Data processing: generic control systems or specific application – Specific application – apparatus or process – Product assembly or manufacturing
Reexamination Certificate
1999-10-19
2004-05-04
Picard, Leo (Department: 2125)
Data processing: generic control systems or specific application
Specific application, apparatus or process
Product assembly or manufacturing
C702S042000
Reexamination Certificate
active
06731996
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention generally relates to the shaping of deformable materials, and more particularly to a method of forming metal sheet into useful articles, wherein tooling is designed using mathematical models that rely on finite element analysis (FEA) techniques to optimize forming operations, tooling design and product performance in the formed articles.
2. Description of Related Art
Many articles are made by stamping, pressing or punching a base material so as to deform it into a piece or part having a useful shape and function. The present invention is concerned with mathematical modelling of the mechanics of such material flow and deformation, and is particularly concerned with the deformation of metal (e.g., aluminum) sheet using tools and dies, to produce a wide variety of products, from beverage cans to components for automotive applications.
When designing the shape of a product, such as a beverage can, it is important to understand how the deformation process will affect the blank of sheet metal. Finite element analysis codes, available from a variety of companies, can be used to analyze plasticity, flow and deformation to optimize forming operations, tooling design and product performance in product designs. These models may result in tooling which improves the quality of a product as well as reducing its cost. The predictive capability of such finite element models is determined to a large extent by the way in which the material behavior is described therein.
In order to appreciate the complexities of modelling the deformation process, it is helpful to understand some basic concepts of mechanical metallurgy, including the concepts of yield stress, workhardening, and strain path.
When some type of external loading device, such as a tensile test machine deforms a metal, the initial response is elastic, with a linear relationship between stress and strain. At some value of the stress, determined by the microstructure of the metal, plastic deformation begins and the response is non-linear, and comprises elastic plus plastic deformation. The yield stress defines the strength of the metal at the condition where plastic deformation is initiated. Deformation beyond the yield stress is characterized by workhardening which causes the stress to increase at an ever decreasing rate until a failure mechanism intervenes and the sample breaks. Thus, the yield stress value and the workhardening curve are the two fundamental entities that define the plastic deformation of metals.
The forming of metal sheet into industrial or consumer products (e.g., cans and automotive components) occurs under multi-axial straining conditions, not the simple uniaxial path described above (tensile testing). In such cases the deformation is described by the strain path. The strain path is defined by the plastic strain tensor.
A tensor is a mathematical entity that is useful in describing various physical properties. Most physical properties can be expressed as either a scalar, a vector, or a tensor. A scalar quantity is one which can be specified with a single number (e.g., temperature or mass), while a vector quantity is one which requires two values, such as magnitude and direction (e.g., velocity or force). A tensor quantity is a higher-order entity that requires more than two values, i.e., more than a single magnitude and direction. For example, a stress tensor is a 3×3 array, each term of which is defined by the stress acting on a given plane, in a given direction. As two direction cosines are required for transformations, the stress tensor is a second order tensor.
The plastic strain (or strain rate) tensor is a second order tensor, which can be expressed as a 3×3 matrix and, in principal axes, has the form:
a
0
0
0
b
0
0
0
c
Common strain paths and their associate values for the plastic strain tensor components are given below:
STRAIN PATH
a
b
c
Uniaxial tension
−0.5
1
−0.5
Uniaxial Compression
0.5
−1
0.5
Biaxial tension
0.5
0.5
−1
Plane Strain Tension
0
1
−1
Plane Strain Compression
0
−1
1
The concepts of the uniaxial stress-strain curve are extended to multi-axial plasticity by defining an effective stress and an effective strain, &sgr;
eff
and &egr;
eff
, which are functions of the components of the stress and plastic strain tensors. The concepts of the yield stress and workhardening are then extended to multi-axial conditions through the use of &sgr;
eff
and &egr;
eff
in place of the &sgr; and &egr; of the uniaxial case. Specifically, the effective stress is given by the second invariant of the stress tensor, and plasticity is referred to as either J
2
or von Mises.
For an isotropic sheet of metal, the plasticity properties do not depend on direction or strain path, and the uniaxial stress-strain curve is all that is required to characterize the forming of sheet into a product. When aluminum sheet is rolled, however, it is anisotropic, meaning that some of the mechanical properties will not be the same in all directions. Because rolled sheet is anisotropic, yield stress as well as workhardening depend on both direction in the sheet and strain path. For example, in aluminum can body stock, the stress-strain curve for a sample cut with its tensile axis in the rolling direction lies below that for a sample cut in the transverse direction. Under multi-axial stress conditions one must now replace the concept of a yield point with that of a yield surface which, in multi-dimensional stress space, defines the boundary between elastic and plastic response. Workhardening manifests itself as an increase in the distance from the origin of stress to a point on the yield surface. One must also allow for the possibility that the workhardening rate may depend on the strain path. Thus, workhardening changes not only the size of the yield surface, but also its shape.
The anisotropy of sheet is determined by crystallographic texture, that is, by the orientations of the crystals that make up the sheet. As single crystal properties are highly anisotropic, the anisotropy of sheet depends on the distribution of orientations of the crystals that comprise it. Thus the orientation distribution function (ODF) is a fundamental property of sheet. There are various types of analysis programs that use crystallographic texture.
The crystallographic texture of sheet, in the form of pole figures, is obtained experimentally using X-ray or neutron diffraction. The ODF and the weights table are calculated from the pole figure data. The latter is particularly important as it defines the volume fraction of crystals having a particular orientation. Typically, the weights for at least 600 discrete orientations are determined by analysis of experimental diffraction data and provide the crucial input for crystal plasticity calculations.
Crystal plasticity theory allows a stress-strain response for a material to be calculated using a given crystallographic texture and specified strain path. A material point simulator (MPS) is an analysis technique that incorporates crystal plasticity theory. Under crystal plasticity theory, the response of a small amount of material subject to a specified strain path is calculated. The response of the aggregate is calculated from the weighted responses to each of the crystals contained in it. Single crystal yield stress and workhardening parameters are determined by an iterative procedure to match prediction from the simulation to a measured stress-strain curve (generally uniaxial tension or compression). Having so determined the single ctystal properties, the stress strain behavior for any desired strain path can be calculated. In addition to conventional workhardening, the calculationsusually include the evolution of texture during deformation along the strain path. In fact, comparison of measured and predicted textures after deformation provides the principal means of validation of material point simuators.
A further analysis technique that is used to model the forming and performance of p
MacEwen Stuart R.
Wu Pei-Dong
Alcan International Limited
Bracewell & Patterson L.L.P.
Picard Leo
Shechtman Sean P
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