Next: Hashin and Rosen
Up: Exact solutions for
Previous: Intrinsic properties for
When thermal strains (eigenstrains) are present, there are several other sets of exact results available to check the output of the programs THERMAL3D.F and THERMAL2D.F. Suppose first that for all n phases, each phase has the same elastic moduli but different isotropic thermal strains, with the thermal strain in each direction being ei for the i'th phase. Then the overall system strain in any direction will be simply ε = Σ ci ei , i = 1, n. This result is known in the physics community as Vegard's law [26]. This result is true for any microstructure, in both two and three dimensions. Actually, the thermal strain tensor need not be isotropic. The overall thermal strain tensor will be the sum of the volume fraction-weighted phase thermal strain tensors.
A related result, not as well-known, by Goodier [27] applies to this same case where there are only two isotropic phases. In this case, the trace of the strain in the inclusion phase (3-D) is simply
The trace of the stress in the inclusion is then also constant, and is simply given by (3-D)
It is known that if an elliptical (2-D) or ellipsoidal (3-D) inclusion is introduced into a material, where the inclusion and matrix have the same elastic moduli but different thermal strains, the individual components of the stress tensor are uniform inside the inclusion [7,27]. Recently, a conjecture has been made that, in 2-D, if the inclusion is in the shape of a equilateral 5-pointed star, the stress tensor components inside the star will also be uniform [28]. The Goodier result [27] of course predicts that at least the trace of the stress tensor will be uniform in any inclusion.
This conjecture about a 5-pointed star can be tested using the program THERMAL2D.F. A digital image can be made of an inclusion in a matrix, where the inclusion has arbitrary shape. With the properties assigned, the system can be relaxed and a stress picture can be made, along with a picture of the trace of the stress tensor. In most case, it should be obvious whether a quantity is uniform within the inclusion or not. In addition, the average stress and its standard deviation can be computed within the inclusion. If a quantity is indeed uniform, then the standard deviation should be much smaller than the average, and reflect only things like finite size effects, and using square pixels to approximate curved and angular boundaries.
Figure 11: Thermal stress for
inclusions with the same elastic moduli as the matrix, but different
thermal eigenstrain. Stress is high (top)
to low (bottom) in color bar).
Left: −σxx, right: negative of the trace of the
stress tensor. Images from top:
Ellipse, 5-pointed star, 6-pointed star.
The top images in Fig. 11 show an elliptical inclusion, with an aspect ratio of three. The stress map on the left is of −σxx, where the stress levels, from high to low, are red-green-white-black (top to bottom in the accompanying color bar). This component of the stress appears to be uniform inside the inclusion, as it is supposed to be. The 2-D moduli used were: E2 = 1, ν2 = 0.3, K2 = 5/7, and G2 = 5/13, in arbitrary units. The inclusion had a thermal strain of exx = eyy = 1, and the matrix had zero thermal strain. The right top image shows the negative trace of the stress tensor, which is also clearly uniform inside the inclusion. The stress in the inclusion is negative (compressive), since a positive thermal strain, constrained by the outer matrix, results in a compressive stress inside the inclusion (see Section 2.4).
The left image in the 2nd row from the top of Fig. 11 shows a map of −σxx, with the same choice of elastic parameters, for a 5-pointed star-shaped inclusion. This component of the stress tensor is clearly non-uniform in the inclusion. The right hand image shows the negative trace of the stress tensor, which certainly appears uniform, according to Goodier's result. With the choice of elastic parameters used, the trace of the stress tensor is predicted to be, inside the inclusion, equal to −1. For a 512 x 512 unit cell, and the area fraction of the star being 7.6%, the trace of the average stress tensor inside the inclusion is 0.93, in error by only 7%. This difference reflects the finite extent of the system, since the Goodier result is for an infinite system, and errors at the star boundary, whose straight lines are approximated by a digital boundary. A similar finite element computation was also carried out recently [29], with similar results. The remaining row in Fig. 11 shows analogous computations for a six-pointed star, with results similar to that of the 5-pointed star.
Next: Hashin and Rosen Up: Exact solutions for Previous: Intrinsic properties for
