发布时间 : 星期五 文章粘弹性分析更新完毕开始阅读
Temperature effects温度效应
The effect of temperature, , on the material behavior is introduced through the dependence of the instantaneous stress, , on temperature and through a reduced time concept. The expression for the linear-elastic shear stress is rewritten as
where the instantaneous shear modulus
, and
is temperature dependent,
is the reduced time, defined by
where is a shift function at time t. This reduced time concept for
temperature dependence is usually referred to as thermo-rheologically simple (TRS) temperature dependence. Often the shift function is approximated by the Williams-Landell-Ferry (WLF) form:
where ,
is the reference temperature at which the relaxation data are given and
,
are calibration constants obtained at this temperature. If
deformation changes will be elastic, based on the instantaneous moduli.
The reduced time concept is also used for the volumetric behavior and the large-strain formulation. For additional information on the WLF equation, see “Viscoelasticity,” Section 4.8.1 of the ABAQUS Theory Manual.
Numerical implementation 数值执行结果
ABAQUS assumes that the viscoelastic material is defined by a Prony series expansion of the dimensionless relaxation modulus:
where N,
, and
,
, are material constants. Substitution in
the small-strain expression for the shear stress yields
where
The
are interpreted as state variables that control the stress relaxation, and
is the “creep” strain: the difference between the total mechanical strain and the instantaneous elastic strain (the stress divided by the instantaneous elastic modulus). In ABAQUS/Standard
is available as the creep strain output variable CE
(“ABAQUS/Standard output variable identifiers,” Section 4.2.1).
A similar Prony series expansion is used for the volumetric response, which is valid for both small- and finite-strain applications:
where
ABAQUS assumes that .
The Prony series expansion, in combination with the finite-strain expression for the shear stress, produces the following large-strain shear model:
where
The
are interpreted as state variables that control the stress relaxation.
If the instantaneous material behavior is defined by linear elasticity or hyperelasticity, the bulk and shear behavior can be defined
independently. However, if the instantaneous behavior is defined by the hyperfoam model, the deviatoric and volumetric constitutive behavior are coupled and it is necessary to use the same relaxation data for both behaviors.
In all of the above expressions temperature dependence is readily introduced by replacing
by
and
by
.
Determination of viscoelastic material parameters定义粘弹性材料参数
The above equations are used to model the time-dependent shear and volumetric behavior of a viscoelastic material. The relaxation parameters can be defined in one of four ways: direct specification of the Prony series parameters, inclusion of creep test data, inclusion of relaxation
test data, or inclusion of frequency-dependent data obtained from
sinusoidal oscillation experiments. Temperature effects are included in the same manner regardless of the method used to define the viscoelastic material.
ABAQUS/CAE allows you to evaluate the behavior of viscoelastic materials by automatically creating response curves based on experimental test data or coefficients. A viscoelastic material can be evaluated only if it is defined in the time domain and includes hyperelastic and/or elastic material data. See “Evaluating hyperelastic and viscoelastic material behavior,” Section 12.4.6 of the ABAQUS/CAE User's Manual. Direct specification（直接参数） The Prony series parameters
,
, and can be defined directly for
each term in the Prony series. There is no restriction on the number of terms that can be used. （应用数目没有限制）If a relaxation time is associated with only one of the two moduli, leave the other one blank or enter a zero.（如果松弛时间仅于两个模量的一个有关联，则另外一个设为1或0） The data should be given in ascending order （升序）of the relaxation time. The number of lines of data given defines the number of terms in the Prony series, N. If this model is used in conjunction with the hyperfoam（超弹泡沫材料模型） material model, the two modulus ratios have to be the same (
).
Input File Usage: *VISCOELASTIC, TIME=PRONY
The data line is repeated as often as needed to define the first, second, third, etc. terms in the Prony series.
ABAQUS/CAE UsProperty module: material editor: Mechanical
ElasticityViscoelastic: Domain: Time and Time:
age: Prony
Enter as many rows of data in the table as needed to define the first, second, third, etc. terms in the Prony series.