On two models in the three-dimensional theory of stability of composites

Abstract

A comparative analysis is made of the infinite-fiber and finite-fiber models in the three-dimensional theory of stability of composites. The results analyzed have been obtained using the three-dimensional linearized theory of stability of deformable bodies. A historical sketch is given of the theory of stability for and approaches used in the mechanics of laminated and fibrous composite materials Keywords: laminated and fibrous composite materials, three-dimensional theory of stability, infinite-fiber model, finite-fiber model, historical sketch Introduction. For nearly half a century now, since the publication [18], researchers have been studying the microbuckling of composites and have been developing the fundamentals of the fracture mechanics of compressed composites assuming that failure begins with microbuckling. Depending on the relative dimensions of reinforcement (mean cross section and length), two essentially different models may be used: an infinite fiber and a finite fiber. The overwhelming majority of recent results have been obtained using the infinite-fiber model and analyzing periodic (along fibers) buckling modes. These results are reported in numerous papers in periodicals and in a number of reviews. Mathematically, the finite-fiber model is certainly more rigorous and general than the infinite-fiber model. In this connection, the finite-fiber model is of importance for the cases where fibers cannot be considered infinitely long.However, the use of the finite-fiber model involves mathematical and computational difficulties because the dimension of the problem cannot be reduced by separating periodic factors dependent on one coordinate reckoned along fibers.For consistency between the infinite-fiber and finite-fiber models, the following obvious conditions should be satisfied. The results predicted by the finite-fiber model should asymptotically tend to the results predicted by the infinite-fiber model as the fiber length tends to infinity.It is only recently that the finite-fiber model has been employed to obtain the results outlined in [7-14, 17, 34-37, etc.]. This makes it possible to compare the two models, which is done in the present paper.It what follows, we will use the terms "infinite-fiber model" and "finite-fiber model." When it comes to laminated composites, one should speak about infinitely long layers and finite-size layers. In this connection, it would be more logical to unite fibrous and laminated composites by using the terms "infinitely long reinforcing elements" and "finite-size reinforcing elements." It however appears that "infinite fiber" and "finite fiber" are more informative terms for experts in composite materials, which is why these terms will be applied below to both fibrous and laminated composites.1. Preliminary Discussion. Here we will discuss the issues in question, including a historical sketch on the studies being conducted and information on modeling fibrous materials in microbuckling studies.1.1. Historical Sketch. It is generally agreed in the modern scientifi...

Stability of composite materials with two parallel short fibers

“…4, curve 2), whereas when E E r m -= 1 300, the magnitude of | | e 11 cr m is less in the former case than in the latter case. No such effect is observed for | | e 11 cr r for all mechanical and geometrical parameters of the composite components (Fig. 3).…”

“…show | | e11 cr as functions of LD -1 for E E Also, curves 1 and 4 correspond to r * = 0.01, curves 2 and 5 to r * = 0.1, and curves 3 and 6 to r * = 1.parameters being the same. Moreover, the behavior of the critical strains changes at some values of these parameters: for…”

“…A comparative contemporary analysis of the results obtained with these models can be found in [11]. It should be noted that the overwhelming majority of results on the three-dimensional theory of stability of laminated and fibrous materials were obtained using the approach of [5,6] in combination with the infinite-fiber model (see [1, 10, 13, etc.]…”

“…For example, | | e 11 cr r was chosen in [11] to compare the models. In the infinite-fiber model, e 11 cr r is the critical compressive strain along the Ox-axis for both reinforcement (fibers) and matrix (binder).…”

Stability of rod-shaped nanoparticles embedded in an elastic matrix

“…Analytical and numerical treatments of such systems were published already 30 years ago, see, e.g., [15]. With the upcoming of composites reinforced by collinear short fibres or whiskers the solution of the stability problem of those reinforcements has obtained a revival within the last decade, see, e.g., the papers by Dekret [3], [5], [4], and Guz and Dekret [9] as well as the literature cited in these papers. It is interesting to note that the problem of magnetic buckling as it occurs due to phase transformations in micromagnetic dots, see [17], can be seen as a special case of the more general situations considered here.…”