Fracture of soft materials

The Neo-Hookean model is generalized to account for the hardening factor:

${\displaystyle W={\frac {\mu }{2b}}\left\{\left[1+{\frac {b}{n}}(I-3)\right]^{n}-1\right\},}$

where b>0 and n>1/2 are material parameters, and ${\displaystyle I=I_{1}}$ is the first invariant of the Cauchy-Green deformation tensor:

${\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2},}$

where ${\displaystyle \lambda _{\alpha }}$are the principle stretches.

Setting n=1, the specific stress-strain function for the neo-Hookean model is derived:

${\displaystyle W={\frac {\mu }{2}}(I-3)}$.

The method of asymptotic analysis consists of analyzing the crack-tip asymptotically to find a series expansion of the deformed coordinates capable to characterize the solution near the crack tip. The analysis is reducible to a nonlinear eigenvalue problem.[3]

The problem is formulated based on a crack in an infinite solid, loaded at infinity with uniform uni-axial tension under condition of plane strain (see Fig.1). As the crack deforms and progresses, the coordinates in the current configuration are represented by ${\displaystyle y_{1}}$ and ${\displaystyle y_{2}}$ in cartesian basis and ${\displaystyle \rho }$ and ${\displaystyle \phi }$ in polar basis. The coordinates ${\displaystyle y_{1}}$ and ${\displaystyle y_{2}}$ are functions of the undeformed coordinates (${\displaystyle r,\theta }$) and near the crack tip, as r→0, can be specified as:

${\displaystyle y_{\alpha }(r,\theta )=r^{m_{\alpha }}\upsilon _{\alpha }(\theta )+r^{p_{\alpha }}q_{\alpha }(\theta )+...}$

${\displaystyle m_{\alpha }<p_{\alpha },}$ ${\displaystyle \alpha =1,2,}$

where ${\displaystyle m_{\alpha }}$, ${\displaystyle p_{\alpha }}$ are unknown exponents, and ${\displaystyle \upsilon _{\alpha }(\theta )}$, ${\displaystyle q_{\alpha }(\theta )}$ are unknown functions describing the angular variation.

In order to obtain the eigenvalues, the equation above is substituted into the constitutive model, which yields the corresponding nominal stress components. Then, the stresses are substituted into the equilibrium equations (the same formulation as in LEFM theory) and the boundary conditions are applied. The most dominating terms are retained resulting in an eigenvalue problem for ${\displaystyle \upsilon _{\alpha }(\theta )}$ and ${\displaystyle m_{\alpha }}$.[4]

For the case of a homogeneous neo-Hookean solid (n=1) under Mode I condition the deformed coordinates for a plane strain configuration are given by[4][5]

${\displaystyle y_{1}=-b_{0}r\sin ^{2}(\theta /2),\quad y_{2}=ar^{1/2}\sin(\theta /2),}$

where a and ${\displaystyle b_{0}}$are unknown positive amplitudes that depends on the applied loading and specimen geometry.

The leading terms for the nominal stress (or first Piola–Kirchhoff stress, denoted by ${\displaystyle \sigma }$ on this page) are:

${\displaystyle \sigma _{11}=\mu b_{0},\quad \sigma _{12}=o(1),}$

${\displaystyle \sigma _{21}=-{\frac {\mu a}{2}}r^{-1/2}\sin(\theta /2),}$

${\displaystyle \sigma _{22}={\frac {\mu a}{2}}r^{-1/2}\cos(\theta /2).}$

Thus, ${\displaystyle \sigma _{11}}$and ${\displaystyle \sigma _{21}}$are bounded at the crack tip and ${\displaystyle \sigma _{21}}$and ${\displaystyle \sigma _{22}}$have the same singularity.

The leading terms for the true stress (or Cauchy stress, denoted by ${\displaystyle \tau }$ on this page),

${\displaystyle \tau _{11}=\mu b_{0}^{2}\sin ^{2}(\theta /2),}$

${\displaystyle \tau _{12}=\tau _{21}=-{\frac {\mu }{2}}ab_{0}r^{-1/2}\sin ^{2}(\theta /2),}$

${\displaystyle \tau _{22}={\frac {\mu }{4}}a^{2}r^{-1}.}$

The only true stress component completely defined by a is ${\displaystyle \tau _{22}}$. It also presents the most severe singularity. With that, it is clear that the singularity differs if the stress is given in the current or reference configuration. Additionally, in LEFM, the true stress field under Mode I has a singularity of ${\displaystyle {r}^{-1/2}}$,[6] which is weaker than the singularity in ${\displaystyle \tau _{22}}$.

While in LEFM the near tip displacement field depends only on the Mode I stress intensity factor, it is shown here that for large deformations, the displacement depends on two parameters (a and ${\displaystyle b_{0}}$ for a plane strain condition).

The crack tip deformation field for a Mode I configuration in a homogeneous material neo-Hookean solid (n=1) is given by[4][5]

${\displaystyle y_{1}=cr\,\cos \theta ,\quad y_{2}=a{\sqrt {r}}\sin(\theta /2),}$

where a and c are positive independent amplitudes determined by far field boundary conditions.

The dominant terms of the nominal stress are

${\displaystyle \sigma _{11}=\mu c,\quad \sigma _{12}=o(1),}$

${\displaystyle \sigma _{21}=-{\frac {\mu a}{2}}r^{-1/2}\sin(\theta /2),}$

${\displaystyle \sigma _{22}={\frac {\mu a}{2}}r^{-1/2}\cos(\theta /2).}$

And the true stress components are

${\displaystyle \tau _{11}=\mu c^{2},\quad \tau _{12}=\tau _{21}=-{\frac {\mu }{2}}acr^{-1/2}\sin(\theta /2),}$

${\displaystyle \tau _{22}={\frac {\mu }{4}}a^{2}r^{-1}.}$

Analogously, the displacement depends on two parameters (a and c for a plane stress condition) and the singularity is stronger in the ${\displaystyle \tau _{22}}$ term.

The distribution of the true stress in the deformed coordinates (as shown in Fig. 1B) can be relevant when analyzing the crack propagation and blunt phenomenon. Additionally, it is useful when verifying experimental results of the deformation of the crack.

The J-integral can be determined by experiments. One common experiment is the pure-shear in an infinite long strip, as shown in Fig. 2. The upper and bottom edges are clamped by grips and the loading is applied by pulling the grips vertically apart by ± ∆.[4] This set generates a condition of plane stress.

Figure 2: Pure-shear experiment.

Under these conditions, the J-integral is evaluated, therefore, as

${\displaystyle J=2h_{0}W(I_{1},I_{2})=2h_{0}\Psi (\lambda ),}$

where ${\displaystyle I_{1}=I_{2}=\lambda ^{2}+\lambda ^{-2}+1,}$ ${\displaystyle \lambda =1+{\frac {\Delta }{h_{0}}},}$

and ${\displaystyle h_{0}}$is the high of the strip undeformed state. The function ${\displaystyle \Psi (\lambda )}$is determined by measuring the nominal stress acting on the strip stretched by ${\displaystyle \lambda }$:

${\displaystyle \Psi =\int _{1}^{\lambda }\sigma (\lambda )d\lambda .}$

Therefore, from the imposed displacement of each grip, ± ∆, it is possible to determine the J-integral for the corresponding nominal stress. With the J-integral, the amplitude (parameter a) of some true stress components can be found. Some other stress components amplitudes, however, depend on other parameters such as c (e.g. ${\displaystyle \sigma _{11}}$ under plane stress condition) and cannot be determined by the pure shear experiment. Nevertheless, the pure shear experiment is very important because it allows the characterization of fracture toughness of soft materials.