In linear elasticity theory, recall that the Cauchy stress field \(\sigma(\textbf x, t)\) is defined as the unique linear transformation that maps any unit vector \(\hat{\textbf n}\) (thought of as being rooted at position \(\textbf x\) in a material at time \(t\)) to the corresponding stress vector \(\boldsymbol{\sigma}_{\hat{\textbf n}}(\textbf x,t):=\sigma(\textbf x, t)\hat{\textbf n}\) acting on the plane normal to \(\hat{\textbf n}\) (in nonlinear elasticity theory, the Cauchy stress field is superseded by the Piola-Kirchoff stress fields which will not be discussed here). Note that in general the stress vector \(\boldsymbol{\sigma}_{\hat{\textbf n}}(\textbf x,t)\) need not be parallel to the normal \(\hat{\textbf n}\) since there are typically both normal and shear stresses (however, when the stress vector \(\boldsymbol{\sigma}_{\hat{\textbf n}}(\textbf x,t)\) is parallel to the normal \(\hat{\textbf n}\), which corresponds physically to the case of only normal stress and so no shear stress, then \(\hat{\textbf n}\) is thus an eigenvector of \(\sigma(\textbf x, t)\) with eigenvalue a principal stress field).
Newton’s Laws in a Continuum
As usual, the key idea is to exploit the system nitpick trick and then apply Newton’s second law to the resultant system to obtain its dynamics. In this case, the right system to look at is an arbitrary comoving control volume \(V(t)\) (e.g. a fluid or solid parcel) whose boundary \(\partial V(t)\) moves with the continuum \(\textbf v_{\partial V(t)}(\textbf x,t)=\textbf v(\textbf x,t)\) for all \(\textbf x\in\partial V(t)\) and times \(t\), and thus whose mass is conserved (note that this is not an if and only if statement; having a comoving control volume is a strictly stronger property than mass conservation). Newton’s second law reads \(\dot{\textbf P}=\textbf F^{\text{ext}}\) for the system \(V(t)\).
First, handle the purely mathematical details of the kinematic (left-hand) side. Clearly, \(\textbf P=\iiint_{\textbf x\in V(t)}\rho(\textbf x,t)\textbf v(\textbf x,t)d^3 x\). Taking \(d/dt\), one can differentiate under the integral sign (see Leibniz integral rule and Reynolds transport theorem) to obtain:
$$\dot{\textbf P}=\iiint_{\textbf x\in V(t)}\frac{\partial}{\partial t}\left(\rho(\textbf x,t)\textbf v(\textbf x,t)\right)d^3 x+\iint_{\textbf x\in\partial V(t)}\rho(\textbf x,t)\textbf v(\textbf x,t)\left(\textbf v_{\partial V}(\textbf x, t)\cdot d^2\textbf x\right)$$
Applying the comoving assumption \(\textbf v_{\partial V(t)}(\textbf x,t)=\textbf v(\textbf x,t)\) and the divergence theorem yields a single volume integral expression (suppressing all the \(\textbf x\) and \(t\) symbols):
$$\dot{\textbf P}=\iiint_{V}\left(\frac{\partial(\rho\textbf v)}{\partial t}+\textbf v\cdot\frac{\partial(\rho \textbf v)}{\partial\textbf x}+\rho\textbf v\left(\frac{\partial}{\partial\textbf x}\cdot\textbf v\right)\right)d^3 x$$
where the advection term can also be expressed via a Jacobian \(\textbf v\cdot\frac{\partial(\rho \textbf v)}{\partial\textbf x}=\frac{\partial(\rho\textbf v)}{\partial\textbf x}\textbf v\). Defining the mass flux \(\textbf J:=\rho\textbf v\), this can be written more compactly via the material derivative as:
$$\dot{\textbf P}=\iiint_V\left(\frac{D\textbf J}{Dt}+\textbf J\left(\frac{\partial}{\partial\textbf x}\cdot\textbf v\right)\right)d^3 x$$
Now turn to the actual physics, the external forces \(\textbf F^{\text{ext}}\) acting on the comoving control volume \(V\). Conceptually, it is helpful to split \(\textbf F^{\text{ext}}=\textbf F^{\text{ext}}_{\partial V}+\textbf F^{\text{ext}}_V\) into external surface forces \(\textbf F^{\text{ext}}_{\partial V}\) acting exclusively on the boundary surface \(\partial V\) of \(V\) and external body forces \(\textbf F^{\text{ext}}_V\) acting throughout \(V\) (right now these forces are all external relative to the system \(V\), but once the entire solid or fluid is taken to be the system, then some of these forces will instead become internal forces). Clearly, each of these components of the total external force are given by:
$$\textbf F^{\text{ext}}_{\partial V}=\iint_{\partial V}\boldsymbol{\sigma}d^2 x$$
$$\textbf F^{\text{ext}}_{V}=\iiint_V\textbf f^{\text{ext}}_Vd^3 x$$
where \(\boldsymbol{\sigma}\) is the stress vector as before and \(\textbf f^{\text{ext}}_V\) the external body force density. Finally, substitute \(\boldsymbol{\sigma}=\sigma\hat{\textbf n}\) and note that \(d^2\textbf x=\hat{\textbf n}d^2x\) so applying the divergence theorem converts it too into a volume integral:
$$\textbf F^{\text{ext}}=\iiint_V\left(\frac{\partial}{\partial\textbf x}\cdot\sigma+\textbf f^{\text{ext}}_V\right)d^3x$$
This finally leads to the differential form (because the control volume \(V\) was arbitrary):
$$\frac{D\textbf J}{Dt}+\textbf J\left(\frac{\partial}{\partial\textbf x}\cdot\textbf v\right)=\frac{\partial}{\partial\textbf x}\cdot\sigma+\textbf f^{\text{ext}}$$
For Mode I plane stress crack loading, an external stress \(\boldsymbol{sigma}_{\text{ext}}\) is applied which induces a uniform internal stress field \(\sigma\) in the material. A crack of “semi-major axis” \(a\) will grow/propagate iff \(a>a^*=\frac{EV’_I}{\pi\sigma^2_{\text{ext}}}\). For brittle materials, \(\mathcal G^*=2\gamma\) (I don’t like the notation, but it’s so common I’ll just use it; maybe think Gibbs free energy is also a sort of “energy released”?), but for ductile materials the potential energetic penalty \(\mathcal G^*\) is generally far greater due the formation of a plastic zone about the crack tip \(\sigma_{\text{int}}=\sigma^*\) which requires external work \(W_{\text{ext}}=\frac{(\sigma^*)^2}{2E}\) to form (since it’s plastic deformation just like doing any other kind of deformation! Or think about the dislocations?). This blunts the crack tip curvature. The punchline is that ductile = tough! Also, fracture toughness \(\mathcal K^*:=\sqrt{E\mathcal G^*}\) (idk why the K notation). So basically, shit breaks when you drop it on the floor for instance because at the moment of collision the high stress \(\sigma_{\text{ext}}\) lowered the critical crack size \(a^*\propto\frac{1}{\sigma_{\text{ext}}^2}\) so much that the natural cracks \(a\) always present as defects in the material (dislocations are the other key kind of defect) exceeded \(a^*\), and therefore by the Griffith criterion it was energetically favorable for them to grow/propagate, leading to fracture! (although depends on brittle vs. ductile again).
A material is said to be viscoelastic iff it exhibits a duality between behaving as a Newtonian fluid (with some dynamic viscosity \(\eta\), hence the visco part of “viscoelastic”) and a linear elastic solid (with some Young’s modulus \(E\), hence the elastic part of “viscoelastic”) (cf. with wave-particle duality in quantum mechanics). There are various ways to make this duality precise, for instance the simplest is to model a viscoelastic material via the Maxwell model, namely a viscous damper (also called a dashpot) of dynamic viscosity \(\eta\) in series with a linear elastic spring of Young’s modulus \(E\). By Newton’s second law (analog of Kirchoff’s laws for electric circuits), this gives rise to a constitutive relation between the external stress \(\sigma_{\text{ext}}\) applied to the viscoelastic Maxwell material and its total elastic strain \(\varepsilon\) as:
$$\dot{\varepsilon}=\frac{\dot{\sigma}_{\text{ext}}}{E}+\frac{\sigma_{\text{ext}}}{\eta}$$
This is not really a differential equation, it’s just saying if you know a priori what the applied external stress \(\sigma_{\text{ext}}=\sigma_{\text{ext}}(t)\) is, then by time integration one can recover the corresponding elastic strain \(\varepsilon(t)\) experienced by the sample. This viscoelastic duality is directly responsible for loading-unloading hysteresis of materials (i.e. the damper dissipates heat energy, corresponding to the enclosed area on a hysteresis loop).