Compatibility equations

The equations of compatibility have the same form as in Elasticity, namely 

In the above argument, it is assumed that the material is homogeneous in space and time. Under non-isothermal conditions in particular, this assumption breaks down. The resulting equations are more complicated than (1.9.26). They will not be discussed in general. A special case is considered in Sect. 6.1, using polar coordinates. 

Because the six components of the strain tensor are functions of three displacements i, they cannot all be independent, for then different portions of the material would share the same coordinate points or there would be voids (gaps), thus violating the continuity of the material. Strictly speaking, the three-dimensional problem is equivalent to determining the compatibility of six equations with three unknown variables (4). 

Let us first consider the two-dimensional case. According to Eq. (4.29), the components of the strain are given by 

Once the derivatives of the displacements are eliminated, the following expression is obtained  

It can be demonstrated that for simply connected domains these conditions are sufficient. 

BETWEEN THE STRESS AND STRAIN TENSORS IN IDEAL ELASTIC SYSTEMS 

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Using the stress-strain relation and the equilibrium conditions, we obtain the Beltrami-Michell form of the compatibility equation ... 

The St. Venant compatibility equations [13,14,15] follow immediately from the strain tensor definition of Equation (2), and are... 

This compatibility equation is exactly equivalent to (1.33) when we take into account the expression (1.35) for R(z) and its dependence on dn/ and on . We find that there is a complete correspondence between the quantum-mechanical and classical theories as to the equation determining the elementary excitations. 

Boundary conditions allow us to obtain specific results for each three-dimensional viscoelastic problem. If the stresses on the surface of the body are stated (first boundary problem), then the system of 15 basic equations is reduced to one of only six independent differential equations containing the six independent stress components. The strategy to follow implies the formulation of the compatibility equations in terms of the stress (Beltrami-Michell compatibility equations). 

The use of cylindrical coordinates is particularly suitable in the solution of axisymmetrical problems. It is worth noting that for a non-simply connected cross section, as occurs in the case of a hollow cylinder, the compatibility equations are not sufficient to guarantee single-valued displacements. In this situation, the displacements themselves must be considered. 

In fact, the solution for a plane stress problem can be determined from the solution of the corresponding plane strain problem and vice versa. Note that in contrast to the plane strain case, the remaining stresses in the plane stress are not required to be independent of z. In fact, the three-dimensionality of plane stress is closely linked to the fact that the conditions fulfilled by the stresses no longer lead to a single nontrivial compatibility equation. In other words, if the remaining stresses ctyy, and <5xy are functions of only X and y, the strain-displacement equations cannot in general be satisfied. 

Thus, the solution of two-dimensional elastostatic problems reduces to the integration of the equations of equilibrium together with the compatibility equation, and to satisfy the boundary conditions. The usual method of solution is to introduce a new function (commonly known as Airy s stress function), and is outlined in the next subsections. 

The compatibility equation may now be written in terms of Airy s stress function through the use of the stress-strain relationships as follows ... 

In addition to the mechanical balance of momentum, the mechanical constitutive equations and compatibility equation, some additional equations are used for coupled calculations, as follows. 

Another factor of anisotropic design analysis is greater dependence of stress distributions on materials properties. For isotropic materials, whether elastic, viscoelastic, etc., static values often result in stress fields which are independent of material stiffness properties. In part, this is due to the fact that Poisson s ratio is the only material parameter appearing in the compatibility equations for stress. This parameter does not vary widely between materials. However, the compatibility equations in stress for anisotropic materials depend on ratios of Young s moduli for different material axes, and this can introduce a strong dependence of stress on material stiffness. This approach can be used in component design, but the product and material design analysis become more closely related. 

The other four are obtained by permuting the subscripts 1, 2, 3. The compatibility equations are important in obtaining elastic solutions and will be discussed further in Chapter 4. 

As a result of the vanishing stresses and strains for plane stress and strain, the compatibility equations (Eqs. (2.31) and (2.32)) simplify to a single equation... 

F7 /8 y = Oij in the form of Euler-Lagrange equations (ojk is the stress tensor) along with the boundary conditions (see e.g. Refs. [47, 58, 59]). This system of differential equations should be solved along with the equations of mechanical equilibrium daij x) /9x, = 0 and compatibility equations equivalent to the mechanical displacement vector , continuity [100]. 

All the symbols in Table II have been explained in Section I, except ixi i = 1,..., k), which is the chemical potential of component Y,. In Table II electrical energy is separated from mechanical, heat, and chemical forms of energy, since its extensive variable q (charge) is not independent of mole numbers rii and its introduction as a variable of state would require the use of compatibility equations. To summarize, the set of variables of state (p, r, i,..., jt) or any other set obtained by replacing one of the variables by its conjugate variable can be used to express the internal energy of the system as a function of its state. Only two functions are relevant in thermodynamics ... 

The purpose of this particular network is to compute the other fluid properties at these points and all fluid properties at points 3 and 6. All that is done by the usual compatibility equations as indicated on the... 

It may be of some interest to give more details for the computation of the state at point 3, which must be calculated at last because it is based upon the known states 11, 4, 1, 5, and 6 and also implicitly the states 2 and 12.The unknown values are U3, p3, and a3. First, the compatibility equation between 3 and 8 is used. Second, the compatibility equation between 3 and 18 is used. As a third equation, the mass integral is used, under the next form... 

The compatibility equation referring to Cq between 5 and 3 gives the value of as. 

This case is also a usual one, and only one compatibility equation is used Cl between points 1 and 3. The three Hugoniot relations are joined to determine the four unknown values of U3, P3, a3, and X3. See Fig. 5. 

Then using Hugoniot equations and the two compatibility equations between 1 and 2 and between 1 and 3 the computation of the state at point 1 is easily accomplished, according to the next scheme.See Fig. 11. 

The governing equations of the structure consist of the equilibrium equations, the compatibility equations and the constitutive equations. The equilibrium equations including momentum effects are ... 

For p/pr 1, the linear relationship (5.81) between wo and p is unchanged because, in this regime of behavior, membrane and bending effects are not coupled to each other. For p/pr 1, the membrane analysis leading to (5.87) is unchanged by the presence of the residual stress except that the compatibility equation (5.85) must be replaced by... 

Another popular and useful approach for many practical engineering problems that can be reduced to two dimensional plane strain or plane stress approximations involves an auxiliary stress potential. In this approach, a bi-harmonic equation is developed based on the stresses (in terms of the potential) satisfying both the equilibrium equation and the compatibility equations. The result is that stresses derived from potentials satisfying the biharmonic equation automatically satisfy the necessary field equations and only the boundary conditions must be verified for any given problem. A rich set of problems may be solved in this manner and examples can be found in many classical texts on elasticity. In conjunction with the use of the stress potential, the principle of superposition is also often invoked to combine the solutions of several relatively simple problems to solve quite complex problems. 

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