Mathematical Physics - Volume II - Numerical Methods

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

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Figure 5.13: Material model with strain-softening implemented into the MCM SPH code and FE code DYNA3D.

The evolution of the damage variable was defined to give linear stress-strain behaviour following the onset of damage growth. In the model, E is the Young’s modulus of a virgin material, which defines material linear elastic behaviour. Material stiffness in the softening regime is defined by the tangent stiffness (softening part of the stress strain curve), E t . Material loading/unloading response in the softening regime was defined using the secant stiffness, denoted as ˜ E = E . The secant stiffness is equivalent to virgin material Young’s modulus. The parameters ε i and ε f define the initiation and critical failure strains, respectively, and ε ∗ denotes the current strain. The stiffness of the softening material was defined by the slope of the strain-softening part of the stress-strain curve:

E ε i ε f − ε i

E t = −

(5.98)

.

Using equation (5.80) damage variable ω can then be expressed as:

ε f ( ε ∗ − ε i ) ε ∗ ( ε f − ε i .

ω =

(5.99)

For ε ∗ = ε i , and for ε ∗ = ε f , i.e. at the point of material failure ω f = 1. A number of numerical tests were performed to illustrate the inherent nonlocal prop erties of the SPH method, where the smoothing length, in addition to its interpolation meaning, represents material characteristic length. The bar, used in the numerical experiments, has an overall length of 2 ℓ = 200 [ mm ] and a square cross section. In the SPH models symmetry planes were used to properly enforce the boundary conditons on the long edges. The origin of the coordinate system was located in the centre of the bar with the x axis aligned with the bar. All degrees of freedom except for the longitudinal direction are restricted in order to ensure uniaxial strain conditions. The material input data used in the simulations is given in Table 5.3.