An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials

Abstract

An elastic analysis of an internal crack with bridging fibers parallel to the free surface in an infinite orthotropic elastic plane is studied. An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials is presented for analyzing the distributions of stress and displacement with the internal asymmetrical crack under the loading conditions of an applied non-homogenous stress and the traction forces on crack faces yielded by the bridging fiber pull-out model. Thus the fiber failure is determined by maximum tensile stress, resulting in fiber rupture and hence the crack propagation would occur in a self-similarity manner. The formulation involves the development of a Riemann-Hilbert problem. Analytical solution of an asymmetrical propagation crack of unidirectional composite materials under the conditions of two moving loads given is obtained, respectively. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be obtained.

Appendix

In this appendix we intend to do the necessary algebra involved in D and D ₁ for orthotropic anisotropy, and also for isotropy, changing the notation in this latter case in order to make a direct comparison with Broberg [25]. We also intend to show D ₁≡0 when T ₁=−T ₂ as mentioned in paper, and also that D/D ₁ is purely imaginary for the possible crack velocities involved in the problem.

Calculation of D,D ₁ for orthotropy (for the subsonic speeds)

Now, in order to illuminate these representations for universal orthotropy, we are going to refer to (7) in literatures [16, 20, 25–27, 42, 70, 71], where η is replaced by τ, and we write it as

$$ MT^{4} + NT^{2} + P = 0$$

(A.1)

where

(A.2)

Now write down certain properties of the roots, i.e. the sums and products, etc. From (A.1) we will obtain

(A.3)

Write $C_{r} = \sqrt{C_{66}/\rho}$, $C_{d} = \sqrt{C_{11}/\rho}$, a=(C ₆₆−ρτ ²), b=(C ₁₁−ρτ ²).

At $\tau^{2} > C_{d}^{2}$, when a<0,b<0; presumed C ₆₆<C ₁₁, from (A.2) we will obtain

(A.4)

Evidently 0<N ²−4PM<N ², and N<0, reduce $- N\pm\sqrt{N^{2} - 4PM} > 0$, this denotes $T_{1}^{2}$, $T_{2}^{2}$ are both positive real, therefore all of the four roots of (A.1) are real. This tests that for $\tau^{2} > C_{d}^{2}$, we will write  Im[D(τ)/D ₁(τ)]=0, which indicates that the disturbance of elastic wave cannot overrun C _d t.

Then putting (6), (8) into (13) in literature [16, 20, 25–27, 42, 70, 71], there results

(A.5)

It is not difficult to test that for |τ|<C _r ,D/D ₁ is purely imaginary for the subsonic speeds. We can give

Case (1). For C _r <|τ|<C _d , remembering that we have taken the positive square root, then we will obtain

$$\mathop{\mathrm{Im}} [D/D_{1}] =\frac{Mg\sqrt{P'/M}(C_{12} + C_{66})[C_{12}^{2} -C_{22}(C_{11} - \rho \tau ^{2})] + MhS'}{\sqrt{N^{2} - 4PM} \cdot (C_{12} + C_{66})(C_{11} - \rho \tau ^{2})} $$

(A.6)

where

Case (2). For |τ|<C _r <C _d , then we can find

(A.7)

Case (3). For isotropy. Isotropy is regarded a special example as orthotropy, from isotropy, we will have

$$ \begin{array}{@{}l}C_{11} = C_{22} = \rho C_{1}^{2}\\[4pt]C_{66} = 0.5(C_{11} - C_{12}) = \rho C_{2}^{2}\end{array}$$

(A.8)

where C ₁,C ₂ are the wave velocities in the isotropic medium, simply gives D/D ₁ as a function of τ, then substituting them into (5) in literature [16, 20, 25–27, 42, 70, 71] and (A.6), we can obtain:

(A.9)