_{1}

^{*}

Goal: In the process of exploitation of ceramic composites often we encounter not only high mechanical stresses but also thermal loads and air-thermal shocks. These loads are transformed into failure/rupture stress energy, when strength of work-pieces is less than loads, which develops pluck from the crack top, resulting in destruction of objects. Considering such extreme operation conditions computation of energies which contribute to materials catastrophe seems rather interesting. Method: The formula parameters were selected on the basis of study and generalization of micro- and macro-mechanical characteristics of ceramic materials. Results: The formula covers the process of creation of energies as a result of mechanical and thermal loads affecting the work-piece and analyses of mechanisms of impact of these energies on the cracks existing in the material; results of energies affecting the existing cracks as a result of such loads and results of starting of mechanisms of spreading of energies developed inside the work piece, which lead material to the catastrophe. Conclusion: On the basis of crack development mechanisms the universal relationship of total energy of the work-piece and its mass was established considering crack developing speed under critical stress conditions. Failure stress energy formula has been offered.

Material strength implies maximum resistance to external and inner tensions. Distances between atoms in the crystalline lattice and interaction forces between atoms determine theoretical strength of materials. Strength of aluminum oxide destruction at bending, according to the power of bonds between atoms, approximately equals to 50,000 MPa, while majority of technical versions of strength shows 300 - 350 MPa and materials characterized by extremely high hardness show 1000 MPa [

Such great discrepancy between theoretical and practical strengths is conditioned by the fact that ceramic materials reveal extreme sensitivity to non-homogeneities and structure defects, such as: inclusions, pores, macro- and micro-cracks. It should be stated that even in case of high technologies it is a very hard problem to avoid such structural flaws. At the morphological study of materials a significant role is attributed to crystalline phase, which is the component that intensifies most the structure. Rearrangement of crystalline phase in the lattice and its sizes exert significant impact on mechanical properties. Fine dispersion dimensions below 5 mcm and their homogeneous spreading increase mechanics of any type ceramic, while their big sizes such as 20 - 40 mcm and higher and unequal spreading in the material mass, decrease not only mechanical but also exploitation properties such as resistance to thermal and air-thermal shocks, electro- and magnetic properties, durability and exploitation at heavy stress conditions, e.g. on power transmission lines, reduction of stress threshold values and others [

Similarly, porous phase exerts complex impact on various properties of ceramics. Great significance is attributed to dimension of pores, their volumetric share, form, closed and round form, through or semi-through form.

The more fine-dispersion are pores which are homogeneously spread in the lattice, the stronger is the material; material is durable and resistant to thermal, air-thermal shock, mechanical stress and shocks. Pores markedly affect complex properties of work-pieces. Their rounded forms, small size pores, less than 5 mcm and their equal arrangement increase resistance of ceramics to external loads. When open porosity according to water absorption equals to zero, volumetric share of closed pores in the lattice varies within 0.5 - 9 vol. %. As to the invisible micro cracks, according to Griffiths they exist in any thermally treated material [

Under the load a body with a crack reaches marginal state of equilibrium, when the crack starts movement or it can move from the place at the slight increase of the given load. In this case stress intensity coefficient will be limiting, critical for this material at the given load conditions. In the elastic body with a crack, if axis is perpendicular to the fault direction, SIC in asymptotic approximation will be expressed by K = lim 2 π r σ y where Kic is a value, that characterizes viscous decomposition of material [

At the impact of forces of interaction between atoms the crystalline structures are created with strictly ordered distances between atoms. In equilibrium state sum of repulsion and attraction between two neighbor atoms equals to zero (

Displacement of atoms from balanced state becomes possible only at the impact of external forces. _{0} inter-space characteristic for such atoms. In case of displacement by ∆a distance, effect of counteraction/resistance is formed that corresponds to the forces for attraction―repulsion. If displacement is small, ratio of the resistance forces and distance between atoms is linear. After removal of load the balanced state is restored, deformation is reversible that is

recoverable, relaxed. Thus, most types of materials are linearly elastic till definite limits [

For consideration of excess stress and stress displacement around the pore the elliptic form of the pore in the plate (^{2}/a. At this spot stress σ_{s} in the direction of the loaded force σ are aches maximum:

σ S / σ a = 1 + 2 ( a / ρ ) 1 / 2 (1)

According to the above given equation, in case of circular pore at the excess of stress we receive factor 3. Stresses will be higher when curvature radius at elliptic edge is sharp. In case of marginal stress in the ellipsis a crack with the curvature radius ρ = 0 will be created. Stresses in this case will be infinitely large. In reality, such mathematical case can’t be encountered since minimal curvature radius equals to the distance between the atoms. In reality immediately at the crack top extremely high stresses are formed, which are close to the theoretical hardness [

Crack curvature radius ρ at the crack top is very small and because of it (a/ρ)^{1}^{/2} will be rather high. Thus in the Equation (1) the summand 1 may become insignificant. Excess of pressure at the crack top is approximately proportional of external pressure σ_{a} (distant field of compression) on the crack-containing material as well as of a root from crack length a. This law leads us to determination of the stress intensity coefficient Ki, as tension measure at the crack top section.

Ki = σ a Υ (2)

Geometrical factor Υ considers geometrical dimensions of crack-containing work-pieces and stress distribution in it. This factor is computed by means of numerical method according to various type loads and it is given in standard tables. Stress intensity factor index shows crack load regime, where the most risky section for material decomposition is loading state that is expressed by modus 1. In distinct from it, load at the crack sides is given by modus II, while load as a result of twisting―is given by modus III. II and III modi in special cases are of technical significance (e.g. composite materials or coatings). In case of monolith materials practically only modus I is used [

Stress spreading near the crack top is given by polar coordinates (

Cracks start to grow slowly when stress intensity factor acquires critical value, that is, when the crack tolerance in the process of loading and resistance energy are minimal and when even minimal increase of any load will turn into decomposition energy. Such process of loading results in increase of crack sizes. This phenomenon is called preceding critical period of crack increase [_{th} threshold value is not registered. At the increase of loading, initially we observe slow acceleration of crack increase (mostly at 10^{−12} m/sec), which alongside with the increase of the stress intensity factor in the first section it is increased within the frames of ratio regularities [

V = AK 1 n = A ∗ ( K i K ic ) (3)

where K_{IC} is coefficient of stress intensity factor. Parameters A (respectively A*) and n depend on the material, temperature and environment. Correspondingly,

for crack propagation speed at the 1 section, we consider environment diffusion at the crack top and the available stress. For most ceramics the exponent n > 15. For some materials section II is acceptable, when crack spreading speed stops growing. Crack speed is so high, that increase of the stress intensity factor does no more affect crack speeding, and the diffusion, as the process that determines acceleration (that runs very slowly) slows down.

When stress intensity factor overcomes crack resistance (section III), the stress disintegration energy develops and rapid growth of the crack occurs by its detachment and the process practically reaches the top speed. It is namely in this moment that crack spreading speed develops that equals to approximately 2000 m/sec (

Materials, which operate in transitional/critical temperature fields and under high mechanical loads, are characterized by thermal stresses and deformations in the conditions, when their mechanical and chemical properties don’t suffer significant changes. The problem is complicated since as a rule, alongside with thermal stresses mechanical stresses are acting from external loads, which contribute to development of high energies in materials. In this case developed energies exert marked impact on cracks and where there is a weak section in the lattice and there are cracks there, their development occurs via detachment plucking. In such cases catastrophic destruction of material is inevitable [

1) Zero order stress is formed in the system of reciprocally connected bodies, due to unequal thermal impact on those bodies or because of different linear temperature expansion coefficients.

2) First order (microscopic) stresses, which are balanced in the spheres of body sizes, are conditioned by non-homogeneity of temperature fields or by body properties.

3) Second order (microscopic) stresses can be formed in case of absence of the first order stresses and they have independent values.

4) Third order defects (sub-microscopic) can’t be called stresses, since stresses are measured in the least zones spheres and they are insignificant. Such defects are revealed in the distances between atoms and their study is difficult.

Identification of faults in separate grains or at their borders and then complete destruction of the material is conditioned by accumulation of defects, which are formed at the effect of thermo-structural stresses as a result of multiple cycles of high temperature impacts or during high mechanical stresses (

Dynamic micro-hardness and elasticity module of the obtained Si-Al-O-N material were determined on the modern dynamic micro-hardness tester “rome DUH-211S” according to the demands of ISO-14577 International standards, used for determination of mechanical characteristics of solid body surfaces (micro-hardness, elasticity module). Results are offered in

Phase composition (%) of Si-Al-O-N composite according to the morphological pattern is: Si-Al-O-N −62.6; silicon carbide −28.0; aluminum oxide −6.2; porous phase −3.2. Composite was obtained at the first stage by metal-thermal and nitriding processes at 1450˚C. Then the composite was dispensed in Teflon mill and was hot-compressed at 1620˚C.

Borders of indents taken from silicon carbide grains are sharp (Figures 5(a)-(c)). Crack that is formed as a result of indenter load on the grain doesn’t

Test mode | Load-unload | ||
---|---|---|---|

Sample name | SiAlon-zv | Sample No. | #1 |

Test force | 200.000 [gf] | Minimum force | 0.200 [gf] |

Loading speed | 1.0 (7.1448 [gf/sec]) | Hold time at load | 5 [sec] |

Hold time at unload | 3 [sec] | Test count | 21 |

Parameter name | Temp | Parameter | 20 |

Comment | 21.06.17-SiAlon-zv-200; DHV5-3 | ||

Poisson’s ratio | 0.190 | ||

Cf-Ap, As Correction | ON | Indenter type | Vickers |

Read times | 2 | Objective lens | 50 |

Indenter elastic | 1.140e+006 [N/mm^{2}] | Indenter poisson’s ratio | 0.070 |

SEQ | Fmax | hmax | hp | hr | DHV-1 | DHV-2 | Eit | Length | HV | Data name |
---|---|---|---|---|---|---|---|---|---|---|

[gf] | [um] | [um] | [um] | [N/mm^{2}] | [um] | |||||

1 | 200.710 | 4.7107 | 1.9264 | 3.1017 | 442.157 | 2643.803 | 7.211e+004 | 15.792 | 1492.537 | SiAlon-200(2) |

2 | 200.786 | 4.2612 | 1.6795 | 2.7414 | 540.546 | 3479.868 | 8.707e+004 | 14.621 | 1741.886 | SiAlon-200(4) |

3 | 200.800 | 4.9636 | 1.7638 | 3.3296 | 398.419 | 3155.263 | 6.588e+004 | 16.959 | 1294.659 | SiAlon-200(5) |

4 | 200.674 | 4.5307 | 1.7788 | 3.0421 | 477.884 | 3100.234 | 8.083e+004 | 15.644 | 1520.484 | SiAlon-200(6) |

5 | 200.675 | 4.3294 | 2.1587 | 2.9575 | 523.381 | 2105.199 | 9.024e+004 | 15.498 | 1549.415 | SiAlon-200(7) |

6 | 200.662 | 3.5295 | 1.5855 | 2.1773 | 787.444 | 3902.198 | 1.254e+005 | 16.595 | 1351.275 | SiAlon-200(8) |

7 | 200.661 | 3.6147 | 1.8441 | 2.4494 | 750.723 | 2884.448 | 1.349e+005 | 17.179 | 1260.907 | SiAlon-200(9) |

8 | 200.738 | 3.0333 | 1.1085 | 1.7530 | 1066.516 | 7985.353 | 1.660e+005 | 12.866 | 2248.651 | SiAlon-200(10) |

9 | 200.959 | 2.8595 | 1.0929 | 1.5884 | 1201.396 | 8224.728 | 1.857e+005 | 12.134 | 2531.125 | SiAlon-200(11) |

10 | 200.866 | 3.0653 | 1.3375 | 2.0446 | 1045.024 | 5488.768 | 1.924e+005 | - | - | SiAlon-200(12) |

11 | 200.737 | 3.1154 | 1.3372 | 2.0317 | 1011.028 | 5488.160 | 1.790e+005 | - | - | SiAlon-200(13) |

12 | 200.960 | 2.5787 | 1.1425 | 1.5447 | 1477.302 | 7525.888 | 2.536e+005 | 12.135 | 2530.738 | SiAlon-200(14) |

13 | 200.923 | 2.7215 | 1.1113 | 1.5055 | 1326.134 | 7952.513 | 2.077e+005 | 11.989 | 2592.358 | SiAlon-200(16) |

14 | 200.501 | 2.8549 | 1.0966 | 1.5509 | 1202.544 | 8150.998 | 1.824e+005 | 12.135 | 2524.953 | SiAlon-200(17) |

15 | 200.497 | 3.4966 | 1.3136 | 2.2145 | 801.640 | 5679.626 | 1.320e+005 | - | - | SiAlon-200(18) |

16 | 200.702 | 2.9626 | 1.1801 | 1.6771 | 1117.798 | 7044.719 | 1.729e+005 | 12.428 | 2409.746 | SiAlon-200(19) |

17 | 200.589 | 3.4541 | 1.4444 | 2.0858 | 821.888 | 4700.234 | 1.288e+005 | 14.474 | 1775.634 | SiAlon-200(20) |

18 | 201.195 | 3.0666 | 1.0307 | 1.5932 | 1045.886 | 9257.288 | 1.515e+005 | 11.698 | 2726.384 | SiAlon-200(21) |

Average | 200.757 | 3.5082 | 1.4407 | 2.1882 | 890.984 | 5487.183 | 1.449e+005 | 14.143 | 1970.050 | |

Std.Dev. | 0.174 | 0.738 | 0.346 | 0.611 | 324.195 | 2330.548 | 52250.109 | 2.028 | 548.126 | |

CV | 0.087 | 21.043 | 23.994 | 27.907 | 36.386 | 42.473 | 36.057 | 14.341 | 27.823 |

spread beyond the grain limits. Matrix, due to its high mechanical properties and energy dissipation, subdues crack spreading and composite hardness retains its value. Such large grains are few and speaking about mechanical properties of the material according to such grains should not be relevant, since increase of their dispersion rate is not a problem, while it gives interesting picture for the purposes of investigation. Especially interesting is

Dynamic hardness (DH) is determined according to the indenter load value and the depth of its indentation in the material in the process of testing and its value is computed by the formula DH = a × F / h 2 ; where a―is a constant value and depends on indenter form; for Vicker’s indenter a = 3.8584. Advantage of the method compared to measuring of common static, that is, linear dimensions of indents (diagonal) is that it covers plastic, as well as elastic components. Results of measuring don’t depend on indent sizes, loads and non-homogeneity of elastic recovery.

Dynamic hardness was determined in load-unload regime before elastic relaxation took place. For each concrete load seven readings were taken, two marginal values were discarded and remaining five values were averaged. Relevant micro-hardness values were determined automatically. Hold-time at maximum loading equaled to 5 sec, at the end of unloading―3 sec. (

Indentation was performed in sample matrix, which consisted of B-SiAlON. As a result of testing its average dynamic hardness equaled to, DHV = 8.9 GPa this is a rather high value.

From load-unload dependence graph (

realized from the crack top by the energy that exceeded critical stress intensity. The same was proved at the application of test force (200 g approximately equals to 2 N) at taking indent depth (

Thermo-structural stresses in the process of changes of temperature fields can be induced: by thermal expansion anisotropy of even only one phase, by anisotropy of dilatation of non-cubic crystalline lattice of materials, by difference in thermal linear expansion of adjacent phases in heterogeneous system, phase transformations, which are accompanied by specific volumetric changes of phases. Methods of quantitative evaluation of resistance to thermal shocks provide mainly determination of resistances to the first order thermo-elastic stresses, which are formed in elastic bodies and in which there are no plastic deformations and phase transformations [

R = δ a / a ′ Ε (4)

where R―is the factor of resistance to thermal shocks, δ―is mechanical strength of the material at bending, a―is thermal coefficient of linear expansion, E―Young’s module, a ′ − λ / y C ―material’s temperature conductivity, λsd―heat conductivity, y―density, C―specific heat capacity.

The formula doesn’t consider sample form, size and dependence of resistance to thermal shock. For real evaluation it is necessary to consider impact of body dimensions.

R = δ / ( Ε a β ( Tk − To ) ) (5)

where Tk and To―are final and starting temperatures of a body, β = r m h / λ and β = δ h / 2 λ ―the sphere and infinite plate, h―heat transmission coefficient between the body surface and environment. According to Kingery’s opinion, physical properties of material don’t suffer changes at alteration of temperature in ΔT_{destr}. Interval and he recommends the following definition for thermal-shock factor:

R ′ = 2 δ destr ⋅ ( 1 − μ ) / Ea = Δ T destr / B (6)

where B factor depends on body geometry, δ_{destr}∙―depends on mechanical strength of the material at stretching and μ―Poisson coefficient.

Kingery, Buessem and others [

R = δ bend ( 1 − μ ) / 3 Ea (7)

when relaxation periods significantly exceed possible service period of work-pieces, no significant thermal stresses occur and selection of materials is based on their elasticity properties, that is on the criteria [

R ′ = δ ( 1 − μ ) / Ea R ″ = R ′ λ R ‴ = E / δ 2 ( 1 − μ )

where R ′ ―is the criterion, which defines material resistance to the formation of cracks in the process of thermal shocks, R ″ ―is the criterion, which evaluates ability of the material to resist thermal loads, at small values of heat-exchange coefficient; R ‴ ―is the criterion, which defines material resistance to propagation of already formed cracks.

The offered data enable us to conclude that high mechanical strength and heat conduction together with low elasticity properties and thermal expansion provide high resistance of materials to crack formation and such ceramics can work at high temperature gradients and at high temperatures.

For electro-technical products synthesized by us we used Harcourt’s method [

Analysis of the table shows that barium-containing electroceramic synthesized on the base of barite and geopolymers enables us to conclude that celsianceramic B1, with 93% celsian phase, sintered at various temperatures, in definite regimes, with zero open porosity, is resistant and stable than B3. B1 is characterized by high value of thermal resistance―480˚C. Apparently it is associated with the low coefficient of dilate, which conditions high resistance of work-pieces to thermal shock and correspondingly resistance to stress distribution. In this material concentration of barium and aluminum silicates is low and it amounts to only some percents and their negative impact on resistance to thermal shocks is insignificant. Apparently lower rate of resistance to thermal shocks that is shown by B3 ceramic is conditioned by its multiphase composition: celsian BaO∙Al_{2}Os∙2SiO_{2}, mulite―3Al_{2}O_{3}∙2SiO_{2}, corundum―Al_{2}O_{3} and vitreous phase, which is more in B1.

Various expansion coefficients of the above referred phases factually exert negative impact on thermal resistance of the material (

Index of materials, and burn temperature, T ˚C | Open porosity, % | True porosity, % | Compactness ρ g/sm^{3} | σ, MPa. | A, 10^{−6} ˚C^{−1 } 20 - 900 | E, GPa. | μ | R ′ τ | R ″ BT/μ | R ‴ 10^{−4} M^{2}/kg |
---|---|---|---|---|---|---|---|---|---|---|

B^{1} 1410 | 0 | 14.2 | 2.99 | 66.4 | 4.1 | 71.32 | 0.283 | 163 | 210.3 | 225.6 |

B^{1} 1450 | 0 | 7.8 | 3.03 | 78.7 | 4.1 | 74.59 | 0.212 | 161 | 164.3 | 220.0 |

B^{1} 1500 | 0.1 | 9.8 | 2.96 | 59.3 | 4.1 | 72.20 | 0.280 | 144 | 185.8 | 285.2 |

B^{3} 1410 | 0.0 | - | 2.60 | 78τ.6 | 4.1 | 69.35 | 0.233 | 220 | 420.2 | 172.9 |

B^{3} 1450 | 0.0 | - | 2.52 | 56.0 | 4.1 | 79.22 | 0.258 | 128 | 244.4 | 340.0 |

different phases [

According to the above stated, while discussing the properties of ceramic materials and composites, integration of rupture stress issues is a very hard problem because of coincidence of complex processes in materials. This is why we considered proper to offer the characteristic of rupture stress which will comprise all

those conditions and properties which will take place in the process at critical loading of the work-piece and the rupture will be inevitable. It can occur in the cases of external energy charge on the work-piece and as a result, accumulation of energy in the piece, because of thermo-structural changes and external mechanical stress distribution. The formula of decomposition stress energy offered by us is:

E td = m ∗ a c .p .

where E_{td}―is energy decomposition tension; m-mass; a-speed of crack propagation.

Decomposition stress energy ^{1}E_{td} equals to the product of m-mass and crack propagation speed―a_{c.p}.

In our case a sample of electro-technical material had a form of a rod, and it had the following sizes:

Length l = 110 mm; width b―20 mm; height a = 10 mm. Mass of a rod of such sizes that was sintered at 1450˚C―equaled to 45.5 g. The work-piece was fully consolidated by zero open pores. If we admit that in the process of decomposition velocity of crack propagation by its dependence on stress intensity factor (from critical state) on the v-k curve is in the third section (

E td = 45.5 × 2000 = 91 kj

Thus, if we define universal connection between total energy of the work-piece and its mass, we’ll receive formula of rupture stress energy offered by us.

^{1}Energy (from Greek energies?action) of the common measure unit for various forms of material movements, which is considered in physics. Various types of energies are used for quantitative characterization of qualitatively different forms of motion and for relevant interactions, such as: mechanical, internal, gravitational, electro-magnetic, and nuclear and others. In international system of units―SI energy is expressed in Joules.

The formula [

Correlative relation of transformation of external stresses of energies into inner energies of the material was determined which determines propagation of defects existing in the work-piece at high energies exceeding the critical, when stress intensity factor overcomes crack resistance and rupture stress energy develops. Thus, universal relation between total energy developed in the work-piece and work-piece mass is established with respect to external energy of rupture stress.

I am grateful to Ilia Vekua Institute of Applied Mathematics of IvaneJavakhishvili Tbilisi State University for assistance in the research of Structural and Micro Mechanical Characteristics.

The author declares no conflicts of interest regarding the publication of this paper.

Kovziridze, Z. (2018) Failure Stress Energy Formula. Journal of Electronics Cooling and Thermal Control, 8, 31-47. https://doi.org/10.4236/jectc.2018.83003