Maxwell or Kelvin Models for Creep Only List the Continuity Criteria

Maxwell Bodies

Rock Mechanics

Howard J. Pincus , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.B Rheology

Idealized elastoplastic behavior is shown in Fig. 4. The rheological model for this is a St. Venant material consisting of a spring (elastic element) and the frictional contact between a weight and the underlying surface (plastic element) in series. The body is deformed elastically (segment OT in Fig. 4) until the yield strength is reached, at which point plastic behavior takes over (segment TU).

FIGURE 4. Idealized elastoplastic behavior. OT and TU represent elastic and plastic behavior, respectively. The stress coordinate at T represents the yield strength.

A more realistic stress–strain curve for rock with elastic–plastic characteristics is shown in Fig. 5. Point Q is approximately at the yield point, marking the transition from elastic to plastic behavior. The OP segment, commonly referred to as the knee of the curve, represents a combination of elastic behavior and the closing of voids. The PQ segment, which is approximately linear, represents the deformation of the solid mineral matter of the rock after the voids have been closed. The QR segment can be interpreted as representing both elastic and plastic behavior, the elastic component decreasing and the plastic component increasing as the curve climbs to the ultimate stress at R. The negative slope of the RS segment indicates decreasing capability to sustain applied stress as strain increases.

FIGURE 5. Typical stress–strain curve for rock with elastoplastic characteristics. OP reflects closing of voids as the rock is compressed; PQ is approximately linear because solid mineral matter is being compressed; QR and RS are, respectively, the ductile and brittle portions of the curve. The stress coordinate of R is termed the ultimate strength, or strength.

The stress–strain behavior of rock is a function not only of the properties of the rock substance itself but also of other factors such as environment (temperature, confining pressure, interstitial fluid), rate of loading, previous loading history, size and shape of the unit being deformed, and spacing, orientation, surface characteristics, and infillings of discontinuities.

The time-dependent deformation characteristics of rock are often shown in creep curves, in which strain is plotted against time for constant stress. In the most common representations of creep curves, it is assumed that the stress is applied instantaneously and is then held constant.

In Fig. 6, such curves are shown as follows:

FIGURE 6. Strain versus time at constant stress (creep curve) for a Burgers body (B), its Kelvin (K) and Maxwell (M) components, and the Newton (N) and Hooke (H) components of the Maxwell body. The Newton (viscous) components are represented by dashpots and the Hooke (elastic) components by springs. Values used in strain calculation were: stress   =   1   ×   10⁸  Pa, E _m  =   6   ×   10¹⁰  Pa, ν_m  =   1.5   ×   10¹³  N · s/m², E _k  =   4   ×   10¹⁰  Pa, and ν_k  =   1.2   ×   10¹³  N · s/m².

H—Hooke body, or elastic element (spring).

N—Newton body, or viscous element (dashpot).

M—Maxwell body (spring and dashpot in series). In this illustration, the Maxwell body is composed of the Hooke and Newton bodies for which curves H and N are shown. Thus, the ordinates for M equal the sums of the corresponding ordinates for H and N.

K—Kelvin body (spring and dashpot in parallel). In this illustration, the Kelvin body is composed of Hooke and Newton bodies different from those in the Maxwell body.

B—Burgers body (Maxwell and Kelvin bodies in series). The ordinates for B equal the sums of the corresponding ordinates for M and K.

The shape of the Burgers curve is much like the creep curves obtained for many rocks in laboratory experiments run over weeks or months. Both loading and unloading are shown in Fig. 7; the loading stress in A is double that in B. Instantaneous loading and unloading are represented by the vertical segments at 0 and 1500   s, respectively; the elastic component of the Maxwell body is responsible for this time-independent behavior.

FIGURE 7. Creep curves for loading of a Burgers body at two stress levels, with unloading at 1500   s. Values used in calculation: (A) stress   =   1.50   ×   10⁸  Pa, (B) stress   =   0.75   ×   10⁸  Pa, E _m  =   6   ×   10¹⁰  Pa, ν_m  =   1.5   ×   10¹³ N · s/m², E _k  =   4   ×   10¹⁰  Pa, and ν_k  =   1.2   ×   10¹³ N · s/m².

Immediately following the initial elastic deformation, the slope of the creep curve decreases and the curve subsequently becomes almost linear. The portion of the curve with decreasing slope, sometimes called the decelerating stage, represents deformation dominated by the behavior of the Kelvin body. The next stage, in which the curve is approaching linearity, is called the steady-state stage; here, the deformation is dominated by the viscous component of the Maxwell body, and in fact the curve is approaching an asymptote with slope equal to that of the Newton component of the Maxwell body.

The decay curve following elastic unloading is simply the unloading curve for the Kelvin body. This time-dependent recovery is referred to as anelastic behavior. The decay curve approaches a horizontal asymptote, the intercept of which on the strain axis is the permanent deformation after unloading. This permanent deformation is merely the irreversible strain of the viscous (Newton) component of the Maxwell body.

The equations pertaining to the foregoing discussion of the Burgers body are presented below. The notation used is as follows: E _m is the elastic (Hooke) component, Maxwell body (pascals); v _m the viscous (Newton) component, Maxwell body (newton seconds per square meter); E _k the elastic (Hooke) component, Kelvin body (pascals); v _k the viscous (Newton) component, Kelvin body (newton seconds per square meter); ϵ the strain; t the time (seconds); and S ₀ the stress (pascals), applied instantaneously at t  =   0 and relieved instantaneously at t  = t _u. The equations are as follows.

Burgers:

(1) ${ϵ}_{\text{B}}={\text{S}}_0\left\{1/{\text{E}}_{\text{m}}+\text{t}/3{\nu }_{\text{m}}+1/{\text{E}}_{\text{k}}\left[1-\text{exp}\left(-{\text{E}}_{\text{k}}\text{t}/3{\nu }_{\text{k}}\right)\right]\right\}.$

[The first two terms inside the curly brackets in Eq. (1) represent the Maxwell component and the third term represents the Kelvin component. The first and second terms of the Maxwell are its Hooke (elastic) and Newton (viscous) components.]

Instantaneous elastic strain (loading and unloading):

(2) ${ϵ}_{\text{I}}={\text{S}}_0/{\text{E}}_{\text{m}}.$

Asymptote to loading curve:

(3) ${ϵ}_{\text{L}}={\text{S}}_0\left\{1/{\text{E}}_{\text{m}}+1/{\text{E}}_{\text{k}}+\text{t}/3{\nu }_{\text{m}}\right\}.$

Asymptote to unloading curve:

(4) ${ϵ}_{\text{U}}={\text{S}}_0{\text{t}}_{\text{u}}/3{\nu }_{\text{m}}.$

The coefficient 3 appearing with the viscosity terms in Eqs. (1), (3), and (4) derives from the assumption of incompressibility for the two Newtonian fluids in the Burgers model.

Although the Burgers model provides a creep curve similar to that of many rocks, it is deficient in that it does not provide for yield strength. The Bingham–Voigt or Kelvin–Bingham model provides a yield stress component in addition to elastic and viscous components (Y in Fig. 8). The yield strength is modeled by a St. Venant material in which weight remains at rest until pull (or push) on it exceeds the sliding friction. Below the yield stress, the behavior is that of an elastic component in series with a Kelvin body; the creep curve is that of an instantaneous elastic deformation followed by that of a Kelvin body. At the yield stress and above, an additional viscous component acts in series with the other components. If this viscous body has a linear response, the overall behavior is that of a Burgers body; the Bingham–Voigt model allows for nonlinear behavior of this viscous body.

FIGURE 8. (a) Bingham–Voigt or Kelvin–Bingham body. Y represents the yield strength component. The Kelvin component consists of N₁ and H₁, and the Bingham component consists of N₂, Y, and H₂. N₂ may represent nonlinear viscosity in modeling rock behavior. (b) Creep curve if the stress level is smaller than the yield strength. (c) Creep curve if the stress level is equal to or greater than the yield strength.

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Paper: Effects of Moisture and Temperature

D.F. Caulfield , A.H. Nissan , in Encyclopedia of Materials: Science and Technology, 2001

4.1 Models for Viscoelasticity

The study of the viscoelastic behavior of materials may be approached on many levels. For much of its history, one approach to viscoelasticity has been in terms of models. Springs and dashpots coupled as Maxwell bodies and/or Voigt (Kelvin) bodies and, arranged in a multitude of arrays, have been used as helpful analogs for different viscoelastic materials ( Alfrey 1948) including paper. Dashpots containing non-Newtonian fluids have often been found necessary to more accurately represent the behavior of the model to a real viscoelastic material (Andersson et al. 1949). In all cases, however, the use of spring/dashpot models, combining elastic and viscous elements, is essentially an empirical curve-fitting technique, and far removed from actual molecular behavior.

Aside from the thermodynamic approach to rheology discussed later, viscoelasticity may be described from a fundamental phenomenological approach using only mathematics and a continuum theory. To some purists, this is the only sound way that viscoelasticity can be understood. However justified that viewpoint may appear, the fact remains that matter (and paper certainly) is not a continuum but consists of molecules, and the differences in properties of different materials must be due essentially to the differences in their molecular structures and behavior.

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The governing theory of elasticity imaging

Salavat R. Aglyamov , in Tissue Elasticity Imaging, 2020

10.1 Viscoelastic tissue response

If biological tissue is considered as an elastic body, it is assumed that tissues have a capacity to store mechanical energy with no dissipation, i.e., the deformation is completely reversible. However, such assumption is correct only if deformation occurs with infinitesimal speed, where thermodynamic equilibrium is established at every moment of time. If the speed of deformation is finite, mechanical energy dissipates and the deformation process is irreversible [1,11]. Under isothermal conditions, energy dissipation is a result of the internal friction, or viscosity, and such mechanical behavior is called viscoelasticity. A viscoelastic body combines properties of a solid body to store mechanical energy and a viscous fluid, which does not store energy, but only dissipates it. The stress-strain relation of an elastic material does not depend on time, i.e., the material does not have a "memory," whereas the mechanical response of a viscoelastic material is not only determined by the current stress and strain but also defined by the history of deformation. Therefore, the viscoelastic stress-strain relation is a function of time and usually it is characterized through several typical viscoelastic responses to the time-dependent load. Most biological tissues demonstrate viscoelastic behavior including creep, relaxation, and hysteresis.

If the load is suddenly applied to the tissue, and is held constant thereafter, the tissue continues to deform and this phenomenon is called strain creep. If a tissue is suddenly deformed, and this deformation is maintained constant thereafter, the resulting stress decreases with time and this viscoelastic response is called stress relaxation. Here, the term "suddenly" means fast enough to induce a viscous response but not fast enough to induce elastic wave propagation in tissue, i.e., inertial terms are neglected for creep and relaxation phenomena. If a tissue is subject to cycles of loading and unloading and the response for each cycle is different, then it is said to display hysteresis. All these phenomena are found in most biological tissues and are used to characterize tissue viscoelasticity.

To describe viscoelastic stress-strain relation, many different models are used, but most of them are based on two one-dimensional rheologic models: the Maxwell model and the Kelvin-Voigt model (see Fig. 2.7A and B). These models can be represented as a combination of simple linear elements such as the linear spring (ideal elastic body obeying Hooke's law) and the linear dashpot (Newtonian viscous fluid obeying Newton's law).

Figure 2.7. Rheologic models of viscoelastic body: (A) Maxwell model, (B) Kelvin-Voigt model, (C) Standard linear model, (D) Creep function of a Maxwell body, (E) Creep function of a Kelvin-Voigt model, (F) Creep function of a standard linear model, (G) Relaxation function of a Maxwell body, (H) Relaxation function of a Kelvin-Voigt body, and (I) Relaxation function of a standard linear model.

Consider, for example, a state of a pure shear deformation shown in Fig. 2.2C, with nonzero stress and strain components σ ₂₃ and ε ₂₃, respectively. As discussed previously, the ideal elastic body reacts instantly to the load, such that the stress is linearly proportional to strain at any moment of time t: σ ₂₃(t)   =   2με ₂₃(t). Stress in the viscous element is proportional to the strain rate: σ ₂₃(t)   =   2ηdε ₂₃(t)/dt, where η is the shear viscosity coefficient.

Combining springs and dashpots, we can obtain different models of viscoelastic behavior. Viscoelastic behavior can be characterized through the nature of the tissue response to suddenly applied stress and strain, i.e., creep and relaxation responses. The unit step function is defined as

(2.46) $h\left(t\right)=\{\begin{array}{c}0,t<0,\\ 1/2,t=0,\\ 1,t>0.\end{array}$

If stress or strain are step functions of time, i.e., σ ₂₃ = σ ₀ h(t) and ε ₂₃ = ε ₀ h(t), for creep and relaxation, respectively, we can define the creep function J(t) and relaxation function G(t) as

(2.47) ${\epsilon }_{23}=J\left(t\right){\sigma }_0,$

${\sigma }_{23}=G\left(t\right){\epsilon }_0.$

The resulting strain and stress are directly related to the creep and relaxation functions.

10.1.1 Maxwell model

The Maxwell model combines elastic and viscous elements arranged in series, as shown in Fig. 2.7A. If stress is applied to the Maxwell model, this stress produces strain in the spring, and the flow in the dashpot, such that total strain rate is

(2.48) $\frac{d{\epsilon }_{23}}{dt}=\frac{1}{2\mu }\frac{d{\sigma }_{23}}{dt}+\frac{1}{2\eta }{\sigma }_{23}.$

Solutions of Eq. (2.48) for creep and relaxation functions have the respective forms

(2.49) $J\left(t\right)=\frac{1}{2}\left(\frac{1}{\mu }+\frac{t}{\eta }\right)h\left(t\right),$

$G\left(t\right)=2\mu e^{-\left(\mu /\eta \right)t}h\left(t\right).$

Creep and relaxation functions for the Maxwell model are shown in Fig. 2.7D and G, respectively. As the creep response is unlimited for t → ∞, the Maxwell body is referred to as the Maxwell fluid. The time constant τ _σ   = η/μ is called a relaxation time and it characterizes the rate of relaxation for the viscous element. A smaller relaxation time corresponds to a faster relaxation process.

10.1.2 Kelvin-Voigt model

For the Kelvin-Voigt model, elastic and viscous elements are connected in a parallel fashion, as shown in Fig. 2.7B. Both elements have the same strain, and the total stress is a sum of the stresses in both elements:

(2.50) ${\sigma }_{23}=2\mu {\epsilon }_{23}+2\eta \frac{d{\epsilon }_{23}}{dt}.$

After the solution of Eq. (2.39), creep and relaxation functions are

(2.51) $J\left(t\right)=\frac{1}{2\mu }\left(1-e^{-\left(\frac{\mu }{\eta }\right)t}\right)h\left(t\right),$

$G\left(t\right)=2\eta \delta \left(t\right)+2\mu h\left(t\right),$

where δ(t) is a Dirac delta function. As shown in Fig. 2.7E, a sudden application of stress produces no immediate deflection because dashpot does not move immediately. The dashpot strain decreases exponentially and the creep function increases with time but with decreasing slope. As time becomes large, all the stress is carried by the spring and we have ε ₂₃=σ ₀/2μ. For the relaxation function (see Fig. 2.7H) a sudden strain requires an infinite stress at the initial moment; however, after that moment, the relaxation function exhibits no stress relaxation. The time τ _ε   = η/μ is referred to as the retardation time and is analogous in meaning to the relaxation time: the time required for the rate of creep to approach nearly zero.

10.1.3 Standard linear body model

The Kelvin-Voigt and the Maxwell models are thus the simplest viscoelastic models. The Maxwell model corresponds to a viscoelastic fluid, whereas the Kelvin-Voigt represents a solid. Therefore the Kelvin-Voigt model is the most widely used viscoelastic model to describe soft tissue behavior under dynamic load. However, both models usually fail to represent real tissue behavior. For example, the Maxwell model demonstrates an unrealistic creep function, while the Kelvin-Voigt model shows no stress relaxation. To improve the representation of viscoelastic behavior, more general models can be considered by combining a number of springs and dashpots, increasing the number of viscoelastic parameters, and adding more exponential terms to the creep and relaxation functions. An example of such generalization is shown in Fig. 2.7C, F, and I, in which the standard linear body model is presented. This model is obtained by adding a spring in parallel to the Maxwell model and enables a realistic representation of both creep and relaxation processes. Two elastic parameters μ ₁ and μ ₂ are needed to represent the instantaneous and large-time elastic response, respectively.

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Crust and Lithosphere Dynamics

S. Cloetingh , P.A. Ziegler , in Treatise on Geophysics, 2007

Appendix 1 Dynamic Model for Slow Lithospheric Extension

To study lithosphere extension, a 2-D finite element model is used. The program is based on a Lagrangian formulation, which makes it possible to track (material) boundaries, like the Moho, in time and space. A drawback of the Lagrangian method is that it is not suitable for solving very large grid deformation problems. This is a problem in analyzing of extension of the lithosphere, which is often accompanied by large deformations. The elements might become too deformed to yield accurate or stable solutions. In order to overcome this problem the finite element grid is periodically remeshed (see Van Wijk and Cloetingh (2002) and references therein).

In the numerical model the base of the lithosphere is defined by the 1300   °C isotherm. Under such conditions, approximately the upper half of the thermal lithosphere behaves elastically on geological time scale, while in the lower half stresses are relaxed by viscous deformation. This viscoelastic behaviour is well described by a Maxwell body, resulting in the following constitutive equation for a Maxwell viscoelastic material:

[1] $\stackrel{.}{\varepsilon }=\frac{1}{2\mu }\sigma +\frac{1}{E}\frac{\text{d}\sigma }{\text{d}t}$

in which ɛ̇ is strain rate, μ is dynamic viscosity, σ is stress, and E is Young's modulus. For a Newtonian fluid the dynamic viscosity μ is constant. In the lithosphere, however, nonlinear creep processes prevail and the relation between stress and strain rate can be described by

[2] $\stackrel{.}{\varepsilon }=A{\sigma }^{n}exp\frac{-Q}{RT}$

where A, n, and Q are experimentally derived material constants, n is the power law exponent, Q is activation energy, R is the gas constant, and T is temperature.

The state of stress is constrained by the force balance:

[3] $\nabla \cdot \sigma +\rho g=0$

where g is gravity and ρ is density.

In this model it is assumed that mass is conserved and the material is incompressible. The continuity equation following from the principle of mass conservation for an incompressible medium is

[4] $\nabla \mathbf{v}=0$

In the model, the density is dependent on the temperature following a linear equation of state:

[5] $\rho ={\rho }_0\left(1-\alpha T\right)$

where ρ₀ is the density at the surface, α is the thermal expansion coefficient, and T is temperature.

Besides viscoelastic behaviour, processes of fracture and plastic flow play an important role in deformation of the lithosphere. This deformation mechanism is active when deviatoric stresses reach a critical level. Here the Mohr–Coulomb criterion is used as a yield criterion to define this critical stress level. The Mohr–Coulomb strength criterion is defined as

[6] $\left|{\tau }_{\mathrm{n}}\right|\le c-{\sigma }_{\mathrm{n}}\cdot \tan \vartheta$

where τ_n is the shear stress component, σ_n is the normal stress component, c is the cohesion of the material, and ϑ is the angle of internal friction. Stresses are adjusted at each time step at which this criterion is reached. Frictional sliding and fault movement are not explicitly described by this criterion.

The displacement field is obtained by solving eqns [1]–[6]. A total of 2560 straight-sided seven-node triangular elements were used with a 13-point Gaussian integration scheme. As the time discretization schemes used are fully implicit, the system is unconditionally stable. However, as the accuracy of the solution remains dependent on the dimension of the time steps, the Courant criterion was implemented.

Processes like sedimentation and erosion are not incorporated in the modeling, although they affect the evolution of a rift basin and rift shoulders, and can change the strength of the lithosphere.

The temperature field in the lithosphere is calculated for each time step using the heat flow equation:

[7] $\rho c_p\frac{\text{d}T}{\text{d}t}={\partial }_{j}k{\partial }_{j}T+H$

where the density ρ is defined by eqn [5], c _p is specific heat, k is conductivity, and H is crustal heat production. Temperatures are calculated on the same grid as the velocity field, and advection of heat is accounted for by the nodal displacements.

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Nonequilibrium physics: From complex fluids to biological systems III. Living systems

Pramod A. Pullarkat Pablo A. Fernández Albrecht Ott , in Physics Reports, 2007

Often viscoelastic media like cells are described as a combination of viscous (dashpot) or elastic (spring) elements. The so-called Maxwell-body consists of a spring in series with a dashpot, when these are in parallel one talks of a Kelvin–Voigt body. As an example, the constitutive equation of the Maxwell body reads $\sigma +\lambda \dot{\sigma }=\eta \dot{\epsilon }$ . Where $\epsilon ={\epsilon }_1+{\epsilon }_2$ is the total strain related to the total stress $\sigma =G{\epsilon }_1=\eta \dot{{\epsilon }_2}$ . $\lambda =\eta /G$ is the relaxation time at constant strain. A Maxwell body in series with a dashpot appears to fit the single cell response to a step stretch reasonably well [52]. Creep responses measured over short periods of time have also been modelled using finite viscous and elastic elements [70]. However, a deeper look at live cell mechanics reveals a very broad range of relaxation times as discussed in Section 3 . This could in principle be represented by a broad distribution of Maxwell bodies with different time constants. However, how the observed (passive) viscoelastic cell response can be related to the composition and structure of the cytoskeleton remains mostly unanswered.

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