- Benchmark timing documents - Error status reports - ABAQUS documentation price list - Training seminar schedule - Newsletters # Anonymous ftp site For users connected to the Internet, HKS maintains useful documents on an anonymous ftp account on the computer ftp.abaqus.com. Simply ftp to ftp.abaqus.com. Login as user anonymous, and type your e-mail address as your password. Directions will come up automatically upon login. # Writing to technical support Address of HKS Headquarters: Hibbitt, Karlsson & Sorensen, Inc. 1080 Main Street Pawtucket, RI 02860-4847, USA Attention: Technical Support Addresses for other offices and representatives are listed in the front of each manual. # Support for academic institutions Under the terms of the Academic License Agreement we do not provide support to users at academic institutions unless the institution has also purchased technical support. Please see the ABAQUS Home Page, or contact us for more information. # Training All HKS offices offer regularly scheduled public training classes. The Introduction to ABAQUS/Standard and ABAQUS/Explicit seminar covers basic usage and nonlinear applications, such as large deformation, plasticity, contact, and dynamics. Workshops provide as much practical experience with ABAQUS as possible. The Introduction to ABAQUS/CAE seminar discusses modeling, managing simulations, and viewing results with ABAQUS/CAE. "Hands-on" workshops are complemented by lectures. Advanced seminars cover topics of interest to customers with experience using ABAQUS, such as engine analysis, metal forming, fracture mechanics, and heat transfer. We also provide training seminars at customer sites. On-site training seminars can be one or more days in duration, depending on customer requirements. The training topics can include a combination of material from our introductory and advanced seminars. Workshops allow customers to exercise ABAQUS on their own computers. For a schedule of seminars see the ABAQUS Home Page, or call HKS or your local HKS representative. # Documentation The following documentation and publications are available from HKS, unless otherwise specified, in printed form and through our online documentation server. For more information on accessing the online books, refer to the discussion of execution procedures in the user's manuals. In addition to the documentation listed below, HKS publishes two newsletters on a regular schedule: ABAQUS/News and ABAQUS/Answers. ABAQUS/News includes topical information about program releases, training seminars, etc. ABAQUS/Answers includes technical articles on particular topics related to ABAQUS usage. These newsletters are distributed at no cost to users who wish to subscribe. Please contact your local ABAQUS support office if you wish to be added to the mailing list for these publications. They are also archived in the Reference Shelf on the ABAQUS Home Page. # Training Manuals Getting Started with ABAQUS/Standard: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Standard for static and dynamic stress analysis simulations. It contains a number of fully worked examples that provide practical guidelines for performing structural analyses with ABAQUS. Getting Started with ABAQUS/Explicit: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Explicit. It begins with the basics of modeling in ABAQUS, so no prior knowledge of ABAQUS is required. A number of fully worked examples provide practical guidelines for performing explicit dynamic analyses, such as drop tests and metal forming simulations, with ABAQUS/Explicit. Lecture Notes: These notes are available on many topics to which ABAQUS is applied. They are used in the technical seminars that HKS presents to help users improve their understanding and usage of ABAQUS (see the "Training" section above for more information about these seminars). While not intended as stand-alone tutorial material, they are sufficiently comprehensive that they can usually be used in that mode. The list of available lecture notes is included in the Documentation Price List. # User's Manuals ABAQUS/Standard User's Manual: This volume contains a complete description of the elements, material models, procedures, input specifications, etc. It is the basic reference document for ABAQUS/Standard. ABAQUS/Explicit User's Manual: This volume contains a complete description of the elements, material models, procedures, input specifications, etc. It is the basic reference document for ABAQUS/Explicit. ABAQUS/CAE User's Manual: This reference document for ABAQUS/CAE includes three comprehensive tutorials as well as detailed descriptions of how to use ABAQUS/CAE for model generation, analysis, and results evaluation. ABAQUS/Viewer User's Manual: This basic reference document for ABAQUS/Viewer includes an introductory tutorial as well as a complete description of how to use ABAQUS/Viewer to display your model and results. ABAQUS/ADAMS User's Manual: This document describes how to install and how to use ABAQUS/ADAMS, an interface program that creates ABAQUS models of ADAMS components and converts the ABAQUS results into an ADAMS modal neutral file that can be used by the ADAMS/Flex program. It is the basic reference document for the ABAQUS/ADAMS program. ABAQUS/CAT User's Manual: This document describes how to install and how to use ABAQUS/CAT, an interface program that creates an ABAQUS input file from a CATIA model and postprocesses the analysis results in CATIA. It is the basic reference document for the ABAQUS/CAT program. ABAQUS/C-MOLD User's Manual: This document describes how to install and how to use ABAQUS/C-MOLD, an interface program that translates finite element mesh, material property, and initial stress data from a C-MOLD analysis to an ABAQUS input file. ABAQUS/Safe User's Manual: This document describes how to install and how to use ABAQUS/Safe, an interface program that calculates fatigue lives and fatigue strength reserve factors from finite element models. It is the basic reference document for the ABAQUS/Safe program. The theoretical background to fatigue analysis is contained in the Modern Metal Fatigue Analysis manual (available only in print). Using ABAQUS Online Documentation: This online manual contains instructions on using the ABAQUS online documentation server to read the manuals that are available online. ABAQUS Release Notes: This document contains brief descriptions of the new features available in the latest release of the ABAQUS product line. ABAQUS Site Guide: This document describes how to install ABAQUS and how to configure the installation for particular circumstances. Some of this information, of most relevance to users, is also provided in the user's manuals. # Examples Manuals ABAQUS Example Problems Manual: This volume contains more than 75 detailed examples designed to illustrate the approaches and decisions needed to perform meaningful linear and nonlinear analysis. Typical cases are large motion of an elastic-plastic pipe hitting a rigid wall; inelastic buckling collapse of a thin-walled elbow; explosive loading of an elastic, viscoplastic thin ring; consolidation under a footing; buckling of a composite shell with a hole; and deep drawing of a metal sheet. It is generally useful to look for relevant examples in this manual and to review them when embarking on a new class of problem. ABAQUS Benchmarks Manual: This volume (available online and, if requested, in print) contains over 200 benchmark problems and standard analyses used to evaluate the performance of ABAQUS; the tests are multiple element tests of simple geometries or simplified versions of real problems. The NAFEMS benchmark problems are included in this manual. ABAQUS Verification Manual: This online-only volume contains more than 5000 basic test cases, providing verification of each individual program feature (procedures, output options, MPCs, etc.) against exact calculations and other published results. It may be useful to run these problems when learning to use a new capability. In addition, the supplied input data files provide good starting points to check the behavior of elements, materials, etc. # Reference Manuals ABAQUS Keywords Manual: This volume contains a complete description of all the input options that are available in ABAQUS/Standard and ABAQUS/Explicit. ABAQUS Theory Manual: This volume (available online and, if requested, in print) contains detailed, precise discussions of all theoretical aspects of ABAQUS. It is written to be understood by users with an engineering background. ABAQUS Command Language Manual: This online manual provides a description of the ABAQUS Command Language and a command reference that lists the syntax of each command. The manual describes how commands can be used to create and analyze ABAQUS/CAE models, to view the results of the analysis, and to automate repetitive tasks. It also contains information on using the ABAQUS Command Language or C++ as an application programming interface (API). ABAQUS Input Files: This online manual contains all the input files that are included with the ABAQUS release and referred to in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual, and the ABAQUS Verification Manual. They are listed in the order in which they appear in the manuals, under the title of the problem that refers to them. The input file references in the manuals hyperlink directly to this book. Quality Assurance Plan: This document describes HKS's QA procedures. It is a controlled document, provided to customers who subscribe to either HKS's Nuclear QA Program or the Quality Monitoring Service. # 1. Introduction and Basic Equations # 1.1 Introduction # 1.1.1 Introduction: general The ABAQUS system includes ABAQUS/Standard, a general-purpose finite element program; ABAQUS/Explicit, an explicit dynamics finite element program; and ABAQUS/Viewer, an interactive postprocessing program that provides displays and output lists from output database files written by ABAQUS/Standard and ABAQUS/Explicit. This manual describes the theories used in ABAQUS. Many sections in this manual apply to both ABAQUS/Standard and ABAQUS/Explicit. Certain sections obviously apply only to either ABAQUS/Standard or ABAQUS/Explicit; for example, all sections in the chapter on procedures apply to ABAQUS/Standard, except the section discussing the explicit dynamic integration procedure, which applies to ABAQUS/Explicit. If it is not obvious to which program a section applies, it is clearly indicated. ABAQUS/Standard includes several added-cost options. The ABAQUS/Aqua option includes features specifically designed for the analysis of beam-like structures installed underwater and subject to loading by water currents and wave action. The ABAQUS/Design option enables the user to parametrize input file quantities and write Python scripts to perform parametric studies. The ABAQUS/USA option allows the Underwater Shock Analysis program originally developed by Lockheed's Research Laboratory and supported by Unique Software Applications to be used within ABAQUS/Standard to study the coupled problem of acoustic shock wave loading of underwater structures. Certain aspects of the theory behind these options are described in this manual. The options are available only if the user's license includes them. The objective of this manual is to define the theories used in ABAQUS that are generally not available in the standard textbooks on mechanics, structures, and finite elements but are well known to the engineer who uses ABAQUS. The manual is intended as a reference document that defines what is available in the code. Nevertheless, it is written in such a way that it can also be used as a tutorial document by a reader who needs to obtain some background in an unfamiliar area. The material is presented in a way that should make it accessible to any user with an engineering background. Some of the theories may be relatively unfamiliar to such a user; for example, few engineering curricula provide extensive background in plasticity, shell theory, finite deformations of solids, or the analysis of porous media. Yet ABAQUS contains capabilities for all of these models and many others. The manual is far from comprehensive in its coverage of such topics: in this sense it is only a reference volume. The user is strongly encouraged to pursue topics of interest through texts and papers. Chapter 7, "References," at the end of this manual lists references that should provide a starting point for obtaining such information. (HKS does not supply copies of papers that have appeared in publications other than those of HKS. EPRI reports can be obtained from Research Reports Center ( RRC), Box 50490, Palo Alto, CA 94303.) Chapter 1, "Introduction and Basic Equations," discusses the notation used in the manual, some basic concepts of kinematics and mechanics--such as rotations, stress, and equilibrium--as well as the basic equations of nonlinear finite element analysis. Chapter 2, "Procedures," describes the various analysis procedures (nonlinear static stress analysis, dynamics, eigenvalue extraction, etc.) that are available in ABAQUS. Chapter 3, "Elements," describes the element formulations. Chapter 4, "Mechanical Constitutive Theories," describes the mechanical constitutive theories. Chapter 5, "Interface Modeling," discusses the most important aspects of the contact/interaction formulation in ABAQUS/Standard. Chapter 6, "Loading and Constraints," describes the formulation of some of the more complicated load types and multi-point constraints. If you are reading this book through the online documentation, it is recommended that you enlarge the book window so that the equations and figures are clearly visible. Refer to Chapter 3, "Printing from an online book," of Using ABAQUS Online Documentation for instructions on printing from the online documentation. Be sure to toggle on Print graphics and equations, or the graphics and equations will not appear in the printed copy. The equations in this manual may appear different in the printed output from the way they appear online; bold terms are sometimes output incorrectly (see the Status Reports on the HKS Home Page, http://www.abaqus.com, for details). To obtain a bound printed copy of this manual, contact your local HKS office or representative. # 1.2 Notation # 1.2.1 Notation 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. # Basic quantities 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
a vector	a or [a]
with the transpose	aT or {a}
a second-order tensor or matrix	a or [a]
with the transpose	aT or [a]T
and
a fourth-order tensor	A

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 a or as a "row" vector aT . 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, $\mathbf { e } _ { \alpha } , \alpha = 1 , 2 , 3$ , 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 a can then be written $$ \mathbf {a} = a ^ {1} \mathbf {e} _ {1} + a ^ {2} \mathbf {e} _ {2} + a ^ {3} \mathbf {e} _ {3}, $$ where the numbers $a ^ { 1 } , a ^ { 2 }$ , and $a ^ { 3 }$ are the components of a associated with $\mathbf { e } _ { 1 } , \mathbf { e } _ { 2 }$ , and $\mathbf { e } _ { 3 }$ . In actual cases the ${ \bf e } _ { \alpha }$ are chosen for convenience (for example, see \`\`Conventions,'' Section 1.2.2 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual, for a description of how base vectors are chosen for surface elements in ABAQUS), and then the $a ^ { \alpha }$ 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 $$ \mathbf {a} = a ^ {\alpha} \mathbf {e} _ {\alpha}. $$ Likewise, the component form of a matrix will be $$ \mathbf {a} = \mathbf {e} _ {\alpha} a ^ {\alpha \beta} \mathbf {e} _ {\beta} = a ^ {\alpha \beta} \mathbf {e} _ {\alpha} \mathbf {e} _ {\beta}, $$ or, written out, $$ \mathbf {a} = \mathbf {e} _ {1} a ^ {1 1} \mathbf {e} _ {1} + \mathbf {e} _ {1} a ^ {1 2} \mathbf {e} _ {2} + \mathbf {e} _ {1} a ^ {1 3} \mathbf {e} _ {3} $$ $$ \mathbf {e} _ {2} a ^ {2 1} \mathbf {e} _ {1} + \mathbf {e} _ {2} a ^ {2 2} \mathbf {e} _ {2} + \mathbf {e} _ {2} a ^ {2 3} \mathbf {e} _ {3} $$ $$ \mathbf {e} _ {3} a ^ {3 1} \mathbf {e} _ {1} + \mathbf {e} _ {3} a ^ {3 2} \mathbf {e} _ {2} + \mathbf {e} _ {3} a ^ {3 3} \mathbf {e} _ {3}. $$ Similarly, a fourth-order tensor can be written in component form as $$ \mathbf {A} = A ^ {\alpha \beta \gamma \delta} \mathbf {e} _ {\alpha} \mathbf {e} _ {\beta} \mathbf {e} _ {\gamma} \mathbf {e} _ {\delta}. $$ 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 $\mathbf { e } _ { \alpha }$ are orthogonal unit vectors. To distinguish this particular case, we will use Latin indices instead of Greek indices. Thus, $\mathbf { e } _ { \alpha }$ are a set of general base vectors; while $\mathbf { e } _ { i }$ are rectangular Cartesian base vectors; and $a ^ { \alpha }$ is the component of the vector a along a general base vector, while $a ^ { i } , i = 1 , 2 , 3$ , is the component of a along the ith Cartesian direction. Vector and tensor concepts and their representation are discussed in many textbooks--see Flugge (1972), for example. # Basic operations The usual matrix and vector operators are indicated in this manual as follows: Dot product of two vectors: $$ a = \mathbf {b} \cdot \mathbf {c} $$ (The dot symbol defines this operation completely, regardless of whether b or c is transposed--i.e., $\mathbf { b } \cdot \mathbf { c } = \mathbf { b } ^ { T } \cdot \mathbf { c } . )$ Cross product of two vectors: $$ \mathbf {a} = \mathbf {b} \times \mathbf {c} $$ Matrix multiplication: $$ \mathbf {a} = \mathbf {b} \cdot \mathbf {c} $$ (It is implicitly assumed that b and c are dimensioned correctly, as needed for the operation to make sense; in addition, if b is a nonsymmetric tensor, $\mathbf { b } ^ { T } \cdot \mathbf { c } \neq \mathbf { b } \cdot \mathbf { c } . )$ Scalar product of two matrices: $$ a = \mathbf {b}: \mathbf {c} $$ This operation means that corresponding conjugate components of the two matrices are multiplied as pairs and the products summed. Thus, for instance, if b is the stress matrix, ¾, and c the conjugate rate of strain matrix, d", then ¾ : d" would give the rate of internal work per volume, $d W ^ { I }$ . It is also necessary to define the dyadic product of two vectors: $$ \mathbf {a} = \mathbf {b c} \qquad \mathrm{or} \qquad \mathbf {a} = \mathbf {b c} ^ {T} $$ This operation creates a second-order tensor (or dyad) out of two vectors. In component notation this notation is equivalent to $a ^ { i j } = b ^ { i } c ^ { j }$ . A matrix of derivatives, $$ \frac {\partial \mathbf {a}}{\partial \mathbf {b}}, $$ means $$ d \mathbf {a} = \left(\frac {\partial \mathbf {a}}{\partial \mathbf {b}}\right) \cdot d \mathbf {b}. $$ Throughout this manual it will be assumed implicitly that, when a derivative is taken with respect to time, we mean the material time derivative; that is, the change in a variable with respect to time whilst looking at a particular material particle. When this is not the case for a particular equation, it will be stated explicitly when the equation appears. Provided that we are careful about interpreting $\left( \partial \mathbf { a } / \partial \mathbf { b } \right)$ in the manner illustrated above, standard concepts of elementary calculus clearly hold; for example, if a is a vector-valued function of the vector-valued function b, which in turn is a vector-valued function of c, that is ${ \bf a } = { \bf a } ( { \bf b } ( { \bf c } ) )$ , then $$ d \mathbf {a} = \frac {\partial \mathbf {a}}{\partial \mathbf {b}} \cdot \frac {\partial \mathbf {b}}{\partial \mathbf {c}} \cdot d \mathbf {c}, $$ or, $\operatorname { i f } \mathbf { a } ( \mathbf { b } , \mathbf { c } )$ : $$ d \mathbf {a} = \frac {\partial \mathbf {a}}{\partial \mathbf {b}} \cdot d \mathbf {b} + \frac {\partial \mathbf {a}}{\partial \mathbf {c}} \cdot d \mathbf {c}. $$ Due to these properties many useful results can be obtained quickly and expressed in a compact, easily understood, form. # Components of a vector or a matrix in a coordinate system In the previous section we introduced the idea that a vector a or a matrix a can be written in terms of components associated with some conveniently chosen set of base vectors, $\mathbf { e } _ { \alpha }$ . We now show how the # Introduction and Basic Equations components $a ^ { \alpha } \left( \ o { \mathrm { o r } } a ^ { \alpha \beta } \right)$ are obtained. We can do so using the dot product. For each of the three base vectors, ${ \bf e } _ { \alpha }$ , we define a conjugate base vector $\mathbf { e } ^ { \alpha }$ , as follows. Choose ${ \boldsymbol { \mathbf { e } } } ^ { 1 }$ as normal to $\mathbf { e } _ { 2 }$ and $\mathbf { e } _ { 3 }$ , such that the dot product ${ \mathbf e } ^ { 1 } \cdot { \mathbf e } _ { 1 } = 1$ . Similarly, choose ${ \tt e } ^ { 2 }$ normal to $\mathbf { e } _ { 3 }$ and $\mathbf { e } _ { 1 }$ , such that ${ \mathbf e } ^ { 2 } \cdot { \mathbf e } _ { 2 } = 1$ ; and $\mathbf { e } ^ { 3 }$ normal to $\mathbf { e } _ { 1 }$ and $\mathbf { e } _ { 2 }$ , such that $\mathbf { e } ^ { 3 } \cdot \mathbf { e } _ { 3 } = 1$ . Thus, $$ \mathbf {e} ^ {1} \cdot \mathbf {e} _ {1} = 1, \mathbf {e} ^ {1} \cdot \mathbf {e} _ {2} = 0, \mathbf {e} ^ {1} \cdot \mathbf {e} _ {3} = 0 $$ $$ \mathbf {e} ^ {2} \cdot \mathbf {e} _ {1} = 0, \mathbf {e} ^ {2} \cdot \mathbf {e} _ {2} = 1, \mathbf {e} ^ {2} \cdot \mathbf {e} _ {3} = 0 $$ $$ \mathbf {e} ^ {3} \cdot \mathbf {e} _ {1} = 0, \mathbf {e} ^ {3} \cdot \mathbf {e} _ {2} = 0, \mathbf {e} ^ {3} \cdot \mathbf {e} _ {3} = 1 $$ We can write this compactly as $$ \mathbf {e} ^ {\alpha} \cdot \mathbf {e} _ {\beta} = \delta_ {\beta} ^ {\alpha}, $$ where $\begin{array} { r } { \delta _ { \beta } ^ { \alpha } = 1 \operatorname { i f } \alpha = \beta } \end{array}$ , and $\delta _ { \beta } ^ { \alpha } = 0$ , otherwise. $( \delta _ { \beta } ^ { \alpha }$ is called the "Kronecker delta.") In matrix notation $\delta _ { \beta } ^ { \alpha }$ is the unit matrix I: we can also write the above equation defining $\mathbf { e } ^ { 1 } , \mathbf { e } ^ { 2 }$ , and $\mathbf { e } ^ { 3 }$ in matrix form as $$ \left\{ \begin{array}{l} \left\lfloor \mathbf {e} ^ {1} \right\rfloor \\ \left\lfloor \mathbf {e} ^ {2} \right\rfloor \\ \left\lfloor \mathbf {e} ^ {3} \right\rfloor \end{array} \right\}. \left\lfloor \left\{\mathbf {e} _ {1} \right\} \left\{\mathbf {e} _ {2} \right\} \left\{\mathbf {e} _ {3} \right\} \right\rfloor = \mathbf {I}, $$ so that, if one set of base vectors--ei, 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 a. Then $\mathbf { a } \cdot \mathbf { e } ^ { \alpha } = a ^ { \beta } \mathbf { e } _ { \beta } \cdot \mathbf { e } ^ { \alpha }$ (writing a in component form, using the basis vectors $\mathbf { e } _ { \beta } )$ , and since $\mathbf { e } _ { \beta } \cdot \mathbf { e } ^ { \alpha } = \delta _ { \beta } ^ { \alpha } = 1$ , only if $\alpha = \beta$ , $$ \begin{array}{l} \mathbf {a} \cdot \mathbf {e} ^ {\alpha} = a ^ {\beta} \mathbf {e} _ {\beta} \cdot \mathbf {e} ^ {\alpha} \\ = a ^ {\beta} \delta_ {\beta} ^ {\alpha} = a ^ {\alpha}. \\ \end{array} $$ In exactly the same way we could have written $$ a _ {\alpha} = \mathbf {a} \cdot \mathbf {e} _ {\alpha} $$ by expressing a as components associated with the $\mathbf { e } ^ { \alpha }$ base vectors, $\mathbf { a } = a _ { \alpha } \mathbf { e } ^ { \alpha }$ . Similarly, for a matrix, $$ a ^ {\alpha \beta} = (\mathbf {e} ^ {\alpha}) ^ {T} \cdot \mathbf {a} \cdot \mathbf {e} ^ {\beta} = \mathbf {e} ^ {\alpha} \cdot \mathbf {a} \cdot \mathbf {e} ^ {\beta}, $$