The Stroh Formalism for Anisotropic Elasticity with Applications to Composite Materials

1991 ◽  
Author(s):  
T. C. Ting
Keyword(s):  
Composite Materials ◽  
Anisotropic Elasticity ◽  
Stroh Formalism

Anisotropic Elasticity

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Composite Materials ◽  
Elastic Constants ◽  
Piezoelectric Materials ◽  
Anisotropic Materials ◽  
Anisotropic Elasticity ◽  
Stress Singularities ◽  
Comprehensive Survey ◽  
The Subject ◽  
Key Topics ◽  
First Time

Anisotropic Elasticity offers for the first time a comprehensive survey of the analysis of anisotropic materials that can have up to twenty-one elastic constants. Focusing on the mathematically elegant and technically powerful Stroh formalism as a means to understanding the subject, the author tackles a broad range of key topics, including antiplane deformations, Green's functions, stress singularities in composite materials, elliptic inclusions, cracks, thermo-elasticity, and piezoelectric materials, among many others. Well written, theoretically rigorous, and practically oriented, the book will be welcomed by students and researchers alike.

The Interface Crack Between Dissimilar Anisotropic Composite Materials

1983 ◽  
Vol 50 (1) ◽  
pp. 169-178 ◽  
Author(s):  
S. S. Wang ◽  
I. Choi
Keyword(s):  
Composite Materials ◽  
Interface Crack ◽  
Hilbert Problem ◽  
Stress Singularity ◽  
Anisotropic Elasticity ◽  
Crack Model ◽  
Closed Crack ◽  
Anisotropic Composite ◽  
Interlaminar Crack ◽  
As Stress

The fundamental nature of an interface crack between dissimilar, strongly anisotropic composite materials under general loading is studied. Based on Lekhnitskii’s stress potentials and anisotropic elasticity theory, the formulation leads to a pair of coupled governing partial differential equations. The case of an interlaminar crack with fully opened surfaces is considered first. The problem is reduced to a Hilbert problem which can be solved in a closed form. Oscillatory stress singularities are observed in the asymptotic solution. To correct this unsatisfactory feature, a partially closed crack model is introduced. Formulation of the problem results in a singular integral equation which is solved numerically. The refined model exhibits an inverse square-root stress singularity for commonly used advanced fiber-reinforced composites such as a graphite-epoxy system. Extremely small contact regions are found for the partially closed interlaminar crack in a tensile field and, therefore, a simplified model is proposed for this situation. Physically meaningful fracture mechanics parameters such as stress intensity factors and energy release rates are defined. Numerical examples for a crack between θ and −θ graphite-epoxy composites are examined and detailed results are given.

The Stroh Formalism

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
General Solution ◽  
Elastic Body ◽  
Three Dimensional ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Two Dimensional ◽  
Stroh's Formalism ◽  
Elasticity Problems ◽  
Anisotropic Elastic ◽  
Stroh’S Formalism

In this chapter we study Stroh's sextic formalism for two-dimensional deformations of an anisotropic elastic body. The Stroh formalism can be traced to the work of Eshelby, Read, and Shockley (1953). We therefore present the latter first. Not all results presented in this chapter are due to Stroh (1958, 1962). Nevertheless we name the sextic formalism after Stroh because he laid the foundations for researchers who followed him. The derivation of Stroh's formalism is rather simple and straightforward. The general solution resembles that obtained by the Lekhnitskii formalism. However, the resemblance between the two formalisms stops there. As we will see in the rest of the book, the Stroh formalism is indeed mathematically elegant and technically powerful in solving two-dimensional anisotropic elasticity problems. The possibility of extending the formalism to three-dimensional deformations is explored in Chapter 15.

Hamiltonian interpretation of the Stroh formalism in anisotropic elasticity

2007 ◽  
Vol 463 (2088) ◽  
pp. 3073-3087 ◽  
Author(s):  
Y.B Fu
Keyword(s):  
Plate Theory ◽  
Hamiltonian Formulation ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Elastic Plates ◽  
Spatial Variables ◽  
The Matrix ◽  
Anisotropic Elastic ◽  
Incompressible Elasticity ◽  
Steady Motions

Stroh's sextic formalism for static problems or steady motions in anisotropic elasticity is a formulation in which the equation of equilibrium/motion is written as a system of first-order differential equations for the displacement and traction in terms of one of the spatial variables. The so-called fundamental elasticity matrix N appearing in this formulation has the property that, when partitioned as a 2×2 block matrix, its 12- and 21-blocks are symmetric matrices and its 11-block is the transpose of its 22-block. This property gives rise to a large number of orthogonality and closure relations and is fundamental to the success of the Stroh formalism in solving a large variety of problems in general anisotropic elasticity. First, we show that the matrix N is guaranteed to have the above property by the fact that the Stroh formulation is in fact a Hamiltonian formulation with one of the spatial variables acting as the time-like variable. This interpretation provides a much desired guide in dealing with other problems for which the governing equations are different, such as incompressible elasticity and problems associated with anisotropic elastic plates as described by the Kirchhoff plate theory. We show that for the last two problems the Hamiltonian interpretation simplifies the derivations significantly, leading to a Stroh formulation in each case which is equivalent to, but much simpler than, what is available in the existing literature.

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