Indentation method for measuring the viscoelastic kernel function of nonlinear viscoelastic soft materials

Abstract

The indentation method of nonlinear viscoelastic materials is investigated through combined numerical and experimental efforts to reveal the correlation between the viscoelastic kernel function and the indentation responses. It is shown that the viscoelastic kernel function of a nonlinear viscoelastic solid with viscous response characterized by a linear rate constitutive equation scales with the normalized relaxation load in an indentation relaxation test. This scaling relation does not depend on the geometry of the indented solid and the profile of the indenter. Therefore, it may serve as a fundamental relation for characterizing the viscoelastic properties of some biological soft tissues and artificial soft materials with regular/irregular surface morphology.

Appendix

In the uniaxial tension test of a homogeneous solid, the motion is described by:

$${x_1} = {{\lambda }}\left( t \right){X_1},{x_2} = {J^{1/2}}{{\lambda }}{\left( t \right)^{ - 1/2}}{X_2},{x_3} = {J^{1/2}}{{\lambda }}{\left( t \right)^{ - 1/2}}{X_3}\quad,$$

((A1))

where x_i (i = 1, 2, 3) and X_I (I = 1, 2, 3) represent the coordinates in the spatial and reference configuration, respectively, λ(t) is the stretch ratio in the direction of the uniaxial tension, and J is the determinant of the deformation gradient. The deformation gradient has the diagonal form:

$$F\left( t \right) = {\text{Diag}}\left[ {{{\lambda }}\left( t \right),{J^{1/2}}{{\lambda }}{{\left( t \right)}^{ - 1/2}},{J^{1/2}}{{\lambda }}{{\left( t \right)}^{ - 1/2}}} \right]\quad.$$

((A2))

We consider a tensile relaxation test, where the stretch ratio is described via a Heaviside step function:

$${{\lambda }}\left( t \right) = \left\{ { {{{{\lambda }}_0}\left( {t \geqslant 0} \right)} \\ {1\left( {t < 0} \right)} \\ \quad.} \right.$$

((A3))

The instantaneous Kirchhoff stress τ₀(t) is the function of the material parameters, the stretch ratio, and the determinant of the deformation gradient J. When t ≥ 0, the stretch ratio λ(t) in a tensile relaxation test and J are constant, and hence, the Kirchhoff stress τ₀(t) also does not change, which is written as τ₀ here. In this case, it is obvious that ${{{\bar F}}_t}\left( {t - t'} \right)$ is a second-order unit tensor. Therefore, from Eq. (9), we have:

$${{{\tau }}^D}\left( t \right) = {{\tau }}_0^D + {{\tau }}_0^D\int\limits_0^t {\frac{{\partial g}}{{\partial t'}}dt'} \quad,$$

((A4))

$${{{\tau }}^H}\left( t \right) = {{\tau }}_0^H + {{\tau }}_0^H\int\limits_0^t {\frac{{\partial g}}{{\partial t'}}dt'} \quad.$$

((A5))

Further, the Kirchhoff stress τ(t) is:

$${{\tau }}\left( t \right) = {\tau ^H}\left( t \right){ + ^H}\left( t \right) = {{{\tau }}_0}\left( {1 + \int\limits_0^t {\frac{{\partial g}}{{\partial t'}}dt'} } \right)\quad.$$

((A6))

Recall that g(0) = 1, Eq. (A6) can be further reduced to:

$${{\tau }}\left( t \right) = {\tau _0}g\left( t \right)\quad.$$

((A7))

In the current configuration, the Cauchy stress can be expressed as:

$$\sigma \left( t \right) = J{{\tau }}\left( t \right)\quad.$$

((A8))

Then, the uniaxial tensile load may be written as:

$${P_u}\left( t \right) = {{{\sigma }}_{11}}\left( t \right)A\left( 0 \right)/{{{\lambda }}_0}\quad,$$

((A9))

where A(0) represents the area of the initial cross section. σ₁₁ is the Cauchy stress in the tensile direction. Equations (A7)–(A9) give:

$${P_u}\left( t \right) = J{{{\tau }}_{{\text{011}}}}g\left( t \right)A\left( 0 \right)/{{{\lambda }}_0}\quad,$$

((A10))

where τ₀₁₁ is the instantaneous Kirchhoff stress in the tensile direction. It is of notice that g(0) = 1 and P_u(0) = Jτ₀₁₁A(0)/λ₀, from Eq. (A10), we get the following correlation between the viscoelastic kernel function g(t) and the relaxation load:

$$g\left( t \right) = {\bar P_u}\left( t \right) = \frac{{{P_u}\left( t \right)}}{{{P_u}\left( 0 \right)}}\quad,$$

((A11))

where P_u(0) is the tension load at the starting point of relaxation.