Continuum mechanics

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Continuum mechanics
Conservation of mass
Conservation of momentum
Navier–Stokes equations
Tensors
Solid mechanics
Solids · Stress · Deformation · Finite strain theory · Infinitesimal strain theory · Elasticity · Linear elasticity · Plasticity · Viscoelasticity · Hooke's law · Rheology
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Fluids · Fluid statics
Fluid dynamics · Viscosity · Newtonian fluids
Non-Newtonian fluids
Surface tension
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Newton · Stokes · Navier · Cauchy· Hooke
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Classical mechanics
\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})Newton's Second Law
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Energy · Momentum
Formulations
Newtonian mechanics
Lagrangian mechanics
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Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and mechanical behavior of materials modeled as a continuum, e.g., solids and fluids (i.e., liquids and gases). A continuum concept assumes that the substance of the body is distributed throughout — and completely fills — the space it occupies.

The continuum concept ignores the fact that matter is made of atoms, is not continuous, and that it commonly has some sort of heterogeneous microstructure, allowing the approximation of physical quantities, such as energy and momentum, at the infinitesimal limit. Differential equations can thus be employed in solving problems in continuum mechanics. Some of these differential equations are specific to the materials being investigated and are called constitutive equations, while others capture fundamental physical laws, such as conservation of mass (continuity equation), the conservation of momentum (equations of motion and equilibrium), and energy (first law of thermodynamics).

Continuum mechanics deals with physical quantities of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical quantities are then represented by tensors, which are mathematical objects that are independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience.

In fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made.

Contents

The continuum concept

Materials, such as solids, liquids and gases, are composed of molecules separated by empty space. Additionally, in a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modeled assuming materials as a continuum, i.e. the matter in the body is continuously distributed filling all the region of space it occupies. A continuum is a body that can be continually sub-divided into infinitesimal small elements with properties being those of the bulk material.

The concept of continuum is a macroscopic physical model, and its validity depends on the type of problem and the scale of the physical phenomena under consideration. A material may be assumed as a continuum when the distance between the real physical particles is very small compared to the dimension of the problem. For example, such is the case when analyzing the deformation behavior of soil deposits, i.e. settlement under a foundation, in soil mechanics. A given volume of soil is generally formed by discrete solid particles (grains) of minerals which are packed in a certain manner leaving voids between them, i.e. granular media. In this sense, soils defeat the definition of a continuum. However, in order to simplify the deformation analysis of the soil, the volume of soil can be assumed as a continuum knowing that the dimensions of particular grain particles are very small compared with the scale of the problem, i.e. the size of the foundation and the volume of the soil mass that is influenced by the foundation load (meters) is greater than the particular soil particles (millimeters).

The validity of the continuum assumption needs to be verified with experimental testing and measurements on the real material under consideration and under similar loading conditions.

Mathematical modeling of a continuum

Figure 1. Configuration of a continuum body Figure 1. Configuration of a continuum body \mathcal B\ \kappa_t(\mathcal B)
\ \mathbf{x}=\kappa_t(\mathbf X)
\ \kappa_t(\cdot)

Kinematics: deformation and motion

Figure 2. Motion of a continuum body. Figure 2. Motion of a continuum body. \ \kappa_0(\mathcal B)

The motion of a continuum body is a continuous time sequence of displacements. Thus, the material body will occupy different configurations at different times so that a particle occupies a series of points in space which describe a pathline.

There is continuity during deformation or motion of a continuum body in the sense that:

\ t=0

When analyzing the deformation or motion of solids, or the flow of fluids, it is necessary to describe the sequence or evolution of configurations throughout time. One description for motion is made in terms of the material or referential coordinates, called material description or Lagrangian description.

Lagrangian description

\ t=0\ \chi(\cdot)
\ \mathbf x=\chi(\mathbf X, t)
\kappa_0(\mathcal B)\ P_{ij\ldots}\ P_{ij\ldots}\ P_{ij\ldots}
\ \frac{d}{dt}[P_{ij\ldots}(\mathbf X,t)]=\frac{\partial}{\partial t}[P_{ij\ldots}(\mathbf X,t)]
\ \mathbf x
\ \mathbf v = \mathbf \dot x =\frac{d\mathbf x}{dt}=\frac{\partial \chi(\mathbf X,t)}{\partial t}

Similarly, the acceleration field is given by

\ \mathbf a= \mathbf \dot v = \mathbf \ddot x =\frac{d^2\mathbf x}{dt^2}=\frac{\partial^2 \chi(\mathbf X,t)}{\partial t^2}
\chi(\cdot)

Eulerian description

\chi(\cdot)\kappa_t(\mathcal B)

Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function

\mathbf X=\chi^{-1}(\mathbf x, t)
\mathbf x

A necessary and sufficient condition for this inverse function to exist is that the determinant of the Jacobian should be different from zero. Thus,

\ J=\left | \frac{\partial \chi_i}{\partial X_J} \right |=\left | \frac{\partial x_i}{\partial X_J} \right |\neq0
\ P_{ij\ldots}
\ P_{ij \ldots}=P_{ij\ldots}(\mathbf X,t)=P_{ij\ldots}[\chi^{-1}(\mathbf x,t),t]=p_{ij\ldots}(\mathbf x,t)
\ P_{ij \ldots}\ p_{ij\ldots}(\mathbf x,t)
\ \frac{d}{dt}[p_{ij\ldots}(\mathbf x,t)]=\frac{\partial}{\partial t}[p_{ij\ldots}(\mathbf x,t)]+ \frac{\partial}{\partial x_k}[p_{ij\ldots}(\mathbf x,t)]\frac{dx_k}{dt}
\ p_{ij\ldots}(\mathbf x,t)\ \mathbf x

Displacement Field

\ P

A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field, In general, the displacement field is expressed in terms of the material coordinates as

\ \mathbf u(\mathbf X,t) = \mathbf b+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J

or in terms of the spatial coordinates as

\ \mathbf U(\mathbf x,t) = \mathbf b+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,
\ \alpha_{Ji}
\ \mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}
\ u_i
\ u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i

Knowing that

\ \mathbf e_i = \alpha_{iJ}\mathbf E_J

then

\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)
\ \mathbf b=0
\ \mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}

Thus, we have

\ \mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J

or in terms of the spatial coordinates as

\ \mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J


Fundamental laws

Conservation of mass

Conservation of momentum

Conservation of energy

Constitutive equations

Applications

Continuum mechanics Solid mechanics is the study of the physics of continuous solids with a defined rest shape. Elasticity (physics) describes materials that return to their rest shape after removal of an applied force.
Plasticity describes materials that permanently deform (change their rest shape) after a large enough applied force. Rheology: Given that some materials are viscoelastic (exhibiting a combination of elastic and viscous properties), the boundary between solid mechanics and fluid mechanics is blurry.
Fluid mechanics (including Fluid statics and Fluid dynamics) deals with the physics of fluids. An important property of fluids is viscosity, which is the force generated by a fluid in response to a velocity gradient. Non-Newtonian fluids
Newtonian fluids