free fixed

free

1.0

r/b

z rz

f τ b

1.2

-0.3

-0.03

0.4 -0.5

0.6

0.01 0.8

0.1 -1.2 0.0

0.3 0.2

-0.1 -0.2 -0.7

-0.01

0.0 0.2 0.4 0.6 0.8 1.0

free

r/b

z z

θb τ

(a)

(b)

Fig. 13. (a) The full-ﬁeld distribution of shear stress srzfor a composite sharp wedge with an interface crack subjected to a concentrated force located atð0:5b; 150°Þ. (b) The full-ﬁeld distribution of shear stress shzfor a composite sharp wedge with an interface crack subjected to a concentrated force located atð0:5b; 150°Þ.

Fig. 14. (a) The full-ﬁeld distribution of shear stress srzfor a composite sharp wedge with an interface crack subjected to a concentrated force located atð0:5b; 150°Þ. (b) The full-ﬁeld distribution of shear stress shz for a composite sharp wedge with an interface crack subjected to a concentrated force located atð0:5b; 150°Þ.

Appendix A. The full-ﬁeld solutions w(r; h) and s_rz(r; h) of composite sharp wedges with ﬁnite radius (concentrated loads)

A.1. Free–free–free boundary condition w^ð1Þ¼ fz

Fig. 15. (a) The full-ﬁeld distribution of shear stress srzfor a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:5b subjected to self-equilibrium forces located atðb; 45°Þ and ð0:7b; 0°Þ. (b) The full-ﬁeld distribution of shear stress shzfor a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:5b subjected to self-equilibrium forces located at ðb; 45°Þ and ð0:7b; 0°Þ.

s^ð1Þ_rz ¼nf_z

Fig. 16. (a) The full-ﬁeld distribution of shear stress srz for a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:5b subjected to a concentrated force located atð0:8b; 45°Þ. (b) The full-ﬁeld distribution of shear stress shzfor a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:5b subjected to a concentrated force located at ð0:8b; 45°Þ.

s^ð2Þ_rz ¼ nl₂f_z A.2. Free–free–ﬁxed boundary condition

w^ð1Þ¼ f_z

Fig. 17. (a) The full-ﬁeld distribution of shear stress srzfor a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:5b subjected to a concentrated force located atð0:8b; 45°Þ. (b) The full-ﬁeld distribution of shear stress shzfor a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:5b subjected to a concentrated force located at ð0:8b; 45°Þ.

s^ð1Þ_rz ¼nfz

Fig. 18. (a) The full-ﬁeld distribution of shear stress srz for a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:4b subjected to a screw dislocation located atð0:6b; 45°Þ. (b) The full-ﬁeld distribution of shear stress shzfor a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:4b subjected to a screw dislocation located at ð0:6b; 45°Þ.

s^ð2Þ_rz ¼ nl₂f_z

pðl₁þ l₂Þr½Cððr=dÞⁿ; nðh aÞÞ þ Cððr=dÞⁿ; nðh þ aÞÞ Cððrd=b²Þⁿ; nðh aÞÞ

Cððrd=b²Þⁿ; nðh þ aÞÞ ðA:8Þ

A.3. Fixed–ﬁxed–free boundary condition w^ð1Þ¼ f_z

Fig. 19. (a) The full-ﬁeld distribution of shear stress srzfor a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:4b subjected to a screw dislocation located atð0:6b; 45°Þ. (b) The full-ﬁeld distribution of shear stress shzfor a composite annular wedge with an apex angle b¼ 120° and a ¼ 0:4b subjected to a screw dislocation located at ð0:6b; 45°Þ.

s^ð1Þ_rz ¼nf_z

Fig. 20. (a) Image force Frexerted on a screw dislocation in a composite wedge with apex angle b¼ 120° and ﬁnite radius r ¼ b. (b) Image force Fhexerted on a screw dislocation in a composite wedge with apex angle b¼ 120° and ﬁnite radius r ¼ b.

s^ð2Þ_rz ¼ nl₂f_z

pðl1þ l2Þr½Cððr=dÞⁿ; nðh aÞÞ Cððr=dÞⁿ; nðh þ aÞÞ þ Cððrd=b²Þⁿ; nðh aÞÞ

Cððrd=b²Þⁿ; nðh þ aÞÞ ðA:12Þ

A.4. Fixed–ﬁxed–ﬁxed boundary condition w^ð1Þ¼ f_z

Fig. 21. (a) Image force Frexerted on a screw dislocation in a composite wedge with apex angle b¼ 120° and ﬁnite radius r ¼ b. (b) Image force Fhexerted on a screw dislocation in a composite wedge with apex angle b¼ 120°and ﬁnite radius r ¼ b.

s^ð1Þ_rz ¼nf_z

Fig. 22. (a) Image force Frexerted on a screw dislocation in an annular wedge with apex angle b¼ 120°. (b) Image force Fhexerted on a screw dislocation in an annular wedge with apex angle b¼ 120°.

Appendix B. The full-ﬁeld solutions w(r; h) and s_rz(r; h) of composite sharp wedges with ﬁnite radius (screw dislocations)

B.1. Free–free–free boundary condition w^ð1Þ¼bz

2p ½Wððr=dÞⁿ; nðh aÞÞ Wððr=dÞⁿ; nðh þ aÞÞ þ kW^þððr=dÞⁿ; nðh aÞÞ

kW^þððr=dÞⁿ; nðh þ aÞÞ Wððrd=b²Þⁿ; nðh aÞÞ þ Wððrd=b²Þⁿ; nðh þ aÞÞ

kW^þððrd=b²Þⁿ; nðh aÞÞ þ kW^þððrd=b²Þⁿ; nðh þ aÞÞ ðB:1Þ

s^ð1Þ_rz ¼nl₁bz

2pr ½Hððr=dÞⁿ; nðh aÞÞ Hððr=dÞⁿ; nðh þ aÞÞ kH^þððr=dÞⁿ; nðh aÞÞ þ kH^þððr=dÞⁿ; nðh þ aÞÞ Hððrd=b²Þⁿ; nðh aÞÞ þ Hððrd=b²Þⁿ; nðh þ aÞÞ

þ kH^þððrd=b²Þⁿ; nðh aÞÞ kH^þððrd=b²Þⁿ; nðh þ aÞÞ ðB:2Þ

w^ð2Þ¼ l1b_z

pðl₁þ l₂Þ½Wððr=dÞⁿ; nðh aÞÞ Wððr=dÞⁿ; nðh þ aÞÞ Wððrd=b²Þⁿ; nðh aÞÞ

þ Wððrd=b²Þⁿ; nðh þ aÞÞ ðB:3Þ

s^ð2Þ_rz ¼ nl₁l₂b_z

pðl₁þ l₂Þr½Hððr=dÞⁿ; nðh aÞÞ Hððr=dÞⁿ; nðh þ aÞÞ Hððrd=b²Þⁿ; nðh aÞÞ

þ Hððrd=b²Þⁿ; nðh þ aÞÞ ðB:4Þ

B.2. Fixed–ﬁxed–free boundary condition w^ð1Þ¼bz

2p ½Wððr=dÞⁿ; nðh aÞÞ þ Wððr=dÞⁿ; nðh þ aÞÞ kW^þððr=dÞⁿ; nðh aÞÞ

kW^þððr=dÞⁿ; nðh þ aÞÞ Wððrd=b²Þⁿ; nðh aÞÞ Wððrd=b²Þⁿ; nðh þ aÞÞ

þ kW^þððrd=b²Þⁿ; nðh aÞÞ þ kW^þððrd=b²Þⁿ; nðh þ aÞÞ ðB:5Þ

s^ð1Þ_rz ¼nl₁b_z

2pr ½Hððr=dÞⁿ; nðh aÞÞ þ Hððr=dÞⁿ; nðh þ aÞÞ þ kH^þððr=dÞⁿ; nðh aÞÞ þ kH^þððr=dÞⁿ; nðh þ aÞÞ Hððrd=b²Þⁿ; nðh aÞÞ Hððrd=b²Þⁿ; nðh þ aÞÞ

kH^þððrd=b²Þⁿ; nðh aÞÞ kH^þððrd=b²Þⁿ; nðh þ aÞÞ ðB:6Þ

w^ð2Þ¼ l1b_z

pðl₁þ l₂Þ½Wððr=dÞⁿ; nðh aÞÞ þ Wððr=dÞⁿ; nðh þ aÞÞ Wððrd=b²Þⁿ; nðh aÞÞ

Wððrd=b²Þⁿ; nðh þ aÞÞ ðB:7Þ

s^ð2Þ_rz ¼ nl₁l₂b_z

pðl₁þ l₂Þr½Hððr=dÞⁿ; nðh aÞÞ þ Hððr=dÞⁿ; nðh þ aÞÞ Hððrd=b²Þⁿ; nðh aÞÞ

Hððrd=b²Þⁿ; nðh þ aÞÞ ðB:8Þ

Appendix C. The full-ﬁeld solutions w(r; h) and s_rz(r; h) of composite annular wedge (concentrated loads) C.1. Free–free–free–free boundary condition

w^ð1Þ¼ f_z

C.2. Fixed–ﬁxed–free–free boundary condition

w^ð1Þ¼ f_z

C.3. Free–free–ﬁxed–free boundary condition

w^ð1Þ¼ f_z

s^ð2Þ_rz ¼ nl₂f_z

Appendix D. The full-ﬁeld solutions w(r; h) and s_rz(r; h) of composite annular wedge (screw dislocations) D.1. Free–free–free–free boundary condition

w^ð1Þ¼bz

D.2. Fixed–ﬁxed–free–ﬁxed boundary condition

w^ð1Þ¼b_z

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在文檔中 Theoretical Full-Field Analysis of Dissimilar Isotropic Composite Annular Wedges under Anti-Plane Deformations (頁 27-40)