# The The intersections of these three planes of

The design of thick walled cylinders for operation at very high
temperatures and  pressure is a complex
problem involving various considerations including definition of the operating
and permissible stress levels, criteria of failure, material behavior etc. Generally
problems involving cylinders have been widely used due to their practical
importance. In order to improve the safety and reliability of cylinder under various
thermo-mechanical loads it is really important for engineers to optimize the
design.

Over the last decade, finite element analysis (FEA) stopped being
regarded only as an analyst’s tool and entered the practical world of design
engineering. CAD software now comes with built-in FEA capabilities and design
engineers use FEA as an everyday design tool in support of the product design
process. However, until recently, most FEA applications undertaken by design
engineers were limited to linear analysis. Such linear analysis provides an
acceptable approximation of real-life characteristics for most problems design
engineers encounter. Nevertheless, occasionally more challenging problems
arise, problems that call for a nonlinear approach.  Non-linear behavior of solids takes two forms:
material non-linearity and geometric non-linearity. The simplest form of non-linear
material behavior is that of elasticity for which the stress is not linearly proportional
response of the material is different. Typical here is the case of classical
elastic– plastic behavior. The term “stiffness” defies the fundamental
difference between linear and nonlinear analysis. Stiffness is a property of a
part or assembly that characterizes its response to the applied load.

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The materials are classified based on the behavior for a particular loading
condition. These are Anisotropic materials, Monoclinic materials, Orthotropic
materials, Transversely isotropic materials and Isotropic materials. Orthotropic
materials are materials whose properties are directionally dependent .Here
Yong’s Modulus change with the direction along the object. There are three
mutually orthogonal planes of material property symmetry in an orthotropic material.
The intersections of these three planes of symmetry are called the principal
material directions. The material behavior is called as especially orthotropic,
when the normal stresses are applied in the principal material directions.
Otherwise, it is called as general orthotropic which behaves almost equivalent
to anisotropic material. Orthotropic materials are a subset of anisotropic
materials, because their properties change when measured from different
directions. Examples of orthotropic materials are wood, many
crystals, and rolled metals. Because of good heat resistance orthotropic bonded
material may work at super high temperatures or under high temperature
difference field. Real materials are not perfectly isotropic. In case of
isotropic material mechanical properties are constant in all direction.
Directionally dependent physical properties of orthotropic bonded materials are
significant due to the affects it has how materials behaves. For example in the
case of fracture mechanics, the way the microstructure of the material oriented
will affect the strength and stiffness of the material in various direction of
crack growth.

Thick walled cylinders are broadly used in chemical, petroleum, military
industries as well as in nuclear power plants .They are usually subjected to
high pressure & temperatures which may be constant or cycling. Industrial
problems often witness ductile fracture of materials due to some discontinuity
in geometry or material characteristics. The conventional elastic analysis of
thick walled cylinders to final radial & hoop stresses is applicable for
the internal pressure up to yield strength of material. General application of
Thick- Walled cylinders include, high pressure reactor vessels used in
metallurgical operations, process plants, air compressor units, pneumatic
reservoirs, hydraulic tanks, storage for gases like butane LPG etc. 1

1.1  Isotropic
Material:

In an isotropic material, properties are the same in all directions
(axial, lateral, and in between). Thus, the material contains an infinite
number of planes of material property symmetry passing through a point. i.e.,
material properties are directionally independent. So, there are two independent
elastic constants. Isotropic materials are those which have the same material properties
in all directions, and normal loads create only normal strains. By comparison, anisotropic
materials have different material properties in all directions at a point in the
body. There are no material planes of symmetry, and normal loads create both
normal strains and shear strains. A material is isotropic if the properties are
independent of direction within the material. For example, consider the element
of a material. If the material is loaded along its 0°, 45°, and 90° directions
and for isotropic material, the modulus of elasticity (E) is the same in
each direction (E0° = E45° = E90°). However, if the
material is anisotropic, it has properties that vary with direction within the
material. In this example, the moduli are different in each direction (E0°
? E45° ? E90°). 2

1.2
Transversely isotropic
Material:

If
a material has axes of symmetry in its longitudinal axis and all directions
perpendicular to its longitudinal axis (i.e., more than three mutually
perpendicular axes of symmetry) then such material is transversely isotropic. (e.g., unidirectional
composites). There are five independent elastic constants for these materials.

1.3
Monoclinic Material:

It
has a single plane of material property symmetry. If xy plane (i.e.; 1-2 plane)
is considered as the plane of material symmetry then, there are 13 independent
elastic constants in the stiffness matrix as given below. As there is a single
plane of material property symmetry, shear stresses from the planes in which
one of the axis is the perpendicular axis of the plane of material symmetry
(i.e.; 2-3 and 3-1 planes) will contribute only to the shear strains in those
planes. And normal stresses will not contribute any shear strains in these planes.

1.4  Orthotropic
Material:

There
are three mutually orthogonal planes of material property symmetry in an
orthotropic material. Fiber-reinforced composites, in general, contain the
three orthogonal planes of material property symmetry and are classified as
orthotropic materials. The intersections of these three planes of symmetry are called
the principal material directions. The material behavior is called as specially orthotropic, when the
normal stresses are applied in the principal material directions. Otherwise, it
is called as general orthotropic which behaves almost equivalent to anisotropic
material. There
are nine independent elastic constants in the stiffness matrix for a specially
orthotropic material. From the stress-strain relationship it is clear that normal
stresses applied in one of the principal material directions on an orthotropic
material cause elongation in the direction of the applied stresses and
contractions in the other two transverse directions. However, normal stresses
applied in any directions other than the principal material directions create
both extensional and shear deformations

In this project the effect of internal
and external pressure and temperature on thick walled cylinder, how radial
stress & hoop Stress will vary with change of radius will be investigated.