A new visco–elasto-plastic model via time–space fractional derivative

To characterize the visco–elasto-plastic behavior of metals and alloys we propose a new constitutive equation based on a time–space fractional derivative. The rheological representative of the model can be analogous to that of the Bingham–Maxwell model, while the dashpot element and sliding friction element are replaced by the corresponding fractional elements. The model is applied to describe the constant strain rate, stress relaxation and creep tests of different metals and alloys. The results suggest that the proposed simple model can describe the main characteristics of the experimental observations. More importantly, the model can also provide more accurate predictions than the classic Bingham–Maxwell model and the Bingham–Norton model.     

服从分数导数Kelvin本构模型的粘弹性阻尼器的阻尼性能分析及试验研究

从分数导数的定义出发,提出了在经典粘弹性模型理论中采用Abel粘壶取代传统牛顿粘壶的新观点.将高分子粘弹性阻尼器在MTS831.10材料试验机上进行动态力学行为试验,对试验结果用分数导数模型进行拟合.结果表明,分数导数Kelvin模型可以同时精确地拟合高分子材料的存储模量和损耗模量随频率变化的曲线,而且其形式简单、统一.在计算过程中需要调整的参数很少.最后将分数导数模型引入静态特性公式,得出了圆筒状粘弹性阻尼器动态刚度与阻尼的表达式.     

分数算子描述的粘弹性材料的本构关系研究

从分数导数的定义出发,提出了在经典粘弹性模型理论中采用Abel粘壶取代传统牛顿粘壶的新观点.将有机硅高分子材料在MTS831.10材料试验机上进行动态力学行为试验,对试验结果分别用经典粘弹性模型和分数导数模型进行拟合.结果表明,分数导数Kelvin模型可以同时精确地拟合高分子材料的存储模量和损耗模量随频率变化的曲线,而且其形式简单、统一,在计算过程中需要调整的参数很少.     

Polynomial operators,stieltjes convolution,and fractional calculus in hereditary mechanics

Summary Fractional calculus is used to describe the general behavior of materials with memory. An expression for the fractional derivative or the fractional integral is developed in terms of the Stieltjes convolution and the Riesz distribution. The general fractional calculus polynomial operator constitutive equation is reduced to a Stieltjes convolution. A constitutive equation which depends on a memory parameter for an isotorpic viscoelastic material is presented. The proposed creep compliance has an initial response, a primary creep region, a secondary creep region and a tertiary creep region. The corresponding relaxation modulus has a glassy region, a leathery region, a rubbery region and a liquid region.     

Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results

In capturing visco-elastic behavior, experimental tests play a fundamental rule, since they allow to build up theoretical constitutive laws very useful for simulating their own behavior. The main challenge is representing the visco-elastic materials through simple models, in order to spread their use. However, the wide used models for capturing both relaxation and creep tests are combinations of simple models as Maxwell and/or Kelvin, that depend on several parameters for fitting both creep and relaxation tests. This paper, following Nutting and Gemant idea of fitting experimental data through a power law function, aims at stressing the validity of fractional model. In fact, as soon as relaxation test is well fitted by power law decay then the fractional constitutive law involving Caputo’s derivative directly appears. It will be shown that fractional model is proper for studying visco-elastic behavior, since it may capture both relaxation and creep tests, requiring the identification of two parameters only. This consideration is assessed by the good agreement between experimental tests on creep and relaxation and the fractional model proposed. Experimental tests, here reported are performed on two polymers having different chemical physical properties such that the fractional model may cover a wide range of visco-elastic behavior.     

黏土蠕变非线性特性及其分数阶导数蠕变模型

针对黏土蠕变的非线性性质,以成都黏土为研究对象展开蠕变试验,发现黏土变形包括瞬时弹性变形、衰减蠕变变形、稳态蠕变变形和加速蠕变变形;黏土长期弹性模量随时间和应力的增加非线性软化;黏滞系数随应力的增加非线性软化,随时间的增加非线性硬化。基于流变学理论、分数阶微积分理论和Harris衰减函数,分别构建了分数阶导数元件、非线性弹性元件和非线性黏滞元件,从而建立了形式简单、参数较少和概念清晰的非线性分数阶导数蠕变模型。将非线性分数阶导数蠕变模型和Burgers蠕变模型进行对比拟合分析,发现非线性分数阶导数蠕变模型各阶段的拟合结果更好,对黏土非线性蠕变的描述更合理,可准确地反映黏土蠕变全过程,表明了所建立非线性分数阶导数蠕变模型的科学合理性。     

饱和分数导数型粘弹性土层竖向振动放大效应

将土骨架视为具有分数阶导数本构关系的粘弹性体,采用Biot动力固结方程,在频率域内研究了饱和分数导数粘弹性土层的土骨架粘性、土层厚度等对竖向振动放大系数的影响。通过动力控制方程解耦和边界条件束缚,给出了经典弹性饱和土、分数导数型粘弹性饱和土和经典粘弹性饱和土三种情况下饱和土层的位移、应力和孔压解析表达式。考察了饱和土各物理和几何参数对竖向振动放大系数的影响,结果表明:在不同土层厚度时,经典弹性饱和土、分数导数型粘弹性饱和土及经典粘弹性饱和土的竖向振动放大系数各不相同;分数导数模型的材料参数对振动放大系数有较大影响。     

分数阶导数线性流变固体模型及其应用

采用Koeller 弹壶元件替代标准线性固体模型中的Newton黏壶, 得到分数阶导数线性流变固体模型, 给出了表征模型动态黏弹特性的存储模量、损耗模量和损耗因子以及表征模型静态黏弹特性的蠕变柔量和松弛模量。采用分数阶导数线性流变固体模型、标准线性固体模型和五参量固体模型对聚丙烯材料应力松弛特性进行分析。讨论了Mittag-Leffler函数的求和截断误差。结果表明分数阶导数线性流变固体模型能更准确描述聚丙烯材料的应力松弛行为。     

Toward an equation of state for solid materials with memory by use of the half-order derivative

Summary The time derivative operator does not depend upon the difference between the current time and the past times; however, the fractional time derivative operator does. Thus, it is reasonable to expect that the fractional derivative would be useful in describing the mathematical theory of the behavior of materials with memory. An equation of state is proposed for solid materials with memory by introducing the half-order fractional calculus derivative in order to relate to the empirical expression used in the fundamental work of Tobolsky and Catsiff. This theory replaces the three empirical functions used by Tobolsky and Catsiff in reducing their experimental data for the low temperature glassy region, the transition region and the quasi-static rubbery plateau region. The square root differential operator with respect to time, D ^1/2, has built in memory since the kernel of this operator depends upon the difference between the current time and the past time. D ^1/2 is a special case of the Abel operator, which is used in the theory of integral equations. The present theory introduces integrals into the standard linear solid resulting in an integral differential equation governing the stress, strain and temperature. It is shown that this proposed linear equation of state for a solid material, which undergoes a second order transition, requires only four phenomenological constants to completely determine the behavior of the solid material. These four phenomenological constants are two relaxation times and two creep times, both of which are functions of the temperature.     

沥青砂混合料粘弹塑力学特性研究

在0.1MPa、0.15MPa、0.2MPa、0.25MPa和0.3MPa下进行了沥青砂试样单轴压缩和蠕变实验,分析了其压缩和蠕变性质,根据变形机理提出了粘弹塑本构模型可由粘弹性和粘塑性的两个子模型串联构成,通过对粘塑性子模型中粘性系数进行改进,理论推导了模型蠕变本构方程,确定了模型参数,并求得模型参数与加载应力函数关系。进行模型预测与实验结果对比,结果表明:该模型能够描述沥青砂试样在不同应力下蠕变变形的3个阶段,反映了沥青砂混合料粘弹塑变形特点。     

竖向集中力作用下分数导数型半无限体粘弹性地基变形分析

研究成果表明地基土体具有粘弹性性质,在长期荷载作用下会产生蠕变变形。为了准确预测地基变形,必须建立精确的土体本构模型。分数导数粘弹性模型具有精确度高,确定模型所需的实验参数少,应用范围广的特点。针对半无限体粘弹性地基,采用分数导数粘弹性本构模型模拟土的力学行为,运用弹性-粘弹性对应原理分析了粘弹性地基的变形。研究发现,采用分数导数粘弹性本构模型得到的地基沉降量较经典粘弹性模型要小,经典粘弹性模型不能很好地反映集中力对周围地基沉降的影响以及地基的蠕变性质。     

Nonlinear creep damage constitutive model for soft rocks

In some existing nonlinear creep damage models, it may be less rigorous to directly introduce a damage variable into the creep equation when the damage variable of the viscous component is a function of time or strain. In this paper, we adopt the Kachanov creep damage rate and introduce a damage variable into a rheological differential constitutive equation to derive an analytical integral solution for the creep damage equation of the Bingham model. We also propose a new nonlinear viscous component which reflects nonlinear properties related to the axial stress of soft rock in the steady-state creep stage. Furthermore, we build an improved Nishihara model by using this new component in series with the correctional Nishihara damage model that describes the accelerating creep, and deduce the rheological constitutive relation of the improved model. Based on superposition principle, we obtain the damage creep equation for conditions of both uniaxial and triaxial compression stress, and study the method for determining the model parameters. Finally, this paper presents the laboratory test results performed on mica-quartz schist in parallel with, or vertical to the schistosity direction, and applies the improved Nishihara model to the parameter identification of mica-quartz schist. Using a comparative analysis with test data, results show that the improved model has a superior ability to reflect the creep properties of soft rock in the decelerating creep stage, the steady-state creep stage, and particularly within the accelerating creep stage, in comparison with the traditional Nishihara model.     

Generalised constitutive equations for densification of metal matrix coated fibre composites

Abstract

The present paper completes a study of constitutive equations for the consolidation processing of continuous fibre reinforced metal matrix composite materials. It builds on an earlier paper in which physically based constitutive equations were derived for the case of symmetrical, isostatic loading. In the present paper, constitutive equations are developed for in plane, general stress states. The total deformation of the consolidating composite is expressed as the sum of a conventional deviatoric creep term, together with a dilatational term, which was derived using a variational method previously published. The equations contain only two material parameters, which are the conventional creep coefficient and exponent for the fibre coating material (in this case, Ti-6Al-4V). The resulting equations have been implemented into finite element software enabling the simulation of practical consolidation processes. The model has been verified by comparing predicted results with those obtained from independent micromechanical models. A number of experimental tests have been carried out, and the model is used to predict the rates of densification for a range of experimental pressure and temperature histories. Good comparisons have been achieved.     

一种新的聚合物基复合材料应力松弛经验模型

视黏弹性元件弹簧和粘壶的特性参数为时间的函数,建立双变参Maxwell模型,求解其本构方程及松弛函数,探讨了其松弛函数与经验KWW函数的关系,提出了一种新的应力松弛经验模型。运用最小二乘原理,建立该模型参数的确定方法。开展了玻璃纤维增强聚合物(GFRP)复合材料的长期应力松弛实验,并通过实验数据的分析验证了该模型的正确性。研究表明,新应力松弛经验模型拟合曲线与实测曲线吻合较好,相关系数达到96%以上,说明了该模型适合描述GFRP复合材料的黏弹性松弛特性。     

Treatment of anisotropic damage development within a scalar damage formulation

This paper is concerned with describing a damage mechanics formulation which provides for non-isotropic effects using a scalar damage variable. An investigation has been in progress for establishing the constitutive behavior of rock salt at long times and low to moderate confining pressures in relation to the possible use of excavated rooms in rock salt formations as repositories for nuclear waste. An important consideration is the effect of damage manifested principally by the formation of shear induced wing cracks which have a stress dependent orientation. The analytical formulation utilizes a scalar damage parameter, but is capable of indicating the non-isotropic dependence of inelastic straining on the stress state and the confining pressure. Also, the equations indicate the possibility of volumetric expansions leading to the onset of tertiary creep and eventually rupture if the damage variable reaches a critical value.     

分数导数黏弹性准饱和土中球空腔振动特性

在频率域内用解析方法研究分析了简谐轴对称荷载和流体压力作用下分数导数黏弹性准饱和中球空腔的稳态响应问题。将土骨架等效为具有分数阶导数本构关系的黏弹性体,基于Biot两相饱和介质模型,通过势函数推导求得了边界部分透水时分数导数粘弹性准饱和土中球空腔的位移、应力和孔压等的解析解。根据界面连续性条件,确定了待定系数的表达式。在此基础上,考察了准饱和土各参数对动力响应的影响,结果表明：轴对称荷载和流体压力两种情况时,相对渗透系数对动力响应的影响有较大的差异。分数导数本构模型更合理地描述了准饱和土中球空腔的振动特性。     

Determining the Form of a Hydrodynamic Flow upon Creep of an Amorphous Cobalt-Based Metal Alloy in a Variable Temperature Field

Deformation of an amorphous cobalt alloy of strip configuration in creep tests in a variable temperature field is studied. An analytical form of the creep curve is found, and a prediction calculation of the resistivity by means of the proposed creep function is carried out. The physical meaning of the parameters of the creep equation is established, and their dimensionality is found. It is shown that, under the action of a variable temperature field, the creep of the alloy is not Newtonian and the material under strain exhibits pseudoplastic properties. It was found that, in the absence of heating, the viscosity of amorphous strip alloys can be considered within the Bingham plastic model.     

Three-dimensional, finite deformation, viscoplastic constitutive models for polymeric materials

A general methodology for developing three-dimensional. finite deformation, viscoplastic constitutive models for polymeric materials is presented. The development begins with the presentation of a one-dimensional spring and dashpot construction which exhibits behavior typical of polymeric materials, namely strain-rate dependence, stress relaxation, and creep. The one-dimensional construction serves as a starting point for the development of a three-dimensional, finite deformation, viscoplastic constitutive model which also exhibits typical polymeric behavior. Furthermore, the three-dimensional constitutive model may be easily generalized to incorporate an arbitrary number of inelastic processes, representing (inelastic) microstructural deformation mechanisms operating on different time scales. Strain-rate dependence, stress relaxation, and creep phenomena are discussed in detail for a simple version of the constitutive model. Test data for a particular polymer is used to validate the simple model. It is concluded that the methodology provides a flexible approach to modeling polymeric materials over a wide range of loading conditions.     

Stability analysis of quasi-brittle materials – creep under multiaxial loading

The aim of this paper is to develop a simple time-dependent Continuum Damage Mechanics model applied to quasi-brittle materials such as rock or concrete. The three-dimensional constitutive visco-damage model describes phenomena like relaxation, creep and rate-dependent loading using a unified framework. A material stability analysis devoted to creep tests highlights a general creep stress stability domain. This convex domain is connected to the property of the associated time-independent Continuum Damage Mechanics model. More particularly, the boundary of this domain in the creep stress space coincides with the invertibility condition of the constitutive matrix considering infinitely slow loading. Phenomenon as creep failure under high-sustained load is explained quite simply within stability theory. Creep failure appears as the manifestation of a saddle-node bifurcation phenomenon.