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.

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A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size. 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 pathline. It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not be one that the body will ever occupy.

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.

In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time.

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An observer standing in the frame of reference observes the changes in the position and physical properties as the material body moves in space as time progresses. This description is normally used in solid mechanics. The material derivative is also known as the substantial derivative , or comoving derivative , or convective derivative. It can be thought as the rate at which the property changes when measured by an observer traveling with that group of particles. Thus, we have. Therefore, the flow velocity field of the continuum is given by. Continuity in the Lagrangian description is expressed by the spatial and temporal continuity of the mapping from the reference configuration to the current configuration of the material points.

All physical quantities characterizing the continuum are described this way.

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In this case the description of motion is made in terms of the spatial coordinates, in which case is called the spatial description or Eulerian description, i. This approach is conveniently applied in the study of fluid flow where the kinematic property of greatest interest is the rate at which change is taking place rather than the shape of the body of fluid at a reference time. Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function.

A necessary and sufficient condition for this inverse function to exist is that the determinant of the Jacobian Matrix , often referred to simply as the Jacobian, should be different from zero. The second term of the right-hand side is the convective rate of change and expresses the contribution of the particle changing position in space motion.

## Continuum Mechanics for Engineers by G. Thomas Mase

Continuity in the Eulerian description is expressed by the spatial and temporal continuity and continuous differentiability of the flow velocity field. A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field, In general, the displacement field is expressed in terms of the material coordinates as.

Continuum mechanics deals with the behavior of materials that can be approximated as continuous for certain length and time scales. The equations that govern the mechanics of such materials include the balance laws for mass , momentum , and energy. Kinematic relations and constitutive equations are needed to complete the system of governing equations.

Physical restrictions on the form of the constitutive relations can be applied by requiring that the second law of thermodynamics be satisfied under all conditions. In the continuum mechanics of solids, the second law of thermodynamics is satisfied if the Clausius—Duhem form of the entropy inequality is satisfied. The balance laws express the idea that the rate of change of a quantity mass, momentum, energy in a volume must arise from three causes:. If there are internal boundaries in the body, jump discontinuities also need to be specified in the balance laws. If we take the Eulerian point of view, it can be shown that the balance laws of mass, momentum, and energy for a solid can be written as assuming the source term is zero for the mass and angular momentum equations.

With respect to the reference configuration the Lagrangian point of view , the balance laws can be written as. The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by. The Clausius—Duhem inequality can be used to express the second law of thermodynamics for elastic-plastic materials. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.

Just like in the balance laws in the previous section, we assume that there is a flux of a quantity, a source of the quantity, and an internal density of the quantity per unit mass. The quantity of interest in this case is the entropy. Then the entropy inequality may be written as. Under the assumption of incrementally isothermal conditions, we have. In terms of the Cauchy stress and the internal energy, the Clausius—Duhem inequality may be written as.

From Wikipedia, the free encyclopedia. Continuum mechanics Laws. Solid mechanics. Fluid mechanics.

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