On the finite displacement problem of a hollow cylinder under internal and external pressures

Using the method of successive approximations we find of this boundary-value problem the first-and second-order solutions. And then we obtain the formulae in the second approximation for the displacement, strain, and stress fields. Also, our results show that after deformation (i) a cross-section of the cylinder must be displaced into a plane section perpendicular to the central axis of the cylinder; and (ii) neither the sum of the strain components E _RR ⁽²⁾ and E ⁽²⁾ nor the sum of the stress components E _RR ⁽¹⁾ and E ⁽¹⁾ maintains contant throughout the cylinder. The latter effect, which is absent from classical elasticity, bears respensibility for the presence of the E _ZZ ⁽¹⁾ . Moreover, there exhibits a linear relation between E _ZZ ⁽¹⁾ and (E _RR ⁽²⁾ +E _OO ⁽¹⁾ ), with the proportionality coefficients depending only on the material of the cylinder.     

An analysis on three-dimensional effects near the tip of a crack in an elastic plate

An infinite plate containing a finite through crack under tensile loading is analysed by Fourier transform based on the Kane-Mindlin kinematic assumptions for the quasi-three-dimensional deformation of plates in extension. The asymptotic expressions of stress and displacement fields near the crack tip, the variation of the stress intensity factor with the plate-thickness and the three-dimensional deformation zone near the crack tip are investigated. The results of the analysis show that, (a) the crack-tip stress and displacement fields accounting for the plate-thickness effects are different from the plane stress solutions and this is true even for extremely small parameter (=1–vh/6 a). In a very small region near the crack tip, plane strain solutions prevail; (b) the ratio of the stress intensity factor K_I to the corresponding plane stress one K_I, K_I/K _I ^o , approaches 1/(1–v²) as tends to zero; (c) plane stress solutions can give satisfactory results for points a distance from the crack tip greater than about three-fourths of the plate-thickness; (d) the linear elastic result for the zone of three-dimensional effects is approximately valid for an elasto-plastic material with linear strain-hardening when the plastic tangential mudulus E_t is not very small.The Project Supported by National Natural Science Foundation of China.     

Molecular dynamics simulation of deformation and failure of nanocrystals of bcc metals

Simulations of uniaxial and hydrostatic tension of Fe and Mo nanocrystal are made by molecular dynamics method. Stress versus strain are obtained while regularities of lattice rearrangement during nanocrystal plastic deformation are considered. Local instability of nanocrystal lattice, which is the cause for transition from elastic to plastic deformation of nanocrystal, is found. It is shown that local shear stresses is a driving force of nanocrystal lattice rearrangements under the conditions of both uniaxial and hydrostatic tension, so, local instability of nanocrystal of bcc metals should be considered as shear instability. Realization of “orthorhombic” path of deformation at 1 0 0 tension of Mo nanocrystal is specific case of above effect. It is demonstrated that unlike covalent nanocrystal, metallic nanocrystals display “heterogeneous” mechanism of crack nucleation, which essence is that cracks nucleate not in homogeneous elastically deformed lattice but in shear bands or near their boundaries, i.e., after non-homogeneous plastic deformation of nanocrystal.     

预估在非比例加载下薄金属板成型极限的损伤基力学模型

探讨了用损伤基力学模型研究应变路径对薄金属板塑性失稳的影响,这种力学模型考虑了材料损伤作用．基于这种模型,在等效应变空问建立了考虑损伤的塑性失稳判据,并用以预估在比例或非比例加载下薄金属板成型极限曲线(FLC)．借助这种理论模型和方法,预估薄金属板的理论成型极限曲线与Graf和Hosford的实验结果一致。     

Analysis of transformation superplastic deformation in pure iron

Summary Transformation superplastic deformation in pure iron is analysed. The transformation phenomenology is first studied when the material is exposed to the constant-rate cyclic temperature histories between 20°C and 1100°C. Transformation diagrams (T-T-A, T-T-T, C-H-T and C-C-T diagrams) are shown to be constructed by means of the transformation kinetics employed. Change in the fraction of austenite, or of ferrite, during the cyclic process is calculated. The deformation in thin-walled tubular specimen is then estimated when it is exposed under a constant tensile or torsional stress to the temperature cyclings. Accumulation of the superplastic deformation due to A ₃ transformation is shown to be well simulated by the theory presented here.

---

Deformationsanalysis der Umwandlungsplastizität bei Eisen

Übersicht Die superplastische Deformation während der allotropen Umwandlung bei Eisen, A ₃ (Austenit Ferrit)-Umwandlung, wird analysiert. Die Umwandlungsphänomenologie wird theoretisch diskutiert, wenn das Material einer zyklischen Temperaturgeschichte zwischen 20°C und 1100°C ausgesetzt wird. Es wird gezeigt, daß man Umwandlungsdiagramme mit Hilfe der vorgeschlagenen Umwandlungskinetik gut konstruieren kann. Die Austenitbildung oder die Ferritbildung während des zyklischen Prozesses wird berechnet. Die Deformation von dünnwandigen, rohrförmigen Proben wird dann analysiert, wenn sie bei konstanter Zugspannung oder Torsionsspannung gleichzeitig durch ein zyklisches Temperaturfeld belastet werden. Die Ergebnisse zeigen, daß die akkumulierende Superplastizitätsdeformation von der Theorie gut beschrieben werden kann.

     

Shear-induced aggregation and break up of fibril clusters close to the percolation concentration

To probe the behaviour of fibrillar assemblies of ovalbumin under oscillatory shear, close to the percolation concentration, c_p (7.5%), rheo-optical measurements and Fourier transform rheology were performed. Different results were found close to c_p (7.3%), compared to slightly further away from c_p (6.9 and 7.1%). For 6.9 and 7.1%, a decrease in complex viscosity, and a linear increase in birefringence, n, with increasing strain was observed, indicating deformation and orientation of the fibril clusters. For 7.3%, a decrease in complex viscosity was followed by an increase in complex viscosity with increasing strain, which coincided with a strong increase in n, dichroism, n, and the intensity of the normalized third harmonic (I₃/I₁). This regime was followed by a second decrease in complex viscosity, where n,n and I₃/I₁ decreased. In the first regime where the viscosity was decreasing with increasing strain, deformation and orientation of existing clusters takes place. At higher oscillatory shear, a larger deformation occurs and larger structures are formed, which is most likely aggregation of the clusters. Finally, at even higher strains, the clusters break up again. An increase in complex viscosity, n, n and I₃/I₁ was observed when a second strain sweep was performed 30 min after the first. This indicates that the shear-induced cluster formation and break up are not completely reversible, and the initial cluster size distribution is not recovered after cessation of flow.     

High‐Velocity Penetration of Plane Shaped‐Charge Jets into Nonlinear Media

This paper reports results of metallographic analysis of metal samples cut from targets penetrated by plane shapedcharge jets. It is shown that the plastic deformation due to penetration has a turbulent nature and, in some cases, it occurs in metals with fractal structure formed after passage of the shock wave running ahead of the jet. A penetration model is proposed that takes into account the nonlinear behavior of the target material and the fractality of its structure.     

Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage

Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic–plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice’s localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic–plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results – e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.     

Test for sheet-metal formability with coupon and circular specimens

Formability in sheet metal is defined as the locus of the limiting strain. The initiation of localized thinning, which manifests itself as a groove, is taken as the criterion of failure. The deformation of circular and coupon specimens is discussed with particular reference to the effect of the process parameters on the strain ratios at the critical section. Curves are presented for the formability of aluminum sheets with failure in both the direction of rolling and that perpendicular to it.     

Evaluation of the VDA 238–100 Tight Radius Bend Test for Plane Strain Fracture Characterization of Automotive Sheet Metals

Accurate characterization of the fracture limit in plane strain tension of automotive sheet metals is critical for the design and crash performance of structural components. Plane strain bending using the VDA 238–100 V-bend test has potential for proportional fracture characterization by avoiding a tensile instability. The VDA 238–100 V-bend test was evaluated using DIC strain measurement to characterize the plane strain fracture limit under proportional plane stress loading and to evaluate the effect of the VDA pre-straining methodology for ductile alloys upon the material response. The load-based failure criterion of the V-bend test was evaluated with DIC to monitor the development of surface cracking. The influence of the non-linear strain path imposed by the pre-straining procedure for ductile materials was then evaluated for three automotive alloys: an advanced high strength dual phase steel, DP1180, a rare-earth magnesium, ZEK100, and an AA5182 aluminum. A fracture criterion based on the load threshold was reasonable for the three alloys considered. Pre-straining in uniaxial tension prior to plane strain bending affected each alloy differently. The DP1180 was not affected by the non-linear strain path whereas the cumulative equivalent strain for the AA5182 and ZEK100 increased by strains of 0.07 and 0.05 strain, respectively. The non-linear strain path within the VDA pre-straining methodology creates ambiguity in comparing the fracture limits of different materials. The plane strain fracture limit for proportional loading can be readily obtained in the V-bend test with DIC strain measurement.

     

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.     

The origin of stress-oscillation damping during startup and reversal of torsional shearing of nematics

Using a controlled-temperature shear cell mounted on a polarizing microscope, we observe the behavior of nematic 4,4-n-octyl-cyanobiphenyl (8CB) during start-up and reversal of shearing in a torsional parallel-plate geometry and correlate this behavior with rheological measurements. During the start-up, a sequence of birefringent rings, or twist walls, are observed that originate at the sample edge and propagate radially inward. Each twist wall is a thin region in which the director is twisted out of the plane of the velocity and velocity-gradient directions. The radial variation of in-plane orientation can be explained by the variation of strain in the parallel-plate device. A high Ericksen-number solution of the Leslie-Ericksen equations predicts a damped oscillatory shear stress response which agrees quantitatively with the measured stress oscillations out to an edge strain of around 50. The damping of the stress oscillations is due to the nonuniformity of strain in the parallel-plate geometry. On reversal of the flow, if the strain, , is smaller than about 500 units, the damping of stress oscillations is reversed; this correlates with an outward radial migration of twist walls. When > 500, disclinations nucleate and spoil the reversibility of stress damping.Deceased     

Collision between an ethanol drop and a water drop

The collision between a water drop and an ethanol drop was studied. In a binary collision between unlike miscible drops with a large surface tension difference, an unbalanced-surface-force (USF) deformation on the drop of larger surface tension, i.e., the water drop, occurs during the first stage of the collision. This deformation may squeeze out small satellites from the water drop at low-impact-parameter collisions or split the water drop at high-impact-parameter collisions. The later stages of the collision behavior, namely, coalescence and separation, resemble those of the drops of the smaller surface tension, i.e., the ethanol drop.     

Post-critical plastic deformation in incrementally nonlinear materials

The formation of multiple macroscopic shear bands is investigated as a mechanism of advanced plastic flow of polycrystalline metals. The overall deformation pattern and material characteristics are determined beyond the critical instant of ellipticity loss, without the need of introducing an internal length scale. This novel approach to the modelling of post-critical plastic deformation is based on the concept of a representative nonuniform solution in a homogeneous material. The indeterminacy of a post-critical representative solution is removed by eliminating unstable solution paths with the help of the energy criterion of path instability. It is shown that the use of micromechanically based, incrementally nonlinear corner theories of time-independent plasticity leads then to gradual concentration of post-critical plastic deformation. The volume fraction occupied by shear bands is found to have initially a well-defined, finite value insensitive to the mesh size in finite element calculations. Further deformation depends qualitatively on details of the constitutive law. In certain cases, the volume fraction of active bands decreases rapidly to zero, leading to material instability of dynamic type. However, for physically hardening materials with the yield-vertex effect, the localization volume typically remains finite over a considerable deformation range. At later stages of the plane strain simulation, differently aligned secondary bands are formed in a series of bifurcations.     

Rigid-plastic and damage behavior in metal forming: Part II—model prediction

Indeterminacy of the Lévy-Mises relations in plane strain rigid-plasticity is overcome by considering change in plastic strain rate due to triaxiality. Better agreement is achieved on the coincidence of uniaxial data with the effective stress and effective strain in addition to the removal of ill conditions associated with necking stresses derived from other theories of rigid plasticity. The formulation accounts for material compressibility and makes use of the variational principle when applied to discrete locations of the continuum. These physical refinements provide the necessary accuracy for predicting potential damage sites and establishing design limits on metal forming products. Developed are damage thresholds for defining the integrity of rigid-plastic materials with nonlinear behavior. Distortion and dilatation of the material elements are mutually interactive and their proportion changes with the local strain rates in the metal forming process. Only the stationary values of the volume energy density could automatically account for the nonlinear relationship between distortion and dilation. Completed in Part I is the development of a modified theory of rigid-plasticity that better describes the yield and fracture behavior of metals. Numerical results for the permanent deformation of sheet metals are provided in Part II; they are compared with those obtained from other theories of rigid-plasticity.     

A decomposition depended on directions for nonsingular linear transformation

ＡＤＥＣＯＭＰＯＳＩＴＩＯＮＤＥＰＥＮＤＥＤＯＮＤＩＲＥＣＴＩＯＮＳＦＯＲＮＯＮＳＩＮＧＵＬＡＲＬＩＮＥＡＲＴＲＡＮＳＦＯＲＭＡＴＩＯＮＺｈａｎｇＳｈｅｎｘｕｅ（张慎学）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ．ＪｉｌｉｎＵｎｉｖｅｒｓｉｔｙ...     

The site model theory and the standard linear solid

Summary The site model theory (SMT) is shown to lead to the same deformation behaviour as that displayed by the standard linear solid (SLS), group I, for all loading conditions. If a second deformation mechanism (inter-molecular slip) is introduced the result is the same as that obtained with the standard linear solid, group II, and models the behaviour of a polymer melt near to the solidification temperature.

---

Zusammenfassung Es wird gezeigt, daß ein einfaches Platzwechsel-Modell (site model theory) bei allen Belastungsbedingungen das gleiche Deformationsverhalten voraussagt wie der lineare Drei-Parameter-Festkörper (standard linear solid, group I). Wenn ein weiterer Deformationsmechanismus (zwischenmolekulare Gleitung) eingeführt wird, entspricht das Verhalten dagegen demjenigen einer linearen Drei-Parameter-Flüssigkeit (standard linear solid, group II), welche das Verhalten einer Polymerschmelze in der Nähe der Schmelztemperatur beschreibt.

---

a = ₁₂ ⁰ + ₂₁ ⁰ , see eq. [1] - b =N ₁ ⁰ ₁₂ ⁰ (V ₁₂ +V ₂₁), see eq. [1] - c = 2N _s ⁰ V _s see eq. [6] - k Boltzmann constant - t time - E,E ₁,E ₂ spring constants, see figures 1 and 3 - E _u unrelaxed modulus - N ₁ ⁰ site 1 equilibrium population in the unstressed state - N _s number of units available for slip - N(t) decrease in site 1 population - N _s (t) net number of slip jumps in the stressaided direction - T temperature (K) - V _i,j activation volume for jumps in directioni j - V _s activation volume for the slip process - strain - strain rate - incremental change in strain per unit change in site population - µ,µ ₁,µ ₂ dashpot constants, see figures 1 and 3 - applied stress - ₀ initial applied stress, (stress relaxation) =(t) (creep) - incremental change in stress per unit change in site population - ⁰ jump rate for slip in the unstressed state - _i,j ⁰ jump rate in the directioni j in the unstressed state With 3 figures and 3 tables     

Gas flow from plane vessels with large angles of inclination of the walls

Numerical solutions are obtained for problems of steady supersonic gas flow from infinite plane vessels with angles of inclination of the walls to the plane of symmetry _c on the interval 90° < _c 180°. The problems are posed and solved in the hodograph plane. It is shown that starting from a certain _c* the flow choking mechanism is determined not by the arrival of the limiting characteristic at the edge of the opening (classical choking mechanism) but by the interaction of the jet with the outside surface of the vessel wall. The effect of _c and the ambient pressure on the local and integral flow characteristics is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 144–151, January–February, 1991.The author wishes to thank A. N. Kraiko for discussing the results and N. I. Tillyaev for assisting with the work.     

About the origin of sharkskin

Summary Some new results of experimental study of linear polyethylene melt flow instabilities are given. Two effects, the main flow instability and the sharkskin phenomenon, are presented in terms of relaxation oscillator theory, and permit an understanding of the influence of experimental parameters on the period of the oscillatory flow. Sharkskin is interpreted to be the same phenomenon as the other instability, but to occur in the die entry region rather than in the whole die itself. Various aspects of sharkskin characteristics are considered.

---

Zusammenfassung Es werden einige neue Ergebnisse einer experimentellen Untersuchung betreffend die Instabilität des Fließverhaltens von Polyäthylen hoher Dichte in geschmolzenem Zustand mitgeteilt. Zwei Effekte, die Haupt-Fließinstabilität und das sogenannte Sharkskin-Phänomen, werden mit Hilfe der Theorie der Relaxations-Schwingungen erklärt. Diese Theorie erlaubt es, die Wirkung der experimentellen Parameter auf die Fließschwingungsdauer zu verstehen. Man findet, daß Sharkskin und Haupt-Fließinstabilität im wesentlichen das gleiche Phänomen darstellen, daß aber Sharkskin im Düseneintritt entsteht statt in der ganzen Düse. Die Intensität von Sharkskin wird zu der Schwingungsperiode in Beziehung gesetzt.

---

A cross section of the barrel - C capacity - l length of the real die - l ₁ length of the equivalent die corresponding to die entry region - l ₂ length of that part of the die for which the pressure gradient is constant - n flow index - P pressure in the reservoir - Q _ex exit flow rate or flow rate within the die - Q _in inlet flow rate - Q ₁ ,Q ₂ upper and lower boundary of the oscillation of flow rate - Q _SS flow rate at which the sharkskin phenomenon occurs - Q _MFI flow rate at which the main flow instability occurs - Q flow rate at which the main flow instability disappears - r ₁ radius of the equivalent die corresponding to die entry region - r ₂ orr radius of the real die - R resistance - S _p plunger speed - S wall shear stress - S _ss critical die wall shear stress at which sharkskin first occurs - S _MF critical die wall shear stress at which the main flow instability first occurs - T oscillation period - T ₀ die auto-oscillation period - t time - t _eq time necessary for reaching equilibrium conditions - t _c that part of the period during which pressure is increasing - t _d that part of the period during which pressure is decreasing - z distance between die and plunger - mass density - apparent shear viscosity - bulk compressibility With 11 figures and 3 tables     

Numerical study for influences of material parameters on hydrostatic bulging of metal sheet

In this paper, the influences of various material parameters, the hardening exponent (n), the rate sensitivity (m). the thickness anisotropy parameter (R) and the index M in the Hosford and Hill yield function, on the hydrostatic bulging of a circular clamped sheet of ductile metal materials are analysed by introducting a rigid-viscoplastic finite element method. By numerical studies, an empirical relationship within the average limit thickness strain – ₃ ^* and the material parameters (n andm) is obtained. Besides, it has been found that the influences of surface shapes of the yield function on the average limit thickness strain can be reflected by the Barlat'sP value which represents the effects ofR andM values.This work is supported by the National Natural Science Foundation and Natural Science Foundation of the Youth of China.