An application of M-integral to cracks in dissimilar elastic materials

1982 ◽  
Vol 20 (1) ◽  
pp. R27-R30 ◽  
Author(s):  
S. Kubo
Keyword(s):  
Elastic Materials ◽  
M Integral

The M-Integral Analysis for a Nano-Inclusion in Plane Elastic Materials Under Uni-Axial or Bi-Axial Loadings

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Tong Hui ◽  
Yi-Heng Chen
Keyword(s):  
Surface Effect ◽  
Continuum Theory ◽  
Elastic Materials ◽  
Hard Inclusion ◽  
Surface Parameters ◽  
Integral Analysis ◽  
Axial Tension ◽  
Soft Inclusion ◽  
Tension Loading ◽  
M Integral

This paper deals with the M-integral analysis for a nano-inclusion in plane elastic materials under uni-axial or bi-axial loadings. Based on previous works (Gurtin and Murdoch, 1975, “A Continuum Theory of Elastic Material Surfaces,” Arch. Ration. Mech. Anal., 57, pp. 291–323; Mogilevskaya, et al., 2008, “Multiple Interacting Circular Nano-Inhomogeneities With Surface/Interface Effects,” J. Mech. Phys. Solids, 56, pp. 2298–2327), the surface effect induced from the surface tension and the surface Lamé constants is taken into account, and an analytical solution is obtained. Four kinds of inclusions including soft inclusion, hard inclusion, void, and rigid inclusions are considered. The variable tendencies of the M-integral for each of four nano-inclusions against the loading or against the inclusion radius are plotted and discussed in detail. It is found that in nanoscale the surface parameters for the hard inclusion or rigid inclusion have a little or little influence on the M-integral, and the values of the M-integral are always negative as they would be in macroscale, whereas the surface parameters for the soft inclusion or void yield significant influence on the M-integral and the values of the M-integral could be either positive or negative depending on the loading levels and the surface parameters. Of great interest is that there is a neutral loading point for the soft inclusion or void, at which the M-integral transforms from a negative value to a positive value, and that the bi-axial loading yields similar variable tendencies of the M-integral as those under the uni-axial tension loading. Moreover, the bi-axial tension loading increases the neutral loading point, whereas the bi-axial tension-compression loading decreases it. Particularly, the magnitude of the negative M-integral representing the energy absorbing of the soft inclusion or void increases very sharply as the radius of the soft inclusion or void decreases from 5 nm to 1 nm.

Surface Effect and Size Dependence on the Energy Release Due to a Nanosized Hole Expansion in Plane Elastic Materials

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
Q. Li ◽  
Y. H. Chen
Keyword(s):  
Energy Release ◽  
Surface Stress ◽  
Size Dependence ◽  
Surface Effect ◽  
High Surface ◽  
Plane Elasticity ◽  
Elastic Materials ◽  
Residual Surface Stress ◽  
Surface Lamé Constants ◽  
M Integral

This paper deals with the surface effect and size dependence on the M-integral representing the energy release due to a nanodefect expansion in plane elasticity. Due to the high surface-to-volume ratio for reinforcing particles in the nanometer scale, the surface effect along the nanosized hole may be induced from the residual surface stress and the surface Lamé constants. The invariant integrals such as the Jk-integral vector and the M-integral customarily used in macrofracture mechanics are extended to treat plane elastic materials containing a nanosized hole. It is concluded that both components of the Jk-integral vanish when the contour selected to calculate the integral encloses the whole nanosized hole. This leads to the independence of the M-integral from the global coordinate shift. It is concluded that the surface effect and the size dependence on the energy release due to the nanohole expansion are significant especially when the hole size is less than 40 nm. This present study reveals that the discrepancies of the M-integral value with the surface effect from the referenced value M0 without the surface effect are mainly induced from the residual surface stress τ0 rather than from the surface Lamé constants μs and λs.

Two State M-Integral Analysis for a Nano-Inclusion in Plane Elastic Materials Under Uni-Axial or Bi-Axial Loadings

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Tong Hui ◽  
Yi-Heng Chen
Keyword(s):  
Surface Tension ◽  
Dominant Role ◽  
Elastic Materials ◽  
Integral Analysis ◽  
Surface Lamé Constants ◽  
M Integral

In this paper, the two state M-integral is extended from macrofacture to nanodefect mechanics. The question as to why the M-integral for a nanovoid or a soft nano-inclusion might be negative is clarified. It is concluded that the surface tension plays a dominant role in evaluating the M-integral, whereas the surface Lamè constants yield much less influence than the surface tension. Their influence on the M-integral for a nanovoid or a soft nano-inclusion could be neglected.

Mechanical power and hyperelasticity

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Mechanical Power ◽  
Elastic Materials ◽  
Cyclic Motion

This chapter covers the notion of hyperelasticity—the concept that stress is derived from a strain—energy function–by invoking an analogy between elastic materials and springs. Alternatively, it can be derived by invoking a work inequality; the notion that work is required to effect a cyclic motion of the material.

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