6. Strain Assoc.Prof.Dr. Ahmet Zafer Şenalp Mechanical Engineering Department Gebze Technical University ME 612 Metal Forming and Theory of Plasticity

Figure 6.1 Definition of the uniaxial strain (a) Tensile and (b) Compressive. Lo is the original length and ΔL the length change after the load application. Engineering strain: True or logarithmic strain: Dr. Ahmet Zafer Şenalp ME 612 2Mechanical Engineering Department, GTU 6. Strain 6.1. Uniaxial Strain (6.1)(6.2)

In addition to the above normal nominal strain, one can define the engineering shear strain as the change of angle as shown in Fig For small angle change, we can write: Figure 6.2. Shear strains are used to define change of angles upon application of forces. Dr. Ahmet Zafer Şenalp ME 612 3Mechanical Engineering Department, GTU 6. Strain 6.1. Uniaxial Strain (6.3)

Figure 6.3. Plane strain involving small distortions To simplify the presentation we only discuss the definition of the two-dimensional strain components but an extension to 3D will be apparent. In Figure 6.3 an infinitesimally small cube is given (before and after deformation) with edge lengths dx and dz. Here the deformation is only in xz plane and the deformation is a function of x and z. Dr. Ahmet Zafer Şenalp ME 612 4Mechanical Engineering Department, GTU 6. Strain 6.2. Two-Dimensional Engineering Strain

At point A small strain components are defined. With the assumption of small deformation; A’PA’C’ and tan(PA’C’) is then PA’C’. From Figure 6.3 With a similar analysis If 3D case is analyzed: Here u,v,w are diplacements in x,y,z directions. Dr. Ahmet Zafer Şenalp ME 612 5Mechanical Engineering Department, GTU 6. Strain 6.2. Two-Dimensional Engineering Strain (6.4) (6.5) (6.6) (6.7)

Shear strains is associated with angular distortions shown as angles RA’B’ and PA’C’. Again with small deformations, PA’C’ angle arctan As PA’C’ angle. A similar analysis for RA’B’ angle: RA’B’ angle. Dr. Ahmet Zafer Şenalp ME 612 6Mechanical Engineering Department, GTU 6. Strain 6.2. Two-Dimensional Engineering Strain (6.8) (6.9) (6.10) <<1; (6.11)

Total shear strain is the sum of these angles or For 3D case; It is important to realize that the g form of shear strains given in Eq. 6.12, 6.13 and 6.14 is equivalent to simple shear strain as measured in a torsion test. Dr. Ahmet Zafer Şenalp ME 612 7Mechanical Engineering Department, GTU 6. Strain 6.2. Two-Dimensional Engineering Strain (6.12) (6.13) (6.14)

Like stress tensor a similar form can be used for strains: The tensor shear strain is equal to half of shear strain given above; Dr. Ahmet Zafer Şenalp ME 612 8Mechanical Engineering Department, GTU 6. Strain 6.3. The Strain Tensor (6.15) = (6.16)

Figure 6.4. Illustration showing that pure shear, (a) and (b) is related to simple shear (c) by a rotation (d). Dr. Ahmet Zafer Şenalp ME 612 9Mechanical Engineering Department, GTU 6. Strain 6.3. The Strain Tensor

Average rotation of infinitesimally small cube is defined by w j. For j=x,y,z the equalities are given as; Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.3. The Strain Tensor (6.17) (6.19) (6.18)

Figure 6.5. Examples of strain states (a) Uniaxial tension for an isotropic material (b) equal hydrostatic tension in the three Cartesian axes and (c) shear. Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.3. The Strain Tensor

The 9 components of strain tensor are necessary to define the deformation status of the cube. Strain tensor is symmetric. Ex: e xy =e yx. Generally e x is used instead of e xx. Principal strain indices are shown by 1,2,3 and thus principal strains are e 1,e 2,e 3. Always; Term is correct. Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.3. The Strain Tensor (6.20)

Consider a unit cube (dimensions 1 × 1 × 1)) along the principal strain directions. Under loading, the cube will deform to another cube of dimensions (1 + e 1 ) × (1 + e 2 ) × (1 + e 3 ). dilation; is defined as the relative volume change. Note that if the deformation preserves volume (incompressible deformation), then; Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.4. Relative Volume Change in Terms of Strain Components (6.21) (6.22) (6.24) (6.23)

Figure 6.6. Deformation of a small element with sides originally parallel to x and y axes. u and ν are here the displacements of point O in the directions of the axes x and y, respectively. Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.5. Transformation of Strain Components in Plane Strain Conditions

Similarly to the transformation equations derived for the stress components, we can derive transformation equations for the strain components. Using the notation of Fig. 6.6, we define the strains as follows: Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.5. Transformation of Strain Components in Plane Strain Conditions (6.25) (6.26) (6.27)

The final strain transformation equations have the following form: The principal strain directions (where  xy = 0) are found from: Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.5. Transformation of Strain Components in Plane Strain Conditions (6.28) (6.29) (6.30) (6.31)

Similarly, the magnitudes of the principal strains are: The maximum shearing strains are found on planes 45 ◦ relative to the principal planes and are given by: Note that the above transformation equations are only valid for small strain. We will not need the transformation equations for the logarithmic strain as we will always try to work on principal strain axes!! Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.5. Transformation of Strain Components in Plane Strain Conditions (6.32) (6.33)

Figure 6.7. The Mohr circle for plane strain problems. Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.6. Mohr’s Circle for Small Strain

Because we have concluded that the transformation properties of stress and strain are identical, it is apparent that a Mohr’s circle for strain may be drawn and that the construction technique does not differ from that of Mohr’s circle for stress(Figure 6.7). In Mohr’s circle for strain, the normal strains are plotted on the horizontal axis, positive to the right. When the shear strain is positive, the point representing the x axis strains is plotted a distance γ/2 below the e line, and the y axis points a distance γ/2 above the e line, and vice versa when the shear strain is negative. Dr. Ahmet Zafer Şenalp ME Mechanical Engineering Department, GTU 6. Strain 6.6. Mohr’s Circle for Small Strain