Three-dimensional Photoelastic Analysis of a Fiber-reinforced Composite Model Standard stress-freezing technique is used to determine the stress distributions in the matrix of an unidirectionally fiber-reinforced composite model subjected to matrix shrinkage and normal transverse loading by R. H. Marloff and I. M. Daniel ABSTRACT--A three-dimensional photoelastic analysis using the "stress-freezing" technique was conducted to determine the stress distributions in the matrix of a unidirectionally fiber-reinforced composite model sub- jected to matrix shrinkage and normal transverse load- ing. The model, consisting of a square array of poly- carbonate rods in an epoxy matrix, simulated a boron- filament-reinforced plastic composite with a fiber-volume fraction of 0.50 at the critical temperature of the matrix epoxy. The effects of matrix shrinkage were separated from those of external loading by analyzing two identical models, one loaded and the other unloaded. The Lam~- Maxwell equations of equilibrium were used to separate stresses along axes of symmetry on interior transverse slices. Axial stress components were obtained by sub- slicing. Results are presented in dimensionless form by dividing the stresses by the average stress through the section. A comparison with theoretical results for a boron-epoxy composite shows excellent agreement, although Poisson's ratio of the model matrix is appre- ciably different from that of the prototype (0.5 compared to 0.35). One significant result was that the maximum stress occurs in the middle of the matrix section between fibers which is at variance with the theoretical prediction of maximum stress at the interface. Stress-concentra- tion factors vary from 1.80 at the interface to 2.0 at the midpoint of the matrix section between fibers. Introduction The structural behavior of a composite material is intimately related to the internal stress level and distribution and to the load transfer between the constituent parts. The field of micromechanics comprises the study of these internal stresses and the mechanics of internal reactions and interactions of the constituent parts due to imposed forces. Knowledge of the internal state of stress serves two main purposes; first, it contributes to the evalu- ation of average (macroscopic) response and, secondly, it provides the basis for understanding failure modes and establishing failure criteria. Stresses in the matrix are of great importance when loads act in a direction normal to the fibers, since in this case initial failure may be governed by R. H. Marloff is Research Engineer, Westinghouse R & D Center, Pitts- burgh, Pa., formerly Associate Research Engineer, IIT Research Institute, Chicago, Ill. I. M. Daniel is Manager of Stress Analysis Section, IIT Research Institute, Chicago, Ill. Paper was presented at 1968 SESA Fall Meeting held in San Francisco, Calif., on October 28-November 1. stress and strain concentration in the matrix. Most of the work in this area is theoretical. Kies 1 conducted simple analyses and calculated the strain magnifications existing in the matrix of a unidirec- tionally reinforced composite under loads trans- verse to the filament directions. Hermann and Pister 2 obtained an exact elasticity solution using numerical methods. Detailed solutions of the same problem have been given by Wilson and Hill 3 using a comformal mapping approach, Foye 4 who used a discrete element method and Adams and Doner 5 who followed a finite-difference procedure. Related experimental work is very limited and it has consisted primarily of two-dimensional model studies. Shrinkage stresses around inclusions have been analyzed by Daniel and Durelli. s, ~ Samp- son s and one of the authors 9 have studied stress distributions in two-dimensional photoelastic speci- mens containing arrays of inclusions and subjected to external load and resin shrinkage. Durelli, et al., 1~ recently conducted many two- and three- dimensional photoelastic studies for the determina- tion of shrinkage and mechanical loading stresses in matrices with various types of inserts. They used a low-temperature-curing epoxy cast around Plexi- glas or other epoxy inserts. This paper describes a three-dimensional photo- elastic study of a fiber-reinforced composite model. The standard stress-freezing technique was used to determine stress distributions in the matrix of a unidirectionally reinforced composite model sub- jected to matrix shrinkage and normal transverse loading. The model consisted of reinforcing and matrix components having a modulus ratio simu- lating that of a boron-reinforced plastic. Experimental Procedure Design of Models Prior to design of the actual models, considerable effort was devoted to the problem of shrinkage stresses induced during the curing of an epoxy around an insert. If these stresses could be made negligible, the rather lengthy process of subtracting 156 I April 1969