A Research of Analytical Solution on Tubular Structure which is Subjected to an Axis Expansion in Different Contributions

: In this research paper, we proposed a classical mechanical study of the behavior of a tubular structure which is subjected to an expansion following the axis of the material. The first part, is focused on kinematic and dynamic aspects, Gradient tensor, left Cauchy-Green tensor ant isotropic invariants are used to proceed to find an analytical exact logarithmic solution function of the radius, of the problem at the limits through a system of partial differential equations in equilibrium condition. The simulations show that in the case of a sinusoidal, logarithmic and exponential function of time, the solution has a strong dependence to the radius. As a results, sinusoidal function of time has a big influence in the shape of the solution graphic which is negative or positive depending to the value of the time and radius. But logarithmic and exponential contribution has not a big influence in the arc shape of the solution graphic but a great influence in the values and it remains positive whatever the value of the radius.


Introduction
In recent years, researchers seem better equipped to discuss the different methods of finding analytical solutions in mathematics, physics or elasticity.This is the case, for example, for irregular problems such as large-scale bodies or areas, where analytical solutions are often found.However, these must be considered as approximate since they are valid only «far from certain edges» that is to say, the edges where are laid conditions to the overall limits of the solid body considered according to the Saint Venant' principle.Analytical solutions provide a better understanding of the essential characteristics of finite transformations.The usual reflexes of superposition of solutions and intuition resulting from linearity or nonlinearity must be abandoned in favor of a complete and rigorous approach of analysis of transformation and behavior.In addition, the choice of a model of behavior of a real material is not always easy.The behaviour of real materials is often complex.Even for structures as common as steel, many aspects of behaviour remain poorly understood and it is even difficult to develop a model representing the behaviour of a given material in all circumstances.In each mechanical or physical problem, it is necessary to choose the simplest model leading to satisfactory results for the intended use. the belief that no progress could be made in the development of theories of nonlinear models unless a completely explicit constitutive equation could be written.Such equations were generally chosen on the basis of an alleged simplicity of constitutive equations.One of the difficulties of this approach lies in the fact that simplicity is very subjective, depending considerably on the choice of variables according to which the relationship is expressed.The transition to the constitutive equation, expressed in phenomenological terms, is generally very difficult.It cannot be done without a related model and complex mathematical considerations.The most modern approach stems largely from the realization that it is possible to write fairly general constitutive equations from phenomenological or geometric considerations.This awareness is already involved in the theory of finite elasticity.Here, the constitutive equation is given by a statement that the strain energy function must depend on the strain gradient.

Suggested Citation
In this paper, we propose the study of the behavior of a structure in tubular form.It is subjected to an expansion following the axis of the material that we assume cylindrical and to an internal pressure.The first part of this study concerns the kinematic and dynamic aspects related to the behavior of the model.We will then proceed to an analytical solution, exact of the problem at the limits resulting from the setting of equations through a system of partial differential equations.We will then simulate the axial behavior and extension-swelling due to pressure.

Mathematical Considerations
A tube of circular section, of axis  �⃗  of inner radius   of outer radius   and of thickness = It is subjected to an  �⃗  axis expansion and an  internal pressure.
Under the action of these stresses, it is considered that the initial tube is transformed into a circular base cylinder, of axis  ⃗  inner radius  , outer radius   and thicknessℎ.
The kinematics of the transformation, kinematically permissible, is defined as: Where  is a function of .The  gradient of this transformation and the left Cauchy-Green tensor =   take the forms: To obtain a behavior relationship that describes the nonlinear hyperelastic mechanical behavior is to define a behavior relationship linking constraints and deformations.To do this, Spencer introduced the second symmetrical Lagrangian tensor of Piola-Kirchoff constraints From the kinematic data associated with the transformation, the Eulerian tensor deformation of Cauchy-Green dilations was characterized.
The   ,  = 1,2,3are the three elementary invariants of tensor  defined by: The stress state for an incompressible isotropic hyperelastic behaviour of energy  is written: where   is the identity matrix of order In incompressible state  = 1, so then we obtain: Considering the equalities (1), ( 2) and ( 3), the components of the Cauchy stress tensor (4), in a system of cylindrical coordinates, with the expression, and considering the nature of the kinematics defined in (1) and the components of the Cauchy tensor in the equations of motion are reduced to the system with the condition ( 1 −  2 ) ≠ 0, we find the two relations of the system give respectively: The second equation of the relation (6) gives a solution of the form: And the first relation of ( 6) gives the expression of (), which finally yields: The existential condition of the solution is given by:

Simulation and Interpretation
For the simulation of the solution, we will choose three kind function of () wich are respectively sinusoidal, logarithmic and exponential to see the bahavior of that solution.

Sinusoidal Contribution
Let choose now a sinusoidal contribution of the the expression () defined by: So then: In the case of a sinusoidal contribution depending to the time with the respect of the boundaries conditions that avoid the infinite path behavior when the time goes to infinity, we see a strong dependence of our solution to the radius.
The solution has here a plane behavior which more look like a tale more the radius increases.
The sinusoidal contribution of our solution has a big influence in the shape of the solution graphic which is negative or positive depending to the value of the contribution.

Logarithmic Contribution
So then: In the case of a logarithmic contribution depending to the time with the respect of the boundaries conditions that avoid the infinite path behavior when the time goes to infinity, we see that the path stays in curve shape with a strong dependence of our solution to the radius.The solution has a behavior which more look like an arc more the radius increases.Here we can say that the contribution of our solution has not a big influence in the arc shape of the solution graphic and it remains positive whatever the value of the radius.In the case of a exponential contribution depending to the time with the respect of the boundaries conditions that avoid the infinite path behavior when the time goes to infinity, we see that the path stays in curve shape with a strong dependence of our solution to the radius as in the logarithmic contribution.The solution has a behavior which more look like an arc more the radius increases.Here we can say that the contribution of our solution has not also big influence in the arc shape of the solution graphic and it remains also positive whatever the value of the radius.

Conclusion
In this paper, we proposed the study of the behavior of a tubular structure which is subjected to an expansion following the axis of the material in classical mechanic.The first part of this study concerns the kinematic and dynamic aspects related to the behavior of the model.Gradient tensor, left Cauchy-Green tensor ant isotropic invariants are calculated and then proceed to an analytical exact solution of the problem at the limits resulting from the setting of equations through a system of partial differential equations from the equilibrium condition.The simulations show that in the case of a sinusoidal contribution depending to the time, the solution has a strong dependence to the radius.The solution has here a plane behavior which more look like a tale more the radius increases.The sinusoidal contribution of our solution has a big influence in the shape of the solution graphic which is negative or positive depending to the value of the contribution.In the case of a logarithmic contribution depending to the time, the solution has a strong dependence to the radius.The solution has a behavior which more look like an arc more the radius increases.
Here we can say that the contribution of our solution has not a big influence in the arc shape of the solution graphic and it remains positive whatever the value of the radius.In the case of a exponential contribution depending to the time, the solution has a strong dependence of our solution to the radius as in the logarithmic contribution.The solution has a behavior which more look like an arc more the radius increases.
Here we can say that the contribution of our solution has not also big influence in the arc shape of the solution graphic and it remains also positive whatever the value of the radius.