ABAQUS Analysis User's Manual
17.7.1 Time domain viscoelasticity
Products: ABAQUS/Standard ABAQUS/Explicit ABAQUS/CAE
References
?“Material library: overview,” Section 16.1.1
?“Elastic behavior: overview,” Section 17.1.1
?“Frequency domain viscoelasticity,” Section 17.7.2
?*VISCOELASTIC
?*SHEAR TEST DATA
?*VOLUMETRIC TEST DATA
?*COMBINED TEST DATA
?*TRS
Overview
The time domain viscoelastic material model:
?describes isotropic rate-dependent material behavior for materials in which dissipative losses primarily caused by “viscous”
(internal damping) effects must be modeled in the time domain;
?assumes that the shear (deviatoric) and volumetric behaviors are independent in multiaxial stress states (except when used for an elastomeric foam);
?can be used only in conjunction with “Linear elastic behavior,”
Section 17.2.1; “Hyperelastic behavior of rubberlike materials,”
Section 17.5.1; or “Hyperelastic behavior in elastomeric foams,”
Section 17.5.2, to define the elastic material properties;
?is active only during a transient static analysis (“Quasi-static analysis,” Section 6.2.5), a transient implicit dynamic analysis (“Implicit dynamic analysis using direct integration,” Section
6.3.2), an explicit dynamic analysis (“Explicit dynamic analysis,”
Section 6.3.3), a steady-state transport analysis (“Steady-state transport analysis,” Section 6.4.1), a fully coupled
temperature-displacement analysis (“Fully coupled thermal-stress analysis,” Section 6.5.4), or a transient (consolidation) coupled pore fluid diffusion and stress analysis (“Coupled pore fluid
diffusion and stress analysis,” Section 6.7.1);
?can be used in large-strain problems; and
?can be calibrated using time-dependent creep test data, time-dependent relaxation test data, or frequency-dependent cyclic test data.
Defining the shear behavior
Time domain viscoelasticity is available in ABAQUS for small-strain applications where the rate-independent elastic response can be defined with a linear elastic material model and for large-strain applications where the rate-independent elastic response must be defined with a hyperelastic or hyperfoam material model.
注释：小应变简化为弹性，大应变超弹性或超弹泡沫
Small strain（小应变）
Consider a shear test at small strain, in which a time varying shear strain,
, is applied to the material. The response is the shear stress . The viscoelastic material model defines as
where is the time-dependen t “shear relaxation modulus”（剪切松弛模量）that characterizes the material's response. This constitutive behavior can be illustrated by considering a relaxation test in which a strain is suddenly applied to a specimen
and then held constant for a long time. The beginning of the experiment, when the strain is suddenly applied, is taken as zero time, so that （t=0,初始应变突然施加）
where is the fixed strain. The viscoelastic material model is “long-t erm elastic”
（长期弹性）in the sense that, after having been subjected to a constant strain for a
very long time, the response settles down to a constant stress; i.e., as
.
The shear relaxation modulus can be written in dimensionless form:
where is the instantaneous shear modulus（瞬时剪切模量）, so that the expression for the stress takes the form
The dimensionless relaxation function has the limiting values and
.
Large strain（大应变）
The equation for the stress can be transformed by using integration by parts:
It is convenient to write this equation in the form
where is the instantaneous shear stress at time t. This form allows a
straightforward generalization（简单概括）to nonlinear elastic deformations by
replacing the linear elastic relation with the nonlinear elasticity relation
. This generalization yields a linear viscoelasticity model, in the sense that
the dimensionless stress relaxation function is independent of the magnitude of the
deformation.
In the above equation the instantaneous stress, , applied at time influences the stress, , at time t. Therefore, to create a proper finite-strain formulation, it is necessary to map the stress that existed in the configuration at time into the configuration at time t. In ABAQUS this is done by means of a mixed “push-forward” transformation
with the relative deformation gradient :
To ensure that the stress remains symmetric, ABAQUS uses the integral form:
where is the deviatoric part of the Kirchhoff stress.基尔莫夫应力偏量
The finite-strain theory is described in more detail in “Finite-strain viscoelasticity,” Section 4.8.2 of the ABAQUS Theory Manual.
Defining the volumetric behavior定义体
The volumetric behavior can be written in a form that is similar to the shear behavior:
where p is the hydrostatic pressure,（静水压力）is the instantaneous elastic
bulk modulus, is the dimensionless bulk relaxation modulus, and is the volume strain.
The above expansion applies to small as well as finite strain since the volume strains are generally small and there is no need to map the pressure from time to time t.
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 is temperature dependent,
, and 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 Us age: Property module: material editor: Mechanical Elasticity Viscoelastic: Domain: Time and Time: Prony
Enter as many rows of data in the table as needed to define the first, second, third, etc. terms in the Prony series.
Creep test data
If creep test data are specified, ABAQUS will calculate the terms in the Prony series automatically. The normalized shear and bulk compliances are
defined as
where
is the shear compliance, is the total shear strain, and
is the constant shear stress in a shear creep test; is the
volumetric compliance,
is the total volumetric strain, and is the constant
pressure in a volumetric creep test. At time
, . The creep data are converted to relaxation data through the convolution
integrals
ABAQUS then uses the normalized shear modulus
and normalized bulk
modulus
in a nonlinear least-squares fit to determine the Prony series
parameters.
Using the shear and volumetric test data consecutively The shear test data and volumetric test data can be used consecutively to define the normalized shear and bulk compliances as functions of time.
A separate least-squares fit is performed on each data set; and the two derived sets of Prony series parameters, and , are merged into one set of parameters,
. Input File Usage:
Use the following three options. The first option is required. Only one of the second and third options is
required.
*VISCOELASTIC , TIME=CREEP TEST DATA *SHEAR TEST DATA
*VOLUMETRIC TEST DATA
ABAQUS/CAE Us age: Property module: material editor: Mechanical Elasticity Viscoelastic: Domain: Time and Time: Creep test data
In addition, select one or both of the following:
Test Data Shear Test Data
Test Data Volumetric Test Data
Using the combined test data
Alternatively, the combined test data can be used to specify the normalized shear and bulk compliances simultaneously as functions of time.
A single least-squares fit is performed on the combined set of test data to determine one set of Prony series parameters, .
Input File Usage: Use both of the following options:
*VISCOELASTIC, TIME=CREEP TEST DATA
*COMBINED TEST DATA
ABAQUS/CAE Us age: Property module: material editor: Mechanical Elasticity Viscoelastic: Domain: Time, Time: Creep test data, and Test Data Combined Test Data
Relaxation test data
As with creep test data, ABAQUS will calculate the Prony series parameters automatically from relaxation test data.
Using the shear and volumetric test data consecutively
Again, the shear test data and volumetric test data can be used consecutively to define the relaxation moduli as functions of time. A separate nonlinear least-squares fit is performed on each data set; and
the two derived sets of Prony series parameters, and , are merged into one set of parameters, .
Input File Usage: Use the following three options. The first option is
required. Only one of the second and third options is
required.
*VISCOELASTIC, TIME=RELAXATION TEST DATA
*SHEAR TEST DATA
*VOLUMETRIC TEST DATA
ABAQUS/CAE Us age: Property module: material editor: Mechanical Elasticity Viscoelastic: Domain: Time and Time: Relaxation test data
In addition, select one or both of the following:
Test Data Shear Test Data
Test Data Volumetric Test Data
Using the combined test data
Alternatively, the combined test data can be used to specify the relaxation moduli simultaneously as functions of time. A single least-squares fit is performed on the combined set of test data to
determine one set of Prony series parameters, .
Input File Usage: Use both of the following options:
*VISCOELASTIC, TIME=RELAXATION TEST DATA
*COMBINED TEST DATA
ABAQUS/CAE Us age: Property module: material editor: Mechanical Elasticity Viscoelastic : Domain: Time , Time:
Relaxation test data , and Test Data Combined Test
Data
Frequency-dependent test data
The Prony series terms can also be calibrated using frequency-dependent test data. In this case ABAQUS uses analytical expressions that relate the Prony series relaxation functions to the storage and loss moduli. The expressions for the shear moduli, obtained by converting the Prony series terms from the time domain to the frequency domain by making use of Fourier
transforms, can be written as follows:
where
is the storage modulus, is the loss modulus, is the
angular frequency , and N is the number of terms in the Prony series. These
expressions are used in a nonlinear least-squares fit to determine the Prony series
parameters from the storage and loss moduli cyclic test data obtained at M frequencies
by minimizing the error function :
where
and
are the test data and
and , respectively, are the
instantaneous and long-term shear moduli. The expressions for the bulk moduli,
and , are written analogously .
The frequency domain data are defined in tabular form by giving the real and imaginary parts of and —where is the circular frequency —as functions of frequency in cycles per time. is the Fourier transform
of the nondimensional shear relaxation function
. Given the frequency-dependent storage and loss moduli , , , and , the real and imaginary parts
of and are then given
as
where
and are the long-term shear and bulk moduli determined from the elastic or hyperelastic properties.
Input File Usage: *VISCOELASTIC , TIME=FREQUENCY DATA
ABAQUS/CAE Us age: Property module: material editor: Mechanical Elasticity Viscoelastic : Domain: Time and Time:
Frequency data
Calibrating the Prony series parameters
You can specify two optional parameters related to the calibration of Prony series parameters for viscoelastic materials: the error tolerance and . The error tolerance is the allowable average root-mean-square error of data points in the least-squares fit, and its default value is 0.01. is the maximum number of terms N in the Prony series, and its default (and maximum) value is 13. ABAQUS will perform the least-squares fit from to until convergence is achieved for the lowest N with respect to the error tolerance.
The following are some guidelines for determining the number of terms in the Prony series from test data. Based on these guidelines, you can
choose
.
?
There should be enough data pairs for determining all the parameters in the Prony series terms. Thus, assuming that N is the number of Prony series terms, there should be a total of at least data points in shear test data, data points in volumetric test
data, data points in combined test data, and data points in the
frequency domain.
? The number of terms in the Prony series should be typically not more
than the n umber of logarithmic “decades” spanned by the test data.
The number of logarithmic “decades” is defined as , where and are the maximum and minimum
time in the test data, respectively.
Input File Usage: *VISCOELASTIC , ERRTOL=error_tolerance , NMAX=
ABAQUS/CAE Us age: Property module: material editor: Mechanical Elasticity Viscoelastic : Domain: Time ; Time:
Creep test data , Relaxation test data , or Frequency
data ;
Maximum number of terms in the Prony series:
; and Allowable average root-mean-square
error: error_tolerance
Temperature effects
Regardless of the method used to define the viscoelastic behavior, thermo-rheologically simple temperature effects can be included by specifying the method used to define the shift function. The shift function can be defined by the Williams-Landell-Ferry (WLF)
approximation.
Input File Usage:
In ABAQUS/Standard use the following option:
*TRS , DEFINITION=WLF
In ABAQUS/Explicit use the following option: *TRS
ABAQUS/CAE Us age: Property module: material editor: Mechanical Elasticity Viscoelastic : Domain: Time , Time: any
method , and Suboptions Trs : Shift function: WLF
User subroutine specification in ABAQUS/Standard In ABAQUS/Standard the shift function can alternatively be specified in user subroutine UTRS .
Input File Usage: *TRS, DEFINITION=USER
ABAQUS/CAE Us age: Property module: material editor: Mechanical Elasticity Viscoelastic: Domain: Time, Time: any method, and Suboptions Trs: Shift function: User subroutine UTRS
Defining the rate-independent part of the material response
In all cases elastic moduli must be specified to define the
rate-independent part of the material behavior. Small-strain linear elastic behavior is defined by an elastic material model (“Linear elastic behavior,” Section 17.2.1), and large-deformation behavior is defined by
a hyperelastic (“Hyperelastic behavior of rubberlike materials,” Section
17.5.1) or hyperfoam (“Hyperelastic behavior in elastomeric foams,” Section 17.5.2) material model. The rate-independent elasticity for any of these models can be defined in terms of either instantaneous elastic moduli or long-term elastic moduli. The choice of defining the elasticity in terms of instantaneous or long-term moduli is a matter of convenience only; it does not have an effect on the solution.
The effective relaxation moduli are obtained by multiplying the instantaneous elastic moduli with the dimensionless relaxation functions as described below.
Linear elastic materials
For linear elastic material behavior
and
where and are the instantaneous shear and bulk moduli determined from
the values of the user-defined instantaneous elastic moduli and :
and .
If long-term elastic moduli are defined, the instantaneous moduli are determined from
Hyperelastic materials
For hyperelastic material behavior the relaxation coefficients are applied either to the constants that define the energy function or directly to the energy function. For the polynomial function and its particular cases (reduced polynomial, Mooney-Rivlin, neo-Hookean, and Yeoh)
for the Ogden function
for the Arruda-Boyce and V an der Waals functions
and for the Marlow function
For the coefficients governing the compressible behavior of the polynomial
models and the Ogden model
for the Arruda-Boyce and V an der Waals functions
and for the Marlow function
If long-term elastic moduli are defined, the instantaneous moduli are determined from
while the instantaneous bulk compliance moduli are obtained from
for the Marlow functions we have
Elastomeric foams
For elastomeric foam material behavior the instantaneous shear and bulk relaxation coefficients are assumed to be equal and are applied to the material constants in the energy function:
If only the shear relaxation coefficients are specified, the bulk relaxation coefficients are set equal to the shear relaxation coefficients and vice versa. If both shear and bulk relaxation coefficients are specified and they are unequal, ABAQUS issues an error message.
If long-term elastic moduli are defined, the instantaneous moduli are determined from
Material response in different analysis procedures
The time-domain viscoelastic material model is active during the following procedures:
?transient static analysis (“Quasi-static analysis,” Section
6.2.5),
?transient implicit dynamic analysis (“Implicit dynamic analysis using direct integration,” Section 6.3.2),
?explicit dynamic analysis (“Explicit dynamic analysis,” Section
6.3.3),
?steady-state transport analysis (“Steady-state transport analysis,” Section 6.4.1),
?fully coupled temperature-displacement analysis (“Fully coupled thermal-stress analysis,” Section 6.5.4), and
?transient (consolidation) coupled pore fluid diffusion and stress analysis (“Coupled pore fluid diffusion and stress analysis,”
Section 6.7.1).
Viscoelastic material response is always ignored in a static analysis. It can also be ignored in a coupled temperature-displacement analysis or in a soils consolidation analysis by specifying that no creep or viscoelastic response is occurring during the step even if creep or viscoelastic material properties are defined (see “Fully coupled thermal-stress analysis,” Section 6.5.4, or “Coupled pore fluid diffusion and stress analysis,” Section 6.7.1). In these cases it is assumed that
the loading is applied instantaneously, so that the resulting response corresponds to an elastic solution based on instantaneous elastic moduli.
ABAQUS/Standard also provides the option to obtain the fully relaxed long-term elastic solution directly in a static or steady-state transport analysis without having to perform a transient analysis. The long-term value is used for this purpose. The viscous damping stresses (the internal stresses associated with each of the Prony-series terms) are increased gradually from their values at the beginning of the step to their long-term values at the end of the step if the long-term value is specified.
Material options
The viscoelastic material model must be combined with an elastic material model. It is used with the isotropic linear elasticity model (“Linear elastic behavior,” Section 17.2.1) to define classical, linear,
small-strain, viscoelastic behavior or with the hyperelastic (“Hyperelastic behavior of rubberlike materials,” Section 17.5.1) or hyperfoam (“Hyperelastic behavior in elastomeric foams,” Section 17.5.2) models to define large-deformation, nonlinear, viscoelastic behavior. The elastic properties defined for these models can be temperature dependent.
Viscoelasticity cannot be combined with any of the plasticity models or with the Mullins effect material model. See “Combining material behaviors,” Section 16.1.3, for more details.
Elements
The time domain viscoelastic material model can be used with any stress/displacement or coupled temperature-displacement element in ABAQUS.
Output
In addition to the standard output identifiers available in ABAQUS (“ABAQUS/Standard output variable identifiers,” Section 4.2.1, and “ABAQUS/Explicit output variable identifiers,” Section 4.2.2), the
following variables have special meaning in ABAQUS/Standard if viscoelasticity is defined:
EE Elastic strain corresponding to the stress state at time t and the instantaneous elastic material properties.
CE Equivalent creep strain defined as the difference between the total strain and the elastic strain.
These strain measures are used to approximate the strain energy, SENER, and the viscous dissipation, CENER. These approximations may lead to underestimation of
the strain energy and overestimation of the viscous dissipation since the effects of internal stresses on these energy quantities are neglected. This can be particularly noticeable in the case of non-monotonic loading.
For the case of large-strain viscoelasticity, ABAQUS/Explicit does not perform the computation of the viscous dissipation for performance reasons. Instead, the contribution of viscous dissipation is included in the strain energy output, SENER; and CENER is output as zero. Consequently, special care must be exercised when interpreting strain energy results of large-strain viscoelastic materials in ABAQUS/Explicit since they include viscous dissipation effects.
第二节高分子材料的高弹性和粘弹性 本章第二、三节介绍高分子材料力学性能。力学性能分强度与形变两大块，强度指材料抵抗破坏的能力，如屈服强度、拉伸或压缩强度、抗冲击强度、弯曲强度等；形变指在平衡外力或外力矩作用下，材料形状或体积发生的变化。对于高分子材料而言，形变可按性质分为弹性形变、粘性形变、粘弹性形变来研究，其中弹性形变中包括普通弹性形变和高弹性形变两部分。 高弹性和粘弹性是高分子材料最具特色的性质。迄今为止，所有材料中只有高分子材料具有高弹性。处于高弹态的橡胶类材料在小外力下就能发生100-1000%的大变形，而且形变可逆，这种宝贵性质使橡胶材料成为国防和民用工业的重要战略物资。高弹性源自于柔性大分子链因单键内旋转引起的构象熵的改变，又称熵弹性。粘弹性是指高分子材料同时既具有弹性固体特性，又具有粘性流体特性，粘弹性结合产生了许多有趣的力学松弛现象，如应力松弛、蠕变、滞后损耗等行为。这些现象反映高分子运动的特点，既是研究材料结构、性能关系的关键问题，又对正确而有效地加工、使用聚合物材料有重要指导意义。 一、高弹形变的特点及理论分析 （一）高弹形变的一般特点 与金属材料、无机非金属材料的形变相比，高分子材料的典型高
弹形变有以下几方面特点。 1、小应力作用下弹性形变很大，如拉应力作用下很容易伸长100%~1000%（对比普通金属弹性体的弹性形变不超过1%）；弹性模量低，约10-1~10MPa（对比金属弹性模量，约104~105MPa）。 2、升温时，高弹形变的弹性模量与温度成正比，即温度升高，弹性应力也随之升高，而普通弹性体的弹性模量随温度升高而下降。 3、绝热拉伸（快速拉伸）时，材料会放热而使自身温度升高，金属材料则相反。 4、高弹形变有力学松弛现象，而金属弹性体几乎无松弛现象。 高弹形变的这些特点源自于发生高弹性形变的分子机理与普弹形变的分子机理有本质的不同。 （二）平衡态高弹形变的热力学分析 取原长为l0的轻度交联橡胶试样，恒温条件下施以定力f，缓慢拉伸至l0+ d l 。所谓缓慢拉伸指的是拉伸过程中，橡胶试样始终具有热力学平衡构象，形变为可逆形变，也称平衡态形变。 按照热力学第一定律，拉伸过程中体系内能的变化d U为： dU- = dQ dW （4-13） 式中d Q为体系吸收的热量，对恒温可逆过程，根据热力学第二定律有， dQ= TdS （4-14）
粘弹性功能梯度有限元法 摘要：有效离散的问题域的能力，使一个有吸引力的仿真技术的有限元方法造型复杂的边界值问题，如沥青混凝土路面材料非均匀性。专门―分级元素‖已被证明是提供高效，准确的功能梯度材料的模拟工具。以前的研究一直局限于功能梯度材料数值模拟弹性材料的行为。因此，当前的工作重点是对功能梯度材料的粘弹性材料有限元分析。在执行分析，使用弹性-粘弹性对应原理，和粘弹性材料的级配占内的元素广义ISO参数化配方。本文强调粘弹性沥青混凝土路面和几个例子的行为，包括核查问题领域的大规模应用，提交证明本办法的特点。DOI: 10.1061/_ASCE_MT.1943-5533.0000006 CE数据库标题：粘弹性;沥青路面混凝土路面;有限元方法。 关键词：粘弹性功能梯度材料，沥青路面，有限元法;通信原则。 概况 功能梯度材料(FGMs_)的特点是空间创建非均匀分布的各种微观结构巩固阶段将具有不同属性的大小和形状、，以及，通过转乘的加固作用和连续的方式(Suresh 和莫滕森基质材料)。他们通常被设计成产生财产渐变旨在优化下不同类型的结构响应加载条件(thermal，机械、电气、光学、etc)。(Cavalcante et al.2007)。这些属性渐变是在生产几种方法，例如通过循序渐进的含量变化相对于另metallic)，采用热的一个阶段ceramic障涂层，或通过使用数量足够多具有不同的属性(Miyamoto et al 的构成阶段。1999_可以根据定制设计器粘弹性FGMs (VFGMs)符合设计要求等作用下粘弹性柱轴向和热加载(Hilton 2005)。最近，Muliana(2009_)提出了黏弹性细观力学模型FGMs 的行为。除了设计或量身定制的功能梯度材料，几个土木工程材料的自然表现出梯度材料的性能。席尔瓦等人。(2006)已研究和仿真竹子，这是一个自然发生的梯度材料。除了自然发生，各种材料和结构呈现非均质物质的分布和构成属性层次生产或建设的做法，老龄化的结果，不同金额暴露恶化代理商，等沥青混凝土路面是一个这样的例子，即老龄化和温度变化产量连续分级的非齐次构性质。老化和温度引起的财产梯度已经有据可查的一些研究人员沥青路面1995年_garrick领域;米尔扎和witczak的1996年，2006年apeagyei; chiasson等。2008_。目前沥青路面粘弹性模拟状态限于要么忽视非均质财产梯度2002年_kim和buttlar;萨阿德等。2006年，2006年BAEK和AL-卡迪;戴夫等。，2007_或者他们考虑通过分层的方法，例如，在美国的关联模型国家公路和运输官员_aashto_机械经验路面设计指南_mepdg_ _araINC。，EC。2002_。精度从使用的重大损失沥青路面层状弹性分析方法有被证明_buttlar等。2006_。广泛的研究已经进行了高效，准确地模拟功能梯度材料。例如，cavalcante等人。_2007_，张和保利诺_2007_，arciniega雷迪_2007_，歌曲和保利诺_2006_都报道功能梯度材料的有限元模拟。然而，大多数的以前的研究一直局限于弹性材料行为。一各种土木工程材料，如聚合物，沥青混凝土，水泥混凝土等，表现出显著的速率和历史影响。这些类型的材料的精确模拟必须使用粘弹性本构模型。1postdoctoral副研究员，DEPT。土木与环境工程大学。伊利诺伊大学厄巴纳- 香槟分校，分校，IL 61801_corresponding author_。工程，系2donald BIGGAR威利特教授。公民权利和环境工程，大学。在厄巴纳香槟分校，伊利诺伊州，IL 61801。3professor和narbey哈恰图良的教师学者，部。民间 与环境工程，大学。位于Urbana-Champaign的伊利诺斯州，分校，IL 61801。 注意:这个手稿于2009年4月17日完成，2009年10月15日提交了批准，2010年2月5日在线发表。直到2011年6月1日，讨论期间打开，必须提交单独讨论个别文件。本文是在民事部分的材料杂志 工程，第一卷。23，没有。1，2011年1月1日起，。ASCE，ISSN 0899-1561 /2011/1-39-48 / $ 25.00。土木工程材料杂志?ASCE / 2011年1月/ 39到2012年，下载03 61.178.77.85。再分配受ASCE许可证或版权。访问https://www.doczj.com/doc/b26789506.html,当前工作提出有限元_fe_的制定专为粘弹性功能梯度材料的分析，特别是沥青混凝土。Paulino和金_2001_探索elasticviscoelastic对应范围内的原则_cp_功能梯度材料。在目前已使用制定基于CP-结合广义的ISO参数制定的研究_gif_金保利诺_2002_。本文提出了有限元的制定，验证，和沥青的详情路面模拟的例子。除了模拟沥青人行道，目前的做法也可以被用于其他工程系统表现出梯度的粘弹性分析行为。这种系统的例子包括金属和在高温_billotte等金属复合材料。二零零六年; koric和托马斯的2008_;聚合物和塑料的系统，经过氧化和/或紫外线硬化_hollaender等。1995年海尔等。1997_和分级纤维增强水泥混凝土结构。分级粘弹性的其他应用领域分析包括精确的模拟接口层之间的接口，如粘弹性材料之间不同的沥青混凝土升降机或模拟的
ANSYS 粘弹性材料 1.1 ANSYS 中表征粘弹性属性问题 粘弹性材料的应力响应包括弹性部分和粘性部分，在载荷作用下弹性部分是即时响应的，而粘性部分需要经过一段时间才能表现出来。一般的，应力函数是由积分形式给出的，在小应变理论下，各向同性的粘弹性本构方程可以写成如下形式： () ()0 02t t de d G t d I K t d d d σττττττ ?=-+-?? (1) 其中 σ＝Cauchy 应力 ()G t ＝为剪切松弛核函数 ()K t ＝为体积松弛核函数 e ＝为应变偏量部分（剪切变形） ?＝为应变体积部分（体积变形） t ＝当前时间 τ＝过去时间 I ＝为单位张量。 该式是根据松弛条件本构方程(1)，通过将一点的应变分解为应变球张量（体积变形）和应变斜张量（剪切变形）两部分，推导而得的。这里不再敖述，可参考相关文献等。 ANSYS 中描述粘弹性积分核函数()G t 和()K t 参数表示方式主要有两种，一种是广义Maxwell 单元（VISCO88 和 VISCO89）所采用的Maxwell 形式，一种是结构单元所采用的Prony 级数形式。实际上，这两种表示方式是一致的，只是具体数学表达式有一点点不同。 1.2 Prony 级数形式 用Prony 级数表示粘弹性属性的基本形式为： ()1exp G n i G i i t G t G G τ∞=?? =+- ??? ∑ (2) ()1exp K n i K i i t K t K K τ∞=?? =+- ??? ∑ (3) 其中，G ∞和i G 是剪切模量，K ∞和i K 是体积模量，G i τ和K i τ是各Prony 级数分量的松弛时间(Relative time)。再定义下面相对模量(Relative modulus) 0G i i G G α= (4) 0K i i K K α= (5) 其中，0G ，0K 分别为粘弹性材质的瞬态模量，并定义式如下：
第7章聚合物的粘弹性 7.1基本概念 弹：外力→形变→应力→储存能量→外力撤除→能量释放→形变恢复 粘：外力→形变→应力→应力松驰→能量耗散→外力撤除→形变不可恢复 理想弹性： 服从虎克定律 σ＝E·ε 应力与应变成正比，即应力只取决于应变。 理想粘性：服从牛顿流体定律 应力与应变速率成正比，即应力只取决于应变速率。 总结：理想弹性体理想粘性体 虎克固体牛顿流体 能量储存能量耗散 形状记忆形状耗散 E＝E(σ.ε.T) E＝E(σ.ε.T.t) 聚合物是典型的粘弹体，同时具有粘性和弹性。 E＝E(σ.ε.T.t) 但是高分子固体的力学行为不服从虎克定律。当受力时，形变会随时间逐渐发展，因此弹性模量有时
间依赖性，而除去外力后，形变是逐渐回复，而且往往残留永久变形(γ∞)，说明在弹性变形中有粘流形变发生。 高分子材料(包括高分子固体，熔体及浓溶液)的力学行为在通常情况下总是或多或少表现为弹性与粘性相结合的特性，而且弹性与粘性的贡献随外力作用的时间而异，这种特性称之为粘弹性。粘弹性的本质是由于聚合物分子运动具有松弛特性。 7.2聚合物的静态力学松弛现象 聚合物的力学性质随时间的变化统称为力学松弛。高分子材料在固定应力或应变作用下观察到的力学松弛现象称为静态力学松弛，最基本的有蠕变和应力松弛。 (一)蠕变 在一定温度、一定应力的作用下，聚合物的形变随时间的变化称为蠕变。 理想弹性体：σ＝E·ε。 应力恒定，故应变恒定，如图7-1。 理想粘性体，如图7－2， 应力恒定，故应变速率为常数，应变以恒定速率增加。
图7-3 聚合物随时间变化图 聚合物：粘弹体，形变分为三个部分; ①理想弹性，即瞬时响应：则键长、键角提供； ②推迟弹性形变，即滞弹部分：链段运动 ③粘性流动：整链滑移 注：①、②是可逆的，③不可逆。 总的形变: (二)应力松弛 在一定温度、恒定应变的条件下，试样内的应力随时间的延长而逐渐减小的现象称为应力松弛。 理想弹性体：，应力恒定，故应变恒定 聚合物： 由于交联聚合物分子链的质心不能位移，应力只能松弛到平衡值。
在国内较早地采用粘弹力学手段与方法开展道路用沥青及沥青混合料力学性能研究，成果先后二次获国家科技进步二等奖，省部级科技进步一等奖。在沥青路面材料与路面结构工程特性研究方面取得较大进展，提出的沥青路面低温开裂能量判据、沥青混合料与沥青路面疲劳损伤的能量判据、沥青混合料CAVF设计法、FAC沥青混合料设计技术等在国内具有较大影响。近年来，倡导采用图像数字技术和离散颗粒单元法研究沥青混合料内部结构与力学性能、探索沥青混合料的计算机虚拟力学试验方法研究，具有显著的技术创新性。此外，为了丰富公路与城市道路建设技术，正在组织展开公路景观评价与设计、铺装景观、预防性养护与路面资产管理等方面的 研究。 为祖国铺发展大道为人民筑致富之路 记沥青路面专家张肖宁教授 柳絮恒王辉 2003年，我国政府宣布，中国高速公路里程跃居世界第二位，高速公路建设规模与速度举世瞩目。沥青路面是高速公路最主要的结构类型，是直接影响高速公路使用寿命与性能的重要结构组成。我国目前沥青混合料年产量约为1.5亿吨，及道路石油沥青生产炼制，年产值约为800亿元。沥青混合料生产与沥青路面铺设已经成为国民经济的重要组成领域，并且逐年迅猛增长。 1983年，张肖宁师从著名的沥青路面专家、日本北海道大学教授菅原照雄，是这位性格古怪的教授指导的最后一位外国留学生。在日学习两年，使张肖宁深知，沥青与沥青混合料是典型的粘弹性材料，研究这些用于特殊结构的材料在自然环境与汽车荷载作用下的复杂力学行为，从中找出规律来指导沥青路面材料设计、性能评价和寿命预测，不啻是一条无比艰难的不归路。 20年过去了，在应用与发展粘弹性力学理论与方法研究沥青与沥青混合料力学行为和路用性能的科研领域，张肖宁一路汗水，一路收获。1991年，他出版了我国第一部以具体工程材料作为研究对象的流变学专著《实验粘弹原理》。他先后参加了国家七五、八五重点科技项目，承担完成国家自然科学基金项目4项，国家教委优秀青年教师基金资助项目1项。他先后两次获得国家科技进步二等奖，获得多项省部级科技进步奖励。在《力学学报》、日本土木工程学会论文报告集等国内外重要学术刊物上发表论文150余篇。1992年破格晋升为教授。1998年，日本长冈技术科学大学以论文形式授予他工学博士学位。 上个世纪90年代中期，我国进入高速公路建设的加速阶段，沥青路面建设领域暴露出许多亟待解决的技术难题。与此同时，美国公布了公路战略研究计划（StrategicHighwayResearchProgram，简称SHRP）的主要研究成果，SHRP始终坚持采用粘弹性力学的方法与手段研究沥青及沥青
这学期新学了一门课：粘弹性力学。以前在本科阶段没有接触过有关弹性和粘弹性力学方面的知识，学起来感觉有些抽象。弹性力学和我们之前所学过的材料力学、结构力学的任务一样，都是分析各种结构或其构件在弹性阶段的应力和位移，校核它们是否具有所需的强度、刚度和稳定性，并且寻求或改进它们的计算方法。然而，它们还是略有不同的。 在以前所学的材料力学中，研究对象主要是杆状构件。材料力学的主要研究内容是这种杆状构件在拉压、剪切、弯曲、扭转作用下的应力和位移。而结构力学则是在材料力学内容的基础上研究由杆状构件所组成的结构，诸如桁架、钢架等。若研究一些非杆状构件，此时就需要运用弹性力学的知识，当然，弹性力学同样适用于杆状构件的研究计算。 虽然材料力学和弹性力学都可以对杆状构件进行分析，但两者的研究方法却是不大相同的。在材料力学的研究中，除了从静力学、几何学、物理学三方面进行分析外，大都会引用一些关于构件的形变状态或者应力分布的假定，这种假定就使得数学推演变得简化了，所以有时得到的答案只是近似解而不是精确解。这种假定在弹性力学中一般是不引用的，在我们这学期所学的有关弹性力学的知识中，只用精确的数学推演而不引用关于形变状态或应力分布的假定，所以结果较材料力学而言更为精确。 通过对以前学过的力学课程对比，能够更好地了解到弹性力学的一些特点，下面我将说一些自己对弹性力学的了解。 在这学期的弹性力学课程中，我们主要从认识弹性力学出发，然后学习了一些基本理论。比如平面应力与平面应变、平衡微分方程、几何方程、物理方程以及边界条件等。然后由这些基本理论出发，对直角坐标系和极坐标系下的平面问题进行解答，了解到了在平面问题中弹性力学的运用。继而学习到了空间问题的一些基本理论 弹性力学主要运用到的基本概念有外力、应力、形变和位移。作用于物体的外力可分为体积力和表面里，可简称为体力和面力。其中体力是分布在物体体积内的力，如重力和惯性力。面力则是分布在物体表面上的力，如流体压力和接触力。物体受到了外力的作用或者由于温度有所改变，物体内部将会发生内力。而应力，其作用在截面的法向量和切向量，也就是正应力和切应力，是和物体的形
得分：_______ 南京林业大学 研究生课程论文2013 ～2014 学年第一学期 课程号：43311 课程名称：高级木材学（含竹材） 论文题目：木材粘弹性 专业：材料学 学号：3130161 姓名：王礼建 任课教师：张耀丽（教授） 二0一三年十二月
木材的粘弹性 王礼建 （南京林业大学木材工业学院，江苏南京210037） 摘要：粘弹性是天然高分子材料固有的一种性质，本文通过对粘弹性过程的概述，分析了温度、含水率等对木材蠕变、应力松弛、滞后、内耗的影响。分析了木材粘弹性产生的原因和研究方法。 关键词：木材、粘弹性、蠕变、应力松弛、滞后、内耗 The study on the viscoelasticity of Wood WANG Li-jian (College of Wood Science and Technology, Nanjing Forestry University, 210037 Nanjing, China) Abstract：Viscoelasticity is a natural inherent nature of polymer materials, this paper outlines the process for viscoelasticity analysis of temperature, moisture content and any other factors of wood creep, stress relaxation, hysteresis, friction effects. The causes of wood viscoelasticity and research methods were analyzed. Key words: wood, viscoelasticity, creep, stress relaxation, hysteresis, friction 前言 木材作为一种生物质高分子材料同时具有弹性和粘性两种不同机理的变形，木材的这种性质称为粘弹性。粘弹性可分为静态粘弹性和动态粘弹性，静态粘弹性包括蠕变和应力松弛；动态粘弹性是指在交变的应力、应变作用下发生的滞后现象和力学损耗[1]。木材在加工或使用过程中往往受到交变应力或交变应变的作用，如木材作为结构件在枕木和桥梁的使用中、木材作为乐器的面板在弹奏中、木材作为减震阻尼材料在吸振中。木材的粘弹性影响着它的加工条件和使用条件，还可以用它来评价木材的减震特性。本文通过对粘弹性过程的概述，分析了温度、含水率等对木材蠕变、应力松弛、滞后、内耗的影响。分析了木材粘弹性产生的原因和研究方法。 1 粘弹性 粘弹性[2]材料在外力条件下将产生应变。理想弹性固体（虎克弹性体）的行为服从虎克定律，应力与应变呈线性关系。受外力时平衡应变瞬时到达，去除外
土体动本构模型的研究现状 土体实际动本构关系是极其复杂的,它在不同的荷载条件、土性条件及排水条件下表现出极不相同的动本构特性. 要建立一个能适用于各种不同条件的动本构模型的普遍形式是不切实际的,其切实的方法是对于不同的工程问题,应该根据土体的不同要求和具体条件,有选择地舍弃部分次要因素,保留所有主要因素,建立一个能反映实际情况的动本构模型. 目前,具体建立的动本构模型已达数十个,大致可分为两大类,即粘弹性模型和弹塑性模型.曲线模型,均属于等效线性模型[2 ] 。Masing 类模型以曲线Hardin Drnevich 或Ram2berg Osgood 曲线等为骨干,改用瞬时剪切模量代替前面的平均剪切模量。为使这类动本构模型更接近实测的动应力应变曲线,很多学者做了大量的工作,以使其能够描述不规则循环荷载作用下土的动本构关系[3 ] 。Iwan 用一系列具有不同屈服水平的理想弹塑性元件来描述土的动本构关系,它分串联型和并联型2 种构成方式。串联型和并联型的伊万模型所描述的动应力应变特性基本上一致,只是前者以应变为自变量,后者以应力为自变量[4 ] 。郑大同在伊万模型的基础上,提出了一个新物理模型,该模型的骨架曲线可为加工硬化状,也可为加工软化状,骨架曲线与滞回曲线的2 个分支既可相同,也可不同[5 ] 。一般的粘弹性模型不能计算永久变形(残余变 形) ,在主要为弹性变形的情况下比较合适。但实际上,土在往复荷载作用下还会因土粒相互滑移,形成新的排列而产生不可恢复的永久变形。为此,Mar2tin 等人根据等应变反复单剪试验结果,提出了循环荷载作用下永久体积应变的增量公式[6 ] 。后来,日本学者八木、大冈和石桥等分别由等应力动单剪试验及扭剪试验各自提出了计算永久体积应变增量的经验公式。国内的姜朴、徐亦敏、娄炎根据动三轴试验应变与破坏振次的关系式。沈珠江[7 ] 对等价粘 弹性模型进行了较全面的研究,认为一个完整的粘弹性模型应该包含4 个经验公式: (1) 平均剪切模量; (2) 阻尼比; (3) 永久体积应变增量和永久剪切应变增量; (4) 当饱和土体处于完全不排水或部分排水条件下,还需给出孔隙水压力增长和消散模型。粘弹性理论是目前应用中的主流,但存在多方面的不足,如不能考虑应变软化,不能考虑应力路径的影响,不能考虑土的各向异性以及大应变时误差大等,但它是试验结果的归纳,形式上直观简单,经过处理改进后,结合有限元程序,就可以计算出循环荷载作用下土工构造物的孔隙水压力和永久变形的 平均发展过程。 211 粘弹性理论 人们早在生产实践中认识到土体的应力—应变关系是非线性的,但实际工程中常用线性理论对这种非线性关系进行简化。自Seed 提出用等价线性方法近似考虑土的非线性以来,粘弹性理论已有了较大的发展。在土体的动力反应分析中,常用的粘弹性理论有等效线性模型和曼辛型非线性模型2 大类。前者把土体视为粘弹性材料,不寻求滞回曲线(即描述卸载与再加载时应力应变规律的曲线) 的具体数学表达式,而是给出等效弹性模量和等效阻尼比随剪应变幅值和有效应力状态变化的表达式,即以G 和λ作为它的动力特性指标引入实际计算;后者则根据不同的加载条件、卸载和再加载条件直接给出动应力应变的表达式。在给出初始加载条件下的动应力应变关系式(骨干曲线方程) 后,再利用曼辛二倍法得出卸荷和再加荷条件下的动应力应变关系,以构成滞回曲线方程[1 ] 。Hardin Drnevich 模型、Ramberg Osgood 模型、双线性模型及一些组合 基于阻尼的地震循环荷载作用下黏土非线性模型 尚守平刘方成王海东 ( 湖南大学, 湖南长沙410082) 摘要: 提出一种基于阻尼比的黏土动应力应变模型, 通过在滞回曲线中显示地引入代表阻尼比大小的形状系数,使得理论滞回曲线真实地反应土体的滞回阻尼性能。首先推导在等幅对称
ANSYS中粘弹性材料的参数意义： 我用的材料知道时温等效方程（W.L.F.方程），ANSYS 中的本构模型用MAXWELL模型表示。 １．活化能与理想气体常数的比值（Tool-Narayanaswamy Shift Function）或者时温方程的第一个常数。 ２．一个常数当用Tool-Narayanaswamy Shift Function的方程描述，或者是时温方程第２个常数 ３．定义体积衰减函数的MAXWELL单元数（在时温方程中用不到） ４．时温方程的参考温度 ５．决定１、２、３、４参数的值 ６－１５定义体积衰减函数的系数， １６－２５定义fictive temperature的松弛时间 这２０个数最终用来定义fictive temperature（在理论手册中介绍，不用在时温方程中） ２６－３０和３１－３５分别定义了材料在不同物理状态时的热扩散系数 ３６－４５用来定义fictive temperature的fictive temperature的一些插值一类的数值，时温方程也用不到 ４６剪切模量开始松弛的值 ４７松弛时间无穷大的剪切模量的值 ４８体积模量开始松弛的值 ４９松弛时间无穷大的体积模量的值 ５０描述剪切松弛模量的MAXWELL模型的单元数 ５１－６０拟合剪切松弛模量的prony级数的系数值 ６１－７０拟合剪切松弛模量的prony级数的指数系数值（形式参看理论手册） ７１描述体积松弛模量的MAXWELL模型的单元数 ７６－８５拟合体积松弛模量的prony级数的系数值 ８５－９５拟合体积松弛模量的prony级数的指数系数值（形式参看理论手册） 进入ansys非线性粘弹性材料有两项：