Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics

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Besides, the text and formulae have thoroughly been reexamined and improved where necessary. In spite of a somewhat di?

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I wish the same for the reader of the book. Although the present book can be referred to as a textbook one? I tried to explain the matter in a brief way, nevert- lessgoinginto detailwherenecessary. Ialsoavoidedtediousintroductions and lengthy remarks about the signi? A reader - terested in tensor algebra and tensor analysis but preferring, however, words instead of equations can close this book immediately after having read the preface.

FourthOrder Tensors. Analysis of Tensor Functions. Vector and Tensor Analysis in Euclidean Space.

Tensor Algebra and Tensor Analysis for Engineers (5th ed.)

Every set of more than n vectors is linearly dependent. The proof of this theorem is similar to the preceding one. Then, the vectors 1. Excluding a vector gk we obtain a set of vectors, say G , with the property that every vector in V is a linear combination of the elements of G. Hence, any set of more than n vectors is linearly dependent. The representation 1. The latter one is likewise impossible because these vectors form a basis of V. The summation of the form 1. For this reason it is usually represented without the summation symbol in a short form by Accordingly, the summation is implied if an index appears twice in a multiplicative term, once as a superscript and once as a subscript.

Such a repeated index called dummy index takes the values from 1 to n the dimension of the vector space in consideration. The sense of the index changes from superscript to subscript or vice versa if it appears under the fraction bar. An n-dimensional vector space furnished by the scalar product with properties C.

Thus, the elements of an orthonormal basis represent pairwise orthog- onal unit vectors. Of particular interest is the question of the existence of an orthonormal basis. In other words, starting from linearly independent vectors x1, x2,. However, the latter result contradicts the fact that the vectors x1 and x2 are linearly independent. Since these vectors are non-zero and mutually orthogonal, they are linearly independent see Exercise 1. By virtue of 1.

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The next important question is whether the dual basis is unique. Then, Thus, we have proved the following theorem. To every basis in an Euclidean space En there exists a unique dual basis. Relation 1. However, it can also be obtained without any orthonormal basis. The components of a vector with respect to the dual bases are suitable for calculating the scalar product. Dual basis in E3. Indeed, with the aid of 1. On the other hand, we can write again using 1.

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Linearity of the mapping 1. Setting in 1. Vector product in E3. Representation of a rotation by a second-order tensor. A rotation of a vector a in E3 about an axis yields another vector r in E3. The Cauchy stress tensor as a linear mapping of the unit surface normal into the Cauchy stress vector.

In other words, the Cauchy stress vector is the same for all surfaces through P which have n as the nor- mal in P. On the other hand, we obtain the same result also by Further, we show that the vector yA, satisfying condition 1.

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Since the order of mappings in 1. Indeed, by virtue of B. In the following, we show that a basis of Linn can be constructed with the aid of the tensor product 1. The dimension of the vector space Linn is thus n2. Thus, it is seen that condition 1. Of special importance is the so-called identity tensor I. In view of 1. For this reason, the identity tensor is frequently called metric tensor. With respect to an orthonor- mal basis relation 1. Let x be a vector and A a second-order tensor.

According to 1. Similarly we obtain by virtue of 1. The transformation rules 1. The composition of tensors 1. Besides, the composition of tensors is characterized by the following properties see Exercise 1. For the tensor product 1. Powers, polynomials and functions of second-order tensors.

On the basis of the composition 1. Indeed, every tensor A in Linn can be represented with respect to the tensor product of the basis vectors in En in the form 1. Hence, considering 1. It does not, however, hold for the mixed components 1. The transposition operation 1. Inserting 1. Mapping both sides of this vector inequality by A and taking 1. Then, using the vector identity see Exercise 1. We prove for example the property D.

To this end, we represent the tensor A with respect to an orthonormal basis 1. Keeping 1. For three arbitrary tensors A, B and C given in the form 1. Indeed, the axioms A. It follows from 1.

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Obviously, symmetric and skew-symmetric tensors are mutually orthogo- nal such that see Exercise 1. Just like symmetric and skew-symmetric tensors, spherical and deviatoric tensors form orthogonal subspaces of Linn. The tensors of the third order can likewise be repre- sented with respect to a basis in Linn e. In other words, linearly independent vectors are all non-zero. Prove that any non-empty subset of linearly independent vectors x1, x2,. Prove that a set of mutually orthogonal non-zero vectors is always linearly independent.

Compare the result with the solution of problem b. Verify that the vectors 1. Prove identity 1. Prove relations 1. Verify the following identities involving the permutation symbol 1. Prove formula 1. Prove relation 1. Prove 1. Prove identities 1. Find commutative pairs of tensors.

Let A and B be two commutative tensors. Evaluate exp 0 and exp I. Prove by means of 1. They are referred to as the derivatives of the vector- and tensor-valued functions x t and A t , respectively. For example, for the derivative of the composition of two second-order tensors 2. A coordinate system is a one to one correspondence between vectors in the n-dimensional Euclidean space En and a set of n real numbers x1 , x2 ,.

These numbers are called coordinates of the corresponding vectors. Cylindrical coordinates in E3. The cylindrical coordinates Fig. Indeed, according to Theorem 1. The compo- nents xi 2. These functions describe the transformation of the coordinate systems. Inserting one relation 2. Theorem 2.

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One can verify that the tangent vectors are linearly independent and form thus a basis of En. Conversely, let the vectors 2.

By means of 2. The trans- formation rules of the form 2. Covariant and contravariant variables are denoted by lower and upper indices, respectively. The co- and contravariant rules can also be recognized in the transforma- tion of the components of vectors and tensors if they are related to tangent vectors.

The transformation rules 2. Transformation of linear coordinates into cylindrical ones 2. Due to the one to one correspondence 2. Gradient of the scalar function r. Comparing this result with 2. In order to evaluate the above gradients 2.

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