Chapter 4
Stress–Strain Relations
4.1 Introduction
In the previous chapters, the state of stress at a point was defined in terms of six
components of stress, and in addition, three equilibrium equations were developed
to relate the internal stresses and the applied forces. These relationships were independent of the deformations (strains) and the material behaviour. Hence, these equations
are applicable to all types of materials.
Also, the state of strain at a point was defined in terms of six components of strain.
These six strain–displacement relations and compatibility equations were derived in
order to relate uniquely the strains and the displacements at a point. These equations
were also independent of the stresses and the material behaviour and hence are
applicable to all materials.
Irrespective of the independent nature of the equilibrium equations and strain–
displacement relations, in practice, it is essential to study the general behaviour of
materials under applied loads including these relations. This becomes necessary due
to the application of a load, stresses, deformations and hence strains will develop in
a body. Therefore in a general three-dimensional system, there will be 15 unknowns,
namely 3 displacements, 6 strains and 6 stresses. In order to determine these 15
unknowns, we have only 9 equations such as 3 equilibrium equations and 6 strain–
displacement equations. It is important to note that the compatibility conditions as
such cannot be used to determine either the displacements or strains. Hence, the
additional six equations have to be based on the relationships between six stresses
and six strains. These equations are known as “Constitutive equations” because they
describe the macroscopic behaviour of a material based on its internal constitution.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
T. G. Sitharam and L. Govindaraju, Theory of Elasticity,
https://doi.org/10.1007/978-981-33-4650-5_4
99
Stress–Strain Relations
4.1 Introduction
In the previous chapters, the state of stress at a point was defined in terms of six
components of stress, and in addition, three equilibrium equations were developed
to relate the internal stresses and the applied forces. These relationships were independent of the deformations (strains) and the material behaviour. Hence, these equations
are applicable to all types of materials.
Also, the state of strain at a point was defined in terms of six components of strain.
These six strain–displacement relations and compatibility equations were derived in
order to relate uniquely the strains and the displacements at a point. These equations
were also independent of the stresses and the material behaviour and hence are
applicable to all materials.
Irrespective of the independent nature of the equilibrium equations and strain–
displacement relations, in practice, it is essential to study the general behaviour of
materials under applied loads including these relations. This becomes necessary due
to the application of a load, stresses, deformations and hence strains will develop in
a body. Therefore in a general three-dimensional system, there will be 15 unknowns,
namely 3 displacements, 6 strains and 6 stresses. In order to determine these 15
unknowns, we have only 9 equations such as 3 equilibrium equations and 6 strain–
displacement equations. It is important to note that the compatibility conditions as
such cannot be used to determine either the displacements or strains. Hence, the
additional six equations have to be based on the relationships between six stresses
and six strains. These equations are known as “Constitutive equations” because they
describe the macroscopic behaviour of a material based on its internal constitution.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
T. G. Sitharam and L. Govindaraju, Theory of Elasticity,
https://doi.org/10.1007/978-981-33-4650-5_4
99