Governing Equations

Solid Mechanics studies the displacement of a solid continuum under prescribed loads and constraints. The fundamental laws that govern the mechanics of solids are the same laws that describe the mechanics of fluids, namely, the conservation of mass, linear momentum, angular momentum, and energy.

The conservation laws, Eqn. (654), Eqn. (655), and Eqn. (658) are written in Eulerian form. In Solid Mechanics, it is more natural to express the conservation laws using a Lagrangian approach, where the observer follows the solid material as it moves through space and time.

Conservation of Mass

In the Lagrangian approach, mass is always conserved. The mass that is contained in any deformed volume is the same mass that was originally contained in the undeformed volume:

Figure 1. EQUATION_DISPLAY

M = \int_{V} ρ (t) d V = \int_{V_{0}} ρ_{0} d V = c o n s t

(4456)

Since the mass within the volume is conserved, volume changes result in density changes. In fact, this leads to a slightly different interpretation of the material density specified in Simcenter STAR-CCM+. The specified density is the material density in the undeformed configuration, at a reference temperature. If $M_{0}$ is the mass contained in a volume $V_{0}$ , with density $ρ_{0}$ , the density in the deformed configuration is:

Figure 2. EQUATION_DISPLAY

ρ (V, T) = \frac{M_{0}}{V (T)} = \frac{\int_{V_{0}} ρ_{0} d V}{\int_{V (T)} d V}

(4457)

where $T$ is the current temperature.

For example, for a linear isotropic elastic material (see Eqn. (4511)) and infinitesimal strain $ε$ (Eqn. (4444)), the change of volume relative to the deformed volume is:

Figure 3. EQUATION_DISPLAY

\frac{V - V_{0}}{V} = ε = α (T - T_{r e f}) + \frac{σ_{m}}{K}

(4458)

where $K$ is the bulk modulus (Eqn. (4512)) and $σ_{m}$ is the mean stress (Eqn. (4438)). The density is:

Figure 4. EQUATION_DISPLAY

ρ (σ_{m}, T) = ρ_{0} (1 - α (T - T_{r e f}) - \frac{σ_{m}}{K})

(4459)

where $T_{r e f}$ is the reference temperature.

Conservation of Momentum: Equation of Motion

The motion of a solid body is governed by Cauchy's equilibrium equation, which expresses the conservation of linear momentum for a continuum. In the Lagrangian approach, the convective term in Eqn. (655) vanishes and the time derivative of the velocity reduces to the partial second derivative of the displacement:

Figure 5. EQUATION_DISPLAY

ρ \ddot{u} - \nabla\cdot σ - b = 0

(4460)

where:

$u$ is the displacement of the solid body
$b$ is the total body force per unit volume
the Cauchy stress tensor $σ$ is symmetric (see Eqn. (657))

When the solid structure has prescribed rotations and translations, Simcenter STAR-CCM+ also accounts for the elastic deformation caused by inertia forces. For more information, see Conservation Equations with Mesh Motion.

Eqn. (4460) is subject to Dirichlet and Neumann boundary conditions, which in Solid Mechanics are referred to as constraints and loads:

Figure 6. EQUATION_DISPLAY

\begin{array}{l} u = \bar{u} & Dirichlet b.c. (constraint) \\ τ = σ \cdot n = \bar{τ} & Neumann b.c. (load) \end{array}

(4461)

where $\bar{u}$ is a prescribed displacement on a surface $Γ_{u}$ , and $\bar{τ}$ is a prescribed traction vector on a surface $Γ_{τ}$ , with surface normal $n$ . In Simcenter STAR-CCM+, you can also apply constraints and forces at points.

Principle of Virtual Work

An alternative formulation of the equilibrium problem is the principle of virtual work, which is particularly suitable for discretization by the Finite Element method. To derive the principle of virtual work, Eqn. (4460) is multiplied by a test function $δ u$ and integrated over the space domain:

Figure 7. EQUATION_DISPLAY

δ Π = \int_{V} δ u \cdot (ρ \ddot{u} - \nabla\cdot σ - b) d V = 0

(4462)

This formulation uses as test function the virtual displacement, $δ u$ , which must satisfy the Dirichlet boundary conditions. After integrating by parts and introducing Neumann boundary conditions, Eqn. (4462) reduces to:

Figure 8. EQUATION_DISPLAY

δ Π = \int_{V} δ u \cdot ρ \ddot{u} d V - \int_{V} δ u \cdot b d V + \int_{V} δ e : σ d V - \int_{Γ_{τ}} δ u \cdot \bar{τ} d Γ = 0

(4463)

where

δ e : σ

is the variation of the strain energy per unit volume induced by stress. Eqn. (4463) assumes the following boundary conditions:

Dirichlet boundary condition (that is, prescribed displacement) $δ u = 0$ at the surface $Γ_{u}$ .
Neumann boundary condition (that is, prescribed surface traction) $τ = \bar{τ}$ at the surface $Γ_{τ}$ . For simplicity, Eqn. (4463) only includes surface boundary conditions. Neumann boundary conditions at points are included in the discretized equation, Eqn. (4556).
The remaining surfaces $Γ_{r} = Γ - Γ_{u} - Γ_{τ}$ are free from loads and constraints.

The differential equation, Eqn. (4460), is called a strong form. The principle of virtual work, Eqn. (4463), is called a weak form, as the equations are satisfied in an integral, or weak, sense.

Eqn. (4463) is written in the current configuration. Most Solid Mechanics formulations use a total Lagrangian approach, where all quantities are specified in the initial configuration. In the infinitesimal strain approximation, there is no distinction between the initial and current configuration.

The stress-induced variation of the strain energy, $δ e : σ$ , can be expressed in the initial configuration as $δ E : S$ , where $δ E$ is the variation of the Green-Lagrange strain and $S$ is the 2nd Piola-Kirchhoff stress (see Eqn. (4453)).

In the initial configuration, Eqn. (4463) becomes:

Figure 9. EQUATION_DISPLAY

δ Π = \int_{V_{0}} δ u \cdot ρ_{0} \ddot{u} d V - \int_{V_{0}} δ u \cdot b d V + \int_{V_{0}} δ E : S d V - \int_{Γ_{τ}} δ u \cdot \bar{τ} d Γ = 0

(4464)

where:

$ρ_{0}$ and $V_{0}$ are the initial density and the initial undeformed volume
the surface integral is left in the current configuration, for convenience

Eqn. (4464) satisfies conservation of linear and angular momentum. The Lagrangian approach also ensures that mass is conserved.