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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 continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy
Augustin-Louis Cauchy
was the first to formulate such models in the 19th century.

Contents

1 Explanation 2 Concept of a continuum 3 Car traffic as an introductory example

3.1 Conservation derives a PDE (Partial differential equation) 3.2 Observation closes the problem

4 Major areas 5 Formulation of models 6 Forces in a continuum 7 Kinematics: deformation and motion

7.1 Lagrangian description 7.2 Eulerian
Eulerian
description 7.3 Displacement field

8 Governing equations

8.1 Balance laws 8.2 Clausius–Duhem inequality

9 Applications 10 See also 11 Notes 12 References 13 External links

Explanation[edit]

Part of a series of articles about

Classical mechanics

F →

= m

a →

displaystyle vec F =m vec a

Second law of motion

History Timeline

Branches

Applied Celestial Continuum Dynamics Kinematics Kinetics Statics Statistical

Fundamentals

Acceleration Angular momentum Couple D'Alembert's principle Energy

kinetic potential

Force Frame of reference Inertial frame of reference Impulse Inertia / Moment of inertia Mass

Mechanical power Mechanical work

Moment Momentum Space Speed Time Torque Velocity Virtual work

Formulations

Newton's laws of motion

Analytical mechanics

Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton–Jacobi equation Appell's equation of motion Udwadia–Kalaba equation Koopman–von Neumann mechanics

Core topics

Damping (ratio) Displacement Equations of motion Euler's laws of motion Fictitious force Friction Harmonic oscillator

Inertial / Non-inertial reference frame Mechanics
Mechanics
of planar particle motion

Motion (linear) Newton's law of universal gravitation Newton's laws of motion Relative velocity Rigid body

dynamics Euler's equations

Simple harmonic motion Vibration

Rotation

Circular motion Rotating reference frame Centripetal force Centrifugal force

reactive

Coriolis force Pendulum Tangential speed Rotational speed

Angular acceleration / displacement / frequency / velocity

Scientists

Galileo Huygens Newton Kepler Horrocks Halley Euler d'Alembert Clairaut Lagrange Laplace Hamilton Poisson Daniel Bernoulli Johann Bernoulli Cauchy

v t e

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 constitutive relations. Continuum mechanics
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[edit] 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 sub-divided into infinitesimal elements with properties being those of the bulk material. The validity of the continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the microstructure 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[edit] Consider car traffic on a highway, with just one lane for simplicity. Somewhat surprisingly, and in a tribute to its effectiveness, continuum mechanics effectively models the movement of cars via a partial differential equation (PDE) for the density of cars. The familiarity of this situation empowers us to understand a little of the continuum-discrete dichotomy underlying continuum modelling in general. To start modelling define that:

x

displaystyle x

measure distance (in km) along the highway;

t

displaystyle t

is time (in minutes);

ρ ( x , t )

displaystyle rho (x,t)

is the density of cars on the highway (in cars/km in the lane); and

u ( x , t )

displaystyle u(x,t)

is the flow velocity (average velocity) of those cars 'at' position

x

displaystyle x

. Conservation derives a PDE (Partial differential equation)[edit] Cars do not appear and disappear. Consider any group of cars: from the particular car at the back of the group located at

x = a ( t )

displaystyle x=a(t)

to the particular car at the front located at

x = b ( t )

displaystyle x=b(t)

. The total number of cars in this group

N =

a ( t )

b ( t )

ρ ( x , t )

d x

displaystyle 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)

d N

/

d t = 0

displaystyle dN/dt=0

. But via the Leibniz integral rule

d N

d t

=

d

d t

a ( t )

b ( t )

ρ ( x , t )

d x

=

a

b

∂ ρ

∂ t

d x + ρ ( b , t )

d b

d t

− ρ ( a , t )

d a

d t

=

a

b

∂ ρ

∂ t

d x + ρ ( b , t ) u ( b , t ) − ρ ( a , t ) u ( a , t )

=

a

b

[

∂ ρ

∂ t

+

∂ x

( ρ u )

]

d x

displaystyle 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]dxend array

This integral being zero holds for all groups, that is, for all intervals

[ a , b ]

displaystyle [a,b]

. The only way an integral can be zero for all intervals is if the integrand is zero for all

x

displaystyle x

. Consequently, conservation derives the first order nonlinear conservation PDE

∂ ρ

∂ t

+

∂ x

( ρ u ) = 0

displaystyle 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[edit] The previous PDE is one equation with two unknowns, so another equation is needed to form a well posed problem. Such an extra equation is typically needed in continuum mechanics and typically comes from experiments. For car traffic it is well established that cars typically travel at a speed depending upon density,

u = V ( ρ )

displaystyle u=V(rho )

for some experimentally determined function

V

displaystyle V

that 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

u = V ( ρ ) = 27.5 ln ⁡ ( 142

/

ρ )

displaystyle u=V(rho )=27.5ln(142/rho )

(km/hr for density in cars/km).[2] Thus the basic continuum model for car traffic is the PDE

∂ ρ

∂ t

+

∂ x

[ ρ V ( ρ ) ] = 0

displaystyle frac partial rho partial t + frac partial partial x [rho V(rho )]=0

for the car density

ρ ( x , t )

displaystyle rho (x,t)

on the highway. Major areas[edit]

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. Non-Newtonian 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 stress-strain relationship. The elastomer is a true continuum, but a homogeneous distribution of voids gives it unusual properties.[3] Formulation of models[edit]

Figure 1. Configuration of a continuum body

Continuum mechanics
Continuum mechanics
models begin by assigning a region in three-dimensional Euclidean space
Euclidean space
to the material body

B

displaystyle mathcal B

being modeled. The points within this region are called particles or material points. Different configurations or states of the body correspond to different regions in Euclidean space. The region corresponding to the body's configuration at time

  t

displaystyle t

is labeled

 

κ

t

(

B

)

displaystyle kappa _ t ( mathcal B )

. A particular particle within the body in a particular configuration is characterized by a position vector

 

x

=

i = 1

3

x

i

e

i

,

displaystyle mathbf x =sum _ i=1 ^ 3 x_ i mathbf e _ i ,

where

e

i

displaystyle 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

X

displaystyle mathbf X

in some reference configuration, for example the configuration at the initial time, so that

x

=

κ

t

(

X

) .

displaystyle mathbf x =kappa _ t (mathbf X ).

This function needs to have various properties so that the model makes physical sense.

κ

t

( ⋅ )

displaystyle 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, orientation-preserving, as transformations which produce mirror reflections are not possible in nature.

For the mathematical formulation of the model,

 

κ

t

( ⋅ )

displaystyle 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[edit] See also: Stress (mechanics)
Stress (mechanics)
and Cauchy stress tensor Continuum mechanics
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

F

C

displaystyle mathbf F _ C

and body forces

F

B

displaystyle mathbf F _ B

.[4] Thus, the total force

F

displaystyle mathcal F

applied to a body or to a portion of the body can be expressed as:

F

=

F

B

+

F

C

displaystyle 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 (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.[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]

T

(

n

,

x

, t )

displaystyle mathbf T (mathbf n ,mathbf x ,t)

that represents this distribution in a particular configuration of the body at a given time

t

displaystyle t,!

. It is not a vector field because it depends not only on the position

x

displaystyle mathbf x

of a particular material point, but also on the local orientation of the surface element as defined by its normal vector

n

displaystyle mathbf n

.[6] Any differential area

d S

displaystyle dS,!

with normal vector

n

displaystyle mathbf n

of a given internal surface area

S

displaystyle S,!

, bounding a portion of the body, experiences a contact force

d

F

C

displaystyle dmathbf F _ C ,!

arising from the contact between both portions of the body on each side of

S

displaystyle S,!

, and it is given by

d

F

C

=

T

(

n

)

d S

displaystyle dmathbf F _ C =mathbf T ^ (mathbf n ) ,dS

where

T

(

n

)

displaystyle 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 frame-indifferent vector (see Euler-Cauchy's stress principle). The total contact force on the particular internal surface

S

displaystyle S,!

is then expressed as the sum (surface integral) of the contact forces on all differential surfaces

d S

displaystyle dS,!

:

F

C

=

S

T

(

n

)

d S

displaystyle mathbf F _ C =int _ S mathbf T ^ (mathbf 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.[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

b

(

x

, t )

displaystyle mathbf b (mathbf x ,t)

(per unit of mass), which is a frame-indifferent vector field. In the case of gravitational forces, the intensity of the force depends on, or is proportional to, the mass density

ρ

(

x

, t )

displaystyle mathbf rho (mathbf x ,t),!

of the material, and it is specified in terms of force per unit mass (

b

i

displaystyle b_ i ,!

) or per unit volume (

p

i

displaystyle p_ i ,!

). These two specifications are related through the material density by the equation

ρ

b

i

=

p

i

displaystyle 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

F

B

=

V

b

d m =

V

ρ

b

d V

displaystyle 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

M

displaystyle mathcal M

about the origin is given by

M

=

M

B

+

M

C

displaystyle 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] 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

F

=

V

a

d m =

S

T

d S +

V

ρ

b

d V

displaystyle mathcal F =int _ V mathbf a ,dm=int _ S mathbf T ,dS+int _ V rho mathbf b ,dV

M

=

S

r

×

T

d S +

V

r

× ρ

b

d V

displaystyle mathcal M =int _ S mathbf r times mathbf T ,dS+int _ V mathbf r times rho mathbf b ,dV

Kinematics: deformation and motion[edit]

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 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

 

κ

0

(

B

)

displaystyle kappa _ 0 ( mathcal B )

to a current or deformed configuration

 

κ

t

(

B

)

displaystyle 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

  t = 0

displaystyle t=0

is considered the reference configuration,

 

κ

0

(

B

)

displaystyle kappa _ 0 ( mathcal B )

. The components

 

X

i

displaystyle X_ i

of the position vector

 

X

displaystyle 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[edit] 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

displaystyle t=0

. An observer standing in the frame of reference observes the changes in the position and physical properties as the material body moves in space as time progresses. The results obtained are independent of the choice of initial time and reference configuration,

κ

0

(

B

)

displaystyle 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

  χ ( ⋅ )

displaystyle chi (cdot )

(Figure 2),

 

x

= χ (

X

, t )

displaystyle mathbf x =chi (mathbf X ,t)

which is a mapping of the initial configuration

κ

0

(

B

)

displaystyle kappa _ 0 ( mathcal B )

onto the current configuration

κ

t

(

B

)

displaystyle kappa _ t ( mathcal B )

, giving a geometrical correspondence between them, i.e. giving the position vector

 

x

=

x

i

e

i

displaystyle mathbf x =x_ i mathbf e _ i

that a particle

  X

displaystyle X

, with a position vector

 

X

displaystyle mathbf X

in the undeformed or reference configuration

κ

0

(

B

)

displaystyle kappa _ 0 ( mathcal B )

, will occupy in the current or deformed configuration

κ

t

(

B

)

displaystyle kappa _ t ( mathcal B )

at time

  t

displaystyle t

. The components

 

x

i

displaystyle x_ i

are called the spatial coordinates. Physical and kinematic properties

 

P

i j …

displaystyle P_ ijldots

, i.e. thermodynamic properties and flow velocity, which describe or characterize features of the material body, are expressed as continuous functions of position and time, i.e.

 

P

i j …

=

P

i j …

(

X

, t )

displaystyle P_ ijldots =P_ ijldots (mathbf X ,t)

. The material derivative of any property

 

P

i j …

displaystyle P_ ijldots

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

i j …

displaystyle P_ ijldots

is simply the partial derivative with respect to time, and the position vector

 

X

displaystyle mathbf X

is held constant as it does not change with time. Thus, we have

 

d

d t

[

P

i j …

(

X

, t ) ] =

∂ t

[

P

i j …

(

X

, t ) ]

displaystyle frac d dt [P_ ijldots (mathbf X ,t)]= frac partial partial t [P_ ijldots (mathbf X ,t)]

The instantaneous position

 

x

displaystyle mathbf x

is a property of a particle, and its material derivative is the instantaneous flow velocity

 

v

displaystyle mathbf v

of the particle. Therefore, the flow velocity field of the continuum is given by

 

v

=

x

˙

=

d

x

d t

=

∂ χ (

X

, t )

∂ t

displaystyle mathbf v = dot mathbf x = frac dmathbf x dt = frac partial chi (mathbf X ,t) partial t

Similarly, the acceleration field is given by

 

a

=

v

˙

=

x

¨

=

d

2

x

d

t

2

=

2

χ (

X

, t )

t

2

displaystyle 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

χ ( ⋅ )

displaystyle chi (cdot )

and

 

P

i j …

( ⋅ )

displaystyle P_ ijldots (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
Eulerian
description[edit] Continuity allows for the inverse of

χ ( ⋅ )

displaystyle chi (cdot )

to trace backwards where the particle currently located at

x

displaystyle mathbf x

was located in the initial or referenced configuration

κ

0

(

B

)

displaystyle 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
Eulerian
description, i.e. the current configuration is taken as the reference configuration. The Eulerian
Eulerian
description, introduced by d'Alembert, focuses on the current configuration

κ

t

(

B

)

displaystyle 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

X

=

χ

− 1

(

x

, t )

displaystyle mathbf X =chi ^ -1 (mathbf x ,t)

which provides a tracing of the particle which now occupies the position

x

displaystyle mathbf x

in the current configuration

κ

t

(

B

)

displaystyle kappa _ t ( mathcal B )

to its original position

X

displaystyle mathbf X

in the initial configuration

κ

0

(

B

)

displaystyle 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,

  J =

χ

i

X

J

=

x

i

X

J

≠ 0

displaystyle J=left frac partial chi _ i partial X_ J right=left frac partial x_ i partial X_ J rightneq 0

In the Eulerian
Eulerian
description, the physical properties

 

P

i j …

displaystyle P_ ijldots

are expressed as

 

P

i j …

=

P

i j …

(

X

, t ) =

P

i j …

[

χ

− 1

(

x

, t ) , t ] =

p

i j …

(

x

, t )

displaystyle P_ ijldots =P_ ijldots (mathbf X ,t)=P_ ijldots [chi ^ -1 (mathbf x ,t),t]=p_ ijldots (mathbf x ,t)

where the functional form of

 

P

i j …

displaystyle P_ ijldots

in the Lagrangian description is not the same as the form of

 

p

i j …

displaystyle p_ ijldots

in the Eulerian
Eulerian
description. The material derivative of

 

p

i j …

(

x

, t )

displaystyle p_ ijldots (mathbf x ,t)

, using the chain rule, is then

 

d

d t

[

p

i j …

(

x

, t ) ] =

∂ t

[

p

i j …

(

x

, t ) ] +

x

k

[

p

i j …

(

x

, t ) ]

d

x

k

d t

displaystyle frac d dt [p_ ijldots (mathbf x ,t)]= frac partial partial t [p_ ijldots (mathbf x ,t)]+ frac partial partial x_ k [p_ ijldots (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

i j …

(

x

, t )

displaystyle p_ ijldots (mathbf x ,t)

occurring at position

 

x

displaystyle 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
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

displaystyle mathbf x

. Displacement field[edit] The vector joining the positions of a particle

  P

displaystyle P

in the undeformed configuration and deformed configuration is called the displacement vector

 

u

(

X

, t ) =

u

i

e

i

displaystyle mathbf u (mathbf X ,t)=u_ i mathbf e _ i

, in the Lagrangian description, or

 

U

(

x

, t ) =

U

J

E

J

displaystyle mathbf U (mathbf x ,t)=U_ J mathbf E _ J

, in the Eulerian
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

 

u

(

X

, t ) =

b

+

x

(

X

, t ) −

X

or

u

i

=

α

i J

b

J

+

x

i

α

i J

X

J

displaystyle 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

 

U

(

x

, t ) =

b

+

x

X

(

x

, t )

or

U

J

=

b

J

+

α

J i

x

i

X

J

displaystyle 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

 

α

J i

displaystyle alpha _ Ji

are the direction cosines between the material and spatial coordinate systems with unit vectors

 

E

J

displaystyle mathbf E _ J

and

e

i

displaystyle mathbf e _ i

, respectively. Thus

 

E

J

e

i

=

α

J i

=

α

i J

displaystyle mathbf E _ J cdot mathbf e _ i =alpha _ Ji =alpha _ iJ

and the relationship between

 

u

i

displaystyle u_ i

and

 

U

J

displaystyle U_ J

is then given by

 

u

i

=

α

i J

U

J

or

U

J

=

α

J i

u

i

displaystyle u_ i =alpha _ iJ U_ J qquad text or qquad U_ J =alpha _ Ji u_ i

Knowing that

 

e

i

=

α

i J

E

J

displaystyle mathbf e _ i =alpha _ iJ mathbf E _ J

then

u

(

X

, t ) =

u

i

e

i

=

u

i

(

α

i J

E

J

) =

U

J

E

J

=

U

(

x

, t )

displaystyle 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

 

b

= 0

displaystyle mathbf b =0

, and the direction cosines become Kronecker deltas, i.e.

 

E

J

e

i

=

δ

J i

=

δ

i J

displaystyle mathbf E _ J cdot mathbf e _ i =delta _ Ji =delta _ iJ

Thus, we have

 

u

(

X

, t ) =

x

(

X

, t ) −

X

or

u

i

=

x

i

δ

i J

X

J

displaystyle 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

 

U

(

x

, t ) =

x

X

(

x

, t )

or

U

J

=

δ

J i

x

i

X

J

displaystyle 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[edit] Continuum mechanics
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
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

Ω

displaystyle Omega

be the body (an open subset of Euclidean space) and let

∂ Ω

displaystyle partial Omega

be its surface (the boundary of

Ω

displaystyle Omega

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

x

=

χ

(

X

) =

x

(

X

)

displaystyle mathbf x = boldsymbol chi (mathbf X )=mathbf x (mathbf X )

where

X

displaystyle mathbf X

is the position of a point in the initial configuration and

x

displaystyle mathbf x

is the location of the same point in the deformed configuration. The deformation gradient is given by

F

=

x

X

= ∇

x

  .

displaystyle boldsymbol F = frac partial mathbf x partial mathbf X =nabla boldsymbol mathbf x ~.

Balance laws[edit] Let

f (

x

, t )

displaystyle f(mathbf x ,t)

be a physical quantity that is flowing through the body. Let

g (

x

, t )

displaystyle g(mathbf x ,t)

be sources on the surface of the body and let

h (

x

, t )

displaystyle h(mathbf x ,t)

be sources inside the body. Let

n

(

x

, t )

displaystyle mathbf n (mathbf x ,t)

be the outward unit normal to the surface

∂ Ω

displaystyle partial Omega

. Let

v

(

x

, t )

displaystyle 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

∂ Ω

displaystyle partial Omega

is moving be

u

n

displaystyle u_ n

(in the direction

n

displaystyle mathbf n

). Then, balance laws can be expressed in the general form

d

d t

[

Ω

f (

x

, t )  

dV

]

=

∂ Ω

f (

x

, t ) [

u

n

(

x

, t ) −

v

(

x

, t ) ⋅

n

(

x

, t ) ]  

dA

+

∂ Ω

g (

x

, t )  

dA

+

Ω

h (

x

, t )  

dV

  .

displaystyle 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 (

x

, t )

displaystyle f(mathbf x ,t)

,

g (

x

, t )

displaystyle g(mathbf x ,t)

, and

h (

x

, t )

displaystyle 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
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)

ρ ˙

+ ρ  

v

= 0

Balance of Mass

ρ  

v

˙

σ

− ρ  

b

= 0

Balance of Linear Momentum
Momentum
(Cauchy's first law of motion)

σ

=

σ

T

Balance of Angular Momentum
Momentum
(Cauchy's second law of motion)

ρ  

e ˙

σ

: (

v

) +

q

− ρ   s

= 0

Balance of Energy.

displaystyle begin aligned dot rho +rho ~ boldsymbol nabla cdot mathbf v &=0&&qquad text Balance of Mass
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
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 aligned

In the above equations

ρ (

x

, t )

displaystyle rho (mathbf x ,t)

is the mass density (current),

ρ ˙

displaystyle dot rho

is the material time derivative of

ρ

displaystyle rho

,

v

(

x

, t )

displaystyle mathbf v (mathbf x ,t)

is the particle velocity,

v

˙

displaystyle dot mathbf v

is the material time derivative of

v

displaystyle mathbf v

,

σ

(

x

, t )

displaystyle boldsymbol sigma (mathbf x ,t)

is the Cauchy stress tensor,

b

(

x

, t )

displaystyle mathbf b (mathbf x ,t)

is the body force density,

e (

x

, t )

displaystyle e(mathbf x ,t)

is the internal energy per unit mass,

e ˙

displaystyle dot e

is the material time derivative of

e

displaystyle e

,

q

(

x

, t )

displaystyle mathbf q (mathbf x ,t)

is the heat flux vector, and

s (

x

, t )

displaystyle 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

ρ   det (

F

) −

ρ

0

= 0

Balance of Mass

ρ

0

 

x

¨

P

T

ρ

0

 

b

= 0

Balance of Linear Momentum

F

P

T

=

P

F

T

Balance of Angular Momentum

ρ

0

 

e ˙

P

T

:

F ˙

+

q

ρ

0

  s

= 0

Balance of Energy.

displaystyle begin aligned rho ~det( boldsymbol F )-rho _ 0 &=0&&qquad text Balance of Mass
Mass
\rho _ 0 ~ ddot mathbf x - boldsymbol nabla _ circ cdot boldsymbol P ^ T -rho _ 0 ~mathbf b &=0&&qquad text Balance of Linear Momentum
Momentum
\ boldsymbol F cdot boldsymbol P ^ T &= boldsymbol P cdot boldsymbol F ^ T &&qquad text Balance of Angular Momentum
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,

P

displaystyle boldsymbol P

is the first Piola-Kirchhoff stress tensor, and

ρ

0

displaystyle rho _ 0

is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor
Piola-Kirchhoff stress tensor
is related to the Cauchy stress tensor by

P

= J  

σ

F

− T

 

where

  J = det (

F

)

displaystyle boldsymbol P =J~ boldsymbol sigma cdot boldsymbol F ^ -T ~ text where ~J=det( boldsymbol F )

We can alternatively define the nominal stress tensor

N

displaystyle boldsymbol N

which is the transpose of the first Piola-Kirchhoff stress tensor such that

N

=

P

T

= J  

F

− 1

σ

  .

displaystyle boldsymbol N = boldsymbol P ^ T =J~ boldsymbol F ^ -1 cdot boldsymbol sigma ~.

Then the balance laws become

ρ   det (

F

) −

ρ

0

= 0

Balance of Mass

ρ

0

 

x

¨

N

ρ

0

 

b

= 0

Balance of Linear Momentum

F

N

=

N

T

F

T

Balance of Angular Momentum

ρ

0

 

e ˙

N

:

F ˙

+

q

ρ

0

  s

= 0

Balance of Energy.

displaystyle begin aligned rho ~det( boldsymbol F )-rho _ 0 &=0&&qquad text Balance of Mass
Mass
\rho _ 0 ~ ddot mathbf x - boldsymbol nabla _ circ cdot boldsymbol N -rho _ 0 ~mathbf b &=0&&qquad text Balance of Linear Momentum
Momentum
\ boldsymbol F cdot boldsymbol N &= boldsymbol N ^ T cdot boldsymbol F ^ T &&qquad text Balance of Angular Momentum
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

v

=

i , j = 1

3

v

i

x

j

e

i

e

j

=

v

i , j

e

i

e

j

  ;    

v

=

i = 1

3

v

i

x

i

=

v

i , i

  ;    

S

=

i , j = 1

3

S

i j

x

j

 

e

i

=

σ

i j , j

 

e

i

  .

displaystyle 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

v

displaystyle mathbf v

is a vector field,

S

displaystyle boldsymbol S

is a second-order tensor field, and

e

i

displaystyle mathbf e _ i

are the components of an orthonormal basis in the current configuration. Also,

v

=

i , j = 1

3

v

i

X

j

E

i

E

j

=

v

i , j

E

i

E

j

  ;    

v

=

i = 1

3

v

i

X

i

=

v

i , i

  ;    

S

=

i , j = 1

3

S

i j

X

j

 

E

i

=

S

i j , j

 

E

i

displaystyle 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

v

displaystyle mathbf v

is a vector field,

S

displaystyle boldsymbol S

is a second-order tensor field, and

E

i

displaystyle mathbf E _ i

are the components of an orthonormal basis in the reference configuration. The inner product is defined as

A

:

B

=

i , j = 1

3

A

i j

 

B

i j

= trace ⁡ (

A

B

T

)   .

displaystyle boldsymbol A : boldsymbol B =sum _ i,j=1 ^ 3 A_ ij ~B_ ij =operatorname trace ( boldsymbol A boldsymbol B ^ T )~.

Clausius–Duhem inequality[edit] 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

ρ

displaystyle rho

and an internal specific entropy (i.e. entropy per unit mass)

η

displaystyle eta

in the region of interest. Let

Ω

displaystyle Omega

be such a region and let

∂ Ω

displaystyle partial Omega

be its boundary. Then the second law of thermodynamics states that the rate of increase of

η

displaystyle eta

in this region is greater than or equal to the sum of that supplied to

Ω

displaystyle Omega

(as a flux or from internal sources) and the change of the internal entropy density

ρ η

displaystyle rho eta

due to material flowing in and out of the region. Let

∂ Ω

displaystyle partial Omega

move with a flow velocity

u

n

displaystyle u_ n

and let particles inside

Ω

displaystyle Omega

have velocities

v

displaystyle mathbf v

. Let

n

displaystyle mathbf n

be the unit outward normal to the surface

∂ Ω

displaystyle partial Omega

. Let

ρ

displaystyle rho

be the density of matter in the region,

q ¯

displaystyle bar q

be the entropy flux at the surface, and

r

displaystyle r

be the entropy source per unit mass. Then the entropy inequality may be written as

d

d t

(

Ω

ρ   η  

dV

)

∂ Ω

ρ   η   (

u

n

v

n

)  

dA

+

∂ Ω

q ¯

 

dA

+

Ω

ρ   r  

dV

.

displaystyle 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

q ¯

= −

ψ

(

x

) ⋅

n

displaystyle bar q =- boldsymbol psi (mathbf x )cdot mathbf n

. Under the assumption of incrementally isothermal conditions, we have

ψ

(

x

) =

q

(

x

)

T

  ;     r =

s

T

displaystyle boldsymbol psi (mathbf x )= cfrac mathbf q (mathbf x ) T ~;~~r= cfrac s T

where

q

displaystyle mathbf q

is the heat flux vector,

s

displaystyle s

is an energy source per unit mass, and

T

displaystyle T

is the absolute temperature of a material point at

x

displaystyle mathbf x

at time

t

displaystyle t

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

d

d t

(

Ω

ρ   η  

dV

)

∂ Ω

ρ   η   (

u

n

v

n

)  

dA

∂ Ω

q

n

T

 

dA

+

Ω

ρ   s

T

 

dV

.

displaystyle 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

ρ  

η ˙

≥ −

(

q

T

)

+

ρ   s

T

.

displaystyle 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

ρ   (

e ˙

− T  

η ˙

) −

σ

:

v

≤ −

q

T

T

.

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

Applications[edit]

Continuum Mechanics

Solid mechanics Fluid mechanics

Engineering

Civil engineering Mechanical engineering Aerospace engineering Biomedical engineering Chemical engineering

See also[edit]

Bernoulli's principle Cauchy elastic material Configurational mechanics Curvilinear coordinates Equation of state Finite deformation tensors Finite strain theory Hyperelastic material Lagrangian and Eulerian
Eulerian
specification of the flow field Movable cellular automaton Peridynamics
Peridynamics
(a non-local continuum theory leading to integral equations) Stress (physics) Stress measures Tensor
Tensor
calculus Tensor
Tensor
derivative (continuum mechanics) Theory of elasticity

Notes[edit]

^ Ostoja-Starzewski, M. (2008). "7-10". Microstructural randomness and scaling in mechanics of materials. CRC Press. ISBN 1-58488-417-7.  ^ A. J. Roberts, A one-dimensional 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 0-582-44282-6. 

References[edit]

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

Bertram, Albrecht (2012). Elasticity and Plasticity of Large Deformations - An Introduction (Third ed.). Springer. ISBN 978-3-642-24615-9. 

Chandramouli, P.N (2014). Continuum Mechanics. Yes Dee Publishing Pvt Ltd. ISBN 9789380381398. 

Eringen, A. Cemal (1980). Mechanics
Mechanics
of Continua (2nd ed.). Krieger Pub Co. ISBN 0-88275-663-X. 

Chen, Youping; James D. Lee; Azim Eskandarian (2009). Meshless Methods in Solid Mechanics
Mechanics
(First ed.). Springer New York. ISBN 1-4419-2148-6. 

Dill, Ellis Harold (2006). Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity. Germany: CRC Press. ISBN 0-8493-9779-0. 

Dimitrienko, Yuriy (2011). Nonlinear Continuum Mechanics
Mechanics
and Large Inelastic Deformations. Germany: Springer. ISBN 978-94-007-0033-8. 

Hutter, Kolumban; Klaus Jöhnk (2004). Continuum Methods of Physical Modeling. Germany: Springer. ISBN 3-540-20619-1. 

Fung, Y. C. (1977). A First Course in Continuum Mechanics
Mechanics
(2nd ed.). Prentice-Hall, Inc. ISBN 0-13-318311-4. 

Gurtin, M. E. (1981). An Introduction to Continuum Mechanics. New York: Academic Press.  Lai, W. Michael; David Rubin; Erhard Krempl (1996). Introduction to Continuum Mechanics
Mechanics
(3rd ed.). Elsevier, Inc. ISBN 978-0-7506-2894-5. Archived from the original on 2009-02-06. 

Lubarda, Vlado A. (2001). Elastoplasticity Theory. CRC Press. ISBN 0-8493-1138-1. 

Lubliner, Jacob (2008). Plasticity Theory (Revised Edition) (PDF). Dover Publications. ISBN 0-486-46290-0. Archived from the original (PDF) on 2010-03-31. 

Malvern, Lawrence E. (1969). Introduction to the mechanics of a continuous medium. New Jersey: Prentice-Hall, Inc. 

Mase, George E. (1970). Continuum Mechanics. McGraw-Hill Professional. ISBN 0-07-040663-4. 

Mase, G. Thomas; George E. Mase (1999). Continuum Mechanics
Mechanics
for Engineers (Second ed.). CRC Press. ISBN 0-8493-1855-6. 

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 0-521-83979-3. 

Ostoja-Starzewski, Martin (2008). Microstructural Randomness and Scaling in Mechanics
Mechanics
of Materials. Boca Raton, FL: Chapman & Hall/CRC Press. ISBN 978-1-58488-417-0. 

Rees, David (2006). Basic Engineering
Engineering
Plasticity - An Introduction with Engineering
Engineering
and Manufacturing Applications. Butterworth-Heinemann. ISBN 0-7506-8025-3. 

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

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moment of inertia angular momentum tensor spin tensor Cauchy stress tensor stress–energy tensor EM tensor gluon field strength tensor Einstein tensor metric tensor (GR)

Mathematicians

Leonhard Euler Carl Friedrich Gauss Augustin-Louis Cauchy Hermann Grassmann Gregorio Ricci-Curbastro Tullio Levi-Civita Jan Arnoldus Schouten Bernhard Riemann Elwin Bruno Christoffel Woldemar Voigt Élie Cartan Hermann Weyl Albert Einstein

Authority control

GND: 40322

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