *k* =1 and α=1; relation (20) becomes

In case critical characteristics *Yi*,*cr* feature deterministic values, critical participation *Pcr*

lifetimes *t* ′

8

**of loadings**

*<sup>l</sup>* <sup>&</sup>lt;*tl* <sup>&</sup>lt;*<sup>t</sup>* ″

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*<sup>l</sup>* (Figure 3).

body, in connection with some phenomenon or process.

especially if the effect of loading depends on the rate of applied load.

cumulation means successive load actions in time.

**4.1. Applications in mechanical engineering**

*T*

*P*

that

which yields unique, precise values in the three cases analogous to Figure 2. One obtains

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

**4. PCE application to solving problems of superposition or/and cumulation**

The principle of critical energy generally refers to the effect of energy cumulation in a physical

Load superposition means the simultaneous load actions upon the physical body. *Load*

The discrimination between the superposition of effects and their cumulation is essential,

**a.** One examines the superposition of effects under loading combining tensile stress, σ,

bending stress, σ*b* and torsion stress τ*t* (Figure 4): σ~ *F* ; σ*<sup>b</sup>* ~*Mb* and τ*<sup>t</sup>* ~*Mt*.

**Figure 4.** A tubular specimen loaded by axial force, *F*, bending moment, *Mb* and torsional moment, *Mt*

s

d

1 1 1

+ + +

 s

 d

*cr b t cr*

s

æ ö æö æ ö = ×+ × + ç ÷ ç÷ ç ÷ ç ÷ è ø èø è ø

sst

 a

the critical values of the stresses corresponding in the numerator. Generally ασ =α*<sup>b</sup>* =α*<sup>t</sup>* =α, so

,

 t

*b t b b t*

.

a

The total participation is calculated with the general relation (12),

s a

Since σ>0 (traction), δσ =1. In a stretched fibre, where σ*<sup>b</sup>* >0, δσ*<sup>b</sup>*

s (0)=1,

.

=1, the denominators represent

(19)

$$P\_{T,\text{max}} = \left(\frac{\sigma}{\sigma\_Y}\right)^2 + \left(\frac{\sigma\_b}{\sigma\_{b,Y}}\right)^2 + \left(\frac{\tau}{\tau\_{t,Y}}\right)^2,\tag{21}$$

where σ*y*, σ*b*,*<sup>y</sup>* and τ*t*,*<sup>y</sup>* is the yield stress under tensile, bending and torsional stress, respec‐ tively.

Consider the particular case when loading occurs only under bending and torsional stress (σ=0). With the aid of the law of equivalence of processes and phenomena [2, 12, 18, 19] one obtained from relation (20), the equivalent bending stress:

$$
\sigma\_{b,ech} = \left(\sigma\_b^{a \ast 1} + K \cdot \tau\_t^{a \ast 1}\right)^{\frac{1}{\left(a \ast 1\right)}}.\tag{22}
$$

*K* = σ*b*,*cr* α+1 <sup>τ</sup>*t*,*cr* <sup>α</sup>+1 is a ratio of some mechanical characteristics of the material. In the linear elastic case, (α=1) Eq. (22) becomes the known relation recommended by literature [20],

$$
\sigma\_{b,cch} = \left(\sigma\_b^2 + K \cdot \pi\_t^2\right)^{0.5} \text{ . \tag{23}
$$

where *K* depends on the theory of strength used, not on the nature and behaviour of the material.

**b.** For engineering structures with cracks, PCE connects the mechanics of the deformable solids to fracture mechanics.

At present, the strength analysis of structures with cracks is done by using fracture mechanics concepts that are different from those of the mechanics of deformable solids. In the mechanics

of the deformable solids one makes use of such concepts as normal stress, shear stress, strain, etc.

In the mechanics of the deformable solids, the strength condition is expressed by inequality:

$$
\sigma\_{eq} \le \sigma\_{al'} \tag{24}
$$

where σ*eq* is the equivalent stress at the tip of the crack and σ*al* , the allowable stress, calculated —generally—with the relationship:

$$\sigma\_{sl} = \min \left( \frac{\sigma\_u}{c\_u}; \frac{\sigma\_y}{c\_y} \right) \tag{25}$$

where σ*u* is the ultimate stress; σ*y* is the yield stress, while *cu* >1 and *cy* >1 are safety coefficients.

In fracture mechanics one resorts to the concepts of stress intensity factor, *Ki* , crack tip opening displacement, δ*<sup>i</sup>* and the integral *Ji* , where *i* =I, II, III, corresponds to the three accepted modes of failure (I, opening; II, sliding; III, tearing). The strength condition is expressed:

$$K\_i \le K\_{i, \text{al}} \quad \text{or} \quad \delta\_i \le \delta\_{i, \text{al}} \text{ or } \mathbf{f}\_i \le \mathbf{f}\_{i, \text{al}}.\tag{26}$$

where *Ki*,*al* , δ*i*,*al* and *Ji*,*al* are the allowable values of the variables *Ki* , δ*<sup>i</sup>* and *Ji* .

By introducing the concepts of *deterioration or damage*, with the aid of PCE there have been established the following relations of the critical stresses (ultimate stress, yield stress or allowable stress) for the structures with cracks that have characteristic dimensions *a* and *2c:*

$$\begin{aligned} \sigma\_{cr}\left(a;c\right) &= \sigma\_{cr} \cdot \left[1 - D\_{\sigma}\left(a;c\right)\right]^{\frac{1}{a+1}} \\ \sigma\_{cr}\left(a;c\right) &= \tau\_{cr} \cdot \left[1 - D\_{r}\left(a;c\right)\right]^{\frac{1}{a+1}} \end{aligned} \tag{27}$$

where σ*cr* and τ*cr* are the critical normal and shear stresses of the structure without cracks; *D*σ(*a*; *c*) and *D*<sup>τ</sup> (*a*; *c*) are the deterioration due to the crack (*a, c*) in the field of normal and shear stresses, respectively.

On the basis of relationships (27) one can experimentally determine the value of the deterio‐ ration. For example, Table 1 lists the values of deterioration *D*σ(*a*; *c*) calculated on the basis of the first relation (27), for some steel specimens:

of the deformable solids one makes use of such concepts as normal stress, shear stress, strain,

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

In the mechanics of the deformable solids, the strength condition is expressed by inequality:

£ (24)

, where *i* =I, II, III, corresponds to the three accepted modes

*J J* (26)

, δ*<sup>i</sup>* and *Ji*

.

, the allowable stress, calculated

(25)

(27)

, crack tip opening

, *eq al* s s

min ; , *<sup>u</sup> <sup>y</sup>*

æ ö <sup>=</sup> ç ÷ ç ÷ è ø

*u y c c* s s

where σ*u* is the ultimate stress; σ*y* is the yield stress, while *cu* >1 and *cy* >1 are safety coefficients.

where σ*eq* is the equivalent stress at the tip of the crack and σ*al*

and the integral *Ji*

*al*

In fracture mechanics one resorts to the concepts of stress intensity factor, *Ki*

of failure (I, opening; II, sliding; III, tearing). The strength condition is expressed:

( ) ( )

*ac D ac*

( ) ( )

*ac D ac*

; 1;

*cr cr*

 s

 t

s

t

*cr cr*

, ,, or or , *K K i i al i i al i i al* £ ££ d d

are the allowable values of the variables *Ki*

By introducing the concepts of *deterioration or damage*, with the aid of PCE there have been established the following relations of the critical stresses (ultimate stress, yield stress or allowable stress) for the structures with cracks that have characteristic dimensions *a* and *2c:*

; 1 ; ,

t

s

<sup>ï</sup> = ×- é ù ë û ïþ

where σ*cr* and τ*cr* are the critical normal and shear stresses of the structure without cracks;

<sup>ü</sup> = ×- é ù ë û ïï

+

ý

a

a +

(*a*; *c*) are the deterioration due to the crack (*a, c*) in the field of normal and shear

s

—generally—with the relationship:

displacement, δ*<sup>i</sup>*

where *Ki*,*al*

*D*σ(*a*; *c*) and *D*<sup>τ</sup>

stresses, respectively.

, δ*i*,*al* and *Ji*,*al*

etc.

10

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$$D\_{\sigma} \left( a; c \right) = 1 - \left( \frac{\sigma\_{\alpha} \left( a; c \right)}{\sigma\_{\alpha}} \right)^{a+1}.$$


**Table 1.** The deterioration *D*(*a*,*c*) due to crack of some steel specimens elongational loaded. The ultimate stress of the crackless specimen σ*u*=455.934 [MPa]

Analogously, for the critical load of the structure with cracks one established the general relation:

$$Y\_{i,cr}\left(a;c\right) = Y\_{i,cr} \cdot \left[1 - D\_{Y\_i}\left(a;c\right)\right]^{\frac{1}{\alpha+1}}\,\tag{28}$$

where *Yi*,*cr* is the critical load for the structure without cracks; *DYi* (*a*; *c*) is the deterioration due to the crack within the load range (force, bending moment, torque, pressure, etc.).

The strength condition of a structure with cracks, after using PCE,

$$
\sigma\_{\text{eq}} \le \sigma\_{\text{al}}(a; c), \tag{29}
$$

which is similar to the relationship (25) from the mechanics of deformable solids; it differs from Eq. (25) in that the allowable stress depends on the crack characteristic parameters, through the concept of deterioration,

$$
\sigma\_{al}\left(a;c\right) = \frac{\sigma\_{cr}\left(a;c\right)}{c\_{\sigma}},\tag{30}
$$

where

Dimensions in mm <sup>2</sup>*c*, mm <sup>σ</sup>*u*(*a*;*c*),

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

**Table 1.** The deterioration *D*(*a*,*c*) due to crack of some steel specimens elongational loaded. The ultimate stress of the

Analogously, for the critical load of the structure with cracks one established the general

α 1

ë û (28)

*eq al* £ (*a c*; ,) (29)

= (30)

(*a*; *c*) is the deterioration due

( ) ( ) <sup>1</sup>

, , ; 1 ;, *<sup>i</sup> Y ac Y D ac i cr i cr <sup>Y</sup>*

to the crack within the load range (force, bending moment, torque, pressure, etc.).

where *Yi*,*cr* is the critical load for the structure without cracks; *DYi*

The strength condition of a structure with cracks, after using PCE,

s s

*al*

s

*a c*

<sup>+</sup> = ×- é ù

which is similar to the relationship (25) from the mechanics of deformable solids; it differs from Eq. (25) in that the allowable stress depends on the crack characteristic parameters, through

( ) ( ; ) ; , *cr*

s

*a c*

*c*s

Sample with around crack

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12

crackless specimen σ*u*=455.934 [MPa]

the concept of deterioration,

relation:

[MPa]

29 330.447 0.6250

*D*(*a*,*c*)

$$
\sigma\_{\alpha}\left(a;\mathcal{c}\right) = \sigma\_{\mu}\left(a;\mathcal{c}\right) = \sigma\_{\mu}\cdot\left[1 - D\_{\sigma}\left(a;\mathcal{c}\right)\right]^{\frac{1}{\alpha+1}}.\tag{31}
$$

The equivalent stress is calculated in the same way as in relationship (25).

The use of relations (29)–(31) requires the determination of deterioration value *D*σ(*a*; *c*) as it was done in works [7, 15–17, 21–24].

The method of calculating presented may replace the calculation based on the concepts of fracture mechanics (26). The connection with fracture mechanics lies only in the calculation of deterioration based on crack geometry.
