Assume that you apply a load to a polymer, some part of the applied load is dissipated by the energy dissipation mechanisms ( such as segmental motions) in the bulk of polymer , and the other part of the load is stored in the material and will be released upon removal of the load (such as the elastic response of a spring!). The former is represented by the loss modulus in the viscoelastic models which the later is shown by the storage modulus. 

On the other hand, Tan delta represents the ratio of the viscous to elastic response of a viscoelastic material or in another word the energy dissipation potential of the material. Therefore, larger Tan delta indicates that your material has more energy dissipation potential so the greater the Tan delta, the more dissipative your material acts under applying loads. So, Tan delta (which is also known as damping factor) shows the energy dissipation potential of your material.

On the other hand, you can compare damping potential of two samples by loss modulus only when their storage modulus is equal (comparable).

Conclusion: Loss modulus is a model parameter which represents the viscous part of viscoelastic materials BUT CANNOT show the damping potential of the material by itself. Tan delta is the ratio of loww to elastic modulus and can be used to compare the damping behavior (energy dissipation potential) of two materials.

Assume that you apply a load to a polymer, some part of the applied load is dissipated by the energy dissipation mechanisms ( such as segmental motions) in the bulk of polymer , and the other part of the load is stored in the material and will be released upon removal of the load (such as the elastic response of a spring!). The former is represented by the loss modulus in the viscoelastic models which the later is shown by the storage modulus. 

On the other hand, Tan delta represents the ratio of the viscous to elastic response of a viscoelastic material or in another word the energy dissipation potential of the material. Therefore, larger Tan delta indicates that your material has more energy dissipation potential so the greater the Tan delta, the more dissipative your material acts under applying loads. So, Tan delta (which is also known as damping factor) shows the energy dissipation potential of your material.

On the other hand, you can compare damping potential of two samples by loss modulus only when their storage modulus is equal (comparable).

Conclusion: Loss modulus is a model parameter which represents the viscous part of viscoelastic materials BUT CANNOT show the damping potential of the material by itself. Tan delta is the ratio of loww to elastic modulus and can be used to compare the damping behavior (energy dissipation potential) of two materials.

Thank you for the insightful answers, Ebrahim.

In your idea, do you think applying damping coefficient or energy dissipation fraction is a good way in numerically simulating progressive damage behavior of composites, usually accompanied by suddenly internal energy loss? Also it is my question "how and where those loss energy goes during progressive damage". Looking forward your ideas.

First of all, I do not have much personal experience in the field of modeling progressive failure. But here is what comes to my mind:

You should condier that damping is the energy dissipation under cyclic load (Dynamic tests). Also, you should keep in mind that most of the dynamic tests are done in Linear Viscoelastic (LVE) Region or in another word in very low starins and low frequencies while during tests which causes damage in the material, the material experiences much higher strains and greater frequencies which causes the failure. The later region is known as non-linear viscoelasticn region.

Another thing that you need to consider is that, the value of loss, storage and damping factor depends on the testing method. Obviously, the loss modulus in flexural test is not the same as compression test. The compexity of the analysis raise from the fact that for example in studying progressive fracture, the deformation of the polymer matrix at the crack tip does not only necessarily consist of one mode of deformation (stretching, bending, compression) and any comination of them could exist which makes it more difficult to decide which parameter should be used to represent the bahvior of the material.

Apart from that, as I mentioned, the loss modulus represents the dissipation mechanisms (again consider that it is measured in cyclic tests) and the damping shows energy dissipation potential. So in homogenous polymer matrices, in my opinion, it is likely to expect the energy dissipation coefficient at the crack tip to be greater for the polymers with higher damping factor.

The loss modulus assumes a homogenous value relative to a single constitutive law, formulated with the same assumptions than for the description of Young moduli.

It assumes that the the dissipation in a material is proportional to the strain energy it stores. It is typically written in the frequency domain E(w)*[1+i eta(w)] with E the youdng modulus, w the pulsation and eta the loss factor.

Composite materials are in fact complex heterogenous assemblies of sub components made off homogeneous materials. That is to say you could define several constitutive laws, one for each homogenous sub-part in a refined model, or use some homogenization technique to obtain an equivalent homogenous material at your system level (usually for specific ranges of application).

Like for the Young modulus, loss modulus is valid for linear problems, it depends on the frequency, and any large transformation or large displacement effect will not be represented in general.

 Your application seems rather complex, and will likely require a study of damping sources and localization (e.g. material or interfacial friction between sub-components ? ...) to obtain something representative.