## Abstract

An initial stress within a solid can arise to support external loads or from processes such as thermal expansion in inert matter or growth and remodelling in living materials. For this reason, it is useful to develop a mechanical framework of initially stressed solids irrespective of how this stress formed. An ideal way to do this is to write the free energy density *Ψ* in terms of initial stress *τ* and the elastic deformation gradient *F*, so we write *Ψ*=*Ψ*(*F*,*τ*). In this paper, we present a new constitutive condition for initially stressed materials, which we call the initial stress symmetry (ISS). We focus on two consequences of this condition. First, we examine how ISS restricts the possible choices of free energy densities *Ψ*=*Ψ*(*F*,*τ*) and present two examples of *Ψ* that satisfy the ISS. Second, we show that the initial stress can be derived from the Cauchy stress and the elastic deformation gradient. To illustrate, we take an example from biomechanics and calculate the optimal Cauchy stress within an artery subjected to internal pressure. We then use ISS to derive the optimal target residual stress for the material to achieve after remodelling, which links nicely with the notion of homeostasis.

## 1. Introduction

When all loads are removed, a body can still hold a significant amount of internal stress, called the *residual stress*. In manufacturing, residual stress has long been noted to be detrimental to, or enhance, the performance of a material. For biological tissues, residual stress is used to self-regulate stress and strain, and ultimately preserve ideal mechanical conditions for the tissue [1,2]. In geophysics, due to gravity and active tectonics, the Earth has developed extremely high initial stresses within its crust, which greatly influence the propagation of seismic waves.

Here, we use the term *initial stress* to broadly mean the internal stress of some reference configuration, irrespective of how the stress was formed or the boundary conditions. In this sense, residual stress is a type of initial stress.

The initial stress felt by any region of a material is due to the push and pull of the surrounding regions. If any region were to be cut out from the material, the stress on its newly formed boundary would be zero, thus reducing the potential energy in the bulk. Based on this concept, Hoger developed constitutive laws for residually stressed materials (e.g. [3–5]). Hoger showed that by taking this idea to its limit and cutting the material into possibly an infinite number of disconnected regions, the material may be relieved of all of its internal stress in a configuration called the *virtual stress-free state*. From that configuration, a hyperelastic energy can be defined as a function of the strain from the virtual state to the current configuration.

Though the use of this virtual state is technically sound, it leads to challenging calculations even for simple deformations and rarely yields analytic results, unless great simplifications are assumed. Moreover, the experimental identification of the virtual state requires cutting the material, which is not always suitable, especially for living organisms. However, using a virtual stress-free reference is routinely seen as the only viable option; quoting Chuong & Fung [6]: ‘To characterize the arterial wall or any other biological soft tissue, we need a stress-free state’. Conversely, we believe that by developing tools to work directly with initially stressed reference configurations, without the need for a stress-free state, will be very useful, especially in biomechanics.

An ideal way to account for the initial stress would be to have a free energy density function *Ψ*=*Ψ*(** F**,

**) written explicitly in terms of the deformation gradient**

*τ***and the initial stress**

*F***, without any**

*τ**a priori*restrictions. For the development of constitutive laws, there is no need to distinguish between residual stresses and initial stresses, a view which is shared with Merodio

*et al.*[7]. The initial stress

**could then be determined from elastic wave speeds [8–10] or by solving the linear equations of momentum balance.**

*τ*Having the free energy in the form *Ψ*(** F**,

**) has been used to investigate how residual stress affects elastic deformations [7,11] and for a mass-growth framework [12]. Wang**

*τ**et al.*[13] found that including one residual stress invariant in the free energy density was a simple way to model the effects of residual stress in the myocardium. Shams

*et al.*[15,14] made progress towards a general framework for initially stressed solids. However, a major obstacle still remains: how to write

*Ψ*(

**,**

*F***) explicitly in terms of the combined invariants of**

*τ***and**

*F***? (Even when the source of anisotropy is only due to the initial stress, the free energy still depends on 10 independent invariants [14].)**

*τ*Johnson & Hoger [5] developed representations for the Cauchy stress response ** F** and

**by first assuming a stress-free virtual state and then inverting numerically the residual stress–strain equation. However, this approach often requires solving numerical non-linear implicit equations. In [5,16] it was applied to a material with virtual state composed by a Mooney–Rivlin strain energy.**

*τ*In this work, we introduce a new constitutive requirement on *Ψ*(** F**,

**) called the initial stress symmetry (ISS). ISS restricts the constitutive form of the stress**

*τ***and**

*F***. To our knowledge, this symmetry has never been discussed before.**

*τ*The ISS also helps answer an important question: how much can the Cauchy stress be altered by adjusting the initial stress? From a modelling perspective, initial stress has been used to make the material more or less compliant [16], to control the Poynting effect [7] or to maintain an ideal internal stress [1]. Given a *Ψ*(** F**,

**) that satisfies ISS, then**

*τ***, there will exist an initial stress**

*σ***that supports**

*τ***.**

*σ*The basic equations for an elastic material subject to initial stress are summarized in §2, and then the ISS is presented. ISS is satisfied automatically if the initial stress is due to an elastic deformation of a stress-free configuration, which we demonstrate in §2b. For this reason, we develop in §3 an example for *Ψ*(** F**,

**) that satisfies ISS, by deforming an incompressible neo-Hookean material from a stress-free virtual state.**

*τ*In §4, we express ISS, in all generality, as nine scalar equations for an incompressible material, written in terms of *Ψ*, two undetermined scalars *p* and *p*_{τ}, and the invariants of ** F** and

**. This form of ISS makes it easier to select appropriate representations for**

*τ**Ψ*(

**,**

*F***). For example, we show, with a minor adjustment to the equations, how to use the scalar equations of ISS to deduce an example of**

*τ**Ψ*(

**,**

*F***) for a compressible material.**

*τ*It is commonly thought that arteries attempt to maintain a homogeneous stress gradient within their walls [17]. In §5, we calculate this optimal Cauchy stress for a simplified arterial wall model, and then show how by using ISS we can calculate the residual stress that exactly supports this optimal Cauchy stress. We finally compare these results against those of the commonly used opening-angle method [6].

## 2. Initially stressed elastic materials

A common approach to model the effect of *residual stress* is to consider a *virtual stress-free configuration* *Ψ* can then be defined as a function of the *deformation gradient*

In this paper, we write the free energy density *Ψ* as a function of the initial stress ** τ** and of the deformation gradient

*Ψ*=

*Ψ*(

**,**

*F***). Note that**

*τ*We assume that ** F** is a purely elastic deformation, but we do not make any assumptions about the origins of the initial stress

**, except that**

*τ***affects the stored energy density**

*τ**Ψ*. Assuming that the body is incompressible, i.e.

**reads [19]:**

*σ**p*is the Lagrange multiplier associated with the constraint of incompressibility and

**is the identity. If the material is compressible, then**

*I**p*is replaced by −2

*I*

_{3}∂

*Ψ*/∂

*I*

_{3}, which completely expresses the dependency of

*Ψ*on

*I*

_{3}; for clarity see equation (2.6). Note that we have and will omit the possible dependence of

*Ψ*on the position

**=**

*F***and**

*I***=**

*σ***; thus, we require that**

*τ**p*when

**=**

*F***. We call the above equation the**

*I**initial stress compatibility*.

The presence of initial stress generally leads to an anisotropic response of the material in reference to *Ψ* can be written as a function of all the independent invariants generated by ** τ** and

**=**

*C*

*F*^{T}

**, the right Cauchy–Green deformation tensor. Following Shams**

*F**et al.*[15], we take the following complete set of 10 independent invariants:

**=**

*B*

*F*

*F*^{T}is the left Cauchy–Green deformation tensor, and

*Ψ*

_{I1},

*Ψ*

_{I2},

*Ψ*

_{J1},

*Ψ*

_{J2},

*Ψ*

_{J3},

*Ψ*

_{J4}are the partial derivatives of

*Ψ*with respect to

*I*

_{1},

*I*

_{2},

*J*

_{1},

*J*

_{2},

*J*

_{3},

*J*

_{4}, respectively. There are no partial derivatives of

*Ψ*with respect to

*I*

_{τ1},

*I*

_{τ2}and

*I*

_{τ3}appearing in (2.6) because

**does not depend on**

*τ***.**

*F*The initial stress compatibility equation (2.2) becomes

Another important physical restriction that can be imposed is the *strong-ellipticity* condition, which is satisfied when the fourth-order tensor of instantaneous moduli
** n**⋅

**=0. Imposing strong ellipticity implies that plane waves may propagate in every direction with a real valued speed [20], and other physically expected behaviour [21]. For a representation of**

*m***and**

*τ***, see [15].**

*F*The issue we address now is how to write *Ψ* explicitly in terms of the invariants (2.3)–(2.5)? We advocate three criteria. First, the free energy density *Ψ* should satisfy the initial stress compatibility (2.7). Second, it should satisfy strong ellipticity (2.8) for all deformations in which the material is expected to be stable [22], and we call the third criterion the ISS requirement.

### (a) Initial stress symmetry

For convenience, let the response function ** F** and

**within the elastic domain of the material. The scalar**

*τ**p*is arbitrary if the material is incompressible, and is

*p*=−2

*I*

_{3}

*Ψ*

_{I3}if the material is compressible.

The ISS states that *p*_{τ}. For a compressible material *p*_{τ}=−2*I*_{3}*Ψ*_{I3}, where ** F** is replaced with

*F*^{−1}

**. In a more precise form, ISS can be stated as**

*τ***and**

*F***, such that**

*τ***=**

*τ*

*τ*^{T}and

*p*and

*p*

_{τ}are, respectively, given by

*p*through its dependence on

**, and analogously for**

*σ**p*

_{τ}. The ISS agrees with the initial stress compatibility, i.e. the condition (2.2), when we have

**.**

*τ*Another way to view the ISS requirement is to assume that ** τ** and

**are due to the elastic deformation of a virtual stress-free state, which we demonstrate in §2b. Hoger emphasized many times that using the virtual stress-free state does not restrict how the residual stress was formed. The same can be said about ISS, as only**

*σ***corresponds to an elastic deformation. Moreover, the ISS is not restricted to elasticity and should hold, under suitable modifications, for other constitutive equations such as those encountered in viscoelasticity.**

*F*One practical outcome from the ISS is to restrict the possible constitutive choices for

A second practical feature arising from ISS is that we can write the initial stress as a function of the Cauchy stress (2.11). This proves useful when ** σ** is known

*a priori*, as will be discussed in §5, where we determine

**from the principle of**

*σ**homeostasis*and then use equation (2.11) to derive

**. The alternative of choosing the initial stress**

*τ***first can be far more complicated.**

*τ*Before developing further the implications of ISS in §4, we introduce in the next section an example of stored energy *Ψ*(** F**,

**) that satisfies ISS, stress compatibility (2.2) and strong ellipticity (2.8).**

*τ*### (b) Stress-free configuration implies initial stress symmetry

Referring to figure 1, the virtual stress-free configuration guarantees that, for any ** τ** and

**, there exists an**

*F**p*and

*p*depends on the boundary conditions on

**=**

*τ***0**in equation (2.6), we see that the right side depends only on

*FF*^{T}and

*p*.

Johnson & Hoger [5] demonstrated that it is possible to determine ** τ**, together with appropriate boundary conditions on

As equation (2.13)_{2} is valid for any ** F** and

_{2}becomes

**as the unique solution. Analogously, equation (2.13)**

*σ*_{1}becomes

This means that, when we make the substitutions (2.15) in equation (2.14), we should also swap ** τ** and

**, so that**

*σ***and the right-hand side is**

*τ***given by ISS (2.12). Finally, the above condition is identically true, since this result is valid for every**

*τ***and**

*F***, so equation (2.16) holds for every**

*τ***and**

*F***.**

*τ*## 3. Initially stressed neo-Hookean material

Here, we derive a simple constitutive equation for an initially stressed body by assuming that both ** τ** and

**arise from deforming an incompressible neo-Hookean material. The result will be an explicit representation of**

*σ**Ψ*in terms of

**and**

*F***, given by equation (3.11) below, that automatically satisfies ISS, the stress compatibility (2.7) and strong ellipticity (2.8). Both ISS and stress compatibility hold because**

*τ***arises due to an elastic deformation from a stress-free state (see §2b), and strong ellipticity is satisfied because the material is a neo-Hookean solid [23].**

*τ*Referring to figure 1, as the material is stress-free in *μ*>0 is the constant shear modulus and ** C**=

*F*^{T}

**, which leads to**

*F***by evaluating equation (3.2) at**

*τ***=**

*F***,**

*I***=**

*σ***,**

*τ***, we require that the deformation**

*τ**T*

_{1}=0, then

*τ*

_{1},

*τ*

_{2}and

*τ*

_{3}of

**, we can write**

*τ*so that, when *T*_{1}=0, we have that *τ*_{1}=*τ*_{2}=*τ*_{3}. From the above, we see that *T*_{1}≥0 and ^{1} *T*_{2}<0, then clearly *T*_{2}≥0 then *Ψ*, and therefore *τ*_{1}=*τ*_{2}=*τ*_{3}, the only viable solution for _{1}. If ** τ** moves into a region where (3.6)

_{2}or (3.6)

_{3}becomes real, then

_{1}to (3.6)

_{2}or (3.6)

_{3}because it can be shown that (3.6)

_{1}does not equal (3.6)

_{2}or (3.6)

_{3}for any

To represent ** τ** and

**, we multiply each side of equation (3.4) on the right with**

*C***and take the trace to get**

*C*_{1}. Note that in the absence of initial stress

**=**

*τ***0**,

*J*

_{1}=

*I*

_{1}, and then

*Ψ*reduces to the classical neo-Hookean model, as expected. Equation (3.11) represents the general extension of the neo-Hookean strain energy function to an initially stressed material, resulting in a function of only five of the nine independent invariants of

**and**

*C***.**

*τ*Substituting (3.11) in the Cauchy stress equation (2.1), we arrive at the constitutive relation
** σ**, the Cauchy stress becomes

**and equation (3.12) becomes**

*τ***for**

*τ***in equation (3.6)**

*σ*_{1}, and

*p*

_{τ}is an undetermined scalar.

Substituting (3.12) in (3.13), we find the connection
** C**, we conclude that

*p*in terms of

**or**

*τ***.**

*F*### (a) Plane strain

The free energy density (3.11) is simplified when the initial strain has only planar components. Accordingly, let us assume *τ*_{13}=*τ*_{23}=0 and *B*_{P}, *C*_{P}, *I*_{P}, *σ*_{P} and *τ*_{P} be ** B**,

**,**

*C***,**

*I***and**

*σ***restricted to the (**

*τ**x*

_{1},

*x*

_{2}) plane, respectively. From equation (3.4), we get

To obtain *τ*_{P}, we take the determinant of each side of equation (3.16), giving
** τ**. The solution

*τ*_{P}=

**0**, while equation (3.16) shows that

*τ*_{P}=

**0**. So for

*τ*_{P}=

**0**, the only viable solution is

*τ*_{P}for every

*τ*_{P}, we should completely discard the solution

For simplicity, we assume *C*_{32}=*C*_{31}=0; then equation (3.3) reads
** F**=

**, we have**

*I***=**

*σ***and**

*τ*## 4. The initial stress symmetry equations

In this section, we show how to express the ISS condition (2.12) as a set of scalar equations that relates the free energy density *Ψ*; *p* and *p*_{τ} from equation (2.12); and the invariants of ** τ** and

**.**

*C*Let us first consider a simple example, namely assuming *Ψ*=*I*_{1}*J*_{1}/2. One can check that this strain energy satisfies the compatibility equations (2.2) with ** τ** and

**, we substitute**

*F***into equation (4.2), which after some rearranging becomes**

*σ**tr*(

*B*^{−1})=

*I*

_{2}, which can be shown by applying the Cayley–Hamilton theorem to

**with**

*B**I*

_{3}=1 (due to incompressibility). In order to satisfy equation (4.3) for every

**and**

*F***, the coefficients of**

*τ***,**

*τ*

*C*^{−1}and

**must all be identically zero (this is shown rigorously in the electronic supplementary material). For the coefficient of**

*I***to be zero, the identity**

*τ**I*

_{1}

*I*

_{2}=1 must hold for every

**, which is obviously impossible: so we conclude that**

*F**Ψ*=

*I*

_{1}

*J*

_{1}/2 does not correctly furnish the Cauchy stress for every reference configuration.

In the electronic supplementary material, we reduce the ISS to nine scalar equations, following a procedure similar to the one above. They are compactly written as
*n* sums over 0,1,2,3,4, *Ψ*_{J0}:=1,
*k*∈{1,2} and *m*∈{1,2,3,4} with ** σ** given by (2.6),

*Ψ*,

*p*and

*p*

_{τ}.

We now look at special cases of *Ψ* which simplify the ISS equations and lead to more practical representations for the free energy density.

### (a) Free energy independent of *J*_{3} and *J*_{4}

When *Ψ* is independent of *J*_{3} and *J*_{4}, equation (4.5) is identically zero and only the first three terms of equation (4.4) are non-zero. The fourth scalar equation of (4.4) becomes ** F** and

**, only when**

*τ**Ψ*does not depend on

*J*

_{2}or does not depend on

*J*

_{1}. We investigate the first case in more detail below.

#### (i) Assuming *Ψ* independent of *J*_{2}, *J*_{3} and *J*_{4}

If *Ψ* does not depend on *J*_{2}, *J*_{3} and *J*_{4}, then equation (4.4) reduces to
_{1} to derive the other three equations. We can check if these equations are consistent with the compatibility equations (2.7) by letting ** F**=

**,**

*I***=**

*σ***and**

*τ**Ψ*

_{I2}(1±1)=0, which only satisfies equations (2.7) if

*Ψ*

_{I2}=0 for

**=**

*F***.**

*I*Let us apply equations (4.10) to the following example:

used by Merodio *et al.* [7]. In this case, equations (4.10)_{1} and (4.10)_{2} are satisfied while equations (4.10)_{3} and (4.10)_{4} become *p*=*p*_{τ}=*μ*. However, restricting *p* to be constant, when it is normally determined from boundary and/or initial conditions, would result in physically unexpected behaviour. This is best seen with an example.

### Example 4.1

For the free energy density expressed in equation (4.11), the Cauchy stress from equation (2.6) becomes
*τ*_{ij}=*τ*_{i}*δ*_{ij}, aligned with the axes of the cube. For the current configuration, we impose two clamped conditions that fix *F*_{11}=λ_{1} and *F*_{33}=λ_{3}, then *F*_{22}=λ_{2}=(λ_{1}λ_{3})^{−1} due to incompressibility, with all the other components of ** F** being zero. With this imposed deformation, the Cauchy stress is diagonal with components

*σ*

_{ij}=

*δ*

_{ij}

*σ*

_{i}. We can also prescribe the stress

*σ*

_{2}, as this will not alter λ

_{1}and λ

_{3}because they are kept fixed. We choose

*σ*

_{2}=

*τ*

_{2}, to ensure compatibility with the reference configuration when λ

_{1}=λ

_{3}=1, which results in

*p*to satisfy the above and be fixed at

*p*=

*μ*so that the ISS is also satisfied. This means that the free energy density equation (4.11) either results in non-physical behaviour or does not satisfy the ISS, which implies that the constitutive relation (4.12) does not hold for every reference configuration.

The choice of *Ψ* should satisfy equations (4.10) without restricting the deformation ** F**, initial stress

**or**

*τ**p*, otherwise the material is likely to exhibit non-physical behaviour. In this respect, the free energy densities

*μ*and

*Ψ*for a compressible material, inspired by the neo-Hookean example in §3, that satisfies ISS without any unphysical restrictions.

For a compressible material, we substitute *p* with −2*I*_{3}*Ψ*_{I3} and *p*_{τ} with *f* and *g* are arbitrary scalar functions.

The corresponding Cauchy stress (2.6) is
_{1} is satisfied as _{2} is satisfied as 2*Ψ*_{J1}=1 when ** F**=

**. By substituting**

*I**p*for −2

*I*

_{3}

*Ψ*

_{I3}, the remaining compatibility equation (2.7)

_{1}becomes

*f*

_{I3}(

**,1)=**

*τ**tr*

**/3−3**

*τ**g*

_{I3}(

**,1); then equation (4.16) and equation (4.14) together imply that**

*τ***=**

*σ***when**

*τ***=**

*F***.**

*I*The ISS equations (4.10)_{3} and (4.10)_{4} now read, respectively,

for every ** F** and

**, where**

*τ**f*

_{I3}and

*g*

_{I3}are, respectively, the partial derivatives of

*f*and

*g*with respect to

*I*

_{3}. Note that the above two equations are equivalent as equation (4.17)

_{1}becomes equation (4.17)

_{2}when we substitute

**for**

*F*

*F*^{−1}and

**for**

*τ***. For this reason, it is sufficient to satisfy equation (4.17)**

*σ*_{1}. If

**=**

*F***, then equation (4.17)**

*I*_{1}is satisfied as it reduces to equation (4.16). The rest of this section will focus on showing that equation (4.17)

_{1}is satisfied for every

**and**

*F***.**

*τ*By rearranging equation (4.14) and taking the determinant on either side, we get
*g*(** τ**,

*I*

_{3}) such that

*k*is a constant, then the right-hand side of equation (4.18) is also equal to

*k*, thus

**must satisfy**

*σ***for**

*τ***and**

*σ***for**

*F*

*F*^{−1}, equation (4.19) becomes

*g*(

**,**

*τ**I*

_{3}) to satisfy equation (4.19), there are three possible solutions given by replacing

*μ*

^{3}with

*g*(

**,**

*τ**I*

_{3}) with

_{1}, (3.6)

_{2}and (3.6)

_{3}, which we, respectively, denote by

*g*

_{1}(

**,**

*τ**I*

_{3}),

*g*

_{2}(

**,**

*τ**I*

_{3}) and

*g*

_{3}(

**,**

*τ**I*

_{3}). We highlighted in §3 that only

*g*(

**,**

*τ**I*

_{3})=

*g*

_{1}(

**,**

*τ**I*

_{3}) is a physically viable solution. So if we choose

*g*(

**,**

*τ**I*

_{3})=

*g*

_{1}(

**,**

*τ**I*

_{3}), then for equation (4.17)

_{1}to be satisfied, we need

**=**

*F***, equation (4.22) is true. We can also conclude from equations (4.20) and (4.21) that −2**

*I**I*

_{3}

*Ψ*

_{I3}will equal either

**. Since −2**

*F**I*

_{3}

*Ψ*

_{I3}should be a continuous function of

**and**

*F***, it cannot change from**

*τ***and**

*σ**I*

_{3}. For this reason, if

*g*

_{1}(

**,1)=−2**

*τ**Ψ*

_{I3}for

**=**

*F***, then**

*I***.**

*F*In conclusion, the model (4.13) satisfies ISS as long as *g*(** τ**,

*I*

_{3})=

*g*

_{1}(

**,**

*τ**I*

_{3}), where

*g*

_{1}(

**,**

*τ**I*

_{3}) is given by replacing

*μ*

^{3}with

_{1}.

## 5. Homogeneous stress gradient in a hollow cylinder

It is now well acknowledged that residual stresses in living materials are vital to maintain ideal mechanical conditions. When an external load changes, growth and remodelling alter the residual stress to best adapt to the new load. An excellent example is how arteries remodel in response to the internal pressure. The residual stress in arteries is thought to protect the arterial wall against strain concentration [24] or stress concentration [1,17]. Here, we adopt the most accepted hypothesis—that the residual stress in the artery acts to minimize the stress gradient [1,25]. The reasoning behind this hypothesis is that if a given tissue grows in response to stress, then homeostasis is only possible if the tissue is under similar stress conditions throughout. So by minimizing the stress gradient, we are selecting the most homogeneous stress possible. We will also consider a simplified artery with only one layer and no shear stress applied to the interior of the artery.

Our strategy is to first choose an optimal Cauchy stress field ** σ**, and then to derive the residual stress from the ISS equation

We finally compare the results with the corresponding ones obtained using the opening angle method [6].

### (a) Plane strain cylinder

To describe the arterial wall, we use cylindrical coordinates (*r*,*θ*,*z*) with the components of ** σ** written in terms of unit basis vectors. We assume that the arterial wall retains its cylindrical symmetry when the internal pressure is removed and that there is no shear stress at the inner wall. These assumptions imply that, when a pressure is applied in the cylinder, the resulting deformation is an inflation.

Due to radial symmetry, the Cauchy stress ** σ** is independent of

*θ*and

*z*, so that in the absence of body forces the equilibrium equations reduce to

*a*and

*b*are the inner and outer radius of the loaded artery, respectively. The equilibrium equations (5.1) together with the boundary conditions lead to

*σ*

_{θr}=0 and

*σ*

_{zr}=0 for all

*r*. As neither

*σ*

_{zθ}nor

*σ*

_{zz}appear in the equations of equilibrium, they are only restricted by the constitutive choice and boundary conditions on the cross-section of the artery. We use this degree of freedom to assume there is no axial torsion

*σ*

_{zθ}=0. Note that, for the constitutive choice equation (2.6),

*τ*

_{ZZ}can be chosen so that

*σ*

_{zz}is constant, while for the simpler choice of plane strain equation (3.21) (with

*σ*

_{zz}=

*σ*

_{33}),

*σ*

_{zz}is determined by

*p*. Finally, the remaining equation of equilibrium is equation (5.2).

### (b) Minimal stress gradient

Here, we develop a method to minimize the stress gradient fields *σ*_{rr}′ and *σ*_{θθ}′, where the prime denotes differentiation with respect to *r*. We make no assumptions about any reference configuration nor do we make any constitutive choice. We only make use of the assumptions from the section above which result in both *σ*_{rr} and *σ*_{θθ} being independent of the coordinates *θ* and *z*; *σ*_{θr}=*σ*_{zr}=*σ*_{zθ}=0; and, for simplicity, we will not consider *σ*_{zz}.

Once *σ*_{θθ} and *σ*_{rr} are determined, we use ISS equation (2.11) to write the residual stresses *τ*_{ΘΘ} and *τ*_{RR} in terms of *σ*_{θθ}, *σ*_{rr} and the deformation gradient in §5c.

Using the equilibrium equation (5.2), we write *σ*_{θθ} in terms of *σ*_{rr} as

As aortas and veins are of many different sizes, for the sake of generality, let us introduce the following dimensionless variables:

Our aim is to minimize the stress gradient density
*α*≡*a*/(*b*−*a*).

Using calculus of variations [26], we find that *φ*′ must satisfy the Euler–Lagrange equation

Realistic values for *α*=*a*/(*b*−*a*) can be obtained from *in vivo* measurements of the lumen radius (inner radius) *a* and total artery thickness *b*−*a*. For the descending thoracic aorta of healthy men around 51 years old, the mean lumen radius is 20 mm [27] and the mean thickness is 1.4 mm [28]. These estimates give *α*^{−1}≈0.07, making it appropriate to expand *φ*′ and λ as a power series in *α*^{−1}, which gives
*φ*′ in equations (5.5), (5.6) and (5.7) to obtain

We plot the above Cauchy stress for *α*^{−1} from 0.05 to 0.10, which is within the physiological range, in figure 3. A graph for *σ*_{rr}/*P* would show all the curves bunched along the same straight line, so instead we have depicted the curves for *σ*_{rr}/*P*−*ξ*. Figure 3 reveals that, as the relative thickness of the arterial wall increases (shading from blue towards red), the circumferential stress *σ*_{θθ} decreases while the radial stress *σ*_{rr} increases and becomes less homogeneous.

A well-established method to quantify the residual stresses within arteries is the opening angle method [6]. For a neo-Hookean material (3.1), we will use the opening angle method to minimize the circumferential stress component
*ϕ*, restricted to the boundary conditions (5.3) and (5.4). The only two parameters with a unit of time and mass are *P* and *μ*, so by rewriting *P*=*P*_{0}*μ* the stress will not depend on *μ*. The results for *P*_{0}=0.05, *P*_{0}=0.20 and *P*_{0}=0.35 are compared with the optimal stress equations (5.18) and (5.19) with parameters for the descending thoracic aorta in figure 4. We see that *σ*_{θθ} for the opening angle method converges to the optimal stress *σ*_{θθ} as *P*/*μ* tends to zero, while the plots for *σ*_{rr} all overlap. Note however that *σ*_{zz} for the opening angle method is not necessarily homogeneous in *r*.

We can now invoke finite elasticity to derive the residual stress ** τ** in the unloaded state necessary to sustain the optimal homogeneous Cauchy stress, and then compare our results with the residual stress predicted by the opening angle method. To do so, we let

*Ψ*be a function of both stress and strain and use the ISS, in the form of equation (2.11), to determine

**as a function of**

*τ***and**

*σ***.**

*F*### (c) Residual stress

Here, we use finite elasticity to connect the current state with the unloaded state. Since the cylindrical symmetry of the artery is maintained when the internal pressure is removed, we describe the unloaded state with the cylindrical coordinates (*R*,*Θ*,*Z*). Then the deformation gradient for the unit basis vectors *e*_{r}, *e*_{θ}, *e*_{z}, *E*_{R}, *E*_{Θ} and *E*_{Z} becomes
_{z} be a constant. Assuming the material is incompressible, we get

Let the reference configuration be a hollow cylinder with inner and outer radius *A* and *B*, respectively, so that
*r*/∂*R* wherever it appears. We will assume that the residual stress ** τ** is homogeneous in

*Θ*and

*Z*, as the deformation, the Cauchy stress from equations (5.18) and (5.19), and the boundary conditions (5.25) and (5.26) are all homogeneous in

*Θ*and

*Z*.

The equilibrium equations now reduce to
*Ψ*, the residual stresses *τ*_{RR} and *τ*_{ΘΘ} will be completely determined from ** F**,

**and**

*σ**p*

_{τ}due to ISS (2.11).

#### (i) *Ψ* independent of *I*_{2}, *J*_{2}, *J*_{3} and *J*_{4}

To simplify, we assume *Ψ* independent of *I*_{2}, *J*_{2}, *J*_{3} and *J*_{4}, so the Cauchy stress (2.6) becomes
** τ**,

**and**

*F**p*with

**,**

*σ*

*F*^{−1}and

*p*

_{τ}, respectively, to get

**from equation (5.21) into the above equation, we arrive at**

*F**τ*

_{ZZ}has not been derived since we have not taken into consideration

*σ*

_{zz}in §5b.

To compare unloaded arteries of different sizes, we write *r* and *R* in terms of the dimensionless radius *ξ*,
*ξ* and using the unloaded boundary condition (5.25) and (5.26), we reach
_{2} is independent of *p*_{τ} and can be used to solve for one of the parameters *P*, *a*, *b*, *A*, λ_{z} or *μ* (note that *B* depends on the other parameters because equation (5.23) evaluated at *R*=*B* gives *B*^{2}=*A*^{2}+λ_{z}(*b*^{2}−*a*^{2})). The only two parameters with a unit of time or mass are *P* and *μ*, so by rewriting *P*=*P*_{0}*μ* equations (5.34) will not depend on *μ*.

To illustrate, we adopt the neo-Hookean material model (3.20), which results in ** τ** for

**restricted to (**

*σ**r*,

*θ*) in

To calculate the residual stress that supports the optimal Cauchy stress, we substitute *σ*_{θθ} and *σ*_{rr} from equations (5.18) and (5.19) into *τ*_{ΘΘ} and *τ*_{RR} in equations (5.29) and (5.30). We then use equation (5.34)_{2} to determine the unloaded geometry from the loaded geometry. To illustrate, we take the parameters for the descending thoracic aorta, as used in the previous section, *a*=20 mm, *b*=1.4 mm and λ_{z}=1. We can then determine the unloaded inner radius *A* for different values of *P*_{0}, which is shown in figure 5. Surprisingly, the unloaded geometry given by the opening angle method is approximately the same as shown in figure 5.

We can also investigate the residual stress in the unloaded state. Once *A* is determined from equation (5.34)_{2}, we can use equation (5.34)_{1} to determine *τ*_{RR}, and then *τ*_{ΘΘ} from equation (5.24). To compare with the opening angle method, we choose *P*/*μ*=0.05,0.20 and 0.35, which all give a reasonable unloaded geometry (figure 5). The results are shown in figure 6.

Biological tissues are very energy efficient. It is a common line of reasoning that biological systems adapt so as to minimize their potential energy. So it is possible that biological materials remodel their residual stress to lower their potential energy. Figure 7*a* shows the free energy density *Ψ* for the opening angle method and for the model (3.20) with the optimal stress equations (5.18) and (5.19). In all cases, *Ψ* is approximately a straight line that decreases as *ξ* increases away from the pressured boundary *ξ*=0 (*r*=*a*). Figure 7*b* shows the difference between *Ψ*/*P* from the opening angle method minus *Ψ*/*P* from the model (3.20), which we denote as Δ*Ψ*/*P*. The integral of Δ*Ψ*/*P* over *ξ*∈[0,1] is positive if the opening angle method has on average a larger free energy density than the model (3.20), with the optimal stress equations (5.18) and (5.19). We found that as *P*/*μ* increased, so did the total free energy in the cylinder cross-section ^{−4} for *P*/*μ*=0.05, 2.3710^{−3} for *P*/*μ*=0.20, and 4.2510^{−3} for *P*/*μ*=0.35.

Essentially, we can see from figures 4 and 7 that the optimal stress equations (5.18) and (5.19) with the neo-Hookean material (3.20) produce a more homogeneous stress and a lower free energy than the opening angle method, though the two methods gave very similar results. The advantage in using the ISS (2.12) for a homeostasis hypothesis for the Cauchy stress ** σ** is twofold. First, we have the freedom to choose

**so as to satisfy the homeostasis hypothesis, which with ISS becomes a separate step from choosing a constitutive equation. Second, it can be relatively straightforward to use ISS (2.12) to quantify the residual stress needed to support the chosen**

*σ***.**

*σ*## 6. Conclusion

In order to quantify the initial stress within a solid using non-destructive experimental techniques, it is advantageous to write the free energy density in terms of the deformation gradient ** F** and the initial stress

**as**

*τ**Ψ*=

*Ψ*(

**,**

*F***).**

*τ*In this article, we presented a new constitutive condition, the initial stress symmetry (ISS), that aids in proposing suitable constitutive relations for *Ψ*=*Ψ*(** F**,

**) by providing nine scalar equations (equations (4.4) and (4.5)). One immediate result is that guessing a functional dependence for**

*τ**Ψ*=

*Ψ*(

**,**

*F***) is not a trivial task. In fact all choices for**

*τ**Ψ*(

**,**

*F***) used in the literature so far do not satisfy ISS. Conversely, using ISS, we proposed two simple choices for**

*τ**Ψ*=

*Ψ*(

**,**

*F***), one incompressible equation (3.11) and one compressible equation (4.13), which can be viewed as initially stressed neo-Hookean materials.**

*τ*One consequence of ISS is that the initial stress can be derived from the Cauchy stress. Furthermore, ISS suggests that it is possible to first choose the Cauchy stress, and second to make a constitutive choice and then use equation (2.11) to determine the corresponding initial stress. We illustrated this method in §5 for a simplified arterial wall model, where we chose the ideal Cauchy stress by using a minimal stress gradient hypothesis as the homeostatic condition, and then we calculated the optimal residual stresses of the unloaded remodelled artery. It is reasonable that other tissues with smooth muscle cells may also remodel to minimize their stress gradient. This means that the approach we developed for the arterial wall could be used to help test this hypothesis for other tissues, such as those found in the heart.

Since initial stresses are widespread in both inert and living matter, the proposed constitutive restriction can be used in many applications towards a non-destructive quantification of initial tensions in solids. Future research includes determining a wider class of material behaviours, e.g. taking into account natural material anisotropy, and linking the initial stress to elastic wave speeds.

## Data accessibility

The material supporting this article has been uploaded as part of the electronic supplementary material.

## Authors' contributions

All the ideas and the mathematics were developed by all three authors. P.C. and M.D. edited and revised the manuscript and the electronic supplementary material. A.L.G. wrote the first drafts, generated the figures and drafted the electronic supplementary material. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

Partial funding by the Irish Research Council, the Royal Society, the Hardiman Scholarship Programme at the National University of Ireland Galway and Regione Lombardia, through the ‘Start-up Packages and PhD Program’ project, is gratefully acknowledged.

## Footnotes

- Received June 30, 2015.
- Accepted October 1, 2015.

- © 2015 The Author(s)