1.3 Assumptions of Linear Elasticity
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neglect some of the influential factors of minor importance. The following are the
assumptions in classical elasticity.
The Body is Continuous
Here the whole volume of the body is considered to be filled with continuous matter,
without any void. Only under this assumption, can the physical quantities in the
body, such as stresses, strains and displacements, be continuously distributed and
thereby expressed by continuous functions of coordinates in space. However, these
assumptions will not lead to significant errors so long as the dimensions of the body
are very large in comparison with those of the particles and with the distances between
neighbouring particles.
The Body is Perfectly Elastic
The body is considered to be wholly obeys Hooke’s law of elasticity, which shows
the linear relations between the stress components and strain components. Under this
assumption, the elastic constants will be independent of the magnitudes of the stress
and strain components.
The Body is Homogenous
In this case, the elastic properties are the same throughout the body. Thus, the elastic
constants will be independent of the location in the body. Under this assumption, one
can analyse an elementary volume isolated from the body and then apply the results
of analysis to the entire body.
The Body is Isotropic
Here the elastic properties in a body are the same in all directions. Hence, the elastic
constants will be independent of the orientation of coordinate axes.
The Displacements and Strains are Small
The displacement components of all points of the body during deformation are very
small in comparison with its original dimensions, and the strain components and the
rotations of all line elements are much smaller than unity. Hence, when formulating
the equilibrium equations relevant to the deformed state, the lengths and angles
of the body before deformation are used. In addition, when geometrical equations
involving strains and displacements are formulated, the squares and products of
the small quantities are neglected. Therefore, these two measures are necessary to
linearize the algebraic and differential equations in elasticity for their easier solution.
1.4 Applications of Linear Elasticity
The very purpose of application of elasticity is to analyse the stresses and displacements of elements within the elastic range and thereby to check the sufficiency of
their strength, stiffness and stability.
3
neglect some of the influential factors of minor importance. The following are the
assumptions in classical elasticity.
The Body is Continuous
Here the whole volume of the body is considered to be filled with continuous matter,
without any void. Only under this assumption, can the physical quantities in the
body, such as stresses, strains and displacements, be continuously distributed and
thereby expressed by continuous functions of coordinates in space. However, these
assumptions will not lead to significant errors so long as the dimensions of the body
are very large in comparison with those of the particles and with the distances between
neighbouring particles.
The Body is Perfectly Elastic
The body is considered to be wholly obeys Hooke’s law of elasticity, which shows
the linear relations between the stress components and strain components. Under this
assumption, the elastic constants will be independent of the magnitudes of the stress
and strain components.
The Body is Homogenous
In this case, the elastic properties are the same throughout the body. Thus, the elastic
constants will be independent of the location in the body. Under this assumption, one
can analyse an elementary volume isolated from the body and then apply the results
of analysis to the entire body.
The Body is Isotropic
Here the elastic properties in a body are the same in all directions. Hence, the elastic
constants will be independent of the orientation of coordinate axes.
The Displacements and Strains are Small
The displacement components of all points of the body during deformation are very
small in comparison with its original dimensions, and the strain components and the
rotations of all line elements are much smaller than unity. Hence, when formulating
the equilibrium equations relevant to the deformed state, the lengths and angles
of the body before deformation are used. In addition, when geometrical equations
involving strains and displacements are formulated, the squares and products of
the small quantities are neglected. Therefore, these two measures are necessary to
linearize the algebraic and differential equations in elasticity for their easier solution.
1.4 Applications of Linear Elasticity
The very purpose of application of elasticity is to analyse the stresses and displacements of elements within the elastic range and thereby to check the sufficiency of
their strength, stiffness and stability.