Viscoelastic models

SDT-visc Contents Functions

2.2 Viscoelastic models

This section details models of structures used to account more precisely for the constitutive behavior of various viscoelastic materials.

For frequency response computations, section section ?? shows how the complex dynamic stiffness is built as a weighted sum of constant matrices associated with the various materials.

For eigenvalue computations or time responses on the full model, the introduction of state space models (section ??) or second order models with internal states (section ??) allow constant matrix computations. Such formulations could be used for frequency domain solutions but they are higher order and the increase in DOF count limits their usefulness. For the case of fractional derivative models (section ??), only modeshape computations are accessible.

2.2.1 Frequency domain representation with variable coefficients

For a structure composed of elastic and viscoelastic materials frequency domain computations only require the knowledge of the complex modulus E_i(s,T,σ₀) for each material. By using the fact that stiffness matrices depend linearly on the constitutive law coefficients, one can represent the dynamic stiffness of a viscoelastic model as a linear combination of constant matrices (for independent complex moduli in the same material their may be more than one matrix associated to a given material)

(2.29)

This representation is the basis for the development of solvers adapted for structures with viscoelastic materials.

For frequency domain computations, it is rather inefficient to reassemble Z at each operating point E_i,s. Two solutions can be implemented easily. On can store the various M, K_e, K_ei, C, K_vi matrices and evaluate the weighted sum (??) at each operating point, or store element matrices and reassemble with a weighing coefficient associated with the material property of each element.

2.2.2 State-space representations

One discusses here state space representations associated with analytical representations of the complex modulus discussed in section ??.

State space models are first order differential equations, assumed here with constant coefficients, with the standard form

(2.30)

The matrices are called : A transfer, B input, C observation and D direct feed-trough. The first equation is the evolution equation while the second is called the observation equation.

The usual technique for time integration of mechanical models is to define a state vector combining displacements and velocities, leading to a model of the form

(2.31)

This model is rarely used as such because M is often singular or, for a consistent mass, leads to matrices M⁻¹K and M⁻¹C that are full. When using generalized coordinates, one can however impose a unit mass matrix and this easily use the model form. Some authors [] also prefer this form to define eigenvalue problems but never build the matrices explicitly so that the full matrices are not built.

The other standard representation is the generalized state-space model

(2.32)

which preserves symmetry and allow the definition of simple orthogonality conditions on complex modes.

One will now introduced generalized state space models for the case of viscoelastic constitutive laws. A rational fraction, that does not go to infinity at high frequencies and having distinct poles, can be represented by a sum of first order rational fractions

(2.33)

By introducing the intermediate (relaxation) field q_vⁱ=−E^∞gⁱ/(s+ωⁱ)q, one can rewrite (??) as a state-space model of higher size

(2.34)

One saw in the previous chapter that real constitutive laws could be represented with rational fractions of relatively high order. This representation is practical only for reduced models (M_R = T^TMT, ...) where the strategy for the selection of reduction basis T will be detailed in section ??.

In practice for compatibility with non-linear hyperelastic behavior, it is preferable to use a stress rate relaxation equation, where the evolution equation is given by

(2.35)

which, in linearized matrix form, using a rank N_v decomposition of K_v=[B_v][B_v]^T, the viscous force given by F_vⁱ=[B_v] q_vⁱ and the viscous states q_vⁱ evolution equation

(2.36)

Assuming a standard mass normalized reduction basis, so that for the chosen states the mass is equal to identity, the resulting state space model is

(2.37)

which has 2 N_q + N_cell× N_v states. Building of such state-space models is implemented in nl_solve Reduc2ss which requires as SDT-nlmodal token.

For time domain representations, SDT-nlsim (see more details in see sdtweb nlfu#uMaxw ) allows the use of stress or strain relaxation to provide adapted non-linear coupling.

2.2.3 Second order models with internal states

If the state space form is more compact a priori, the operators available in a given FEM code may make its manipulation more difficult. A classical solution is thus to build a second order model of the form usual in mechanics and thus easily manipulated with a mechanically oriented code. The implementation of internal states, however requires the definition of multiple fields at the same node which is not easily implemented in all software packages.

The Anelastic Displacement field []) method considers a modulus representation of the form (??), which leads to a model of the form

(2.38)

The absence of a mass associated with internal states q_vi can lead to problems with certain solvers. An alternative is the GHM [] method which represents the modulus as

(2.39)

and defines internal states by q_vj=αⁱ/s²+2ζⁱωⁱs+(ωⁱ)²q. One will note that not all rational fractions can be represented in the form (??).

The introduction of q_vi fields in the previous section corresponds to the classical thermodynamics theory of materials with augmented potential including internal states. The first row in equation (??) indeed corresponds to the representation of stress in viscoelastic materials in the form

(2.40)

In practice, the q_v are only non zero for viscoelastic elements. The direct use of matrices assembled following (??) must thus be done with solvers capable of eliminating unused DOFs. When using reduced models, the problem does not normally occur since reduced basis vectors are typically non zero over the whole structure and thus lead to none zero internal states.

In the time domain, this formalism is more easily dealt with because the time evolution of internal states can easily be computed by time integration of the relation between the q_vi and q. For a model of form (??), the evolution of the internal state is thus given by

(2.41)

which can be easily integrated (this approach is used in ABAQUS [] for example). This formalism corresponds to the separate treatment of bloc rows in (??) or (??) which is simple in the time domain but leads to a non-linear problem in the frequency domain, unless operators are defined implicitly as in Ref [].

2.2.4 Fractional derivatives

The internal state formalism can also be used to represent fractional derivative constitutive laws if one uses non integer but rational derivatives. For a common denominator p, one will use a modulus of the form

(2.42)

and build a state space model of the form

(2.43)

where the state vector will combine fractional derivatives of the displacement s^k/pq where k=1:2p−1 and of the internal state q_vk=−E_k/(s^k/p+ω_k)q (see [] for example).

In practice, the number of blocs in the state vector being proportional to p, constant matrix representations are thus limited to small values of p. This limits practical uses of fractional derivative models to frequency domain response and non-linear eigenvalue computations (see section ??).