On the Changing Order of Singularity at a Crack Tip

1977 ◽  
Vol 44 (4) ◽  
pp. 625-630 ◽  
Author(s):  
R. J. Nuismer ◽  
G. P. Sendeckyj
Keyword(s):  
Closed Form ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Crack Tip ◽  
Form Solution ◽  
Square Root ◽  
Rigid Line Inclusion ◽  
Rigid Line ◽  
Line Inclusion ◽  
Crack Tip Stress

The nature of the transition in the crack tip stress singularity from an inverse square root to an inverse fractional power as a crack tip reaches a phase boundary or a geometrical discontinuity for interface cracks is examined. This is done by analyzing the simple closed-form solution to the problem of a rigid line inclusion with one side partially debonded for the case of antiplane deformation. For this example, the crack tip stress singularity changes from an inverse square root to an inverse three-quarters power as the crack tips approach the inclusion tips (i.e., when one face of the rigid line inclusion is completely debonded). A detailed analysis, based on series expansions of the closed-form solution, is used to show how the singularity transition occurs. Moreover, the expansions indicate difficulties that may be encountered when solving such problems by approximate methods.

Green’s functions for soft materials containing a hard line inhomogeneity

2019 ◽  
Vol 24 (11) ◽  
pp. 3614-3631 ◽  
Author(s):  
Pengyu Pei ◽  
Guang Yang ◽  
Cun-Fa Gao
Keyword(s):  
Thermal Expansion ◽  
Heat Source ◽  
Closed Form ◽  
Hilbert Problem ◽  
Closed Form Solution ◽  
Soft Materials ◽  
Form Solution ◽  
Soft Material ◽  
Elastic Plane ◽  
Rigid Line

The linear elastic plane deformation of a soft material containing a rigid line inhomogeneity subjected to a concentrated force, a concentrated moment, and a point heat source was studied. Distinct from the existing rigid line inhomogeneity model which neglects the deformation of the inhomogeneity induced by both the mechanical stresses and thermal expansion, the current model allows for the thermal expansion-induced stretch and rotation of the inhomogeneity. In this context, we derive the closed-form solution for the full stress field in the soft material by solving the corresponding Riemann–Hilbert problem. In particular, our solution can serve as the Green’s function to establish other analytical solutions for more practical and complicated problems in this area. Several numerical examples are presented to illustrate our closed-form solution corresponding to the thermal loading. It is found that the presence of the heat source contributes significantly to the rigid rotation of inhomogeneity, and the thermal expansion-induced stretch of the inhomogeneity has a great impact on the stress intensity factors at the inhomogeneity tips.

Orthotropy Rescaling for the Fracture Problem in Anisotropic Piezoelectric Materials

2002 ◽  
Author(s):  
William S. Oates ◽  
Christopher S. Lynch
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Work Done ◽  
Elastic Dielectric

To date, much of the work done on ferroelectric fracture assumes the material is elastically isotropic, yet there can be considerable polarization induced anisotropy. More sophisticated solutions of the fracture problem incorporate anisotropy through the Stroh formalism generalized to the piezoelectric material. This gives equations for the stress singularity, but the characteristic equation involves solving a sixth order polynomial. In general this must be accomplished numerically for each composition. In this work it is shown that a closed form solution can be obtained using orthotropy rescaling. This technique involves rescaling the coordinate system based on certain ratios of the elastic, dielectric, and piezoelectric coefficients. The result is that the governing equations can be reduced to the biharmonic equation and solutions for the isotropic material utilized to obtain solutions for the anisotropic material. This leads to closed form solutions for the stress singularity in terms of ratios of the elastic, dielectric, and piezoelectric coefficients. The results of the two approaches are compared and the contribution of anisotropy to the stress intensity factor discussed.

Analysis of Conducting Rigid Inclusion at the Interface of Two Dissimilar Piezoelectric Materials

1998 ◽  
Vol 65 (1) ◽  
pp. 76-84 ◽  
Author(s):  
Wei Deng ◽  
S. A. Meguid
Keyword(s):  
Function Theory ◽  
Piezoelectric Materials ◽  
Electric Displacement ◽  
Stress Singularity ◽  
Equilibrium Conditions ◽  
Rigid Line Inclusion ◽  
Rigid Line ◽  
Elastic Mismatch ◽  
Square Root Singularity ◽  
Line Inclusion

This paper is concerned with the electro-elastic analysis of a conducting rigid line inclusion at the interface of two bonded piezoelectric materials. By combining the analytic function theory and the Stroh formalism, we were able to obtain closed-form expressions for the field variables. Both the mechanical stresses and the electric displacement are shown to have at least one of the following behaviors: (i) traditional square root singularity; (ii) nonsquare root singularity; and (iii) oscillatory singularity, which depend upon the electro-elastic mismatch at the interface. By using the static equilibrium conditions, the rigid rotation vector of the inclusion is determined and the extended stress singularity factors (ESSF) are evaluated.

Explicit Solutions For the Magnetoelastic Fields with a Rigid Line Inclusion

2002 ◽  
Vol 18 (4) ◽  
pp. 199-205 ◽  
Author(s):  
Chun-Bo Lin ◽  
Hsien-Mou Lin
Keyword(s):  
Hilbert Problem ◽  
Crack Problem ◽  
Stress Singularity ◽  
Stress Fields ◽  
Variable Theory ◽  
Rigid Line Inclusion ◽  
Complex Variable Theory ◽  
Rigid Line ◽  
Square Root Singularity ◽  
Line Inclusion

ABSTRACTA general solution to the magnetoelastic problem with a rigid line inclusion is presented. Based upon the complex variable theory, the proposed analysis dealing with sectionally holomorphic functions can be reduced to find the solution of the Hilbert problem. It is indicated that the magnetoelastic stress fields near the inclusion tip possess a square root singularity just like that of the corresponding crack problem. The stress singularity coefficients which are defined in this study to characterize the near tip fields are similar to the stress intensity factors for crack problem. Numerical results of the stress distribution in the vicinity of inclusion tip are also displayed in graphic form to elucidate the effect of various parameters.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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