损伤粘弹性基本理论及其结构的静、动力学行为分析
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摘要
本文利用粘弹性材料的本构理论及带微孔的弹性微结构理论，对损伤粘弹性力学的基本理论及其结构的线性和非线性的拟静态和动力学行为进行了比较系统的理论分析和数值模拟，获得了一些新的理论结果和数值计算结果。主要的工作如下：
1．应用带空隙的线弹性微结构理论和粘弹性理论，从粘弹性的微分型和积分型本构关系出发，分别给出了损伤粘弹性固体的两种线性本构方程。
2．从粘弹性材料的Boltzmann迭加原理和带空洞材料的线弹性本构关系出发，给出了一种具有广义力场的损伤粘弹性材料的本构模型；应用变积方法得到了损伤粘弹性材料以卷积形式表示的泛函，同时建立了相应的广义变分原理和广义势能原理。
3．给出了小变形假设条件下损伤粘弹性Timoshenko梁的运动微分方程，应用变积方法，建立了损伤粘弹性Timoshenko梁的广义变分原理。利用Laplace变换和Laplace数值逆变换，研究了两端简支损伤粘弹性Timoshenko梁在阶跃载荷作用下的准静态力学行为。考察了材料的粘性和损伤对梁的力学行为的影响。
4．应用损伤粘弹性材料积分型本构关系和Timoshenko梁的假设，在有限变形条件下，给出了粘弹性基础上损伤粘弹性Timoshenko梁运动微分方程组。在数值上分析了粘弹性基础上损伤粘弹性Timoshenko梁的动力学行为。考察基础对梁的力学行为的影响。
5．从损伤粘弹性材料的卷积型本构关系出发，推导了在小挠度和大挠度条件下，损伤粘弹性薄板的运动微分方程，建立了损伤粘弹性薄板小挠度问题的广义变分原理，讨论了损伤粘弹性简支矩形薄板在阶跃载荷的作用下的准静态力学行为。同时，在有限变形下，讨论了损伤粘弹性薄板在谐载荷作用下的动力学行为。
6．从厚板的Timoshenko几何变形假设和损伤粘弹性材料的积分型本构关系出发，推导了带损伤粘粘性中厚板含剪切、挤压及转动惯性效应的动力学方程。并给出了损伤粘弹性中厚板的广义变分原理。应用Galerkin方法和非线性动力学中的数值分析方法，揭示出四边简支矩形损伤粘弹性中厚板具有丰富的非线性动力学行为。考察了载荷参数、结构几何参数和材料参数对损伤粘弹性中厚板动力学行为
摘要
的影响。为了比较，分析了小变形情况下损伤粘弹性中厚板的动力学行为和无损
伤粘弹性中厚板的动力学特性，考察了损伤对厚板的动力学稳定性的影响。
7.采用积分型的损伤粘弹材料本构关系，给出了损伤粘弹性梁一柱的运动微分方程，
建立了损伤粘弹性梁一柱的广义变分原理。应用Galerkin方法和非线性动力学中
的数值分析方法，揭示出两端简支损伤粘弹性梁一柱具有丰富的动力学行为，同
时考察了材料参数对系统响应的影响。最后，比较了两种不同端部条件下损伤粘
弹性梁一柱的动力学特性。
8.桩基看成是具有损伤的粘弹性材料组成的Timoshenko梁，并将地基视作一种线
性粘弹性材料。在有限变形条件下，应用损伤粘弹性卷积型的本构理论，给出了
损伤粘弹性桩基的静、动力学行为分析的初边值问题。应用Galerkin方法和非线
性动力学的数值分析方法，研究了具有适当端部条件的桩基动力学行为。可以看
出在横向简谐力作用下，损伤粘弹性桩具有丰富的动力学性质。考察了各种参数
对桩基动力学行为的影响，特别考察了损伤对桩基力学行为的影响。比较了l-
阶和2一阶Galerkin截断系统的动力学性质。
In this dissertation, according to the theory of viscoelastic materials and the micro- structure theory for linear elastic materials with damage, theoretical analyses of viscoelastic materials with damage and numerical simulations of quasi-static and dynamical analysis for viscoelastic structures with damage are systematically studied. New theoretical and numerical results are obtained. The main results contain as follows:
1 .Applying the micro-structure theory for linear elastic materials with voids and the constitutive laws of viscoelastic materials, two kinds of linear constitutive equations of viscoelastic solids with damage are given by the differential-type and integral-type constitutive laws of linear viscoelastic materials.
2.From the Boltzmann's constitutive law of viscoelastic materials and the linear theory of elastic material with voids, a constitutive model of generalized force fields for viscoelastic solids with voids is given. The convolution-type functional ispresented, the generalized variational principles and potential energy principle of viscoelastic solids with voids are presented by using the variational integral method.
3.Under the case of small deflections, the generalized differential equations of motion for Timoshenko beams with damage are derived. By using the variational integral method, the generalized variational principle of viscoelastic Timoshenko beams with damage is presented. The quasi-static behaviors of the viscoelastic Timoshenko beam with two sides of the beam are simply supported, and under step loading are analyzed by using Laplace transformation and the numerical inverse Laplace transform. The influences of material parameters and damage on the quasi-static behavior of the beam are considered in detail.
4.From convolution-type constitutive equations for linear viscoelastic materials with damage and the hypotheses of Timoshenko beams with large deflections, the differential equations of motion governing nonlinear dynamical behavior of Timoshenko beams with damage on viscoelastic foundation are given. The dynamic behaviors of viscoelastic Timoshenko beams with damage on viscoelastic foundation are numerically analyzed. At the same time, the influence of the foundation on the dynamic behaviors of beam is also studied.
5.From the constitutive model expressed by convolution method for viscoelastic solids with damage, initial-boundary-value problems analyzing static-dynamic behaviors of viscoelastic thin plates with damage are all formulated under the case of small and finite deflections. Under the case of small deflections, the generalized variational principle of viscoelastic thin plates with damage is established. The quasi-static behaviors of the viscoelastic rectangular thin plate with damage under step loading are analyzed when the boundary of plate is simply supported. At the same time, the dynamical response of the viscoelastic thin plate with damage subjected to a periodic excitation is studied.
6.Based on Timoshenko geometry deformation hypotheses of thick plates and the integral-type contitutive model of viscoelastic solids with damage, the nonlinear governing equations are derived for dynamic analysis of viscoelastic thick plates with damage. The generalized variational principle of viscoelastic thick plates with damage is presented. Applying the Galerkin method and numerical methods in nonlinear dynamics, the dynamical behaviors of viscoelastic plates with simply supported edges are discussed in detail. The influences of the load, geometry and material
parameters on the dynamical behaviors of the viscoelastic plate with damage are considered. The dynamical behavior of viscoelastic plates with damage under small deformations is also analyzed. To consider the effect of damage on the dynamical behavior of plate, we compare dynamical properties of plates with damage and without damage.
7.According to the constitutive model expressed by convolution method for viscoelastic solids with damage, the differential equations of motion governing dynamical behaviors
1．应用带空隙的线弹性微结构理论和粘弹性理论，从粘弹性的微分型和积分型本构关系出发，分别给出了损伤粘弹性固体的两种线性本构方程。
2．从粘弹性材料的Boltzmann迭加原理和带空洞材料的线弹性本构关系出发，给出了一种具有广义力场的损伤粘弹性材料的本构模型；应用变积方法得到了损伤粘弹性材料以卷积形式表示的泛函，同时建立了相应的广义变分原理和广义势能原理。
3．给出了小变形假设条件下损伤粘弹性Timoshenko梁的运动微分方程，应用变积方法，建立了损伤粘弹性Timoshenko梁的广义变分原理。利用Laplace变换和Laplace数值逆变换，研究了两端简支损伤粘弹性Timoshenko梁在阶跃载荷作用下的准静态力学行为。考察了材料的粘性和损伤对梁的力学行为的影响。
4．应用损伤粘弹性材料积分型本构关系和Timoshenko梁的假设，在有限变形条件下，给出了粘弹性基础上损伤粘弹性Timoshenko梁运动微分方程组。在数值上分析了粘弹性基础上损伤粘弹性Timoshenko梁的动力学行为。考察基础对梁的力学行为的影响。
5．从损伤粘弹性材料的卷积型本构关系出发，推导了在小挠度和大挠度条件下，损伤粘弹性薄板的运动微分方程，建立了损伤粘弹性薄板小挠度问题的广义变分原理，讨论了损伤粘弹性简支矩形薄板在阶跃载荷的作用下的准静态力学行为。同时，在有限变形下，讨论了损伤粘弹性薄板在谐载荷作用下的动力学行为。
6．从厚板的Timoshenko几何变形假设和损伤粘弹性材料的积分型本构关系出发，推导了带损伤粘粘性中厚板含剪切、挤压及转动惯性效应的动力学方程。并给出了损伤粘弹性中厚板的广义变分原理。应用Galerkin方法和非线性动力学中的数值分析方法，揭示出四边简支矩形损伤粘弹性中厚板具有丰富的非线性动力学行为。考察了载荷参数、结构几何参数和材料参数对损伤粘弹性中厚板动力学行为
摘要
的影响。为了比较，分析了小变形情况下损伤粘弹性中厚板的动力学行为和无损
伤粘弹性中厚板的动力学特性，考察了损伤对厚板的动力学稳定性的影响。
7.采用积分型的损伤粘弹材料本构关系，给出了损伤粘弹性梁一柱的运动微分方程，
建立了损伤粘弹性梁一柱的广义变分原理。应用Galerkin方法和非线性动力学中
的数值分析方法，揭示出两端简支损伤粘弹性梁一柱具有丰富的动力学行为，同
时考察了材料参数对系统响应的影响。最后，比较了两种不同端部条件下损伤粘
弹性梁一柱的动力学特性。
8.桩基看成是具有损伤的粘弹性材料组成的Timoshenko梁，并将地基视作一种线
性粘弹性材料。在有限变形条件下，应用损伤粘弹性卷积型的本构理论，给出了
损伤粘弹性桩基的静、动力学行为分析的初边值问题。应用Galerkin方法和非线
性动力学的数值分析方法，研究了具有适当端部条件的桩基动力学行为。可以看
出在横向简谐力作用下，损伤粘弹性桩具有丰富的动力学性质。考察了各种参数
对桩基动力学行为的影响，特别考察了损伤对桩基力学行为的影响。比较了l-
阶和2一阶Galerkin截断系统的动力学性质。
In this dissertation, according to the theory of viscoelastic materials and the micro- structure theory for linear elastic materials with damage, theoretical analyses of viscoelastic materials with damage and numerical simulations of quasi-static and dynamical analysis for viscoelastic structures with damage are systematically studied. New theoretical and numerical results are obtained. The main results contain as follows:
1 .Applying the micro-structure theory for linear elastic materials with voids and the constitutive laws of viscoelastic materials, two kinds of linear constitutive equations of viscoelastic solids with damage are given by the differential-type and integral-type constitutive laws of linear viscoelastic materials.
2.From the Boltzmann's constitutive law of viscoelastic materials and the linear theory of elastic material with voids, a constitutive model of generalized force fields for viscoelastic solids with voids is given. The convolution-type functional ispresented, the generalized variational principles and potential energy principle of viscoelastic solids with voids are presented by using the variational integral method.
3.Under the case of small deflections, the generalized differential equations of motion for Timoshenko beams with damage are derived. By using the variational integral method, the generalized variational principle of viscoelastic Timoshenko beams with damage is presented. The quasi-static behaviors of the viscoelastic Timoshenko beam with two sides of the beam are simply supported, and under step loading are analyzed by using Laplace transformation and the numerical inverse Laplace transform. The influences of material parameters and damage on the quasi-static behavior of the beam are considered in detail.
4.From convolution-type constitutive equations for linear viscoelastic materials with damage and the hypotheses of Timoshenko beams with large deflections, the differential equations of motion governing nonlinear dynamical behavior of Timoshenko beams with damage on viscoelastic foundation are given. The dynamic behaviors of viscoelastic Timoshenko beams with damage on viscoelastic foundation are numerically analyzed. At the same time, the influence of the foundation on the dynamic behaviors of beam is also studied.
5.From the constitutive model expressed by convolution method for viscoelastic solids with damage, initial-boundary-value problems analyzing static-dynamic behaviors of viscoelastic thin plates with damage are all formulated under the case of small and finite deflections. Under the case of small deflections, the generalized variational principle of viscoelastic thin plates with damage is established. The quasi-static behaviors of the viscoelastic rectangular thin plate with damage under step loading are analyzed when the boundary of plate is simply supported. At the same time, the dynamical response of the viscoelastic thin plate with damage subjected to a periodic excitation is studied.
6.Based on Timoshenko geometry deformation hypotheses of thick plates and the integral-type contitutive model of viscoelastic solids with damage, the nonlinear governing equations are derived for dynamic analysis of viscoelastic thick plates with damage. The generalized variational principle of viscoelastic thick plates with damage is presented. Applying the Galerkin method and numerical methods in nonlinear dynamics, the dynamical behaviors of viscoelastic plates with simply supported edges are discussed in detail. The influences of the load, geometry and material
parameters on the dynamical behaviors of the viscoelastic plate with damage are considered. The dynamical behavior of viscoelastic plates with damage under small deformations is also analyzed. To consider the effect of damage on the dynamical behavior of plate, we compare dynamical properties of plates with damage and without damage.
7.According to the constitutive model expressed by convolution method for viscoelastic solids with damage, the differential equations of motion governing dynamical behaviors
引文
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