An analytical expression for the J<sub>2</sub>‐integral of an interfacial crack in orthotropic bimaterials

Tafreshi, Azam

doi:10.1111/ffe.12587

Cited by 8 publications

(10 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For contour s 1 , z 1 = z 2 = z , ( n 2 ) ds = − dz , and for contour s 2 , z 1 = − iμ 1 z , z 2 = − iμ 2 z , ( n 1 ds ) = − idz . Based on Equation , the displacement derivative vectors for each material are

{[u_{j, 1}]}^{(n)} = 2 Re {\{A [\begin{matrix} center \\ f_{1}^{'} (z) \\ f_{2}^{'} (z) \end{matrix}]\}}^{(n)}, {[u_{j, 2}]}^{(n)} = 2 Re {\{AP [\begin{matrix} center \\ f_{1}^{'} (z_{1}) \\ f_{2}^{'} (z_{2}) \end{matrix}]\}}^{(n)},

where

{[\begin{matrix} center \\ f_{1}^{'} (z_{1}) \\ f_{2}^{'} (z_{2}) \end{matrix}]}^{(n)} = \frac{1}{4 \sqrt{2 π}} \times \frac{1}{\sqrt{- i}} {[\begin{matrix} center \\ \frac{1}{\sqrt{μ_{1}} (μ_{2} - μ_{1})} & 0 \\ 0 & \frac{1}{\sqrt{μ_{2}} (μ_{2} - μ_{1})} \end{matrix}]}^{(n)} ([], \begin{array}{c} ([], \overset{(n)}{ψ_{1} ((), z)} + i \overset{(n)}{μ_{2}} β_{21} \overset{(n)}{ϕ_{1} ((), z)}) & ([], \overset{(n)}{μ_{2}} ψ_{1} ((), z)) \end{array})

…”

Section: Review Of the Analytical Evaluation Of The Jk‐integrals For mentioning

confidence: 94%

“…Equation can be written as

\overset{(n)}{G ((), z)} = \frac{1}{4 \sqrt{2 π}} ([], \begin{array}{c} \overset{(n)}{ψ} ((), z) & i β_{12} \overset{(n)}{ϕ} ((), z) \\ i β_{21} \overset{(n)}{ϕ} ((), z) & \overset{(n)}{ψ} ((), z) \end{array}) ([], \begin{array}{c} K_{2} \\ K_{1} \end{array}) = \frac{1}{4 \sqrt{2 π}} \overset{(n)}{L} K .

…”

Section: Interfacial Crack and Interface Sifsmentioning

confidence: 99%

“…Combining Equations and , we have

{[f^{'} (z)]}^{(n)} = \frac{1}{4 \sqrt{2 π}} \times \frac{1}{{\overset{((), n)}{μ}}_{2} - {\overset{((), n)}{μ}}_{1}} ([], \begin{array}{c} ([], \overset{(n)}{ψ ((), z)} + i {\overset{((), n)}{μ}}_{2} β_{21} \overset{(n)}{ϕ ((), z)}) K_{2} + ([], {\overset{((), n)}{μ}}_{2} \overset{(n)}{ψ ((), z)} + i β_{12} \overset{(n)}{ϕ ((), z)}) K_{1} \\ - ([], \overset{(n)}{ψ ((), z)} + i {\overset{((), n)}{μ}}_{1} β_{21} \overset{(n)}{ϕ ((), z)}) K_{2} - ([], {\overset{((), n)}{μ}}_{1} \overset{(n)}{ψ ((), z)} + i β_{12} \overset{(n)}{ϕ ((), z)}) K_{1} \end{array}) .

…”

Section: Interfacial Crack and Interface Sifsmentioning

confidence: 99%

“…The following briefly demonstrates the previous study of this author for the theoretical derivation of J k ‐integrals for a straight interfacial flaw between 2 aligned orthotropic materials using the complex function method. The same procedure will be used for the evaluation of the J 2 ‐integral of a cubic symmetry/isotropic bimaterial.…”

Section: Review Of the Analytical Evaluation Of The Jk‐integrals For mentioning

confidence: 99%

“…In a recent study by this author, an analytical relation between J 2 ‐integral of a straight interfacial crack in aligned orthotropic/orthotropic bimaterials using the complex function method was derived. In this study, numerical J k values in conjunction with this analytical expression can be used to estimate SIFs of these materials.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A unified and integrated numerical‐analytical approach for evaluation of J_k‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

Tafreshi

2018

Fatigue Fract Eng Mat Struct

Self Cite

View full text Add to dashboard Cite

A new unified and integrated method for the numerical‐analytical calculation of Jk‐integrals of an in‐plane traction free interfacial crack in homogeneous orthotropic and isotropic bimaterials is presented. The numerical algorithm, based on the boundary element crack shape sensitivities, is generic and flexible. It applies to both straight and curved interfacial cracks in anisotropic and/or isotropic bimaterials. The shape functions of semidiscontinuous quadratic quarter point crack tip elements are correctly scaled to adapt the singular oscillatory near tip field of tractions. The length of crack is designated as the design variable to compute the strain energy release rate precisely. Although an analytical equation relating J1 and stress intensity factors (SIFs) exists, a similar relation for J2 in debonded anisotropic solids for decoupling SIFs is not available. An analytical expression was recently derived by this author for J2 in aligned orthotropic/orthotropic bimaterials with a straight interface crack. Using this new relation and the present computed Jk values, the SIFs can be decoupled without the need for an auxiliary equation. Here, the aforementioned analytical relation is reconstructed for cubic symmetry/isotropic bimaterials and used with the present numerical algorithm. An example with known analytical SIFs is presented. The numerical and analytical magnitudes of Jk for an interface crack in orthotropic/orthotropic and cubic symmetry/isotropic bimaterials show an excellent agreement.

show abstract

{[u_{j, 1}]}^{(n)} = 2 Re {\{A [\begin{matrix} center \\ f_{1}^{'} (z) \\ f_{2}^{'} (z) \end{matrix}]\}}^{(n)}, {[u_{j, 2}]}^{(n)} = 2 Re {\{AP [\begin{matrix} center \\ f_{1}^{'} (z_{1}) \\ f_{2}^{'} (z_{2}) \end{matrix}]\}}^{(n)},

where

{[\begin{matrix} center \\ f_{1}^{'} (z_{1}) \\ f_{2}^{'} (z_{2}) \end{matrix}]}^{(n)} = \frac{1}{4 \sqrt{2 π}} \times \frac{1}{\sqrt{- i}} {[\begin{matrix} center \\ \frac{1}{\sqrt{μ_{1}} (μ_{2} - μ_{1})} & 0 \\ 0 & \frac{1}{\sqrt{μ_{2}} (μ_{2} - μ_{1})} \end{matrix}]}^{(n)} ([], \begin{array}{c} ([], \overset{(n)}{ψ_{1} ((), z)} + i \overset{(n)}{μ_{2}} β_{21} \overset{(n)}{ϕ_{1} ((), z)}) & ([], \overset{(n)}{μ_{2}} ψ_{1} ((), z)) \end{array})

…”

Section: Review Of the Analytical Evaluation Of The Jk‐integrals For mentioning

confidence: 94%

“…Equation can be written as

\overset{(n)}{G ((), z)} = \frac{1}{4 \sqrt{2 π}} ([], \begin{array}{c} \overset{(n)}{ψ} ((), z) & i β_{12} \overset{(n)}{ϕ} ((), z) \\ i β_{21} \overset{(n)}{ϕ} ((), z) & \overset{(n)}{ψ} ((), z) \end{array}) ([], \begin{array}{c} K_{2} \\ K_{1} \end{array}) = \frac{1}{4 \sqrt{2 π}} \overset{(n)}{L} K .

…”

Section: Interfacial Crack and Interface Sifsmentioning

confidence: 99%

“…Combining Equations and , we have

{[f^{'} (z)]}^{(n)} = \frac{1}{4 \sqrt{2 π}} \times \frac{1}{{\overset{((), n)}{μ}}_{2} - {\overset{((), n)}{μ}}_{1}} ([], \begin{array}{c} ([], \overset{(n)}{ψ ((), z)} + i {\overset{((), n)}{μ}}_{2} β_{21} \overset{(n)}{ϕ ((), z)}) K_{2} + ([], {\overset{((), n)}{μ}}_{2} \overset{(n)}{ψ ((), z)} + i β_{12} \overset{(n)}{ϕ ((), z)}) K_{1} \\ - ([], \overset{(n)}{ψ ((), z)} + i {\overset{((), n)}{μ}}_{1} β_{21} \overset{(n)}{ϕ ((), z)}) K_{2} - ([], {\overset{((), n)}{μ}}_{1} \overset{(n)}{ψ ((), z)} + i β_{12} \overset{(n)}{ϕ ((), z)}) K_{1} \end{array}) .

…”

Section: Interfacial Crack and Interface Sifsmentioning

confidence: 99%

Section: Review Of the Analytical Evaluation Of The Jk‐integrals For mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A unified and integrated numerical‐analytical approach for evaluation of J_k‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

Tafreshi

2018

Fatigue Fract Eng Mat Struct

Self Cite

View full text Add to dashboard Cite

show abstract

Mixed Finite Element for Kinking Crack Analysis in an Orthotropic Media

Derouiche

Bouziane

Bouzerd

2021

Procedia Structural Integrity

View full text Add to dashboard Cite

Analytical fracture parameters of two unequal collinear interface cracks in an orthotropic bimaterial

Tafreshi¹

2022

Theoretical and Applied Fracture Mechanics

View full text Add to dashboard Cite

An analytical expression for the J₂‐integral of an interfacial crack in orthotropic bimaterials

Abstract: A Tafreshiorcid

Cited by 8 publications

References 54 publications

A unified and integrated numerical‐analytical approach for evaluation of J_k‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A unified and integrated numerical‐analytical approach for evaluation of J_k‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

Mixed Finite Element for Kinking Crack Analysis in an Orthotropic Media

Analytical fracture parameters of two unequal collinear interface cracks in an orthotropic bimaterial

Contact Info

Product

Resources

About

An analytical expression for the J2‐integral of an interfacial crack in orthotropic bimaterials

Abstract: A Tafreshiorcid

Cited by 8 publications

References 54 publications

A unified and integrated numerical‐analytical approach for evaluation of Jk‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A unified and integrated numerical‐analytical approach for evaluation of Jk‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

Mixed Finite Element for Kinking Crack Analysis in an Orthotropic Media

Analytical fracture parameters of two unequal collinear interface cracks in an orthotropic bimaterial

Contact Info

Product

Resources

About

An analytical expression for the J₂‐integral of an interfacial crack in orthotropic bimaterials

A unified and integrated numerical‐analytical approach for evaluation of J_k‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A unified and integrated numerical‐analytical approach for evaluation of J_k‐integrals of an interfacial crack in orthotropic and isotropic bimaterials