Fourth order tensor

Here g [ ] may be called the elastic compliance tensor, andl [-]maybe called the inelastic compliance tensor. Note that g is a fourth-order tensor which shares the symmetries of t. Again, (5.16) may be written as... 

The stiffness and compliances in stress-strain and strain-stress relations are fourth-order tensors because they relate two second-order tensors ... 

Contracted notation is a rearrangement of terms such that the number of indices is reduced although their range increases. For second-order tensors, the number of indices is reduced from 2 to 1 and the range increased from 3 to 9. The stresses and strains, for example, are contracted as in Table A-1. Similarly, the fourth-order tensors for stiffnesses and compliances in Equations (A.42) and (A.43) have 2 instead of 4 free indices with a new range of 9. The number of components remains 81 (3 = 9 ). 

A fourth-order tensor can be written as a 9x9 array in analogy to Equation (A.51) but, by use of contracted notation, is sometimes drastically simplified to a 6 x 6 symmetric array. 

Ctjki is a fourth order tensor that linearly relates a and e. It is sometimes called the elastic rigidity tensor and contains 81 elements that completely describe the elastic characteristics of the medium. Because of the symmetry of a and e, only 36 elements of Cyu are independent in general cases. Moreover only 2 independent rigidity constants are present in Cyti for linear homogeneous isotropic purely elastic medium Lame coefficient A and /r have a stress dimension, A is related to longitudinal strain and n to shear strain. For the purpose of clarity, a condensed notation is often used... 

The requirement of isotropy permits the representation of the fourth-order tensor in terms of two material functions in such a way that the stress-strain relationship becomes... 

Here, A is a fourth-order tensor that must be symmetric in its first two indices,... 

The algebra of multi-way arrays is described in a field of mathematics called tensor analysis, which is an extension and generalization of matrix algebra. A zero-order tensor is a scalar a first-order tensor is a vector a second-order tensor is a matrix a third-order tensor is a three-way array a fourth-order tensor is a four-way array and so on. The notions of addition, subtraction and multiplication of matrices can be generalized to multi-way arrays. This is shown in the following sections [Borisenko Tarapov 1968, Budiansky 1974],... 

The fourth order tensor C is the derivative of the elastic tensor with respect to damage variable ... 

How many components are there in a fourth-order tensor How many components would be independent if the tensor was symmetric ... 

Momentum space averages Fourth-order tensor with components -F E Transpose of the tensor A... 

Similarly, it is possible to define a fourth-order tensor... 

The evolution Eq. 5.9 contains a fourth order tensor ayki that is also unknown. This equation is evidently not a elosed form. To close the set of evolution equations, a closure approximation has to be introduced to represent the fourth order tensor in term of the seeond order tensor. There are various closure approximations available. 

Given two second-order tensors A (Ay) and B (Bij), we have different types of products the dyadic product, the single dot product and the double dot product. The dyadic product is a fourth-order tensor, written as... 

Since Ci and 8 are symmetric tensors, each of them has 6 independent components, the fourth order stiffness tensor Cyia contains at most 36 independent constants, such that it can be displayed as a 6 x 6 matrix of components using contracted notation, Cto), where m,n — 1,2,3,4,5,6. There is a unique correspondence between of and Cijki- The index m is related to ij, and n is related to Id, as shown in Table B.l. For instance, Cu22 = C12, C1323 = C54- Since C = C the number of independent constants is generally 21. For orthotropic materials, the the number of independent constants further reduces to 9. When the fourth order tensor is transformed to the principal axes, all Qj — 0, except for Cn, C22, C33, C12. 13. 23. 44, 55, and... 

Here Lijki is a fourth order tensor describing the linear relation between strain rate and stress rate and Nij represents the nonlinear part. 

A third-order tensor is represented by a coordinate cube with 3x3x3 = 27 components. This scheme can be extended to arbitrarily high orders. A fourth-order tensor, having 3 = 81 components, cannot be imagined geometrically. Nevertheless, it is of great importance in material science (see section 2.4.2). 

The parameters of the Kelvin-Voigt model and the internal are fourth order tensors, while the strain and the stress are second order tensors. An imdetermined number of Kelvin-Voigt elements give flexibility to the model without increasing the complexity of the constitutive law as it will be discussed later in this section. Assuming a virgin material, having no p>ermanent strain due to earlier where J is the instantaneous elastic strain, where... 

It is worth recalling here that each tensor has an order (I, II, III, IV, etc.). Tensor order reflects the physical properties of a tensor and is determined by the power of the direction cosines product, that is, the power of the product of linear transformation coefficients. The tensor order physically reflects the possibility of visualizing the various properties of a field or a body from different viewpoints. Tensor order is also an indicator of the different ways in which spatial anisotropy is revealed. Scalar quantities, that is, temperature, mass, and amount of heat, are zeroth-order tensors the vectors of velocity or force are the first-order tensors mechanical stresses and strains are second-order tensors, while the elasticity modulus is a fourth-order tensor, as will be shown in the following text. 

In accordance with the experiment, the transverse compression strains have an inverse sign to the longitudinal extensions, and their contribution, v, is on the order of 0.2-0.3. These two parameters, that is, the modulus with the units of stress (Pa), [E] = [o], and dimensionless Poisson ratio v are sufficient for describing arbitrary deformations of an isotropic solid. It is worth pointing out here that two parameters (E and v, G and v, or G and E) are the smallest pieces of fourth order tensors that are sufficient for the description of strains. It is not possible for one to get away with using a single parameter. Eor an arbitrary principal stress tensor. 

Here, summation takes place over the four repeating indices. The coefficients a constitute the direction cosines to the fourth power the parameters (or represent a fourth-order tensor, referred... 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

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