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Source: Fracture of soft materials

The fracture of soft materials involves large deformations and crack blunting before propagation of the crack can occur. Consequently, the stress field close to the crack tip is significantly different from the traditional formulation encountered in the Linear elastic fracture mechanics. Therefore, fracture analysis for these applications requires a special attention.
The Linear Elastic Fracture Mechanics (LEFM) and K-field (see Fracture Mechanics) are based on the assumption of infinitesimal deformation, and as a result are not suitable to describe the fracture of soft materials. However, LEFM general approach can be applied to understand the basics of fracture on soft materials.
The solution for the deformation and crack stress field in soft materials considers large deformation and is derived from the finite strain elastostatics framework and hyperelastic material models.
Soft materials (Soft matter) consist of a type of material that e.g. includes soft biological tissues as well as synthetic elastomers, and that is very sensitive to thermal variations. Hence, soft materials can become highly deformed before crack propagation.

Hyperelastic material models

soft

materials

= Generalized neo-Hookean (GNH)

Finite strain crack tip solutions (under large deformation)

= Method of asymptotic analysis

α
=
1
,
2
,

{\displaystyle \alpha =1,2,}

where

m

α

{\displaystyle m_{\alpha }}

,

p

α

{\displaystyle p_{\alpha }}

are unknown exponents, and

υ

α

(
θ
)

{\displaystyle \upsilon _{\alpha }(\theta )}

,

q

α

(
θ
)

{\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

υ

α

(
θ
)

{\displaystyle \upsilon _{\alpha }(\theta )}

and

m

α

{\displaystyle m_{\alpha }}

.

Deformation and stress field in a plane strain crack

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

y

1

=
−

b

0

r

sin

2

⁡
(
θ

/

2
)
,

y

2

=
a

r

1

/

2

sin
⁡
(
θ

/

2
)
,

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

where a and

b

0

{\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

σ

{\displaystyle \sigma }

on this page) are:

σ

11

=
μ

b

0

,

σ

12

=
o
(
1
)
,

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

σ

21

=
−

μ
a

2

r

−
1

/

2

sin
⁡
(
θ

/

2
)
,

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

σ

22

=

μ
a

2

r

−
1

/

2

cos
⁡
(
θ

/

2
)
.

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

Thus,

σ

11

{\displaystyle \sigma _{11}}

and

σ

21

{\displaystyle \sigma _{21}}

are bounded at the crack tip and

σ

21

{\displaystyle \sigma _{21}}

and

σ

22

{\displaystyle \sigma _{22}}

have the same singularity.
The leading terms for the true stress (or Cauchy stress, denoted by

τ

{\displaystyle \tau }

on this page),

τ

11

=
μ

b

0

2

sin

2

⁡
(
θ

/

2
)
,

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

τ

12

=

τ

21

=
−

μ
2

a

b

0

r

−
1

/

2

sin

2

⁡
(
θ

/

2
)
,

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

τ

22

=

μ
4

a

2

r

−
1

.

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

The only true stress component completely defined by a is

τ

22

{\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

r

−
1

/

2

{\displaystyle {r}^{-1/2}}

, which is weaker than the singularity in

τ

22

{\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

b

0

{\displaystyle b_{0}}

for a plane strain condition).

Deformation and stress field in a plane stress crack

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

y

1

=
c
r

cos
⁡
θ
,

y

2

=
a

r

sin
⁡
(
θ

/

2
)
,

{\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

σ

11

=
μ
c
,

σ

12

=
o
(
1
)
,

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

σ

21

=
−

μ
a

2

r

−
1

/

2

sin
⁡
(
θ

/

2
)
,

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

σ

22

=

μ
a

2

r

−
1

/

2

cos
⁡
(
θ

/

2
)
.

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

And the true stress components are

τ

11

=
μ

c

2

,

τ

12

=

τ

21

=
−

μ
2

a
c

r

−
1

/

2

sin
⁡
(
θ

/

2
)
,

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

τ

22

=

μ
4

a

2

r

−
1

.

{\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

τ

22

{\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.

J-integral

The J-integral represents the energy that flows to the crack, hence, it is used to calculate the energy release rate, G. Additionally, it can be used as a fracture criterion. This integral is found to be path independent as long as the material is elastic and damages to the microstructure are not occurring.
Evaluating J on a circular path in the reference configuration yields

J
=
π
A

(

2
n
−
1

2
n

)

2
n
−
1

n

2
−
n

a

2
n

,

{\displaystyle J=\pi A\left({\frac {2n-1}{2n}}\right)^{2n-1}n^{2-n}a^{2n},}

for plane strain Mode I, where a is the amplitude of the leading order term of

y

2

{\displaystyle y_{2}}

and A and n are material parameters from the strain-energy function.
For plane stress Mode I in a neo-Heookean material J is given by

J
=

μ
π

2

(

b
n

)

n
−
1

(

2
n
−
1

2
n

)

2
n
−
1

n

1
−
n

a

2
n

,

{\displaystyle J={\frac {\mu \pi }{2}}\left({\frac {b}{n}}\right)^{n-1}\left({\frac {2n-1}{2n}}\right)^{2n-1}n^{1-n}a^{2n},}

where b and n are material parameters of GNH solids. For the specific case of a neo-Hookean model, where n=1, b=1 and

A
=
μ

/

2

{\displaystyle A=\mu /2}

, the J-integral for plane stress and plane strain in Mode I are the same:

J
=

μ
π

a

2

4

.

{\displaystyle J={\frac {\mu \pi a^{2}}{4}}.}

= J-integral in the pure-shear experiment

=
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 ± ∆. This set generates a condition of plane stress.

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

J
=
2

h

0

W
(

I

1

,

I

2

)
=
2

h

0

Ψ
(
λ
)
,

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

where

I

1

=

I

2

=

λ

2

+

λ

−
2

+
1
,

{\displaystyle I_{1}=I_{2}=\lambda ^{2}+\lambda ^{-2}+1,}

λ
=
1
+

Δ

h

0

,

{\displaystyle \lambda =1+{\frac {\Delta }{h_{0}}},}

and

h

0

{\displaystyle h_{0}}

is the high of the strip undeformed state. The function

Ψ
(
λ
)

{\displaystyle \Psi (\lambda )}

is determined by measuring the nominal stress acting on the strip stretched by

λ

{\displaystyle \lambda }

:

Ψ
=

∫

1

λ

σ
(
λ
)
d
λ
.

{\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.

σ

11

{\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.

Interface cracks

To approach the interaction of adhesion between soft adhesives and rigid substrates, the asymptotic solution for an interface crack problem between a GNH material and a rigid substrate is specified. The interface crack configuration considered here is shown in Fig.3 where the lateral slip is disregarded.
For the special neo-Hookean case with n=1, and

v

1

¯

=

v

2

¯

=
cos
⁡
θ

{\displaystyle {\overline {v_{1}}}={\overline {v_{2}}}=\cos \theta }

, the solution for the deformed coordinates is

y

1

=

a

1

r

1
2

sin
⁡

(

θ
2

)

+
r
cos
⁡
θ
,

{\displaystyle y_{1}=a_{1}r^{\frac {1}{2}}\sin \left({\frac {\theta }{2}}\right)+r\cos \theta ,}

y

2

=

a

2

r

1
2

sin
⁡

(

θ
2

)

,

{\displaystyle y_{2}=a_{2}r^{\frac {1}{2}}\sin \left({\frac {\theta }{2}}\right),}

which is equivalent to

y

1

=

a

1

a

2

y

2

−

(

y

2

a

2

)

2

.

{\displaystyle y_{1}={\frac {a_{1}}{a_{2}}}y_{2}-\left({\frac {y_{2}}{a_{2}}}\right)^{2}.}

According to the above equation, the crack on this type of interface is found to open with a parabolic shape. This is confirmed by plotting the normalized coordinates

y

1

/

a

2

2

{\displaystyle y_{1}/a_{2}^{2}}

vs

y

2

/

a

2

2

{\displaystyle y_{2}/a_{2}^{2}}

for different

a

1

/

a

2

{\displaystyle a_{1}/a_{2}}

ratios (see Fig. 4).
To go through the analysis of the interface between two GNH sheets with the same hardening characteristics, refer to the model described by Gaubelle and Knauss.