On the long-term response of elastic-perfectly plastic solids to dynamic cyclic loads

It is shown that the long-term response of an elastic-perfectly plastic solid subjected to dynamic actions cyclically varying in time is characterized by stresses, plastic strain rates and velocities that are all periodic with the same period of the external actions, and are in perfect analogy with the quasi-static case; on the other hand, plastic strains and displacements are in general nonperiodic (except in case of alternating plasticity) and may increase indefinitely (except when elastic or plastic shakedown occurs). Besides, the work performed by the external actions in the steady cycle equals the work performed by the elastic stresses (i.e. pertaining to the elastic response of the body to the same actions) through the plastic strain rates.

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Sommario Per un solido elastico perfettamente plastico soggetto ad azioni cicliche dinamiche si mostra che la risposta a lungo termine è caratterizzata da tensioni, deformazioni plastiche incrementali e velocità tutte periodiche con lo stesso periodo delle azioni esterne, in analogia di quanto avviene nel caso quasi-statico; per contro le deformazioni plastiche e gli spostamenti sono in generale non periodici (tranne nel caso di plasticità alternata) e possono crescere indefinitamente (tranne nel caso di adattamento elastico o plastico). Inoltre il lavoro compiuto dalle azioni esterne in un ciclo stazionario risulta eguale al lavoro delle tensioni elastiche (cioè ottenute come risposta puramente elastica del solido alle stesse azioni) attraverso le deformazioni plastiche incrementali.

     

Shakedown analysis of shape memory alloy structures

Phase transformational shakedown of a structure refers to a status that plastic strains cease developing after a finite number of loading cycles, and subsequently the structure undergoes only elastic deformation and alternating phase transformations with limited magnitudes. Due to the intrinsic complexity in the constitutive relations of shape memory alloys (SMA), there is as yet a lack of effective methods for modeling the mechanical responses of SMA structures, especially when they develop both phase transformation and plastic deformation. This paper is devoted to present an algorithm for analyzing shakedown of SMA structures subjected to cyclic or varying loads within specified domains. Based on the phase transformation and plastic yield criteria of von Mises-type and their associated flow rules, a simplified three-dimensional phenomenological constitutive model is first formulated accounting for different regimes of elastic–plastic deformation and phase transformation. Different responses possible for SMA bodies exposed to varying loads are discussed. The classical Melan shakedown theorem is extended to determine a lower bound of loads for transformational shakedown of SMA bodies without necessity of a step-by-step analysis along the loading history. Finally, a simple example is given to illustrate the application of the present theory as well as some basic features of shakedown of SMA structures. It is interesting to find that phase transformation may either increase or decrease the load-bearing capacity of a structure, depending upon its constitutive relations, geometries and the loading mode.     

THE PRECISE AND NEW ANALYSIS FOR A MODE Ⅲ GROWING CRACK IN AN ELASTIC-PERFECTLY PLASTIC SOLID

The near crack line field analysis method has been used to investigate into ModeⅢ quasistatically propagating crack in an elastic-perfectly plastic material.Thesignificance of this paper is that the usual small scale yielding theory has been brokenthrough.By obtaining the general solutions of the stresses and the displacement rate ofthe near crack line plastic region,and by matching the general solutions with theprecise elastic fields(not the usual elastic K-dominant fields)at the elastic-plasticboundary,the precise and new solutions of the stress and deformation fields,the sizeof the plastic region and the unit normal vector of the elastic-plastic boundary havebeen obtained near the crack line.The solutions of this paper are sufficiently precisenear the crack line region because the roughly qualitative assumptions of the smallscale yielding theory have not been used and no other roughly qualitative assumptionshave been taken,either.The analysis of this paper shows that the assumingly“steady-state cas     

Stress across an elastoplastic boundary of a mode I crack: parabolic to hyperbolic plasticity transition

In previous work, the stresses of a mode I elastic–plastic fracture mechanics problem were analytically continued across a prescribed elastoplastic boundary for plane stress loading conditions involving a linear elastic/perfectly plastic material obeying the Tresca yield condition. Immediately across the elastic-plastic boundary, a nonlinear parabolic partial differential equation governs the plastic stress field. The present solution deals with stresses extending beyond the parabolic region into the hyperbolic region of the plastic zone. This analytical solution is obtained through a tranformation of the original system of nonlinear partial differential equations into a linear system with constant coefficients. The solution, so obtained, is expressible in terms of elementary transcendental functions. It also exhibits a limiting line which passes through the crack tip. This feature of the solution suggests the formation of a plastic hinge in the material.     

Strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic–plastic material

An analytical solution is presented for an internally pressurized thick-walled spherical shell of an elastic strain-hardening plastic material. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. The solution gives explicit expressions for the stress, strain and displacement components. The inner radius of the shell enters these expressions not only in non-dimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect. The classical plasticity-based solution of the same problem is shown to be a special case of the present solution. Numerical results for the maximum effective stress in the shell wall are also provided to illustrate applications of the newly derived solution. The new solution can be used to construct improved expanding cavity models in indentation mechanics that incorporate both the strain-hardening and indentation size effects.     

EXACT SOLUTIONS OF NEAR CRACK LINE FIELDS FOR MODE I CRACK UNDER PLANE STRESS CONDITION IN AN ELASTIC－PERFECTLY PLASTIC SOLID

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Wavelet Analysis of Structures: Statics, Dynamics and Damage Identification

Applications of the wavelet transform in solid mechanics are presented herein. The analysis problem is addressed first where the objective is to derive an adaptive wavelet-based method for the static and dynamic analysis of structures. This is done via a collocation scheme. The damage identification problem is investigated next making use of the space-localized properties of the wavelet transform: the regularity of the solution is detected by looking at the amplitude of the coefficients of the wavelet decomposition of the response. Open problems are finally outlined that will be the object of future work.     

Why the stress trajectories in the Dean-Hutchinson plastic sector of the growing mode III crack are an unfocused fan

The plastic zone of the growing mode III crack in an elastic perfectly plastic solid consists of two sectors in contact with each other. The sector closer to the crack plane, first studied analytically by Chitaley and McClintock (CM), consists of a fan of straight maximum shear stress trajectories that are focused on the crack tip. The other sector, first analyzed numerically by Dean and Hutchinson (DH), is a ‘radial’ fan of straight lines that are not focused at the crack tip or at another common point. In this paper it is shown with use of the dislocation density field that the need that the stress magnitude in the plastic wake be below the yield stress requires the existence of an unfocused fan in the DH sector. It appears unlikely that this result can be obtained without explicit use of dislocations.     

One plasticity model for problems of plastic metal working

Scalar and tensor models of plastic flow of metals extending plasticity theory are considered over a wide range of temperatures and strain rates. Equations are derived using the physico-phenomenological approach based on modern concepts and methods of the physics and mechanics of plastic deformation. For hardening and viscoplastic solids, a new mathematical formulation of the boundary-value plasticity problem taking into account loading history is obtained. Results of testing of the model are given. A numerical finite-element algorithm for the solution of applied problems is described. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 159–169, November–December, 2008.     

Refined Dugdale plastic zones of an external circular crack

The refined Dugdale-type plastic zones ahead of an external circular crack, subjected to a uniform displacement at infinity, are evaluated both analytically and numerically. The analytical method utilizes potential theory in classical linear elasticity with emphasis on the contrast from the internal crack problem. A closed-form solution to the mixed boundary problem is obtained to predict the length of the plastic zone for a Tresca yield condition. The analytical solution is also used to benchmark the results obtained from the numerical method, which show good agreement. Through an iterative scheme, the numerical technique is able to estimate the size of crack tip plasticity zone, which is governed by the non-linear von Mises criterion. The relationships between the applied displacement and the length of the plastic zone are compared for the different yielding conditions. Computational modeling has demonstrated that the plastic constraint effect based on the true yield condition can significantly influence the load-bearing capacity. It is also discovered from the comparative study that the stress components predicted by the three different yield conditions may differ notably; however, the stress triaxiality in the ligament region has only small deviations. The proposed study may find applications in predicting the plastic flow in a circumferentially notched round bars under tension.     

Investigation of three-dimensional aspects of grain-scale plastic surface deformation of an aluminum oligocrystal

This is a study of plastic strain localization, surface roughening and of the origin of these phenomena in polycrystals. An oligocrystal aluminum sample with a single quasi-2D layer of coarse grains is plastically deformed under uniaxial tensile loading. During deformation, the history of strain localization, surface roughening, microstructure and in-grain fragmentation is carefully recorded. Using a crystal plasticity finite element model, corresponding high-resolution simulations are conducted. A series of comparisons identifying aspects of good and of less good match between model predictions and experiments is presented. The study suggests that the grain topology and microtexture have a significant influence on the origin of strain heterogeneity. Moreover, it suggests that the final surface roughening profiles are related both to the macro strain localization and to the intra-grain interaction. Finally slip lines observed on the surface of the samples are used to probe the activation of slip systems in detail. The study concludes with an assessment of the limitations of the crystal plasticity model.     

船用加筋板架爆炸载荷下动态响应数值分析

针对船用加筋板架复杂结构在爆炸冲击波作用下的动态响应 ,采用商用高动态非线性有限元程序MSC/Dytran ,讨论了大尺寸加强结构板架迎爆承载问题 ,提出了复杂板架结构爆炸冲击波作用下动态响应的有限元计算方法 ,并进行了模型试验。试验结果与计算结果吻合较好 ,验证了应用程序及计算模型参数的稳定性和可靠性。对加筋板架两种承载形式 (大尺寸加强构件迎爆或背爆设置 )在爆炸冲击波作用下的动态响应 (板架中心挠度和塑性分布 )差异的分析研究表明 ,大尺寸骨架 (纵骨和肋骨 )背向爆炸冲击波设置将分散爆炸冲击波的冲击作用、减小板架变形、增强其抵抗爆炸冲击波冲击的能力 ,使结构偏于安全。     

On the numerical solution of structures with fractal geometry: The FE approach

The scope of the present paper is to present the numerical aspects of the theory developed in [1]. The fractal geometry of structure(s) is approximated either through the IFS (iterated function system) method or through the FI (fractal interpolation) method. These approximations of the fractal through classical curves and surfaces are combined with the FEM in order to get numerical results for important technical problems, which cannot be satisfactorily treated by other methods.

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Sommario Lo scopo del lavoro é quello di discutere gli aspetti numerici della teoria sviluppata in [1]. La geometria frattale della/e struttura é approssimata sia attraverso il metodo IFS (iterated function system) che il metodo FI (fractal interpolation). Queste approssimazioni frattali, attraverso curve e superfici classiche, sono combinate con il metodo degli elementi finiti, onde potere ottenere risultati numerici per importanti problemi tecnici che non potrebbero venire soddisfacentemente affrontati con altri metodi.

     

Influences of microscope size effect of matrix on plastic potential and void growth of porous materials

Based on approximate theoretical analyses on a typical spherical cell containing a spherical microvoid, the influences of matrix materials' microscopic scale on the macroscopic constitutives potential theory of porous material and microvoid growth have been investigated in detail. By assuming that the plastic deformation behavior of matrix materials follows the strain gradient (SG) plastic theory involving the stretch and rotation gradients, the ratio (λ=l/a) of the matrix materials' intrinsic characteristic lengthl to the microvoid radiusa is introduced into the plastic constitutives potential and the void growth law. The present results indicate that, when the radiusa of microvoids is comparable with the intrinsic characteristic lengthl of the matrix materials, the influence of microscopic size effect on neither the constitutive potential nor the microvoid evolution predicted can be ignored. And when the void radiusa is much lager than the intrinsic characteristic lengthl of the matrix materials, the present model can retrogress automatically to the improved Gurson model that takes into account the strain hardening effect of matrix materials. Project supported by the National Natural Science Foundation of China (No. A10102006).     

The Effect of Multiple Buckling Modes on the Postbuckling Behavior of Plane Elastic Frames. Part II. Asymmetric Frames

The postbuckling behavior of an asymmetric one-bay, two-storey frame with clamped edges is analyzed. Columns have different bending stiffnesses and are pairwise of the same length. By assuming columns to be inextensible and shear undeformable, and beams to be rigid, two buckling modes are possible which are described by the sidesway of the lower floor with a rigid horizontal displacement of the upper floor and a sidesway of the upper floor, the lower floor undergoing no displacement. By properly selecting the ratios EI/h² (EI being the bending stiffness of a column and h its length) the two buckling modes may occur simultaneously. A third buckling mode is also possible which is characterized by no displacement of the horizontal beams and local deflection of one or more columns in the shape of a beam with fixed edges. This third case will not be considered in this paper. The Koiter general nonlinear theory of elastic stability recast in a form convenient for the development of finite elment models along lines similar to the recent presentation by Budiansky has been employed in the analysis. Nonlinear constraints on the field variable φ (φ being the cross-section rotation) are accounted for by means of Lagrangian multipliers. Results show that the postbuckling behavior of a single buckling mode is always asymmetric unstable and depends both on the degree of asymmetry of the structure and on the ratio h/l, l being the frame span. The occurrence of simultaneous buckling modes exacerbates the imperfection sensitivity of the structure.     

On the problem of crack system with an elliptic hole

The problem of cracks with an elliptic hole in an infinite plane is investigated. By introducing the fictitious loads on the hole edge and using the Muskhelishvili method, the problem is reduced to solving a system of mixed-type integral equations in which some are Fredholm equations but others Cauchy-type singular ones. A numerical method is suggested and can be used for the treatment of other similar equations. The numerical results for some typical examples are given, showing that the method is really effective.     

圆柱壳的塑性膨胀和变形

根据材料不可压缩假设,适当选取材料塑性本构关系,将圆柱壳塑性膨胀运动简化为以圆柱壳内 半径为因变量的常微分方程初值问题。对3种初始内外半径比纯铜厚壁圆柱壳膨胀过程的计算表明,应变率 硬化阻碍圆柱壳的塑性膨胀运动,应变硬化和温度效应的影响可以忽略,而与圆柱壳的厚度无关。     

The role of dislocation walls for nanoindentation to shallow depths

Dislocation events are seen as excursions or pop-in events in the load–displacement curve of nanoindentation experiments. Two nanoindenters have been used to examine the difference between quasi-static and dynamic loading during indentation. Yield excursions were present in the load–displacement curves of both the statically and dynamically loaded single crystal nickel samples. Only one major excursion occurred in each quasi-static indent, nominally loaded at 100 μN/s while staircase yielding was observed under dynamic loading indentation with a 45 Hz oscillation of 2 nm superimposed on a 60 μN/s loading rate. Thermal activation analysis is used to explain the arrest and reinitiation of the yielding with activation volumes being modeled. For nanoindentation experiments differences between quasi-static and dynamic loading are described by the models presented. It is proposed that insight into the plastic deformation mechanisms associated with such plastic instabilities will provide one of the keys to length scale effects necessary to understanding nanostructures.