Elastic-Plastic Plane-Strain Solutions With Separable Stress Fields

1969 ◽  
Vol 36 (3) ◽  
pp. 528-532
Author(s):  
D. B. Bogy
Keyword(s):  
Stress Field ◽  
Plane Strain ◽  
Yield Criterion ◽  
A Priori ◽  
Equivalent Plastic Strain ◽  
Uniaxial Stress ◽  
Stress Fields ◽  
Isotropic Hardening ◽  
Flow Rule ◽  
Elastic Plastic

A stress field σij(χ, t) depending on position χ and time t will be called separable if the time-dependence enters only through a scalar multiplier; i. e., if σij(χ,t)=s(χ,t)σˆij(χ). It is shown here that elastic-plastic plane-strain solutions in the infinitesimal (flow) theory of plasticity satisfying Tresca’s yield criterion and associated flow rule with linear isotropic hardening, based on either equivalent plastic strain or total plastic work, can occur with separable stress fields only in the following instances: (a) solutions with uniaxial stress fields, as in bending, (b) solutions with stress fields such that the entire domain changes from elastic to plastic at the same time, and (c) solutions with stress fields for which the elastic-plastic boundary coincides with a principal shear stress trajectory. Whether or not a plasticity solution has a separable stress field can be determined a priori by examining the corresponding elasticity solution.

scholarly journals Finite Plane Strain Bending under Tension of Isotropic and Kinematic Hardening Sheets

Materials ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1166
Author(s):  
Stanislav Strashnov ◽  
Sergei Alexandrov ◽  
Lihui Lang
Keyword(s):  
Plane Strain ◽  
Kinematic Hardening ◽  
Tensile Force ◽  
Plastic Region ◽  
Equivalent Plastic Strain ◽  
Isotropic Hardening ◽  
Finite Plane ◽  
Present Solution ◽  
Bending Under Tension ◽  
Elastic Unloading

The present paper provides a semianalytic solution for finite plane strain bending under tension of an incompressible elastic/plastic sheet using a material model that combines isotropic and kinematic hardening. A numerical treatment is only necessary to solve transcendental equations and evaluate ordinary integrals. An arbitrary function of the equivalent plastic strain controls isotropic hardening, and Prager’s law describes kinematic hardening. In general, the sheet consists of one elastic and two plastic regions. The solution is valid if the size of each plastic region increases. Parameters involved in the constitutive equations determine which of the plastic regions reaches its maximum size. The thickness of the elastic region is quite narrow when the present solution breaks down. Elastic unloading is also considered. A numerical example illustrates the general solution assuming that the tensile force is given, including pure bending as a particular case. This numerical solution demonstrates a significant effect of the parameter involved in Prager’s law on the bending moment and the distribution of stresses at loading, but a small effect on the distribution of residual stresses after unloading. This parameter also affects the range of validity of the solution that predicts purely elastic unloading.

scholarly journals Finite Pure Plane Strain Bending of Inhomogeneous Anisotropic Sheets

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 145
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Yeong-Maw Hwang
Keyword(s):  
Plane Strain ◽  
Yield Criterion ◽  
Large Strains ◽  
Rigid Plastic ◽  
Plastic Properties ◽  
Elastic Plastic ◽  
Starting Point ◽  
The Difference ◽  
Inside Radius ◽  
Elastic Unloading

The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary for solving transcendent equations and evaluating ordinary integrals. The solution’s starting point is a transformation between Eulerian and Lagrangian coordinates that is valid for a wide class of constitutive equations. The symmetric distribution relative to the center line of the sheet is separately treated where it is advantageous. It is shown that this type of symmetry simplifies the solution. Hill’s quadratic yield criterion is adopted. Both elastic/plastic and rigid/plastic solutions are derived. Elastic unloading is also considered, and it is shown that reverse plastic yielding occurs at a relatively large inside radius. An illustrative example uses real experimental data. The distribution of plastic properties is symmetric in this example. It is shown that the difference between the elastic/plastic and rigid/plastic solutions is negligible, except at the very beginning of the process. However, the rigid/plastic solution is much simpler and, therefore, can be recommended for practical use at large strains, including calculating the residual stresses.

Plane-Strain Crack-Tip Fields for Pressure-Sensitive Dilatant Materials

1990 ◽  
Vol 57 (1) ◽  
pp. 40-49 ◽  
Author(s):  
F. Z. Li ◽  
J. Pan
Keyword(s):  
Plane Strain ◽  
Effective Stress ◽  
Hydrostatic Stress ◽  
Yield Criterion ◽  
Crack Tip ◽  
Flow Rule ◽  
Pressure Sensitivity ◽  
Crack Tip Fields ◽  
Pressure Sensitive ◽  
Plane Strain Crack

Plane-strain crack-tip stress and strain fields are presented for materials exhibiting pressure-sensitive yielding and plastic volumetric deformation. The yield criterion is described by a linear combination of the effective stress and the hydrostatic stress, and the plastic dilatancy is introduced by the normality flow rule. The material hardening is assumed to follow a power-law relation. For small pressure sensitivity, the plane-strain mode I singular fields are found in a separable form similar to the HRR fields (Hutchinson, 1968a, b; Rice and Rosengren, 1968). The angular distributions of the fields depend on the material-hardening exponent and the pressure-sensitivity parameter. The low-hardening solutions for different degrees of pressure sensitivity are found to agree remarkably with the corresponding perfectly-plastic solutions. An important aspect of the effects of pressure-sensitive yielding and plastic dilatancy on the crack-tip fields is the lowering of the hydrostatic stress and the effective stress directly ahead of the crack tip, which may contribute to the experimentally-observed enhancement of fracture toughness in some ceramic and polymeric composite materials.

A Compressible Elastic, Perfectly Plastic Wedge

1958 ◽  
Vol 25 (2) ◽  
pp. 239-242
Author(s):  
D. R. Bland ◽  
P. M. Naghdi
Keyword(s):  
Plane Strain ◽  
Yield Criterion ◽  
The State ◽  
Hardening Material ◽  
Included Angle ◽  
Elastic Plastic ◽  
Numerical Example ◽  
Perfectly Plastic ◽  
Elastic Perfectly Plastic

Abstract This paper is concerned with a compressible elastic-plastic wedge of an included angle β < π/2 in the state of plane strain. The solution, deduced for an isotropic nonwork-hardening material, employs Tresca’s yield criterion and the associated flow rules. By means of a numerical example the solution is compared with that of an incompressible elastic-plastic wedge in one case (β = π/4) for various positions of the elastic-plastic boundary.

Elastic-Plastic Stress Distribution in a Plastically Anisotropic Rotating Disk

2004 ◽  
Vol 71 (3) ◽  
pp. 427-429 ◽  
Author(s):  
N. Alexandrova ◽  
S. Alexandrov
Keyword(s):  
Stress Distribution ◽  
Yield Criterion ◽  
Plastic Anisotropy ◽  
Rotating Disk ◽  
Flow Rule ◽  
Plane State ◽  
Annular Disk ◽  
Elastic Plastic ◽  
State Of Stress ◽  
Associated Flow Rule

The plane state of stress in an elastic-plastic rotating anisotropic annular disk is studied. To incorporate the effect of anisotropy on the plastic flow, Hill’s quadratic orthotropic yield criterion and its associated flow rule are adopted. A semi-analytical solution is obtained. The solution is illustrated by numerical calculations showing various aspects of the influence of plastic anisotropy on the stress distribution in the rotating disk.

Elastic-plastic stress distributions and limit angular velocities in rotating hyperbolic annular discs

2007 ◽  
Vol 221 (2) ◽  
pp. 137-142 ◽  
Author(s):  
N N Alexandrova ◽  
P M M Vila Real
Keyword(s):  
Rotational Speed ◽  
Yield Criterion ◽  
Circumferential Stress ◽  
Radial Pressure ◽  
Thickness Variation ◽  
Flow Rule ◽  
Stress Distributions ◽  
Elastic Plastic ◽  
Angular Velocities ◽  
Hyperbolic Form

Plastic analytical stress analysis of a rotating annular disc with its contours being free from the radial pressure and with specifically variable thickness is presented in terms of the Mises-yield criterion and its associated flow rule. The hyperbolic form of thickness variation is considered and optimized towards the maximum rotational speed and favourable stress combinations. Radial and circumferential stress distributions in the disc both in the intermediate elastic-plastic and in the limit plastic states are obtained. As a particular case, limit elastic angular velocity parameter is derived. The influences of rotational speed as well as the disc's thickness profile on the plastic solution and size of elastic-plastic zone are demonstrated and discussed. The results obtained may be used for the correct implementation of numerical codes and preliminary engineering design.

A nonlinear simulation of a bi-failure specimen through improved discretisation methods: A validation study

2018 ◽  
Vol 53 (8) ◽  
pp. 616-629 ◽  
Author(s):  
Behzad V Farahani ◽  
Jorge Belinha ◽  
Paulo J Tavares ◽  
Pedro MGP Moreira
Keyword(s):  
Digital Image Correlation ◽  
Digital Image ◽  
Numerical Study ◽  
Interpolation Method ◽  
Yield Criterion ◽  
Image Correlation ◽  
Uniaxial Tensile ◽  
Isotropic Hardening ◽  
Flow Rule ◽  
Nonlinear Simulation

A robust and efficient scheme is rendered to elastoplastically study the material nonlinearity of structural components. In this investigation, a specimen manufactured from the aluminium alloy AA6061-T6 is considered. It is mechanically loaded under a uniaxial tensile state and the experimental strain datum is analysed by three-dimensional digital image correlation. Due to specific specimen geometry, complex stress states will occur. However, the specimen yields due to an approximated uniaxial stress state. The obtained remote stress/strain from experimental data is used to validate the computational solutions using advanced discretisation approaches. Therefore, as a preliminary numerical study, the model is simulated through the finite element method formulation. Afterwards, another numerical strategy is adopted – the radial point interpolation method. The Newton–Raphson initial stiffness method is thereby adapted to complete the nonlinear solutions algorithm. Furthermore, the elastoplastic demeanour of aluminium alloys is determined with the von Mises yield criterion, an isotropic hardening rule and an associative flow rule. Obtained computational results fit the experimental digital image correlation solution, which allow to conclude that the proposed meshless methodology is efficient and reliable.

Plane-Strain Tension Tests on Aluminum Alloy Sheet

1995 ◽  
Vol 117 (2) ◽  
pp. 168-171 ◽  
Author(s):  
Fadi Taha ◽  
Alejandro Graf ◽  
William Hosford
Keyword(s):  
Aluminum Alloy ◽  
Plane Strain ◽  
Yield Criterion ◽  
Alloy Sheet ◽  
Isotropic Hardening ◽  
Aluminum Alloy Sheet ◽  
Strain Measurements ◽  
Plane Strain Tension ◽  
Tension Tests ◽  
High Exponent

A simple way of making plane-strain tension tests on sheet specimens has been developed. This method was used to test sheets of aluminum alloy 2008 T4 and the results were analyzed in terms of a high exponent yield criterion and isotropic hardening. Experimentally measured forces agreed with those calculated from strain measurements using uniaxial tension test curves.

The Study of Elastic-Plastic Fatigue Behavior of Nigerian-Rolled NST 37-2 Steel

2010 ◽  
Vol 3 ◽  
pp. 28-41
Author(s):  
B.O. Malomo ◽  
S.A. Ibitoye ◽  
L.O. Adekoya
Keyword(s):  
Numerical Study ◽  
Plastic Material ◽  
Yield Criterion ◽  
Kinematic Hardening ◽  
Fatigue Behavior ◽  
Numerical Results ◽  
Test Material ◽  
Flow Rule ◽  
Static Fatigue ◽  
Elastic Plastic

The NST 37-2 steel represents about 75% volume of Nigerian-produced steel which is yet to be fully characterized for its fatigue behavior. Thus, its suitability for many applications is questionable. This paper presents a framework based on the theory of elasto-plasticity in order to make appropriate recommendations in this regard. Experimentally, tensile tests were carried out on test specimens to establish the baseline material properties of the steel in annealed, as-rolled, normalized and hardened/tempered conditions. Fatigue tests were then conducted at 60% Su; 70% Su and 80% Su of the test material and fractographic examinations on the test specimens were subsequently carried out. The frequency harmonic fatigue analysis was implemented in the ANSYS software environment for the numerical study. The elastic-plastic material property was described by the von Mises yield criterion, the flow rule of Prandtl-Reuss, and the kinematic hardening rule of Prager. The numerical results indicate with respect to rate-dependence fatigue behavior that the annealed test specimen is most resilient under cyclic deformation as compared with the normalized, hardened/tempered and as-rolled specimens respectively. The experimental and numerical results were found to be in close agreement and based on the general performance, the steel material is recommended for use in low cycle, quasi-static fatigue applications.

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