where au is the ultimate (or tensile) strength of the material. For materials that exhibit brittle behavior in tension, but ductile behavior in confined compression (e.g., concrete, rocks, and soils), the Rankine criterion is sometimes combined with the Tresca or von Mises criterion to describe the failure behavior of the materials. If used in this context, the criterion is referred to as the Tresca or von Mises criterion with a tension cut-off.

2. Mohr-Coulomb criterion. This criterion is often used to describe the shear failure of soil. Failure is said to occur when a limiting shear stress reaches a value as defined by an envelope, which is expressed as a function of normal stress, soil cohesion, and friction angle. If the principal stresses are such that sP1 > sP2 > sP3, the Mohr-Coulomb criterion can be written as

where c is the cohesion and f is the angle of internal friction. 3. Drucker-Prager criterion. This criterion is an extension of the von Mises criterion, where the influence of hydrostatic stress on failure is incorporated by the addition of the term aI1, where

I1 is the first stress invariant as defined in Equations 1.13 (note that s11 + s22 + s33 = sp1 +

where a and k are material constants to be determined by curve-fitting of the above equation to experimental data.

If yielding does not signify failure of a material (which is often the case for ductile materials), the postyield behavior of the material is described by the use of a flow rule. A flow rule establishes the relative magnitudes of the components of plastic strain increment d6ipj and the direction of the plastic strain increment in the strain space. It is written as d6p = d10| (1-48)

where dl is a positive scalar factor of proportionality, g is a plastic potential in stress space, and 0g/0s,j is the gradient, which represents the direction of a normal vector to the surface defined by the plastic potential at point s,j. Equation 1.48 implies that dep is directed along the normal to the surface of the plastic potential. If the plastic potential g is equal to the yield function f, Equation 1.48 is called the associated flow rule. Otherwise, it is called the nonassociated flow rule.

Using the elastic stress-strain relationship expressed in Equation 1.39, the flow rule expressed in Equation 1.48 with g=f (i.e., associated flow rule), the consistency condition for an elastic-perfectly plastic material given by df = fdsj = 0 (1-49)

and the following relationship among total, elastic, and inelastic (plastic) strains, deij = de| + dep (1.50)

it has been shown (Chen and Han 1988) that an incremental stress-strain relationship for an elastic-perfectly plastic material that follows the associated flow rule can be written as dsj = Dip deH (1.51)

where D^kl is the incremental elastic-perfectly plastic material stiffness matrix given by

Djkl — D"kl (0f/0Srs )Drstu (0f/0Sfu) (1.52)

where Dijki (or Dijmn, Dpqki, etc.) is the indicial form of DI given in Equation 1.38.

1.4.4 Hardening Rules

If a material exhibits work-hardening behavior in which a state of stress beyond yield can exist, then in addition to the initial yield surface f a new yield surface, called subsequent yield or loading surface F, needs to be defined. Like the initial yield surface, the loading surface demarcates elastic behavior from inelastic behavior. If the stress point moves on or within the loading surface, no additional plastic strain will be induced. If the stress point is on the loading surface and the loading condition is such that it pushes the stress point out of the loading surface, additional plastic deformations will occur. When this happens, the configuration of the loading surface will change. The condition of loading and unloading for a multiaxial stress state is mathematically defined as follows.

If the stress point is on the loading surface (i.e., if F = 0), loading occurs if

and unloading occurs if rfdaij < 0 (1.54)

where nF represents a component of a unit vector that is normal to the loading surface F, that is,

For the special case when nFda j = 0, that is, the loading vector da,j is perpendicular to the corresponding component of the unit normal vector nF, a state of neutral loading is said to have occurred. Note that additional plastic strain is induced only during loading, but not during neutral loading or unloading.

According to the incremental or flow theory of plasticity, the configuration of the loading surface when loading occurs can be described by the use of a hardening rule. A hardening rule establishes a relationship between the subsequent yield stress of a material and the inelastic deformation accumulated during prior excursion into the inelastic regime. A number of hardening rules have been proposed over the years. They can often be classified into or associated with one of the following:

1. Isotropic hardening. This hardening rule assumes that during plastic deformations, the loading surface is merely an expansion, without distortion, of the initial yield surface. Mathematically, this surface is represented by the equation

where k is a constant, which is a function of the total (i.e., cumulated) plastic strain ep. Although this is one of the simplest hardening rules, it has a serious drawback in that it cannot be used to account for the Bauschinger effect, which states that the occurrence of an initial plastic deformation in one direction (e.g., in tension) will cause a reduction in material resistance to a subsequent plastic deformation in the opposite direction (e.g., in compression). Since the Bauschinger effect is present in most structural materials, the use ofisotropic hardening should be limited to problems that involve only monotonic loading in which no stress reversals will occur.

2. Kinematic hardening. This hardening rule (Prager 1955, 1956) assumes that during plastic deformation, the loading surface is formed by a simple rigid body translation (with no change in size, shape, and orientation) of the initial yield surface in stress space. Thus, the equation of the loading surface takes the form

where k is a constant to be determined experimentally and Zij are the coordinates of the centroid of the loading surface, which changes continuously throughout plastic deformation. It should be noted that contrary to isotropic hardening, kinematic hardening takes full account of the Bauschinger effect, so much so that the amount of ''loss'' of material resistance in one direction during subsequent plastic deformation is exactly equal to the amount of initial plastic deformation the material experiences in the opposite direction, which may or may not be truly reflective of real material behavior.

3. Mixed hardening. As the name implies, this hardening rule (Hodge 1957) contains features of both the isotropic and the kinematic hardening rules described above. It has the form

where Zj and k are as defined in Equations 1.56 and 1.57. In mixed hardening, the loading surface is defined by a translation (as described by the term zij) and expansion (as measured by the term k(ep)), but no change in shape, of the initial yield surface. The advantage of using the mixed hardening rule is that one can conveniently simulate different degrees of the Bauschinger effect by adjusting the two hardening parameters (z,j- and k) of the model.

1.4.5 Effective Stress and Effective Plastic Strain

Effective stress and effective plastic strain are variables that allow the hardening parameters contained in the above hardening models to be correlated with an experimentally obtained uniaxial stress-strain curve of the material. The effective stress has unit of stress, and it should reduce to the stress Sn in a uniaxial stress condition. Table 1.1 summarizes the equations for the effective stress and hardening parameter for two materials modeled using the isotropic hardening rule. The equations shown in Table 1.1 can also be used for materials modeled using the kinematic or mixed hardening rule provided that the effective stress oe is replaced by a reduced effective stress o^, computed using a reduced stress tensor given by oij - oij - Zj (1-59)

Effective plastic strain increment de^can be defined in the context of plastic work per unit volume in the form dWp = Oe dep (1.60)

By using Equation 1.48 in conjunction with a material model, it can be shown (Chen and Han 1988) that for a von Mises material d£P = ^§d£P d£P (1.61)

and for a Drucker-Prager material dep =4±£/p/L /¡Tû (1.62)

The effective stress and effective plastic strain are related by the incremental stress-strain equation doe = Hp dep (1.63)

where Hp is the plastic modulus, which is obtained as the slope of the uniaxial stress-plastic strain curve at the current value of oe.

Using the concept of effective plastic strain, flow rule, consistency condition, relationship between total, elastic, and plastic strains, elastic stress-strain relationship, and a hardening rule, it can be shown (Chen and Han 1988) that an incremental stress-strain relationship for an elastic-work-hardening material can be written in the form of Equation 1.51 with

Dep _ D _ Dijmn(Ç>g/Ç>°mn) (9F/9°W)DWH n ,

TABLE 1.1 Effective Stress

Material model

Effective stress, se

Hardening parameter, k

von Mises

0 0

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