Issue 31

E.M. Nurullaev et alii, Frattura ed Integrità Strutturale, 31 (2015) 120-126; DOI: 10.3221/IGF-ESIS.31.09 121 T HE THEORY OF CALCULATING THE MECHANICAL FRACTURE ENERGY echanical characteristics of polymeric composite materials (PCMs) based on the 3D cross-linked plasticized elastomeric matrix filled with solid silica particles significantly affect the service life of these materials. In this regard, the most important formulation (structural) parameters are the molecular structure of the polymeric matrix of the binder, its type and degree of plasticization, maximum volume filling, depending on the shape and fractional composition of dispersed filler particles, as well as the physical and chemical interaction at the binder-filler interface [5]. Direct and indirect optimization problems of developing new types of similar PCMs with the desired combination of stress-strain properties can be solved by means of a structural-mechanical dependence which was obtained earlier [6, 7]. Let us explore a variant of maintaining the integrity of РСМ until the sample breaks (Poisson's ratio → 0.5), which is of the most practical interest for increasing its service life, including working conditions of the rocket engine charge under domestic pressure. Earlier, in connection with PCMs, we presented a physical and mathematical description of the dependence of relative (related to the initial cross-section) stress (σ (MPa)) from stretch at break (αb(mm)) of the filled elastomer with account for its basic structural parameters [7]:       2 1 2 3 1 1 2 3 1 29 0.225 10 1 1.25 1 m ch r g m RT exp T T a                                          (1) where 3 (  / ) ch c mol cm M    is the concentration of transverse chemical bonds in the polymer binder matrix; ρ ( g/cm 3 ) – density of the polymer binder; c M – average internodal molecular weight of the 3D cross-linked polymer; φ r ( vol. fraction ) = (1 – φ sw ) – polymer volume fraction in the binder; φ sw ( vol. fraction ) – plasticizer volume fraction in the binder; R (J/K·mol) – universal gas constant; T ∞ ( К ) – equilibrium temperature, at which intermolecular interaction (the concentration of transverse physical bonds – ν ph (mol/cm 3 ) ) in the binder is negligible ( ν ph (mol/cm 3 )→ 0); T ( К )– sample test temperature; T g ( К )– structural glass-transition temperature of the polymer binder; a   ( с -1 ) – velocity shift ratio; φ ( vol. fraction ) – volume fraction of the dispersed filler; φ m ( vol. fraction ) – maximum permissible volume filler fraction depending on the shape and fractional composition of the filler particles. In practice, we can estimate the structural parameter value c M by means of a molecular graph [7]:       32 3 32 2 21 1 12 2 23 3 32 2 2 c n M f R f R f R f R f R f                        where R 1 and R 2 are molecular chains of two rubber substances with two kinds of reactive end groups f 1 and f 2 ; R 3 – a molecule of the cross-linking agent with three antipodal reactive groups – f 3 ; combinations of subscripts at f denote the chemical reaction products of i-th and j-th antipodal reactive end groups of R 1 -type and R 2 -type bifunctional polymers, as well as R3-type cross-linking trifunctional agent. Taking into account that usually R 3 << R 1 and R 2 , we can assume that c M is proportional to increment addition of the molar fraction of linear polymerization   1 12 2 n R f R    according to the molecular graph. The value of φ m ( vol. fraction ) can be calculated [8] or determined by a viscosimetric method [9]. Relative elongation ( α (mm)) is connected with deformation ( ε (%)) by a well-known ratio rating: α = 1 + ε / 100. On conditions that the integrity of the material is maintained, true stress (σ ver ( MPa )) is equal to the product of σ·α , but in practice it is more convenient to use conventional stress for comparison with research results of PCMs that do not remain integral until the sample breaks [7]. On the basis of the Eq. (1) we can write the following relation:         2 1 2 3 1 1 2 3 1 1 1 29 0.225 10     1 1.25 1 b b m ch r g m W d RT exp T T a                                                 (2) where W ( J ) , is energy (work) of mechanical fracture at uniaxial tension with dimensions: MPа  elongation ( mm ) = 1·10 3 J. The latter are the equivalent of the time during which the polymeric composite material resists increasing tensile stress. M