dissertation global and local fracture properties of metal matrix ...

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Section 3 one made of the matrix material which has an appropriate misfit strain, such that the stress field is the same as for the actual inclusion. In Figure 3.7, a region (the inclusion) is cut from the unstressed elastically homogeneous material, and is then imagined to undergo a shape change (the transformation strain, εεεε T ) free from the constraining material. The inclusion cannot now be replaced directly back into the hole from it came. Instead surface tractions are first applied in order to return it to its original shape. Once back in position, the two regions are then welded together, i.e. there is no movement or sliding along the interface, and the surface tractions are then removed. Equilibrium is then reached between the matrix and the inclusion at a constrained strain, ε c , of the inclusion relative to its initial shape before removal. Since the inclusion is strained uniformly throughout, the stress within it can be calculated using Hooke’s Law in terms of the elastic strain (ε c - ε T ) and the stiffness tensor of the material, Cm. σσσσI = Cm(εεεε c - εεεε T ) (3.4) Eshelby found that ε c can be obtained from ε T by means of a tensor termed the Eshelby tensor “S”, which can be calculated in terms of the shape (aspect ratio) of the inclusion and of the Poisson’s ratio of the matrix, see e.g. [2, 60] εεεε c = Sεεεε T (3.5) The tensor S thus expresses the relationship between the final constrained inclusion shape and the shape of the inclusion after its cutting out and deformation. For the case of the spherical inclusions (aspect ratio is equal to 1), the only non-zero components of the Eshelby tensor are [61] m 7 − 5ν S( 1, 1) = S( 2, 2) = S( 3, 3) = m 15( 1−ν ) m 5ν −1 S( 1, 2) = S( 2, 1) = S( 1, 3) = S( 3, 1) = S( 2, 3) = S( 3, 2) = m 15( 1−ν ) m 2 ⋅ ( 4 − 5ν ) S( 4, 4) = S( 5, 5) = S( 6, 6) = m 15( 1−ν ) 23
Section 3 The Eshelby tensor does not depend on the size of the inclusion. Therefore, micromechanical methods based on the Eshelby tensor do not have an intrinsic length scale, i.e. the results do not depend on the size of inclusions. 3.3.2.2. Some general mean-field relations It should be noted that the Eshelby theory is developed for the case when both constituents the matrix and the reinforcements are elastic. However, the MMCs usually have elastic reinforcements and elastic-plastic matrix. A mean field approach based on the Eshelby theory is used to estimate stresses in the particles and in the matrix for such materials. Mean field approaches operate on the basis of averaged stress and strain fields in the constituent phases. Some general relations apply for linking the homogeneous meso-fields to the micro-fields [62] by employing so called ‘concentration tensors’ as 24 (3.6a) (3.6b) Here, B p and B m denote the stress concentration tensors, and σ p and σ m are the (averaged) micro-stress tensors for the particle and matrix phases, respectively; σ is the meso-stress tensor (Fig. 3.6). Equivalent relations hold for the strains. Note that Eqs. (3.6a,b) are tensorial relations taking into account the full 3D stress state. In our case, the meso-stress tensor σ is determined by the HRR-theory (Eq. 3.1). Following Benveniste’s [63] interpretation of the Mori-Tanaka approach, the phase concentration tensors are evaluated as ξ is the particle volume fraction, I the unit tensor, and concentration tensor, which can be written according to [62] as B p dil σ σ p = m = B B p m σ σ [ ] 1 p − ( 1 − ξ ) I + p p B B dil ξ = B dil [ ] 1 p − ( 1− ξ ) I + m B ξ = B dil [ ] 1 -1 -1 − m p m I + Es ( I − S)( E − s ) = E , (3.7a) . (3.7b) p B dil the dilute particle stress , (3.8)
Page 1 and 2: MONTANUNIVERSITÄT MONTANUNIVERSIT
Page 3 and 4: TABLE OF CONTENTS 1. Introduction 5
Page 5 and 6: 7.4.4. The effect of ECAP on the lo
Page 7 and 8: Section 1 The main results of this
Page 9 and 10: Section 2 2.2. The influence of the
Page 11 and 12: Section 2 high strength, fine reinf
Page 13 and 14: Section 2 etched to mark the inclus
Page 15 and 16: Section 2 (4.) The specimen prepara
Page 17 and 18: Section 3 Fig. 3.1. The digital ele
Page 19 and 20: height [µm] height [µm] 10 0 -10
Page 21 and 22: Section 3 Fig. 3.5. The observation
Page 23: Section 3 3.3.2. The determination
Page 27 and 28: equivalent stress Section 3 Fig. 3.
Page 29 and 30: Assume σ m eq Calculate matrix sec
Page 31 and 32: Section 4 Table 4.1. Chemical compo
Page 33 and 34: 4.1.1.3. Fracture mechanics tests S
Page 35 and 36: Section 4 Fig. 4.4. A view of a com
Page 37 and 38: Section 4 Table 4.3. Mechanical pro
Page 39 and 40: εfr [%] εfr [%] N 16 12 8 4 0 150
Page 41 and 42: Section 4 4.2. The effect of the he
Page 43 and 44: Section 4 It should be noted that n
Page 45 and 46: CODi [µm] 40 30 20 10 0 Section 4
Page 47 and 48: Section 4 Table 4.4. Results of the
Page 49 and 50: CODi [µm] COD i [µm] CODi [µm] 2
Page 51 and 52: Cast MMCs 10%-RT 10%-8h 10%-24h 10%
Page 53 and 54: COD vi [µm] COD vi [µm] CODvi [µ
Page 55 and 56: CODvi [µm] CODvi [µm] 80 60 40 20
Page 57 and 58: Section 4 Table 4.6. The values of
Page 59 and 60: σ p max [MPa] σ p max [MPa] σ p
Page 61 and 62: σ p max [MPa] σ p max [MPa] 1200
Page 63 and 64: σ interf max [MPa] σ interf max [
Page 65 and 66: Section 4 4.5. The relation between
Page 67 and 68: Section 4 Table 4.9. Conditions for
Page 69 and 70: Ln(Ln(1/(1-F σ))) Ln(Ln(1/(1-F σ)
Page 71 and 72: Section 5 diameter of 2…8 µm and
Page 73 and 74: MnS-inclusion r θ MnS-inclusion Se
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σ interf [MPa] σ interf [MPa] σ
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Section 6 where n is the number of
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Ji [kN/m] J i [kN/m] 25 20 15 10 5
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Section 7 7.2. A model by Tan and Z
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Section 7 its initial length, li. O
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1 µm Section 7 5 µm Fig. 7.3. A s
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Section 7 a) b) c) Fig. 7.4. Micros
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Section 7 It is important that no p
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Section 7 between the Rtot and the
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Section 7 Fig. 7.8. The fracture su
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10 z [mm] 5 -10 -15 Section 7 0 -20
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Section 7 Table 7.3. Data on measur
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8. Summary Section 8 This work deal
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Section 8 The effect of the homogen
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Table 9.1. Data on stress for Speci
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Table 9.11. Data on stress for Spec
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Table 9.17. Results of the stereoph
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1 2 1 2 3 3 4 5 6 5 9 119 7 7 8 9 6
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8 9 10 6 10 122 11 12 13 14 11 12 1
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9 10 11 12 13 124 50 µm Fig. 9.6.
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1 2 1 2 3 3 4 4 126 5 5 6 9 7 50 µ
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1 1 2 2 3 4 3 4 5 5 128 6 7 6 7 50
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Fig. 9.12. Fracture surface with ma
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1 2 3 1 2 3 4 5 6 7 8 9 10 4 5 132
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Fig. 12.36. Fracture surface of the
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[14] Lewandowski JJ, Liu C, Hunt WH
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[42] Rice JR, Rosengren GF. Plane s
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[70] ASTM E1820-01, “Standard Tes
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