13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- spatial orientation β r . For volume element V it is possible to write: c Sf V V V = + , where cV is a part of volume occupied by solid, not cracked, material, and Sf V is a total volume of all micro-cracks in V . Let us introduce Sf / V V η= – the fraction of volume which is occupied by cracks. Growth of the total micro-cracks volume with rate Sf V & at the fixed total element volume V leads to decrease of the solid material volume with the rate c Sf V V V η =− =− ⋅ & & &. If h R << , than this change of solid material volume occurs predominantly due to solid material deformation in the β r direction. Corresponding deformation of substance in auxiliary coordinate system is characterized by a sole strain component: с c 1 1 V const dW V d dt V dt ββ η η = = =− − & , Transformation of this strain tensor into lab coordinates is the next: 1 1 ik i k dW d dt dt η ββ η ⎛ ⎞ =− ⎜ ⎟ ⎝ − ⎠ , (4) where ikW is symmetrical strain tensor. The use of ikW as an additional strain of solid material allows considering of the tensile stresses relaxation caused by cracks nucleation and growth. Assuming that all micro-cracks in volume element have the same size, it is possible to write for the micro-cracks volume fraction f n V η= ⋅ . The adjacent micro-cracks can coalesce forming a main crack or fractionized zone of material. We suppose that separate micro-cracks develop solitary until their diameter 2R reaches the value of an average distance between the cracks, which is equal to 1/3 n− . If the relation 1/3 2R n− ≥ is satisfied in any volume element, then material of this element is assumed to be completely fractured, and all stress components are set equal to zero in it. In the present report a uniaxial deformation of substance along Oz-axis is considered with the constant macroscopic strain rate / const zz d dt ε = . The maximal tensile stress operates along Oz-axis: ze β= r r , zz βσ σ = . The continuum mechanics equations in this case are reduced to the next set: zz zz d d dW dt dt dt ε ρ ρ ⎛ ⎞ =− ⎜ + ⎟ ⎝ ⎠ , (5) zz zz zz zz zz d dW dw dU S dt dt dt dt ε ρ σ ⎛ ⎞ = + + ⎜ ⎟ ⎝ ⎠ , (6) ( , ) zz zz P U S σ ρ =− + , (7) 4 2 3 zz zz zz zz dS d dW dw G G dt dt dt dt ε ⎛ ⎞ = + − ⎜ ⎟ ⎝ ⎠ , (8) ( ) 1 1 zz dW d dt dt η η =− − , (9) 3 2 zz n R G σ η π ⎛ ⎞ = ⋅ ⎜ ⎟ ⎝ ⎠ , (10) where zz w is the plastic distortion due to the dislocations movement; U is the specific internal energy, P is the pressure; zz S is deviator of stresses. All variables here are averaged over physically small volumes. The Eqs (5)-(10) have to be completed by the Eq. (1) and Eq. (3) for the
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