NEW STRAIN GRADIENT THEORY AND ANALYSIS

A new strain gradient theory which is based on energy nonlocal model is proposed in this paper, and the theory is applied to investigate the size effects in thin metallic wire torsion, ultra-thin beam bending and micro-indentation of polycrystalline copper. First, an energy nonlocal model is suggested. Second, based on the model, a new strain gradient theory is derived. Third, the new theory is applied to analyze three representative experiments.     

Determination of material intrinsic length and strain gradient hardening in microbending process

Microbending experiments of pure aluminum show that the springback angles increase with the decrease of foil thickness, which indicates obvious size effects and attributes to plastic strain gradient hardening. Then a constitutive model, taking into accounts both plastic strain and plastic strain gradient hardening, is proposed to analyze the microbending process of thin foil. The model is based on the relationship between shear yield stress and dislocation density, in which the material intrinsic length is related to material properties and average grain numbers along the characteristic scale direction of part. It is adopted in analytical model to calculate the non-dimensional bending moment and predict the springback angle after microbending. It is confirmed that the predictions by the proposed hardening model agree well with the experimental data, while those predicted by the classical plasticity model cannot capture such size effects. The contribution of plastic strain gradient increases with the decrease of foil thickness and is independent on the bending angle.     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

SIMULATIONS OF MECHANICAL BEHAVIOR OF POLYCRYSTALLINE COPPER WITH NANO-TWINS

Mechanical behavior and microstructure evolution of polycrystalline copper with nano-twins were investigated in the present work by finite element simulations. The fracture of grain boundaries are described by a cohesive interface constitutive model based on the strain gradient plasticity theory. A systematic study of the strength and ductility for different grain sizes and twin lamellae distributions is performed. The results show that the material strength and ductility strongly depend on the grain size and the distribution of twin lamellae microstructures in the polycrystalline copper.     

Size effects and dislocation patterning in two-dimensional bending

We perform atomistic Monte Carlo simulations of bending a Lennard-Jones single crystal in two dimensions. Dislocations nucleate only at the free surface as there are no sources in the interior of the sample. When dislocations reach sufficient density, they spontaneously coalesce to nucleate grain boundaries, and the resulting microstructure depends strongly on the initial crystal orientation of the sample. In initial yield, we find a reverse size effect, in which larger samples show a higher scaled bending moment than smaller samples for a given strain and strain rate. This effect is associated with source-limited plasticity and high strain rate relative to dislocation mobility, and the size effect in initial yield disappears when we scale the data to account for strain rate effects. Once dislocations coalesce to form grain boundaries, the size effect reverses and we find that smaller crystals support a higher scaled bending moment than larger crystals. This finding is in qualitative agreement with experimental results. Finally, we observe an instability at the compressed crystal surface that suggests a novel mechanism for the formation of a hillock structure. The hillock is formed when a high angle grain boundary, after absorbing additional dislocations, becomes unstable and folds to form a new crystal grain that protrudes from the free surface.     

Analysis of plate in second strain gradient elasticity

The bending analysis of a thin rectangular plate is carried out in the framework of the second gradient elasticity. In contrast to the classical plate theory, the gradient elasticity can capture the size effects by introducing internal length. In second gradient elasticity model, two internal lengths are present, and the potential energy function is assumed to be quadratic function in terms of strain, first- and second-order gradient strain. Second gradient theory captures the size effects of a structure with high strain gradients more effectively rather than first strain gradient elasticity. Adopting the Kirchhoff’s theory of plate, the plane stress dimension reduction is applied to the stress field, and the governing equation and possible boundary conditions are derived in a variational approach. The governing partial differential equation can be simplified to the first gradient or classical elasticity by setting first or both internal lengths equal to zero, respectively. The clamped and simply supported boundary conditions are derived from the variational equations. As an example, static, stability and free vibration analyses of a simply supported rectangular plate are presented analytically.     

A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this paper, a novel size-dependent functionally graded(FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton's principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.     

A size-dependent Reddy–Levinson beam model based on a strain gradient elasticity theory

A size-dependent Reddy–Levinson beam model is developed based on a strain gradient elasticity theory. Governing equations and boundary conditions are derived by using Hamilton’s principle. The model contains three material length scale parameters, which may effectively capture the size effect in micron or sub-micron. This model can degenerate into the modified couple stress model or even the classical model if two or all material length scale parameters are taken to be zero respectively. In addition, the present model recovers the micro scale Timoshenko and Bernoulli–Euler beam models based on the same strain gradient elasticity theory. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Reddy–Levinson beam are solved respectively; the results are compared with the reduced models. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Reddy–Levinson models are getting larger as the beam thickness is comparable to the material length scale parameters. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. This study may be helpful to characterize the mechanical properties of small scale beam-like structures for a wide range of potential applications.     

基于广义应变梯度理论的纳米梁挠曲电效应研究

挠曲电效应是应变梯度与电极化的耦合,它存在于所有的电介质材料中。在纳米电介质结构的挠曲电效应研究中,应变梯度弹性对挠曲电响应的影响一直以来被低估甚至被忽略了。根据广义应变梯度理论,应变梯度弹性中独立的尺度参数只有三个,而文献中所采用的一个或两个尺度参数的应变梯度理论只是它的简化形式。基于该理论,论文建立了考虑广义应变梯度弹性的三维电介质结构的理论模型,并以一维纳米梁为例研究了其弯曲问题的挠曲电响应及其能量俘获特性。结果表明,纳米梁的挠曲电响应存在尺寸效应,并且弹性应变梯度会影响结构挠曲电的尺寸效应,特别是当结构的特征尺寸低于尺度参数时。论文的工作为更进一步理解纳米尺度下的挠曲电机理和能量俘获特性提供理论基础和设计依据。     

Size-effects in plane strain sheet-necking

A finite strain generalization of the strain gradient plasticity theory by Fleck and Hutchinson (J. Mech. Phys. Solids 49 (2001a) 2245) is proposed and used to study size effects in plane strain necking of thin sheets using the finite element method. Both sheets with rigid grips at the ends and specimens with shear free ends are analyzed. The strain gradient plasticity theory predicts delayed onset of localization when compared to conventional theory, and it depresses deformation localization in the neck. The sensitivity to imperfections is analyzed as well as differently hardening materials.     

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.     

高速冲击表面处理对金属材料力学性能和组织结构的影响

高速冲击表面处理过程中的应变率对金属材料的宏观力学性能和微观组织结构都具有重要影响。根据当前应变率效应的研究成果,从宏观与微观相结合的角度出发,综述了高速冲击表面处理过程中应变率对金属材料强度和塑性的影响规律,并重点阐述了不同应变率下金属材料内部微观组织结构的演变规律,主要包括晶粒结构、绝热剪切带、相变、位错组态和析出相以及变形孪晶等。此外,还分析了组织结构随应变率的演化和微观变形机制的转变对材料力学性能的强化和弱化机理。最后,对高速冲击表面处理梯度组织的变形特点进行了总结。提出了不同组织结构对材料性能影响的综合效应模型,以期为应变率效应的深入研究奠定基础。     

Size-effects on yield surfaces for micro reinforced composites

Size effects in heterogeneous materials are studied using a rate independent higher order strain gradient plasticity theory, where strain gradient effects are incorporated in the free energy of the material. Numerical studies are carried out using a finite element method, where the components of the plastic strain tensor appear as free variables in addition to the displacement variables. Non-conventional boundary conditions are applied at material interfaces to model a constraint on plastic flow due to dislocation blocking. Unit cell calculations are carried out under generalized plane strain conditions. The homogenized response of a material with cylindrical reinforcing fibers is analyzed for different values of the internal material length scale and homogenized yield surfaces are presented. While the main focus is on initial yield surfaces, subsequent yield surfaces are also presented. The center of the yield surface is tracked under uniaxial loading both in the transverse and longitudinal directions and an anisotropic Bauschinger effect is shown to depend on the size of the fibers. Results are compared to conventional predictions, and size-effects on the kinematic hardening are accentuated.     

Chiral effects in uniformly loaded rods

Examples of chiral materials include some auxetic materials, bones, some honeycomb structures, as well as composites with inclusions. The chiral effects cannot be described within classical elasticity. In the context of the linear theory of Cosserat elastic solids, we investigate the deformation of a chiral rod subjected to tractions on the lateral surface, to body loads, and to resultant forces and moments on the ends. The work is motivated by the recent interest in the using of the Cosserat elastic solid as model for auxetic composites, carbon nanotubes and bones. The three-dimensional problem is reduced to the study of some generalized plane strain problems. New chiral effects are presented. In the case of cylinders of arbitrary cross-section, the flexure produced by a transversal force, in contrast with the case of achiral materials, is accompanied by extension and bending by terminal couples. The body loads and the tractions on the lateral surface produce extension, flexure, torsion, bending by terminal couples and a plane strain. It is shown that a uniform pressure acting on the lateral surface of a chiral circular cylinder does not produce bending effects.     

晶界结构及其对力学性质的影响(Ⅱ)

<正> 3 晶界结构对力学性质的影响 晶界是固体材料中的一种面缺陷.由于晶界和位错以及其他缺陷和杂质之间的相互作用      

能量非局部模型和新的应变梯度理论

提出一种新的基于能量非局部模型的应变梯度理论,并应用此理论对多晶铜以及薄膜基底的微压痕硬度进行理论预测和数值分析. 首先,提出了能量非局部模型,并由此模型,得出新应变梯度理论的本构关系;其次,由变分原理,得出相应的有限元公式;再次,给出了微压痕硬度的有限元分析方法;最后,将该理论预测结果与经典理论预测结果以及实验结果进行了对比. 结果表明,计算结果与实验结果相符;而经典理论的预测结果远低于实验结果.      

A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory

A size-dependent Kirchhoff micro-plate model is developed based on the strain gradient elasticity theory. The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. The static bending, instability and free vibration problems of a rectangular micro-plate with all edges simple supported are carried out to illustrate the applicability of the present size-dependent model. The results are compared with the reduced models. The present model can predict prominent size-dependent normalized stiffness, buckling load, and natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.     

基于应变率阶跃测试的晶体铜压入应变率敏感性研究

纳米压入测试可以原位获取材料的诸多力学性能,包括弹性模量,硬度,屈服应力,应变率敏感指数等。本文利用应变率阶跃测试技术对多晶铜试样的应变率敏感性进行测试分析,硬度-位移曲线表明压头下方所存在的变形梯度对各阶跃应变率下的硬度值存在明显影响;采用基于晶体细观机制的塑性应变梯度理论对压入变形梯度效应予以修正,比较了修正与未修正数据所得的应变率敏感指数,在有效剔除压入变形梯度影响的基础上,应变率阶跃测试可实现单次压入下材料应变率敏感性的测试表征。     

多晶石墨烯拉伸力学性能及其影响因素的灵敏度分析

晶粒尺寸、温度和应变率等对纳米材料的力学性能有重要影响。本文通过分子动力学（MD）数值模拟,分析了不同晶粒尺寸多晶石墨烯在不同温度、拉伸应变率下的杨氏弹性模量、极限应力、极限应变等拉伸力学性能。结果表明,晶粒尺寸、温度和拉伸应变率对拉伸力学性能有较大影响。利用正交实验法,分别分析了杨氏弹性模量、极限应力和极限应变对晶粒尺寸、温度和拉伸应变率的敏感程度。结果表明,杨氏弹性模量和极限应力对影响因素的敏感程度由大到小依次为晶粒尺寸、温度和拉伸应变率;极限应变对影响因素的敏感程度由大到小依次为晶粒尺寸、拉伸应变率和温度。研究结果可为多晶石墨烯的理论研究和工程应用提供参考。     

Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations

The higher-order stress work-conjugate to slip gradient in single crystals at small strains is derived based on the self-energy of geometrically necessary dislocations (GNDs). It is shown that this higher-order stress changes stepwise as a function of in-plane slip gradient and therefore significantly influences the onset of initial yielding in polycrystals. The higher-order stress based on the self-energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5-32] and applied to single-slip-oriented 2D and 3D model crystal grains of size D. It is thus found that the self-energy of GNDs gives a D^-1-dependent term for the averaged resolved shear stress in such a model grain under yielding. Using published experimental data for several polycrystalline metals, it is demonstrated that the D^-1-dependent term successfully explains the grain size dependence of initial yield stress and the dislocation cell size dependence of flow stress in the submicron to several-micron range of grain and cell sizes.