PSI - Issue 23

I.S. Nikitin et al. / Procedia Structural Integrity 23 (2019) 125–130 Author name / StructuralIntegrity Procedia 00 (2019) 000 – 000

126

2

(layered, block media) and with nonlinear (viscous-plastic type) slip conditions at contact boundaries can be obtained with the method of the asymptotic homogenization (Burago et al., 2019) or with the discrete variant of the slippage theory (Nikitin, 2008;Nikitin et al., 2017). In all of these cases, in the constitutive system of equations the non-linear free term and the small stress relaxation time are included. For the stable numerical solution of differential equations, the explicit-implicit method with the explicit approximation of motion equations and the implicit approximation of constitutive equations containing small parameter in the denominator of the free term was proposed. A set of effective formulas for the stress tensor correcting at the “elastic” step with f irst and second order of approximation was obtained. 2. Introduction Non-linear interaction conditions between contacting structural elements may be formulated. In the Cartesian coordinate system i x ( i =1,2,3) the unbounded elastic medium with the oriented system of periodic parallel slip planes is considered. The orientation is defined with the unit vector of normal n .The distance between slip planes is constant and equal to  . The density of the material  and Lame moduli  and  are known constants. The stress state of the medium is described with the stress tensor σ .The shear stress vector at the slip plane is equal to ,    τ =σ n-(n σ n)n the normal stress is equal to n     n σ n .We define the slip velocity vector γ and the delamination velocity vector   ω n , based on the discontinuity of tangential    V and normal   n V velocity at contact boundaries:   /   γ V = ,   / n V    . We assume the presence of thick viscous-plastic interlayers between elastic layers    , however we will not take it into account explicitly, but take it into account by using slip conditions at compressed layers boundaries. At the contact boundaries the special conditions are specified. The non-linear viscous-plastic slip is realized when 0 n   :   2 2 / 1 s F       γ τ τ , 0   Here   n u   – the normalized discontinuity of normal displacements at the contact boundary, defining by the equation   , / ( )     ,  – the viscosity coefficient, ( ) ( ) ( ) F y F y H y   - the non-linear function, that is non-zero outside the region of the yield strength s   τ , ( ) H y – the Heaviside step function, ( ) H y =0 at 0 y  , ( ) H y =1 at 0 y  . The function ( ) F  is commonly approximated with: 1 ( ) q F     , 0 q  , 2 2 / 1 s    τ . The contact plane with the interaction condition defined is called the slip-delamination plane. To construct the continuum model with a set of these slip-delamination planes we are going to deal with γ and ω as discontinuous functions of space and time. Also we will use main relationships from the slippage theory as many other authors. It allows us to take into account contributions from γ and ω into non-elastic strain tensors  e and  e respectively: ( ) / 2      e n γ γ n , 0   γ n The delamination condition is 0  , 0 n   τ = .

  ω n

(         e n ω ω n n n , ) / 2

n

If the normal to contact boundaries n is oriented along the x 2 , then its components are

s j  -

2   , where

j

j

Kronecker's symbol, s, j =1,2,3. The full strain tensor e equals to the sum of elastic and non-elastic parts