Plane Strain Problems

A plane strain state is defined as a state of strain where the components of the vector displacement take the form 

These equations imply that the strains y, and are zero. From the stress-strain relationships, Eq. (4.75), 

The equilibrium equations are reduced by two, since g z is a function of only X and y, as Eq. (4.75) immediately shows. Consequently, no body force exists in the z direction in a plane strain state. Problems of this type are two-dimensional and therefore are governed by only eight equations, which let us find eight unknowns Gyy, Uxy, y x Yyyi, Yxy and Uy. As will be 

In practice, the body must be cylindrical or prismatic with uniform cross section and fixed ends. An alternative condition to the last one is that the tractions on the lateral surface are normal to the axis of the system and are functions of only x and y. Obviously, these conditions are only approximately fulfilled in actual situations. 

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. 

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Let us assume that stress gradient in axial direction is present but smooth. Then we can use a perturbation method and expand the solution of equation (30) in a series. The first term of this expansion will be a solution of the plane strain problem and potential N will be equal to zero. The next terms of the stress components will contain potential N also. 

The first case considered is solute desorption during unconfined compression. We consider a two dimensional plane strain problem, see Fig. 1. A sinusoidal strain between 0 and 15 % is applied at 0.001 Hz, 0.01 Hz, 0.1 Hz and 1 Hz. To account for microscopic solute spreading due to fluid flow a dispersion parameter is introduced. Against the background of the release of newly synthesized matrix molecules the diffusion parameter is set to the value for chondroitin sulfate in dilute solution Dcs = 4 x 10 7 cm2 s-1 [4] The dispersion parameter Dd is varied in the range from 0 mm to 1 x 10 1 mm. The fluid volume fraction is set to v = 0.9, the bulk modulus k = 8.1 kPa, the shear modulus G = 8.9 kPa and the permeability K = lx 10-13m4 N-1 s-1 [14], The initial concentration is normalized to 1 and the evolution of the concentration is followed for a total time period of 4000 s. for the displacement and linear discontinuous. For displacement and fluid velocity a 9 noded quadrilateral is used, the pressure is taken linear discontinuous. 

Figure 8.2. Stress components in a plane-strain problem.

Consider a plane-strain problem in the x-z plane, as shown in Fig. 8.2. The stress tensor, expressed in Cartesian coordinates, takes the form... 

For a plane strain problem (see Chap. 16), = 0, and the only nonzero... 

Let us consider now the deformation and stresses of a cylindrical pipe under two different boundary conditions (Fig. 16.2). In both eases the length of the pipe is considered constant according to the requirements for a plane strain problem. The external and internal radii are R2 and R, respectively. If the applied forces and the displacements are also uniform, the deformation is purely radial, and in cylindrical coordinates = u r). According to the Navier equations, rot u = 0. Hence, Vdiv u = 0, which implies... 

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. 

Similar parameters may be defined for plane strain problems. The previous results have shown that the solution of crack problems in FGMs is not very sensitive to the Poisson s ratio. Thus, v is assumed to be constant throughout the medium. In this study it is further assumed that in the graded materials the variations in En, E22 and G12 are proportional. Referring to... 

Without showing the algebra here, we note that whereas eqn. (12.6a) suffices for the cylindrically symmetrical problem, for the plane-strain problem we use three second derivatives from the full set of five ... 

The nature of a BMT study is well demonstrated with BMT3 of the DECOVALEX I project. It was a problem associated with a near-field repository model, set up as a two-dimensional plane-strain problem in which a tunnel with a deposition hole was located in a fractured rock mass. The model is 50 X 50 m in size, and situated at 500 m below the ground level (Figure 1). The fracture network is a two-dimensional realization of 6,580 fractures from a realistic three-dimensional fracture network model of the Stripa Mine, Sweden (Figure 2). The problem is set up as a fully coupled THM near-field repository problem, with thermal effects caused by heat release from radioactive waste in the deposition hole (the heater). Heat output decreases... 

In plane strain problems, the displacements that exist in a particular direction are assumed to be zero. If this direction is Xy it follows from the definition of strain (Eq. (2.14)) that e,3=e23=e33=0, i.e., the strains are two-dimensional. As an example, consider the problem shown in Fig. 4.13 a knife edge indenting a thick block of material. Most of the displacements are occurring in the x and directions, i.e., the material is being pushed downwards or sideways. The only exceptions are in the vicinity of the front and back surfaces, where displacements in the Xj direction are possible. Overall, the components of the displacement vector at any point can be assumed to be independent of Xj. From Hooke s Law, the assumption that 3=e23 33 plies that <7 3=cr23=0. As with plane stress, only the stress components and are needed to define the... 

In Chapter 2, stress and strain were defined, the compatibility and equilibrium equations were introduced and the relationship between stress and strain was defined. Thus, any solution that satisfies all these equations and the appropriate boundary conditions will be the solution that gives the stress and strain distribution for a particular loading geometry. For the most general problems, the scientific process can be difficult but for plane stress and plane strain problems in elastically isotropic bodies the solution involves a single differential equation. 

Armero, F. 2000. On the locking and stability of finite elements in finite deformation plane strain problems , Computers Structures 75 261-290. 

The increment of volumetric strain for a two-dimensional plane strain problem is... 

It can be rigorously shown using Eqs. (A.22) and (A.23) that for both plane stress and plane strain problems, the following relations hold in the absence of body forces ... 

The thick wall cylinder shown in Fig. 9.1 is a good example of a problem which is solved using the stress function approach to the solution of two dimensional plane stress or plane strain problems of engineering importance. In this approach, stress fields are derived from a set of potentials, ,... 

In bonded materials that are assumed to be piecewise homogeneous, from the viewpoint of mechanical failure,a fundamental problem is that of a crack lying parallel to an interface. This, of course, includes the case of interfacial zones modelled as a homogeneous elastic continuum. Even though for a realistic analysis the particular component geometry and loading conditions may be very important, the characteristic features of the problem may be captured by considering the plane strain problem for two bonded half spaces. 

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