Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuum. The French mathematician Augustin Louis Cauchy was the first to formulate such models in the 19th century, but research in the area continues today.

Modeling an object as a continuum assumes that the substance of the object completely 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. Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the particular material studied is added through a constitutive relation.

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

The concept of a continuum

Materials, such as solids, liquids and gases, are composed of molecules separated by empty space. On a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modeled assuming the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire region of 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 modeling objects as continua depends on the type of problem and the scale of the physical phenomena under consideration. A material may be assumed to be a continuum when the distance between the physical particles is very small compared to the overall size of the problem. For example, such is the case when analyzing the deformation behavior of soil deposits in soil mechanics. A given volume of soil is composed of discrete solid particles (called grains) of minerals that are packed together with voids in between them. In this sense, soils are not continuous. To simplify the deformation analysis of the soil, the volume of soil can be assumed to be a continuum because the grain particles are very small compared to the scale of the problem.

The validity of the continuum assumption may be verified by

• experimental testing and measurements on the real material under consideration to check that the material satisfies the Hill condition,
• by a theoretical analysis, in which some clear periodicity or homogeneity in the microscale exists over which the material may be averaged. This leads to the concept of a Representative Volume Element (RVE), which describes the length scale at which the continuum assumption breaks down.

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

Mathematical modeling of a continuum

Figure 1. Configuration of a continuum body

In continuum mechanics, a material body ${\mathcal {B}}$ is a set of infinitesimal volumetric elements $\ X$, called particles or material points. A material body is expressed as a continuum by assuming that at any configuration, or geometrical state of the body, there is a region $\ R$ in a three dimensional euclidean space ${\mathcal {E}}$ such that every point of that region is occupied by a material point $\ X$, i.e there is a one-to-one correspondence between material points and space points.

The configuration $\ \kappa _{t}({\mathcal {B}})$, or geometrical state of the material body ${\mathcal {B}}$ at a particular time $\ t$ is characterized by the position vector $\ \mathbf {x} =x_{i}\mathbf {e} _{i}$ of all particles at that time with respect to an arbitrary frame of reference (Figure 1). Mathematically, this is expressed by the mapping function

$\ \mathbf {x} =\kappa _{t}(\mathbf {X} )$

where $\ \kappa _{t}(\cdot )$ is a continuous function, i.e. uniquely invertible and differentiable as many times as necessary.

Kinematics: deformation and motion

Figure 2. Motion of a continuum body.

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 consist 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 $\ \kappa _{0}({\mathcal {B}})$ to a current or deformed configuration $\ \kappa _{t}({\mathcal {B}})$ (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 pathline.

There is continuity during deformation or motion 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 to be one the body actually will ever occupy. Often, the configuration at $\ t=0$ is considered the reference configuration , $\ \kappa _{0}({\mathcal {B}})$. The components $\ X_{i}$ of the position vector $\ \mathbf {X}$ of a particle, taken with respect to the reference configuration, are called the material or reference coordinates.

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

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 referential 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, $\kappa _{0}({\mathcal {B}})$. This description is normally used in solid mechanics.

In the Lagrangian description, the motion of a continuum body is expressed by the mapping function $\ \chi (\cdot )$ (Figure 2),

$\ \mathbf {x} =\chi (\mathbf {X} ,t)$

which is a mapping of the initial configuration $\kappa _{0}({\mathcal {B}})$ onto the current configuration $\kappa _{t}({\mathcal {B}})$, giving a geometrical correspondence between them, i.e. giving the position vector $\ \mathbf {x} =x_{i}\mathbf {e} _{i}$ that a particle $\ X$, with a position vector $\ \mathbf {X}$ in the undeformed or reference configuration $\kappa _{0}({\mathcal {B}})$, will occupy in the current or deformed configuration $\kappa _{t}({\mathcal {B}})$ at time $\ t$. The components $\ x_{i}$ are called the spatial coordinates.

Physical and kinematic properties $\ P_{ij\ldots }$, i.e. thermodynamic properties and velocity, which describe or characterize features of the material body, are expressed as continuous functions of position and time, i.e. $\ P_{ij\ldots }=P_{ij\ldots }(\mathbf {X} ,t)$.

The material derivative of any property $\ P_{ij\ldots }$ 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 $\ P_{ij\ldots }$ is simply the partial derivative with respect to time, and the position vector $\ \mathbf {X}$ is held constant as it does not change with time. Thus, we have

$\ {\frac {d}{dt}}[P_{ij\ldots }(\mathbf {X} ,t)]={\frac {\partial }{\partial t}}[P_{ij\ldots }(\mathbf {X} ,t)]$

The instantaneous position $\ \mathbf {x}$ is a property of a particle, and its material derivative is the instantaneous velocity $\ \mathbf {v}$ of the particle. Therefore, the velocity field of the continuum is given by

$\ \mathbf {v} =\mathbf {\dot {x}} ={\frac {d\mathbf {x} }{dt}}={\frac {\partial \chi (\mathbf {X} ,t)}{\partial t}}$

Similarly, the acceleration field is given by

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

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 $\chi (\cdot )$ and $\ P_{ij\ldots }(\cdot )$ are single-valued and continuous, with continuous derivatives with respect to space and time to whatever order is required, usually to the second or third.

Eulerian description

Continuity allows for the inverse of $\chi (\cdot )$ to trace backwards where the particle currently located at $\mathbf {x}$ was located in the initial or referenced configuration$\kappa _{0}({\mathcal {B}})$. 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 $\kappa _{t}({\mathcal {B}})$, 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 of fluid flow 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

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

which provides a tracing of the particle which now occupies the position $\mathbf {x}$ in the current configuration $\kappa _{t}({\mathcal {B}})$ to its original position $\mathbf {X}$ in the initial configuration $\kappa _{0}({\mathcal {B}})$.

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

$\ J=\left|{\frac {\partial \chi _{i}}{\partial X_{J}}}\right|=\left|{\frac {\partial x_{i}}{\partial X_{J}}}\right|\neq 0$

In the Eulerian description, the physical properties $\ P_{ij\ldots }$ are expressed as

$\ P_{ij\ldots }=P_{ij\ldots }(\mathbf {X} ,t)=P_{ij\ldots }[\chi ^{-1}(\mathbf {x} ,t),t]=p_{ij\ldots }(\mathbf {x} ,t)$

where the functional form of $\ P_{ij\ldots }$ in the Lagrangian description is not the same as the form of $\ p_{ij\ldots }$ in the Eulerian description.

The material derivative of $\ p_{ij\ldots }(\mathbf {x} ,t)$, using the chain rule, is then

$\ {\frac {d}{dt}}[p_{ij\ldots }(\mathbf {x} ,t)]={\frac {\partial }{\partial t}}[p_{ij\ldots }(\mathbf {x} ,t)]+{\frac {\partial }{\partial x_{k}}}[p_{ij\ldots }(\mathbf {x} ,t)]{\frac {dx_{k}}{dt}}$

The first term on the right-hand side of this equation gives the local rate of change of the property $\ p_{ij\ldots }(\mathbf {x} ,t)$ occurring at position $\ \mathbf {x}$. 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 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 $\ \mathbf {x}$.

Displacement Field

The vector joining the positions of a particle $\ P$ in the undeformed configuration and deformed configuration is called the displacement vector $\ \mathbf {u} (\mathbf {X} ,t)=u_{i}\mathbf {e} _{i}$, in the Lagrangian description, or $\ \mathbf {U} (\mathbf {x} ,t)=U_{J}\mathbf {E} _{J}$, 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

$\ \mathbf {u} (\mathbf {X} ,t)=\mathbf {b} +\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=\alpha _{iJ}b_{J}+x_{i}-\alpha _{iJ}X_{J}$

or in terms of the spatial coordinates as

$\ \mathbf {U} (\mathbf {x} ,t)=\mathbf {b} +\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=b_{J}+\alpha _{Ji}x_{i}-X_{J}\,$

where $\ \alpha _{Ji}$ are the direction cosines between the material and spatial coordinate systems with unit vectors $\ \mathbf {E} _{J}$ and $\mathbf {e} _{i}$, respectively. Thus

$\ \mathbf {E} _{J}\cdot \mathbf {e} _{i}=\alpha _{Ji}=\alpha _{iJ}$

and the relationship between $\ u_{i}$ and $\ U_{J}$ is then given by

$\ u_{i}=\alpha _{iJ}U_{J}\qquad {\text{or}}\qquad U_{J}=\alpha _{Ji}u_{i}$

Knowing that

$\ \mathbf {e} _{i}=\alpha _{iJ}\mathbf {E} _{J}$

then

$\mathbf {u} (\mathbf {X} ,t)=u_{i}\mathbf {e} _{i}=u_{i}(\alpha _{iJ}\mathbf {E} _{J})=U_{J}\mathbf {E} _{J}=\mathbf {U} (\mathbf {x} ,t)$

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in $\ \mathbf {b} =0$, and the direction cosines become Kronecker deltas, i.e.

$\ \mathbf {E} _{J}\cdot \mathbf {e} _{i}=\delta _{Ji}=\delta _{iJ}$

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

$\ \mathbf {U} (\mathbf {x} ,t)=\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=\delta _{Ji}x_{i}-X_{J}$

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 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:

1. the physical quantity itself flows through the surface that bounds the volume,
2. there is a source of the physical quantity on the surface of the volume, or/and,
3. there is a source of the physical quantity inside the volume.

Let $\Omega$ be the body (an open subset of Euclidean space) and let $\partial \Omega$ be its surface (the boundary of $\Omega$).

Let the motion of material points in the body be described by the map

$\mathbf {x} ={\boldsymbol {\chi }}(\mathbf {X} )=\mathbf {x} (\mathbf {X} )$

where $\mathbf {X}$ is the position of a point in the initial configuration and $\mathbf {x}$ is the location of the same point in the deformed configuration.

The deformation gradient is given by

${\boldsymbol {F}}={\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}={\boldsymbol {\mathbf {x} }}\cdot \nabla ~.$

Balance Laws

Let $f(\mathbf {x} ,t)$ be a physical quantity that is flowing through the body. Let $g(\mathbf {x} ,t)$ be sources on the surface of the body and let $h(\mathbf {x} ,t)$ be sources inside the body. Let $\mathbf {n} (\mathbf {x} ,t)$ be the outward unit normal to the surface $\partial \Omega$. Let $\mathbf {v} (\mathbf {x} ,t)$ be the velocity of the physical particles that carry the physical quantity that is flowing. Also, let the speed at which the bounding surface $\partial \Omega$ is moving be $u_{n}$ (in the direction $\mathbf {n}$).

Then, balance laws can be expressed in the general form

${\cfrac {d}{dt}}\left[\int _{\Omega }f(\mathbf {x} ,t)~{\text{dV}}\right]=\int _{\partial \Omega }f(\mathbf {x} ,t)[u_{n}(\mathbf {x} ,t)-\mathbf {v} (\mathbf {x} ,t)\cdot \mathbf {n} (\mathbf {x} ,t)]~{\text{dA}}+\int _{\partial \Omega }g(\mathbf {x} ,t)~{\text{dA}}+\int _{\Omega }h(\mathbf {x} ,t)~{\text{dV}}~.$

Note that the functions $f(\mathbf {x} ,t)$, $g(\mathbf {x} ,t)$, and $h(\mathbf {x} ,t)$ 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 Lagrangian point of view, it can be shown that the balance laws of mass, momentum, and energy for a solid can be written as

${\begin{aligned}{\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\cdot \mathbf {v} &=0&&\qquad {\text{Balance of Mass}}\\\rho ~{\dot {\mathbf {v} }}-{\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}-\rho ~\mathbf {b} &=0&&\qquad {\text{Balance of Linear Momentum}}\\{\boldsymbol {\sigma }}&={\boldsymbol {\sigma }}^{T}&&\qquad {\text{Balance of Angular Momentum}}\\\rho ~{\dot {e}}-{\boldsymbol {\sigma }}:({\boldsymbol {\nabla }}\mathbf {v} )+{\boldsymbol {\nabla }}\cdot \mathbf {q} -\rho ~s&=0&&\qquad {\text{Balance of Energy.}}\end{aligned}}$

In the above equations $\rho (\mathbf {x} ,t)$ is the mass density (current), ${\dot {\rho }}$ is the material time derivative of $\rho$, $\mathbf {v} (\mathbf {x} ,t)$ is the particle velocity, ${\dot {\mathbf {v} }}$ is the material time derivative of $\mathbf {v}$, ${\boldsymbol {\sigma }}(\mathbf {x} ,t)$ is the Cauchy stress tensor, $\mathbf {b} (\mathbf {x} ,t)$ is the body force density, $e(\mathbf {x} ,t)$ is the internal energy per unit mass, ${\dot {e}}$ is the material time derivative of $e$, $\mathbf {q} (\mathbf {x} ,t)$ is the heat flux vector, and $s(\mathbf {x} ,t)$ is an energy source per unit mass.

With respect to the reference configuration, the balance laws can be written as

${\begin{aligned}\rho ~\det({\boldsymbol {F}})-\rho _{0}&=0&&\qquad {\text{Balance of Mass}}\\\rho _{0}~{\ddot {\mathbf {x} }}-{\boldsymbol {\nabla }}_{\circ }\cdot {\boldsymbol {P}}^{T}-\rho _{0}~\mathbf {b} &=0&&\qquad {\text{Balance of Linear Momentum}}\\{\boldsymbol {F}}\cdot {\boldsymbol {P}}^{T}&={\boldsymbol {P}}\cdot {\boldsymbol {F}}^{T}&&\qquad {\text{Balance of Angular Momentum}}\\\rho _{0}~{\dot {e}}-{\boldsymbol {P}}^{T}:{\dot {\boldsymbol {F}}}+{\boldsymbol {\nabla }}_{\circ }\cdot \mathbf {q} -\rho _{0}~s&=0&&\qquad {\text{Balance of Energy.}}\end{aligned}}$

In the above, ${\boldsymbol {P}}$ is the first Piola-Kirchhoff stress tensor, and $\rho _{0}$ is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by

${\boldsymbol {P}}=J~{\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}~{\text{where}}~J=\det({\boldsymbol {F}})$

We can alternatively define the nominal stress tensor ${\boldsymbol {N}}$ which is the transpose of the first Piola-Kirchhoff stress tensor such that

${\boldsymbol {N}}={\boldsymbol {P}}^{T}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}~.$

Then the balance laws become

${\begin{aligned}\rho ~\det({\boldsymbol {F}})-\rho _{0}&=0&&\qquad {\text{Balance of Mass}}\\\rho _{0}~{\ddot {\mathbf {x} }}-{\boldsymbol {\nabla }}_{\circ }\cdot {\boldsymbol {N}}-\rho _{0}~\mathbf {b} &=0&&\qquad {\text{Balance of Linear Momentum}}\\{\boldsymbol {F}}\cdot {\boldsymbol {N}}&={\boldsymbol {N}}^{T}\cdot {\boldsymbol {F}}^{T}&&\qquad {\text{Balance of Angular Momentum}}\\\rho _{0}~{\dot {e}}-{\boldsymbol {N}}:{\dot {\boldsymbol {F}}}+{\boldsymbol {\nabla }}_{\circ }\cdot \mathbf {q} -\rho _{0}~s&=0&&\qquad {\text{Balance of Energy.}}\end{aligned}}$

The operators in the above equations are defined as such that

${\boldsymbol {\nabla }}\mathbf {v} =\sum _{i,j=1}^{3}{\frac {\partial v_{i}}{\partial x_{j}}}\mathbf {e} _{i}\otimes \mathbf {e} _{j}=v_{i,j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}~;~~{\boldsymbol {\nabla }}\cdot \mathbf {v} =\sum _{i=1}^{3}{\frac {\partial v_{i}}{\partial x_{i}}}=v_{i,i}~;~~{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}=\sum _{i,j=1}^{3}{\frac {\partial S_{ij}}{\partial x_{j}}}~\mathbf {e} _{i}=\sigma _{ij,j}~\mathbf {e} _{i}~.$

where $\mathbf {v}$ is a vector field, ${\boldsymbol {S}}$ is a second-order tensor field, and $\mathbf {e} _{i}$ are the components of an orthonormal basis in the current configuration. Also,

${\boldsymbol {\nabla }}_{\circ }\mathbf {v} =\sum _{i,j=1}^{3}{\frac {\partial v_{i}}{\partial X_{j}}}\mathbf {E} _{i}\otimes \mathbf {E} _{j}=v_{i,j}\mathbf {E} _{i}\otimes \mathbf {E} _{j}~;~~{\boldsymbol {\nabla }}_{\circ }\cdot \mathbf {v} =\sum _{i=1}^{3}{\frac {\partial v_{i}}{\partial X_{i}}}=v_{i,i}~;~~{\boldsymbol {\nabla }}_{\circ }\cdot {\boldsymbol {S}}=\sum _{i,j=1}^{3}{\frac {\partial S_{ij}}{\partial X_{j}}}~\mathbf {E} _{i}=S_{ij,j}~\mathbf {E} _{i}$

where $\mathbf {v}$ is a vector field, ${\boldsymbol {S}}$ is a second-order tensor field, and $\mathbf {E} _{i}$ are the components of an orthonormal basis in the reference configuration.

The inner product is defined as

${\boldsymbol {A}}:{\boldsymbol {B}}=\sum _{i,j=1}^{3}A_{ij}~B_{ij}=A_{ij}~B_{ij}~.$

The Clausius–Duhem inequality

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, and an internal entropy density per unit mass ($\eta$) in the region of interest.

Let $\Omega$ be such a region and let $\partial \Omega$ be its boundary. Then the second law of thermodynamics states that the rate of increase of $\eta$ in this region is greater than or equal to the sum of that supplied to $\Omega$ (as a flux or from internal sources) and the change of the internal entropy density due to material flowing in and out of the region.

Let $\partial \Omega$ move with a velocity $u_{n}$ and let particles inside $\Omega$ have velocities $\mathbf {v}$. Let $\mathbf {n}$ be the unit outward normal to the surface $\partial \Omega$. Let $\rho$ be the density of matter in the region, ${\bar {q}}$ be the entropy flux at the surface, and $r$ be the entropy source per unit mass. Then the entropy inequality may be written as

${\cfrac {d}{dt}}\left(\int _{\Omega }\rho ~\eta ~{\text{dV}}\right)\geq \int _{\partial \Omega }\rho ~\eta ~(u_{n}-\mathbf {v} \cdot \mathbf {n} )~{\text{dA}}+\int _{\partial \Omega }{\bar {q}}~{\text{dA}}+\int _{\Omega }\rho ~r~{\text{dV}}.$

The scalar entropy flux can be related to the vector flux at the surface by the relation ${\bar {q}}=-{\boldsymbol {\psi }}(\mathbf {x} )\cdot \mathbf {n}$. Under the assumption of incrementally isothermal conditions, we have

${\boldsymbol {\psi }}(\mathbf {x} )={\cfrac {\mathbf {q} (\mathbf {x} )}{T}}~;~~r={\cfrac {s}{T}}$

where $\mathbf {q}$ is the heat flux vector, $s$ is a energy source per unit mass, and $T$ is the absolute temperature of a material point at $\mathbf {x}$ at time $t$.

We then have the Clausius–Duhem inequality in integral form:

${{\cfrac {d}{dt}}\left(\int _{\Omega }\rho ~\eta ~{\text{dV}}\right)\geq \int _{\partial \Omega }\rho ~\eta ~(u_{n}-\mathbf {v} \cdot \mathbf {n} )~{\text{dA}}-\int _{\partial \Omega }{\cfrac {\mathbf {q} \cdot \mathbf {n} }{T}}~{\text{dA}}+\int _{\Omega }{\cfrac {\rho ~s}{T}}~{\text{dV}}.}$

We can show that the entropy inequality may be written in differential form as

${\rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}.}$

In terms of the Cauchy stress and the internal energy, the Clausius–Duhem inequality may be written as

${\rho ~({\dot {e}}-T~{\dot {\eta }})-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \leq -{\cfrac {\mathbf {q} \cdot {\boldsymbol {\nabla }}T}{T}}.}$

Applications

• Mechanics
  • Solid mechanics
  • Fluid mechanics
• Engineering
  • Mechanical engineering
  • Civil engineering
  • Aerospace engineering

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

See also

• Finite deformation tensors
• Finite strain theory
• Stress (physics)
• Stress measures
• Hyperelastic material
• Cauchy elastic material
• Equation of state
• Theory of elasticity
• Bernoulli's principle
• Peridynamics (a non-local continuum theory leading to integral equations)
• Tensor calculus
• Curvilinear coordinates
• Tensor derivative (continuum mechanics)

References

• Batra, R. C. (2006). Elements of Continuum Mechanics. Reston, VA: AIAA.

• Fung, Y. C. (1977). A First Course in Continuum Mechanics (2nd edition ed.). Prentice-Hall, Inc. ISBN 0133183114. {{cite book}}: |edition= has extra text (help)

• Dill, Ellis Harold (2006). Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity. Germany: CRC Press. ISBN 0849397790.

• Hutter, Kolumban (2004). Continuum Methods of Physical Modeling. Germany: Springer. ISBN 3540206191. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Gurtin, M. E. (1981). An Introduction to Continuum Mechanics. New York: Academic Press.

• Lai, W. Michael (1996). Introduction to Continuum Mechanics (3rd edition ed.). Elsevier, Inc. ISBN 978-0-7506-2894-5. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Lubarda, Vlado A. (2001). Elastoplasticity Theory. CRC Press. ISBN 0849311381.

• Lubliner, Jacob (2008). Plasticity Theory (Revised Edition) (PDF). Dover Publications. ISBN 0486462900.

• Mase, George E. (1970). Continuum Mechanics. McGraw-Hill Professional. ISBN 0070406634.

• Mase, G. Thomas (1999). Continuum Mechanics for Engineers (Second Edition ed.). CRC Press. ISBN 0-8493-1855-6. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Maugin, G. A. (1999). The Thermomechanics of Nonlinear Irreversible Behaviors: An Introduction. Singapore: World Scientific.

• Nemat-Nasser, Sia (2006). Plasticity: A Treatise on Finite Deformation of Heterogeneous Inelastic Materials. Cambridge: Cambridge University Press. ISBN 0521839793.

• Ostoja-Starzewski, Martin (2008). Microstructural Randomness and Scaling in Mechanics of Materials. Boca Raton, FL: Chapman & Hall/CRC Press. ISBN 9781-1-58488-417-0. {{cite book}}: Check |isbn= value: length (help)

• Rees, David (2006). Basic Engineering Plasticity - An Introduction with Engineering and Manufacturing Applications. Butterworth-Heinemann. ISBN 0750680253.

• Wright, T. W. (2002). The Physics and Mathematics of Adiabatic Shear Bands. Cambridge, UK: Cambridge University Press.