Continuum mechanics
Updated: 20170831T03:48Z
Continuum mechanics  

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Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician AugustinLouis Cauchy was the first to formulate such models in the 19th century.
Contents
Explanation
Classical mechanics 

Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \vec{F} = m\vec{a} 
Branches

Core topics 
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 interatomic 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 constitutive relations.
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.
Concept of a continuum
Materials, such as solids, liquids and gases, are composed of molecules separated by space. On a microscopic 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 subdivided 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 exists. 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.^{[1]}
When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than that of the representative volume element (RVE) size, one employs a statistical volume element (SVE), which, in turn, leads to 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. The RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous.
Specifically for fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made.
Car 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 continuumdiscrete dichotomy underlying continuum modelling in general.
To start modelling define that: Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): x
measure distance (in km) along the highway; Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): tis time (in minutes); Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \rho(x,t)is the density of cars on the highway (in cars/km in the lane); and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): u(x,t)is the flow velocity (average velocity) of those cars 'at' position Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): x
.
Conservation derives a PDE
Cars do not appear and disappear.Consider any group of cars: from the particular car at the back of the group located at Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): x=a(t)
to the particular car at the front located at Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): x=b(t)
.The total number of cars in this group Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): N=\int_{a(t)}^{b(t)} \rho(x,t)\,dx.Since cars are conserved (if there is overtaking, then the `car at the front \ back' may become a different car) Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): dN/dt=0.But via the Leibniz integral rule
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \begin{array}{rcl}\frac{dN}{dt}&=&\frac{d}{dt}\int_{a(t)}^{b(t)} \rho(x,t)\,dx \\&=&\int_{a}^{b} \frac{\partial\rho}{\partial t}\,dx +\rho(b,t)\frac{db}{dt}\rho(a,t)\frac{da}{dt} \\&=&\int_{a}^{b} \frac{\partial\rho}{\partial t}\,dx +\rho(b,t)u(b,t)\rho(a,t)u(a,t) \\&=&\int_{a}^{b} \left[ \frac{\partial\rho}{\partial t}+ \frac{\partial}{\partial x}(\rho u) \right] dx \end{array}
This integral being zero holds for all groups, that is, for all intervals Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): [a,b].The only way an integral can be zero for all intervals is if the integrand is zero for all Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): x.Consequently, conservation derives the first order nonlinear conservation PDE
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \frac{\partial\rho}{\partial t}+ \frac{\partial}{\partial x}(\rho u)=0
for all positions on the highway.
This conservation PDE applies not only to car traffic but also to fluids, solids, crowds, animals, plants, bushfires, financial traders, and so on.
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, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): u=V(\rho)
for some experimentally determined function Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): Vthat is a decreasing function of density.
For example, experiments in the Lincoln Tunnel, New York, found that a good fit (except at low density) is obtained by Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): u=V(\rho)=27.5\ln(142/\rho)
(km/hr for density in cars/km).^{[2]}
Thus the basic continuum model for car traffic is the PDE
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \frac{\partial\rho}{\partial t}+ \frac{\partial}{\partial x}[\rho V(\rho)]=0
for the car density Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \rho(x,t)
on the highway.
Major areas
Continuum mechanics The study of the physics of continuous materials  Solid mechanics The study of the physics of continuous materials with a defined rest shape.  Elasticity Describes materials that return to their rest shape after applied stresses are removed.  
Plasticity Describes materials that permanently deform after a sufficient applied stress.  Rheology The study of materials with both solid and fluid characteristics.  
Fluid mechanics The study of the physics of continuous materials which deform when subjected to a force.  NonNewtonian fluids do not undergo strain rates proportional to the applied shear stress.  
Newtonian fluids undergo strain rates proportional to the applied shear stress. 
An additional area of continuum mechanics comprises elastomeric foams, which exhibit a curious hyperbolic stressstrain relationship. The elastomer is a true continuum, but a homogeneous distribution of voids gives it unusual properties.^{[3]}
Formulation of models
Continuum mechanics models begin by assigning a region in threedimensional Euclidean space to the material body Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ tis labeled Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \kappa_t(\mathcal B)
.
A particular particle within the body in a particular configuration is characterized by a position vector
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf x =\sum_{i=1}^3 x_i \mathbf e_i,
where Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf e_i
are the coordinate vectors in some frame of reference chosen for the problem (See figure 1). This vector can be expressed as a function of the particle position Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf Xin some reference configuration, for example the configuration at the initial time, so that
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{x}=\kappa_t(\mathbf X).
This function needs to have various properties so that the model makes physical sense. Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \kappa_t(\cdot)
needs to be:
 continuous in time, so that the body changes in a way which is realistic,
 globally invertible at all times, so that the body cannot intersect itself,
 orientationpreserving, as transformations which produce mirror reflections are not possible in nature.
For the mathematical formulation of the model, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \kappa_t(\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 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf F_C
and body forces Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf F_B
.^{[4]} Thus, the total force Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathcal F
applied to a body or to a portion of the body can be expressed as:
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathcal F = \mathbf F_B + \mathbf F_C
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 (EulerCauchy'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.^{[5]}
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 ^{[4]} Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf T(\mathbf n, \mathbf x, t)
that represents this distribution in a particular configuration of the body at a given time Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): t\,\!
. It is not a vector field because it depends not only on the position Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf x
of a particular material point, but also on the local orientation of the surface element as defined by its normal vector Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf n
.^{[6]}
Any differential area Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): dS\,\!
with normal vector Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf nof a given internal surface area Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): S\,\!
, bounding a portion of the body, experiences a contact force Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): d\mathbf F_C\,\!
arising from the contact between both portions of the body on each side of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): S\,\!
, and it is given by
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): d\mathbf F_C= \mathbf T^{(\mathbf n)}\,dS
where Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf T^{(\mathbf n)}
is the surface traction,^{[7]} also called stress vector,^{[8]} traction,^{[9]} or traction vector.^{[10]} The stress vector is a frameindifferent vector (see EulerCauchy's stress principle).
The total contact force on the particular internal surface Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): S\,\!
is then expressed as the sum (surface integral) of the contact forces on all differential surfaces Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): dS\,\!
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf F_C=\int_S \mathbf T^{(\mathbf n)}\,dS
In continuum mechanics a body is considered stressfree if the only forces present are those interatomic 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.^{[10]}^{[11]} 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^{[12]} 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.^{[7]} 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,^{[13]} i.e. acting on every point in it. Body forces are represented by a body force density Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf b(\mathbf x, t)
(per unit of mass), which is a frameindifferent vector field.
In the case of gravitational forces, the intensity of the force depends on, or is proportional to, the mass density Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf \rho (\mathbf x, t)\,\!
of the material, and it is specified in terms of force per unit mass (Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): b_i\,\!
) or per unit volume (Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): p_i\,\!). These two specifications are related through the material density by the equation Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \rho b_i = p_i\,\!. Similarly, the intensity of electromagnetic forces depends upon the strength (electric charge) of the electromagnetic field.
The total body force applied to a continuous body is expressed as
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf F_B=\int_V\mathbf b\,dm=\int_V \rho\mathbf b\,dV
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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathcal M
about the origin is given by
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathcal M= \mathbf M_B + \mathbf M_C
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 body moments and couple stresses^{[14]}^{[15]} (surface couples,^{[12]} contact torques^{[13]}). Body moments, or body couples, are moments per unit volume or per unit mass applied to the volume of the body. Couple stresses are moments per unit area applied on a surface. 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.^{[8]}^{[9]}^{[12]}
Materials that exhibit body couples and couple stresses in addition to moments produced exclusively by forces are called polar materials.^{[9]}^{[13]} Nonpolar 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 nonpolar 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
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathcal F = \int_V \mathbf a\,dm = \int_S \mathbf T\,dS + \int_V \rho\mathbf b\,dV
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathcal M = \int_S \mathbf r \times \mathbf T\,dS + \int_V \mathbf r \times \rho\mathbf b\,dV
Kinematics: deformation and motion
A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigidbody displacement and a deformation. A rigidbody 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \kappa_0(\mathcal B)
to a current or deformed configuration Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \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 be one that the body will ever occupy. Often, the configuration at Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ t=0
is considered the reference configuration, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \kappa_0 (\mathcal B)
. The components Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ X_i
of the position vector Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf Xof 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ 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, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \chi(\cdot)
(Figure 2),
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf x=\chi(\mathbf X, t)
which is a mapping of the initial configuration Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \kappa_0(\mathcal B)
onto the current configuration Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \kappa_t(\mathcal B)
, giving a geometrical correspondence between them, i.e. giving the position vector Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf{x}=x_i\mathbf e_i
that a particle Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ X
, with a position vector Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf X
in the undeformed or reference configuration Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \kappa_0(\mathcal B)
, will occupy in the current or deformed configuration Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \kappa_t(\mathcal B)
at time Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ t
. The components Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ x_i
are called the spatial coordinates.
Physical and kinematic properties Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ P_{ij\ldots}, 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. Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ P_{ij\ldots}=P_{ij\ldots}(\mathbf X,t).
The material derivative of any property Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ P_{ij\ldots}
is simply the partial derivative with respect to time, and the position vector Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf Xis held constant as it does not change with time. Thus, we have
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \frac{d}{dt}[P_{ij\ldots}(\mathbf X,t)]=\frac{\partial}{\partial t}[P_{ij\ldots}(\mathbf X,t)]
The instantaneous position Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf x
is a property of a particle, and its material derivative is the instantaneous flow velocity Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf vof the particle. Therefore, the flow velocity field of the continuum is given by
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf v = \dot{\mathbf x} =\frac{d\mathbf x}{dt}=\frac{\partial \chi(\mathbf X,t)}{\partial t}
Similarly, the acceleration field is given by
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf a= \dot{\mathbf v} = \ddot{\mathbf 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \chi(\cdot)
and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ P_{ij\ldots}(\cdot)are singlevalued 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \chi(\cdot)
to trace backwards where the particle currently located at Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf xwas located in the initial or referenced configuration Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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.^{[16]}
Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf X=\chi^{1}(\mathbf x, t)
which provides a tracing of the particle which now occupies the position Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf x
in the current configuration Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \kappa_t(\mathcal B)to its original position Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf Xin the initial configuration Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \kappa_0(\mathcal B)
.
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,
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ J=\left  \frac{\partial \chi_i}{\partial X_J} \right =\left  \frac{\partial x_i}{\partial X_J} \right \neq0
In the Eulerian description, the physical properties Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ P_{ij\ldots}
are expressed as
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ P_{ij \ldots}
in the Lagrangian description is not the same as the form of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ p_{ij \ldots}in the Eulerian description.
The material derivative of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ p_{ij\ldots}(\mathbf x,t), using the chain rule, is then
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \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 righthand side of this equation gives the local rate of change of the property Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ p_{ij\ldots}(\mathbf x,t)
occurring at position Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf x
. The second term of the righthand 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf x.
Displacement field
The vector joining the positions of a particle Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ P
in the undeformed configuration and deformed configuration is called the displacement vector Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf u(\mathbf X,t)=u_i\mathbf e_i
, in the Lagrangian description, or Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \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
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \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
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \alpha_{Ji}
are the direction cosines between the material and spatial coordinate systems with unit vectors Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf E_Jand Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf e_i
, respectively. Thus
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}
and the relationship between Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ u_i
and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ U_Jis then given by
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i
Knowing that
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf e_i = \alpha_{iJ}\mathbf E_J
then
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf b=0, and the direction cosines become Kronecker deltas, i.e.
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}
Thus, we have
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \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
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \ \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:
 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega
be the body (an open subset of Euclidean space) and let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \partial \Omega be its surface (the boundary of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega
).
Let the motion of material points in the body be described by the map
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{x} = \boldsymbol{\chi}(\mathbf{X}) = \mathbf{x}(\mathbf{X})
where Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{X}
is the position of a point in the initial configuration and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{x}is the location of the same point in the deformed configuration.
The deformation gradient is given by
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \nabla \boldsymbol{\mathbf{x}} ~.
Balance laws
Let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): f(\mathbf{x},t)
be a physical quantity that is flowing through the body. Let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): g(\mathbf{x},t)be sources on the surface of the body and let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): h(\mathbf{x},t)be sources inside the body. Let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{n}(\mathbf{x},t)be the outward unit normal to the surface Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \partial \Omega
. Let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{v}(\mathbf{x},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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \partial \Omega is moving be Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): u_n(in the direction Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{n}
).
Then, balance laws can be expressed in the general form
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): f(\mathbf{x},t), Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): g(\mathbf{x},t), and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 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 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)
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): { \begin{align} \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 (Cauchy's first law of motion)} \\ \boldsymbol{\sigma} & = \boldsymbol{\sigma}^T & & \qquad\text{Balance of Angular Momentum (Cauchy's second law of motion)} \\ \rho~\dot{e}  \boldsymbol{\sigma}:(\boldsymbol{\nabla}\mathbf{v}) + \boldsymbol{\nabla} \cdot \mathbf{q}  \rho~s & = 0 & & \qquad\text{Balance of Energy.} \end{align} }
In the above equations Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \rho(\mathbf{x},t)
is the mass density (current), Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \dot{\rho}is the material time derivative of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \rho
, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{v}(\mathbf{x},t)
is the particle velocity, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \dot{\mathbf{v}}is the material time derivative of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{v}
, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{\sigma}(\mathbf{x},t)
is the Cauchy stress tensor, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{b}(\mathbf{x},t)is the body force density, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): e(\mathbf{x},t)is the internal energy per unit mass, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \dot{e}is the material time derivative of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): e
, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{q}(\mathbf{x},t)
is the heat flux vector, and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): s(\mathbf{x},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
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): { \begin{align} \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{align} }
In the above, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{P}
is the first PiolaKirchhoff stress tensor, and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \rho_0is the mass density in the reference configuration. The first PiolaKirchhoff stress tensor is related to the Cauchy stress tensor by
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{P} = J~\boldsymbol{\sigma}\cdot\boldsymbol{F}^{T} ~\text{where}~ J = \det(\boldsymbol{F})
We can alternatively define the nominal stress tensor Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{N}
which is the transpose of the first PiolaKirchhoff stress tensor such that
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{N} = \boldsymbol{P}^T = J~\boldsymbol{F}^{1}\cdot\boldsymbol{\sigma} ~.
Then the balance laws become
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): { \begin{align} \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{align} }
The operators in the above equations are defined as such that
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{v}
is a vector field, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{S}is a secondorder tensor field, and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{e}_iare the components of an orthonormal basis in the current configuration. Also,
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{v}
is a vector field, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{S}is a secondorder tensor field, and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{E}_iare the components of an orthonormal basis in the reference configuration.
The inner product is defined as
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{A}:\boldsymbol{B} = \sum_{i,j=1}^3 A_{ij}~B_{ij} = \operatorname{trace}(\boldsymbol{A}\boldsymbol{B}^T) ~.
Clausius–Duhem inequality
The Clausius–Duhem inequality can be used to express the second law of thermodynamics for elasticplastic 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 (Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \eta) in the region of interest.
Let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega
be such a region and let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \partial \Omega be its boundary. Then the second law of thermodynamics states that the rate of increase of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \etain this region is greater than or equal to the sum of that supplied to Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \partial \Omega
move with a flow velocity Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): u_nand let particles inside Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omegahave velocities Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{v}
. Let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{n}
be the unit outward normal to the surface Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \partial \Omega
. Let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \rho
be the density of matter in the region, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \bar{q}be the entropy flux at the surface, and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): rbe the entropy source per unit mass.
Then the entropy inequality may be written as
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \cfrac{d}{dt}\left(\int_{\Omega} \rho~\eta~\text{dV}\right) \ge \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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \bar{q} = \boldsymbol{\psi}(\mathbf{x})\cdot\mathbf{n}. Under the assumption of incrementally isothermal conditions, we have
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \boldsymbol{\psi}(\mathbf{x}) = \cfrac{\mathbf{q}(\mathbf{x})}{T} ~;~~ r = \cfrac{s}{T}
where Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{q}
is the heat flux vector, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): sis an energy source per unit mass, and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): Tis the absolute temperature of a material point at Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathbf{x}at time Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): t
.
We then have the Clausius–Duhem inequality in integral form:
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): { \cfrac{d}{dt}\left(\int_{\Omega} \rho~\eta~\text{dV}\right) \ge \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
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): { \rho~\dot{\eta} \ge  \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
 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): { \rho~(\dot{e}  T~\dot{\eta})  \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le  \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}. }
Applications
See also
 Bernoulli's principle
 Cauchy elastic material
 Configurational mechanics
 Curvilinear coordinates
 Equation of state
 Finite deformation tensors
 Finite strain theory
 Hyperelastic material
 Lagrangian and Eulerian specification of the flow field
 Movable cellular automaton
 Peridynamics (a nonlocal continuum theory leading to integral equations)
 Stress (physics)
 Stress measures
 Tensor calculus
 Tensor derivative (continuum mechanics)
 Theory of elasticity
Notes
 ^ OstojaStarzewski, M. (2008). "710". Microstructural randomness and scaling in mechanics of materials. CRC Press. ISBN 1584884177.
 ^ A. J. Roberts, A onedimensional introduction to continuum mechanics, World Scientific, 1994
 ^ Dienes, J. K.; Solem, J. C. (1999). "Nonlinear behavior of some hydrostatically stressed isotropic elastomeric foams". Acta Mechanica. 138: 155–162.
 ^ ^{a} ^{b} Smith & Truesdell p.97
 ^ Slaughter
 ^ Lubliner
 ^ ^{a} ^{b} Liu
 ^ ^{a} ^{b} Wu
 ^ ^{a} ^{b} ^{c} Fung
 ^ ^{a} ^{b} Mase
 ^ Atanackovic
 ^ ^{a} ^{b} ^{c} Irgens
 ^ ^{a} ^{b} ^{c} Chadwick
 ^ Maxwell pointed out that nonvanishing body moments exist in a magnet in a magnetic field and in a dielectric material in an electric field with different planes of polarization. Fung p.76.
 ^ Couple stresses and body couples were first explored by Voigt and Cosserat, and later reintroduced by Mindlin in 1960 on his work for Bell Labs on pure quartz crystals. Richards p.55.
 ^ Spencer, A.J.M. (1980). Continuum Mechanics. Longman Group Limited (London). p. 83. ISBN 0582442826.
References
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 Bertram, Albrecht (2012). Elasticity and Plasticity of Large Deformations  An Introduction (Third ed.). Springer. ISBN 9783642246159.
 Chandramouli, P.N (2014). Continuum Mechanics. Yes Dee Publishing Pvt Ltd. ISBN 9789380381398.
 Eringen, A. Cemal (1980). Mechanics of Continua (2nd ed.). Krieger Pub Co. ISBN 088275663X.
 Chen, Youping; James D. Lee; Azim Eskandarian (2009). Meshless Methods in Solid Mechanics (First ed.). Springer New York. ISBN 1441921486.
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 Lubliner, Jacob (2008). Plasticity Theory (Revised Edition) (PDF). Dover Publications. ISBN 0486462900. Archived from the original (PDF) on 20100331.
 Malvern, Lawrence E. (1969). Introduction to the mechanics of a continuous medium. New Jersey: PrenticeHall, Inc.
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 Mase, G. Thomas; George E. Mase (1999). Continuum Mechanics for Engineers (Second ed.). CRC Press. ISBN 0849318556.
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