Elasticity Theories with Higher-order Gradients of Inertia and Stiffness for the Modelling of Wave Dispersion in Laminates

Dispersive wave propagation is simulated with a continuum elasticity theory that incorporates gradients of strain and inertia. The additional parameters are the Representative Volume Element (RVE) sizes in statics and dynamics, respectively. For the special case of a periodic laminate, expressions for these two RVE sizes can be provided based on the properties of the two components. The fourth-order governing equations are rewritten in two sets of coupled second-order equations, whereby the two sets of unknowns are the macroscopic displacements and the microscopic displacements. The resulting formulation is thus a true multi-scale continuum. In a numerical wave propagation example it is shown that the higher-order continuum model provides an excellent approximation of an explicit model of the heterogeneous laminate.     

Enforcing Boundary Conditions in Micro-Macro Transition for Second Order Continuum

In recent years the multiscale computational homogenisation has been extensively developed. Such micro-macro modelling does not require any constitutive assumptions at the macro-level. The multi-scale computational homogenisation has also been extended for the second order continuum at the macro level Kouznetsova V.G., Geers M.G.D., and Brekelmans V.A.M (2004). The second-order framework is based on incorporation of the gradient of macroscopic deformation in micro to macro multiscale transition. The introduction of the secondorder continuum at macro-scale takes into account the size effect and gives more accurate results in case of insufficient scale separation. The general framework of computational homogenisation has been presented in Kouznetsova V.G (2002).     

A study of size effects and length scales in fracture and fatigue of metals by second gradient modelling*

As the dimensions of structures are scaled down to the micro‐ and nano‐domains, the mechanical behaviour becomes size dependent and thus, we cannot expect the classical elasticity solutions to hold. In particular, recent experimental investigations of fatigue strength of metals show pronounced strengthening due to the influences of small geometrical dimensions. Based on second gradient elasticity framework as particularized on beams, closed form solutions to idealized problems of elastic cantilever bending, elastic three‐point bending and elasto‐plastic torsion have been found, showing considerable stiffening, toughening and hardening, respectively, compared to the classical theory predictions. In these models, the intrinsic material length scale was taken to be constant. Furthermore, we describe a gradient solid with a characteristic length which is not a fixed material parameter but depends on the amount of plastic effective strain amplitude, as obtained from cyclic strain hardening. A respective evolution law is suggested and discussed.     

A new formulation and 𝒞0‐implementation of dynamically consistent gradient elasticity

In his article a special form of gradient elasticity is presented that can be used to describe wave dispersion. This new format of gradient elasticity is an appropriate dynamic extension of the earlier static counterpart of the gradient elasticity theory advocated in the early 1990s by Aifantis and co‐workers. In order to capture dispersion of propagating waves, both higher‐order inertia and higher‐order stiffness contributions are included, a fact which implies (and is denoted as) dynamic consistency. The two higher‐order terms are accompanied by two associated length scales. To facilitate finite element implementations, the model is rewritten such that ??⁰‐continuity of the interpolation is sufficient. An auxiliary displacement field is introduced which allows the original fourth‐order equations to be split into two coupled sets of second‐order equations. Positive‐definiteness of the kinetic energy requires that the inertia length scale is larger than the stiffness length scale. The governing equations, boundary conditions and the discretized system of equations are presented. Finally, dispersive wave propagation in a one‐dimensional bar is considered in a numerical example. Copyright © 2007 John Wiley & Sons, Ltd.     

Intrinsic material length,Theory of Critical Distances and Gradient Mechanics: analogies and differences in processing linear‐elastic crack tip stress fields

The Theory of Critical Distances (TCD) is a bi‐parametrical approach suitable for predicting, under both static and high‐cycle fatigue loading, the non‐propagation of cracks by directly post‐processing the linear‐elastic stress fields, calculated according to continuum mechanics, acting on the material in the vicinity of the geometrical features being assessed. In other words, the TCD estimates static and high‐cycle fatigue strength of cracked bodies by making use of a critical distance and a reference strength which are assumed to be material constants whose values change as the material microstructural features vary. Similarly, Gradient Mechanics postulates that the relevant stress fields in the vicinity of crack tips have to be determined by directly incorporating into the material constitutive law an intrinsic scale length. The main advantage of such a method is that stress fields become non‐singular also in the presence of cracks and sharp notches. The above idea can be formalized in different ways allowing, under both static and high‐cycle fatigue loading, the static and high‐cycle fatigue assessment of cracked/notched components to be performed without the need for defining the position of the failure locations a priori. The present paper investigates the existing analogies and differences between the TCD and Gradient Mechanics, the latter formalized according to the so‐called Implicit Gradient Method, when such theories are used to process linear‐elastic crack tip stress fields.     

气辅挤出胀大比与滑移段长度的关系

采用参数渐变法和Thompson变换,对粘弹性高分子熔体在不同气体辅助挤出口模内的流动进行了数值模拟研究。考察了体积流量、松弛时间和滑移段长度对挤出物挤出胀大比的影响。研究表明,熔体在滑移段的停留时间与材料松弛时间之比与挤出胀大比之间存在指数衰减关系,其实质是熔体在滑移段处于形变衰减过程。理论分析与数值模拟具有高度的一致性,表明该方程可用于指导气辅口模设计。