ABAQUS应变塑性模型

ε is strain, σ is stress, E is Young's modulus, and K and n are constants that depend on the material being considered. The first term on the right side, the strain, while the second term,

, is equal to the elastic part of

, accounts for the plastic

part, the parameters K and n describing the hardening behavior of the material. Introducing the yield strength of the material, σ0, and defining a new parameter, α, related to K as

, it is

convenient to rewrite the term on the extreme right side as follows:

Replacing in the first expression, the Ramberg-Osgood equation can be written as

[edit] Hardening behavior and Yield offset

In the last form of the Ramberg-Osgood model, the hardening behavior of the material depends on the material constants and . Due to the power-law relationship between stress and plastic strain, the

Ramberg-Osgood model implies that plastic strain is present even for very low levels of stress. Nevertheless, for low applied stresses and for the commonly used values of the material constants α and n, the plastic strain remains negligible compared to the elastic strain. On the other hand, for stress levels higher than σ0, plastic strain becomes progressively larger than elastic strain.

The value can be seen as a yield offset, as shown in figure 1. This

, when

.

comes from the fact that Accordingly (see Figure 1):

elastic strain at yield = plastic strain at yield =

= yield offset

Commonly used values for are ~5 or greater, although more precise values are usually obtained by fitting of tensile (or compressive) experimental data. Values for can also be found by means of fitting to experimental data, although for some materials, it can be fixed in order to have the yield offset equal to the accepted value of strain of 0.2%, which means:

Figure 1: Generic representation of the Stress-Strain curve by means of the Ramberg-Osgood equation. Strain corresponding to the yield point is the sum of the elastic and plastic components.

[edit] Sources

1. ^ Ramberg, W., & Osgood, W. R. (1943). Description of stress-strain curves by three parameters. Technical Note No. 902, National Advisory Committee For Aeronautics, Washington DC. [1] 11.2.13 Deformation plasticity

Products: ABAQUS/Standard ABAQUS/CAE

References

“Material library: overview,” Section 9.1.1  “Inelastic behavior,” Section 11.1.1  *DEFORMATION PLASTICITY



Overview

The deformation theory Ramberg-Osgood plasticity model:

is primarily intended for use in developing fully plastic solutions for fracture mechanics applications in ductile metals; and

 cannot appear with any other mechanical response material models since it completely describes the mechanical response of the material.



One-dimensional model

In one dimension the model is

where

is the stress;

is the strain;

is Young's modulus (defined as the slope of the stress-strain curve at zero stress);

is the “yield” offset;

is the yield stress, in the sense that, when and

,

;

n

is the hardening exponent for the “plastic” (nonlinear) term:

.

The material behavior described by this model is nonlinear at all stress levels, but for commonly used values of the hardening exponent ( or more) the nonlinearity becomes significant only at stress magnitudes approaching or exceeding .

Generalization to multiaxial stress states

The one-dimensional model is generalized to multiaxial stress states using Hooke's law for the linear term and the Mises stress potential and associated flow law for the nonlinear term:

where

is the strain tensor,

is the stress tensor,

is the equivalent hydrostatic stress,

is the Mises equivalent stress,

is the stress deviator, and

is the Poisson's ratio.

The linear part of the behavior can be compressible or incompressible, depending on the value of the Poisson's ratio, but the nonlinear part of the behavior is incompressible (because the flow is normal to the Mises stress potential). The model is described in detail in “Deformation plasticity,” Section 4.3.9 of the ABAQUS Theory Manual.

You specify the parameters , , , n, and directly. They can be defined as a tabular function of temperature.

Input File Usage: *DEFORMATION PLASTICITY

ABAQUS/CAE Usage: Property module: material editor: Mechanical

Deformation Plasticity

Typical applications

The deformation plasticity model is most commonly applied in static loading with small-displacement analysis, where the fully plastic

solution must be developed in a part of the model. Generally, the load is ramped on until all points in the region being monitored satisfy the condition that the “plastic strain” dominates and, hence, exhibit fully plastic behavior, which is defined as

or

You can specify the name of a particular element set to be monitored in a static analysis step for fully plastic behavior. The step will end when the solutions at all constitutive calculation points in the element set are fully plastic, when the maximum number of increments specified for the step is reached, or when the time period specified for the static step is exceeded, whichever comes first. Input File Usage: *STATIC, FULLY PLASTIC=ElsetName

ABAQUS/CAE Usage: Step module: Create Step: General: Static, General:

Other: Stop when region region is fully plastic.

Elements

Deformation plasticity can be used with any stress/displacement element in ABAQUS/Standard. Since it will generally be used for cases when the deformation is dominated by plastic flow, the use of “hybrid” (mixed formulation) or reduced-integration elements is recommended with this material model.

*DEFORMATION PLASTICITY

Specify the deformation plasticity model.

This option is used to define the mechanical behavior of a material as a deformation theory Ramberg-Osgood model. Product: ABAQUS/Standard Type: Model data Level: Model

Reference:



“Deformation plasticity,” Section 11.2.13 of the ABAQUS Analysis User's Manual

There are no parameters associated with this option.

Data lines to define deformation plasticity:

First line:

1. Young's modulus, . 2. Poisson's ratio, . 3. Yield stress, . 4. Exponent, .

5. Yield offset, . 6. Temperature.

Repeat this data line as often as necessary to define the dependence of the deformation plasticity parameters on temperature.