Efthymios S. Folias

Professor of Mechanics & Mathematics (emeritus)

Department of Mathematics

University of Utah

Ph.D. Aeronautics (Solid Mechanics), California Institute of Technology, 1963

M.S. Mathematics, University of New Hampshire, 1960

B.S. Electrical Engineering, University of New Hampshire, 1959

Professor of Mechanics & Mathematics (emeritus)

Department of Mathematics

University of Utah

Ph.D. Aeronautics (Solid Mechanics), California Institute of Technology, 1963

M.S. Mathematics, University of New Hampshire, 1960

B.S. Electrical Engineering, University of New Hampshire, 1959

— RESEARCH ACTIVITIES OF E.S. FOLIAS —

1. Failure in Composite Materials

The phenomenon of failure in composite material systems is not well understood as it is in the case of metals. One of the many reasons for this is that failure is a localized effect that can usually be traced to a stress riser or a material imperfection. Moreover, edge effects, residual stresses, and delaminations at interfaces complicate matters even further. While some of the fundamental questions are well defined, their complete understanding is still lacking. This is because such problems (i) are mathematically 3D in nature and (ii) deal with a material that is anisotropic. Be that as it may, accurate solutions are essential for a complete understanding of the damage tolerance characteristics of the material in the presence of cracks, holes, or discontinuities.

In the last ten years, the thrust of our research activities has been along the following three directions:

(i)

*The 3D stress analysis of composite laminates in the presence of circular cutouts.*At first, we considered the analytical investigation of isotropic layers. Subsequently, the analysis was extended to also include transversely isotropic layers. As a practical matter, we are trying to predict the initial stages of damage in the vicinity of a hole. Moreover, we are studying the interlaminar stresses at a bonded interface and the effect that various lay-ups have on the stress distribution. Subsequently, we will derive a criterion for failure. The work thus far has been supported by AFSOR.

(ii)

*A micromechanics approach to the modeling of certain composite material systems.*The fibers in this approach are assumed to be cylindrical inclusions which are periodically embedded into a continuous matrix. The stress field is then examined near the interfaces for possible debonding and how the load transfer characteristics of a broken fiber are transferred to the remaining portion.

(iii)

*An investigation of the effect which stress waves, traveling parallel as well as perpendicular to the laminae, have on preexisting cracks either in a lamina or a bonded interface.*In this analysis a [0°/90°] stacking sequence was considered.

2. Pressure Vessel Design

A thorough investigation was undertaken in order to assess the role that an initial curvature plays on the catastrophic failure of metal sheets possessing initial flaws. This extensive study of the subject has led to a series of publications from which we were ultimately able to derive a very general and reliable design criterion against failure that also incorporates a correction factor for plasticity. The criterion is presently used by ASTM and industry in order to predict catastrophic failures in pressurized vessels of an arbitrary shell geometry (e.g., cylindrical, spherical, conical, etc.) by knowing only material properties, type of loading, geometry of the structure, and geometry of the flaw. A comparison with experimental data available in the literature, as well as with experimental work carried out at the University of Utah, substantiates its validity and use. Such a criterion finds numerous applications to the field of pressure vessel technology (e.g., pipes, airplanes, submarines, space station, nuclear reactors, rockets, missiles, etc.).

Finally, because a plate is a special case of a shell, we were successful in establishing an appropriate correlation function relating the fracturing characteristics of a flat plate with those of an initially curved sheet. In experimental work on brittle fracture, for example, considerable time and money can be shaved for one may now predict the response behavior of curved sheets from agreement for metals and very good agreement for pressurized graphite/epoxy cylinders. Japanese and European designers were the first to capitalize on the practical advantages of this correlation function.

Recently, we are extending this analytical work to also include what effects do the (i) impact and (ii) thickness have on the failing characteristics of a pressurized vessel.

3. Effect of Specimen Thickness (

*3D Analytical Stress Analysis*)

Despite careful design, practically every structure contains stress concentrations due to holes. Bolt holes and rivet holes are necessary components for structural joints, e.g., connection holes for pipes, access holes, etc. It is not surprising, therefore, that the majority of service cracks nucleate in the area of stress concentrations at the edge of a hole. In seeking appropriate and reliable design criteria, the knowledge of the effect of the thickness on the stress concentration factor is a prerequisite.

Our extensive efforts in this area are beginning to pay off. A few years ago, we were able to construct the a general 3D analytical solution for the equilibrium of a linear elastic homogeneous and isotropic layer. This general solution was then used successfully in order to determine the effect of specimen thickness on the stress concentration factor. We were also able to extract the explicit behavior of the stress field in the neighborhood of the intersection of the hole and the free surface, thus providing us with further insight of the 3D effects of the problem. The results explain the experimental observations that, for relatively thick plates, a fatigue crack almost always manifests itself approximately one radius distance away from the free surface. The analysis also revealed that (i) in general the location of the fatigue crack is a function of the diameter to thickness ratio and that (ii) the transition between thick and thin plates takes place in the region where 1/10 < (2a/2h) < 10.

The above results encouraged us to look further into the effect that specimen thickness has on the stress intensity factor and to see some definite answers to the phenomenon of 3D fracture. While it is true that such an investigation does not include the nonelastic behavior of the material at the crack tip per se, it (i) can evince many characteristics of the actual behavior of a cracked plate, including those due to thickness, and (ii) can assist us in understanding better the 3D elasto-plastic analogue (e.g., by providing us with information regarding a discretized element mesh, etc.).

4. Fracture of Highway Pavements

Another research effort has been the investigation of the cracking of concrete and asphalt pavements which is due to the constant application of heavy, repetitive loads. Although the initial effects of the crack on the riding quality are minor, intrusion of water very quickly causes appreciable swelling on certain sub-grades with resulting bumps and a very rough riding pavement. Also the high stress concentration at the tip of the crack, plus the frost heave, can result in additional cracking and spauling with potholes as the final result. Cracking, therefore, is considered to be economically very serious, and it represents a major problem that pavement designers must face.

The problem of course is extremely difficult because of the many parameters which are involved. However, the application of modern fracture mechanics concepts may lead to a better understanding of this complex phenomenon and, ultimately, to a practical solution.

In this investigation, we looked into the fracturing characteristics of different pavement models for which we were able to derive appropriate design criteria. The first analysis was based on a Winkler and Zimmerman type of foundation; the second was based on the theory of Vlasov, which accounts for the vertical shear stress (good model for concrete highway pavements); and the third was based on a viscoelastic layered model. The activities in this area, however, were finally terminated due to unavailability of research funds.

*E.S. Folias*

Salt Lake City, Utah

September 2002

Salt Lake City, Utah

September 2002