General Analytic Solutions and Complementary Variational Principles for Large Deformation Nonsmooth Mechanics

This paper presents a nonlinear dual transformation method and general complementary energy principle for solving large deformation theory of elastoplasticity governed by nonsmooth constitutive laws. It is shown that by using this method and principle, the nonconvex and nonsmooth total potential energy is dual to a smooth complementary energy functional, and fully nonlinear equilibrium equations in finite deformation problems can be converted into certain tensor equations. The algebraic relation between the first and the second Piola–Kirchhoff stresses are revealed. A closed form solution for general three-dimensional large deformation boundary value problems is obtained. The properties of this general solution are clarified by a triality extremum principle. This triality theory reveals an important phenomenon in nonconvex variational problems. Applications are illustrated by nonlinear, nonsmooth equilibrium problems in Hencky's plasticity, 3D cylindrical structures and post buckling analysis of elastoplastic bar with jumping and hardening effects. The idea and methods presented in this paper can be used and generalized to solve many nonlinear boundary value problems in finite deformation theory.Sommario. Il lavoro presenta un metodo di trasformazione duale nonlineare ed un principio generale di energia complementare per la soluzione di problemi di teoria elastoplastica in grandi deformazioni governati da leggi costitutive con discontinuità. Si mostra come, usando il metodo ed il principio proposti, l' energia potenziale totale discontinua e nonconvessa duale di un funzionale energia complementare continuo, e le equazioni di equilibrio nonlineare in problemi di deformazione finita, possano essere convertite in equazioni tensoriali. Vengono mostrate le relazioni algebriche fra il primo ed il secondo tensore delle tensioni di Piola–Kirchhoff. Si ottiene una soluzione in forma chiusa per problemi al contorno generali tridimensionali in grandi deformazioni. Le proprietà di tale soluzione generale vengono chiarite per mezzo di un principio estremale di trialità. La eoria della trialità evidenzia un fenomeno importante in problemi variazionali nonconvessi. Vengono presentate applicazioni a problemi di equilibrio nonlineare con discontinuità in situazioni di plasticità alla Hencky, strutture cilindriche in 3D, e nell'analisi postcritica di una barra elastoplastica con effetti hardening e di jumping. L'idea ed i metodi presentati in questo lavoro possono essere usati e generalizzati per risolvere molti problemi al contorno nonlineari nella teoria delle deformazioni finite.     

基于最小加速度原理的刚塑性动力问题

应用有限变形的最小加速度原理，推导出了小变形或有限变形构件受到刚性飞射物撞击时的刚塑性运动方程，指出该方法具有直接、简捷和可靠的特点。     

Influence of deformation on irreversible and reversible crystallization of poly(ethylene‐co‐1‐octene)

The effect of uniaxial deformation and subsequent relaxation at ambient temperature on irreversible and reversible crystallization of homogeneous poly(ethylene‐co‐1‐octene) with 38 mol % 1‐octene melt‐crystallized at 10 K min was explored by calorimetry, X‐ray scattering, and Fourier transform infrared spectroscopy. At 298 K, the enthalpy‐based crystallinity of annealed specimens increased irreversibly by stress‐induced crystallization from initially 15% to a maximum of, at least, 19% when a permanent set of more than 200% was attained. The crystallinity increased by formation of crystals of pseudohexagonal structure at the expense of the amorphous polymer, and as a result of destruction of orthorhombic crystals. The stress‐induced increase of crystallinity was accompanied by an increase in the apparent specific heat capacity from 2.44 to about 2.59 J g^?1 K^?1, which corresponds to an increase of the total reversibility of crystallization from, at least, 0.10 to 0.17% K^?1. The specific reversibility calculated for 100% crystallinity increased from 0.67 to 0.89% K^?1 and points to a changed local equilibrium at the interface between the crystal and amorphous phases. The deformation resulted in typical changes of the phase structure and crystal morphology that involve orientation and destruction of crystals as well as the formation of fibrils. The effect of the decrease of the entropy of the strained melt on the reversibility of crystallization and melting is discussed. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1223–1235, 2002     

Molecular deformation mechanisms of isotactic polypropylene in α‐ and β‐crystal forms by FTIR spectroscopy

The orientation behavior of isotactic polypropylene (iPP) in α‐ and β‐crystal form was investigated by rheo‐optical Fourier transformed infrared (FTIR) spectroscopy. This method enabled quantification of the degree of orientation as a feature of structural changes during uniaxial elongation in not only the crystalline phase but also the amorphous one. Molecular orientation mechanisms can be successfully derived from experimental results. Generally, three mechanisms were detected for iPP: (1) interlamellar separation in the amorphous phase, (2) interlamellar slip and lamellar twisting at small elongations, and (3) intralamellar slip at high elongations. The third mechanism was favored by α‐PP, whereas β‐PP favored the second mechanism, which, in fact, was responsible for the different mechanical properties of both materials at the macroscopic level. On the other hand, crystallization conditions may have significantly affected the amorphous orientation. Nevertheless, for both iPP types the chains in the amorphous phase always oriented less than did those in the crystalline phase. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 4478–4488, 2004     

Flange earing and its control on deepdrawing of anisotropy circular sheets

The Hill's quadric anisotropy yield function and the Barlat-Lian anisotropy yield function describing well anisotropy sheet metal with stronger texture are introduced into a quadric-flow corner constitutive theory of elastic-plastic finite deformation suitable for deformation localization analysis. And then, the elastic-plastic large deformation finite element formulation based on the virtual power principle and the discrete Kirchhoff shell element model including the yield functions and the constitutive theory are established. The focus of the present research is on the numerical simulation of the flange earing of the deep-drawing of anisotropy circular sheets, based on the investigated results, the schemes for controlling the flange earing are proposed. Supported by NSFC(No. 19832020) and National Automobile Dynamic Simulation Laboratory of China.     

Diffusion effects on the interfacial tension of immiscible polymer blends

Most methods of measuring the interfacial tension between two immiscible polymers are based on the analysis of the shape that a drop of one polymer immersed in the other one exhibits under the action of flow or gravity. In such a situation, the small, yet nonzero mutual solubility between the two polymers acts toward mass transfer between the drop and the surrounding fluid. In this work, diffusion effects on the interfacial tension of the pair polyisobutylene/polydimethylsiloxane have been investigated by drop deformation under shear flow. When the drop was made of polyisobutylene, drop size decreased with time due to diffusion. Drop shrinkage was associated with a significant increase in interfacial tension, until an apparent plateau value was reached. The effect was attributed to a selective migration of molecular weights, which would act to enrich the drop with higher molar mass material. To support such an interpretation, drop viscosity was evaluated by drop shape analysis and it was actually found to increase with time. In some cases, the ratio between drop and continuous phase viscosity became higher than the critical value for drop breakup in shear flow. Upon inverting the phases (i.e., when the drop was made of polydimethylsiloxane), no significant transient effects were observed. In the light of these results, the problem of what are the correct values of interfacial tension and viscosity ratio for a polymer blend of a certain composition will also be discussed. Received: 25 January 1999 Accepted: 24 May 1999     

On the deformation of linear r-fields

We study the question: For which (r,n) can a linear r-field on the (n-1)-sphere in an n-dimensional real linear space be deformed through a continuous path of linear r-fields into an orthonormal r-field. We provide complete answers for the cases: (r,n)=(2,4),(3,4), and provide several partial results for the cases (r,n)=(2,2m), where m is an even integer satisfying m4. Characterizations of linear r-fields are pivotal in the investigation.     

Security number of grid-like graphs

The security number of a graph is the cardinality of a smallest vertex subset of the graph such that any attack on the subset is defendable. In this paper, we determine the security number of two-dimensional cylinders and tori. This result settles a conjecture of Brigham et al. [R.C. Brigham, R.D. Dutton, S.T. Hedetniemi, Security in graphs, Discrete Appl. Math. 155 (2007) 1708-1714].     

Rheology and microstructures formation of immiscible model polymer blends under steady state and transient flows

Viscoelastic properties of model immiscible blend were studied here under steady state condition at different initial conditions and transient flow conditions. The flow‐induced microstructure has been studied on these model blends. For this system, the elastic properties of the blend are mainly governed by the interface. Measurement of the dynamic modulus and of the first normal stress difference, both reflecting this enhanced elasticity, have been used to prove the blend morphology. The dynamic moduli after cessation of shear flow, the mean diameter of the disperse phase as generated by the shear flow, have been calculated using the model of Palierne. A procedure based on a direct fitting of the dynamic moduli with the model is compared with the one that uses a weight relaxation spectrum. On the other hand, the steady state normal stress data have been related to the morphology of the blend by means of Doi and Ohta model. The specific interfacial area is found to be inversely proportional to the ratio of interfacial tension over shear stress for the blend. The flow behavior during transient shear flow was also discussed. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 3519–3533, 2005