What is this symbol? (MPM notes on conservation of momentum)

Question

I am trying to read some MPM notes. On section 7 there is a formula on the conservation of momentum. $$ R(X, 0) \frac{\partial V}{\partial t} = \nabla^X \cdot P + R(X, 0)g $$

There are 2 things I don't get, the left makes sense, it's mass times velocity, i.e. momentum. But the right is nonsensical to me.

First what is $\nabla^X\cdot$ as an operator? That almost looks like it is computing the divergence relative to the material coordinates $X$, but then I don't understand the equation at all. The left hand side is a vector, the divergence of the pressure is a scalar quantity, so already the types don;t match. And I cannot understand for the life of me what $g$ is supposed to be either.

Answer 1

This is the Lagrangian equation of motion for deformable continuous media. Heuristically, it can by viewed as an expression of Newton's second law of motion per unit volume. Term by term,

• $R(\boldsymbol{X},0)\, \partial_t \boldsymbol{V}$ is the product between the initial mass density $R(\boldsymbol{X},0)$ in Lagrangian coordinates* and the acceleration $\partial_t \boldsymbol{V}$, where $\boldsymbol{V}$ represents the particle velocity. Thus, this quantity describes inertia per unit volume.
• $\nabla^\boldsymbol{X} \cdot \boldsymbol{P}$ is the Lagrangian divergence of the first Piola-Kirchhoff stress tensor $\boldsymbol{P}$. The latter can be represented as a matrix, so that its divergence is a vector. This term is a cohesion force that is internal to the material.
• $R(\boldsymbol{X},0)\, \boldsymbol{g}$ is the product between the initial mass density and the body forces per unit mass $\boldsymbol{g}$ (e.g., the acceleration of gravity). Therefore it is an external force, see Section 7.2 of the lecture notes.

*Here, the symbol $\boldsymbol{X}$ represents the position of a particle in the initial undeformed state. This space coordinate is commonly used in solid mechanics.