Continuum mechanics

Continuum mechanics is a annex of mechanics that deals with the assay of the kinematics and the automated behavior of abstracts modelled as a connected accumulation rather than as detached particles. The French mathematician Augustin Louis Cauchy was the aboriginal to codify such models in the 19th century, but analysis in the breadth continues today.

Modelling an article as a continuum assumes that the actuality of the article absolutely fills the amplitude it occupies. Modelling altar in this way ignores the actuality that amount is fabricated of atoms, and so is not continuous; however, on breadth scales abundant greater than that of inter-atomic distances, such models are awful accurate. Fundamental concrete laws such as the attention of mass, the attention of momentum, and the attention of activity may be activated to such models to acquire cogwheel equations anecdotic the behavior of such objects, and some advice about the accurate actual advised is added through a basal relation.

Continuum mechanics deals with concrete backdrop of debris and fluids which are absolute of any accurate alike arrangement in which they are observed. These concrete backdrop are again represented by tensors, which are algebraic altar that accept the appropriate acreage of getting absolute of alike system. These tensors can be bidding in alike systems for computational convenience.

Concept of a continuum

Materials, such as solids, liquids and gases, are composed of molecules afar by abandoned space. On a arresting scale, abstracts accept cracks and discontinuities. However, assertive concrete phenomena can be modelled bold the abstracts abide as a continuum, acceptation the amount in the physique is continuously broadcast and fills the absolute arena of amplitude it occupies. A continuum is a physique that can be always sub-divided into atomic elements with backdrop getting those of the aggregate material.

The authority of the continuum acceptance may be absolute by a abstract analysis, in which either some bright aeon is articular or statistical accord and ergodicity of the microstructure exists. More specifically, the continuum hypothesis/assumption hinges on the concepts of a adumbrative aggregate aspect (RVE) (sometimes alleged "representative elementary volume") and break of scales based on the Hill–Mandel condition. This action provides a hotlink amid an experimentalist's and a theoretician's angle on basal equations (linear and nonlinear elastic/inelastic or accompanying fields) as able-bodied as a way of spatial and statistical averaging of the microstructure.1

When the break of scales does not hold, or if one wants to authorize a continuum of a bigger resolution than that of the RVE size, one employs a statistical aggregate aspect (SVE), which, in turn, leads to accidental continuum fields. The closing again accommodate a micromechanics base for academic bound elements (SFE). The levels of SVE and RVE hotlink continuum mechanics to statistical mechanics. The RVE may be adjourned alone in a bound way via beginning testing: if the basal acknowledgment becomes spatially homogeneous.

Specifically for fluids, the Knudsen amount is acclimated to appraise to what admeasurement the approximation of chain can be made.

Formulation of models

Continuum mechanics models activate by allotment a arena in three dimensional Euclidean amplitude to the actual physique \mathcal B getting modeled. The credibility aural this arena are alleged particles or actual points. Altered configurations or states of the physique accord to altered regions in Euclidean space. The arena agnate to the body's agreement at time \ t is labeled \ \kappa_t(\mathcal B).

A accurate atom aural the physique in a accurate agreement is characterized by a position vector

\ \mathbf x =\sum_{i=1}^3 x_i \mathbf e_i,

where \mathbf e_i are the alike vectors in some anatomy of advertence called for the botheration (See amount 1). This agent can be bidding as a action of the atom position \mathbf X in some advertence configuration, for archetype the agreement at the antecedent time, so that

\mathbf{x}=\kappa_t(\mathbf X).

This action needs to accept assorted backdrop so that the archetypal makes concrete sense. \kappa_t(\cdot) needs to be:

connected in time, so that the physique changes in a way which is realistic,

globally invertible at all times, so that the physique cannot bisect itself,

orientation-preserving, as transformations which aftermath mirror reflections are not accessible in nature.

For the algebraic conception of the model, \ \kappa_t(\cdot) is aswell affected to be alert continuously differentiable, so that cogwheel equations anecdotic the motion may be formulated.

Forces in a continuum

Continuum mechanics deals with deformable bodies, as against to adamant bodies. A solid is a deformable physique that possesses microburst strength, sc. a solid can abutment microburst armament (forces alongside to the actual apparent on which they act). Fluids, on the added hand, do not sustain microburst forces. For the abstraction of the automated behavior of debris and fluids these are affected to be connected bodies, which agency that the amount fills the complete arena of amplitude it occupies, admitting the actuality that amount is fabricated of atoms, has voids, and is discrete. Therefore, if continuum mechanics refers to a point or atom in a connected physique it does not call a point in the interatomic amplitude or an diminutive particle, rather an arcadian allotment of the physique application that point.

Following the classical dynamics of Newton and Euler, the motion of a actual physique is produced by the activity of evidently activated armament which are affected to be of two kinds: apparent armament \mathbf F_C and physique armament \mathbf F_B.2 Thus, the complete force \mathcal F activated to a physique or to a allocation of the physique can be bidding as:

\mathcal F = \mathbf F_B + \mathbf F_C

Surface armament or acquaintance forces, bidding as force per assemblage area, can act either on the bonds apparent of the body, as a aftereffect of automated acquaintance with added bodies, or on abstract centralized surfaces that apprenticed portions of the body, as a aftereffect of the automated alternation amid the locations of the physique to either ancillary of the apparent (Euler-Cauchy's accent principle). If a physique is acted aloft by alien acquaintance forces, centralized acquaintance armament are again transmitted from point to point central the physique to antithesis their action, according to Newton's additional law of motion of attention of beeline drive and angular drive (for connected bodies these laws are alleged the Euler's equations of motion). The centralized acquaintance armament are accompanying to the body's anamorphosis through basal equations. The centralized acquaintance armament may be mathematically declared by how they chronicle to the motion of the body, complete of the body's actual makeup.3

The administration of centralized acquaintance armament throughout the aggregate of the physique is affected to be continuous. Therefore, there exists a acquaintance force physique or Cauchy absorption acreage 2 \mathbf T(\mathbf n, \mathbf x, t) that represents this administration in a accurate agreement of the physique at a accustomed time t\,\!. It is not a agent acreage because it depends not abandoned on the position \mathbf x of a accurate actual point, but aswell on the bounded acclimatization of the apparent aspect as authentic by its accustomed agent \mathbf n.4

Any cogwheel breadth dS\,\! with accustomed agent \mathbf n of a accustomed centralized apparent breadth S\,\!, bonds a allocation of the body, adventures a acquaintance force d\mathbf F_C\,\! arising from the acquaintance amid both portions of the physique on anniversary ancillary of S\,\!, and it is accustomed by

d\mathbf F_C= \mathbf T^{(\mathbf n)}\,dS

where \mathbf T^{(\mathbf n)} is the apparent traction,5 aswell alleged accent vector,6 traction,7 or absorption vector.8 The accent agent is a frame-indifferent agent (see Euler-Cauchy's accent principle).

The complete acquaintance force on the accurate centralized apparent S\,\! is again bidding as the sum (surface integral) of the acquaintance armament on all cogwheel surfaces dS\,\!:

\mathbf F_C=\int_S \mathbf T^{(\mathbf n)}\,dS

In continuum mechanics a physique is advised stress-free if the abandoned armament present are those inter-atomic armament (ionic, metallic, and van der Waals forces) appropriate to authority the physique calm and to accumulate its appearance in the absence of all alien influences, including gravitational attraction.89 Stresses generated during accomplish of the physique to a specific agreement are aswell afar if because stresses in a body. Therefore, the stresses advised in continuum mechanics are abandoned those produced by anamorphosis of the body, sc. abandoned about changes in accent are considered, not the complete ethics of stress.

Body armament are armament basic from sources alfresco of the body10 that act on the aggregate (or mass) of the body. Saying that physique armament are due to alfresco sources implies that the alternation amid altered locations of the physique (internal forces) are embodied through the acquaintance armament alone.5 These armament appear from the attendance of the physique in force fields, e.g. gravitational acreage (gravitational forces) or electromagnetic acreage (electromagnetic forces), or from inertial armament if bodies are in motion. As the accumulation of a connected physique is affected to be continuously distributed, any force basic from the accumulation is aswell continuously distributed. Thus, physique armament are defined by agent fields which are affected to be connected over the complete aggregate of the body,11 i.e. acting on every point in it. Physique armament are represented by a physique force physique \mathbf b(\mathbf x, t) (per assemblage of mass), which is a frame-indifferent agent field.

In the case of gravitational forces, the acuteness of the force depends on, or is proportional to, the accumulation physique \mathbf \rho (\mathbf x, t)\,\! of the material, and it is defined in agreement of force per assemblage accumulation (b_i\,\!) or per assemblage aggregate (p_i\,\!). These two blueprint are accompanying through the actual physique by the blueprint \rho b_i = p_i\,\!. Similarly, the acuteness of electromagnetic armament depends aloft the backbone (electric charge) of the electromagnetic field.

The complete physique force activated to a connected physique is bidding as

\mathbf F_B=\int_V\mathbf b\,dm=\int_V \rho\mathbf b\,dV

Body armament and acquaintance armament acting on the physique advance to agnate moments of force (torques) about to a accustomed point. Thus, the complete activated torque \mathcal M about the agent is accustomed by

\mathcal M= \mathbf M_B + \mathbf M_C

In assertive situations, not frequently advised in the assay of the automated behavior or materials, it becomes all-important to cover two added types of forces: these are physique moments and brace stresses1213 (surface couples,10 acquaintance torques11). Physique moments, or physique couples, are moments per assemblage aggregate or per assemblage accumulation activated to the aggregate of the body. Brace stresses are moments per assemblage breadth activated on a surface. Both are important in the assay of accent for a polarized dielectric solid beneath the activity of an electric field, abstracts area the atomic anatomy is taken into application (e.g. bones), debris beneath the activity of an alien alluring field, and the break approach of metals.6710

Materials that display physique couples and brace stresses in accession to moments produced alone by armament are alleged arctic materials.711 Non-polar abstracts are again those abstracts with abandoned moments of forces. In the classical branches of continuum mechanics the development of the approach of stresses is based on non-polar materials.

Thus, the sum of all activated armament and torques (with account to the agent of the alike system) in the physique can be accustomed by

\mathcal F = \int_V \mathbf a\,dm = \int_S \mathbf T\,dS + \int_V \rho\mathbf b\,dV

\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 agreement of a continuum physique after-effects in a displacement. The displacement of a physique has two components: a rigid-body displacement and a deformation. A rigid-body displacement consists of a accompanying adaptation and circling of the physique after alteration its appearance or size. Anamorphosis implies the change in appearance and/or admeasurement of the physique from an antecedent or undeformed agreement \ \kappa_0(\mathcal B) to a accepted or askew agreement \ \kappa_t(\mathcal B) (Figure 2).

The motion of a continuum physique is a connected time arrangement of displacements. Thus, the actual physique will absorb altered configurations at altered times so that a atom occupies a alternation of credibility in amplitude which call a pathline.

There is alternation during anamorphosis or motion of a continuum physique in the faculty that:

The actual credibility basic a bankrupt ambit at any burning will consistently anatomy a bankrupt ambit at any consecutive time.

The actual credibility basic a bankrupt apparent at any burning will consistently anatomy a bankrupt apparent at any consecutive time and the amount aural the bankrupt apparent will consistently abide within.

It is acceptable to analyze a advertence agreement or antecedent action which all consecutive configurations are referenced from. The advertence agreement charge not be one that the physique will anytime occupy. Often, the agreement at \ t=0 is advised the advertence configuration, \ \kappa_0 (\mathcal B). The apparatus \ X_i of the position agent \ \mathbf X of a particle, taken with account to the advertence configuration, are alleged the actual or advertence coordinates.

When allegory the anamorphosis or motion of solids, or the breeze of fluids, it is all-important to call the arrangement or change of configurations throughout time. One description for motion is fabricated in agreement of the actual or referential coordinates, alleged actual description or Lagrangian description.

edit Lagrangian description

In the Lagrangian description the position and concrete backdrop of the particles are declared in agreement of the actual or referential coordinates and time. In this case the advertence agreement is the agreement at \ t=0. An eyewitness continuing in the referential anatomy of advertence observes the changes in the position and concrete backdrop as the actual physique moves in amplitude as time progresses. The after-effects acquired are absolute of the best of antecedent time and advertence configuration, \kappa_0(\mathcal B). This description is commonly acclimated in solid mechanics.

In the Lagrangian description, the motion of a continuum physique is bidding by the mapping action \ \chi(\cdot) (Figure 2),

\ \mathbf x=\chi(\mathbf X, t)

which is a mapping of the antecedent agreement \kappa_0(\mathcal B) assimilate the accepted agreement \kappa_t(\mathcal B), giving a geometrical accord amid them, i.e. giving the position agent \ \mathbf{x}=x_i\mathbf e_i that a atom \ X, with a position agent \ \mathbf X in the undeformed or advertence agreement \kappa_0(\mathcal B), will absorb in the accepted or askew agreement \kappa_t(\mathcal B) at time \ t. The apparatus \ x_i are alleged the spatial coordinates.

Physical and kinematic backdrop \ P_{ij\ldots}, i.e. thermodynamic backdrop and velocity, which call or characterize appearance of the actual body, are bidding as connected functions of position and time, i.e. \ P_{ij\ldots}=P_{ij\ldots}(\mathbf X,t).

The actual acquired of any acreage \ P_{ij\ldots} of a continuum, which may be a scalar, vector, or tensor, is the time amount of change of that acreage for a specific accumulation of particles of the affective continuum body. The actual acquired is aswell accepted as the abundant derivative, or comoving derivative, or convective derivative. It can be anticipation as the amount at which the acreage changes if abstinent by an eyewitness traveling with that accumulation of particles.

In the Lagrangian description, the actual acquired of \ P_{ij\ldots} is artlessly the fractional acquired with account to time, and the position agent \ \mathbf X is captivated connected as it does not change with time. Thus, we have

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

The direct position \ \mathbf x is a acreage of a particle, and its actual acquired is the direct dispatch \ \mathbf v of the particle. Therefore, the dispatch acreage of the continuum is accustomed by

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

Similarly, the dispatch acreage is accustomed by

\ \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 bidding by the spatial and banausic alternation of the mapping from the advertence agreement to the accepted agreement of the actual points. All concrete quantities anecdotic the continuum are declared this way. In this sense, the action \chi(\cdot) and \ P_{ij\ldots}(\cdot) are single-valued and continuous, with connected derivatives with account to amplitude and time to whatever adjustment is required, usually to the additional or third.

edit Eulerian description

Continuity allows for the changed of \chi(\cdot) to trace backwards area the atom currently amid at \mathbf x was amid in the antecedent or referenced agreement \kappa_0(\mathcal B). In this case the description of motion is fabricated in agreement of the spatial coordinates, in which case is alleged the spatial description or Eulerian description, i.e. the accepted agreement is taken as the advertence configuration.

The Eulerian description, alien by d'Alembert, focuses on the accepted agreement \kappa_t(\mathcal B), giving absorption to what is occurring at a anchored point in amplitude as time progresses, instead of giving absorption to alone particles as they move through amplitude and time. This access is calmly activated in the abstraction of aqueous breeze area the kinematic acreage of greatest absorption is the amount at which change is demography abode rather than the appearance of the physique of aqueous at a advertence time.14

Mathematically, the motion of a continuum application the Eulerian description is bidding by the mapping function

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

which provides a archetype of the atom which now occupies the position \mathbf x in the accepted agreement \kappa_t(\mathcal B) to its aboriginal position \mathbf X in the antecedent agreement \kappa_0(\mathcal B).

A all-important and acceptable action for this changed action to abide is that the account of the Jacobian Matrix, generally referred to artlessly as the Jacobian, should be altered from zero. Thus,

\ 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 concrete backdrop \ P_{ij\ldots} are bidding as

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

where the anatomic anatomy of \ P_{ij \ldots} in the Lagrangian description is not the aforementioned as the anatomy of \ p_{ij \ldots} in the Eulerian description.

The actual acquired of \ p_{ij\ldots}(\mathbf x,t), application the alternation rule, is then

\ \frac{d}{dt}p_{ij\ldots}(\mathbf x,t)=\frac{\partial}{\partial t}p_{ij\ldots}(\mathbf x,t)+ \frac{\partial}{\partial x_k}p_{ij\ldots}(\mathbf x,t)\frac{dx_k}{dt}

The aboriginal appellation on the right-hand ancillary of this blueprint gives the bounded amount of change of the acreage \ p_{ij\ldots}(\mathbf x,t) occurring at position \ \mathbf x. The additional appellation of the right-hand ancillary is the convective amount of change and expresses the addition of the atom alteration position in amplitude (motion).

Continuity in the Eulerian description is bidding by the spatial and banausic alternation and connected differentiability of the dispatch field. All concrete quantities are authentic this way at anniversary burning of time, in the accepted configuration, as a action of the agent position \ \mathbf x.

edit Displacement field

The agent abutting the positions of a atom \ P in the undeformed agreement and askew agreement is alleged the displacement agent \ \mathbf u(\mathbf X,t)=u_i\mathbf e_i, in the Lagrangian description, or \ \mathbf U(\mathbf x,t)=U_J\mathbf E_J, in the Eulerian description.

A displacement acreage is a agent acreage of all displacement vectors for all particles in the body, which relates the askew agreement with the undeformed configuration. It is acceptable to do the assay of anamorphosis or motion of a continuum physique in agreement of the displacement field, In general, the displacement acreage is bidding in agreement of the actual coordinates as

\ \mathbf u(\mathbf X,t) = \mathbf b+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J

or in agreement of the spatial coordinates as

\ \mathbf U(\mathbf x,t) = \mathbf b+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,

where \ \alpha_{Ji} are the administration cosines amid the actual and spatial alike systems with assemblage vectors \ \mathbf E_J and \mathbf e_i, respectively. Thus

\ \mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}

and the accord amid \ u_i and \ U_J is again accustomed by

\ u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i

Knowing that

\ \mathbf e_i = \alpha_{iJ}\mathbf E_J

then

\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)

It is accepted to blanket the alike systems for the undeformed and askew configurations, which after-effects in \ \mathbf b=0, and the administration cosines become Kronecker deltas, i.e.

\ \mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}

Thus, we have

\ \mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J

or in agreement of the spatial coordinates as

\ \mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J

Governing equations

Continuum mechanics deals with the behavior of abstracts that can be approximated as connected for assertive breadth and time scales. The equations that administer the mechanics of such abstracts cover the antithesis laws for mass, momentum, and energy. Kinematic relations and basal equations are bare to complete the arrangement of administering equations. Concrete restrictions on the anatomy of the basal relations can be activated by acute that the additional law of thermodynamics be annoyed beneath all conditions. In the continuum mechanics of solids, the additional law of thermodynamics is annoyed if the Clausius–Duhem anatomy of the anarchy asperity is satisfied.

The antithesis laws accurate the abstraction that the amount of change of a abundance (mass, momentum, energy) in a aggregate accept to appear from three causes:

the concrete abundance itself flows through the apparent that bound the volume,

there is a antecedent of the concrete abundance on the apparent of the volume, or/and,

there is a antecedent of the concrete abundance central the volume.

Let Ω be the physique (an accessible subset of Euclidean space) and let \partial \Omega be its apparent (the abuttals of Ω).

Let the motion of actual credibility in the physique be declared by the map

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

where \mathbf{X} is the position of a point in the antecedent agreement and \mathbf{x} is the area of the aforementioned point in the askew configuration.

The anamorphosis acclivity is accustomed by

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

edit Antithesis laws

Let f(\mathbf{x},t) be a concrete abundance that is abounding through the body. Let g(\mathbf{x},t) be sources on the apparent of the physique and let h(\mathbf{x},t) be sources central the body. Let \mathbf{n}(\mathbf{x},t) be the apparent assemblage accustomed to the apparent \partial \Omega . Let \mathbf{v}(\mathbf{x},t) be the acceleration of the concrete particles that backpack the concrete abundance that is flowing. Also, let the acceleration at which the bonds apparent \partial \Omega is affective be un (in the administration \mathbf{n}).

Then, antithesis laws can be bidding in the accepted form

\cfrac{d}{dt}\left\int_{\Omega} f(\mathbf{x},t)~\text{dV}\right = \int_{\partial \Omega } f(\mathbf{x},t)u_n(\mathbf{x},t) - \mathbf{v}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x},t)~\text{dA} + \int_{\partial \Omega } g(\mathbf{x},t)~\text{dA} + \int_{\Omega} h(\mathbf{x},t)~\text{dV} ~.

Note that the functions f(\mathbf{x},t), g(\mathbf{x},t), and h(\mathbf{x},t) can be scalar valued, agent valued, or tensor admired - depending on the concrete abundance that the antithesis blueprint deals with. If there are centralized boundaries in the body, jump discontinuities aswell charge to be authentic in the antithesis laws.

If we yield the Lagrangian point of view, it can be apparent that the antithesis laws of mass, momentum, and activity for a solid can be accounting as

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

In the aloft equations \rho(\mathbf{x},t) is the accumulation physique (current), \dot{\rho} is the actual time acquired of ρ, \mathbf{v}(\mathbf{x},t) is the atom velocity, \dot{\mathbf{v}} is the actual time acquired of \mathbf{v}, \boldsymbol{\sigma}(\mathbf{x},t) is the Cauchy accent tensor, \mathbf{b}(\mathbf{x},t) is the physique force density, e(\mathbf{x},t) is the centralized activity per assemblage mass, \dot{e} is the actual time acquired of e, \mathbf{q}(\mathbf{x},t) is the calefaction alteration vector, and s(\mathbf{x},t) is an activity antecedent per assemblage mass.

With account to the advertence configuration, the antithesis laws can be accounting as

{ \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, \boldsymbol{P} is the aboriginal Piola-Kirchhoff accent tensor, and ρ0 is the accumulation physique in the advertence configuration. The aboriginal Piola-Kirchhoff accent tensor is accompanying to the Cauchy accent tensor by

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

We can alternatively ascertain the nominal accent tensor \boldsymbol{N} which is the alter of the aboriginal Piola-Kirchhoff accent tensor such that

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

Then the antithesis laws become

{ \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 aloft equations are authentic as such that

\boldsymbol{\nabla} \mathbf{v} = \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial x_j}\mathbf{e}_i\otimes\mathbf{e}_j = v_{i,j}\mathbf{e}_i\otimes\mathbf{e}_j ~;~~ \boldsymbol{\nabla} \cdot \mathbf{v} = \sum_{i=1}^3 \frac{\partial v_i}{\partial x_i} = v_{i,i} ~;~~ \boldsymbol{\nabla} \cdot \boldsymbol{S} = \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial x_j}~\mathbf{e}_i = \sigma_{ij,j}~\mathbf{e}_i ~.

where \mathbf{v} is a agent field, \boldsymbol{S} is a second-order tensor field, and \mathbf{e}_i are the apparatus of an orthonormal base in the accepted configuration. Also,

\boldsymbol{\nabla}_{\circ} \mathbf{v} = \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial X_j}\mathbf{E}_i\otimes\mathbf{E}_j = v_{i,j}\mathbf{E}_i\otimes\mathbf{E}_j ~;~~ \boldsymbol{\nabla}_{\circ}\cdot\mathbf{v} = \sum_{i=1}^3 \frac{\partial v_i}{\partial X_i} = v_{i,i} ~;~~ \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{S} = \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial X_j}~\mathbf{E}_i = S_{ij,j}~\mathbf{E}_i

where \mathbf{v} is a agent field, \boldsymbol{S} is a second-order tensor field, and \mathbf{E}_i are the apparatus of an orthonormal base in the advertence configuration.

The close artefact is authentic as

\boldsymbol{A}:\boldsymbol{B} = \sum_{i,j=1}^3 A_{ij}~B_{ij} = trace(\boldsymbol{A}\boldsymbol{B}^T) ~.

edit Clausius–Duhem inequality

The Clausius–Duhem asperity can be acclimated to accurate the additional law of thermodynamics for elastic-plastic materials. This asperity is a account apropos the irreversibility of accustomed processes, abnormally if activity amusement is involved.

Just like in the antithesis laws in the antecedent section, we accept that there is a alteration of a quantity, a antecedent of the quantity, and an centralized physique of the abundance per assemblage mass. The abundance of absorption in this case is the entropy. Thus, we accept that there is an anarchy flux, an anarchy source, and an centralized anarchy physique per assemblage accumulation (η) in the arena of interest.

Let Ω be such a arena and let \partial \Omega be its boundary. Again the additional law of thermodynamics states that the amount of access of η in this arena is greater than or according to the sum of that supplied to Ω (as a alteration or from centralized sources) and the change of the centralized anarchy physique due to actual abounding in and out of the region.

Let \partial \Omega move with a acceleration un and let particles central Ω accept velocities \mathbf{v}. Let \mathbf{n} be the assemblage apparent accustomed to the apparent \partial \Omega . Let ρ be the physique of amount in the region, \bar{q} be the anarchy alteration at the surface, and r be the anarchy antecedent per assemblage mass. Again the anarchy asperity may be accounting as

\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 anarchy alteration can be accompanying to the agent alteration at the apparent by the affiliation \bar{q} = -\boldsymbol{\psi}(\mathbf{x})\cdot\mathbf{n}. Beneath the acceptance of incrementally isothermal conditions, we have

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

where \mathbf{q} is the calefaction alteration vector, s is an activity antecedent per assemblage mass, and T is the complete temperature of a actual point at \mathbf{x} at time t.

We again accept the Clausius–Duhem asperity in basic form:

{ \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 appearance that the anarchy asperity may be accounting in cogwheel anatomy as

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

In agreement of the Cauchy accent and the centralized energy, the Clausius–Duhem asperity may be accounting as

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

Applications

Mechanics

Solid mechanics

Fluid mechanics

Engineering

Mechanical engineering

Civil engineering

Aerospace engineering

edit See also