What is Continuum Mechanics?

The area of mechanics known as continuum mechanics studies the deformation and force transmission through materials that are modelled as continuous masses as opposed to discrete particles. The first to develop such models in the 19th century was the French mathematician Augustin-Louis Cauchy.

A continuum model presupposes that the object's material entirely fills the space it resides in. The idea that matter is composed of atoms is disregarded, yet the description of matter on length scales considerably larger than interatomic distances is nonetheless correct enough.

With the use of differential equations that describe how such stuff behaves in accordance with physical rules like mass conservation, momentum conservation, and energy conservation, the concept of a continuous medium enables intuitive analysis of bulk matter.

Constitutive relationships express information about the particular

The physical characteristics of solids and liquids are handled by continuum mechanics independently of the specific coordinate system in which they are seen.

Tensors, which are mathematical objects having the notable property of being independent of coordinate systems, serve as a representation for these qualities.

This enables the specification of physical attributes at any point along the continuum in terms of useful continuous mathematical functions. The principles of continuum mechanics serve as the foundation for the theories of elasticity, plasticity, and fluid mechanics.

The Continuous Idea

The mathematical foundation for analyzing large-scale forces and deformations in materials is based on the idea of a continuum.

Even though materials are made up of discrete atoms and molecules that are separated from one another by void space, microscopic fissures, and crystallographic flaws, physical processes are frequently modelled by taking into account a substance that is dispersed throughout a certain region of space.

A continuum is a body that can continuously be divided into minuscule components with locally specified physical characteristics at any given position. Therefore, continuous functions can be used to characterize the bulk material's properties, and calculus can be used to study how these properties change over time.

Two additional independent assumptions are frequently used in the study of continuum mechanics in addition to the continuity assumption. These are isotropy (assumption of directionally invariant vector qualities) and homogeneity (assumption of identical properties at all places).

To make the analysis simpler, the information may be divided into areas where certain auxiliary assumptions are appropriate even if they are not globally applicable. One or both assumptions can be abandoned in more complicated instances.

The differential equations describing the evolution of material properties in these situations are frequently solved using computational approaches.

Development of Models

A region in three-dimensional Euclidean space is first given to the material body β that will be modelled in continuum mechanics models. These points are referred to as particles or material points.

Different bodily arrangements or states correspond to various places in Euclidean space. Kt (β) denotes the region corresponding to the body's configuration at time t.

X = Kt(X)

For the model to make physical sense, this function must possess several characteristics.

Kt(.) is also assumed to be twice continuously differentiable for the mathematical formulation of the model in order to allow for the construction of differential equations describing the motion.

Continuum-based forces

In contrast to rigid bodies, deformable bodies are the focus of continuum mechanics. A solid is a deformable body with shear strength; shear forces are forces that act parallel to the material surface on which the solid is composed. On the other hand, shear forces are not tolerated by fluids.

The motion of a material body is created by the action of externally applied forces, which are supposed to be of two kinds: surface forces Fc and body forces Fb, according to the classical dynamics of Newton and Euler. As a result, the total force F that is exerted on a body or a part of a body can be stated as:

F = Fc+ Fb

Surface Tension

According to the Euler-Cauchy stress principle, surface forces?also known as contact forces?can act on a body's bounding surface because of mechanical contact with other bodies, or on hypothetical internal surfaces that enclose portions of the body because of mechanical interaction between the body parts on either side of the surface.

According to Newton's third law of motion of conservation of linear momentum and angular momentum (for continuous bodies, these laws are known as the Euler's equations of motion), internal contact forces are transmitted from point to point inside the body to balance the action of external contact forces.

Through constitutive equations, the internal contact forces are connected to the body's deformation. Independent of the body's material composition, the internal contact forces can be quantitatively characterized in terms of how they relate to the body's motion.

In continuum mechanics, a body is said to be stress-free if the only forces acting on it are those necessary to keep it together and maintain its shape in the absence of all external influences, including gravitational attraction (ionic, metallic, and van der Waals forces). When evaluating stresses in a body, the strains created during the manufacturing of the body to a particular configuration are also disregarded.

As a result, only stresses caused by the body's deformation are taken into account in continuum mechanics, and neither absolute nor relative changes in stress are taken into account.

Body pressure

External forces acting on the volume (or mass) of the body are referred to as body forces. Saying that a body's forces come from outside sources indicates that a body's internal forces?its interactions between its many parts?are only visible through contact forces.

These forces result from a body's interaction with force fields, such as the gravitational or electromagnetic fields, or from inertial forces experienced while a body is moving. Any force emanating from the mass of a continuous body is believed to be continuously distributed, just like the mass.

As a result, body forces are described by vector fields that operate on every point in the body and are continuous over the full volume of the body. Body forces are represented by a frame-indifferent vector field called the body force density, or b(x,t) (per unit of mass).

In the case of gravitational forces, the force's intensity is expressed in terms of force per unit mass (bi) or force per unit volume (pi), and it depends on or is proportional to the material's mass density, or p(x,t).

The formula pbi = pi links these two criteria to material density. The strength (electric charge) of the electromagnetic field determines how intense the electromagnetic forces will be.

Motion and Distortion in Kinematics

A displacement happens when a continuum body's configuration changes. A rigid-body displacement and a deformation make up a body's displacement. The simultaneous translation and rotation of a rigid body without altering its size or shape constitutes a rigid-body displacement.

When a body is said to be deformed, it means that it has changed from its initial, or undeformed, configuration, Ko (B), to its present, or deformed, configuration, kt(B).

A continuum body moves in a continuous series of displacements across time. As a result, the material body will take on various forms at various periods in time, causing a particle to occupy a set of positions in space that define a route line. A continuum body experiences continuity while moving or changing shape in the following ways:

• A closed curve will always be formed by the same material points that formed it at any given moment.
• Any time material points create a closed surface, they will always do so later, and the matter inside the closed surface will always stay inside.

Finding a reference configuration or initial condition to which all other settings may be compared is useful. It is not required that the reference configuration be one that the body will ever occupy.

The configuration at time t=0 is frequently referred to as the reference configuration, or kt(B). The parts Xi of a particle's position vector X that are measured in relation to a reference configuration are referred to as the material or reference coordinates.

It is vital to characterize the progression or evolution of configurations through time when analyzing the mobility or deformation of solids or the flow of fluids. The term "material description" or "Lagrangian description" refers to one method of describing motion in terms of the material or referential coordinates.

Description in Lagrangian

The position and physical characteristics of the particles are specified in terms of the material or referential coordinates and time in the Lagrangian description. The setup at t=0 in this situation serves as the reference configuration.

Standing outside the frame of reference, an observer can see how the physical characteristics and position of the material body change over time as it moves across space.

The selection of initial time and reference configuration, ko(B), has no bearing on the outcomes. Solid mechanics typically makes use of this description.

Eulerian explanation

Due to continuity, it is possible to determine where the particle at X was when it was in the original or reference configuration ko(B).

This is referred to as a spatial description or Eulerian description because the current configuration is used as the reference configuration when describing motion in terms of spatial coordinates.

Instead, then concentrating on individual particles as they travel through space and time, the Eulerian description, developed by d'Alembert, concentrates on the current configuration kt(B), paying attention to what is happening at a fixed location in space as time moves forward.

This method is easily used in the study of fluid flow since the rate of change, rather than the shape of the fluid body at a reference moment, is the kinematic attribute of greatest importance.

Regulation equations

For specific length and time scales, the behavior of materials that may be roughly described as continuous is the subject of continuum mechanics. The balance laws for mass, momentum, and energy are included in the equations that control the mechanics of such materials.

The system of controlling equations must also include kinematic relations and constitutive equations. The second law of thermodynamics must always hold true to impose physical limitations on the shape of the constitutive interactions.

If the Clausius-Duhem variant of the entropy inequality is satisfied, the second law of thermodynamics in continuum mechanics of solids is satisfied.

The balancing laws state that there must be three factors involved for a quantity's rate of change in a volume (mass, momentum, or energy):

1. The physical quantity itself passes through the volume's surface.
2. there is a physical amount source on the volume's surface, or/and,
3. The volume contains a source of the physical quantity.

Let partial Omega be its surface (the boundary of Omega), and let Omega be its body (an open subset of Euclidean space).

Contradictory Clausius-Duhem

Contradictory Clausius-Duhem

For elastic-plastic materials, the second rule of thermodynamics can be expressed using the Clausius-Duhem inequality. This imbalance makes a remark about how natural processes are irreversible, especially when energy loss is involved.

We suppose that there is a flux of a quantity, a source of the quantity, and an internal density of the quantity per unit mass, just like in the balance laws in the preceding section.

Entropy is the quantity that is of importance in this situation. As a result, we presumptively assume that the region of interest has an entropy flux, an entropy source, an internal mass density rho, and an internal specific entropy (i.e., entropy per unit mass) eta.

Let Omega serve as such an area Let be the limit of it. Therefore, according to the second law of thermodynamics, the rate of rise of eta in this region is greater than or equal to the sum of that provided to (as a flux or from internal sources) and the change in the internal entropy density pn due to material flowing in and out of the region.

Allow to move with a flow velocity Un and allow v for the internal particle velocities. Let n represent the surface's outward normal. Let rho be the area's density of matter, q the flux of entropy at the surface, and r the source of entropy per mass.

Validity

A theoretical examination that either identifies some distinct periodicity or finds statistical homogeneity and ergodicity of the microstructure can confirm the validity of the continuum assumption.

The continuum hypothesis is more particularly based on the ideas of a representative elementary volume and scale separation based on the Hill-Mandel condition.

This condition offers a means of spatial and statistically averaging the microstructure as well as a link between an experimentalist's and a theoretician's perspective on constitutive equations (linear and nonlinear elastic/inelastic or coupled fields).

A statistical volume element (SVE), which produces random continuum fields, is used when the separation of scales does not hold or when one needs to generate a continuum with a finer resolution than the size of the representative volume element (RVE).

The latter then give stochastic finite elements (SFE) a micromechanics foundation. Continuum mechanics and statistical mechanics are connected by the SVE and RVE levels.

The RVE can only be assessed experimentally when the constitutive reaction is spatially homogeneous.