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
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
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