Continuum mechanics is a branch of

:$\backslash mathbf\; x\; =\; \backslash sum\_^3\; x\_i\; \backslash mathbf\; e\_i,$ where $\backslash mathbf\; e\_i$ are the coordinate vectors in some

"Objectivity in classical continuum mechanics: Motions, Eulerian and Lagrangian functions; Deformation gradient; Lie derivatives; Velocity-addition formula, Coriolis; Objectivity"

by Gilles Leborgne, April 7, 2021

"Part IV Velocity-addition formula and Objectivity"

{{DEFAULTSORT:Continuum Mechanics Classical mechanics

mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...

that deals with the mechanical behavior of materials modeled as a continuous mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...

rather than as discrete particles. The French mathematician Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...

was the first to formulate such models in the 19th century.
Explanation

A continuum model assumes that the substance of the object fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. These models can be used to derive differential equations that describe the behavior of such objects using physical laws, such as mass conservation, momentum conservation, and energy conservation, and some information about the material is provided by constitutive relationships. Continuum mechanics deals with the physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. The physical properties are then represented by tensors, which are mathematical objects with the property of being independent of coordinate systems. Coordinate systems allow these tensors to be expressed computationally.Concept of a continuum

Space separates molecules that make up solids, liquids, and gases. Materials have cracks and discontinuities on a microscopic level. Physical phenomena can, however, be modeled if the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire space it occupies. A continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material. The validity of the continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the microstructure exist. More specifically, the continuum hypothesis/assumption hinges on the concepts of a representative elementary volume and separation of scales based on the Hill–Mandel condition. This condition provides a link between an experimentalist's and a theoretician's viewpoint on constitutive equations (linear and nonlinear elastic/inelastic or coupled fields) as well as a way of spatial and statistical averaging of the microstructure. When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than the size of the representative volume element (RVE), a statistical volume element (SVE) is employed, which results in random continuum fields. The latter then provide a micromechanics basis for stochastic finite elements (SFE). The levels of SVE and RVE link continuum mechanics to statistical mechanics. Experimentally, the RVE can only be evaluated when the constitutive response is spatially homogenousCar traffic as an introductory example

Consider car traffic on a highway, with just one lane for simplicity. Somewhat surprisingly, and in a tribute to its effectiveness, continuum mechanics effectively models the movement of cars via a partial differential equation (PDE) for the density of cars. The familiarity of this situation empowers us to understand a little of the continuum-discrete dichotomy underlying continuum modelling in general. To start modelling define that: $x$ measures distance (in km) along the highway; $t$ is time (in minutes); $\backslash rho(x,t)$ is the density of cars on the highway (in cars/km in the lane); and $u(x,t)$ is theflow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...

(average velocity) of those cars 'at' position $x$.
Conservation derives a PDE ( partial differential equation)

Cars do not appear and disappear. Consider any group of cars: from the particular car at the back of the group located at $x=a(t)$ to the particular car at the front located at $x=b(t)$. The total number of cars in this group is $N\; =\; \backslash int\_^\; \backslash rho(x,t)\; \backslash ,\; dx$. Since cars are conserved (if there is overtaking, then the 'car at the front / back' may become a different car) $dN\; /\; dt\; =\; 0$. But via theLeibniz integral rule
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
\int_^ f(x,t)\,dt,
where -\infty < a(x), b(x) < \infty and the integral are

:$\backslash begin\; \backslash frac\; \&=\; \backslash frac\; \backslash int\_^\; \backslash rho(x,t)\backslash ,dx\; \backslash \backslash \; \&=\backslash int\_^\; \backslash frac\backslash ,dx\; +\; \backslash rho(b,t)\; \backslash frac\; -\; \backslash rho(a,t)\backslash frac\; \backslash \backslash \; \&=\backslash int\_^\; \backslash frac\backslash ,dx\; +\backslash rho(b,t)u(b,t)-\backslash rho(a,t)u(a,t)\; \backslash \backslash \; \&=\backslash int\_^\; \backslash left;\; href="/html/ALL/l/\backslash frac\_+\_\backslash frac(\backslash rho\_u)\_\backslash right.html"\; ;"title="\backslash frac\; +\; \backslash frac(\backslash rho\; u)\; \backslash right">\backslash frac\; +\; \backslash frac(\backslash rho\; u)\; \backslash right$
This integral being zero holds for all groups, that is, for all intervals $;\; href="/html/ALL/l/,b.html"\; ;"title=",b">,b$Observation closes the problem

The previous PDE is one equation with two unknowns, so another equation is needed to form a well-posed problem. Such an extra equation is typically needed in continuum mechanics and typically comes from experiments. For car traffic it is well established that cars typically travel at a speed depending upon density, $u=V(\backslash rho)$ for some experimentally determined function $V$ that is a decreasing function of density. For example, experiments in theLincoln Tunnel
The Lincoln Tunnel is an approximately tunnel under the Hudson River, connecting Weehawken, New Jersey, to the west with Midtown Manhattan in New York City to the east. It carries New Jersey Route 495 on the New Jersey side and unsigned New ...

found that a good fit (except at low density) is obtained by $u=V(\backslash rho)=27.5\backslash ln(142/\backslash rho)$ (km/hr for density in cars/km).
Thus, the basic continuum model for car traffic is the PDE
:$\backslash frac+\; \backslash frac;\; href="/html/ALL/l/rho\_V(\backslash rho).html"\; ;"title="rho\; V(\backslash rho)">rho\; V(\backslash rho)$
for the car density $\backslash rho(x,t)$ on the highway.
Major areas

An additional area of continuum mechanics comprises elastomeric foams, which exhibit a curious hyperbolic stress-strain relationship. The elastomer is a true continuum, but a homogeneous distribution of voids gives it unusual properties.Formulation of models

Continuum mechanics models begin by assigning a region in three-dimensionalEuclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

to the material body $\backslash mathcal\; B$ being modeled. The points within this region are called particles or material points. Different ''configurations'' or states of the body correspond to different regions in Euclidean space. The region corresponding to the body's configuration at time $t$ is labeled $\backslash kappa\_t(\backslash mathcal\; B)$.
A particular particle within the body in a particular configuration is characterized by a position vector :$\backslash mathbf\; x\; =\; \backslash sum\_^3\; x\_i\; \backslash mathbf\; e\_i,$ where $\backslash mathbf\; e\_i$ are the coordinate vectors in some

frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathem ...

chosen for the problem (See figure 1). This vector can be expressed as a function of the particle position $\backslash mathbf\; X$ in some ''reference configuration'', for example the configuration at the initial time, so that
:$\backslash mathbf=\backslash kappa\_t(\backslash mathbf\; X).$
This function needs to have various properties so that the model makes physical sense. $\backslash kappa\_t(\backslash cdot)$ needs to be:
* continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...

in time, so that the body changes in a way which is realistic,
* globally invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that i ...

at all times, so that the body cannot intersect itself,
* orientation-preserving
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed ...

, as transformations which produce mirror reflections are not possible in nature.
For the mathematical formulation of the model, $\backslash kappa\_t(\backslash cdot)$ is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.
Forces in a continuum

Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. A solid is a deformable body that possesses shear strength, ''sc.'' a solid can support shear forces (forces parallel to the material surface on which they act). Fluids, on the other hand, do not sustain shear forces. For the study of the mechanical behavior of solids and fluids these are assumed to be continuous bodies, which means that the matter fills the entire region of space it occupies, despite the fact that matter is made of atoms, has voids, and is discrete. Therefore, when continuum mechanics refers to a point or particle in a continuous body it does not describe a point in the interatomic space or an atomic particle, rather an idealized part of the body occupying that point. Following the classical dynamics of Newton andEuler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in m ...

, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces $\backslash mathbf\; F\_C$ and body forces $\backslash mathbf\; F\_B$. Thus, the total force $\backslash mathcal\; F$ applied to a body or to a portion of the body can be expressed as:
:$\backslash mathcal\; F\; =\; \backslash mathbf\; F\_C\; +\; \backslash mathbf\; F\_B$
Surface forces

'' Surface forces'' or ''contact forces'', expressed as force per unit area, can act either on the bounding surface of the body, as a result of mechanical contact with other bodies, or on imaginary internal surfaces that bound portions of the body, as a result of the mechanical interaction between the parts of the body to either side of the surface ( Euler-Cauchy's stress principle). When a body is acted upon by external contact forces, internal contact forces are then transmitted from point to point inside the body to balance their action, according toNewton's third law of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motio ...

of conservation of linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...

and angular momentum (for continuous bodies these laws are called the Euler's equations of motion). The internal contact forces are related to the body's deformation through constitutive equations
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and appro ...

. The internal contact forces may be mathematically described by how they relate to the motion of the body, independent of the body's material makeup.
The distribution of internal contact forces throughout the volume of the body is assumed to be continuous. Therefore, there exists a ''contact force density'' or ''Cauchy traction field'' $\backslash mathbf\; T(\backslash mathbf\; n,\; \backslash mathbf\; x,\; t)$ that represents this distribution in a particular configuration of the body at a given time $t\backslash ,\backslash !$. It is not a vector field because it depends not only on the position $\backslash mathbf\; x$ of a particular material point, but also on the local orientation of the surface element as defined by its normal vector $\backslash mathbf\; n$.
Any differential area $dS\backslash ,\backslash !$ with normal vector $\backslash mathbf\; n$ of a given internal surface area $S\backslash ,\backslash !$, bounding a portion of the body, experiences a contact force $d\backslash mathbf\; F\_C\backslash ,\backslash !$ arising from the contact between both portions of the body on each side of $S\backslash ,\backslash !$, and it is given by
:$d\backslash mathbf\; F\_C=\; \backslash mathbf\; T^\backslash ,dS$
where $\backslash mathbf\; T^$ is the ''surface traction'', also called ''stress vector'', ''traction'', or ''traction vector''. The stress vector is a frame-indifferent vector (see Euler-Cauchy's stress principle).
The total contact force on the particular internal surface $S\backslash ,\backslash !$ is then expressed as the sum (surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one m ...

) of the contact forces on all differential surfaces $dS\backslash ,\backslash !$:
:$\backslash mathbf\; F\_C=\backslash int\_S\; \backslash mathbf\; T^\backslash ,dS$
In continuum mechanics a body is considered stress-free if the only forces present are those inter-atomic forces ( ionic, metallic, and van der Waals force
In molecular physics, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and ...

s) required to hold the body together and to keep its shape in the absence of all external influences, including gravitational attraction. Stresses generated during manufacture of the body to a specific configuration are also excluded when considering stresses in a body. Therefore, the stresses considered in continuum mechanics are only those produced by deformation of the body, ''sc.'' only relative changes in stress are considered, not the absolute values of stress.
Body forces

'' Body forces'' are forces originating from sources outside of the body that act on the volume (or mass) of the body. Saying that body forces are due to outside sources implies that the interaction between different parts of the body (internal forces) are manifested through the contact forces alone. These forces arise from the presence of the body in force fields, ''e.g.''gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...

(gravitational force
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...

s) or electromagnetic field (electromagnetic force
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...

s), or from inertial forces when bodies are in motion. As the mass of a continuous body is assumed to be continuously distributed, any force originating from the mass is also continuously distributed. Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body, ''i.e.'' acting on every point in it. Body forces are represented by a body force density $\backslash mathbf\; b(\backslash mathbf\; x,\; t)$ (per unit of mass), which is a frame-indifferent vector field.
In the case of gravitational forces, the intensity of the force depends on, or is proportional to, the mass density $\backslash mathbf\; \backslash rho\; (\backslash mathbf\; x,\; t)\backslash ,\backslash !$ of the material, and it is specified in terms of force per unit mass ($b\_i\backslash ,\backslash !$) or per unit volume ($p\_i\backslash ,\backslash !$). These two specifications are related through the material density by the equation $\backslash rho\; b\_i\; =\; p\_i\backslash ,\backslash !$. Similarly, the intensity of electromagnetic forces depends upon the strength (electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respecti ...

) of the electromagnetic field.
The total body force applied to a continuous body is expressed as
:$\backslash mathbf\; F\_B=\backslash int\_V\backslash mathbf\; b\backslash ,dm=\backslash int\_V\; \backslash rho\backslash mathbf\; b\backslash ,dV$
Body forces and contact forces acting on the body lead to corresponding moments of force (torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of t ...

s) relative to a given point. Thus, the total applied torque $\backslash mathcal\; M$ about the origin is given by
:$\backslash mathcal\; M=\; \backslash mathbf\; M\_C\; +\; \backslash mathbf\; M\_B$
In certain situations, not commonly considered in the analysis of the mechanical behavior of materials, it becomes necessary to include two other types of forces: these are ''couple stresses'' (surface couples, contact torques) and ''body moments''. Couple stresses are moments per unit area applied on a surface. Body moments, or body couples, are moments per unit volume or per unit mass applied to the volume of the body. Both are important in the analysis of stress for a polarized dielectric solid under the action of an electric field, materials where the molecular structure is taken into consideration (''e.g.'' bones), solids under the action of an external magnetic field, and the dislocation theory of metals.
Materials that exhibit body couples and couple stresses in addition to moments produced exclusively by forces are called ''polar materials''. ''Non-polar materials'' are then those materials with only moments of forces. In the classical branches of continuum mechanics the development of the theory of stresses is based on non-polar materials.
Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) in the body can be given by
:$\backslash mathcal\; F\; =\; \backslash int\_V\; \backslash mathbf\; a\backslash ,dm\; =\; \backslash int\_S\; \backslash mathbf\; T\backslash ,dS\; +\; \backslash int\_V\; \backslash rho\backslash mathbf\; b\backslash ,dV$
:$\backslash mathcal\; M\; =\; \backslash int\_S\; \backslash mathbf\; r\; \backslash times\; \backslash mathbf\; T\backslash ,dS\; +\; \backslash int\_V\; \backslash mathbf\; r\; \backslash times\; \backslash rho\backslash mathbf\; b\backslash ,dV$
Kinematics: motion and deformation

A change in the configuration of a continuum body results in adisplacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
* Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...

. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration $\backslash kappa\_0(\backslash mathcal\; B)$ to a current or deformed configuration $\backslash kappa\_t(\backslash mathcal\; B)$ (Figure 2).
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 path line.
There is continuity during motion or deformation of a continuum body in the sense that:
* The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
* The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.
It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not be one that the body will ever occupy. Often, the configuration at $t=0$ is considered the reference configuration, $\backslash kappa\_0\; (\backslash mathcal\; B)$. The components $X\_i$ of the position vector $\backslash mathbf\; X$ of a particle, taken with respect to the reference configuration, are called the material or reference coordinates.
When analyzing the motion or deformation 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

In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time. In this case the reference configuration is the configuration at $t=0$. An observer standing in the frame of reference observes the changes in the position and physical properties as the material body moves in space as time progresses. The results obtained are independent of the choice of initial time and reference configuration, $\backslash kappa\_0(\backslash mathcal\; B)$. This description is normally used insolid mechanics
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ...

.
In the Lagrangian description, the motion of a continuum body is expressed by the mapping function $\backslash chi(\backslash cdot)$ (Figure 2),
:$\backslash mathbf\; x=\backslash chi(\backslash mathbf\; X,\; t)$
which is a mapping of the initial configuration $\backslash kappa\_0(\backslash mathcal\; B)$ onto the current configuration $\backslash kappa\_t(\backslash mathcal\; B)$, giving a geometrical correspondence between them, i.e. giving the position vector $\backslash mathbf=x\_i\backslash mathbf\; e\_i$ that a particle $X$, with a position vector $\backslash mathbf\; X$ in the undeformed or reference configuration $\backslash kappa\_0(\backslash mathcal\; B)$, will occupy in the current or deformed configuration $\backslash kappa\_t(\backslash mathcal\; B)$ at time $t$. The components $x\_i$ are called the spatial coordinates.
Physical and kinematic properties $P\_$, i.e. thermodynamic properties and flow velocity, which describe or characterize features of the material body, are expressed as continuous functions of position and time, i.e. $P\_=P\_(\backslash mathbf\; X,t)$.
The material derivative of any property $P\_$ of a continuum, which may be a scalar, vector, or tensor, is the time rate of change of that property for a specific group of particles of the moving continuum body. The material derivative is also known as the ''substantial derivative'', or ''comoving derivative'', or ''convective derivative''. It can be thought as the rate at which the property changes when measured by an observer traveling with that group of particles.
In the Lagrangian description, the material derivative of $P\_$ is simply the partial derivative with respect to time, and the position vector $\backslash mathbf\; X$ is held constant as it does not change with time. Thus, we have
:$\backslash frac;\; href="/html/ALL/l/\_(\backslash mathbf\_X,t).html"\; ;"title="\_(\backslash mathbf\; X,t)">\_(\backslash mathbf\; X,t)$Eulerian description

Continuity allows for the inverse of $\backslash chi(\backslash cdot)$ to trace backwards where the particle currently located at $\backslash mathbf\; x$ was located in the initial or referenced configuration $\backslash kappa\_0(\backslash mathcal\; B)$. In this case the description of motion is made in terms of the spatial coordinates, in which case is called the spatial description or Eulerian description, i.e. the current configuration is taken as the reference configuration. The Eulerian description, introduced by d'Alembert, focuses on the current configuration $\backslash kappa\_t(\backslash mathcal\; B)$, giving attention to what is occurring at a fixed point in space as time progresses, instead of giving attention to individual particles as they move through space and time. This approach is conveniently applied in the study offluid flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...

where the kinematic property of greatest interest is the rate at which change is taking place rather than the shape of the body of fluid at a reference time.
Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function
:$\backslash mathbf\; X=\backslash chi^(\backslash mathbf\; x,\; t)$
which provides a tracing of the particle which now occupies the position $\backslash mathbf\; x$ in the current configuration $\backslash kappa\_t(\backslash mathcal\; B)$ to its original position $\backslash mathbf\; X$ in the initial configuration $\backslash kappa\_0(\backslash mathcal\; B)$.
A necessary and sufficient condition for this inverse function to exist is that the determinant of the Jacobian Matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables a ...

, often referred to simply as the Jacobian, should be different from zero. Thus,
:$J\; =\; \backslash left,\; \backslash frac\; \backslash \; =\; \backslash left,\; \backslash frac\; \backslash \; \backslash neq\; 0$
In the Eulerian description, the physical properties $P\_$ are expressed as
:$P\_=P\_(\backslash mathbf\; X,t)=P\_;\; href="/html/ALL/l/chi^(\backslash mathbf\_x,t),t.html"\; ;"title="chi^(\backslash mathbf\; x,t),t">chi^(\backslash mathbf\; x,t),t$
where the functional form of $P\_$ in the Lagrangian description is not the same as the form of $p\_$ in the Eulerian description.
The material derivative of $p\_(\backslash mathbf\; x,t)$, using the chain rule, is then
:$\backslash frac;\; href="/html/ALL/l/\_(\backslash mathbf\_x,t).html"\; ;"title="\_(\backslash mathbf\; x,t)">\_(\backslash mathbf\; x,t)$
The first term on the right-hand side of this equation gives the ''local rate of change'' of the property $p\_(\backslash mathbf\; x,t)$ occurring at position $\backslash mathbf\; x$. The second term of the right-hand side is the ''convective rate of change'' and expresses the contribution of the particle changing position in space (motion).
Continuity in the Eulerian description is expressed by the spatial and temporal continuity and continuous differentiability of the flow velocity field. All physical quantities are defined this way at each instant of time, in the current configuration, as a function of the vector position $\backslash mathbf\; x$.
Displacement field

The vector joining the positions of a particle $P$ in the undeformed configuration and deformed configuration is called thedisplacement vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a s ...

$\backslash mathbf\; u(\backslash mathbf\; X,t)=u\_i\backslash mathbf\; e\_i$, in the Lagrangian description, or $\backslash mathbf\; U(\backslash mathbf\; x,t)=U\_J\backslash mathbf\; E\_J$, in the Eulerian description.
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
:$\backslash mathbf\; u(\backslash mathbf\; X,t)\; =\; \backslash mathbf\; b+\backslash mathbf\; x(\backslash mathbf\; X,t)\; -\; \backslash mathbf\; X\; \backslash qquad\; \backslash text\backslash qquad\; u\_i\; =\; \backslash alpha\_b\_J\; +\; x\_i\; -\; \backslash alpha\_X\_J$
or in terms of the spatial coordinates as
:$\backslash mathbf\; U(\backslash mathbf\; x,t)\; =\; \backslash mathbf\; b+\backslash mathbf\; x\; -\; \backslash mathbf\; X(\backslash mathbf\; x,t)\; \backslash qquad\; \backslash text\backslash qquad\; U\_J\; =\; b\_J\; +\; \backslash alpha\_x\_i\; -\; X\_J\; \backslash ,$
where $\backslash alpha\_$ are the direction cosines between the material and spatial coordinate systems with unit vectors $\backslash mathbf\; E\_J$ and $\backslash mathbf\; e\_i$, respectively. Thus
:$\backslash mathbf\; E\_J\; \backslash cdot\; \backslash mathbf\; e\_i\; =\; \backslash alpha\_=\backslash alpha\_$
and the relationship between $u\_i$ and $U\_J$ is then given by
:$u\_i=\backslash alpha\_U\_J\; \backslash qquad\; \backslash text\; \backslash qquad\; U\_J=\backslash alpha\_u\_i$
Knowing that
:$\backslash mathbf\; e\_i\; =\; \backslash alpha\_\backslash mathbf\; E\_J$
then
:$\backslash mathbf\; u(\backslash mathbf\; X,t)=u\_i\backslash mathbf\; e\_i=u\_i(\backslash alpha\_\backslash mathbf\; E\_J)=U\_J\backslash mathbf\; E\_J=\backslash mathbf\; U(\backslash mathbf\; x,t)$
It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in $\backslash mathbf\; b=0$, and the direction cosines become Kronecker deltas, i.e.
:$\backslash mathbf\; E\_J\; \backslash cdot\; \backslash mathbf\; e\_i\; =\; \backslash delta\_=\backslash delta\_$
Thus, we have
:$\backslash mathbf\; u(\backslash mathbf\; X,t)\; =\; \backslash mathbf\; x(\backslash mathbf\; X,t)\; -\; \backslash mathbf\; X\; \backslash qquad\; \backslash text\backslash qquad\; u\_i\; =\; x\_i\; -\; \backslash delta\_X\_J$
or in terms of the spatial coordinates as
:$\backslash mathbf\; U(\backslash mathbf\; x,t)\; =\; \backslash mathbf\; x\; -\; \backslash mathbf\; X(\backslash mathbf\; x,t)\; \backslash qquad\; \backslash text\backslash qquad\; U\_J\; =\; \backslash delta\_x\_i\; -\; X\_J$
Governing equations

Continuum mechanics deals with the behavior of materials that can be approximated as continuous for certain length and time scales. The equations that govern the mechanics of such materials include the balance laws formass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...

, momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...

, and energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of h ...

. Kinematic
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fi ...

relations and constitutive equations
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and appro ...

are needed to complete the system of governing equations. Physical restrictions on the form of the constitutive relations can be applied by requiring that the second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...

be satisfied under all conditions. In the continuum mechanics of solids, the second law of thermodynamics is satisfied if the Clausius–Duhem form of the entropy inequality is satisfied.
The balance laws express the idea that the rate of change of a quantity (mass, momentum, energy) in a volume must arise from three causes:
#the physical quantity itself flows through the surface that bounds the volume,
#there is a source of the physical quantity on the surface of the volume, or/and,
#there is a source of the physical quantity inside the volume.
Let $\backslash Omega$ be the body (an open subset of Euclidean space) and let $\backslash partial\; \backslash Omega$ be its surface (the boundary of $\backslash Omega$).
Let the motion of material points in the body be described by the map
:$\backslash mathbf\; =\; \backslash boldsymbol(\backslash mathbf)\; =\; \backslash mathbf(\backslash mathbf)$
where $\backslash mathbf$ is the position of a point in the initial configuration and $\backslash mathbf$ is the location of the same point in the deformed configuration.
The deformation gradient is given by
:$\backslash boldsymbol\; =\; \backslash frac\; =\; \backslash nabla\; \backslash mathbf\; ~.$
Balance laws

Let $f(\backslash mathbf,t)$ be a physical quantity that is flowing through the body. Let $g(\backslash mathbf,t)$ be sources on the surface of the body and let $h(\backslash mathbf,t)$ be sources inside the body. Let $\backslash mathbf(\backslash mathbf,t)$ be the outward unit normal to the surface $\backslash partial\; \backslash Omega$. Let $\backslash mathbf(\backslash mathbf,t)$ be the flow velocity of the physical particles that carry the physical quantity that is flowing. Also, let the speed at which the bounding surface $\backslash partial\; \backslash Omega$ is moving be $u\_n$ (in the direction $\backslash mathbf$). Then, balance laws can be expressed in the general form :$\backslash cfrac\backslash left;\; href="/html/ALL/l/int\_\_f(\backslash mathbf,t)~\backslash text\backslash right.html"\; ;"title="int\_\; f(\backslash mathbf,t)~\backslash text\backslash right">int\_\; f(\backslash mathbf,t)~\backslash text\backslash right$ The functions $f(\backslash mathbf,t)$, $g(\backslash mathbf,t)$, and $h(\backslash mathbf,t)$ can be scalar valued, vector valued, or tensor valued - depending on the physical quantity that the balance equation deals with. If there are internal boundaries in the body, jump discontinuities also need to be specified in the balance laws. If we take the Eulerian point of view, it can be shown that the balance laws of mass, momentum, and energy for a solid can be written as (assuming the source term is zero for the mass and angular momentum equations) :$$ In the above equations $\backslash rho(\backslash mathbf,t)$ is the mass density (current), $\backslash dot$ is the material time derivative of $\backslash rho$, $\backslash mathbf(\backslash mathbf,t)$ is the particle velocity, $\backslash dot$ is the material time derivative of $\backslash mathbf$, $\backslash boldsymbol(\backslash mathbf,t)$ is theCauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completel ...

, $\backslash mathbf(\backslash mathbf,t)$ is the body force density, $e(\backslash mathbf,t)$ is the internal energy per unit mass, $\backslash dot$ is the material time derivative of $e$, $\backslash mathbf(\backslash mathbf,t)$ is the heat flux vector, and $s(\backslash mathbf,t)$ is an energy source per unit mass.
With respect to the reference configuration (the Lagrangian point of view), the balance laws can be written as
:$$
In the above, $\backslash boldsymbol$ is the first Piola-Kirchhoff stress tensor, and $\backslash rho\_0$ is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by
:$\backslash boldsymbol\; =\; J~\backslash boldsymbol\backslash cdot\backslash boldsymbol^\; ~\backslash text~\; J\; =\; \backslash det(\backslash boldsymbol)$
We can alternatively define the nominal stress tensor $\backslash boldsymbol$ which is the transpose of the first Piola-Kirchhoff stress tensor such that
:$\backslash boldsymbol\; =\; \backslash boldsymbol^T\; =\; J~\backslash boldsymbol^\backslash cdot\backslash boldsymbol\; ~.$
Then the balance laws become
:$$
The operators in the above equations are defined as such that
:$\backslash boldsymbol\; \backslash mathbf\; =\; \backslash sum\_^3\; \backslash frac\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\; =\; v\_\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\; ~;~~\; \backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash sum\_^3\; \backslash frac\; =\; v\_\; ~;~~\; \backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol\; =\; \backslash sum\_^3\; \backslash frac~\backslash mathbf\_i\; =\; \backslash sigma\_~\backslash mathbf\_i\; ~.$
where $\backslash mathbf$ is a vector field, $\backslash boldsymbol$ is a second-order tensor field, and $\backslash mathbf\_i$ are the components of an orthonormal basis in the current configuration. Also,
:$\backslash boldsymbol\_\; \backslash mathbf\; =\; \backslash sum\_^3\; \backslash frac\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\; =\; v\_\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\; ~;~~\; \backslash boldsymbol\_\backslash cdot\backslash mathbf\; =\; \backslash sum\_^3\; \backslash frac\; =\; v\_\; ~;~~\; \backslash boldsymbol\_\backslash cdot\backslash boldsymbol\; =\; \backslash sum\_^3\; \backslash frac~\backslash mathbf\_i\; =\; S\_~\backslash mathbf\_i$
where $\backslash mathbf$ is a vector field, $\backslash boldsymbol$ is a second-order tensor field, and $\backslash mathbf\_i$ are the components of an orthonormal basis in the reference configuration.
The inner product is defined as
:$\backslash boldsymbol:\backslash boldsymbol\; =\; \backslash sum\_^3\; A\_~B\_\; =\; \backslash operatorname(\backslash boldsymbol\backslash boldsymbol^T)\; ~.$
Clausius–Duhem inequality

TheClausius–Duhem inequality
The Clausius–Duhem inequality is a way of expressing the second law of thermodynamics that is used in continuum mechanics. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamicall ...

can be used to express the second law of thermodynamics for elastic-plastic materials. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.
Just like in the balance laws in the previous section, we assume that there is a flux of a quantity, a source of the quantity, and an internal density of the quantity per unit mass. The quantity of interest in this case is the entropy. Thus, we assume that there is an entropy flux, an entropy source, an internal mass density $\backslash rho$ and an internal specific entropy (i.e. entropy per unit mass) $\backslash eta$ in the region of interest.
Let $\backslash Omega$ be such a region and let $\backslash partial\; \backslash Omega$ be its boundary. Then the second law of thermodynamics states that the rate of increase of $\backslash eta$ in this region is greater than or equal to the sum of that supplied to $\backslash Omega$ (as a flux or from internal sources) and the change of the internal entropy density $\backslash rho\backslash eta$ due to material flowing in and out of the region.
Let $\backslash partial\; \backslash Omega$ move with a flow velocity $u\_n$ and let particles inside $\backslash Omega$ have velocities $\backslash mathbf$. Let $\backslash mathbf$ be the unit outward normal to the surface $\backslash partial\; \backslash Omega$. Let $\backslash rho$ be the density of matter in the region, $\backslash bar$ be the entropy flux at the surface, and $r$ be the entropy source per unit mass.
Then the entropy inequality may be written as
:$\backslash cfrac\backslash left(\backslash int\_\; \backslash rho~\backslash eta~\backslash text\backslash right)\; \backslash ge\; \backslash int\_\; \backslash rho~\backslash eta~(u\_n\; -\; \backslash mathbf\backslash cdot\backslash mathbf)\; ~\backslash text\; +\; \backslash int\_\; \backslash bar~\backslash text\; +\; \backslash int\_\; \backslash rho~r~\backslash text.$
The scalar entropy flux can be related to the vector flux at the surface by the relation $\backslash bar\; =\; -\backslash boldsymbol(\backslash mathbf)\backslash cdot\backslash mathbf$. Under the assumption of incrementally isothermal conditions, we have
:$\backslash boldsymbol(\backslash mathbf)\; =\; \backslash cfrac\; ~;~~\; r\; =\; \backslash cfrac$
where $\backslash mathbf$ is the heat flux vector, $s$ is an energy source per unit mass, and $T$ is the absolute temperature of a material point at $\backslash mathbf$ at time $t$.
We then have the Clausius–Duhem inequality in integral form:
:$$
We can show that the entropy inequality may be written in differential form as
:$$
In terms of the Cauchy stress and the internal energy, the Clausius–Duhem inequality may be written as
:$$
Applications

* Continuum mechanics **Solid mechanics
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ...

** Fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...

* Engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

** Civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewa ...

** Mechanical engineering
Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, an ...

** Aerospace engineering
** Biomedical engineering
Biomedical engineering (BME) or medical engineering is the application of engineering principles and design concepts to medicine and biology for healthcare purposes (e.g., diagnostic or therapeutic). BME is also traditionally logical sciences ...

** Chemical engineering
Chemical engineering is an engineering field which deals with the study of operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials int ...

See also

*Transport phenomena
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mecha ...

* Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...

* Cauchy elastic material
* Configurational mechanics
* Curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inv ...

* Equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or inter ...

* Finite deformation tensors
* Finite strain theory
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...

* Hyperelastic material
* Lagrangian and Eulerian specification of the flow field
__NOTOC__
In classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an ind ...

* Movable cellular automaton
The movable cellular automaton (MCA) method is a method in Computational mechanics, computational solid mechanics based on the discrete concept. It provides advantages both of classical cellular automaton and discrete element method, discrete ele ...

* Peridynamics (a non-local continuum theory leading to integral equations)
* Stress (physics)
* Stress measures
* Tensor calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields ( tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Le ...

* Tensor derivative (continuum mechanics)
* Theory of elasticity
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and o ...

Explanatory notes

References

Citations

Works cited

* * * * * * * * * * * *General references

* * * * * * * * * * * * * * * * * * *External links

"Objectivity in classical continuum mechanics: Motions, Eulerian and Lagrangian functions; Deformation gradient; Lie derivatives; Velocity-addition formula, Coriolis; Objectivity"

by Gilles Leborgne, April 7, 2021

"Part IV Velocity-addition formula and Objectivity"

{{DEFAULTSORT:Continuum Mechanics Classical mechanics