Continuum mechanics

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Continuum mechanics is the study of how materials deform and transmit forces, treating them as continuous mediums rather than discrete particles. Whether it’s solids or fluids, this field helps us analyze everything from everyday materials to complex engineering challenges! 🧪🛠️

Key concepts include:

• Elasticity: Materials that return to original shape after stress.
• Plasticity: Materials that permanently deform under stress.
• Fluid Mechanics: Understanding how fluids behave when forces are applied. 💧

Let’s explore the fascinating world of materials!

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.

Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. 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. Information about the specific material is expressed in constitutive relationships.

Continuum mechanics treats the physical properties of solids and fluids independently of any particular coordinate system in which they are observed. These properties are represented by tensors, which are mathematical objects with the salient property of being independent of coordinate systems. This permits definition of physical properties at any point in the continuum, according to mathematically convenient continuous functions. The theories of elasticity, plasticity and fluid mechanics are based on the concepts of continuum mechanics.

Concept of a continuum

The concept of a continuum underlies the mathematical framework for studying large-scale forces and deformations in materials. 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. A continuum is a body that can be continually sub-divided into infinitesimal elements with local material properties defined at any particular point. Properties of the bulk material can therefore be described by continuous functions, and their evolution can be studied using the mathematics of calculus.

Apart from the assumption of continuity, two other independent assumptions are often employed in the study of continuum mechanics. These are homogeneity (assumption of identical properties at all locations) and isotropy (assumption of directionally invariant vector properties). If these auxiliary assumptions are not globally applicable, the material may be segregated into sections where they are applicable in order to simplify the analysis. For more complex cases, one or both of these assumptions can be dropped. In these cases, computational methods are often used to solve the differential equations describing the evolution of material properties.

Major areas

Continuum mechanics
The study of the physics of continuous materials

• Solid mechanics
  The study of the physics of continuous materials with a defined rest shape.
  • Elasticity
    Describes materials that return to their rest shape after applied stresses are removed.
  • Plasticity
    Describes materials that permanently deform after a sufficient applied stress.
    • Rheology
      The study of materials with both solid and fluid characteristics.
• Fluid mechanics
  The study of the physics of continuous materials which deform when subjected to a force.
  • Non-Newtonian fluid
    Do not undergo strain rates proportional to the applied shear stress.
    • Rheology
      The study of materials with both solid and fluid characteristics.
  • 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.

Validity

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 exist. More specifically, the continuum hypothesis 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. 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. The latter then provide a micromechanics basis for stochastic finite elements (SFE). The levels of SVE and RVE link continuum mechanics to statistical mechanics. Experimentally, the RVE can only be evaluated when the constitutive response is spatially homogenous.

Applications

• Continuum mechanics
  • Solid mechanics
  • Fluid mechanics
• Engineering
  • Civil engineering
  • Mechanical engineering
  • Aerospace engineering
  • Biomedical engineering
  • Chemical engineering

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