# Paraelasticity

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## Abstract

A 3D hysteretic, fully reversible constitutive model with rate-independent damping and with variable secant stiffness is proposed. A reversible dilatancy–contractancy effect is an optional feature as described in the companion paper (Niemunis et al. in Acta Geotech, 2011). The present paper describes the basic aspects of the model only. A strain path reversal is defined, and a treatment of the past reversals using a stack structure is proposed.

### Keywords

Hysteresis Paraelasticity Stack of reversals### List of symbols

- \(\vec{\bf 1}\)
\(-\frac{1}{\sqrt3} {\rm diag}(1,1,1)\)

- \(\vec{{\bf 1}^*}\)
\(-\frac{1}{\sqrt6} {\rm diag}(-2,1,1)\)

*a*1D double strain amplitude

*A*Current state described by \(A=\{\varvec{\epsilon}^A,\varvec{\sigma}^A,{\bf N}^A,d_{AR},e^a,a_{P1}\}\)

*A*_{L},*A*_{U},*A*_{T}Areas of

*s*−*e*loops*A*^{n}State before the current increment

*A*^{n+1}State after the current increment

**c**Center of the loading circle

*d*_{ARi}Diameter of the circle passing through \(\varvec{\epsilon}^A\) and \(\varvec{\epsilon}^{Ri}\)

*d*_{L}Diameter of the largest PE circle (material parameter)

- \(\dot d_{AR1}\)
Rate of the diameter of the loading circle

- \(\bar d_i\)
Euclidean distance \(\| \; \varvec{\epsilon}^A - \varvec{\epsilon}^{Ri}\; \|\)

*d*_{RRi}Diameter of the circle passing through \(\varvec{\epsilon}^{Ri}\) and \(\varvec{\epsilon}^{Ri+1}\)

*D*Damping ratio

*e*1D strain span

- \(e_P, e_{\bar Q}\)
Isomorphic invariants \(\epsilon_P,\epsilon_{\bar Q}\) of the strain span

**e****e**Strain span \(\varvec{\epsilon}^A - \varvec{\epsilon}^{R1}\)

*e*^{Ri},*e*^{A}Void ratio at the

*i*th reversal*R*_{i}, and at the current state*A*, respectively*E*1D elastic stiffness

- E
Isotropic hypoelastic tangential stiffness

*f*Material constant for PE stiffness

*G*Elastic shear modulus

*H*1D PE secant stiffness

- H
Paraelastic secant stiffness

*i*Index of reversals

- \({\mathsf{I}}\)
Fourth order identity tensor

*K*Elastic bulk modulus

*l*Length of the stack,

*l*≥ 2*L*State at the last indelible reversal described by \(L=\{\varvec{\epsilon}^{L},\varvec{\sigma}^{L},{\bf N}^{L},d_{L},e^{L},P_{RRL}\}\)

*m*_{R}Ratio between the largest PE secant stiffness and the secant stiffness for a larger strain amplitude

**N**^{Ri}Outer normal to the reversal circle at \(\varvec{\epsilon}^{Ri}\)

*P*Isomorphic pressure \(P = -\vec{\bf 1} : \varvec{\sigma}\)

*P*_{ARi}Reversible dilatancy/contractancy effect manifested in terms of pressure

*Q*Isomorphic invariant of stress deviator \( Q = \|\varvec{\sigma}^*\|\)

- \(\bar Q\)
\(\bar Q = \vec{{\bf 1}^*} : \varvec{\sigma}\) note that \(|\bar Q| = Q\) for axially symm. stress

*r*Radius of the loading circle

*R*_{i}State \(R_i=\{\varvec{\epsilon}^{Ri},\varvec{\sigma}^{Ri},{\bf N}^{Ri},d_{Ri},e^{Ri},P_{RRi}\}\) at the

*i*th reversal*s*1D stress span, \({\bf s} = \varvec{\sigma} - \varvec{\sigma}^{R1}\)

- \(s_P, s_{\bar Q}\)
Isomorphic invariants of

**s**analogous to \(P,\bar Q\)**s**Stress span \(\varvec{\sigma}^A - \varvec{\sigma}^{R1}\)

**s**′Tangential stiffness \({\partial {\bf s}}/{\partial {\bf e}}\)

*z*Inclination of the drift \(\dot s_Q/ \dot s_P\)

- \(\sqcup', \sqcup`\)
Abbreviations: \(\sqcup'={\partial \sqcup}/{\partial {\bf e}}\; {\text{and}}\; \sqcup` = {\partial \sqcup}/{\partial {e_Q}}\)

- \(\sqcup^A\)
State variable \(\sqcup\) at the current state

*A*- \(\sqcup^{\rm HP}\)
State variable \(\sqcup\) outside the PE region

- \(\sqcup^n\)
State variable \(\sqcup\) before the current strain increment

- \(\sqcup^{n\!+\!1}\)
State variable \(\sqcup\) after the current strain increment

- \(\sqcup^{\rm PE}\)
State variable \(\sqcup\) within the PE region

- \(\sqcup^{Ri}\)
State variable \(\sqcup\) at the

*i*th reversal*R*_{i}- \(\sqcup^*\)
Deviatoric portion of \(\sqcup\)

- χ
material constant for PE stiffness

- \(\Updelta\sqcup\)
Increment of \(\sqcup\)

- \(\epsilon\)
1D strain

- \(\varvec{\epsilon}\)
Strain tensor (compression negative)

- \(\epsilon_P, \epsilon_{\bar Q}\)
Isomorphic invariants of strain \(\varvec{\epsilon}\) analogous to \(P, \bar Q\) of stress \(\varvec{\sigma}\)

- γ
1D shear strain

- γ
^{ampl} 1D shear strain amplitude at the most recent reversal

- θ
Fraction of the strain increment within the largest PE circle before dragging

- ϑ
_{i} *d*_{ARi}/*d*_{RRi}abbreviation- λ, μ
Lamé parameters

- \({\bf e}\, \varvec{\eta}\)
Dyadic product between

**e**and \(\varvec{\eta}\) (written without \(\otimes\) symbol)- \(\varvec{\eta}\)
Outer normal to the loading circle

- σ
1D stress (compression negative)

- \(\varvec{\sigma}\)
Stress (tension positive)

- τ
1D shear stress

- τ
^{R} 1D shear stress at the most recent reversal

### Material parameters

The following is the list of material parameters required by PE. Some intuitive values are suggested.

- –
κ

_{PEiso}=*P*/*K*= 0.001. Isomorphic PE elastic coefficient (Butterfield)- –
ν = ν

_{PE}= 0.2. PE Poisson number- –
*f*= 304.57- –
χ = 0.956. Exponent (<1) controlling the damping ratio

- –
*d*_{L}= 0.0002. Diameter of the largest PE circle

## 1 Introduction

*strain loop*and a

*strain cycle*, Fig. 1. Both denote closed strain paths i.e. the initial strain and the final strain are identical. A cycle is a special kind of loop which is commenced at a reversal, i.e. at a turning point of the strain path.

^{1}In paraelasticity (PE) strain cycles must lead to closed stress paths but strain loops need not.

A non-unique relation between stress and strain is necessary to capture the damping (energy dissipation). The paraelastic model introduces the concept of a strain span **e** which is the sum of strain increments measured from the most recent reversal. This concept was originally proposed in [6, 12]. PE is presented as a stand-alone model here, but it can improve the small-strain stiffness behavior of other models, in our case the hypoplasticity (HP) [19] or viscohypoplasticity [9, 10].

a modified definition of the distance from the reversal point and the update rules

an additional reversible dilatancy–contractancy behavior [11]

## 2 The stack of reversals and its update rules

^{2}In this section, we describe the reversals and their properties:

condition for a generation and deletion of a strain path reversal

loading direction, strain span, distance between reversals

loading circle, reversal circle

stack of reversals and push, pop, and drag operations defined on the stack

root reversal

splitting of the strain span

### 2.1 Stack of reversals

Like most non-linear constitutive models the paraelastic model works with strain increments \(\Updelta\varvec{\epsilon}\). Starting from the most recent reversal *R*_{1} (defined further), the strain increments are added in a so-called *strain span*\({\bf e} =\varvec{\epsilon}- \varvec{\epsilon}^{R1}\). The model describes *stress span*\({\bf s} = \varvec{\sigma}- \varvec{\sigma}^{R1}\) as a function of the strain span. A *reversal* is established when the distance^{3} (denoted as *d*_{AR1}^{4}) between the current state *A* and the most recent reversal *R*_{1} starts to decrease. The oldest reversal, a so-called *root* reversal *L*, must be specially initialized. The youngest reversal \(R_1 = \{ \varvec{\epsilon}^{R1},\varvec{\sigma}^{R1}, {\bf N}^{R1}, e^{R1}, d_{RR1}, P_{RR1}\}\) consists of the following information: strain \( \varvec{\epsilon}^{R1} \), stress \(\varvec{\sigma}^{R1}\), loading direction **N**^{R1}, void ratio *e*^{R1} corresponding to this reversal, the corrected pressure *P*_{RR1} accounting for dilatancy/contractancy effects, see [11], and the distance *d*_{RR1} from *R*_{1} to the one but last reversal \(R_2 = \{\varvec{\epsilon}^{R2}, \varvec{\sigma}^{R2},{\bf N}^{R2}, e^{RR2},d_{RR2}, P_{RR2} \}\). Older reversals \(R_3 = \{\varvec{\epsilon}^{R3}, \varvec{\sigma}^{R3}, {\bf N}^{R3}, e^{RR3}, d_{RR3}, P_{RR3} \}\) etc. have the same structure. The *root* reversal \(L = \{ \varvec{\epsilon}^{L}, \varvec{\sigma}^{L}, {\bf N}^{L}, e^{L}, d_{L}, P_{RRL}=0 \}\) is indelible and needs a special treatment. Its distance *d*_{L} is a material constant defining the size of the paraelastic region. The current state is stored in an analogous collective variable \(A= \{\varvec{\epsilon},\varvec{\sigma}, {\bf N}, e^{A}, d_{AR1}, P_{AR1} \}\). Sometimes it is necessary to distinguish between the current states \(A^n= \{\varvec{\epsilon}^n,\varvec{\sigma}^n, {\bf N}^n,e^{n}, d_{AR1}^n, P^n_{AR1} \}\) and \(A^{n+1}= \{\varvec{\epsilon}^{n+1}, \varvec{\sigma}^{n+1}, {\bf N}^{n+1}, e^{{n+1}}, d_{AR1}^{n+1}, P^{n+1}_{AR1} \}\) before and after the current strain increment, respectively.

The current state and all reversals are kept in a LIFO ^{5} list \(\{ A,R_1,R_2, \ldots{L}\}\) of length *l* termed the *stack of reversals*.

A turn in the strain path may generate a new reversal (a reversal is *pushed* onto the stack) and during a monotonic (over-)loading the most recent reversals are removed from the stack (*popped*).

In a series of cycles with decreasing amplitude (a shakedown), all reversals are meaningful and the stack may become too large for the computer memory. In order to restrict the memory requirement of the model, a *consolidation of stack* may be performed.

### 2.2 Loading and reversal circles

*loading circle*(a hyper-sphere in the 6D strain space) is uniquely defined by the following conditions, Fig. 2:

it passes through \(\varvec{\epsilon}^{R1}\) and \(\varvec{\epsilon}^{n+1} \)

it has a unit outer normal

**N**^{R1}at \(\varvec{\epsilon}^{R1}\)

**c**and the radius

*r*of the

*loading circle*from the equation system

*distance*between

*A*

^{n+1}and

*R*

_{1}is defined as the diameter

*d*

_{AR1}

^{n+1}= 2

*r*of the loading circle, i.e. it can be calculated from

**e**:

**N**

^{R1}< 0 only and \(d_{AR1}^{n+1}= \infty\) must be used

^{6}otherwise. All

*d*

_{ARi}must be smaller than the diameter of the root circle

*d*

_{L}(a material constant). The

*loading direction*

**N**

^{n+1}is defined as a unit normal with respect to the loading circle

*A*

^{n+1}.

*Unloading* occurs if *d*_{AR1}^{n+1} < *d*_{AR1}^{n}. In such case, the loading circle passing through *A*^{n} becomes a *reversal circle* with the properties \(\{ \varvec{\epsilon}^n, \varvec{\sigma}^n, {\bf N}^n, e_{A}^n, d_{AR1}^n, P_{AR1}^n \}\) memorized as the youngest reversal *R*_{1}. Moreover, the indices of the older reversals in the stack are increased by one. We call it the *push* operation.

*R*

_{i+1},

*R*

_{i}is denoted as

*d*

_{RRi}. It can also be calculated from (2) using \({\bf e} = \varvec{\epsilon}^{R{i+1}} -\varvec{\epsilon}^{Ri} \) and

**N**

^{Ri}. Note that

*d*

_{ARi}satisfies the axioms of distance:

*d*_{AR1}≥ 0 (non-negativity)*d*_{AR1}= 0 iff*A*=*R*_{1}(identity of indiscernibles)symmetry

*d*_{AR1}=*d*_{R1A}*d*_{AR2}+*d*_{AR1}≥*d*_{RR1}(triangle inequality)

### 2.3 Update rules for the stack

The calculation of the paraelastic model is performed applying small ^{7} strain increments \(\Updelta \varvec{\epsilon}\). Given \( \Updelta \varvec{\epsilon}\) we may find the new state *A*^{n+1} using \(\varvec{\epsilon}^{n+1} = \varvec{\epsilon}^n + \Updelta \varvec{\epsilon},\, e^{n+1} = e^n + {\text {tr}}\;(\Updelta\varvec{\epsilon})/(1 + e^n)\), Equations (2) and (3) for *d*_{AR1}^{n+1} and **N**^{n+1} respectively. The stress \(\varvec{\sigma}^{n+1}\) and dilatancy/contractancy pressure *P*_{AR1}^{n+1} are calculated as presented in the next subsections.

*loading*conditions

*unloading condition*

*R*

_{1}and

*A*

^{n}) is memorized as the most recent

*reversal circle*and a new loading circle develops from

*A*

^{n}. Moreover, the indices

*R*

_{i}of the remaining reversals are incremented by one. We say that

*A*

^{n}is

*pushed*onto the stack, Fig. 3. The new (usually very small) loading circle is passing through

*A*

^{n}and

*A*

^{n+1}and has a common outer normal direction with the new reversal circle at \(\varvec{\epsilon}^{R1}\).

*overloading condition*

*R*

_{1}and

*R*

_{2}with diameter

*d*

_{RR}=

*d*

_{RR1}.

The reversal point *R*_{1} is swept out of the material memory and the indices of the remaining reversals are decreased by one, Fig. 3. This update of stack is termed *pop*. If the distance from *A*^{n+1} to the new *R*_{1} is larger than the new *d*_{RR1}, the new *R*_{1} must be popped too. The reversals are popped until the loading condition *d*_{AR1}^{n+1} < *d*_{RR1} is reached or until the root reversal *L* is overloaded (it cannot be popped). Note that unloading cannot occur during multiple pop (when several reversals are *popped* by a single increment \(\Updelta\varvec{\epsilon}\)). It can be shown that *d*_{AR1} remains a continuous function of strain upon a single or multiple pop. In other words, an infinitesimally small increment \(\Updelta\varvec{\epsilon}\) may pop a reversal but *d*_{AR1} with respect to new *R*_{1} remains almost identical as *d*_{AR1} with respect to the old one.

*d*

_{L}of the outermost circle corresponds to the root reversal

*L*which cannot be popped. Therefore, if the current distance

*d*

_{AR1}exceeds

*d*

_{L}, a special

*dragging*procedure is activated. Given a strain increment \(\theta \Updelta\varvec{\epsilon}\) that surpassed the root circle

*d*

_{AR1}

^{n+1}>

*d*

_{L}we distinguish a part \(\theta \Updelta\varvec{\epsilon}\) of this increment that lies within the root circle and another part \((1-\theta) \Updelta\varvec{\epsilon}\) that protrudes outside of it (Fig. 4). The decomposition of the strain increment is termed

*split*.

*A*

^{n}lies inside the root circle and

*A*

^{n+1}lies outside of it, the increment \(\Updelta\varvec{\epsilon}\) must be split into two portions: the inner paraelastic portion \(\Updelta\varvec{\epsilon}^{\rm PE}\) and the protruding dragging portion \(\Updelta\varvec{\epsilon}^{\rm HP}\).

*L*is proposed to move parallel to

*A*with identical increments of stress \(\Updelta\varvec{\sigma} = \Updelta \varvec{\sigma}^L\) and strain \(\Updelta\varvec{\epsilon} = \Updelta\varvec{\epsilon}^L\). The loading direction

**N**remains unchanged while dragging.

*push*operation would add a new state

*A*

^{n+1}on the top of the stack keeping

*A*

^{n}as a reversal

*R*

_{1}with

*d*

_{AR1}, and preserving

*L*with the same size

*d*

_{L}. However, only one reversal point with

*d*

_{L}is allowed for. Therefore,

*L*is removed from the stack and

*R*

_{1}is treated as the new

*L*. This special case is handled by the push algorithm. A Mathematica script that calculates the push, pop, and split procedures for the isomorphic P–Q space is given in Appendix (push, pop and split procedures in P–Q space).

### 2.4 Initialization of the stack

The initialization of the stack should represent the loading history of the soil. The paraelastic stack must consist at least of two components {*A*, *R*_{1} = *L*} which define the root reversal circle and the current loading circle.

Soils subjected to small cyclic perturbations should be initialized with a series of cycles with smaller amplitudes (shakedown) about the *K*_{0} direction. This shakedown condition is illustrated in Fig. 6, right.

Examples of the aforementioned initializations of the stack are written in the form of user’s routines^{8} for Abaqus.^{9}

The initial stress \(\varvec{\sigma}^A\), initial void ratio *e*, and the initial strain \(\varvec{\epsilon} = {\bf 0}\) are usually prescribed. The reversals for the shake down process up to the root reversal are unknown and would have to be calculated backwards. Instead, one may perform a forwards calculation starting from *L* = *A* and then correct the starting position *L* until the desired state *A* is reached.^{10}

## 3 Stress–strain relation and PE stiffness

*d*

_{AR1}

^{n+1}>

*d*

_{AR1}

^{n}, the stress–strain relationship is proposed in the following form

*d*

_{AR1}is given in (2) and \({\mathsf{E}}\) is the isotropic elastic stiffness tensor

^{11}λ and μ. For applications restricted to small

*d*

_{AR1}, the relation (11) is unique and can be inverted. The tangential stiffness (Jacobian) obtained from (11) is

*f*> 0 and χ > 0 must guarantee that both the secant stiffness \( (1 - f d_{AR1}^\chi){\mathsf{E}} \) in (11) and the tangential stiffness (13) are positive. Tensor \(\varvec{\eta}\) is outer normal to the loading circle at

*A*as discussed further, see Fig. 17. For a rough estimation of the restrictions on

*f*and χ, we may use the inequality (52) or (58) derived in Appendix (validity range).

### 3.1 Negative dissipation

The elastic part \({\mathsf{E}}\) is barotropic (pressure dependant) and it has not been derived from any elastic potential. Therefore, one could expect that the model may violate the Second Law of thermodynamics. However, \({\mathsf{E}}\) is based on the pressure in the middle of the root circle and hence it remains constant unless dragging occurs. In other words, in pure PE range the pressure dependence of \({\mathsf{E}}\) cannot lead to a perpetuum mobile of the second kind.

Independently, a non-unique \(\varvec{\sigma}(\varvec{\epsilon})\) relation could also lead to the violation of the Second Law. Numerically, we attempted to construct a cycle (closed strain and stress) corresponding to minimum dissipation (possibly negative). In spite of varying the form of the cycles, the starting position, the sense of circulation, and the number of reversals, we could never obtain negative dissipation of energy. None of the tested cycles could produce mechanical work from heat. A formal mathematical proof of the compliance of PE with the Second Law seems cumbersome.

### 3.2 Smooth transition to a large strain model

We have used a hypoplastic model [19] to examine the problems that appear in the performance of PE in combination with other constitutive models. The hypoplastic (HP) tangential stiffness applied during dragging and the paraelastic tangential stiffness at *d*_{AR1} = *d*_{L} should be similar. By this similarity, we avoid strong kinks in stress–strain curves upon overloading of the *d*_{L} circle. Due to directional dependence of both stiffnesses, a requirement of smooth transition for all directions of loading would be too restrictive. Let us assume, for simplicity \(\nu\) = 0 (λ = 0). Let the HP stiffness be \({\mathsf{E}}^{\rm HP} = {2 \mu} / {m_R} {\mathsf{I}} \), that is *m*_{R}—times smaller than the largest PE stiffness. Moreover, we assume that \({\mathsf{E}}^{\rm HP}\) remains constant for all strain spans **e** in the root circle and for all directions of *loading*\(\dot{\bf e}\). Depending on the span **e** and on the loading direction \(\dot{\bf e}\), the paraelastic stiffness given by (13) varies between \(2 \mu (1 - (1+\chi) f d_{L}^\chi){\mathsf{I}}\) for \(\dot{\bf e} \| {\bf e}\) and \(2 \mu (1 - f d_{L}^\chi){\mathsf{I}}\) for all perpendicular directions. We have assumed ν = 0, hence the stiffness upon *unloading*\({\mathsf{E}} = 2 \mu {\mathsf{I}}\) holds.

*f*should be chosen in accordance with

*m*

_{R}in such way that the HP stiffness lies between the extreme values of the PE stiffness, namely

### 3.3 Properties of the 1D hysteretic cycle

#### 3.3.1 Peak stress

*s*= σ − σ

^{R1}measured from the latest reversal

*R*

_{1}. The constitutive relation (11) takes the form

*E*,

*f*, and χ < 1 are all positive. In the case

*e*> 0 and \( \dot{e} > 0,\) we have |

*e*| =

*e*and differentiating

*s*=

*E*

*e*−

*f*

*E*

*e*

^{χ+1}with respect to

*e*, we obtain the tangential stiffness

*e*< 0 and \( \dot{e} < 0\) we substitute

*e*= −η with |

*e*| = η into (15) and using the chain rule d

*s*/d

*e*= −d

*s*/dη, we arrive at (16) too. The tangential stiffness (16) should be always positive. The condition d

*s*/d

*e*> 0 implies a limitation

#### 3.3.2 Damping and stiffness dependence on amplitude

*D*is defined

^{12}using the area

*A*=

*A*

_{L}+

*A*

_{U}−

*A*

_{T}, Fig. 7, enclosed within the hysteretic cycle and normalized with stress and strain amplitudes

Using the notation introduced in Fig. 7, we derive an expression for the work upon a strain cycle from (15).

*A*

_{L}for d

*e*> 0 in the range from

*e*= 0 to \( e= a = 2 \epsilon^{\rm ampl} \) is

*e*< 0 is identical

*A*

_{U}=

*A*

_{L}and using

*f*can be determined from experimental results. Firstly, the ratio

*m*

_{R}between secant PE stiffness

*H*for two different strain amplitudes must be defined. One of the selected strain amplitudes should coincide with the largest PE stiffness (plateau on the modulus degradation curve), Fig. 9, top.

*m*

_{R}is proposed to be determined from (22) in the range of amplitudes between

*e*= 0.01

*d*

_{L}and 10

*d*

_{L}with

*d*

_{L}= 10

^{−4}, namely

*m*

_{R}≈ 5.0 according to [7].

For a given *m*_{R}, *d*_{L}, and *D*(*d*_{L}), e.g., from Fig. 9, we may numerically solve (23) and (21) for *f* and χ.

Figure 8 shows the variation of the PE secant stiffness *H* and of the damping ratio *D* for different values of *m*_{R}. The curves are calibrated for \(d_L=0.02 {\%} \) and *D*(*d*_{L}) = 0.02 taken from Fig. 9, bottom. The stiffness ratio *m*_{R} is calculated between strain amplitudes 2 × 10^{−6} and 2 × 10^{−3}.

These values coincide with experimental curves [4, 7, 15, 18] for a clean sand, Fig. 9. A residual value of *D* ≈ 0.005 for vanishingly small strain amplitudes is reported in [18]. We decided to disregard this effect here. This small residual value of damping may be caused by an experimental device. We simply shift the experimental damping curve by 0.005 downwards before calibration.

#### 3.3.3 Masing rule

A sequence of cycles with gradually decreasing amplitude until it becomes vanishingly small is termed a shakedown. Let us examine the behavior upon a monotonic loading applied after a 1D shakedown.

^{R}and \(\epsilon^R\) denote stress and strain at the most recent reversal point, and

*F*is the function describing the 1D relation between σ and \(\epsilon\) for the first loading. One can recognize the stress and strain spans appearing in (24).

Various propositions for *F*() can be found in the literature, e.g. [2, 4, 13, 14], but their 3D generalizations are not presented.

## 4 Continuity of stress

*d*

_{AR1}=

*d*

_{R1A}guarantees

*independently of the strain path*. Note that

**s**is a function of the

*current*distance

*d*

_{AR1}and the

*current*strain span

**e**only and therefore our result holds despite possible intermediate (=overloaded) loops, even if they were not closed. The reversibility of stress upon the outer closed cycle is preserved.

The stress \(\varvec{\sigma}(\varvec{\epsilon})\) should be a continuous tensorial function (without jumps) of strain. The continuity of stress must hold in general case. Therefore, we must examine this continuity of stress for *push* and *pop*.

### 4.1 Continuity of stress after *push*

*neutral direction*. Let \(\Updelta \varvec{\epsilon}_a \) cause unloading

*d*

_{AR1}

^{n+1}<

*d*

_{AR1}

^{n}so that \(\varvec{\epsilon}^n\) is pushed on the stack whereas \(\Updelta\varvec{\epsilon}_b\) corresponds to loading with

*d*

_{AR1}

^{n+1}>

*d*

_{AR1}

^{n}(in Fig. 11).

*d*

_{AR1 a}

^{n+1}≈

*d*

_{AR1 b}

^{n+1}≈

*d*

_{AR1}

^{n}. For unloading with \(\Updelta \varvec{\epsilon}_a,\) we have

*R*

_{2}(instead of

*R*

_{1}) to ease the comparison with (26). The stress response to both increments is indeed almost identical, because (26) and (28) are calculated using the same stiffness \({\mathsf{H}}\) (based on the common diameter

*d*

_{AR1}

^{n+1}≈

*d*

_{AR1}

^{n}). Using PE with the Euclidean definition of distance, we would obtain much larger \({\mathsf{H}}\) in the last summand in (27) compared with \({\mathsf{H}}\) in (28) so the difference \(\varvec{\sigma}_a^{n+1} - \varvec{\sigma}_b^{n+1}\) could be substantial.

^{13}causes a rapid change in stiffness which is represented by the kinks in the response envelope for neutral strain directions (i.e. for \({\bf N} : \Updelta \varvec{\epsilon} \approx 0\), see the profiles in Fig. 12). Despite the kinks, the continuity of stress response across the neutral direction is satisfied which is of advantage compared with the definition proposed in [5, 6]. Similar concave kinks are often generated by elastoplasticity.

### 4.2 Continuity of stress after *pop*

*R*

_{1}, Fig. 13. This implies almost equal distances

*d*

_{AR1}

^{n}≈

*d*

_{RR1}≈

*d*

_{AR1}

^{n+1}satisfying

*d*

_{AR1}

^{n}<

*d*

_{RR1}<

*d*

_{AR1}

^{n+1}. The stress \(\varvec{\sigma}^n\) before pop can be calculated from

*R*

_{2}as

*d*

_{AR1}

^{n}≈

*d*

_{RR1}(Fig. 13, left). After the increment, we pop

*R*

_{1}but preserve the numbers of reversals, that is, we do not rename

*R*

_{2}to

*R*

_{1}(Fig. 13, right).

*R*

_{2}as

*d*

_{AR1}

^{n+1}≈

*d*

_{RR1}. Comparing (29) and (30) for infinitesimally small increment that is for

*d*

_{AR1}

^{n}≈

*d*

_{AR1}

^{n+1}and \(\varvec{\epsilon}^n \approx\varvec{\epsilon}^{n+1}\) we conclude \(\varvec{\sigma}^n \approx \varvec{\sigma}^{n+1}\) which means that the continuity requirement is satisfied.

## 5 Drift of stress paths

*Q*-axis. It turns out, however, that the stress path corresponding to the monotonic straining along \(\epsilon_Q\) has a zigzag form quantified with \(\dot {s}_Q/\dot {s}_P = \pm z\). We have not found a remedy against it, and to some extend (say, for

*z*> 4), this zigzag must be accepted. Before a detailed discussion of the zigzag effect, we should mention that a related phenomenon has been observed in laboratory [1, 16, 17], see Fig. 15. Straining along \(\epsilon_Q\) was preceded either by volumetric compression or by volumetric extension. Evidently, the stress path is slightly drifted off the vertical direction depending on the strain history.

*d*

_{AR1}—dependent secant stiffness \({\mathsf{H}} = (1 - f d_{AR1}^\chi){\mathsf{E}} \) proposed in (11). The effect can be understood examining the

*tangential*PE stiffness \({\mathsf{K}}\)

^{14}in Fig. 17. Note that \(\lim_{A \to R } \| \varvec{\eta} \|= \infty\). The stress rate \(\dot{\bf s} = {\mathsf{K}} : \dot{\bf e}\) consists of the basic portion \({\mathsf{H}}:\dot{\bf e}\) and of the drift \( - f \chi d_{AR1}^{\chi-1} {\mathsf{E}} : {\bf e}\; \varvec{\eta}: \dot{\bf e} \) which is responsible for the characteristic drift of the stress path after 90° turn and for the zigzag. The stress rate due to drift is proportional to \({\mathsf{E}}:{\bf e}\) whereas the direction of basic stress rate is proportional to \({\mathsf{E}} : \dot{\bf e}\) like in conventional elasticity. The directions \({\mathsf{E}}:{\bf e}\) and \({\mathsf{E}}:\dot{\bf e}\) can be very different, which explains the deflection of the PE stress path compared the elastic one. The drift is particularly strong for large \( \| \varvec{\eta}\|\) with \(\dot{{\bf e}} \| \varvec{\eta}\) and \({\bf e} \perp \dot{\bf e}\).

*rates*obtained for deviatoric strain rate from the two cases depicted in Fig. 18. Their difference is responsible for the zigzag in Fig. 14. In both cases, we have almost the same \(d_{AR1} \hbox{ and } \varvec{\eta}\). According to (31) the difference between stress rates in these two cases results from different drifts

*e*

_{Q}at

*e*

_{P}=

*const*can be quantified using the PE equation in the P–Q space with

*e*

_{Q}, the sign of inclination \(\dot{s}_P/\dot{s}_Q\) depends on the sign of

*e*

_{P}so switching

*e*

_{P}due to consecutive pops causes the zigzag shown in Fig. 14. For the special case ν = 0, we have 3

*K*= 2

*G*and for

*e*

_{P}≈

*e*

_{Q}≈

*d*

_{AR1}/2 with

*d*′

_{AR1}≈ 2, we obtain an estimation

*z*> 4, (see Fig. 19). This condition defines a rough restriction for the model

*f*≈ 300.0. Other restricting conditions to define the range of applicability of the proposed paraelastic model are shown in Appendix.

## 6 PE with Euclidean distance

*d*

_{AR1}given in (2). They proposed constitutive equations that described the compliance rather than stiffness and did not include dilatancy, cf. [11], but the general approach with spans of strain and stress was similar. However, the Euclidean distance \(\bar{d}_{AR1}\) caused a serious problem of discontinuity in functional \(\varvec{\sigma}(\varvec{\epsilon})\). Two examples are illustrated in Fig. 20.

Let us first consider two strain increments applied along the loading circle \(\bar{d}_{AR1} = {\text{const}}\) (Fig. 20, left). Two strain increments \(\Updelta \varvec{\epsilon}_a\) and \(\Updelta \varvec{\epsilon}_b\) have been chosen almost perpendicularly to **e** so that they lie infinitesimally close to loading circle (depicted as the arc in Fig. 20). The increment \(\Updelta \varvec{\epsilon}_b\) corresponds to loading, and the increment \(\Updelta \varvec{\epsilon}_a\) to unloading with a small \(\bar{d}_{AR1}\) calculated from the new reversal point. Hence, the stiffness in the latter case is much larger and in consequence \(\Updelta \varvec{\sigma}_a\) is greater than \(\Updelta\varvec{\sigma}_b\), which can be seen as discontinuity in functional \(\varvec{\sigma}(\varvec{\epsilon})\). The problem disappears if we use *d*_{AR1} instead of \(\bar{d}_{AR1}\).

Another discontinuity in \(\varvec{\sigma}(\varvec{\epsilon})\) may appear along a monotonous strain path (vertical line) interrupted by a small unloading, Fig. 20, right. If this unloading occurred vertically downwards and were followed by reloading and overloading also along the vertical line the whole event would be wiped out from the material memory. However, we may generate a reversal in the form of a small spiral section constructed in such way that the Euclidean distance \(\bar{d}_{AR1} = \| {\bf e} \| \) upon the spiral is monotonically growing. Along such path, we may return to the original direction of deformation and then continue along the vertical line. An infinitesimally small spiral disturbance may therefore lead to the update of the reversal point and to the substantial change in the stiffness, which can be seen as discontinuity in functional \(\varvec{\sigma}(\varvec{\epsilon})\). Again, the problem disappears if we use *d*_{AR1} instead of \(\bar{d}_{AR1}\).

## 7 Outlook

The presented model has been extended incorporating the description of reversible dilatancy-contractancy effects. This extension as well as the finite element implementation of the extended paraelasticity are presented in a companion paper [11].

## Footnotes

- 1.
More precise definition is given in Sect. 2.

- 2.
The concept of overloading is explained in Sect. 2.3.

- 3.
Definition is given further in this section.

- 4.
At this moment the notation with long indices may seem clumsy, however, it will become natural in more complex cases.

- 5.
Last in, first out.

- 6.
*d*_{AR1}^{n+1}= 10*d*_{L}is sufficient for practical purposes. - 7.
Large increments may lead to inconsistencies in the constitutive description. One can show that within a single large increment a new unloading surface may appear and then be erased. The constitutive consequences of this are visible only if this increment is calculated with several small sub-increments.

- 8.
These Fortran95 routines can be obtained directly from the authors.

- 9.
Abaqus is a reg. trademark of Simulia Inc.

- 10.
In this case, the subroutines SDVINI and SIGINI may use the procedures of the main UMAT routine in Abaqus.

- 11.
The elastic moduli depend on the stress and on the void ratio corresponding to the center of the root circle. These values may change only due to dragging and hence the moduli may be considered as material constants within the paraelastic model.

- 12.
The definition is based on a 1D oscillator with Newtonian damping. The generalization to our frictional damping is based on the energy dissipation per cycle. The generalization for 6d strain space is difficult.

- 13.
As the diameter of the circle connecting the reversal

*R*_{1}and the current state*A*. - 14.Mathematica script to plot the region \(1/\|\varvec{\eta}\|\).

## Notes

### Acknowledgments

The financial support from the German Research Community (DFG-GeoTech 1136) is gratefully acknowledged. The second author also wishes to acknowledge the support from Departamento Administrativo de Ciencia, Tecnología e Innovación COLCIENCIAS, DAAD and University of Los Andes/CEIBA.

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