Introduction to deformable solid mechanics: educational manual

256 Pages·2019·3.168 MB·English

by  Искакбаев А.

Introduction
AL-FARABI KAZAKH NATIONAL UNIVERSITY

Alibay Iskakbayev

INTRODUCTION
TO DEFORMABLE SOLID MECHANICS

Educational manual

Almaty
«Qazaq University»
2019
1
Introduction to Deformable Solid Mechanics
UDC 531 (075)
LBC 222я73
I-85
Recommended for publication by the Scientific Council
of the Faculty of Mechanical Mathematics
and RISO of Al-Farabi Kazakh National University
(Protocol №3 dated 06.02.2019)

Reviewers:
Doctor of technical sciences, professor B.B. Teltaev
Doctor of physical and mathematical sciences, professor G.B. Sheryazdanov

Translated from Russian language edition:
«Введение в механику деформируемого твердого тела»
(published by Palmarium Academic Publishing, German in 2014)
and edited by: A.A. Iskakbayeva, A.Zh. Bibossinov

Iskakbayev Alibay
I-85 Introduction to deformable solid mechanics: educational
manual / Alibay Iskakbayev. – Almaty: Qazaq University,
2019. – 256 p.
ISBN 978-601-04-3929-0

This manual, which is distinguished by its extremely complete ter-
minology on the mechanics of deformable solid body with explanations that
are accessible to perception, will help students to become familiar with the
logic of constructing models of the mechanics of deformable solid body. The
manual contains more than 150 figures and 10 tables, which favorably affects
the students' mastering of the course material of the mechanics of deformable
solid body, regardless of the form of education.
The book meets all modern requirements for the provision of complete and
accessible educational information to students with any form of education.
The educational manual is intended for university students and can be
used in preparation and promotion knowledge in the field.

UDC 531 (075)
LBC 222я73

ISBN 978-601-04-3929-0 © Iskakbayev Alibay, 2019
© Al-Farabi KazNU, 2019
2
Introduction

INTRODUCTION

The main task of the mechanics of a deformable solid (MDTT) is
the modeling of the processes of deformation and fracture.
Destruction refers to one of the types of durability. Violation of
the strength of materials and structural elements may occur as a result
of:
1) excessive (elastic or plastic) deformation;
2) loss of stability;
3) fracture.
Fracture may be partial or complete. In case of partial destruction
in the body, material damage occurs in the form of individual cracks
or in the form of material defects distributed over the volume, resulting
in a change in the mechanical properties of the material. With
complete destruction, the body is divided into parts.
Fracture can also be divided into the following types:
1. Plastic fracture. Occurs after a significant plastic deformation
that occurs throughout (or almost all of) the volume of the body.
A type of plastic failure is a rupture after 100% neck contraction under
tension resulting from the loss of the ability of a material to resist
plastic deformation. This case can also be attributed to one of the types
of durability.
2. Brittle fracture. Occurs as a result of propagation of the main
crack after plastic deformation, concentrated in the area of the
mechanism of destruction.
In case of ideally brittle fracture, there is no plastic deformation,
therefore, after fracture, it is possible to re-compose the body of the
previous size from fragments without gaps between them.
3
Introduction to Deformable Solid Mechanics
In the case of a quasi-brittle fracture, there is a plastic zone in
front of the edge of the crack and riveted material at the crack surface.
The other, and a much larger, volume of the body is in an elastic state.
At present, a fracture, in which the breaking stress in the net
section is higher than the yield strength, but lower than the tensile
strength, is called quasi-brittle.
3. Fatigue damage. Occurs with cyclic (repeated) loading as a
result of accumulation of irreversible damage. The fracture is macro-
scopically brittle, however, the material at the surface of the fracture
is substantially riveted. There are fatigue and low-cycle fatigue.
Fatigue is characterized by nominal stresses, lower yield strength,
repeated loading is macroscopically observed in the elastic region, the
number of cycles before failure is large.
Low-cycle fatigue is characterized by nominal stresses, high yield
strength; macroscopic plastic deformation occurs at each cycle; the
number of cycles before failure is relatively small.
4. Deformation and creep rupture. The sample under load at
elevated temperatures, even under conditions of a uniform stress state,
begins to break down. It forms a system of microscopic cracks along
the grain boundaries.
In fact, at the grain boundaries, pores are first formed, then these
pores merge into cracks with rounded ends, extending within one
grain. Over time, the number of microscopic cracks increases, they
merge into one or several through cracks and destruction occurs. The
question of the movement or equilibrium of a macroscopic crack in
such a formulation is not significant, the main role is played by the
kinetics of accumulation of microcracks. When the density reaches a
critical value, destruction occurs. The position and shape of the
fracture macrocrack are more or less random.
5. Corrosion destruction. There are three main mechanisms of
influence of corrosive environments on the crack resistance of
structural materials: adsorption reduction of strength, hydrogen
embrittlement and corrosion dissolution.
The adsorption of surfactants on the surface of a highly stressed
material in the tip of a crack causes a decrease in the surface energy
and facilitates destruction (the Rehbinder effect). Adsorption effects
can be successfully used to increase the efficiency of metalworking.
4
Introduction
Note that of the three main mechanisms the adsorption effect is
dominant at high values of the stress intensity factor when due to high-
speed subcritical crack growth, other mechanisms do not have time to
occur.
It is well known that the effect of moisture on metals leads to
corrosion and destruction. Corrosion often does not change the
mechanical properties of the material, but leads to a gradual uniform
decrease in the size of the loaded part, for example, due to gradual
dissolution. As a result, the stresses acting in the dangerous section
increase, and when they exceed the permissible level, destruction will
occur. According to modern concepts, the main process that
accelerates subcritical crack growth, leading to accidents, is hydrogen
embrittlement of a small region near the tips of the cracks.
In most cases of corrosion growth of cracks, the processes of
adsorption, hydrogen embrittlement and corrosion dissolution are
mutually interconnected, and one process causes the manifestation of
others. The interrelationship of these processes is complicated by the
influence of the metal structure, the type of stress state, and external
loading conditions. The study of this linkage is the subject of corrosion
fracture mechanics.
This tutorial covers modeling of the processes of deformation and
fracture, which are the main tasks of the mechanics of a deformable
solid body. It contains the basic concepts and models of the theory of
elasticity and viscoelasticity, the theory of strength and cracks, the
theory of stability of deformable bodies, the theory of plasticity, the
mechanics of composite materials and damage mechanics. Students
reveal the mechanisms of violation of the strength of solids.

5
Introduction to Deformable Solid Mechanics

1
SOME INFORMATION FROM CONTINUUM
MECHANICS

1.1. Stress condition

Stress condition is characterized by a symmetric stress tensorat in
any point of a continuous medium

   
 x xy xz
(1.1)
ˆ    ,
xy y yz
 
  
 xz yz z

where  , , are normals,  , , are tangential stresses on
x y z xy yz xz
the areas vertical to x, y, z coordinate axis.

The stress vectorp on an arbitrary orientated area with the unit
ï

normal n (Figure 1.1) is determined by Cauchy formula:

p n  n  n ,
x x x xy y xz z

(1.2)
p  n n  n ,
y xy x y y yz z

p  n  n n ,
 z xz x yz y z z


where n ,n ,n are components of the unit normal vector n.
x y z
6
1. Some information from continuum mechanics

Figure 1.1. Components of stress vector


Calculating vector p on the direction of the normal, we obtain
ï
the normal stress  on this area
n

  n2  n2 n2 
n x x y y z z (1.3)
2 n n 2 n n 2 n n .
xy x y yz y z zx z x

The tangential stress rate is equal to:
n

 p2 p2 p2 21/2. (1.4)
n x y z n

There are three mutually perpendicular planes at any point of a
continuum, on which the tangential stresses are zero. The directions of
the normals to these areas form the main directions of the stress tensor
and do not depend on the initial coordinate system x, y, z. This means
that any stress state at the considered point can be caused by the
extension of the point area in three mutually perpendicular directions.
The corresponding stresses are called the principal normal stresses;
let us denote them by , , and let us enumerate the principal
1 2 3
axes so that

   . (1.5)
1 2 3
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Introduction to Deformable Solid Mechanics
Using equations (1.2)-(1.4) it is easy to find that in the
cross-sections dividing in half angles between the general planes
and passing, respectfully, through the general axes 1, 2, 3, the
1
absolute values of tangential stresses are equal to  ,
2 2 3
1 1
 ,   .
2 3 1 2 1 2

Figure 1.2. Principal tangential stresses

The tangential stresses in these cross-sections reach extremal
values and are called the principal tangential stresses:

    
  2 3,   3 1,   1 2 . (1.6)
1 2 2 2 3 2

The value of actual tangential stress  changes with changes in
n
the area orientation. The biggest value  at a certain point is called
n
the maximal tangential stress  . If condition (1.5) is satisfied, then
max
from (1.6) we find


   1 3.
max 2 2
8
1. Some information from continuum mechanics
The normal stress  at the area does not depend on the choice
n
of the coordinate system and changes only when the area rotates.
The principal invariants of the stress state as the simplest
symmetric functions of arguments in the principal stresses are written
as:
 ˆ   3,
1 1 2 3

 ˆ  , (1.7)
2 1 2 2 3 3 1
  ˆ,

3 1 2 3
Here
1 
    , (1.8)
3 x y z

is called as average (or average hydrostatic) pressure (it is also called
as average normal stress) at the point.
As a rule, materials have different mechanical resistance to the
shear and to the uniform integrated compression, it is convenient to
write the stress tensor as (1.9)-(1.11):

ˆ ˆ ˆ , (1.9)
1 2

 0 0
 
ˆ 0  0, (1.10)
1
 
0 0 
 

    
 x xy xz 
ˆ      , (1.11)
2 xy y yz
 
   
 xz yz z 

here ˆ is a spherical tensor (1.10) corresponding to the average
1
pressure at the point, ˆ is a tensor (1.11) which characterizes
2
9
Introduction to Deformable Solid Mechanics
tangential stresses at a given point and is called a stress deviator. The
main directions of the stress deviator ˆ and stress tensor ˆ coincide.
2
The deviator's invariants are easily obtained from Eq. (1.7) if
, ,  are replaced by  , , :
1 2 3 1 2 3

  ˆ 0,
1 2
 1 
 ˆ    2   2  2 , (1.12)
2 2 6 1 2 2 3 3 1

 ˆ    ,
 3 2 1 2 3

The nonnegative quantity (1.13)

  ˆ 
i 2 2
(1.13)
 1 2 2262 2 21/2
x y y z z x xy yz xz
6

is called intensity of tangential stresses.
Stress intensity is a quantity:

1  
   2  2  2 1/2. (1.14)
i 1 2 2 3 3 1
2

The stress intensity in Eq. (1.13)-(1.14) vanishes to zero only
when the stress state is the hydrostatic pressure.
For a pure shear

 ,  0,  ,
1 2 3

where is a shear stress. So,   .
i
In the case of extension (compression) in the x axis direction

 ;     0;
x 0 y z xy yz zx
10