TAREA PRÁCTICA DE REPASO DE LAS UF1 Y UF

As mentioned in previous section, the equivalent plastic strain is an important parameter in failure analysis of ductile materials. Therefore, it is one of the aims of this study to determine the equivalent plastic strain from the uniaxial micro-tensile test of the SLM Ti-6Al-4V micro-struts. The findings are to be used in the measurement of fracture and prediction of failure in the finite element analysis that will be discussed in Chapter 4.

A typical elastic-linear strain hardening curve is shown in Figure 2.46, where σY is

the yield stress, E is the Young’s modulus, Et is the tangential modulus after

yielding. The figure shows a bilinear curve, as a simplification from reality, which is a continuous curve, as shown in Figure 2.47. Derivations of equations for hardening and flow rule are as in the following discussions [Tan (2009); Lemaitre and Chaboche (1990)].

Figure 2.46: A typical stress (true) – strain (true) curve from a uniaxial test [Tan (2009)]

Figure 2.47: A continuous true stress-strain curve of A316 stainless steel with the identification of linear hardening in tension [Lemaitre and Chaboche (1990)]

The von Mises yield criterion can be expressed in another form;

, [2.19] Where is the equivalent yield stress which is defined as

3 . [2.20] Based on the relation between J2 and the deviatoric stress tensor, one has

. [2.21] The resistance to plastic flow of a deformed solid is known as strain hardening. The yield criterion will change with further development of plastic deformation. It is assumed that the yield criterion that can account for the hardening can be written as

, [2.22] where , called equivalent plastic strain, is a scalar measure of plastic strain tensor defined through time integration of the equivalent plastic strain rate

Where 0 at the time t = 0, and the equivalent plastic strain rate is defined as . [2.24] During plastic deformation, 0 , due to the incompressibility. Therefore, in a uniaxial tension, the plastic strain rate is

0 0

0 0

0 0

, which gives the equivalent plastic strain as

2 3 2 3 1 4 1 4 [2.25] The equivalent yield stress as a function of equivalent plastic strain, , can be obtained from experiment, such as uniaxial test or torsion test, based on the condition that the yield criterion must be satisfied at all times during plastic straining.

In uniaxial stress state, letting axis 1 along the tensile direction, the stress tensor is 0 0

0 0 0

0 0 0 , [2.26]

The deviatoric stress tensor is

0 0

0 0

0 0

, [2.27]

Therefore the equivalent stress is . As shown in Equation 2.25 that the equivalent strain during uniaxial tensile test . Thus, the uniaxial stress- strain can be employed to determine the hardening law for a general stress state.

As in previous Figure 2.46, the relation between the equivalent yield stress Y and the equivalent plastic strain can be derived. From the figure, during the plastic deformation

, [2.28] Where is the initial yield stress, and is the strain at the yield point.

Since , where is the elastic part of the strain and is the plastic part, one has

.

Since for uniaxial tension, the equivalent stress , and the equivalent plastic strain , the relation between the equivalent yield stress and the equivalent plastic strain as assumed in Equation 2.22 is

. [2.29]

Figure 2.48 shows a stress-strain curve of the as-received SLM Ti-6Al-4V micro- strut, with a projected tangent line (dashed line), for the extraction of tangential modulus after yielding Et . The tangent was drawn passing both the yield strength

value and the maximum point of the initial stress-strain curve. A curve which is similar to a typical stress-strain curve (Figure 2.46) is obtained, and shown in Figure 2.49. The initial modulus value E is 45 GPa while the yield stress value σY is 245

MPa. From Figure 2.49, the tangential modulus after yielding Et is determined as 6.7

GPa. By substitution of the required values into Equation 2.29, the relation between the equivalent yield stress and the equivalent plastic strain of the as-received SLM Ti-6Al-4V micro-strut is given by Equation 2.30.

Figure 2.48: Stress-strain curve of the as-received SLM Ti-6Al-4V micro-strut, with a projected tangent line [S(1-15)-35-200-1000-AR]

Figure 2.49: Stress-strain curve of the as-received SLM Ti-6Al-4V micro-strut, similar to the typical curve as in Figure 2.46

0 100 200 300 400 500 600 0 0.01 0.02 0.03 0.04 0.05 True   Stress   [MPa] True Strain

A similar procedure was applied to the stress-strain curve of the heat-treated SLM Ti-6Al-4V micro-strut, as shown in Figure 2.50. A projected tangent line (dashed line) was drawn passing both the yield strength value and the maximum point of the initial stress-strain curve. A similar curve to that of typical stress-strain curve (Figure 2.46) is obtained, and shown in Figure 2.51. The initial modulus value E for the heat- treated SLM Ti-6Al-4V strut is 65 GPa and the corresponding yield stress value σY is

340 MPa. The tangential modulus after yielding Et is determined as 20.1 GPa. After

the substitution of all required values into Equation 2.29, the relation between the equivalent yield stress and the equivalent plastic strain of the heat-treated SLM Ti- 6Al-4V micro-strut is given by Equation 2.31.

0.34 29.098 GPa [2.31]

Figure 2.50: Stress-strain curve of the heat-treated SLM Ti-6Al-4V micro-strut, with a projected tangent line [S(1-9)-35-200-1000-HT(B)]

0 200 400 600 800 1000 1200 1400 0 0.01 0.02 0.03 0.04 0.05 True   Stress   [MPa] True Strain

Figure 2.51: Stress-strain curve of the heat-treated SLM Ti-6Al-4V micro-strut, similar to the typical curve as in Figure 2.46