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Notation is often a serious obstacle that prevents an engineer from using advanced textbooks; for example, general curvilinear tensor analysis and functional analysis are both necessary in some of the theories used in ABAQUS, but the unfamiliar notations commonly used in these areas often discourage the user from pursuing their study. The notation used in most of this manual (direct matrix notation) may be unfamiliar to some readers; but it is not difficult or time consuming to gain enough familiarity with the notation for it to be useful, and it is definitely worthwhile. This notation is commonly used in the modern engineering literature—it is a shorthand version of the familiar matrix notation used in many older engineering textbooks. The notation is appealing—once it is understood—because it allows the equations to be developed concisely, and the physical ideas can be perceived without the distraction of the complexities that arise from the choice of the particular basis system that will eventually be used to express the same concepts in component form. Because the notation has become so standard in the literature, the user who wishes or needs to read textbooks and papers that are related to the use of ABAQUS will find that familiarity with this notation is desirable.
Both direct matrix notation and component form notation are used in the manual. Both notations are described in this section. Direct matrix notation is used whenever possible. However, vectors, matrices, and the higher-order tensors used in the theories must eventually be written in component form to store them as a set of numbers on the computer. Thus, both ways of writing these quantities will be needed in the manual.
The quantities needed to formulate the theory are scalars, vectors, second-order tensors (matrices), and—occasionally—fourth-order tensors (for example, the stress-strain transformation for linear elasticity). In direct matrix notation these are written as:
| a scalar value |  |
| a vector |  or  |
| with the transpose |  or  |
| a second-order tensor or matrix | or  |
| with the transpose | or  |
| and | |
| a fourth-order tensor |  |
Vectors and second-order tensors (matrices) are written in the same way: they are distinguished by the context. In direct matrix notation there is generally no need to indicate that a vector must be transposed. The context determines whether a vector is to be used as a “column” vector or as a “row” vector . In this case the transpose superscript is only used to improve the readability of an expression. On the other hand, for second-order nonsymmetric tensors the addition of a transpose superscript will change the meaning of an expression.
This representation of vectors and tensors is very general and convenient for developing the theory so that the equations can be understood easily in terms of their physical meaning. However, in actual computations we have to work with individual numbers, so vectors and tensors must be expressed in terms of their components. These components are associated with an axis system that defines a set of base vectors at each point in space. The simplest axis system is rectangular Cartesian, because the base vectors are orthogonal unit vectors in the same direction at all points. Unfortunately, we need more generality than this because we will be dealing with shells and beams, where stress, strain, etc. are most conveniently described in terms of directions on the surface of the shell (or associated with the axis of the beam), and these usually change as we move around on the surface. To retain this necessary generality and express vectors and matrices in component form, we introduce a general set of base vectors, 
, 
, which are not necessarily orthogonal or of unit length but are sufficient to define the components of a vector (for this purpose they must not be parallel or have zero length). A vector can then be written
, 
, and 
are the components of associated with 
, 
, and 
.In actual cases the are chosen for convenience (for example, see “Conventions,” Section 1.2.2 of the ABAQUS Analysis User's Manual, for a description of how base vectors are chosen for surface elements in ABAQUS), and then the 
are obtained.
To save writing, we adopt the usual summation convention that a repeated index is summed—in this case over the range 1 to 3—so that the above equation is written
While we will need such completely general base vectors for describing the stresses and strains on shells and beams, in many cases it is convenient to use rectangular Cartesian components so that the are orthogonal unit vectors. To distinguish this particular case, we will use Latin indices instead of Greek indices. Thus, are a set of general base vectors; while 
are rectangular Cartesian base vectors; and is the component of the vector along a general base vector, while 
, 
, is the component of along the 
th Cartesian direction.
Vector and tensor concepts and their representation are discussed in many textbooks—see Flugge (1972), for example.
The usual matrix and vector operators are indicated in this manual as follows:
Dot product of two vectors:
(The dot symbol defines this operation completely, regardless of whether 
or 
is transposed—i.e., 
)
Cross product of two vectors:
Matrix multiplication:
(It is implicitly assumed that and are dimensioned correctly, as needed for the operation to make sense; in addition, if is a nonsymmetric tensor, 
)
Scalar product of two matrices:
This operation means that corresponding conjugate components of the two matrices are multiplied as pairs and the products summed. Thus, for instance, if is the stress matrix, 
, and the conjugate rate of strain matrix, 
, then 
would give the rate of internal work per volume, 
.
It is also necessary to define the dyadic product of two vectors:
This operation creates a second-order tensor (or dyad) out of two vectors. In component notation this notation is equivalent to 
.
A matrix of derivatives,
Provided that we are careful about interpreting 
in the manner illustrated above, standard concepts of elementary calculus clearly hold; for example, if is a vector-valued function of the vector-valued function , which in turn is a vector-valued function of , that is 
, then
: 
Due to these properties many useful results can be obtained quickly and expressed in a compact, easily understood, form.
In the previous section we introduced the idea that a vector or a matrix can be written in terms of components associated with some conveniently chosen set of base vectors, . We now show how the components (or 
) are obtained. We can do so using the dot product. For each of the three base vectors, 
, we define a conjugate base vector 
, as follows. Choose 
as normal to and , such that the dot product 
. Similarly, choose 
normal to and , such that 
; and 
normal to and , such that 
. Thus,
if 
, and 
, otherwise. (
is called the “Kronecker delta.”) In matrix notation 
is the unit matrix 
: we can also write the above equation defining , , and in matrix form as 
, say—is known, the others are easily obtained.With this additional set of base vectors, we can immediately obtain the components of a vector or a matrix as follows.
Consider a vector . Then 
(writing in component form, using the basis vectors 
), and since 
, only if ,
as components associated with the base vectors, 
.Similarly, for a matrix,
These component definitions are particularly convenient for calculating the dot product of two vectors, for we can write
Similarly, the scalar product of two matrices is
and 
arrays, arranged as matrices, and then sum the products.Finally, on the computer we need to store only one form of component: 
, or , . We can always go from one to the other using the “metric tensor,” 
, and its inverse, 
, which are defined as
For
; similarly 
, and, by extension, for matrices, 
The metric tensor and its inverse are symmetric:
The two sets of base vectors and components of vectors or matrices associated with them are named as follows:
| | are covariant base vectors, |
| | are contravariant base vectors, |
 | are covariant components of a vector (or matrix), |
 | are contravariant components of a vector (or matrix). |
, and vice versa. The simplest case is when the basis is a set of orthogonal unit vectors (a rectangular Cartesian system) because then—from the definition 
—we see that 
, and so 
and we need not distinguish the type of component. Whenever possible a rectangular Cartesian system is chosen, so the type of component need not be distinguished. This system is discussed in more detail in the sections on beam elements and shell elements.Consider a vector-valued function, , which is expressed in component form on a basis system, . Let the vector-valued function depend on : 
. Then
associated with a change 
is 
Now suppose is written on a different basis—
, say—so that we store as the components
Typically we would then write
Readers who are familiar with general curvilinear tensor analysis will recognize 
as the covariant derivative of with respect to 
, often written as 
. The advantage of the direct matrix notation is clear: because we can imagine and as vectors in space, we have a physical understanding of what we mean by 
; it is the change in the vector-valued function as a function of another vector-valued function . For computations we must express and in component form. Then
for and for . Typically and will both be the simple rectangular Cartesian bases 
As an example, consider the usual expression for strain rate,
to be evaluated, where 
is the velocity of the material currently flowing through the point 
in space. Let us now derive the components of 
when the basis system for both and is the cylindrical system that we usually choose for axisymmetric problems, with the basis vectors 
We consider position to be defined by the coordinates 
, with
Thus,
and 
, so these results simplify to the familiar expressions 
In summary, direct matrix notation allows us to obtain all our fundamental results without reference to any particular choice of coordinate system. Careful application of the concept of the covariant derivative then allows these general results to be projected into component form for computation.
The concepts of virtual displacements and virtual work are fundamental to the development. Virtual quantities are infinitesimally small variations of physical measures, such as displacement, strain, velocity, and so on. The virtual variation of a scalar quantity is indicated by 
; of a vector or matrix by 
.
We extend this notation to such expressions as
. This notation corresponds to the virtual rate of deformation (a measure of strain rate) if is a virtual velocity field.Most structural problems concern the description of the way a structure behaves as it is loaded and moves from its reference configuration. Thus, we often compare positions of a point in the current (deformed) configuration and a reference configuration that is usually chosen as the configuration when the structure is unloaded or, in the case of geotechnical problems, when the model is subject only to geostatic stresses. To distinguish these configurations, we use lowercase type () to indicate the current position and uppercase type (
) to indicate the initial position of the same material point in the same spatial coordinate frame. In ABAQUS we almost always store the rectangular Cartesian components of and . The exception is in axisymmetric structures, where radial (
) and axial (
) components are stored.
So far we have discussed quantities that are considered to be associated with all points in a model. The finite element approximation is based on assuming interpolations, by which displacement, position, and—often—other variables at any material point are defined by a finite number of nodal variables. In this manual we use uppercase superscripts to refer to individual nodal variables or nodal vectors and adopt the summation convention for these indices.
Hence, the interpolation can be written quite generally as
is some vector-valued function at any point in the structure; 
, 
up to the total number of variables in the problem, is a set of 
vector interpolation functions (these are functions of the material coordinates, 
); and 
, is a set of nodal variables.In some sections in this manual we need to describe operations on nodal variables for the complete system of finite element equations. In these sections we use the classical matrix-vector notation. In this notation represents a column vector containing nodal variables, represents a row vector, and a matrix is written as 
. Common operations are the scalar product between two vectors,
in index notation) and the matrix-vector product
in index notation).