Willam-Warnke yield criterion - AbsoluteAstronomy.com

The Willam-Warnke yield criterion
Yield (engineering)
The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed...

is a function that is used to predict when failure will occur in concrete

Concrete

Concrete is a composite construction material, composed of cement and other cementitious materials such as fly ash and slag cement, aggregate , water and chemical admixtures.The word concrete comes from the Latin word...

and other cohesive-frictional materials such as rock

Rock (geology)

In geology, rock or stone is a naturally occurring solid aggregate of minerals and/or mineraloids.The Earth's outer solid layer, the lithosphere, is made of rock. In general rocks are of three types, namely, igneous, sedimentary, and metamorphic...

, soil

Soil

Soil is a natural body consisting of layers of mineral constituents of variable thicknesses, which differ from the parent materials in their morphological, physical, chemical, and mineralogical characteristics...

, and ceramic

Ceramic

A ceramic is an inorganic, nonmetallic solid prepared by the action of heat and subsequent cooling. Ceramic materials may have a crystalline or partly crystalline structure, or may be amorphous...

s. This yield criterion has the functional form

where

is the first invariant of the Cauchy stress tensor, and

are the second and third invariants of the deviatoric part of the Cauchy stress tensor. There are three material parameters (

- the uniaxial compressive strength,

- the uniaxial tensile strength,

- the equibiaxial compressive strength) that have to be determined before the Willam-Warnke yield criterion may be applied to predict failure.

In terms of

, the Willam-Warnke yield criterion can be expressed as

where

is a function that depends on

and the three material parameters and

depends only on the material parameters. The function

can be interpreted as the friction angle which depends on the Lode angle (

). The quantity

is interpreted as a cohesion pressure. The Willam-Warnke yield criterion may therefore be viewed as a combination of the Mohr-Coulomb

Mohr-Coulomb theory

Mohr–Coulomb theory is a mathematical model describing the response of brittle materials such as concrete, or rubble piles, to shear stress as well as normal stress. Most of the classical engineering materials somehow follow this rule in at least a portion of their shear failure envelope...

and the Drucker-Prager yield criteria.

Willam-Warnke yield function

In the original paper, the three-parameter Willam-Warnke yield function was expressed as

where

is the first invariant of the stress tensor,

is the second invariant of the deviatoric part of the stress tensor,

is the yield stress in uniaxial compression, and

is the Lode angle given by

The locus of the boundary of the stress surface in the deviatoric stress plane is expressed in polar coordinates by the quantity

which is given by

where

The quantities

and

describe the position vectors at the locations

and can be expressed in terms of

The parameter

in the model is given by

The Haigh-Westergaard representation

Yield surface

A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the...

of the Willam-Warnke yield condition can be
written as