## Constitutive Laws: Classification of Materials Mechanical Response

### According to Linearity

The mechanical response of a material can be classified as either linear or nonlinear. For a linear material, the load is directly proportional to the response. In such materials, the plot of a measure of the stress versus a measure of the strain exhibits a linear behaviour during both loading and unloading. A nonlinear material is a material that exhibits any other response.

### According to Energy Dissipation

The mechanical response of a material can be classified as either elastic or inelastic. An elastic material will not dissipate energy through a loading cycle (i.e., the amount of energy stored or released is path independent, and a specimen will return to its initial position once the load is removed). Such materials are also called Green Elastic or Hyperelastic. An inelastic material, on the other hand, will dissipate energy through loading cycles.

It is important to note that elasticity and linearity are two different concepts. Traditional construction materials like concrete and steel have a linear elastic response where the load is proportional to the deformation and energy is not dissipated through loading cycles in the elastic loading stage. Rubber, on the other hand, is an example of a nonlinear elastic material. The energy stored inside the material during loading is fully recoverable upon unloading, provided the load is within a certain limit. Thus, rubber is considered elastic. However, the relationship between the load and the deformation is not necessarily linear.

### According to Direction or Material Symmetries

The mechanical response of a material can be classified as either isotropic or anisotropic. Anisotropic materials could be generally anisotropic, othrotropic, or transversely isotropic. The description of each is as follows:

• #### Isotropic materials

These are materials whose response is independent of the loading orientation; i.e., if a specimen is loaded in any direction, the response would be the same. Examples of materials that behave in this manner are traditional concrete and steel materials.

• #### An anisotropic material

These are materials whose mechanical properties that are dependent on the orientation of the specimen. There are different levels of anisotropy:

• ##### Generally anisotropic material

A generally anisotropic material is one that exhibits no symmetric response along any specified axes.

• ##### Orthotropic material

Such material has a symmetric response along three perpendicular axes (e.g., a material with different fibers oriented in two or three orthogonal directions).

• ##### Transversely isotropic material

The material has a symmetric response along an axis that is normal to a transverse plane of isotropy. If the material is loaded in the transverse plane of isotropy, then the behaviour is isotropic, i.e., the material properties will be independent of the orientation within that plane. Wood materials with fibers oriented in a certain direction are perfect examples of transversely isotropic materials. Many biological tissues possessing internal fibers, such as ligaments, can also be considered transversely isotropic.

### According to Homogeneity

The mechanical response of a material can be classified as either homogenous or nonhomogenous. A homogeneous material has a response that is independent of the specimen used during a mechanical loading experiment. Most traditional engineering materials (e.g., steel) can be considered homogeneous. A non-homogeneous material has material properties that vary according to the location of the specimen used in the experiment.

### According to Time Dependence

The mechanical response of a material can be classified as either viscous or non-viscous. A viscous material is a material whose response is time dependent (also called rate dependent). The mechanical response of a viscous material is dependent on the rate of application of the load or on whether the response varies with time for a static load. Creep and stress relaxation are examples of phenomena exhibited by viscous materials. A viscous material can exhibit creep under a constant load; i.e., under a constant load, the deformation will increase. A viscous material can also exhibit stress relaxation when subjected to a specified deformation; i.e., the load required to hold a specified deformation will decrease with time when that deformation is held constant. Many materials exhibit creep and stress relaxation, but the speed by which such materials creep or relax will dictate whether such viscous response is required to be studied for a certain application of the material. A non-viscous material is a material whose response is time independent (also called rate independent). The response of non-viscous materials is independent of the rate of the application of the load. Most engineering materials can be considered time independent when the rate of load application is relatively low.