The second example is an analytic function but unlike the first analytic
function, eqn. (1), there is a physical basis for this function, hence, there is
an opportunity to relate our function to the geometry and physics of this
particular problem. This function is given below where the strain,
&epsilon_{zz}, is a function of the crack radius, r_{c},
fiber volume fraction, V_{f}, and elastic property ratio, R, taken from
a report Ref.[1] written by **Carmen, Lesko, and Reifsnider.**

(3)

The function given above is derived from the boundary value problem shown below in Fig. 13, where the unidirectional fiber-reinforced composite sustains a crack in the center fiber of a composite subject to a prescribed remote strain and the crack grows towards the adjacent fibers. The strains at the interface of the adjacent fibers are identical due to symmetry and these strains should increase with increasing crack growth. The effect of increasing fiber volume fraction and elastic property ratio on the interface strain is less obvious, but the physics would suggest an increasing strain with increasing fiber volume fraction and elastic property ratio. As before with eqn. (1), trends can not be deduced from eqn. (3) because the complexity of the analytic function does not suggest a possible trend, "it goes like". Here our objective is to better understand the relationship between interfacial strains over a range of parameters that can be physically controlled during fabrication of this composite material system. Hence like the previous experimental data set we are also interested in choosing from a range of possible parameters where this selection process results in a well designed composite material system. Again this selection process can best be accomplished using the compressed-interactive-comparative format.

Figure 13. Physical parameters defined for fiber fracture model.

Again we avoid using the family-of-curves format and immediately construct the
function in the comparative-interactive-compressed format as shown in Fig. 14.
Before moving any planes, we observe the highest strain appears in the lower left
corner where R=1, r_{c} = r_{a}, and V_{f}.

Figure 14. Interfacial strain relationship with material parameters.

As in the previous examples the horizontal r_{c} - V_{f} plane
is the initial plane of
interest but as shown in Fig. 15 movement of this plane reveals little new
information. In Fig. 16, the strain variations observed in the r_{c} - R plane
reveal a minimum strain over a large range of fiber volume fractions. But in
Fig. 17 we observe a minimum strain only for the larger crack lengths. In these
figures we observe that for a given crack length we can choose an optimum fiber
volume fraction and elastic property ratio that will minimize the interfacial
strain at the unbroken neighboring fibers. Note that both the fiber volume
fraction and elastic property ratio can be controlled by the material engineer
who is designing this material system. Hence we can use this visual method to
optimize the fabrication of a material system for a given property: in this
example we have minimized the strain to failure of adjacent fibers. Of course
the same result could have been obtained by using a variational analytic method,
but only if the function is differentiable. In some cases a numerical method is
appropriate when an analytic approach is impossible. In either case it may be
desirable to first observe in a comparative format which parameters are the best
candidates for optimization. Again we find that by using a visual comparative
method first, we can more efficiently use numerical, analytical or experimental
methods which are better at providing quantitative information. Hence visual
methods complement existing methods.

Figure 15. Variation in interfacial strain in the r_{c} -
V_{f} plane at various values for R.

Figure 16. Variation in interfacial strain in the r_{c} -
R plane at various values for V_{f}.

Figure 17. Variation in interfacial strain in the V_{f} -
R plane at various values for r_{c}.

The contents of this web site are © Copyright Ronald D. Kriz. You may print or save an electronic copy of parts of this web site for your own personal use. If used elsewhere in published or unpublished form please cite as, "private communication, R.D. Kriz" and include link to this web address. Permission must be sought for any other use.

Send comments to: rkriz@vt.edu

Ronald D. Kriz, Short Bio

Engineering Science and Mechanics

College of Engineering

Virginia Tech

Blacksburg, Virginia 24061

Created September, 1995

Revised August, 2008

Original: http://www.sv.vt.edu/classes/ESM4714/methods/Carmen.html

Current: http://www.sv.rkriz.net/classes/ESM4714/methods/Carmen.html