The stresses evaluated in the previous section for elastic biceps muscle will now be used to evaluate dynamic stresses for viscoelastic biceps muscle. The elastic model is a simplified model. However, the results derived from the elastic model allow us to evaluate the viscoelastic model, which more closely approximates human tissue.

To determine the dynamic stress responding to the applied dynamic extension for viscoelastic fasciae, a constitutive equation

^{13} was used to address the viscoelastic behavior in biologic soft tissues. This equation can be written as the following:

\[\ T_{\mathrm{ij}}(x,t)={{\int}_{-{\infty}}^{\mathrm{t}}}G_{\mathrm{ijkl}}(t-{\tau})\frac{{\partial}T_{\mathrm{kl}}^{(\mathrm{e}){\ }(x,{\tau})}}{{\partial}{\tau}}d{\tau}\]

In equation (14),

*T*^{e} is the pseudo-elastic stress and a function of strain, which, in turn, is a function of position,

*x*, and time,

*t*;

*G*_{ijkl} (

*i,j,k,l*= 1, 2, 3, 4, 5, 6) is the tensor of relaxation function; τ is a dummy variable for the integration from the beginning (

*t*=0) of the deformation to the present time,

*t*.

At this stage, we can designate *T*_{1}, *T*_{2}... *T*_{6} for *T*_{11}, *T*_{22}, *T*_{33}, *T*_{23}, *T*_{31}, *T*_{12}, and *E*_{1}, *E*_{2}... *E*_{6} for *E*_{11}, *E*_{22}, *E*_{33}, *E*_{23}, *E*_{31}, *E*_{12}.

Then, equation (14) can be rewritten as the following:

\[\ T_{\mathrm{n}}={{\int}_{0}^{\mathrm{t}}}G_{\mathrm{nq}}(t-{\tau})\frac{{\partial}T^{(\mathrm{e})}}{{\partial}{\tau}}d{\tau}{\ }(n,q=1,2,3,4,5,6)\]

In equation (15), the uninvolved diagonal relaxation functions are assumed to be 0, and we retain

*G*_{11},

*G*_{22},

*G*_{33},

*G*_{13}. Then, we get the following:

\[\ \begin{array}{rl}T_{1}&={{\int}_{0}^{\mathrm{t}}}\left(G_{11}(t-{\tau})\frac{{\partial}T_{1}}{{\partial}{\tau}}+G_{13}(t-{\tau})\frac{{\partial}T_{3}}{{\partial}{\tau}}\right)d{\tau}\\T_{2}&={{\int}_{0}^{\mathrm{t}}}\left(G_{22}(t-{\tau})\frac{{\partial}T_{2}}{{\partial}{\tau}}\right)d{\tau}\\T_{3}&={{\int}_{0}^{\mathrm{t}}}\left(G_{31}(t-{\tau})\frac{{\partial}T_{1}}{{\partial}{\tau}}+G_{33}(t-{\tau})\frac{{\partial}T_{3}}{{\partial}{\tau}}\right)d{\tau}\end{array}\]

The final step in developing field equations for viscoelastic skeletal muscle is to determine the relaxation function. Fung

^{14} introduced a one-dimensional standard model to derive a linear reduced relaxation function with a continuous spectrum. This widely accepted reduced relaxation function

^{14} is presented as the following:

\[\ G(t)=\frac{1+{{\int}_{0}^{{\infty}}}S(q)e^{-\mathrm{t}{/}\mathrm{q}}dq}{1+{{\int}_{0}^{{\infty}}}S(q)dq}\]

In equation (17), the relaxation spectrum

*S(q)* denotes the amplitude of the viscous dissipation subject to the frequency 1/

*q*. For many biologic materials that exhibit viscoelastic behavior and low sensitivity to strain rates, the designation of the spectrum with constant amplitude over a range of frequency has been found to fit the experimental results fairly well.

^{14} This spectrum assumes the following form:

\[\ \begin{array}{rl}S(q)&=\frac{C0}{q},\mathrm{for}{\ }q_{1C}{\leqslant}q{\leqslant}q_{2C}\\S(q)&=0,\mathrm{otherwise}\end{array}\]

In equation (18), the parameters

*C*0,

*q*_{1C}, and

*q*_{2C} can be determined by experiments examining the stress relaxation of the tissues under constant strain and by experiments of dynamic loading. In the present study, we adopted the parameters obtained from the study of subcutaneous tissues of rats by Iatridis et al.

^{15} Those researchers

^{15} reported that

*C*0=0.25,

*q*_{1C}=1.86 seconds, and

*q*_{2C}=110.4 seconds. Combining equations (15) through (18), we can numerically resolve the viscoelastic stress-time and stress-strain relations for the biceps muscle.

Because the viscosity is uniform in the muscle tissues, we assume, in equation (18), that *G*_{11}(*t* - τ) = *G*_{22}(*t* - τ) = *G*_{33}(*t* - τ) = *G*_{13}(*t* - τ)= *G*(*t* - τ), and that the values of the parameters *C*0=0.25, *q*_{1C}=1.86 seconds, and *q*_{2C}=110.4 seconds in *G*(*t* - τ) are as in equation (14).