Study of the Effect of Deformation on the Axes of Anisotropy

1991 ◽  
Vol 113 (2) ◽  
pp. 187-191 ◽  
Author(s):  
A. Kumar ◽  
Shyam K. Samanta ◽  
K. Mallick
Keyword(s):  
Theoretical Model ◽  
Metal Forming ◽  
Grain Shape ◽  
Reference State ◽  
Deformation Tensor ◽  
Yield Behavior ◽  
Experimental Scheme ◽  
Principal Stretches ◽  
Principal Directions ◽  
Subsequent Deformation

Many metal forming operations, such as rolling and tube drawing, are known to induce orthotropic anisotropy. The change of axes of orthotropy with subsequent deformation has been studied in this paper. The change in the orthotropy directions is of great importance for understanding and interpreting the subsequent yield behavior of metals. Based on Hill’s hypothesis that the orthotropy axes coincides with the principal directions of stretch, the change in orthotropy directions has been studied theoretically and experimentally. Since the grain shape and its direction of elongation is a good indicator of the principal stretches and its directions, it has been used as an experimental means of determining, not only the directions of principal stretches in an as received material, but also to determine approximately the deformation it has undergone so far from a reference state. A fully annealed isotropic state is chosen as the reference state. The directions of the axes of anisotropy, induced as a result of finite deformation applied to this reference state, are characterized in terms of the principal directions of the Cauchy’s deformation tensor. An experimental scheme has been developed to determine the varying directions of orthotropy for comparison with the theoretical model.

Strain History and Polar Decomposition

1996 ◽  
Author(s):  
Gerhard Oertel
Keyword(s):  
Polar Decomposition ◽  
Transformation Matrix ◽  
Strain History ◽  
Transformation Matrices ◽  
Before And After ◽  
The Matrix ◽  
Principal Stretches ◽  
Principal Directions ◽  
The Right ◽  
Asymmetric Transformation

The effect of two consecutive strains (only two states enter into the calculation of a strain, the states before and after, independently of the actual strain path) can be calculated by premultiplying the transformation matrix of the first strain (its stretch tensor) with that of the second. Unless the two strains are coaxial (their principal directions coincide), however, the resulting cumulative transformation matrix represents not only a strain but also a rigid-body rotation; in that case the matrix is asymmetric. The method of polar decomposition allows one to interpret the combined transformation as if it had come about either by a strain followed by a rotation (right polar decomposition) or by a rotation followed by a strain (left polar decomposition). Let 𝔸 and 𝔹 be two stretch tensors, or transformation matrices, representing each a strain without rotation; and let the strain 𝔹 follow the strain 𝔸. Then the combined transformation matrix 𝔽 is: . . . 𝔹𝔸 = 𝔽 = ℝ𝕌= 𝕍ℝ, (8.1) . . . where 𝔽 results from premultiplication of the earlier stretch 𝔸 with the later 𝔹, where ℝ𝕌 is the “right” and 𝕍ℝ the “left” decomposition of 𝔽, where 𝕌 and 𝕍 are two distinct stretch tensors, and where ℝ is the transformation matrix for a rotation (elements of rotation matrices are indicated by the symbol aij elsewhere in this book). 𝔽 is asymmetric and ℝ differs from the identity matrix (δij) except when 𝔸 and 𝔹 are coaxial. 𝕌 and 𝕍 have the same principal stretches and differ by orientation only. In Problems 120 to 122, false approaches in the search for an appropriate decomposition of an asymmetric transformation were recognized by yielding impossible values for a rotation. Application of eq. (8.1) makes such a trial-and-error approach unnecessary.

scholarly journals Theory and Experiment of Isotropic Electromagnetic Beam Bender Made of Dielectric Materials

2010 ◽  
Vol 150-151 ◽  
pp. 1508-1516 ◽  
Author(s):  
Qi Bo Deng ◽  
Jin Hu ◽  
Zheng Chang ◽  
Xiao Ming Zhou ◽  
Geng Kai Hu
Keyword(s):  
Electromagnetic Wave Propagation ◽  
Dielectric Materials ◽  
Deformation Tensor ◽  
Spatial Transformation ◽  
Low Loss ◽  
Material Parameters ◽  
Electromagnetic Beam ◽  
Reduced Parameters ◽  
Principal Stretches ◽  
Special Case

In this paper, we utilize the deformation transformation optics (DTO) method to design electromagnetic beam bender, which can change the direction of electromagnetic wave propagation as desire. According to DTO, the transformed material parameters can be expressed by deformation tensor of the spatial transformation. For a beam bender, since the three principal stretches at each point induced by the spatial transformation are independent to each other, there are many possibilities to simplify the transformed material parameters of the bender by adjusting the stretches independently. With the DTO method, we show that the reported reduced parameters of the bender obtained by equivalent dispersion relation can be derived as a special case. An isotropic bender is also proposed according to this method, and it is fabricated by stacking dielectric materials in layered form. Experiments validate the function of the designed isotropic bender for a TE wave; it is also shown that the isotropic bender has a broadband with low loss, compared with the metamaterial bender. The isotropic bender has much easier design and fabrication procedures than the metamaterial bender.

Some Consequences of Material Frame Indifference on Engineering Flow Calculations

2004 ◽  
Author(s):  
G. Mompean ◽  
L. Thais ◽  
L. Helin
Keyword(s):  
Angular Velocity ◽  
Deformation Tensor ◽  
Algebraic Models ◽  
Frame Indifference ◽  
Material Frame Indifference ◽  
Flow Through ◽  
Principal Directions ◽  
Viscoelastic Liquids ◽  
Turbulent Fluids ◽  
Material Frame

The principle of material frame indifference (MFI) is a fundamental and controversial principle of continuum mechanics which has been invoked to derive recent nonlinear algebraic models for stresses of viscoelastic liquids (Mompean, Thompson, and Souza Mendes [2003]) and for Newtonian turbulent fluids (Rumsey, Gatski, and Morrison [2000]). The purpose of the present study is to identify regions of a flow field where MFI should be considered. Such regions are identified by computing the angular velocity of the principal directions of the rate-of-deformation tensor in order to obtain a Euclidean objective vorticity tensor. The method is applied to the planar flow through an abrupt 4:1 contraction. The main results are: (i) MFI should be taken into account in regions characterized by the transition between two different viscometric kinematics and significant velocities (ii) MFI can be safely ignored in regions of pure viscometric behavior as well as in recirculation regions.

Interpretation of Diffraction Data from In Situ Stress Measurements during Biaxial Sheet Metal Forming

2013 ◽  
Vol 768-769 ◽  
pp. 441-448
Author(s):  
Thomas Gnäupel-Herold ◽  
Mark Iadicola ◽  
Adam Creuziger ◽  
Tim Foecke ◽  
Lin Hu
Keyword(s):  
Sheet Metal Forming ◽  
Metal Forming ◽  
Lattice Strain ◽  
Hybrid Approach ◽  
Yield Behavior ◽  
Strain Measurements ◽  
X Ray ◽  
Lattice Strains ◽  
Model Predictions

Biaxial yield behavior is determined in-situ through X-ray lattice strain measurements. The distributions of d-spacings in different sample directions is affected both by the changes in diffraction elastic constants (DEC) from evolving texture and by the intergranular (IG) strains. Model predictions were found to be lacking, thus, a hybrid approach was developed based on measurements of DEC and IG strains at selected biaxial deformations. In order to convert measured lattice strains to stress for any given biaxial plastic strain a theoretical approximation was fitted to the experimental data, thus allowing the estimation of the evolution of DEC and IG strains with plastic deformation.

Comparison of Hyper- and Hypoelastic-Based Formulations of Elastoplasticity with Applications to Metal Forming

2013 ◽  
Vol 554-557 ◽  
pp. 2321-2329 ◽  
Author(s):  
Tim Brepols ◽  
Ivaylo N. Vladimirov ◽  
Stefanie Reese
Keyword(s):  
Metal Forming ◽  
Finite Strain ◽  
Fe Simulation ◽  
Kinematic Hardening ◽  
Deformation Gradient ◽  
Deformation Tensor ◽  
Drawing Process ◽  
Metallic Materials ◽  
Multiplicative Decomposition ◽  
Plastic Parts

The aim of this work is to examine two specific finite strain elastoplasticity models in terms of their applicability in metal forming processes, namely a hyperelastic-based model which relies upon a multiplicative decomposition of the deformation gradient into elastic and plastic parts and a hypoelastic-based model which makes use of an additive elastic-plastic split of the rate of deformation tensor. Both models allow for nonlinear isotropic and kinematic hardening and were implemented as user material subroutines (UMAT) into ABAQUS/Standard. Various sample calculations were performed to assess the respective properties and capabilities of the models. The FE simulation of a deep drawing process produced nearly congruent results for both models which suggests that they are equally well-suited for modeling metallic materials in metal forming processes.

Deformation Analysis of Side-Wall Curl in the Sheet-Metal Forming of Flanged Channels

2005 ◽  
Vol 127 (2) ◽  
pp. 369-375 ◽  
Author(s):  
Fuh-Kuo Chen ◽  
Pao-Ching Tseng
Keyword(s):  
Theoretical Model ◽  
Stress Distribution ◽  
Sheet Metal ◽  
Sheet Metal Forming ◽  
Metal Forming ◽  
Numerical Procedure ◽  
Sheet Thickness ◽  
Side Wall ◽  
Forming Process ◽  
Proposed Model

The side-wall curl occurring in the sheet-metal forming process of a flanged channel was examined by a proposed theoretical model in the present study. Since the side-wall curl results from the elastic recovery of the plastically deformed sheet metal, the stress distribution produced in the forming process is examined. In the theoretical model, the deformation of the sheet metal drawn over the die shoulder is assumed to be subjected to bending, sliding, and unbending processes, in which only the sliding process contributes to the frictional force. The governing equations derived from the theoretical model were solved by a numerical procedure, and the stress distribution through the sheet thickness was obtained to calculate the side-wall curl. The proposed model was validated by the finite element simulations both quantitatively and qualitatively, and by the experimental data obtained from the published literature qualitatively. By using the proposed model, the effects of the process parameters on the side-wall curl were investigated.

Design and Experimental Verification during Extrusion of Square Sections from Round Billets through Curved Dies

2011 ◽  
Vol 491 ◽  
pp. 249-256 ◽  
Author(s):  
Akshaya Kumar Rout ◽  
Kali Pada Maity ◽  
Manoja Kumar Parida
Keyword(s):  
Mechanical Properties ◽  
Theoretical Model ◽  
Metal Forming ◽  
Dimensional Accuracy ◽  
Extrusion Process ◽  
Uniform Microstructure ◽  
Dead Metal Zone ◽  
Extruded Product ◽  
Cad Models ◽  
Die Profile

Extrusion is one of the widely used metal forming processes. The extrusion process is carried out conventionally using a shear faced die, but shear faced dies have many practical problems such as a dead metal zone, more redundant work. In the present investigation, the evolution of uniform microstructure in extruded product with improved mechanical properties for quality products to get dimensional accuracy. A mathematically contoured non-linear converging die has been designed for extrusion of square section from round billet. CAD models of die profile have also been developed. The experiments have been conducted to verify the proposed theoretical model. The extrusion test rigs have been fabricated to carry out extrusion through mathematically contoured dies.

Strain Histories and Strain Distributions in a Cup Drawing Operation

1971 ◽  
Vol 93 (2) ◽  
pp. 461-466 ◽  
Author(s):  
T. C. Hsu ◽  
W. R. Dowle ◽  
C. Y. Choi ◽  
P. K. Lee
Keyword(s):  
Sheet Metal ◽  
Metal Forming ◽  
Drawing Process ◽  
Neck Formation ◽  
Forming Process ◽  
Orthogonal Space ◽  
Points Of Interest ◽  
Metal Forming Process ◽  
Principal Directions ◽  
Cup Drawing

In the axisymmetrical cup drawing process, the principal directions of the strains are fixed with respect to the work material at every point and in every stage of the process; in other words, the strains are entirely coaxial ones—if the small strains in simple shear due to friction are ignored. For a properly chosen set of orthogonal space coordinates, therefore, the strains may be plotted in triangular coordinates. In such a coordinate system for strains, the loci for constant penetrations show the strain distributions, and those for constant initial radial positions show the strain histories. In these loci it is easy to see thinning and thickening, circumferential expansion and contraction, neck formation, variation in thickness, and other points of interest to the sheet metal engineer. Typical examples of strain histories and strain distributions in a cup drawing operation are shown. The method is applicable to any axisymmetrical sheet metal forming process.

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