Properties for Design of Composite Structures. Neil McCartney
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to a homogeneous orthotropic plate.
2.1 Introduction
The principal objective of this book is to develop theoretical models and associated software that can predict the deformation behaviour of composite materials for situations where the system is first undamaged, but then develops progressively growing damage as the applied loading is gradually increased. Even though the composite systems are heterogeneous, continuum methods can be applied to each constituent of the composite, and to assist in the development of models for homogenised effectively continuous systems where the details of the reinforcement and damage distribution have been smoothed, and where effective properties may be defined.
The objective of this chapter is to describe the fundamental principles on which theoretical developments will be based. The topics to be covered are the nature of vectors and tensors, the definitions of displacement and velocity vectors, the balance laws for mass, momentum and energy, thermodynamics involving stress and strain state variables, and a thorough treatment of the linear elastic behaviour of anisotropic materials, including a contracted notation that is used widely in the composites field.
2.2 Vectors
A vector v is a mathematical entity that possesses both a magnitude and a direction. The vector is usually a physical quantity that is independent of the coordinate system that will be used to describe its properties. A very convenient approach is first to define an orthonormal set of coordinates (x1,x2,x3). Three axes are drawn in the x1,x2 and x3 directions, which are all at right angles to one another. Such a system is often described as a Cartesian set of coordinates. The positive directions of the x1, x1 and x3 axes are described by unit vectors i1 i2 and i3, respectively, where i1=(1,0,0),i2=(0,1,0),i3=(0,0,1). The unit vectors are such that
The three coordinates (x1,x2,x3) describe the location of a point x that is known as the position vector, which may be written as x=x1i1+x2i2+x3i3. In tensor theory based on Cartesian coordinates, this is written in the shorter form x=xkik where a summation over values k = 1, 2, 3 is implied when a suffix is repeated (k in this example). Any vector v may be written as v=v1i1+v2i2+v3i3, or as v=vkik when using tensor notation. The scalar quantities vk, k = 1, 2, 3, are the components of the vector v with values depending on the choice of coordinates. The magnitude of the vector v is specified by
and its value is independent of the system of coordinates that is selected. The magnitude is, thus, an invariant of the vector.
The unit vector in the direction of the vector v is specified by v/|v|. Examples of vectors that occur in the physical world are forces, displacements, velocities and tractions.
2.3 Tensors
The tensors to be used in the book are either second order or fourth order. Tensors are usually physical quantities that are independent of the coordinate system that is used to describe their properties. For the given coordinate system having unit vectors i1, i2and i3, a second-order tensor t is expressed in terms of the unit vectors as follows:
or, more compactly, using tensor notation in the form
where summation over values 1, 2, 3 is implied by the repeated suffices j and k. The quantities tjk are known as the components of a second-order tensor with values depending on the choice of coordinates. There are three independent invariants of second-order tensors which can be expressed in a variety of forms, the simplest being
A fourth-order tensor T is expressed in terms of the unit vectors of the coordinate system as follows
where summation over values 1, 2, 3, is implied by the repeated suffices i, j, k and l. The quantities Tijkl are known as the components of a fourth-order tensor with values depending on the choice of coordinates.
2.3.1 Fourth-order Elasticity Tensors
Elastic stress-strain equations are often written in the following form (see, for example, (2.153) and (2.154) given later in the chapter which includes thermal terms).
It is clear that
which may be written as
where
The fourth-order tensor Iijmn can be defined by
where δij denotes the Kronecker delta symbol which has the value unity when i = j and the value zero otherwise. Clearly
