Acta Geotechnica

, Volume 6, Issue 2, pp 67–80 | Cite as

Paraelasticity

  • A. Niemunis
  • L. F. Prada-Sarmiento
  • C. E. Grandas-Tavera
Research Paper

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

ALAUAT

Areas of s − e loops

An

State before the current increment

An+1

State after the current increment

c

Center of the loading circle

dARi

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

dL

Diameter of the largest PE circle (material parameter)

\(\dot d_{AR1}\)

Rate of the diameter of the loading circle

\(\bar d_i\)

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

dRRi

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 \(\varvec{\epsilon}^A - \varvec{\epsilon}^{R1}\)

eRi, eA

Void ratio at the ith reversal Ri, 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 \(L=\{\varvec{\epsilon}^{L},\varvec{\sigma}^{L},{\bf N}^{L},d_{L},e^{L},P_{RRL}\}\)

mR

Ratio between the largest PE secant stiffness and the secant stiffness for a larger strain amplitude

NRi

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

P

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

PARi

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

Ri

State \(R_i=\{\varvec{\epsilon}^{Ri},\varvec{\sigma}^{Ri},{\bf N}^{Ri},d_{Ri},e^{Ri},P_{RRi}\}\) at the ith 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 ith reversal Ri

\(\sqcup^*\)

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

γampl

1D shear strain amplitude at the most recent reversal

θ

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

ϑi

dARi/dRRi 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

τR

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.

κPEiso = P/K = 0.001. Isomorphic PE elastic coefficient (Butterfield)

ν = νPE = 0.2. PE Poisson number

f = 304.57

χ = 0.956. Exponent (<1) controlling the damping ratio

dL = 0.0002. Diameter of the largest PE circle

1 Introduction

Purely hysteretic constitutive behavior excludes cumulative effects but allows for a non-unique stress–strain dependence. We distinguish between a 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.
Fig. 1

Closed strain cycle implies closed stress cycle. Closed strain loop does 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].

Compared with [6] the present version, paraelasticity contains two novel elements, namely:
  • a modified definition of the distance from the reversal point and the update rules

  • an additional reversible dilatancy–contractancy behavior [11]

In this first article, the basic concepts of strain path reversals and their update rules are defined and presented in Sect. 2. The performance of paraelasticity in the description of shear degradation and in the simulation of damping ratio curves is discussed in Sect. 3.3. Numerical examples for the 1D model and calibration of material constants are given in Sects. 3.3.2, 3.3.3.

2 The stack of reversals and its update rules

The paraelastic description of the hysteretic strain–stress behavior is based on the strain reversals. They are points in the strain space which may be generated by a sudden turn in the strain increments and which may be deleted by overloading.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

The sequence of reversals describes the history of deformation. The PE stress–strain relation is path dependent because it depends on the stack.

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 R1 (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 distance3 (denoted as dAR14) between the current state A and the most recent reversal R1 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 NR1, void ratio eR1 corresponding to this reversal, the corrected pressure PRR1 accounting for dilatancy/contractancy effects, see [11], and the distance dRR1 from R1 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 dL 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

The 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 NR1 at \(\varvec{\epsilon}^{R1}\)

Given a reversal \(R_1 = \{ \varvec{\epsilon}^{R1}, \varvec{\sigma}^{R1}, {\bf N}^{R1}, e^{R1}, d_{RR1}, P_{RR1} \}\) and the updated current strain \(\varvec{\epsilon}^{n+1} = \varvec{\epsilon}^n + \Updelta\varvec{\epsilon}\) we calculate the center c and the radius r of the loading circle from the equation system
$$ \begin{aligned} {\bf c} &= \varvec{\epsilon}^{R1} - r{\bf N}^{R1} \\ r &= \|\varvec{\epsilon}^{n+1} - {\bf c} \| \end{aligned} $$
(1)
The distance between An+1 and R1 is defined as the diameter dAR1n+1 = 2r of the loading circle, i.e. it can be calculated from
$$ d_{AR1}^{n+1} = - \frac{ {\bf e}:{\bf e} } { {\bf e} : {\bf N}^{R1}} \hbox{ with } {\bf e} = \varvec{\epsilon}^{n+1} - \varvec{\epsilon}^{R1} $$
(2)
rather than from the Euclidean norm \(\| \varvec{\epsilon}^{n+1}- \varvec{\epsilon}^{R1} \|\). Equation (2) is valid for e : NR1 < 0 only and \(d_{AR1}^{n+1}= \infty\) must be used6 otherwise. All dARi must be smaller than the diameter of the root circle dL (a material constant). The loading directionNn+1 is defined as a unit normal with respect to the loading circle
$$ {\bf N}^{n+1} = \left[\varvec{\epsilon}^{n+1} - {\bf c} \right]^\rightarrow = \left[\varvec{\epsilon}^{n+1} - ( \varvec{\epsilon}^{R1} - r {\bf N}^{R1} ) \right]^\rightarrow $$
(3)
at An+1.
Fig. 2

Loading circle (small) and reversal circle (large) in the strain space generated by the strain path with reversals

Unloading occurs if dAR1n+1 < dAR1n. In such case, the loading circle passing through An 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 R1. Moreover, the indices of the older reversals in the stack are increased by one. We call it the push operation.

The distance between any two consecutive reversals Ri+1, Ri is denoted as dRRi. It can also be calculated from (2) using \({\bf e} = \varvec{\epsilon}^{R{i+1}} -\varvec{\epsilon}^{Ri} \) and NRi. Note that dARi satisfies the axioms of distance:
  • dAR1 ≥ 0 (non-negativity)

  • dAR1 = 0 iff A = R1 (identity of indiscernibles)

  • symmetry dAR1 = dR1A

  • dAR2 + dAR1 ≥ dRR1 (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 An+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 dAR1n+1 and Nn+1 respectively. The stress \(\varvec{\sigma}^{n+1}\) and dilatancy/contractancy pressure PAR1n+1 are calculated as presented in the next subsections.

Under loading conditions
$$ d_{AR1}^{n+1} > d_{AR1}^n\quad \hbox{ and } \quad d_{AR1}^{n+1} < d_{RR1} $$
(4)
the diameter of the loading circle is monotonically increasing. No reversals are either generated or deleted.
Under unloading condition
$$ d_{AR1}^{n+1} < d_{AR1}^n $$
(5)
the current loading circle (passing through R1 and An) is memorized as the most recent reversal circle and a new loading circle develops from An. Moreover, the indices Ri of the remaining reversals are incremented by one. We say that An is pushed onto the stack, Fig. 3. The new (usually very small) loading circle is passing through An and An+1 and has a common outer normal direction with the new reversal circle at \(\varvec{\epsilon}^{R1}\).
Fig. 3

The strain path creating new reversal circles upon unloading (3UL) or swallowing them upon loading (3L)

Under overloading condition
$$ d_{AR1}^{n+1} > d_{RR1} $$
(6)
the current loading circle outgrows the most recent reversal circle defined between R1 and R2 with diameter dRR = dRR1.

The reversal point R1 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 An+1 to the new R1 is larger than the new dRR1, the new R1 must be popped too. The reversals are popped until the loading condition dAR1n+1 < dRR1 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 dAR1 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 dAR1 with respect to new R1 remains almost identical as dAR1 with respect to the old one.

The diameter dL of the outermost circle corresponds to the root reversal L which cannot be popped. Therefore, if the current distance dAR1 exceeds dL, a special dragging procedure is activated. Given a strain increment \(\theta \Updelta\varvec{\epsilon}\) that surpassed the root circle dAR1n+1 > dL 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.
Fig. 4

After an increment, the new strain may lie outside the great circle. In such a case, the increment must be split into two portions: the first one (\(\theta\Updelta{\varvec{\epsilon}}\)) corresponds to the portion between \(\varvec{\epsilon}^n\) and the point on the great circle before dragging. The second portion is the protruding part outside the great circle (\((1-\theta)\Updelta\varvec{\epsilon}\)). Dragging takes place along the protruding portion

The conditions for push, pop, and drag operators can be summarized as follows
$$ \left\{\begin{array}{lll}& d_{AR1}^{n+1} < d_{AR1}^n : & \hbox{renumber } R^{n+1}_{i+1} := R^n_i, \\ & & \hbox{ and add } R^{n+1}_1 := A^n \\ & l^n > 2\quad \hbox{ and } \quad d_{AR1}^{n+1} > d^n_{RR1} : & \hbox{erase } R^n_1, \\ & & l^{n+1}=l^n-1, \\ & & \hbox{renumber } R^{n+1}_i := R^n_{i+1} \\ & L = R_1 \!\quad \hbox{ and } \!\quad d_{AR1}^{n+1} > d_L : & \hbox{dragging } \end{array} \right. $$
(7)
The index \(i=1,\ldots,L\) denotes the sequence of reversals stored in the stack. Conditions for push and pop operations are shown schematically in Fig. 5.
Fig. 5

A closed strain cycle with a sub-cycle. Each unloading generates a reversal. Each overloading removes a reversal. Reversals are renumbered as they are created or deleted

If An lies inside the root circle and An+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 \(\Updelta\varvec{\epsilon}^{\rm HP}\).
$$ \Updelta\varvec{\epsilon} = \Updelta\varvec{\epsilon}^{\rm PE} + \Updelta\varvec{\epsilon}^{\rm HP} =\theta \Updelta\varvec{\epsilon} + (1- \theta) \Updelta\varvec{\epsilon} $$
(8)
We may easily determine θ as shown in Appendix (determination of the split parameter θ).
Under dragging condition,
$$ d^{n+1}_{AR1} > d_L $$
(9)
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.
An additional constitutive rule is required for the first unloading after dragging, i.e., for
$$ d^n_{AR1} = d_L \hbox{ and } d^{n+1}_{AR1} < d_L $$
(10)
A usual push operation would add a new state An+1 on the top of the stack keeping An as a reversal R1 with dAR1, and preserving L with the same size dL. However, only one reversal point with dL is allowed for. Therefore, L is removed from the stack and R1 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 {AR1 = L} which define the root reversal circle and the current loading circle.

If the soil is freshly pluviated, as it would be the case of a sample prepared in the laboratory, the normal vectors defining the paraelastic circles ought to be aligned along the vertical direction, as shown in Fig. 6, left.
Fig. 6

Initialization the stack with reversals. Left a freshly pluviated soil can be initialized with a monotonic loading along the vertical (x1) direction. Right shakedown condition defining a series of nested circles generated by vertical (propagating along x1) shear waves with decreasing amplitude

Soils subjected to small cyclic perturbations should be initialized with a series of cycles with smaller amplitudes (shakedown) about the K0 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 routines8 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

In this section, we interrelate the strain span \({\bf e} = \varvec{\epsilon} - \varvec{\epsilon}^{R1}\) and the stress span \({\bf s} = \varvec{\sigma} - \varvec{\sigma}^{R1}\). For any monotonously growing strain path with dAR1n+1 > dAR1n, the stress–strain relationship is proposed in the following form
$$ {\bf s} = {\mathsf{H}} : {\bf e} = (1 - f d_{AR1}^\chi){\mathsf{E}}: {\bf e}, $$
(11)
wherein dAR1 is given in (2) and \({\mathsf{E}}\) is the isotropic elastic stiffness tensor
$$ \mathsf{E} = \lambda {\bf 1 } \,{\bf 1 } + 2 \mu{\mathsf{I}} $$
(12)
with the Lamé constants11 λ and μ. For applications restricted to small dAR1, the relation (11) is unique and can be inverted. The tangential stiffness (Jacobian) obtained from (11) is
$$ {\bf s}^{\prime}=\frac{\partial {{\bf s}}} {\partial {{\bf e}}}= {\mathsf{H}} - f \chi d_{AR1}^{\chi-1} {\mathsf{E}} : {\bf e}\; \varvec{\eta} \quad\hbox{ with } \varvec{\eta} = \frac{\partial {d_{AR1}}} {\partial {{\bf e}}} $$
(13)
The material constants 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 dAR1 = dL should be similar. By this similarity, we avoid strong kinks in stress–strain curves upon overloading of the dL 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 mR—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.

The PE parameter f should be chosen in accordance with mR in such way that the HP stiffness lies between the extreme values of the PE stiffness, namely
$$ (1+\chi) f \ge \frac{m_R - 1 } {m_R d_{L}^\chi } \ge f $$
(14)

3.3 Properties of the 1D hysteretic cycle

3.3.1 Peak stress

In the 1D version, our paraelastic model simplifies to a scalar \(\sigma-\epsilon\) constitutive relation with the definition of distance \( d_{AR1} = | \epsilon - \epsilon^{R1} | = |e|\) and with the scalar spans \(e = \epsilon - \epsilon^{R1}\) and s = σ − σR1 measured from the latest reversal R1. The constitutive relation (11) takes the form
$$ s = (1 - f |e|^\chi) E e \quad\hbox{ with }\quad d_{AR} = |e| $$
(15)
The material constants Ef, and χ < 1 are all positive. In the case e > 0 and \( \dot{e} > 0,\) we have |e| = e and differentiating s = Ee − fEeχ+1 with respect to e, we obtain the tangential stiffness
$$ \frac{ \hbox{d}\sigma} { \hbox{d}\epsilon}= \frac{ \hbox{d}s} { \hbox{d}e}= E - \left( \chi +1\right) f E |e|^{\chi} $$
(16)
For e < 0 and \( \dot{e} < 0\) we substitute e =  −η with |e| = η into (15) and using the chain rule ds/de =  −ds/dη, we arrive at (16) too. The tangential stiffness (16) should be always positive. The condition ds/de > 0 implies a limitation
$$ d_L < \left[ (\chi+1)f \right]^{-1/\chi} $$
(17)
which is less restrictive than (14).

3.3.2 Damping and stiffness dependence on amplitude

The damping ratio D is defined12 using the area A = AL + AU − AT, Fig. 7, enclosed within the hysteretic cycle and normalized with stress and strain amplitudes
$$ D = \frac{A_L + A_U - A_T} {4 \pi \frac{1}{ 2} s ^{\rm ampl} \epsilon^{\rm ampl}} $$
(18)
The area enclosed within the hysteretic cycle can be found from the analytical integration of the constitutive model.
Fig. 7

The dissipated work upon a closed strain cycle. The abbreviation F = fE is introduced

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

The energy supply AL for de > 0 in the range from e = 0 to \( e= a = 2 \epsilon^{\rm ampl} \) is
$$ A_L = \int\limits_{0}^{a} E\,e - e\,f\,E | e |^\chi \hbox{d}e = \frac{1}{ 2} E a^2 - \frac{1}{{2+\chi}}\, f E a^{2+\chi} $$
The complementary energy for de < 0 is identical AU = AL and using
$$ A_T = a \left( 1 - f a^{\chi} \right) E a, $$
(19)
we obtain
$$ A_L + A_U - A_T = \frac{ \chi f E a^{2+\chi} } {2+\chi} $$
(20)
The damping ratio can be calculated for any χ as:
$$ D = \frac{A_L + A_U - A_T} {4 \pi \frac{1}{ 8} A_T}=\frac{2 f \chi a^{1 + \chi} } {\pi (2 + \chi) \left[ a - f a^{1 + \chi} \right] } $$
(21)
The secant stiffness, evaluated at \(\epsilon = a\) and for any value of χ is:
$$ H = \frac{s}{e} = E( 1 - f a^\chi) $$
(22)
Realistic curves of modulus degradation and damping ratio versus shear amplitude obtained with PE are shown in Fig. 8.
Fig. 8

Normalized secant stiffness H/E and damping ratio D(a) curves obtained with PE using \(d_L = 0.02 {\%} \) and D(dL) = 0.02 for mR = 2.0 and mR = 5.0 with the common E. Fitting the experimental curves \(D-\ln\gamma^{\rm ampl}\) and \(G-\ln\gamma^{\rm ampl}\) one should consider \(\gamma^{\rm ampl}/\sqrt2 = \epsilon^{\rm ampl} = a/2\)

The material parameters χ, f can be determined from experimental results. Firstly, the ratio mR 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.
Fig. 9

Modulus degradation curves G/Gmax vs γ and damping ratio D vs γ curves obtained from RC-tests for different mean pressures and relative densities Dr0 [18]. Note that \(\gamma/\sqrt2 = \epsilon = d_{AR}\) holds. The sub-angular natural quartz sand used in the RC-tests has the following characteristics: d50 = 2 mm, Cu = 4, emin = 0.439, emax = 0.728. Ranges of damping ratios from studies documented in the literature are represented with the shadow zones

The ratio mR is proposed to be determined from (22) in the range of amplitudes between e = 0.01dL and 10dL with dL = 10−4, namely
$$ m_R = \frac{1 - f 10^{-2 \chi } d_L ^\chi } {1-f 10^{ \chi} d_L^\chi } $$
(23)
with mR ≈ 5.0 according to [7].

For a given mRdL, and D(dL), 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 mR. The curves are calibrated for \(d_L=0.02 {\%} \) and D(dL) = 0.02 taken from Fig. 9, bottom. The stiffness ratio mR 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.

According to the Masing rule [8], the back-bone curve \(\sigma =F(\epsilon)\) should perfectly fit the unloading curve if scaled by two and rotated by 180°, that is
$$ \frac{\sigma-\sigma^R} {2}= F \left( \frac{\epsilon-\epsilon^R} {2} \right) $$
(24)
wherein σR 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.

The Masing rule is satisfied by the PE model if the number of reversal points during shakedown is sufficiently large, Fig. 10.
Fig. 10

Calculated strain cycles with decreasing amplitude followed by the monotonic loading. Left 34 reversals, right 200 reversals. The backbone curve satisfies the Masing rule

4 Continuity of stress

In the present section, the 3D version of the PE model is discussed. The symmetry of the distance dAR1 = dR1A guarantees
$$ {\bf s}({\bf e}) = -{\bf s}(-{\bf e}) $$
(25)
for monotonous loading and monotonous unloading (only a single change of the latest reversal point). The stress span, which has been built up upon loading, vanishes completely upon closing of the strain cycle independently of the strain path. Note that s is a function of the current distance dAR1 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

Continuity of functional \(\varvec{\sigma}(\varvec{\epsilon})\) means that infinitesimally small changes in strain increment cannot cause finite changes in stress increment. This requirement should be satisfied by any pair of similar strain increments, i.e. \(\Updelta\varvec{\epsilon}_a \approx \Updelta \varvec{\epsilon}_b\), should result in \(\Updelta \varvec{\sigma}_a \approx \Updelta\varvec{\sigma}_b\). Let us consider strain increments close to the neutral direction. Let \(\Updelta \varvec{\epsilon}_a \) cause unloading dAR1n+1 < dAR1n so that \(\varvec{\epsilon}^n\) is pushed on the stack whereas \(\Updelta\varvec{\epsilon}_b\) corresponds to loading with dAR1n+1 > dAR1n (in Fig. 11).
Fig. 11

Continuity of stress after push. The given strain path up to the current point (star) may continue upon unloading \((\dot{d}_{AR1} < 0)\) (with reversal) or loading \((\dot{d}_{AR1} >0)\) with almost the same strain increment \(\Updelta\varvec{\epsilon}_a \approx \Updelta\varvec{\epsilon}_b\). The definition of distance based on the current loading circle guarantees the continuity of stress \(\Updelta\varvec{\sigma}_a \approx \Updelta\varvec{\sigma}_b\)

The increments must be almost neutral and hence the incremented strains \(\varvec{\epsilon}^{n+1}_a\) and \(\varvec{\epsilon}^{n+1}_b\) must lie very close to the circle passing through \(\varvec{\epsilon}^n\) and \(\varvec{\epsilon}^{R1}\). They are practically touching this circle from the inside and from the outside, respectively. In both cases, therefore, the stress is calculated using the same stiffness \({\mathsf{H}}\) corresponding to dAR1 an+1dAR1 bn+1dAR1n. For unloading with \(\Updelta \varvec{\epsilon}_a,\) we have
$$ \varvec{\sigma}^{n+1}_{a} = \varvec{\sigma}^{R1} + {\mathsf{H}}: {\bf e}^{n+1} $$
(26)
$$=\varvec{\sigma}^{R2} + {\mathsf{H}}: ( \varvec{\epsilon}^{R1} - \varvec{\epsilon}^{R2} ) + {\mathsf{H}}: ( \varvec{\epsilon}^{n+1}_{a}- \varvec{\epsilon}^{R1} ) $$
(27)
For loading with \(\Updelta \varvec{\epsilon}_b,\) we obtain
$$ \varvec{\sigma}^{n+1}_{b} = \varvec{\sigma}^{R2} + {\mathsf{H}}: (\varvec{\epsilon}^{n+1}_{b} - \varvec{\epsilon}^{R2}) $$
(28)
In (28), the most recent reversal has been denoted as R2 (instead of R1) 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 dAR1n+1dAR1n). 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.
The continuity of stress response for different directions of stretching can be visualized using polar plots of stiffness, so-called response envelopes or response polars [3]. The definition (2) of distance13 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.
Fig. 12

The response envelopes in the P–Q space generated by strain increments \((\Updelta\epsilon_P)^2+(\Updelta\epsilon_Q)^2=0.005^2\) for loading and unloading at dAR = 0.005 and dL = 0.01. Strain increments in different direction are applied to a constant origin state \((\epsilon_P, \epsilon_Q)^0\) and (P, Q)0. The response envelopes are obtained with dAR defined by (2). Left the origin state was preceded by purely deviatoric deformation (vertical arrow). Right the origin state was preceded by purely volumetric deformation (horizontal arrow)

4.2 Continuity of stress after pop

The continuity of stress should also hold after a reversal is popped. Consider an infinitesimally small strain increment \(\Updelta\varvec{\epsilon} \approx {\bf 0}\) that pops the reversal R1, Fig. 13. This implies almost equal distances dAR1ndRR1dAR1n+1 satisfying dAR1n < dRR1 < dAR1n+1. The stress \(\varvec{\sigma}^n\) before pop can be calculated from R2 as
$$ \varvec{\sigma}^n = \varvec{\sigma}^{R1} + {\mathsf{H}} : {\bf e}^n\\ = \varvec{\sigma}^{R2} + {\mathsf{H}} : ( \varvec{\epsilon}^{R1} - \varvec{\epsilon}^{R2} ) + {\mathsf{H}} : (\varvec{\epsilon}^n - \varvec{\epsilon}^{R1}) $$
(29)
with both stiffnesses \({\mathsf{H}}\) corresponding to dAR1ndRR1 (Fig. 13, left). After the increment, we pop R1 but preserve the numbers of reversals, that is, we do not rename R2 to R1 (Fig. 13, right).
Fig. 13

Stress continuity during pop. Left the strain state just before pop. Right the strain state after pop

The updated stress \(\varvec{\sigma}^{n+1}\) corresponding to \(\varvec{\epsilon}^{n+1}\) is calculated from R2 as
$$ \begin{aligned} \varvec{\sigma}^{n+1} &= \varvec{\sigma}^{R2} + {\mathsf{H}} : ( \varvec{\epsilon}^{n+1} - \varvec{\epsilon}^{R2} ) \hbox{ with } \\ \varvec{\epsilon}^{n+1} - \varvec{\epsilon}^{R2} &= \varvec{\epsilon}^{n+1} - \varvec{\epsilon}^{R1} + \varvec{\epsilon}^{R1}- \varvec{\epsilon}^{R2} \end{aligned} $$
(30)
and with \({\mathsf{H}}\) corresponding to dAR1n+1dRR1. Comparing (29) and (30) for infinitesimally small increment that is for dAR1ndAR1n+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

We examine the performance of the PE upon strain paths in the isomorphic \(\epsilon_P - \epsilon_Q\) space. Let a monotonic strain path along \(\epsilon_Q\) axis be preceded by a 1D shakedown along the \(\epsilon_P\) axis, Fig. 14.
Fig. 14

The continuity of stress provokes changes in the curvature of the stress increments after a shakedown followed by consecutive pops

Judging by the diagonal isotropic elastic 2 × 2 stiffness \({\mathsf{E}} = \hbox{diag}(3K,2G ) \) one could expect that the PE response to \(\epsilon_Q\) straining is a stress path along the 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.
Fig. 15

Influence of the former volumetric deformation on the shape of the stress curve obtained with paraelasticity upon deviatoric strain path

Conversely, in order to follow a purely deviatoric stress path, one needs a small volumetric compression or extension in experiment, depending of the sign of the preceding isotropic deformation [16], Fig. 16.
Fig. 16

Strain vectors observed experimentally after the turning point O of the stress path [16]

The drift of the stress path is caused by the dAR1—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 \({\mathsf{K}}\)
$$ {\mathsf{K}}= \frac{\partial {\varvec{\sigma}}} {\partial {\varvec{\epsilon}}}= \frac{\partial {{\bf s}}} {\partial {{\bf e }}} = {\mathsf{H} } - \chi f d_{AR1}^{\chi-1} {\mathsf{E}} : {\bf e }\; \varvec{\eta} \hbox{ with } $$
(31)
$$ \varvec{\eta}=\frac{\partial {d_{AR1}}} {\partial {{\bf e}}} = \frac{-2 \psi \overset{\rightarrow}{{\bf e}} + {\bf N}^{R1}} {\psi^2} \hbox{ and } \psi = \overset{\rightarrow}{{\bf e}} : {\bf N}^{R1}\\ $$
(32)
It can be shown that \(\|\varvec{\eta}\| = 1/\psi^2\) and that \(\varvec{\eta}\) is perpendicular to the loading circle. The distribution of \(\varvec{\eta}\) on a loading circle is presented: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 \({\bf e} \perp \dot{\bf e}\).
Fig. 17

Distribution of \(\varvec{\eta}\) on a loading circle. Note that \(1/\|\varvec{\eta}\|\) and not infinite \(\| \varvec{\eta}\|\) is plotted

Let us now compare the stress 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
$$ ({\mathsf{K}}^{\rm comp} - {\mathsf{K}}^{\rm ext}) : \dot{\bf e} = - f \chi d_{AR1}^{\chi-1} {\mathsf{E}} : ({\bf e}^{\rm comp} \! - \! {\bf e}^{\rm ext}) (\varvec{\eta}\! : \! \dot{{\bf e}}) $$
(33)
acting in different directions.
Fig. 18

Left deviatoric strain path (ext) preceded by isotropic extension. Right deviatoric strain path (comp) preceded by isotropic compression. The volumetric deformation has too strong impact on the tangential stiffness for large distances dAR1dL

The inclination of the stress path during an increase in eQ at eP = const can be quantified using the PE equation in the P–Q space with
$$ \left\{ \begin{array}{c} s_P \\ s_Q \end{array} \right\} = (1-f d_{AR1}^\chi) \left[ \begin{array}{cc} 3K & 0 \\ 0 & 2G \end{array} \right] \cdot \left\{ \begin{array}{c} e_P \\ e_Q \end{array} \right\} $$
(34)
and the stress rate due to \(\dot{e}_Q\) is
$$ \left\{ \begin{array}{c} \dot{s}_P \\ \dot{s}_Q \end{array} \right\} = - \chi f d_{AR1}^{\chi-1} d^{\prime}_{AR1} \dot{e}_Q \left\{ \begin{array}{c} 3K \; e_P \\ 2G \; e_Q \end{array} \right\} \\ + (1-f d_{AR1}^\chi) \left\{ \begin{array}{c} 0 \\ 2G \end{array} \right\}\dot{e}_Q $$
(35)
wherein \(d^{\prime}_{AR1} = {\partial {d_{AR1}}} / {\partial {e_Q}} \).
The drift of the stress path can be quantified and bounded as a function of the PE constants. For the case of an isotropic compression followed by purely deviatoric shearing, the drift \(\dot{s}_P/ \dot{s}_Q\) is obtained as the ratio
$$ \frac{\dot{s}_P} {\dot{s}_Q}= \frac{- \chi f d_{AR1}^{\chi-1} d^{\prime}_{AR1} 3K e_P } {-\chi f d_{AR1}^{\chi-1} d^{\prime}_{AR1} 2G e_Q + (1-f d_{AR1}^\chi) 2G } $$
(36)
or
$$ \frac{\dot{s}_P} {\dot{s}_Q}= \frac{ 3K e_P } { 2G e_Q + \frac{ d_{AR1}}{ d^{\prime}_{AR1} \chi}\left(1- \frac{1 } {d_{AR1}^{\chi} f }\right) 2G } $$
(37)
Note that for a given eQ, the sign of inclination \(\dot{s}_P/\dot{s}_Q\) depends on the sign of eP so switching eP due to consecutive pops causes the zigzag shown in Fig. 14. For the special case ν = 0, we have 3K = 2G and for ePeQdAR1/2 with dAR1 ≈ 2, we obtain an estimation
$$ \frac{1} {z}=\frac{\dot{s}_P}{\dot{s}_Q}= \frac{1 } {1 + \frac{ 1 } { \chi}\left(1- \frac{1 } {d_{AR1}^{\chi} f}\right) } $$
(38)
with the inclination parameter, say z > 4, (see Fig. 19). This condition defines a rough restriction for the model
$$ d_{L} < \left[ f \left(z + 1 + \frac{1}{ \chi} \right) \chi \right]^{- 1/\chi} \approx 0.03 {\%} $$
(39)
assuming χ ≈ 0.9 and f ≈ 300.0. Other restricting conditions to define the range of applicability of the proposed paraelastic model are shown in Appendix.
Fig. 19

The slope of an increment in the stress span \(\dot{s}_P/\dot{s}_Q\) is quantified as a function of dAR1f, χ. A reduced value of dAR1 can be chosen to reduce the zigzag effect

6 PE with Euclidean distance

In the seventies, Hueckel and Nova [5, 6, 12] proposed a paraelastic model based on the Euclidean distance
$$ \bar{d}_{AR1} = \|{\bf e}\| $$
(40)
which is simpler than our definition of dAR1 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.
Fig. 20

Shortcomings of the definition \(\bar{d}_{AR1}=\| {\bf\ e} \|\) used in [6]. Left the given strain path up to the point R may continue upon unloading \({\dot{\bar{d}}_{AR1}} < 0)\) (with reversal) or loading \((\dot{\bar{d}}_{AR1} > 0)\) with almost the same strain increment \(\Updelta \varvec{\epsilon}_a \approx \Updelta\varvec{\epsilon}_b\). The corresponding stress increments are very different \(\Updelta\varvec{\sigma}_a \gg \Updelta\varvec{\sigma}_b\). Right even an infinitesimally small spiral detour with continuously increasing \(\bar{d}_{AR1}=\| {\bf\ e} \|\) can create a new R

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 dAR1 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 dAR1 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. 1.

    More precise definition is given in Sect. 2.

  2. 2.

    The concept of overloading is explained in Sect. 2.3.

  3. 3.

    Definition is given further in this section.

  4. 4.

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

  5. 5.

    Last in, first out.

  6. 6.

    dAR1n+1 = 10dL is sufficient for practical purposes.

  7. 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. 8.

    These Fortran95 routines can be obtained directly from the authors.

  9. 9.

    Abaqus is a reg. trademark of Simulia Inc.

  10. 10.

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

  11. 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. 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. 13.

    As the diameter of the circle connecting the reversal R1 and the current state A.

  14. 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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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • A. Niemunis
    • 1
  • L. F. Prada-Sarmiento
    • 1
    • 2
  • C. E. Grandas-Tavera
    • 1
  1. 1.Institute of Soil Mechanics and Rock MechanicsKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.School of EngineeringUniversity of Los AndesBogotáColombia

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