Analysis of Multiple Rigid-Line Inclusions for Application to Bio-Materials

2007 ◽  
Author(s):  
Pawan S. Pingle ◽  
Larissa Gorbatikh ◽  
James A. Sherwood
Keyword(s):  
Stress Intensity ◽  
Stress Intensity Factors ◽  
Crack Problem ◽  
Building Blocks ◽  
Duality Principle ◽  
Inclusion Problem ◽  
Multiple Cracks ◽  
Intensity Factors ◽  
Aspect Ratios ◽  
Rigid Line

Hard biological materials such as nacre and enamel employ strong interactions between building blocks (mineral crystals) to achieve superior mechanical properties. The interactions are especially profound if building blocks have high aspect ratios and their bulk properties differ from properties of the matrix by several orders of magnitude. In the present work, a method is proposed to study interactions between multiple rigid-line inclusions with the goal to predict stress intensity factors. Rigid-line inclusions provide a good approximation of building blocks in hard biomaterials as they possess the above properties. The approach is based on the analytical method of analysis of multiple interacting cracks (Kachanov, 1987) and the duality existing between solutions for cracks and rigid-line inclusions (Ni and Nasser, 1996). Kachanov’s method is an approximate method that focuses on physical effects produced by crack interactions on stress intensity factors and material effective elastic properties. It is based on the superposition technique and the assumption that only average tractions on individual cracks contribute to the interaction effect. The duality principle states that displacement vector field for cracks and stress vector-potential field for anticracks are each other’s dual, in the sense that solution to the crack problem with prescribed tractions provides solution to the corresponding dual inclusion problem with prescribed displacement gradients. The latter allows us to modify the method for multiple cracks (that is based on approximation of tractions) into the method for multiple rigid-line inclusions (that is based on approximation of displacement gradients). This paper presents an analytical derivation of the proposed method and is applied to the special case of two collinear inclusions.

Analysis of Branched Interface Cracks Between Dissimilar Anisotropic Media

1989 ◽  
Vol 56 (4) ◽  
pp. 844-849 ◽  
Author(s):  
G. R. Miller ◽  
W. L. Stock
Keyword(s):  
Stress Intensity ◽  
Singular Integral ◽  
Stress Intensity Factors ◽  
Crack Problem ◽  
Singular Integral Equations ◽  
Crack Branching ◽  
Intensity Factors ◽  
Function Solution ◽  
Special Cases ◽  
Branch Crack

A solution is presented for the problem of a crack branching off the interface between two dissimilar anisotropic materials. A Green’s function solution is developed using the complex potentials of Lekhnitskii (1981) allowing the branched crack problem to be expressed in terms of coupled singular integral equations. Numerical results for the stress intensity factors at the branch crack tip are presented for some special cases, including the no-interface case which is compared to the isotropic no-interface results of Lo (1978).

Elastodynamic Fracture Analysis of Multiple Cracks by Laplace Finite Element Alternating Method

1999 ◽  
Vol 67 (3) ◽  
pp. 606-615 ◽  
Author(s):  
W.-H. Chen ◽  
C.-L. Chang ◽  
C.-H. Tsai
Keyword(s):  
Finite Element ◽  
Stress Intensity ◽  
Laplace Transform ◽  
Stress Intensity Factors ◽  
Dynamic Stress ◽  
Multiple Cracks ◽  
Intensity Factors ◽  
Dynamic Stress Intensity Factors ◽  
Alternating Method ◽  
The Laplace Transform

The Laplace finite element alternating method, which combines the Laplace transform technique and the finite element alternating method, is developed to deal with the elastodynamic analysis of a finite plate with multiple cracks. By the Laplace transform technique, the complicated elastodynamic fracture problem is first transformed into an equivalent static fracture problem in the Laplace transform domain and then solved by the finite element alternating method developed. To do this, an analytical solution by Tsai and Ma for an infinite plate with a semi-infinite crack subjected to exponentially distributed loadings on crack surfaces in the Laplace transform domain is adopted. Finally, the real-time response can be computed by a numerical Laplace inversion algorithm. The technique established is applicable to the calculation of dynamic stress intensity factors of a finite plate with arbitrarily distributed edge cracks or symmetrically distributed central cracks. Only a simple finite element mesh with very limited number of regular elements is necessary. Since the solutions are independent of the size of time increment taken, the dynamic stress intensity factors at any specific instant can even be computed by a single time-step instead of step-by-step computations. The interaction among the cracks and finite geometrical boundaries on the dynamic stress intensity factors is also discussed in detail. [S0021-8936(00)02103-6]

Stress Intensity Factors of Multiple Cracked Sheet With Riveted Stiffeners

1991 ◽  
Vol 113 (3) ◽  
pp. 280-284 ◽  
Author(s):  
T. Nishimura
Keyword(s):  
Stress Intensity ◽  
Integral Equations ◽  
Stress Intensity Factors ◽  
Fredholm Integral Equations ◽  
Multiple Cracks ◽  
Basic Solution ◽  
Intensity Factors ◽  
Single Crack ◽  
Numerical Examples ◽  
Unknown Density

A new method is proposed for analyzing the stress intensity factors of multiple cracks in a sheet reinforced with riveted stiffeners. Using the basic solution of a single crack and taking unknown density of surface tractions and fastener forces, Fredholm integral equations and compatibility equations of displacements among the sheet, fasteners, and stiffeners are formulated. After solving the unknown density, the stress intensity factors of multiple cracks in the sheet are determined. Some numerical examples are analyzed.

A Method to Extract Interface Stress Intensity Factors Using Interlayers

2005 ◽  
Author(s):  
Sridhar Santhanam
Keyword(s):  
Stress Intensity ◽  
Interface Crack ◽  
Stress Intensity Factors ◽  
Crack Problem ◽  
Interface Stress ◽  
Mode I ◽  
Universal Relation ◽  
Intensity Factors ◽  
Infinite Blocks

A method is presented here to extract stress intensity factors for interface cracks in plane bimaterial fracture problems. The method relies on considering a companion problem wherein a very thin elastic interlayer is artificially inserted between the two material regions of the original bimaterial problem. The crack in the companion problem is located in the middle of the interlayer with its tip located within the homogeneous interlayer material. When the thickness of the interlayer is small compared with the other length scales of the problem, a universal relation can be established between the actual interface stress intensity factors at the crack tip for the original problem and the mode I and II stress intensity factors associated with the companion problem. The universal relation is determined by formulating and solving a boundary value problem. This universal relation now allows the determination of the stress intensity factors for a generic plane interface crack problem as follows. For a given interface crack problem, the companion problem is formulated and solved using the finite element method. Mode I and II stress intensity factors are obtained using the modified virtual crack closure method. The universal relation is next used to obtain the corresponding interface stress intensity factors for the original interface crack problem. An example problem involving a finite interface crack between two semi-infinite blocks is considered for which analytical solutions exist. It is shown that the method described above provides very acceptable results.

An Integral Technique to Evaluate Opening Mode Stress Intensity Factors for Embedded Planar Cracks of Arbitrary Shape

1985 ◽  
Vol 9 (1) ◽  
pp. 1-10 ◽  
Author(s):  
M.A.A. Khattab ◽  
D.J. Burns ◽  
R.J. Pick ◽  
J.C. Thompson
Keyword(s):  
Stress Intensity ◽  
Stress Intensity Factors ◽  
Arbitrary Shape ◽  
Intensity Factors ◽  
Simple Method ◽  
Aspect Ratios ◽  
Opening Mode ◽  
Integration Techniques ◽  
Mode Stress ◽  
Coarse Grids

In this paper, techniques are developed to handle the integrable singularities of the integral proposed by Burns and Oore for the estimation of opening mode stress intensity factors for embedded planar defects of arbitrary shape. The hybrid numerical-analytical integration techniques developed consider separately two crack front zones and one interior zone of the crack surface. Parameters are established for the sizing of the integration elements within each zone. Studies of elliptical defects with aspect ratios between 1 and 10 demonstrate the accuracy and efficiency of this procedure for computing opening mode stress intensity factors. A simple method which compensates for the quadrature error associated with computationally inexpensive, coarse grids is outlined.

The Mixed-Mode Fracture Analysis of Multiple Embedded Cracks in a Semi-Infinite Plane by an Analytical Alternating Method

2002 ◽  
Vol 124 (4) ◽  
pp. 446-456 ◽  
Author(s):  
Chih-Yi Chang ◽  
Chien-Ching Ma
Keyword(s):  
Stress Intensity ◽  
Stress Intensity Factors ◽  
Mixed Mode ◽  
Fundamental Solutions ◽  
Multiple Cracks ◽  
Mixed Mode Fracture ◽  
Infinite Plane ◽  
Intensity Factors ◽  
Alternating Method ◽  
Mode Stress

An efficient analytical alternating method is developed in this paper to evaluate the mixed-mode stress intensity factors of embedded multiple cracks in a semi-infinite plane. Analytical solutions of a semi-infinite plane subjected to a point force applied on the boundary, and a finite crack in an infinite plane subjected to a pair of point forces applied on the crack faces are referred to as fundamental solutions. The Gauss integrations based on these point load fundamental solutions can precisely simulate the conditions of arbitrarily distributed loads. By using these fundamental solutions in conjunction with the analytical alternating technique, the mixed-mode stress intensity factors of embedded multiple cracks in a semi-infinite plane are evaluated. The numerical results of some reduced problems are compared with available results in the literature and excellent agreements are obtained.

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