Mathematical Physics - Volume II - Numerical Methods

Chapter 5. Review of Development of the Smooth Particle Hydrodynamics (SPH) Method

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The longitudinal displacement function for the linear elastic response is derived from appropriate initial and boundary conditions: u ( x , t ) = − ν t − x + L c e + ν t + x + L c e (5.89) where the expressions in the brackets ⟨· ⟩ need to be positive-definite. The corresponding strain is obtained as: where H ( · ) is the Heaviside step function. The stress state induced by this deformation is: σ x = E ( 1 − ν ) ( 1 − 2 ν )( 1 + ν ) ε x (5.91) This assumption of linear elasticity (5.89) only holds for t < L / c e , before the centre of the bar enters the strain-softening regime at response time t = L / c e , when the two stress step waves meet. At the centre of the bar, the slope of the stress-strain curve (the strain in the bar satisfies the condition: ε p / 2 < ε ≤ ε p ) becomes negative, i.e. F ′ ( ε ) < 0, and the wave speed c becomes imaginary. Consequently, the equation of motion in the softening domain becomes an elliptic PDE: Theoretically, the softening is limited to an area of zero width at x = 0. So, a disconti nuity with a displacement jump develops at this point, giving a difference in magnitude of strain ε = 4 ν ⟨ t − L / c e ⟩ . Strain increases towards infinity and stress drops to zero within the softening zone. Release waves are generated from this point and propagate into the bar. The infinite strain in the softening domain can be expressed by the Dirac Delta function δ ( x ) as: ε x = 4 ν t − L c e δ ( x ) . (5.93) The solution for the strain field outside the softening zone, t > L / c e and x < 0, (consequently in x > 0 part of the bar, due to symmetry of the problem), is then: c 2 e ∂ 2 u ∂ x 2 + ∂ 2 u ∂ t 2 = 0 with c 2 = F ′ ( ε ) ρ . (5.92) ε x = ∂ u ∂ x = ν c e H t − x + L c e + H t + x − L c e (5.90)

ν c e

H t −

c e −

H t −

c e

+ 4 c e t − L ν δ ( x ) .

x + L

L − x

ε x =

(5.94)

Following Bažant and Belytschko [8] this analytical solution was used to derive a comparison between an elastic ( ε < ε p / 2 ) and a strain-softening ( ε p / 2 < ε < ε p ) wave propagation problem. Figure 5.9 to Figure 5.10 show the solutions for longitudinal displacement, strain and stress along the bar at time t = 3 L / 2 c e , for both the elastic and strain-softening responses of a local continuum. The elastic solution represents continuous wave propagation after