The idealized viscous liquid (represented by the lumped-component damper/dashpot in the Maxwell model) has infinite stiffness for instantaneous movements. You state this yourself in your constitutive law relating $\sigma_N(t)$ and $\dot\gamma_N(t)$; a step increase in $\gamma_N$ results in $\sigma_N\to\infty$.

This result is nonsensical because the load on two components in series must be equal, but the spring cannot sustain an infinite load; the other constitutive law implies that the maximum load that can be applied by the spring is $\gamma (0) G$.

Therefore, the spring must take up the entire initial displacement in the case of a step strain, and so $\gamma_N(0)=0$.

(Another interesting case is the configuration of a spring and damper attached in parallel. In that case, the assumption of a step increase in strain becomes completely unviable!)

The idealized viscous liquid (represented by the lumped-component damper/dashpot in the Maxwell model) has infinite stiffness for instantaneous movements. You state this yourself in your constitutive law relating $\sigma_N(t)$ and $\dot\gamma_N(t)$; a step increase in $\gamma_N$ results in $\sigma_N\to\infty$.

This result is nonsensical because the load on two components in series must be equal, but the spring cannot sustain an infinite load; the other constitutive law implies that the maximum load that can be applied by the spring on the damper attached to one of its ends is $\gamma (0) G$.

Therefore, the spring must take up the entire initial displacement in the case of a step strain, and so $\gamma_N(0)=0$.

(Another interesting case is the configuration of a spring and damper attached in parallel. In that case, the assumption of a step increase in strain becomes completely unviable!)

The idealized viscous liquid (represented by the lumped-component damper/dashpot in the Maxwell model) has infinite stiffness for instantaneous movements. You state this yourself in your constitutive law relating $\sigma_N(t)$ and $\dot\gamma_N(t)$; a step increase in $\gamma_N$ results in $\sigma_N\to\infty$. Therefore, the spring must take up the entire initial displacement in the case of a step strain, and so $\gamma_N(0)=0$.

(Another interesting case is the configuration of a spring and damper attached in parallel. In that case, the assumption of a step increase in strain becomes completely unviable!)

The idealized viscous liquid (represented by the lumped-component damper/dashpot in the Maxwell model) has infinite stiffness for instantaneous movements. You state this yourself in your constitutive law relating $\sigma_N(t)$ and $\dot\gamma_N(t)$; a step increase in $\gamma_N$ results in $\sigma_N\to\infty$.

This result is nonsensical because the load on two components in series must be equal, but the spring cannot sustain an infinite load; the other constitutive law implies that the maximum load that can be applied by the spring is $\gamma (0) G$.

Therefore, the spring must take up the entire initial displacement in the case of a step strain, and so $\gamma_N(0)=0$.

(Another interesting case is the configuration of a spring and damper attached in parallel. In that case, the assumption of a step increase in strain becomes completely unviable!)