# continuum mechanics

• It is not a vector field because it depends not only on the position of a particular material point, but also on the local orientation of the surface element as defined by
its normal vector .

• This vector can be expressed as a function of the particle position in some reference configuration, for example the configuration at the initial time, so that This function
needs to have various properties so that the model makes physical sense.

• All physical quantities are defined this way at each instant of time, in the current configuration, as a function of the vector position .

• The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical
laws, such as mass conservation, momentum conservation, and energy conservation.

• If we take the Eulerian point of view, it can be shown that the balance laws of mass, momentum, and energy for a solid can be written as (assuming the source term is zero
for the mass and angular momentum equations) In the above equations is the mass density (current), is the material time derivative of , is the particle velocity, is the material time derivative of , is the Cauchy stress tensor, is the body
force density, is the internal energy per unit mass, is the material time derivative of , is the heat flux vector, and is an energy source per unit mass.

• Therefore, there exists a contact force density or Cauchy traction field[4] that represents this distribution in a particular configuration of the body at a given time .

• With respect to the reference configuration (the Lagrangian point of view), the balance laws can be written as In the above, is the first Piola-Kirchhoff stress tensor, and
is the mass density in the reference configuration.

• A particular particle within the body in a particular configuration is characterized by a position vector where are the coordinate vectors in some frame of reference chosen
for the problem (See figure 1).

• 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).

• 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.

• [5][page needed] Any differential area with normal vector of a given internal surface area , bounding a portion of the body, experiences a contact force arising from the contact
between both portions of the body on each side of , and it is given by where is the surface traction,[6] also called stress vector,[7] traction,[8][page needed] or traction vector.

• 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 or in terms of the spatial coordinates as where are the direction cosines between the material and spatial coordinate systems with unit vectors and , respectively.

• Thus, the total applied torque about the origin is given by In certain situations, not commonly considered in the analysis of the mechanical behavior of materials, it becomes
necessary to include two other types of forces: these are couple stresses[note 1][note 2] (surface couples,[11] contact torques)[12] and body moments.

• [3] Thus, the total force applied to a body or to a portion of the body can be expressed as: Surface forces Surface forces or contact forces, expressed as force per
unit area, can act either on the bounding surface of the body, as a result of mechanical contact with other bodies, or on imaginary internal surfaces that bound portions of the body, as a result of the mechanical interaction between the parts
of the body to either side of the surface (Euler-Cauchy’s stress principle).

• 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.

• 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 Kinematics: motion and deformation A change
in the configuration of a continuum body results in a displacement.

• In the Lagrangian description, the motion of a continuum body is expressed by the mapping function (Figure 2), which is a mapping of the initial configuration onto the current
configuration , giving a geometrical correspondence between them, i.e.

• [14] Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function which provides a tracing of the particle which now occupies
the position in the current configuration to its original position in the initial configuration .

• The total contact force on the particular internal surface is then expressed as the sum (surface integral) of the contact forces on all differential surfaces : 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.

• Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body,[12] i.e.

• Thus, we have or in terms of the spatial coordinates as Governing equations Continuum mechanics deals with the behavior of materials that can be approximated as continuous
for certain length and time scales.

• 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 components of the position vector of a particle, taken with respect to the reference configuration, are called the material or reference coordinates.

• The balance laws express the idea that the rate of change of a quantity (mass, momentum, energy) in a volume must arise from three causes: 1. the physical quantity itself
flows through the surface that bounds the volume, 2. there is a source of the physical quantity on the surface of the volume, or/and, 3. there is a source of the physical quantity inside the volume.

• The Eulerian description, introduced by d’Alembert, focuses on the current configuration , 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.

• Then, balance laws can be expressed in the general form The functions , , and can be scalar valued, vector valued, or tensor valued – depending on the physical quantity that
the balance equation deals with.

• When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than the size of the representative volume element (RVE), a statistical
volume element (SVE) is employed, which results in random continuum fields.

• In this sense, the function and 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.

• The material derivative of any property 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 of , using the chain rule, is then The first term on the right-hand side of this equation gives the local rate of change of the property occurring
at position .

• Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium (also called a
continuum) rather than as discrete particles.

• 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.

• There is continuity during motion or deformation of a continuum body in the sense that: • The material points forming a closed curve at any instant will always form a closed
curve at any subsequent time.

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

• Thus, the material body will occupy different configurations at different times so that a particle occupies a series of points in space which describe a path line.

• 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;
Fluid mechanics: The study of the physics of continuous materials which deform when subjected to a force; 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; Non-Newtonian fluid: Do not undergo strain rates proportional to the applied shear stress; Newtonian fluids: undergo strain rates proportional to the applied shear stress; Rheology: The
study of materials with both solid and fluid characteristics.

• 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.

• Therefore, the flow velocity field of the continuum is given by Similarly, the acceleration field is given by 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.

• giving the position vector that a particle , with a position vector in the undeformed or reference configuration , will occupy in the current or deformed configuration at
time .

• A continuum is a body that can be continually sub-divided into infinitesimal elements with local material properties defined at any particular point.

• Let the motion of material points in the body be described by the map where is the position of a point in the initial configuration and is the location of the same point in
the deformed configuration.

• In the Lagrangian description, the material derivative of is simply the partial derivative with respect to time, and the position vector is held constant as it does not change
with time.

• Thus and the relationship between and is then given by Knowing that then It is common to superimpose the coordinate systems for the undeformed and deformed configurations,
which results in , and the direction cosines become Kronecker deltas, i.e.

• Then the second law of thermodynamics states that the rate of increase of in this region is greater than or equal to the sum of that supplied to (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.

• 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.

• 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 and body forces .

• Although materials are composed of discrete atoms and molecules, separated by empty space or microscopic cracks and crystallographic defects, physical phenomena can often
be modeled by considering a substance distributed throughout some region of space.

• Properties of the bulk material can therefore be described by continuous functions, and their evolution can be studied using the mathematics of calculus.

• A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size.

• This permits definition of physical properties at any point in the continuum, according to mathematically convenient continuous functions.

• The reference configuration need not be one that the body will ever occupy.

• the current configuration is taken as the reference configuration.

• 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.

• One description for motion is made in terms of the material or referential coordinates, called material description or Lagrangian description.

Works Cited

[‘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.[13]
• ^ 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.[citation needed]
Citations
1. ^ Malvern 1969, p. 2.
2. ^ Dienes & Solem 1999, pp. 155–162.
3. ^ Smith 1993, p. 97.
4. ^ Smith
1993.
5. ^ Lubliner 2008.
6. ^ Jump up to:a b Liu 2002.
7. ^ Jump up to:a b Wu 2004.
8. ^ Jump up to:a b c Fung 1977.
9. ^ Jump up to:a b Mase 1970.
10. ^ Atanackovic & Guran 2000.
11. ^ Jump up to:a b c Irgens 2008.
12. ^ Jump up
13. ^ Fung 1977, p. 76.
14. ^ Spencer 1980, p. 83.
Works cited
• Atanackovic, Teodor M.; Guran, Ardeshir (16 June 2000). Theory of Elasticity for Scientists and Engineers. Dover books on physics. Springer Science
• Chadwick, Peter (1 January 1999). Continuum Mechanics: Concise Theory and Problems. Courier Corporation. ISBN 978-0-486-40180-5.
• Dienes, J. K.; Solem, J. C. (1999). “Nonlinear behavior of some hydrostatically
stressed isotropic elastomeric foams”. Acta Mechanica. 138 (3–4): 155–162. doi:10.1007/BF01291841. S2CID 120320672.
• Fung, Y. C. (1977). A First Course in Continuum Mechanics (2nd ed.). Prentice-Hall, Inc. ISBN 978-0-13-318311-5.
• Irgens, Fridtjov
(10 January 2008). Continuum Mechanics. Springer Science & Business Media. ISBN 978-3-540-74298-2.
• Liu, I-Shih (28 May 2002). Continuum Mechanics. Springer Science & Business Media. ISBN 978-3-540-43019-3.
• Lubliner, Jacob (2008). Plasticity
Theory (PDF) (Revised ed.). Dover Publications. ISBN 978-0-486-46290-5. Archived from the original (PDF) on 31 March 2010.
• Ostoja-Starzewski, M. (2008). “7-10”. Microstructural randomness and scaling in mechanics of materials. CRC Press. ISBN
978-1-58488-417-0.
• Spencer, A. J. M. (1980). Continuum Mechanics. Longman Group Limited (London). p. 83. ISBN 978-0-582-44282-5.
• Roberts, A. J. (1994). A One-Dimensional Introduction to Continuum Mechanics. World Scientific.
• Smith, Donald
R. (1993). “2”. An introduction to continuum mechanics-after Truesdell and Noll. Solids mechanics and its applications. Vol. 22. Springer Science & Business Media. ISBN 978-90-481-4314-6.
• Wu, Han-Chin (20 December 2004). Continuum Mechanics and
Plasticity. Taylor & Francis. ISBN 978-1-58488-363-0.
General references
• 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. doi:10.1007/978-3-642-24615-9. ISBN 978-3-642-24615-9. S2CID 116496103.
• Chandramouli, P.N (2014). Continuum Mechanics. Yes Dee Publishing Pvt Ltd. ISBN 9789380381398. Archived from the original on 4 August 2018. Retrieved
24 March 2014.
• Eringen, A. Cemal (1980). Mechanics of Continua (2nd ed.). Krieger Pub Co. ISBN 978-0-88275-663-9.
• Chen, Youping; James D. Lee; Azim Eskandarian (2009). Meshless Methods in Solid Mechanics (First ed.). Springer New York. ISBN
978-1-4419-2148-2.
• Dill, Ellis Harold (2006). Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity. Germany: CRC Press. ISBN 978-0-8493-9779-0.
• Dimitrienko, Yuriy (2011). Nonlinear Continuum 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 978-3-540-20619-4.
• 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 (3rd ed.). Elsevier, Inc. ISBN 978-0-7506-2894-5. Archived from the original on 6 February 2009.
• Lubarda, Vlado A. (2001). Elastoplasticity Theory.
CRC Press. ISBN 978-0-8493-1138-3.
• 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 978-0-07-040663-6.
• Mase,
G. Thomas; George E. Mase (1999). Continuum Mechanics for Engineers (Second ed.). CRC Press. ISBN 978-0-8493-1855-9.
• 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 978-0-521-83979-2.
• Ostoja-Starzewski, Martin (2008). Microstructural Randomness and Scaling in Mechanics
of Materials. Boca Raton, FL: Chapman & Hall/CRC Press. ISBN 978-1-58488-417-0.
• Rees, David (2006). Basic Engineering Plasticity – An Introduction with Engineering and Manufacturing Applications. Butterworth-Heinemann. ISBN 978-0-7506-8025-7.
• Wright,
T. W. (2002). The Physics and Mathematics of Adiabatic Shear Bands. Cambridge, UK: Cambridge University Press.
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