4.2.1 Granular Materials

There are three nonlinear elastic models for granular materials, a standard Grains model, a model according to Boyce, and a model according to Jardine. See §16.2 for background theory. In addition to these models, porosity may be specified, but in practice this is only useful for clay models [§5.1.4]. It is not advisable to use porosity stand-alone.

4.2.1.1 Grains

This is the standard model for granular materials. See §16.2.1 for background theory.

    (syntax)

ELAST

GRAINS specifies the standard Grains model for granular materials.

ELAVAL

are the values to specify the nonlinear elasticity. Value g1 is the reference shear modulus G₁ and k1 is the reference compression modulus K₁ . Value n is the constant n of the degree of nonlinear elasticity. The value of n - 1 is the exponent in the Power Law which describes the dependence of the shear and compression moduli on stresses (16.11).

Value emtens is the linear Young's modulus E_t and nutens is the linear Poisson's ratio $\nu_{{\mathrm{t}}}^{}$ . Both values are for the tensile regime and used if the grain stress becomes tensile. If a $\nu$ < 0 is calculated from the reference shear and compression moduli then DIANA sets $\nu$ = 0 .

4.2.1.2 Boyce

The Boyce model, for granular materials under repeated loading, is based on the following relation between linear and nonlinear stiffness:

$$\left(\vphantom{ 1 + \nu }\right. \nu \left.\vphantom{ 1 + \nu }\right) \nu {\frac{{ 3 K_{1} - 2 G_{1} }}{{ 6 K_{1} + 2 G_{1} }}}$$ (4.4)

The implementation of the Boyce model uses a reference pressure p₀ which by default is -1.0 , but which is automatically set to -1.0 kPa if you specify table 'UNITS' with alternative units for mass, length and time. For a linear elastic analysis, you must also specify E and $\nu$ . The Boyce model can be applied in combination with Tresca, Von Mises, Mohr-Coulomb, and Drucker-Prager plasticity. See §16.2.2 for background theory.

    (syntax)

ELAST

BOYCE specifies the Boyce model for granular materials.

ELAVAL

are the values to specify the nonlinear elasticity. Value g1 is the reference shear modulus G₁ and k1 is the reference compression modulus K₁ . Value n is the constant n of the degree of nonlinear elasticity. The value of n - 1 is the exponent in the Power Law which describes the dependence of the shear and compression moduli on stresses.

    (file.dat)

'MATERI'
  1   YOUNG   638.10314
      POISON  0.282207
      ELAST   BOYCE
      ELAVAL  248.83 488.31 0.56

4.2.1.3 Jardine

The Jardine model is an elastoplastic model. It combines nonlinear elastic behavior with ideal Tresca plasticity (no hardening). The application of this model is restricted to plane strain, axisymmetric and solid elements. The Jardine model is characterized by initially stiff behavior and the stiffness decreases with increasing strain. Laboratory studies by Jardine et al. [51] have shown that even at very small strains, many soils exhibit nonlinear stress-strain behavior. For many geotechnical problems, very small strains occur in a large part of the model. See §16.2.3 for background theory. The Jardine model is defined by the following parameters.

    (syntax)

YOUNG

e is the Young's modulus E . (E > 0 )

POISON

nu is Poisson's ratio $\nu$ . ( 0$\le$$\nu$ < 0.5 ) [$\nu$ = 0.49 ]

For reasons of consistency it is recommended to define Poisson's ratio $\nu$ = 0.49 and Young's modulus E = f₁($\varepsilon_{{\mathrm{eq;min}}}^{}$) (16.21).

ELAST

JARDIN specifies the Jardine model for granular materials.

ELAVAL

are the values to specify the nonlinear elasticity [Fig.16.2]. ( 0 < C < D < E )Values c, d and e are the strains C , D and E , at maximum, medium and minimum stiffness respectively. Value f is the maximum stiffness F . (F > 0 )Value g is the medium stiffness G . (G < F )Values emin and emax are the strain bounds $\varepsilon_{{\mathrm{min}}}^{}$ and $\varepsilon_{{\mathrm{max}}}^{}$ ( C$\le$$\varepsilon_{{\mathrm{min}}}^{}$ < $\varepsilon_{{\mathrm{max}}}^{}$ )of the fitting range.

Tresca plasticity

parameters must be specified without hardening [§5.1.1].

    (file.dat)

'MATERI'
  1   YOUNG   380000.
      ELAST   JARDIN
      ELAVAL  0.0001 0.004 0.008 400000. 80000. 0.0001 0.004 
      YIELD   TRESCA
      YLDVAL  680.0