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 Bending With Isotropic Strain Hardening at Large Strains

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
Sergei Alexandrov ◽  
Yeong-Maw Hwang
Keyword(s):  
Strain Hardening ◽  
Plane Strain ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Form Solution ◽  
Isotropic Hardening ◽  
Large Strains ◽  
Rigid Plastic ◽  
Elastic Plastic ◽  
Hardening Law

Finite deformation elastic-plastic analysis of plane-strain pure bending of a strain hardening sheet is presented. The general closed-form solution is proposed for an arbitrary isotropic hardening law assuming that the material is incompressible. Explicit relations are given for most popular conventional laws. The stage of unloading is included in the analysis to investigate the distribution of residual stresses and springback. The paper emphasizes the method of solution and the general qualitative features of elastic-plastic solutions rather than the study of the bending process for a specific material. In particular, it is shown that rigid-plastic solutions can be used to predict the bending moment at sufficiently large strains.

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.

Plane Strain Rigid Plastic Finite Element Formulation for Sheet Metal Forming Processes

1993 ◽  
Vol 207 (3) ◽  
pp. 167-171 ◽  
Author(s):  
P A F Martins ◽  
M J M Barata Marques
Keyword(s):  
Finite Element ◽  
Plane Strain ◽  
Sheet Metal ◽  
Sheet Metal Forming ◽  
Metal Forming ◽  
Yield Criterion ◽  
Large Strain ◽  
Theoretical Development ◽  
Element Formulation ◽  
Rigid Plastic

A rigid plastic finite element model for analysing two-dimensional plane strain sheet metal forming processes is described. The model is based on the large strain formulation using membrane theory, and the material is assumed to be rigid plastic, work hardening and conforms to Hill's anisotropic yield criterion and associated flow rules. The theoretical development follows the work of Kobayashi and Kim on the axisymmetric modelling of sheet metal forming. An application of the model for plane strain cylindrical punch stretching is presented. The results obtained are compared with those provided through an analytical membrane solution described in this work. The agreement found between both solutions is excellent.

The importance of shear in the yield criterion for the bending of slabs

1971 ◽  
Vol 6 (1) ◽  
pp. 13-19 ◽  
Author(s):  
R H Wood
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Yield Criterion ◽  
Experimental Investigations ◽  
Plastic Shear ◽  
Rigid Plastic ◽  
Elastic Plastic ◽  
Simple Supports ◽  
Shear Strains ◽  
First Time

An impasse has arisen in present-day rigid-plastic theory for bending of slabs, in that exact solutions may not exist for certain combinations of shape, load, and boundary conditions. It is now shown, for the first time, that a complete elastic-plastic solution for a uniformly loaded square slab is unobtainable even with simple supports, in spite of the fact that an exact rigid-plastic solution is already well known. These discrepancies are traced to over-idealization of the yield criterion, in particular to the possibility of missing plastic-shear strains. The need for new experimental investigations is apparent, as well as for changes in the theory.

scholarly journals An Analytic Solution for an Expanding/Contracting Strain-Hardening Viscoplastic Hollow Cylinder at Large Strains and Its Application to Tube Hydroforming Design

Processes ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 2161
Author(s):  
Lihui Lang ◽  
Sergei Alexandrov ◽  
Marina Rynkovskaya
Keyword(s):  
Strain Rate ◽  
Hollow Cylinder ◽  
Yield Criterion ◽  
Tube Hydroforming ◽  
Equivalent Strain ◽  
Large Strains ◽  
Flow Rule ◽  
Rigid Plastic ◽  
Equivalent Strain Rate ◽  
Pressure Independent

This paper presents a semi-analytic rigid/plastic solution for the expansion/contraction of a hollow cylinder at large strains. The constitutive equations comprise the yield criterion and its associated flow rule. The yield criterion is pressure-independent. The yield stress depends on the equivalent strain rate and the equivalent strain. No restriction is imposed on this dependence. The solution is facilitated using the equivalent strain rate as an independent variable instead of the polar radius. As a result, it reduces to ordinary integrals. In the course of deriving the solution above, the transformation between Eulerian and Lagrangian coordinates is used. A numerical example illustrates the solution for a material model available in the literature. A practical aspect of the solution is that it readily applies to the preliminary design of tube hydroforming processes.

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.

The Theoretical Analysis of Metal-Forming Problems in Plane Strain

1952 ◽  
Vol 19 (1) ◽  
pp. 97-103
Author(s):  
E. H. Lee
Keyword(s):  
Plane Strain ◽  
Metal Forming ◽  
Plastic Material ◽  
Large Strains ◽  
Rigid Plastic ◽  
Small Strains ◽  
Plastic Type ◽  
Static Determinacy ◽  
Complete Solutions

Abstract The plastic flow in plane strain of an ideally plastic material subjected to large strains is considered. Elastic strains are negligible and a rigid-plastic type of analysis is adopted. The equations to be satisfied are detailed, and they include stress and velocity equations in the plastic regions, as well as consideration of the stress field in the rigid regions to check the validity of small strains there. Complete solutions satisfying these conditions require the determination of the rigid-plastic boundaries to delineate the regions in which the various conditions must be satisfied. The fallacy of static determinacy of such problems in terms of the stress equations only is emphasized. The study of complete solutions indicates errors in solutions commonly accepted in the literature which are based on the stress equations only. Examples are discussed. The general occurrence of velocities in the boundary conditions of forming problems is pointed out, and the difficulty of setting such problems in terms of boundary stresses only is illustrated by examples.

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.

Improved Compliance Solutions for C(T), SE(B) and Clamped SE(T) Specimens Including Side-Grooves, Varying Thicknesses and 3-D Effects

2014 ◽  
Author(s):  
Gustavo Henrique B. Donato ◽  
Felipe Cavalheiro Moreira
Keyword(s):  
Crack Growth ◽  
Plane Strain ◽  
Plane Stress ◽  
Potential Drop ◽  
Crack Size ◽  
Growth Conditions ◽  
Size Estimation ◽  
Unloading Compliance ◽  
Integrity Assessments ◽  
Elastic Unloading

Fracture toughness and Fatigue Crack Growth (FCG) experimental data represent the basis for accurate designs and integrity assessments of components containing crack-like defects. Considering ductile and high toughness structural materials, crack growing curves (e.g. J-R curves) and FCG data (in terms of da/dN vs. ΔK or ΔJ) assumed paramount relevance since characterize, respectively, ductile fracture and cyclic crack growth conditions. In common, these two types of mechanical properties severely depend on real-time and precise crack size estimations during laboratory testing. Optical, electric potential drop or (most commonly) elastic unloading compliance (C) techniques can be employed. In the latter method, crack size estimation derives from C using a dimensionless parameter (μ) which incorporates specimen’s thickness (B), elasticity (E) and compliance itself. Plane stress and plane strain solutions for μ are available in several standards regarding C(T), SE(B) and M(T) specimens, among others. Current challenges include: i) real specimens are in neither plane stress nor plane strain - modulus vary between E (plane stress) and E/(1-ν2) (plane strain), revealing effects of thickness and 3-D configurations; ii) furthermore, side-grooves affect specimen’s stiffness, leading to an “effective thickness”. Previous results from current authors revealed deviations larger than 10% in crack size estimations following existing practices, especially for shallow cracks and side-grooved samples. In addition, compliance solutions for the emerging clamped SE(T) specimens are not yet standardized. As a step in this direction, this work investigates 3-D, thickness and side-groove effects on compliance solutions applicable to C(T), SE(B) and clamped SE(T) specimens. Refined 3-D elastic FE-models provide Load-CMOD evolutions. The analysis matrix includes crack depths between a/W=0.1 and a/W=0.7 and varying thicknesses (W/B = 4, W/B = 2 and W/B = 1). Side-grooves of 5%, 10% and 20% are also considered. The results include compliance solutions incorporating all aforementioned effects to provide accurate crack size estimation during laboratory fracture and FCG testing. All proposals revealed reduced deviations if compared to existing solutions.

On Thick-Walled Cylinder Under Internal Pressure

2003 ◽  
Vol 125 (3) ◽  
pp. 267-273 ◽  
Author(s):  
W. Zhao ◽  
R. Seshadri ◽  
R. N. Dubey
Keyword(s):  
Internal Pressure ◽  
Strain Curve ◽  
Stress Strain ◽  
Piecewise Linearization ◽  
Elastic Plastic ◽  
Plastic Cylinder ◽  
Parametric Functions ◽  
The Difference ◽  
New Formulation ◽  
Walled Cylinder

A technique for elastic-plastic analysis of a thick-walled elastic-plastic cylinder under internal pressure is proposed. It involves two parametric functions and piecewise linearization of the stress-strain curve. A deformation type of relationship is combined with Hooke’s law in such a way that stress-strain law has the same form in all linear segments, but each segment involves different material parameters. Elastic values are used to describe elastic part of deformation during loading and also during unloading. The technique involves the use of deformed geometry to satisfy the boundary and other relevant conditions. The value of strain energy required for deformation is found to depend on whether initial or final geometry is used to satisfy the boundary conditions. In the case of low work-hardening solid, the difference is significant and cannot be ignored. As well, it is shown that the new formulation is appropriate for elastic-plastic fracture calculations.

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