Elastic Properties

The following elastic models are available for defining your material elastic properties in the Define Material Properties dialog:

• Isotropic

• Transversely Isotropic

• Orthotropic

• Duncan-Chang Hyperbolic

Isotropic

An Isotropic material implies that the material properties do not vary with direction. The elastic properties of an Isotropic material are fully defined by a single value of Young’s Modulus and a single value of Poisson’s Ratio.

Transversely Isotropic

A Transversely Isotropic material has properties which vary with direction. The elastic properties are specified in two orthogonal directions as follows:

• E1 and E2 are the in-plane elastic moduli.

• Ez (the out-of-plane elastic modulus) is assumed to be equal to E1.

• The required Poisson’s ratios are n12 and n1z = nz1 = n. Note that nij implies strain in the j-direction due to strain in the i-direction. Also note that Ei / Ej = nij / nji. Therefore, the plane 1-Z is the isotropic plane

• G12 is the in-plane shear modulus

• The Angle is measured between the x-direction and the direction of E1, as shown in the figure below.

Definition of Angle for Transversely Isotropic or Orthotropic materials

NOTE: you cannot use the Transversely Isotropic elastic model in conjunction with a plasticity analysis in Phase2. If you use the Transversely Isotropic elastic model, then only an elastic analysis can be carried out (i.e. you cannot use Material Type = Plastic in Strength Parameters, in conjunction with Elastic Type = Transversely Isotropic).

Orthotropic

An Orthotropic material has properties which vary with direction. The elastic properties are specified in THREE orthogonal directions.

For an Orthotropic material three Young's moduli are required - E1, E2 and Ez - where the out-of-plane elastic modulus Ez is no longer equal to E1. Three separate Poisson’s ratios are also required, n12, n1z and n2z. The in-plane shear modulus G12 and the Angle definition are the same as for a Transversely Isotropic material.

NOTE: you cannot use the Orthotropic elastic model in conjunction with a plasticity analysis in Phase2. If you use the Orthotropic elastic model, then only an elastic analysis can be carried out (i.e. you cannot use Material Type = Plastic in Strength Parameters, in conjunction with Elastic Type = Orthotropic).

Duncan-Chang Hyperbolic

The Duncan-Chang Hyperbolic constitutive model [ Duncan and Chang, (1970) ] is widely used for the modeling of soil behaviour, and is capable of modeling the non-linear, stress-dependent and inelastic behaviour of cohesive and cohesionless soils.

The Duncan-Chang Hyperbolic model can only be used in conjunction with the Mohr-Coulomb failure criterion in Phase2.

The following input parameters are used to define the Duncan-Chang Hyperbolic model in Phase2:

Modulus Number (Ke) - this dimensionless parameter represents Young's modulus (range of values: 350-1120)

Modulus Exponent (n) - governs the stress dependence of Ke on sigma3 (range of values: 0-1)

Unloading Modulus Number (Ku) - is used to calculate the tangential modulus for unloading/reloading conditions

Failure Ratio (Rf) - defines the shape of the stress-strain curve (range of values: 0.60-0.95)

Atmospheric Pressure (Patm) - used for normalization of stress input

Poisson's Ratio (v) - can be specified as either Constant or Stress Dependent.

• If Poisson's Ratio = Constant, then Poisson's Ratio is entered directly as a constant value.

• If Poisson's Ratio = Stress Dependent, then the Bulk Modulus parameters are entered (see below).

Bulk Modulus Number (Kb) - this dimensionless parameter characterizes the volumetric change (range of values: 200-700)

Bulk Modulus Exponent (m) - governs the stress dependence of Kb on sigma3 (range of values: 0-1)

Based on a hyperbolic stress-strain curve and stress-dependent material properties, the following equations are derived [ Duncan and Chang, (1970) ] for the Duncan-Chang Hyperbolic model.

The tangential modulus (Et) for a given stress condition, is given by Eqn.1.

              Eqn.1

The tangential bulk modulus (Bt) and tangential Poisson's ratio are given by Eqn.2 and Eqn.3 respectively.

                                                    Eqn.2

                                                          Eqn.3