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C-MOLD Shrinkage & Warpage User's Guide
Because a thin-wall part is broken into shell elements, each element is calculated in two dimensions. The physical state of stress at any point can be expressed by three stress components in the elemental local coordinate system, two normal stresses xx, yy, and the shear stress xy. In finite element formulations, these stress components are usually written in a vector form, and the elastic properties are organized correspondingly in a matrix form. This matrix is called the Hook matrix. As mentioned in Appendix A, the shell element consists of uncoupled plate-bending (DKT) and plane-stress membrane (VRT) approximations. The calculation of the stresses are also based on these two approximations. The constitutive equations in the linear-elastic plate-bending (b) and plane-stress (ps) cases are given by:
where M is the vector of internal bending moments, w is the local displacement component perpendicular to the element plane, and are the strain and curvature vectors, and Dps, Db are the Hook matrices, respectively.
For a homogeneous, isotropic material model, the two Hook matrices for the shell element are as follows:
where E is the elastic modulus and v is the Poisson's ratio, h is the thickness of the element. Comparing (C.3) and (C.4) one can see that these two matrices are proportional. This is only true for a transversely homogeneous, isotropic material.
For the case of a homogeneous orthotropic material (transversely isotropic), the matrix for the plane-stress is
where E1, E2 are the elastic moduli in the first and second principal directions, respectively, v12 is the Poisson's ratio for the principal plane, G12 is the shear modulus. In this case, the Hook matrix for the plate-bending remains proportional to the matrix for the plane-stress problem.
The plate-bending stresses vary linearly across the thickness, and their maximum values are given by
The total stresses are the summation of these two contributions,
The strain and curvature calculations are based on the displacement solution, and the element local deformation matrix Bps and Bb, i.e.
and ui, vi and wi (i = 1, 2, 3) are the displacement components in local element coordinate system at the three node locations, xi, yi, zi are the corresponding rotation components.
The expressions of the deformation matrices Bps and Bb can be derived from of the shell element approximations (VRT + DKT), which can be found in reference 2.
The above stress calculations are first calculated at three Gauss integration points in an element, and then averaged into element center for the elemental values of the stresses. All the stress calculations are based on local coordinate system, thus the stress components are expressed in the local coordinate system. In order to avoid the coordinate-dependency of the components, principal stresses are calculated for output.
The stress state at a point can be represented by the well known Mohr's circle as shown in Figure C-1:
Figure C-1. Geometric representation of stress state in the well known Mohr's Circle.
The principal stresses are the extreme values of the normal stresses. Since they characterize the physical state of the stress at a point, they are independent of any coordinates of reference. They are calculated in the following way:
in which 1 represents the maximum normal stress and 2 the minimum normal stress.
The corresponding directions of the principal stresses are called the principal directions. The angle for these is calculated using the following equation:
The maximum shear stress is the extreme value of the shear stress. It is in ±45° from the principal directions. The value of the maximum shear stress is:
The mean stress is calculated by averaging the principal stresses:
The Mises-Hencky stress is calculated by using the following formulation:
The highest of the above mentioned stress values in a part is often used to check against the yield stress, to help the design engineer make sure the part does not fail. The distributions of these stresses are used to predict the performance of the design.
1. Timoshenko, S. and Woinowsky-Krieger, S., Theory of Plates and Shells, Second Edition, 1959
2. Kudryavtsev, L., Structural Analysis of Injection-Molded Parts, CIMP Progress Report No. 17, Feb. 1993
3. Zienkiewicz, O. C., The Finite Element Method, The third edition, 1977
4. Beer, F. P. and Johnson, E. R. Jr., Mechanics of Materials, McGraw-Hill, Inc., 1981, 1979
5. Spotts, M. F., Design of Machine Elements, Prentice-Hall, Inc. Englewood Cliffs, N.J., 1978, 1971, 1961, 1953, 1948
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