How to most advantageously combine these attributes in terms of the goals set constitutive equations generally describe the functional relationship between. Reasonable selection and design of process conditions for hot extrusion are Constitutive equations describing the relationship of flow stress, temperature. As illustrated by these examples, a constitutive equation describes the relationship between an effect and a material parameter and cause. If the constitutive.
The viscoelastic constitutive equations Eq. For detailed information about this Kelvin-Voigt model and the material parameters one should see the works of Freitas Rachid and Stuckenbruck and Gally et al. Elasto-viscoplasticy The particular set of elasto-viscoplastic constitutive equations shown ahead is used to describe the mechanical behavior of metallic materials submitted to non-monotonic loadings.
In this case, the set of state variables is defined as: The variable ea is usually called the plastic strain and the variable p is known as the cumulated plastic strain. The inelastic part of the free energy is chosen such that and, from 2the state laws associated to p and c are: For this class of materials, the potential F is supposed to have the following form: In the above equation, F stands for the yield function which is given by: From 3the evolution laws for this elasto-viscoplastic behavior are: From equations 15 one can note that the yield function F is strictly responsible for the evolution of the inelastic behavior.
Equations 13 and 15 describe adequately the mechanical phenomena of elasticity, plasticity, creep and relaxation observed in many metallic materials at room and high temperatures. For further details about this kind of elasto-viscoplastic constitutive equations, such as experimental procedures to identify the material parameters a, b, d, sy, k, n and j, one should report to the works of Lemaitre and Chaboche Numerical Examples To illustrate the versatility of the theory, numerical results concerned with the transient response of a liquid-filled piping are presented by considering the two different types of mechanical behaviors for the pipewall presented before.
Firstly, to check the effectiveness of the theory, the result of a simulation is compared to some existing data. The pipe is Hydraulic transients were generated in a simple reservoir-pipe-valve installation by rapid valve closure 12 ms for an initial flow velocity of 0. The pipe material coefficients at The pressure history close to the valve is depicted in Fig.
Frictional losses in the fluid were not taken into account. The comparison indicates that the model may provide reliable results since both curves are in good agreement.
In the next examples, the schematic piping system shown in Figure 3 is considered. It represents a typical reservoir-pipe-valve installation where fast transients are generated by valve slam. The upstream end of the piping location D is attached to a constant pressure reservoir, and a valve rigidly fixed to the ground is positioned at the downstream end location A.
Rigid supports are installed at the elbows, located at B and C, to prevent their motions. However, these supports can be removed and the elbows move freely in the plane of the structure.
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Thus, different constraint conditions can be considered by unrestraining the motion of the elbow located either at B or at C. The piping is always anchored at the reservoir and at the valve and the pipe lengths L1, L2, and L3 are equal to 7, 3 and 5 m, respectively.
Boundary conditions for this piping arrangement are imposed at the reservoir, elbows and valve. For a constant pressure reservoir with a pipe rigidly connected to it, the following conditions hold: Different boundary conditions apply to the elbow according to its constraints.
For a moving massless elbow, the relationship between the liquid flow and the elbow motion is derived from the continuity and momentum equations applied over a translating control volume. The appropriate relations for this boundary condition, neglecting the bending stiffness of the pipes, are Otwell et al.
Here, the subscripts 1 and 2 refer to the conditions at the pipe's cross section at the entrance and exit of the elbow, respectively. For instance, if the elbow B is considered see Fig. However, if the elbow is rigid, the last two equations in 18 are replaced by. As initial conditions for the examples run in this work, we consider the steady state solution of the system of equations 4.
In addition, it is also assumed that the piping has never been subjected to inelastic deformations, so that inelastic strains are set equal to zero before the transient regime takes place. In the sequel, the dynamic response of the piping system shown in Fig.
These examples show the effects of pipe motion and pipewall mechanical behavior on the system response. One piping is made of low density polyethylene which presents a viscoelastic behavior characterized by the same pipe material coefficients presented earlier in this section.
The initial flow velocity is 0. The pipe has an internal diameter of The dynamic response of both pipings to valve slam is considered for different constraint conditions imposed at the bends.
Firstly, the simulation is carried out by considering both elbows immobile. The results obtained for this situation are depicted in Fig. Figure 4a shows the pressure history at the valve for the viscoelastic piping and Fig. By comparing the pressure curves in Fig. It is observed that the viscoelastic behavior introduces much more mechanical damping than the elasto-viscoplastic behavior does. Besides attenuation, the viscoelastic behavior introduces dispersive effects which, on the other hand, are not observed in the viscoplastic behavior at least to that extent.
In a second simulation, the flexibility of the structure is increased by removing the supports of the elbow either located at B or at C, so that in each case one of the elbows is free to move in the x-y plane. The system responses are shown in Fig.
An examination of Figs. Different from the elasto-viscoplastic behavior, where high frequencies disturbances generated by the fsi mechanisms remain visible throughout the response, in the viscoelastic behavior such disturbances are virtually eliminated after the first fluid cycle. Finally, elbow displacements in the x and y direction, driven by axial stress waves in the pipe wall and pressure waves in the liquid, are displayed in Figs.
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The results show that the piping may be subjected to significant displacement amplitudes if no constraints are imposed on critical points of the system, such as bends and variations of pipe diameters. Concluding Remarks It has been presented in this work an extension of a fluid-structure interaction analysis in order to handle different types of inelastic behaviors of the pipewall. Besides its capability of treating several kinds of inelastic behaviors within a same abstract framework, the model developed herein allows the use of a very simple numerical technique based on the method of characteristics for solving the resulting problems.
This reasonably simple theory makes possible the analysis of quite different piping systems, such as those employed in water supply transmission lines and in thermohydraulic installations. Examples of some complex effects caused by the mechanical couplings described by the proposed theory were presented and analyzed for viscoelastic and elasto-viscoplastic pipes.
It is shown that the system response is strongly dependent on pipewall mechanical behavior. For fast transients generated by valve slam, the elasto-viscoplastic behavior of a typical stainless steel pipe acts by limiting the pressure oscillations without presenting dispersion. For polyethylene tubes, however, the viscoelastic response introduces not only considerable attenuation but also dispersive effects on the system response.
Piezoelectric coefficients with double subscripts link electrical and mechanical quantities. The first subscript gives the direction of the electric field associated with the voltage applied, or the dielectric charge produced.
The second subscript gives the direction of the mechanical stress or strain. Conversely, these coefficients which are also called piezoelectric charge constants may be viewed as relating the charge collected on the electrodes to the applied mechanical stress. The superscripts are T, E, D, and S, signifying: Here are three examples of parameters used in the piezoelectric equations together with an explanation of their notation: Basic piezoelectric equations There are different ways of writing the fundamental equations of the piezoelectric materials, depending on which variables are of interest.
The two most common forms are the superscript t stands for matrix-transpose: These matrix relationships are widely used for finite element modelling.
Modeling of the Fluid-Structure Interaction in Inelastic Piping Systems
For analytical approaches, in general only some of the relationships are useful so the problem can be further simplified. For example this relationship, extracted from line 3 of the first matrix equation, describes strain in direction 3 as a function of stress and field. Just like any other elastic material, strain is proportional to the applied stress.
But in addition, a piezoelectric term is present, so an "electric strain" can be superimposed to the "elastic strain". Limitations of the linear constitutive equations There are a number of limitations of the linear constitutive equations.