Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium 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 and isotropy. 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
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.Formulation of models
Continuum mechanics models begin by assigning a region in three-dimensional Euclidean space to the material body 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 is labeled.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. 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. 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.
Forces in a continuum
A solid is a deformable body that possesses shear strength, sc. a solid can support shear forces. Fluids, on the other hand, do not sustain shear forces.Following the classical dynamics of Isaac Newton and Leonhard 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. 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. 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. 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.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 that represents this distribution in a particular configuration of the body at a given time. 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.
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, also called stress vector, traction, or traction vector. The stress vector is a frame-indifferent vector.
The total contact force on the particular internal surface is then expressed as the sum 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 required to hold the body together and to keep its shape in the absence of all external influences, including gravitational attraction. 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
Body forces are forces originating from sources outside of the body that act on the volume of the body. Saying that body forces are due to outside sources implies that the interaction between different parts of the body are manifested through the contact forces alone. These forces arise from the presence of the body in force fields, e.g. gravitational field or electromagnetic field, 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, i.e. acting on every point in it. Body forces are represented by a body force density , 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 of the material, and it is specified in terms of force per unit mass or per unit volume. These two specifications are related through the material density by the equation. Similarly, the intensity of electromagnetic forces depends upon the strength of the electromagnetic field.
The total body force applied to a continuous body is expressed as
Body forces and contact forces acting on the body lead to corresponding moments of force relative to a given point. 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 and body moments. Couple stresses are moments per unit area applied on a surface. Body moments, or body couples, are moments per unit volume or per unit mass applied to the volume of the body. 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, solids under the action of an external magnetic field, and the dislocation theory of metals.
Materials that exhibit body couples and couple stresses in addition to moments produced exclusively by forces are called polar materials. 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 in the body can be given by
Kinematics: motion and deformation
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 to a current or deformed configuration .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 path line.
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.
- 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.
When analyzing the motion or deformation 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.