# Metastability Essay

6600 Words
Nov 1st, 2015
27 Pages

Chapter 2

Stress Definition

“Stress tensor” is a tautology since the term tensor (latin tensio) means stress. The pleonasm, however, became naturalized in books dealing with mechanical stress (Hahn 1985, p. 20, fn. 1).

Referring to other second-rank tensors in physics like the inertial tensor, this tautology disappears

This chapter presents the fundamental concept of stress as it is defined from a mathematical, physical and continuum mechanics point of view. The stress tensor defining the state of stress at a point is introduced using the continuum concept of a stress vector (traction) defining the state of stress on a plane (Sect. 2.1). Principal stresses and their orientations are deduced from solving the eigenvalue problem

Stress Definition

“Stress tensor” is a tautology since the term tensor (latin tensio) means stress. The pleonasm, however, became naturalized in books dealing with mechanical stress (Hahn 1985, p. 20, fn. 1).

Referring to other second-rank tensors in physics like the inertial tensor, this tautology disappears

This chapter presents the fundamental concept of stress as it is defined from a mathematical, physical and continuum mechanics point of view. The stress tensor defining the state of stress at a point is introduced using the continuum concept of a stress vector (traction) defining the state of stress on a plane (Sect. 2.1). Principal stresses and their orientations are deduced from solving the eigenvalue problem

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A. Zang, O. Stephansson, Stress Field of the Earth’s Crust,

DOI 10.1007/978-1-4020-8444-7_2, © Springer Science+Business Media B.V. 2010

17

18

2 Stress Definition

Fig. 2.1 Traction vector

σ (n)

σ acting on a hypothetical

¯

(fictitious) slicing plane A with surface normal n within

¯

a deformable body n P

-n

V2

x3

V1

A

x2

σ (-n)

P=P (x1, x2, x3) n=(n1, n2, n3)

x1

dF

A→0 dA

σ (P(¯ ), n) = lim

¯

x ¯

(2.1)

In general, the traction vector σ can vary from point to point, and is therefore a

¯

function of the location of the point P(¯ ). However, at any given point, the traction x will also, in general, be different on different planes that pass through the point.

¯

Therefore, σ will also be a function of n, the outward unit normal vector of the

¯

¯

¯

slicing plane. In summary, σ is a function of two vectors, the position vector x

¯

and the normal vector of the slicing plane n. In 1823, the French mathematician

Augustin Baron Cauchy (1789–1857) introduced the concept of stress by eliminating the difficulty that σ is a function of two vectors, σ (¯ , n) at the price that stress

¯

¯ x ¯ became a second-order tensor (Jaeger et al. 2007).

We have three remarks about Eq. (2.1). First, Eq. (2.1) is an empirical formula,

i.e. is confirmed by experimental

DOI 10.1007/978-1-4020-8444-7_2, © Springer Science+Business Media B.V. 2010

17

18

2 Stress Definition

Fig. 2.1 Traction vector

σ (n)

σ acting on a hypothetical

¯

(fictitious) slicing plane A with surface normal n within

¯

a deformable body n P

-n

V2

x3

V1

A

x2

σ (-n)

P=P (x1, x2, x3) n=(n1, n2, n3)

x1

dF

A→0 dA

σ (P(¯ ), n) = lim

¯

x ¯

(2.1)

In general, the traction vector σ can vary from point to point, and is therefore a

¯

function of the location of the point P(¯ ). However, at any given point, the traction x will also, in general, be different on different planes that pass through the point.

¯

Therefore, σ will also be a function of n, the outward unit normal vector of the

¯

¯

¯

slicing plane. In summary, σ is a function of two vectors, the position vector x

¯

and the normal vector of the slicing plane n. In 1823, the French mathematician

Augustin Baron Cauchy (1789–1857) introduced the concept of stress by eliminating the difficulty that σ is a function of two vectors, σ (¯ , n) at the price that stress

¯

¯ x ¯ became a second-order tensor (Jaeger et al. 2007).

We have three remarks about Eq. (2.1). First, Eq. (2.1) is an empirical formula,

i.e. is confirmed by experimental