Plane-strain

Of particular interest are situations where the strain tensor is essentially two-dimensional such situations are often encountered in cross-sections of solids which can be considered infinite in one dimension, by convention assigned the z axis or the x axis, and due to the symmetry of the problem only the strain on a plane perpendicular to this direction matters. This is referred to as plane strain . We discuss the form that the stress-strain relations take in the case of plane strain for an isotropic solid, using the notation X3 for the special axis and X, X2 for the plane perpendicular to it. 

by assumption we will have 33 = 0, that is, there is no strain in the special direction. Moreover, all physical quantities have no variation with respect to X3. Thus, all derivatives with respect to X3, as well as the stress components 7,3, / = 1,2, are set to zero. With these conditions, the general equilibrium equations, Eq. (E.18), with no external forces, reduce to 

From the general stress-strain relations of an isotropic solid, Eq. (E.31), and the fact that 

Using this last result in the expressions for cn and 622 of Eq. (E.31), taking the second derivatives of these expressions with X2 and Xi, respectively, and the cross-derivative of 12 with Xi and X2, and combining with Eqs. (E.45) and (E.46), we arrive at 

In order to solve the problem of plane strain, all that remains to do is find the form of the Airy function. Inserting the expression of Eq. (E.49) into Eq. (E.48) we obtain 

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Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

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. 

Figure C2.1.17. Stress-strain curve measured from plane-strain compression of bisphenol-A polycarbonate at 25 ° C. The sample was loaded to a maximum strain and then rapidly unloaded. After unloading, most of the defonnation remains.

ASTM E399-90, "Plane Strain Fracture Toughness of Metallic Materials," MnnualBook ofMSTM Standards, ASTM PubHcations, Philadelphia, 1993. 

ASTM D5045-91, "Plane Strain Fracture Toughness and Strain Energy Release Rate of Plastic Materials," A.nnualBook ofyiSTM Standards, ASTM Puhhcations, Philadelphia, 1993. 

Substantial work on the appHcation of fracture mechanics techniques to plastics has occurred siace the 1970s (215—222). This is based on earlier work on inorganic glasses, which showed that failure stress is proportional to the square root of the energy required to create the new surfaces as a crack grows and iaversely with the square root of the crack size (223). For the use of linear elastic fracture mechanics ia plastics, certaia assumptioas must be met (224) (/) the material is linearly elastic (2) the flaws within the material are sharp and (J) plane strain conditions apply ia the crack froat regioa. 

The stress—strain relationship is used in conjunction with the rules for determining the stress and strain components with respect to some angle 9 relative to the fiber direction to obtain the stress—strain relationship for a lamina loaded under plane strain conditions where the fibers are at an angle 9 to the loading axis. When the material axes and loading axes are not coincident, then coupling between shear and extension occurs and... 

W. K. Wilson, Plane Strain Craek Toughness Testing of High Strength Metallic Materials , ASTM STP 410, American Society forTesting and Materials, Philadelphia, 1966, pp. 75 76. 

Plane Strain Fracture Toughness of Metallic Materials, 1982 Annual Book of ASTM Standards, Part 10, Standard No. E399. 

This is an alternative form of equation (2.91) and expresses the fundamental material parameter Gc in terms the applied stress and crack size. From a knowledge of Gc it is therefore possible to specify the maximum permissible applied stress for a given crack size, or vice versa. It should be noted that, strictly speaking, equation (2.96) only applies for the situation of plane stress. For plane strain it may be shown that material toughness is related to the stress system by the following equation. 

Note that the symbol Gic is used for the plane strain condition and since this represents the least value of toughness in the material, it is this value which is usually quoted. Table 2.2 gives values for G c for a range of plastics. 

This is for plane stress and so for the plane strain situation... 

The in-plane strains and the moments and curvatures are all linked in this non-symmetrical lamina and are obtained from... 

In the above situation, if the stresses are such that = 0 then this condition is referred to as plane strain. This is because strains are experienced in only one plane even though there are stresses in all three co-ordinate directions. 

For orthotropic materials, imposing a state of plane stress results in implied out-of-plane strains of... 

Thus, the plate is in a state of generalized plane strain in the x-z plane. 

Note. is the plane strain fracture toughness (see Section 8.9). 

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