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A material model, material or material law, is a quantification of physical material properties. Material properties indicate the ability of a material to react to physical influences (such as forces, heat, or currents). Material models are independent of the shape of a body and are usually experimentally motivated. The aim of a material model is to be able to predict how and to what extent the material reacts to external influences. The models are mathematical models that have a creator (i.e. an author), the model maker. Although they are often referred to as material laws, they do not have the general validity of physical laws, since different creators may present different models for the same material property, differing in their application, dependencies, calculation complexity, accuracy and range of validity.

A reactions of a material is called a material quantity or property, and the influencing variables are constitutive variables. The linking of the constitutive variables and material quantity takes place in an equation, which is called material or constitutive equation. Furthermore, inequalities can also occur which separate the different behavioral modes (such as plastic flow, phase transitions) of the material. The simplest possible relationship between material size and constitutive variable is the proportionality: The material size is equal to a constitutive variable multiplied by a constant. Such a connection often has a defining character for a material property, as the examples show: specific heat capacity, permittivity, permeability (magnetism). Material properties, and therefore also the material constants, are always dependent on the temperature, which can be taken into account with temperature coefficients. If further or more complex dependencies are considered relevant in the calculation case under consideration, the model creator is put into action, which then creates a new model suitable for the case considered (if this does not already exist). Thus the development from the first definition of a property, and quantification of it with a material constant, will be continued, through the consideration of a temperature dependence, towards complex models. Causes of complexity can be nonlinearities, multi-axis dependency or dependency on more /several constitutive variables.

The continuum mechanics has its own knowledge, the materials science, which deals with the classification of materials, the classification of material properties and the creation of material models. Materials science and materials engineering develop material models based on the need to characterize the materials as precisely as possible.

Scope of validity
Material models have a limited validity because they do not take account of the influences exerted by the model operators. When applying the models, make sure that the assumptions underlying the model are correct, for example:


 * Ambient conditions such as temperature or pressure
 * Time span over which the material behavior is to be monitored (long-term or short-term behavior).
 * Change rates of constitutive variables. One differentiates between static, quasi-static, moderate rates or high-speed ranges.
 * Scale of material samples. One differentiates between macro-, meso- or micro-planes.
 * Chemical condition. Materials can change their properties by corrosion.

Physical framework
Materials are, on the one hand, subject to physical laws such as mass, pulse, and energy conservation, or Maxwell's equations. On the other hand, a material sample follows geometric bonds, which are the subject area of ​​the kinematics and which describes the possible movements and resulting deformations and strains. Material models which indicate a quantitative relationship between the variables in these physical and kinematic equations are suitable for calculating the reactions of a body following the natural laws on external influences.

The second main point of thermodynamics has a special status: When building material models, care must be taken that the entropy production of the material is not negative in all possible temporal courses of the constitutive variables.

Simple Materials
Some material properties are so complex that they are presented in more detail using the reaction of test specimens. Examples of this are the impact strength, notched impact strength, or component-Wöhler lines. This is more a matter of component properties because the separation of material properties and specimen properties (shape, size or surface characteristics of the test specimen) can not be achieved reliably or not at reasonable cost. However, with most material properties, it has been found that any part of a material sample has the same properties as the sample itself. For the determination of the material response at one point of the sample, one needs to consider only one (infinitesimal) small neighborhood of the point. The material reacts locally to local influences. Moreover, experience teaches that the material response is entirely dependent on past or present, but not on future influences, that materials are deterministic. The principle of material objectivity states that an observer of any translucent or rotatory motion measures the same material response as an experimentator resting relative to the sample. Materials that are local, deterministic and objective are called simple, and only such are the subject of classical material theory.

Objective sizes
For material variables and constitutive variables, only objective quantities can be used. Being an objective quantity / property means that an observer of any translational or rotatory motion and an experimentator resting relative to the sample, will measure the same value of the quantity /property. Mass, density, temperature, heat, specific internal energy and entropy are scalar objective variables. Objective directed quantities, vectors, are, for example, forces, stress, heat flux and entropy flow vectors. The velocity of a particle, however, is, for example, not an objective quantity because it is judged differently by differently moving observers. Furthermore, tensor quantities occur in some natural laws, the objectivity of which is to be examined in a particular case. An important example is the Cauchy stress tensor, which is an objective quantity.

Types of Material Equations
Equations describing a material can be divided into three classes:

While the members of the first two classes are algebraic in nature, other forms may also occur in the constitutive equations:
 * 1) Those which indicate the material symmetry or directional dependence. In one such, the properties of the material are different in one direction than in the other. A well-known example of this is wood, which is different in fiber direction than transversely thereto.
 * 2) Material constraints prohibit material changes. The best known example of this is the incompressibility in which the volume of a material sample is invariable.
 * 3) Constitutive equations which formulate the functional relationship between material variables and the constitutive variables.


 * Differential Equation,
 * Integral equation,
 * Algebraic equations and
 * inequalities.

Conservative Materials
Conservative materials have a special form of constitutive equations where the material size results from the derivation of a scalar potential according to the constitutive variable. An example of this is Hooke's law, where the potential is the shape-change energy. Here, one has the special properties:
 * 1) Path unrelatedness: The material size always has the same value, given the value of the constitutive variable, regardless of the way the final state was reached.
 * 2) Conservativity: If the potential is an energy, no work is performed along a closed path, else some energy will be consumed. Any work that has been completed is returned by the system to the point of return to the original point.

Material parameters / empirical parameters
For the quantification of the material behavior, the constitutive equations contain material parameters / empirical parameters or - as they are also called - material constants, which allow the model to be adapted / optimized to measured values. It is generally customary to design the model in such a way that the parameters have positive values ​​for real materials. If negative values ​​occur during the adjustment, caution is generally required.

Mechanical solids models
Selection of single-axis material laws: A: elastic. B: ideal elastic. C: plastic. D: ideal linear elastic The following table shows representatives of the four material models of classical continuum mechanics for solids.


 * If the stress-strain profile is identical on loading and unloading, the fabric is elastic or ideally elastic (b in the picture) and not viscous, i.e. Speed-independent. Ideal elastic fabric laws do not necessarily have to be linear, but can also be nonlinear as shown in the picture. If a material law is linear and ideally elastic (d in the image), then one speaks of a linear ideal elastic law. Since this expression is very long, one speaks abbreviated of a linear law of elasticity or simply of a law of elasticity, eg in Hooke's law.
 * If the stress-strain curve is again at the starting point at the relief, but the path differs from that at the load (a in the figure), the material law is viscoelastic and a hysteresis is seen. Rubber-like substances show such behavior. With very slow stress-strain curves, the hysteresis disappears and one sees a course as shown in figure b.
 * If the start and end points are not the same, one speaks of a plastic law of matter (c in the picture). This applies to substances that flow during the load, see also the example below. Here, a hysteresis independent of the strain rate is observed. If the start and end points are not the same, a hysteresis which is dependent on the strain rate, but which does not disappear even in the case of very slow stress-strain curves, is visco-plasticity present. The hysteresis curve for very slow stress-strain curves is called equilibrium hysteresis.

The four classes of material mentioned are therefore dependent on the strain rate or are independent and exhibit an equilibrium hysteresis or not. Creep is a property of viscous substances.

If there is a linear relationship between the voltage and the strain, then one speaks of a linear law of matter (d in the picture). It must also be linear in its thermal expansion. Laws which do not show a linear relationship are called nonlinear laws of matter / constitutive equations.