PSI - Issue 39

D. Fedotova et al. / Procedia Structural Integrity 39 (2022) 419–431 Author name / Structural Integrity Procedia 00 (2019) 000–000

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As a result of a numerical study, the experimental data on crack growth for all used materials are represented in terms of the elastic, classical plastic SIFs and new plastic SIFs CMSGP. Elastic–plastic analysis was based on the classical Hutchinson-Rice-Rosengren theory of plasticity and the CMSG plasticity theory which were implemented in the commercial finite element package ANSYS (2012) by means of a user material subroutine USERMAT. Numerical calculations were performed using the experimental set of forces F (Table 2) for each tested specimen made of 34X and P2M steels, 7050 aluminum, and Ti-6Al-4V alloys. A wide range of values for the intrinsic material length parameter l were used in our computations. This value was varied between 1 and 10 μm, forming the basis for the parametric study. 4. Crack tip stress fields In this section, the behavior of the effective normalized stresses under crack extension is compared for the two theories of plasticity. The illustrate the effects of gradient plasticity in stress fields distributions at the crack tip, a comparison is presented with the classical plasticity model in the form of the Hutchinson-Rice-Rosengren (HRR) solution. The calculation results (Fig. 3) contain radial distributions of effective stresses on Mises in the plane of the crack location, normalized to the yield strength of the material  e /  y under plane strain, depending on the normalized distance r/l , for a given value of the intrinsic of the material length l = 5 µm. The strain gradient theory of plasticity is highlighted with solid lines, and the stipple lines correspond to the classical HRR model. Fig. 3a and 3b are related to pure mode I and pure mode II. Next Fig. 3c represent Point 2 that corresponds to an increment in the crack length of 1 mm after the crack kinked and Fig. 3d represents point 7, which corresponds to the state before the final fracture.

Table 3. Crack tip singularity.

l=1 µm

l=5 µm

l=10 µm Steel P2M

Steel P2M

Ti-6Al-4V

Steel P2M

Ti-6Al-4V

Ti-6Al-4V

Pure Mode I Point 1

-0.647

-0.649

-0.669

-0.656

-0.648

-0.649

Pure Mode II Point 1

-0.562

-0.539

-0.548

-0.516

-0.526

-0.500

Mixed Mode Point 2

-0.676

-0.664

-0.675

-0.666

-0.675

-0.664

Mixed Mode Point 3

-0.674

-0.639

-0.671

-0.625

-0.670

-0.646

Mixed Mode Point 4

-0.681

-0.619

-0.680

-0.642

-0.665

-0.631

Mixed Mode Point 5

-0.678

-0.614

-0.674

-0.635

-0.669

-0.621

Mixed Mode Point 6

-0.673

-0.61

-0.672

-0.608

-0.671

-0.614

Mixed Mode Point 7

-0.678

-0.611

-0.648

-0.610

-0.624

-0.613

From the results presented in Fig. 3, the effective stresses on the crack extension under gradient plasticity are an order of magnitude higher than under the classical HRR-theory of plasticity. As moving away from the crack tip, the effects of gradient plasticity disappear and the solution gradually moves to the state of a classical singularity of the HRR type. The angles of inclination of the stress distribution curves for the gradient plasticity of CMSG and for the classical HRR solution are noticeably different, which indicates a different type of singularity at the crack tip. The trends observed in Fig. 3 persist for different values of the intrinsic parameter l (1, 5 and 10 µm). The asymptotic crack tip singularity indexes obtained are presented in Table 3, for steel P2M and Ti-6Al-4V alloy with an intrinsic material length l = 5 µm and both mode I/II and mixed modes conditions within plane strain state.