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TENSORS/ 3-D STRESS STATE

TENSORS/ 3-D STRESS STATE. Tensors Tensors are specified in the following manner: A zero-rank tensor is specified by a sole component, independent of the system of reference (e.g., mass, density).

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TENSORS/ 3-D STRESS STATE

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  1. TENSORS/ 3-D STRESS STATE

  2. Tensors • Tensors are specified in the following manner: • A zero-rank tensor is specified by a sole component, independent of the system of reference (e.g., mass, density). • A first-rank tensor is specified by three (3) components, each associated with one reference axis (e.g., force). • A second-rank tensor is specified by nine (9) components, each associated simultaneously with two reference axes (e.g., stress, strain). • A fourth-rank tensor is specified by 81 components, each associated simultaneously with four reference axes (e.g., elastic stiffness, compliance).

  3. (4-1) • The Number of Components (N) required for the description of a TENSOR of the nthRank in a k-dimensional space is: N = kn EXAMPLES (a) For a 2-D space, only four components are required to describe a second rank tensor. (b) For a 3-D space, the number of components N = 3n Scalar quantities  30  Rank Zero Vector quantities  31  Rank One Stress, Strain  32  Rank Two Elastic Moduli  34  Rank Four

  4. The indicial (also called dummy suffix) notation will be used. • The number of indices (subscripts) associated with a tensor is equal to its rank. It is noted that: • density () does not have a subscript • force has one (F1, F2, etc.) • stress has two (12, 22, etc.) • The easiest way of representing the components of a second- rank tensor is as a matrix • For the tensor T, we have:

  5. The collection of stresses on an elemental volume of a body is called stress tensor, designated as ij. In tensor notation, this is expressed as: where i and j are iterated over x, y, and z, respectively. (4-2)

  6. Here, two identical subscripts (e.g., xx) indicate a normal stress, while a differing pair (e.g., xy) indicate a shear stress. It is also possible to simplify the notation with normal stress designated by a single subscript and shear stresses denoted by , so: x  xx xy  xy (4-3)

  7. In general, a property T that relates two vectors p = [p1, p2, p3] and q = [q1, q2, q3] in such a way that where T11, T12, ……. T33 are constants in a second rank tensor. (4-4)

  8. (4-5) • (Eqn. 4-4) can be expressed matricially as: • Equation 4-5 can be expressed in indicial notation, where • The symbol is usually omitted, and the Einstein’s summation rule used. (4-6) (4-7) “dummy” Subscript (appears twice) Free Subscript

  9. Transformations X3 p • Transformation of vector p [p1, p2, p3 ] from reference system x1, x2, x3 to reference x’1, x’2, x’3 can be carried out as follows where = X’3 X’2 X2 X1 X’1 Angle between X’iXj Old New

  10. (4-8) • In vector notation: p = p1i1 + p2i2 + p3i3 where i1, i2, and i3 are unit vectors p’ = p1 cos(X’1X1) + p2 cos(X’1X1) + p3 cos(X’1X1) = a11 p1 + a12 p2 + a13 p3 where aij = cos (X’iXj) is the direction cosine between X’i andXj. (4-9) Old New

  11. Old System • The nine angles that the two systems form are as follows: = This is known as the TRANSFORMATION Matrix New System (4-10)

  12. What is the transformation matrix for a simple rotation of 30o about the z-direction? 30o 30o

  13. For any Transformation from p to p’,determine the Transformation Matrix and use as follows: This can be written as: • It is also possible to perform the opposite operation, i.e., new to old (4-11) (4-12) (4-13)

  14. (4-14a) • The Transformation of a second rank Tensor [Tkl] from one reference frame to another is given as: OR, for stress Eqn. 4-14(a) is the transformation law for tensors and the letters and subscripts are immaterial. • Transformation from new to old system is given as: (4-14b) (4-15)

  15. NOTES on Transformation • Transformation does not change the physical integrity of the tensor, only the components are transformed. • Stress/strain Transformation results in nine components. • Each component of the transformed 2nd rank tensor has nine terms. • lij and Tij are completely different, although both have nine components. • Lijis the relationship between two systems of reference. • Tij is a physical entity related to a specific system of reference.

  16. Transformation of the stress tensor ij from the system of axes to the • We use eqn. 4-14: • First sum over j = 1, 2, 3 • Then sum over i = 1, 2, 3 (4-16)

  17. For each value of k and l there will be an equation similar to eqn. 4-16. • To find the equation for the normal stress in the x’1 direction, let m = 1 and n = 1. • Let us determine the shear stress on the x’ plane and the z’ direction, that is ’13 or x’z’ for which m = 1 and n = 3

  18. The General definition of the Transformation of an nth-rank tensor from one reference system to another (i.e., TT’) is given by: T’mno……. = lmilnjlok…………….Tijk……….. Note that aij = lmi (the letters are immaterial) • The transformation does not change the physical integrity of the tensor, only the components are transformed (4-16)

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