Continuum mechanics
Continuum mechanics is a branch of mechanics that studies the motion of deformable bodies in response to external loads. The term deformation here is taken so far that even the flow of a liquid or a gas flows excluded. Accordingly include solid mechanics, fluid mechanics and gas theory to continuum mechanics. In continuum mechanics is the microscopic structure of matter, so for example the lattice structure of crystalline solids and the molecular structure of liquids, aside and approached the object of study as a continuum. The sizes of the density, temperature, and the three components of the velocity at any point in space are present within a body, which makes the continuum mechanics to a field theory. The non- classical theories of continuum mechanics include the relativistic continuum mechanics, the Cosserat continuum in which each material point in addition has three rotational degrees of freedom or the non-local materials.
The theoretical background of continuum mechanics is the physics, the practical application is carried out in different areas of mechanical engineering, the theoretical civil engineering, material science, computer science and the medical in geophysics and other fields of geosciences.
The best known in the field of scientific and technical problems of solid mechanics and most widely used numerical methods, the finite element method solves the equations of continuum mechanics ( approximately) using methods of calculus of variations. In fluid dynamics an equal rank comes to the finite volume method.
- 4.1 Mass Balance
- 4.2 momentum balance
- 4.3 Angular momentum balance
- 4.4 Energy Balance
- 4.5 Clausius Duhem inequality
- 5.1 Simple materials
- 5.2 Material Constraints
- 5.3 Material symmetries
- 5.4 Constitutive equations
- 6.1 footnotes
- 6.2 Literature
- 6.3 External links
Historical Outline
The continuum mechanics is based on the mechanics, physics, differential and integral calculus whose historic career can be looked up there. At this point, the specific continuum mechanical development will be outlined.
Leonardo da Vinci (1452-1519) contributed by many sketches of flow processes to the development of the methodology of fluid mechanics. Galileo Galilei (1564-1642) founded the strength of materials and dealt with hydraulics, Torricelli (1608-1647) and Blaise Pascal (1623-1662) dealt among other things with the hydrostatics and hydrodynamics. Edme Mariotte (1620-1684) provided input on the problems of liquids and gases and put it first on constitutive laws. Robert Hooke (1635-1703) formulated in 1676 was named after him Hooke's Law on the elastic behavior of solids. Pitot (1695-1771) investigated the dynamic pressure in flows. Leonhard Euler (1707-1783) gave considerable impetus to the mechanics of rigid and deformable body and hydro mechanics. The fundamental continuum mechanics in terms of voltage and strain tensor were introduced by Cauchy ( 1789-1857 ). Other contributors were from Poisson (1781-1840), Navier (1785-1836), Stokes (1819-1903), Kirchhoff (1824-1887) and Piola ( 1794-1850 ). Impetus in the field of basic research in continuum mechanics gave Truesdell (1919-2000) and Noll ( born 1925 ).
Statements of continuum mechanics
The continuum mechanics contains two different categories of statements:
Describe the general statements
The individual statements about the material properties of the body are taken in the # material theory. They create the connection between the laws of nature and the deformations of bodies.
The mathematical description allows compact formulation of the laws of nature in balance equations and the material properties in the constitutive equations.
The system of
Is complete and leads to the fundamental predictability of response of bodies to given external loads.
Kinematics
Here only the specific continuum mechanical aspects will be described, more is look up under kinematics. The kinematics are in continuum mechanics transformation equations for variables in the initial configuration to the corresponding variables in the current configuration. In the deformation vector fields are transformed, which is accomplished by tensors. The tensors in continuum mechanics most commonly used are tensors of second stage that map vectors from three-dimensional vector spaces, linear on each other. Therefore applies to these tensors component-wise representation
Wherein a base of the prototype space that the image space and the dyadic product, see # tensor basis. With the trace operator, the scalar product between tensors and the amount of a tensor is defined. The separation of the hydrostatic or spherical component provides the deviator
It is the unit tensor.
The material body
The material body as a carrier of the physical processes fulfilled uniformly parts of the space of our intuition. In continuum mechanics, the body is imaged by means of configurations bijective in a Euclidean vector space, wherein the properties of the body are continuously smeared over the room. With this idealization of the body is described as a point quantity, can be formed in the gradients and integrals.
This has two consequences:
For a body following configurations are used:
The combination of these configurations
Should so often be continuously differentiable, as is necessary in context. The original image is identified with the space occupied by the body in the reference configuration volume and the image space with the space occupied by the body in the current configuration volume.
Material and spatial coordinates
The physical coordinates of a material point are given with the reference configuration:
The instantaneous configuration are the spatial coordinates of the material point in space:
The movement function describes article held the path of a material point through the room.
Due to -one correspondence of the configurations in the description of the material body all a material point associated variables (eg density, temperature and velocity ) can be described as a function of its material or spatial coordinates. In the former case, the motion of a material point is observed around the room what is the material or Lagrangian description manner (after Joseph Louis Lagrange - ) and which is preferred in solid mechanics. In the second case, it is observed that the material points to pass a given point in space and the possible physical quantities there what the spatial or Eulerian description manner designated and which is used in the fluid dynamics. Unless otherwise stated, sizes in the Lagrangian mode of description with uppercase letters or the index and those of Euler are denoted by lowercase letters.
Differential operators
In the continuum mechanics for the material and the spatial approach mainly two differential operators, gradient and divergence, used:
This includes the physical and operators. Similar definitions apply for the spatial operators and in the spatial formulation.
Local and material time derivative
The time derivative of a material point size associated with, for example, the temperature can be evaluated at fixed point in space or held firmly held substantive point. The former is the local time derivative of the latter, the material or Substantial dissipation.
The partial derivative article held spatial point is the local time derivative, i.e. the rate of change which can be observed at a fixed point in space.
The material time derivative in the Lagrangian description
In the Eulerian description, the material time derivative of the local and an additional convective component consists of:
This is the material coordinate of the particle, which is located at the site at the time, the velocity of this particle at the site and represents the convective portion dar.
The deformation gradient
The deformation gradient forms the vectors of the tangent space at a point of the initial configuration in the tangent space in the current configuration from
It is the derivative of the function of movement according to the physical coordinates
And the directional derivative
Calculated what his transformation properties of the line elements clarified.
It also transforms the surface element, multiplied by the surface normal of the sheet piece, and the volume element of the output configuration, in the current configuration:
This is the determinant and the inverse transpose. With these elements, integrals can in the initial and the current configuration ( equivalent: in the material and spatial formulation) be converted into each.
Strain tensors
Using the deformation gradient defined the distortion mass. The polar decomposition of the deformation splits the deformation locally in a pure rotation, mediated by the rotation tensor, and a pure extension, mediated by the symmetric positive definite right and left stretch tensor respectively.
The latter are used to define a variety of strain tensors, for example, the nominal strain
The Hencky strain
(calculated by principal component analysis of, drawing of the logarithms of the diagonal elements and inverse transform ) of the Green- Lagrange strain
And Euler Almansi strains
It is the unit tensor. The latter Dehnungstensoren are motivated from comparing two material line elements and in point:
Distortion speeds
From the time derivative
Derived from the material and the spatial Verzerrungsgeschwindigkeitstensor that exactly disappear when rigid body motions are present.
Geometric linearization
The equations of continuum mechanics for solid state experienced a considerable simplification when small shifts can be accepted. Shifts are the difference
And the Verschiebungsgradient is the tensor
If a characteristic dimension of the body, it is then required as well as of small displacements of both, so that all terms, the higher powers of or include can be neglected. This leads to
This means that all distortion measures übergehehen of small displacements in the linearized strain tensor. A distinction between the Lagrangian and Eulerian description is no longer necessary. This geometrically linear analysis is valid for values up to 3-8%. If there are no small displacements, is spoken of finite or large displacements.
Laws of nature
The in mechanics formulated for extended bodies natural laws are in continuum mechanics expressed as a global integral equations from which with suitable continuity assumptions local (differential ) equations can be derived which must be satisfied at each material point. These equations are referred to as material theory. By means of equivalence transformations of the local equations can further principles are motivated then. Global and local equations can further be related to the instantaneous or the output configuration, so that there are four equivalent formulations for each law.
The basis for the formulation of the balance equations is the term of the stress tensor, which represents the stresses in bodies due to external loads. Newton's second law describes the reaction of a body to an external force. In reality and in continuum mechanics such forces are always initiated surface, that is, on a portion of the surface of the body act voltage vectors (vectors with unit force per unit area ) a propagating as tensions in the body. In addition, can affect distributed volume forces like gravity on each particle of the body. Now you can cut the body mentally, so that form at the cut surfaces of average voltages, however, depend on the orientation of the cut surfaces, ie their normals. The transformation of the normal stress vectors in accomplishing the stress tensor, which is the content of the Cauchy Fundamentaltheorems:
This is the Cauchy stress tensor. When considering the same relationship in the initial configuration or the change of the surface elements must be considered:
The stress tensor is called the first Piola - Kirchoff'scher stress tensor. It represents the voltages based on the output surface. The transpose is called the nominal voltages. For small distortions of these stress tensors coincide.
The balance equations of the mechanism described, the effect of the outside world on a body, and the resulting change of physical quantities. These variables are the mass, momentum, the angular momentum and energy. In a closed system, where, by definition, an interaction is excluded with the outside world, are derived from the balance equations conservation laws. In addition to those known in the mechanics external influences there are in the continuum and internal sources and sinks, such as the gravity is an internal source of friction. The second law of thermodynamics is taken into account in the form of the Clausius -Duhem inequality.
Mass balance
Is the density of the material in the spatial and in the description. Assuming that there are no mass sources any form, means the mass balance, that the mass of a body
Is constant in time:
The local forms are called continuity equation.
Momentum balance
The momentum equation states that the change in momentum is equal to the externally applied forces (volume- distributed or superficial ):
Wherein an acceleration of gravity, surface tension and or the surface of the body at the time, and must be observed.
Angular momentum balance
The angular momentum theorem states that the change of the angular momentum is equal to the externally applied torques (volume- distributed or superficial ):
What is the cross product, any, time- fixed position vector and the second Piola - Kirchhoff stress tensor is.
Energy balance
The thermo-mechanical energy balance indicates that the change in the total energy of a body is equal to the sum of the heat input and output of all external forces. The total energy consists in the Lagrangian description of the internal energy with the specific internal energy and the kinetic energy:
This includes internal heat sources of the body, the heat flow per unit area and on the surface of the body element outwardly normal. The negative sign of the term finally provides a power supply if the heat flow is directed into the body.
In the Eulerian description is called the global energy balance:
The local forms are:
Lagrangian description:
Eulerian description:
Clausius Duhem inequality
The Clausius -Duhem inequality follows from the application of the second law of thermodynamics to solids. With the specific entropy and the entropy flow is the Clausius -Duhem inequality:
In the formulation of Matrialgleichungen that indicate the stresses in relation to distortion, it must be ensured that the local forms of the Clausius -Duhem inequality is satisfied for arbitrary processes.
Material theory
The material theory focuses on the individual characteristics of materials. Destination of a material model is to describe the main aspects of the material behavior, said what is essential is determined by the observer. Fabric or material laws, such as material models are sometimes called, do not have the general validity of physical laws. Centrally in the modeling material, the dependence of the voltages of the strain (or vice versa), to be described in the form of equations, so that the deformation of bodies made of this material can be calculated. The classical continuum mechanics considered simple materials whose properties are reproduced with material constraints, material symmetries and constitutive equations.
Simple materials
The material theory of classical continuum mechanics presupposes determinism, locality and objectivity of the material. Determinism means that the current state of a body is determined completely and uniquely in one of its material points by the last movement of the body. Locality restricts the sphere of influence of the outside world on the current state of stress in a material point to its close surroundings, effects propagate from a material point to its nearest continues. Material objectivity means that the voltages do not change when the movement arbitrary rigid body motions are superimposed. Materials that meet these three requirements, simply hot. With simple materials of degree one, tensions arise in a material point from their past values and the current value of the Green's tensor or derivable therefrom sizes on this point. Materials of higher degree also use higher derivatives with respect to the material coordinates as the first, which account for the deformation gradient.
Material constraints
Material constraints represent kinematic constraints that limit the deformation of a material. The best known of these conditions is the incompressibility that allow the material only volume-preserving deformations, as they show some liquids or rubbery materials. The kinematic constraint is here. The reaction stresses in the material are then obtained from the balance equations and boundary conditions. When incompressibility is for example the reaction power of the pressure in the material. The largest constraint is that which distinguishes the rigid body. Here the stresses are completely determined by the laws of nature and constraints.
Material symmetries
Material symmetries which describe transformations of the deformation gradient are allowed, without changing the voltages. These transformations form the symmetry group of the material. All volume preserving transformations allowed, there is a liquid or a gas. For solids, only rotations are allowed: For isotropic solids are all the twists in transversely isotropic arbitrary rotations about an axis in orthotropic only 180 ° rotations about three mutually orthogonal axes and fully anisotropic, are just " twists " allows to 0 °.
Constitutive equations
The constitutive equations give a relation between the strains and the stresses in the form of integral, differential or algebraic equations. This may not contradict the physical constraints. The following material models give examples of constitutive equations.
The solid models from three to six are representatives of four groups of models of the classical theory of materials, the speed-dependent or -independent behavior with or without ( equilibrium ) describe hysteresis.
Example
With the expansion of a twisted pad under pure train (see picture) are the equations of continuum mechanics are applied. It was oriented in the starting configuration in the global Cartesian coordinate system, the length in the x direction, the width and height in the y direction in the z- direction and is the origin parallel to the coordinate axes. This block will elongated, with set dilations in material X, Y and Z directions respectively, and then rotated by 90 ° about the z- axis.
In the current configuration the material points have the spatial coordinates. The physical lines, and are therefore oriented parallel to the y- axis in the current configuration.
The deformation and Verschiebungsgradient calculated from the derivative
Because of the rotation of 90 ° results in the polar decomposition
From this one gets the strain tensors:
What one sees that for large rotations the geometrically linear rotations are not usable.
Using Hooke's law for large deformations, the second Piola Kirchhoff stresses arise in the Lagrangian picture:
For pure train can be seen and therefore
Wherein the Young's modulus. The first Piola -Kirchhoff tensor is
In the considered here statics of momentum states
What is given because. The Cauchy stress tensor gets the form:
The reference configuration is suitable to the unit cube. So get the material points in the initial configuration, the coordinates
The operating between the reference and the output configuration " deformation gradient " is also referred to as a Jacobian matrix:
The volume integral of the field size is then
And can be calculated in this form with the numerical Gaussian quadrature.