Constitutive equation

A material model, material or substance law, is a quantification of physical properties of materials. Material properties be stated the ability of a material to physical influences ( such as forces, heat supplies or voltages ) to respond. Material models are independent of the shape of a body and are usually motivated by experiment. Destination of a material model is able to predict how and to what extent the material reacts to external influences. As models they form relevant to the model builder relationships from mathematical, are therefore mathematical models, which has copyright, the modeler. Although they are often referred to as the material laws, they do not have the general validity of physical laws, because for the same material may be present, which is the observed dependencies, the computational effort, the accuracy and its scope differ in the application area, different models of different model creators.

The reactions of a material called material sizes and influence quantities Kontitutivvariable. Linking the Kontitutivvariablen and material variables is done in equations that are called material or constitutive. Furthermore, inequalities can also occur that the different modes of behavior (such as plastic flow, phase transitions ) of the material from each other. The simplest connection between the material and size Konstitutivvariable is the proportionality: The material size is equal to a Konstitutivvariable multiplied by a constant. Such a relationship has often definitional character for a material property, as the examples of specific heat capacity, permittivity, permeability ( magnetism ) pointing to the side material constant. Material properties and hence the material constants are always dependent on the temperature, which can be taken into account temperature coefficient. Are considered more and more complex dependencies as relevant in the situation calculated, the modeler enters the picture, then a suitable model for the considered case created (if it does not yet exist ). Thus, the development of the first definition of a property and quantification with a material constants on the consideration of temperature dependence is traced to complex models. Causes of complexity can be nonlinearities multiaxiality or depending on several Konstitutivvariablen.

The continuum mechanics has its own area of ​​knowledge, the material theory, which deals with the classification of materials and material properties and the creation of media models. The materials science and materials engineering material models developed from the need to characterize the materials developed by it as closely as possible.

Scope

Material models have a limited validity because they are not considered by the model builder which are excluded influences. In the application of the models is to pay attention to whether the underlying model assumptions, apply such as:

• Environmental conditions such as temperature or pressure
• Periods over which the material behavior is to be monitored (long-term or short-term performance).
• Rates of change of Konstitutivvariablen. A distinction is static, semi- static, moderate or high rates of speed ranges.
• Size scale of the material samples. One distinguishes macro -, meso- or micro-level of analysis.
• Chemical state. Materials can change their properties due to corrosion.

Physical framework

Materials are on the one hand the physical laws such as mass, momentum and energy conservation or Maxwell's equations. On the other hand follows a material sample geometric bonds, which are to the field of kinematics and describes the possible movement and resulting deformations and expansions. Material models that specify a quantitative relationship between the variables in these physical and kinematic equations are suitable to calculate the responses of the laws of nature following the body to external influences.

The second law of thermodynamics has a special status: In the preparation of material models must be taken to ensure that all possible temporal profiles of the Kontitutivvariablen the entropy of the material is non-negative.

Simple materials

Some material properties are so complex that they are presented more examples based on the response of test specimens. Examples include the impact strength, notched impact strength or component SN curves. This is thus rather to component properties, because the separation of material property and sample property (shape, size or surface finish of the specimen ) is not reliable or not possible with reasonable effort. For most material properties, however, the idea has proved that any part of a material sample has the same properties as the sample itself. To determine the material response at a point of the sample, then you need to consider only one ( infinitesimal ) small neighborhood of the point. The material reacts locally on local influences. Moreover, experience teaches that the material response but not completely dependent on past or present of future influences that are materials are deterministic. The principle of material objectivity states that an arbitrarily translotorisch or rotationally moving observer the same material response measures as a relatively dormant for sample experimenter. Materials that are local, deterministic and objectively called easy and only such are the subject of the classical theory of materials.

Material equations

Lenses sizes

For material sizes and Kontitutivvariable only objective variables can be used, ie such, in the same way one perceives any translotorisch or rotationally moving observer as a relatively dormant for sample experimenter. Mass, density, temperature, heat, specific internal energy and entropy are scalar objective sizes. Lenses directional quantities, vectors are eg forces, voltage, heat flow and Entropieflussvektoren. However, the velocity of a particle, for example, no objective size, because it is perceived differently by different observers moving. Furthermore, occur in some natural laws on tensorial quantities whose objectivity is to be proved in each case. An important example is the Cauchy stress tensor just mentioned, which is an objective quantity.

Types of constitutive equations

Equations which describe the material can be divided into three classes:

While the members of the first two classes are of algebraic nature, in the constitutive equations, other shapes may occur:

• Differential equation
• Integral equation
• Inequality and also
• Algebraic equation.

Conservatives materials

Conservatives materials possess a special form of constitutive equations in which the material size is determined by the derivative of a scalar potential after Kontitutivvariablen. An example of this is the Hooke's law where the potential is the strain energy. Here one has the special properties:

Material parameters

As they are also called - - For the quantification of the material behavior, the constitutive equations of material parameters or contain material constants that allow the model to measured values ​​adapt. It is common practice to design the model so that in real materials, the parameters have positive values. If negative values ​​occur in the adaptation, caution is required in the rule.

Mechanical solid models

The following table shows representatives of the four material models of classical continuum mechanics of solids.

• If the stress-strain curve for loading and unloading the same, so the constitutive law is elastic or perfectly elastic ( b in the figure) and not viscos, ie independent of speed. Ideally elastic material laws do not necessarily linear, but can also be non-linear as shown in the picture. If a substance law and ideal linear elastic ( d in the figure), one speaks of an ideally linear elastic law. Since this expression is very long, spoken abbreviated by a linear elasticity law or just from one material, for example, when Hooke's law.

• If the stress-strain curve for the relief back at the starting point to, but the path is different for relief from the stress at (a in the figure), the constitutive law is visco- elastic and you can see a hysteresis. Rubbery materials exhibit such behavior. At very slow stress-strain curves, the hysteresis disappears and you see a situation as in figure b.

• If start and end points are not equal, then one speaks of a plastic material law ( c in the figure). This applies to substances that flow during exercise, see also the example below. Here observed is independent of the strain rate hysteresis.

• Are start and end points is not equal and is observed depending on the strain rate hysteresis, which is not zero, even at very slow stress-strain curves, there is viscous plasticity. The hysteresis curve at very slow stress-strain curves is called Gleichgewichtshysterese here.

The four classes of materials mentioned are thus dependent or independent of the strain rate and show a Gleichgewichtshysterese or not. The crawl is a property viscoser substances.

There is a linear relationship between the stress and the strain, then one speaks of a linear constitutive law ( d in the figure). It must be also linear with respect to its thermal expansion. Law, do not show a linear relationship, referred to as non-linear material law.