Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium (also called a continuum) rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.
Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships.
Continuum mechanics treats the physical properties of solids and fluids independently of any particular coordinate system in which they are observed. These properties are represented by tensors, which are mathematical objects with the salient property of being independent of coordinate systems. This permits definition of physical properties at any point in the continuum, according to mathematically convenient continuous functions. The theories of elasticity, plasticity and fluid mechanics are based on the concepts of continuum mechanics.
The concept of a continuum underlies the mathematical framework for studying large-scale forces and deformations in materials. Although materials are composed of discrete atoms and molecules, separated by empty space or microscopic cracks and crystallographic defects, physical phenomena can often be modeled by considering a substance distributed throughout some region of space. A continuum is a body that can be continually sub-divided into infinitesimal elements with local material properties defined at any particular point. Properties of the bulk material can therefore be described by continuous functions, and their evolution can be studied using the mathematics of calculus.
Apart from the assumption of continuity, two other independent assumptions are often employed in the study of continuum mechanics. These are homogeneity (assumption of identical properties at all locations) and isotropy (assumption of directionally invariant vector properties). If these auxiliary assumptions are not globally applicable, the material may be segregated into sections where they are applicable in order to simplify the analysis. For more complex cases, one or both of these assumptions can be dropped. In these cases, computational methods are often used to solve the differential equations describing the evolution of material properties.
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.
Continuum mechanics models begin by assigning a region in three-dimensional Euclidean space to the material body
lB
t
\kappat(lB)
A particular particle within the body in a particular configuration is characterized by a position vector
x=
3 | |
\sum | |
i=1 |
xiei,
ei
X
x=\kappat(X).
This function needs to have various properties so that the model makes physical sense.
\kappat( ⋅ )
For the mathematical formulation of the model,
\kappat( ⋅ )
See also: Stress (mechanics) and Cauchy stress tensor. 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.
Following the classical dynamics of Newton and Euler, 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
FC
FB
lF
lF=FC+FB
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 to Newton's third law of motion of conservation of linear momentum 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. 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
T(n,x,t)
t
x
n
Any differential area
dS
n
S
dFC
S
dFC=T(n)dS
where
T(n)
The total contact force on the particular internal surface
S
dS
FC=\intST(n)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 forces) 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 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 (gravitational forces) or electromagnetic field (electromagnetic forces), 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
b(x,t)
In the case of gravitational forces, the intensity of the force depends on, or is proportional to, the mass density
\rho(x,t)
bi
pi
\rhobi=pi
The total body force applied to a continuous body is expressed as
FB=\intVbdm=\intV\rhobdV
Body forces and contact forces acting on the body lead to corresponding moments of force (torques) relative to a given point. Thus, the total applied torque
lM
lM=MC+MB
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
lF=\intVadm=\intSTdS+\intV\rhobdV
lM=\intSr x TdS+\intVr x \rhobdV
A change in the configuration of a continuum body results in a displacement. 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
\kappa0(lB)
\kappat(lB)
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:
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
\kappa0(lB)
Xi
X
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.
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
\kappa0(lB)
In the Lagrangian description, the motion of a continuum body is expressed by the mapping function
\chi( ⋅ )
x=\chi(X,t)
which is a mapping of the initial configuration
\kappa0(lB)
\kappat(lB)
x=xiei
X
X
\kappa0(lB)
\kappat(lB)
t
xi
Physical and kinematic properties
Pij\ldots
Pij\ldots=Pij\ldots(X,t)
The material derivative of any property
Pij\ldots
In the Lagrangian description, the material derivative of
Pij\ldots
X
d | |
dt |
[Pij\ldots
(
| ||||
ij\ldots |
(X,t)]
The instantaneous position
x
v
v=
x | = |
dx | = | |
dt |
\partial\chi(X,t) | |
\partialt |
Similarly, the acceleration field is given by
a=v |
=\ddot{x}=
d2x | = | |
dt2 |
\partial2\chi(X,t) | |
\partialt2 |
Continuity in the Lagrangian description is expressed by the spatial and temporal continuity of the mapping from the reference configuration to the current configuration of the material points. All physical quantities characterizing the continuum are described this way. In this sense, the function
\chi( ⋅ )
Pij\ldots( ⋅ )
Continuity allows for the inverse of
\chi( ⋅ )
x
\kappa0(lB)
The Eulerian description, introduced by d'Alembert, focuses on the current configuration
\kappat(lB)
Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function
X=\chi-1(x,t)
which provides a tracing of the particle which now occupies the position
x
\kappat(lB)
X
\kappa0(lB)
A necessary and sufficient condition for this inverse function to exist is that the determinant of the Jacobian matrix, often referred to simply as the Jacobian, should be different from zero. Thus,
J=\left|
\partial\chii | |
\partialXJ |
\right|=\left|
\partialxi | |
\partialXJ |
\right| ≠ 0
In the Eulerian description, the physical properties
Pij\ldots
Pij=Pij\ldots(X,t)=Pij\ldots[\chi-1(x,t),t]=pij\ldots(x,t)
where the functional form of
Pij
pij
The material derivative of
pij\ldots(x,t)
d | |
dt |
[pij\ldots
(
| (x,t)]+ | ||||
ij\ldots |
\partial | |
\partialxk |
[pij\ldots
(
|
The first term on the right-hand side of this equation gives the local rate of change of the property
pij\ldots(x,t)
x
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
x
See main article: Displacement field (mechanics).
The vector joining the positions of a particle
P
u(X,t)=uiei
U(x,t)=UJEJ
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
u(X,t)=b+x(X,t)-X or ui=\alphaiJbJ+xi-\alphaiJXJ
or in terms of the spatial coordinates as
U(x,t)=b+x-X(x,t) or UJ=bJ+\alphaJixi-XJ
where
\alphaJi
EJ
ei
EJ ⋅ ei=\alphaJi=\alphaiJ
and the relationship between
ui
UJ
ui=\alphaiJUJ or UJ=\alphaJiui
Knowing that
ei=\alphaiJEJ
u(X,t)=uiei=ui(\alphaiJEJ)=UJEJ=U(x,t)
It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in
b=0
EJ ⋅ ei=\deltaJi=\deltaiJ
Thus, we have
u(X,t)=x(X,t)-X or ui=xi-\deltaiJXJ
or in terms of the spatial coordinates as
U(x,t)=x-X(x,t) or UJ=\deltaJixi-XJ
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 for mass, momentum, and energy. Kinematic relations and constitutive equations 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 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:
Let
\Omega
\partial\Omega
\Omega
Let the motion of material points in the body be described by the map
x=\boldsymbol{\chi}(X)=x(X)
X
x
The deformation gradient is given by
\boldsymbol{F}=
\partialx | |
\partialX |
=\nablax~.
Let
f(x,t)
g(x,t)
h(x,t)
n(x,t)
\partial\Omega
v(x,t)
\partial\Omega
un
n
Then, balance laws can be expressed in the general form
\cfrac{d}{dt}\left[\int\Omegaf(x,t)~dV\right]=\int\partialf(x,t)[un(x,t)-v(x,t) ⋅ n(x,t)]~dA+\int\partialg(x,t)~dA+\int\Omegah(x,t)~dV~.
f(x,t)
g(x,t)
h(x,t)
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)
{ \begin{align}
\rho |
+\rho(\boldsymbol{\nabla} ⋅ v)&=0&& BalanceofMass\\ \rho~
v |
-\boldsymbol{\nabla} ⋅ \boldsymbol{\sigma}-\rho~b&=0&& BalanceofLinearMomentum(Cauchy'sfirstlawofmotion)\\ \boldsymbol{\sigma}&=\boldsymbol{\sigma}T && BalanceofAngularMomentum(Cauchy'ssecondlawofmotion)\\ \rho~
e |
-\boldsymbol{\sigma}:(\boldsymbol{\nabla}v)+\boldsymbol{\nabla} ⋅ q-\rho~s&=0 && BalanceofEnergy. \end{align} }
In the above equations
\rho(x,t)
\rho |
\rho
v(x,t)
v |
v
\boldsymbol{\sigma}(x,t)
b(x,t)
e(x,t)
e |
e
q(x,t)
s(x,t)
{ \begin{align} \rho~\det(\boldsymbol{F})-\rho0&=0&& BalanceofMass\\ \rho0~\ddot{x
In the above,
\boldsymbol{P}
\rho0
\boldsymbol{P}=J~\boldsymbol{\sigma} ⋅ \boldsymbol{F}-T~where~J=\det(\boldsymbol{F})
We can alternatively define the nominal stress tensor
\boldsymbol{N}
\boldsymbol{N}=\boldsymbol{P}T=J~\boldsymbol{F}-1 ⋅ \boldsymbol{\sigma}~.
{ \begin{align} \rho~\det(\boldsymbol{F})-\rho0&=0&& BalanceofMass\\ \rho0~\ddot{x
The operators in the above equations are defined as
\boldsymbol{\nabla}v=
3 | |
\sum | |
i,j=1 |
\partialvi | |
\partialxj |
ei ⊗ ej=vi,jei ⊗ ej~;~~ \boldsymbol{\nabla} ⋅ v=
3 | |
\sum | |
i=1 |
\partialvi | |
\partialxi |
=vi,i~;~~ \boldsymbol{\nabla} ⋅ \boldsymbol{S}=
3 | |
\sum | |
i,j=1 |
\partialSij | |
\partialxj |
~ei=\sigmaij,j~ei~.
v
\boldsymbol{S}
ei
\boldsymbol{\nabla}\circv=
3 | |
\sum | |
i,j=1 |
\partialvi | |
\partialXj |
Ei ⊗ Ej=vi,jEi ⊗ Ej~;~~ \boldsymbol{\nabla}\circ ⋅ v=
3 | |
\sum | |
i=1 |
\partialvi | |
\partialXi |
=vi,i~;~~ \boldsymbol{\nabla}\circ ⋅ \boldsymbol{S}=
3 | |
\sum | |
i,j=1 |
\partialSij | |
\partialXj |
~Ei=Sij,j~Ei
v
\boldsymbol{S}
Ei
The inner product is defined as
\boldsymbol{A}:\boldsymbol{B}=
3 | |
\sum | |
i,j=1 |
Aij~Bij=\operatorname{trace}(\boldsymbol{A}\boldsymbol{B}T)~.
The Clausius–Duhem inequality 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
\rho
η
Let
\Omega
\partial\Omega
η
\Omega
\rhoη
\partial\Omega
un
\Omega
v
n
\partial\Omega
\rho
\bar{q}
r
\cfrac{d}{dt}\left(\int\Omega\rho~η~dV\right)\ge \int\partial\rho~η~(un-v ⋅ n)~dA+\int\partial\bar{q}~dA+\int\Omega\rho~r~dV.
\bar{q}=-\boldsymbol{\psi}(x) ⋅ n
\boldsymbol{\psi}(x)=\cfrac{q(x)}{T}~;~~r=\cfrac{s}{T}
q
s
T
x
t
We then have the Clausius–Duhem inequality in integral form:
{ \cfrac{d}{dt}\left(\int\Omega\rho~η~dV\right)\ge \int\partial\rho~η~(un-v ⋅ n)~dA-\int\partial\cfrac{q ⋅ n
{ \rho~
η |
\ge-\boldsymbol{\nabla} ⋅ \left(\cfrac{q
{ \rho~(
e |
-T~
η |
)-\boldsymbol{\sigma}:\boldsymbol{\nabla}v\le-\cfrac{q ⋅ \boldsymbol{\nabla}T}{T}. }
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 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 homogenous.