The formula (6.18) can be reduced to the formula (6.15) for the linear theory of elasticity if only the reference (non-perturbed) state is natural (non-

OF LINEARIZED THEORY OF ELASTICITY

Note 6.3. The formula (6.18) can be reduced to the formula (6.15) for the linear theory of elasticity if only the reference (non-perturbed) state is natural (non-

deformed).

The three first algebraic invariants (6.4) form next group of the linearized relations. They follow from the representation (6.19) and are written therefore through the components of the gradient of displacement

6

1 o, ,

nl n l n l

I G u u

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL(RESIDUAL) STRESSES BECAUSE THENON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES. IN THIS WAY, THE HUGE PERS- PECTIVES AROSE FOR MECHANICS OF MATERIALS IN STUDYING THE EFFECT OF INITIAL STRESSES ON MECHANICAL PROCESSES OF DEFORMATION.

, (6.20)

6

2 2 _kl^o _{nl n l}^o, _{n k},

I H G u u

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL(RESIDUAL) STRESSES BECAUSE THENON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES. IN THIS WAY, THE HUGE PERS- PECTIVES AROSE FOR MECHANICS OF MATERIALS IN STUDYING THE EFFECT OF INITIAL STRESSES ON MECHANICAL PROCESSES OF DEFORMATION.

, (6.21)

6

3 3 _{kl lm}^{o o} _{nm n m}^o, _{n k},

I H H G u u

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL(RESIDUAL) STRESSES BECAUSE THENON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES. IN THIS WAY, THE HUGE PERS- PECTIVES AROSE FOR MECHANICS OF MATERIALS IN STUDYING THE EFFECT OF INITIAL STRESSES ON MECHANICAL PROCESSES OF DEFORMATION.

. (6.22) Note 6.4. The condition of incompressibility

6

3 0

I

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL(RESIDUAL) STRESSES BECAUSE THENON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES. IN THIS WAY, THE HUGE PERS- PECTIVES AROSE FOR MECHANICS OF MATERIALS IN STUDYING THE EFFECT OF INITIAL STRESSES ON MECHANICAL PROCESSES OF DEFORMATION.

is transformed into the equality

6

,

, 0

o o

mk nm n m n k

G G u u

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL(RESIDUAL) STRESSES BECAUSE THENON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES. IN THIS WAY, THE HUGE PERS- PECTIVES AROSE FOR MECHANICS OF MATERIALS IN STUDYING THE EFFECT OF INITIAL STRESSES ON MECHANICAL PROCESSES OF DEFORMATION.

, (6.23) where

6

²

^{det 2}

o o

mk o ij ij

mk mk

G G H

G H

w

w

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL(RESIDUAL) STRESSES BECAUSE THENON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES. IN THIS WAY, THE HUGE PERS- PECTIVES AROSE FOR MECHANICS OF MATERIALS IN STUDYING THE EFFECT OF INITIAL STRESSES ON MECHANICAL PROCESSES OF DEFORMATION.

.

Next important set of linearized characteristics is the set of components of stress tensors. As it follows from representation of the strain tensors through the components of the gradient of displacements, the first candidate on the following using is the Piola-Kirchhoff stress tensor which form the ordered pair just with the gradient of displacements. Therefore, the linearized equations of motion are written in the classical form (3.13)

6

i ik k, i

u t F

U

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL(RESIDUAL) STRESSES BECAUSE THENON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES. IN THIS WAY, THE HUGE PERS-

. (6.24) But it was so happened that the Cauchy-Lagrange stress tensor is usually used in the linearized theory of elasticity and then the formula of the link between the Piola-Kirchhoff and Lagrange- Cauchy stress tensors is needed

6

o,

o ,

ik ij nj n j ij n j

t V G u V u

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT

. (6.25) Then the motion equations (6.24) can be written in the new form that can be considered as the new linearized form

6

,

, ,

o o

i ij nj n j ij n j k i

u u u F

U ^ª¬V G V ^º¼ . (6.26)

FOUNDATIONS OF MECHANICS OF MATERIALS PART 1

SHORT DESCRIPTION OF LINEARIZED THEORY OF ELASTICITY

Thus, the motion equations relative to the perturbances, which describe the deformation in the actual (perturbed state), include the stresses and gradient of displacements of the reference (non-perturbed) state.

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL (RESIDUAL) STRESSES BECAUSE THE NON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES.

IN THIS WAY, THE HUGE PERS-PECTIVES AROSE FOR MECHANICS OF MATERIALS IN STUDYING THE EFFECT OF INITIAL STRESSES ON MECHANICAL PROCESSES OF DEFORMATION.

The partial case was often studied in the linearized theory of elasticity – the case when the initial (basic) state is uniform.

Then the displacements in this state are expressed through the constant principal stretches

6

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL(RESIDUAL) STRESSES BECAUSE THENON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES. IN THIS WAY, THE HUGE PERS- PECTIVES AROSE FOR MECHANICS OF MATERIALS IN STUDYING THE EFFECT OF INITIAL STRESSES ON MECHANICAL PROCESSES OF DEFORMATION.

i const

O (it is assumed sometimes that

6

THIS SITUATION OPENS THE GREAT POSSIBILITIES TO TAKE INTO ACCOUNT THE INITIAL(RESIDUAL) STRESSES BECAUSE THENON-PERTURBED STATE IS ASSUMED NOT NATURAL AND CAN BE CONSIDERED AS THE STATE WITH THE INITIAL STRESSES OR PRINCIPAL STRETCHES. IN THIS WAY, THE HUGE PERS- PECTIVES AROSE FOR MECHANICS OF MATERIALS IN STUDYING THE EFFECT OF INITIAL STRESSES ON MECHANICAL PROCESSES OF DEFORMATION.

Oi are changing slowly in time) as follows

6

1

ko i i ik

u x O G . (6.27)

The expression (6.27) generates a row of important relations

6

, ,

ik 1 2 i i ku k k iu

H O O , (6.28)

6

1 2 2 1

o o

ik i ik

H ^§¨© O ^·¸¹G , (6.29)

6

²

²

²

1 _{k k k}^o, , 2 _k 1 _{k k k}^o, , 3 3 4 _k 1 _{k k k}^o,

I O u I O Ou I O O u , (6.30)

6

2 3

2 2 2

1^o 1 2 _i^o 3 , 2^o 1 4 _i^o 1 , 3^o 1 8 _i^o 1

I §¨ O ·¸ I §¨ O ·¸ I §¨ O ·¸

© ¹ © ¹ © ¹ . (6.31)

Let us return to the presented above equations of motion. They together with the constitutive equations form the basic system of equations of the linearized theory of elasticity. The initial and boundary conditions have the classical form, when the Piola-Kirchhoff stress tensor is used. But the constitutive equations need the special analysis. This will be done after some discussion about the role of small and large strains in the linearized theory of elasticity.

FOUNDATIONS OF MECHANICS OF MATERIALS PART 1

SHORT DESCRIPTION OF LINEARIZED THEORY OF ELASTICITY

First, the distinction between small strains and perturbances should be discussed. So, because the refere- nce (basic) state admits both the large and small strains, then some problems can arise in comparing the small strains and perturbances. It seems to be worthy to cite the note from the book of Guz [11].

Note 6.5. ... in this book, along with the theory of finite strains, the theory of small strains is used. Moreover, perturbances are also considered small quantities.

This circumstance (small deformations and perturbations) can be (with a fuzzy understanding of the problem statement) a source of misunderstandings and errors both in the linearization of the main relations and in interpreting the significance and accuracy of the results obtained in the framework of the theory of strains. In this regard, we note that small deformations are understood to mean strains that are much less than unity (in contrast to the theory of finite strains, where strains can also be of the order of unity and arbitrarily greater than unity), but not arbitrarily small, tending to zero (they tend to zero only when external influences are removed, and then provided that no residual deformation occurs). The perturbations are, according to the statement of the problem, arbitrarily small quantities. Thus, in the linearized mechanics of deformable bodies ... although the initial deformations as deformations of the non-perturbed state are small, they should nevertheless be considered finite quantities with respect to perturbations.

The linearized theory of elasticity differs two cases

of the non-perturbed (initial, basic) states – THE CASES OF LARGE AND SMALL STRAINS.

Case of the large strains.

The large strains are described by use of the nonlinear theory of elasticity. This theory operates with the notions:

“the volumes ^Vo and boundary surfaces ^So of the elastic body in the non-perturbed state”;

“the volumes V and boundary surfaces ^S of the elastic body in the perturbed state”.

They are assumed here the distinguishing states.

Let us return to the characteristics of states.

First, repeat the formulas (6.9) and (6.10):

the reference (non-perturbed) state is described mainly by characteristics

6

1, , , ,2 3 1, , , ,2 3 1, , , ,...,2 3 ,...

o o o o

m nm iik nm

u x x x t H ⁶x x x t V x x x t e H

FOUNDATIONS OF MECHANICS OF MATERIALS PART 1

SHORT DESCRIPTION OF LINEARIZED THEORY OF ELASTICITY

Add now four new characteristics:

for the volume forces

6

m mo m

F F Fc, for the surface forces

6

m om cm

6 6 6 ,

for the ratio of elementary volume of body after and before deformation

6

^{Vafter Vbefore Vafter Vbefore o Vafter Vbefore}

^c,

for the ratio of area of elementary rectangular after and before deformation

6

Sk^after Sk^before Sk^after Sk^{before o} Sk^after Sk^before

c.

Now, the linearized relative to perturbances equations of equilibrium and boundary conditions can be written.

6

^o,

^o , , ^o

^{after before after before}

^o 0

ij nj un j ij n ju k F Vi V F Vi V

V G V

ª º

¬ ¼ , (6.32)

in stresses

6

,

,

o after before after before o o o

i V V i V V ni^ªV Gij nj un j Vij n ju ^º

6 6 ¬ ¼^,, (6.33)

in displacements

6

1, ,2 3 0

k x x x S

u _ . (6.34)

The equations (6.32) can be complemented by the linearized constitutive equations and expressions for the strain tensor. They are shown at the first part of this chapter.

This finished the very short description of the case of large strains.

Note 6.6. When the linearized constitutive equations and expressions for the strain