Collisions in shape memory alloys

TL;DR

This paper develops a predictive model for instantaneous collisions in shape memory alloys, incorporating macroscopic and microscopic velocities, phase changes, and thermodynamics, with proven existence and uniqueness of solutions and numerical simulations.

Contribution

It introduces a novel model for SMA collisions that accounts for microscopic phase change velocities and provides mathematical proofs of solution existence and uniqueness.

Findings

Proven existence and uniqueness of solutions in 2D and 3D.

Numerical simulations demonstrating collision effects in 2D SMA.

Model captures phase change dynamics during collisions.

Abstract

We present here a model for instantaneous collisions in a solid made of shape memory alloys (SMA) by means of a predictive theory which is based on the introduction not only of macroscopic velocities and temperature, but also of microscopic velocities responsible of the austenite-martensites phase changes. Assuming time discontinuities for velocities, volume fractions and temperature, and applying the principles of thermodynamics for non-smooth evolutions together with constitutive laws typical of SMA, we end up with a system of nonlinearly coupled elliptic equations for which we prove an existence and uniqueness result in the 2 and 3 D cases. Finally, we also present numerical results for a SMA 2D solid subject to an external percussion by an hammer stroke.

Equations

$$\frac{d{\beta }_i}{dt},$$

$$\vec{U}=({\vec{U}}^{+},{\vec{U}}^{-}),$$

$$\left[{\beta }_i\right]={\beta }_i^{+}-{\beta }_i^{-}.$$

$$E=\left(\epsilon (\vec{u}),{\beta }_i,\mathbf{grad}{\beta }_i,T\right).$$

$$D(\frac{{\vec{U}}^{+}+{\vec{U}}^{-}}{2}),$$

$$\underline{T}=\frac{T^{+}+T^{-}}{2},$$

$$\delta E=(D(\frac{{\vec{U}}^{+}+{\vec{U}}^{-}}{2}),\left[{\beta }_i\right],\mathbf{grad}\left[{\beta }_i\right],\mathbf{grad}\underline{T}).$$

$$\rho \left[\vec{U}\right]=\mathrm{div}\Sigma ,in\Omega ,$$

$$\Sigma ={\Sigma }^T,in\Omega ,$$

$$-B_i^p+\mathrm{div}{\vec{H}}_i^p=0,in\Omega ,$$

$$\Sigma \vec{N}={\vec{G}}^p,{\vec{H}}_i^p\cdot \vec{N}=0,on\partial \Omega ,$$

$$\left[\rho ({\beta }_1+{\beta }_2+{\beta }_3)\right]=0,$$

$$\left[({\beta }_1+{\beta }_2+{\beta }_3)\right]=0,$$

$${\beta }_{voids}=1-({\beta }_1+{\beta }_2+{\beta }_3),$$

$$\left[\mathcal{E}\right]+\left[\mathcal{K}\right]={\mathcal{T}}_{ext}+\mathcal{C},$$

$$\mathcal{E}={\int }_{\Omega }ed\Omega ,$$

$$\left[\mathcal{E}\right]=-{\mathcal{T}}_{int}+\mathcal{C},$$

$$\underline{T}=\frac{T^{+}+T^{-}}{2}.$$

$$\mathcal{C}={\int }_{\partial \Omega }-\left(T^{+}{\vec{Q}}_p^{+}+T^{-}{\vec{Q}}_p^{-}\right)\cdot \vec{N}d\Gamma +{\int }_{\Omega }{T}^{+}{\mathcal{B}}^{+}+T^{-}{\mathcal{B}}^{-}d\Omega ,$$

$$\left[e\right]$$

$$\left[e\right]=\left[\Psi \right]+\underline{s}\left[T\right]+\underline{T}\left[s\right]$$

$$=\Sigma :D(\frac{{\vec{U}}^{+}+{\vec{U}}^{-}}{2})+B_i^p\left[{\beta }_i\right]+{\vec{H}}_i^p\cdot \mathbf{grad}\left[{\beta }_i\right]$$

$$-\mathrm{div}\left\{\underline{T}\Sigma \left({\vec{Q}}_p\right)+\left[T\right]\Delta \left({\vec{Q}}_p\right)\right\}+\underline{T}\Sigma \left(\mathcal{B}\right)+\left[T\right]\Delta \left(\mathcal{B}\right),$$

$$T^{+}{\mathcal{B}}^{+}+T^{-}{\mathcal{B}}^{-}=\Sigma (T\mathcal{B}),$$

$$\Sigma (T\mathcal{B})=\underline{T}\Sigma \left(\mathcal{B}\right)+\left[T\right]\Delta \left(\mathcal{B}\right),$$

$$\Sigma \left(\mathcal{B}\right)={\mathcal{B}}^{+}+{\mathcal{B}}^{-}and,\Delta \left(\mathcal{B}\right)=\frac{{\mathcal{B}}^{+}-{\mathcal{B}}^{-}}{2}.$$

$$\left[\mathcal{S}\right]={\int }_{\Omega }\left[s\right]d\Omega \ge -{\int }_{\Gamma }\left({\vec{Q}}_p^{+}+{\vec{Q}}_p^{-}\right)\cdot \vec{N}d\Gamma +{\int }_{\Omega }{\mathcal{B}}^{+}+{\mathcal{B}}^{-}d\Omega ,$$

$$\left[s\right]\ge -\mathrm{div}\Sigma \left({\vec{Q}}_p\right)+\Sigma \left(\mathcal{B}\right).$$

$$\left[\Psi \right]+\underline{s}\left[T\right]+\mathrm{div}\left(\left[T\right]\Delta ({\vec{Q}}_p)\right)-\left[T\right]\Delta (\mathcal{B})$$

$$\le \Sigma :D(\frac{{\vec{U}}^{+}+{\vec{U}}^{-}}{2})+B_i^p\left[{\beta }_i\right]+{\vec{H}}_i^p\cdot \mathbf{grad}\left[{\beta }_i\right]-\mathbf{grad}\underline{T}\cdot \Sigma ({\vec{Q}}_p).$$

$$\left[\Psi \right]=\Psi (T^{+},{\beta }_i^{+},\mathbf{grad}{\beta }_i^{+})-\Psi (T^{-},{\beta }_i^{-},\mathbf{grad}{\beta }_i^{-})$$

$$=\Psi (T^{+},{\beta }^{+},\mathbf{grad}{\beta }^{+})-\Psi (T^{+},{\beta }_i^{-},\mathbf{grad}{\beta }_i^{-})$$

$$+\Psi (T^{+},{\beta }_i^{-},\mathbf{grad}{\beta }_i^{-})-\Psi (T^{-},{\beta }_i^{-},\mathbf{grad}{\beta }_i^{-}).$$

$$\Psi (T^{+},{\beta }_i^{-},\mathbf{grad}{\beta }_i^{-})-\Psi (T^{-},{\beta }_i^{-},\mathbf{grad}{\beta }_i^{-})\le -s^{fe}\left[T\right],$$

Full text

**Key words. ** Shape memory alloys, collisions, existence and uniqueness result, numerical examples.

**AMS subject classification. ** 73C02, 73C35, 35B65.

1 Introduction

Collisions of solids produce discontinuities of velocities and discontinuities of temperatures at collision time $t$. We consider a solid made of shape memory alloys. It occupies domain $\Omega$ with boundary $\partial\Omega$. There is a vast literature on shape memory alloys. We mention only the predictive theory which introduces besides the macroscopic velocity and the temperature, velocities at the microscopic level which are responsible for the phase changes between the martensites and austenite phases, [13], [14]. We have chosen to represent at the macroscopic level, the velocities at the microscopic level by the velocities

[TABLE]

of the volumes fractions $\beta_{i}$ of the different phases, $\beta_{3}$ for the austenite phase and $\beta_{1}$, $\beta_{2}$ for the martensite phases, assuming there are two of them. There can be voids in the mixture of the three phases with volume fraction $\beta_{voids}$.

We investigate collisions involving shape memory alloys and assume these collisions are instantaneous, [16]. We denote with subscript -, quantities before collision and subscript +, quantities after collision. For example, we denote

[TABLE]

the actual velocities, $\vec{U}^{-}$ being the actual velocity before the collision and $\vec{U}^{+}$ being the actual velocity after. We denote $\left[X\right]=X^{+}-X^{+}$, the discontinuity of quantity $X$

In collisions, there are rapid variations of the velocities at the microscopic level resulting in rapid variations of the volumes fractions $\beta_{i}$. Thus we assume also the volume fractions are discontinuous [16], [18], [19], [23]

[TABLE]

The collisions being dissipative phenomena, they produce burst of heat which intervene in the thermal evolution. They result in temperature discontinuities and they may produce phase changes. Moreover voids may also appear, [17]. Thus the volume fractions discontinuities and the temperature discontinuities are coupled as they are in smooth evolutions, [15]. Transient and fast but smooth phenomena in shape memory alloys are investigated in [6], [8]. The last paper contains experimental results.

2 The Model

2.1 The State Quantities and the Quantities which Describe the

Evolution

The state quantities are

[TABLE]

In a collision, the small displacement $\vec{u}$ does not change, thus the small deformation, $\varepsilon(\vec{u})$, remains constant. But as already seen, the phase volume fractions and the temperature $T$ do vary in collisions. We have the state quantities before collision, $E^{-}$, and $E^{+}$ after.

The quantities which describe the evolution are the evolution of the velocity of deformation

[TABLE]

where $D$ is the usual deformation operator, and the gradient of the average temperature introduced in collision theory, together with the variation of the volume fractions and their gradients. The average temperature is

[TABLE]

where $T^{+}$ and $T^{-}$ are the temperatures after and before collision. Its gradient is involved in the description of heat diffusion occurring in collisions. Thus the quantities which describe the evolution, $\delta E$ is

[TABLE]

The discontinuity $\left[\beta_{i}\right]$ is the non smooth part of the velocity $d\beta_{i}/dt$. Let us note that $\delta E$ is objective.

2.2 The Equations of Motion

The equations of motion result from the principle of virtual work introducing percussion stress $\Sigma$, percussion work $B_{p}$ and percussion work flux vector $\vec{H}_{p}$, [20]. They are

[TABLE]

and

[TABLE]

where $\vec{G}^{p}$ is the surface external percussion, [20]. It is assumed no body percussion and no external action at the microscopic level. Equation (2) is the angular momentum equation of motion, see [21] p. 248-251.

2.3 The Mass Balance

It is

[TABLE]

or

[TABLE]

assuming the density $\rho$ is constant and the same for each phase. A possible evolution of the time discontinuity of the voids volume fraction

[TABLE]

is given by a constitutive law. Examples are given in [20].

2.4 The First Law of Thermodynamics

The first and second laws of thermodynamics intervene in the derivation of the constitutive laws. We recall them to get the new mechanical and thermal collision constitutive laws.

The first law can be written as

[TABLE]

where

[TABLE]

is the internal energy, $\mathcal{K}$ is the kinetic energy, $\mathcal{C}$ is the thermal impulse received by the solid, and $\mathcal{T}_{ext}$ is actual work of the external forces. With the principle of virtual work where the velocities are the actual velocities, i.e., with the theorem of kinetic energy, the first law gives

[TABLE]

where $\mathcal{T}_{int}$ is actual work of the internal forces. The temperature may be discontinuous, [16], [15]: we have already defined $T^{-}$ the temperature before the collision and $T^{+}$ the temperature after the collision and

[TABLE]

We assume that the external impulse heat is received either at temperature $T^{-}$ or at temperature $T^{+}$

[TABLE]

where $\vec{Q}_{p}$ is the impulsive entropy flux vector and $\mathcal{B}$ the impulsive entropy source. Relationship (6) being true for any subdomain of $\Omega$, we get the energy balance law

[TABLE]

By using the Helmholtz relationship, $e=\Psi+Ts$, we have

[TABLE]

where a sum

[TABLE]

is split in an other sum

[TABLE]

with

[TABLE]

Remark 1**.**

To avoid too many notation, we use letter $\Sigma$ with two meanings: the percussion stress $\Sigma$ which appears in the equation of motion and the sum $\Sigma\left(\mathcal{B}\right)=\mathcal{B}^{+}+\mathcal{B}^{-}$. They appear in different context and the sum $\Sigma\left(\mathcal{B}\right)$ has always an argument.

2.5 The Second Law of Thermodynamics

The Second Law may be stated as

[TABLE]

which gives

[TABLE]

Combining relationships (8) and (9), we get

[TABLE]

Let us note that the right hand side of (10) is a scalar product between internal forces and related evolution quantities whereas the left hand side is not a scalar product. Let us try to relate $\left[\Psi\right]$ to a scalar product. We have

[TABLE]

Because the free energy is a concave function of temperature $T$, we have

[TABLE]

with

[TABLE]

where $\hat{\partial}\Psi_{T}$ is the set of the uppergradients of the concave function

[TABLE]

We assume $\Psi$ is a convex function of $(\vec{\beta},\mathbf{grad}$ $\vec{\beta})$. Thus we have

[TABLE]

with

[TABLE]

where $\partial\Psi_{\beta,\mathbf{grad}\beta}$ is the subdifferential set of convex function $\Psi$ of $(\vec{\beta},\mathbf{grad}$ $\vec{\beta})$. The internal forces $\left(B_{i}^{fe},\vec{H}_{i}^{fe}\right)$ depend on the future state $(T^{+},\beta_{i}^{+},\mathbf{grad}$ $\beta_{i}^{+})$, in agreement with our idea that the constitutive laws sum up what occurs during the discontinuity of the state quantities. It results

[TABLE]

As usual, we assume no dissipation with respect to $\left[T\right]$, [16], [18], [15] and have

[TABLE]

Thus

[TABLE]

with $S=s^{fe}-\bar{s}$. This relationship splits either the received heat impulse, $T^{+}\mathcal{B}^{+}+T^{-}\mathcal{B}^{-}$, or the received entropy impulse, $\mathcal{B}^{+}+\mathcal{B}^{-}$, between the two temperatures, $T^{+}$ and $T^{-}$. Let us note that this relationship depends on the future state via the average entropy $\bar{s}$.

We may choose a pseudo-potential of dissipation

[TABLE]

and constitutive laws

[TABLE]

It results from this choice that the internal forces satisfy inequality

[TABLE]

and the second law is satisfied

Theorem 2**.**

If constitutive laws (12), (14) and (15) are satisfied, then the second law is satisfied.

Proof.

If relationship (15) is satisfied, inequality (16) is satisfied. Then it is easy to prove that the inequality (10) which is equivalent to the second law is satisfied.

Remark 3**.**

The discontinuity $\left[\Psi\right]$ may be split in a different manner

[TABLE]

We get

[TABLE]

[TABLE]

If $\Psi$ it is a convex function of $(\beta,\mathbf{grad}$ $\beta)$

[TABLE]

[TABLE]

The internal forces $\left(\hat{B}^{fe},\hat{H}^{fe}\right)$ depend entirely on the past state $(T^{-},\beta_{i}^{-},\mathbf{grad}\beta_{i}^{-})$. Because we think that the constitutive laws sum up what occurs during the discontinuity, it is mandatory that the internal forces depend on the future state $(T^{+},\beta_{i}^{+},\mathbf{grad}$ $\beta_{i}^{+})$. Thus this splitting of the free energy does not seem as good as the one we have chosen.

2.6 The Free Energy

A shape memory alloy is considered as a mixture of the martensite and austenite phases with volume fractions $\beta_{i}$. The volume free energy of the mixture we choose is

[TABLE]

where the $\Psi_{i}$’s are the volume free energies of the ${i}$ phases and $h$ is a free energy describing interactions between the different phases. We have assumed that internal constraints are physical properties, hence, we decide to choose properly the two functions describing the material, i.e., the free energy $\Psi$ and the pseudo-potential of dissipation $\Phi$, in order to take these constraints into account. Since, the pseudo-potential describes the kinematic properties (i.e., properties which depend on the velocities) and the free energy describes the state properties, obviously the internal constraints

[TABLE]

and

[TABLE]

because voids may appear, are to be taken into account with the free energy $\Psi$, [17].

For this purpose, we assume the $\Psi_{i}$’s are defined over the whole linear space spanned by $\beta_{i}$ and the free energy is defined by

[TABLE]

We choose the very simple interaction free energy

[TABLE]

where $\vec{\beta}=(\beta_{1},\beta_{2},\beta_{3})$ and $I_{C}$ is the indicator function of the convex set

[TABLE]

Moreover, by $(k/2)\left|\mathbf{grad}\vec{\beta}\right|^{2}$ we mean the product of two tensors $\mathbf{grad}\vec{\beta}$ multiplied by the interfacial energy coefficient $(k/2)>0$. The terms $I_{C}(\vec{\beta})+(k/2)\left|\mathbf{grad}\vec{\beta}\right|^{2}$ may be seen as a mixture or interaction free-energy.

The only effect of $I_{C}(\vec{\beta})$ is to guarantee that the proportions $\beta_{1}$, $\beta_{2}$ and $\beta_{3}$ take admissible physical values, i.e. they satisfy constraints (18) and (19) (see also 20). The interaction free energy term $I_{C}(\vec{\beta})$ is equal to zero when the mixture is physically possible ($\vec{\beta}\in C$) and to $+\infty$ when the mixture is physically impossible ($\vec{\beta}\notin C$).

Let us note even if the free energy of the voids phase is [math], the voids phase has physical properties due to the interaction free energy term $(k/2)\left|\mathbf{grad}\vec{\beta}\right|^{2}$ which depends on the gradient of $\vec{\beta}$. It is known that this gradient is related to the interfaces properties: $\mathbf{grad}{\beta_{1}}$, $\mathbf{grad}{\beta_{2}}$ describes properties of the voids-martensites interfaces and $\mathbf{grad}{\beta_{3}}$ describes properties of the voids-austenite interface. In this setting, the voids have a role in the phase change and make it different from a phase change without voids. The model is simple and schematic but it may be upgraded by introducing sophisticated interaction free energy depending on $\vec{\beta}$ and on $\mathbf{grad}\vec{\beta}$.

Remark 4**.**

A slightly more sophisticated interfacial energy is

[TABLE]

with different interaction phase parameters $k_{i}$.

For the sake of simplicity, we choose the volume free energies, [13], [14], [18]

[TABLE]

where $K$ is the volume elastic tensor and $C$ the volume heat capacities of the phases and the quantity $l_{a}$ is the latent heat martensite-austenite volume phase change at temperature $T_{0}$, $\mathbf{1}$ is the unit tensor.

Concerning the function $\tau(T)$, we assume the schematic simple expression

[TABLE]

with $\overline{\tau}\leq 0$ and assume the temperature $T_{c}$ is greater than $T_{0}$. With those assumptions, it results

[TABLE]

Remark 5**.**

Depending on the sign of $\mathbf{I}:\varepsilon(\vec{u})=\mathop{\mathrm{d}iv}\vec{u}$, free energy $\Psi$ is either a concave or a convex function of temperature $T$. As explained in [18] (see Remark 5.3 page 72), it is easy to overcome this difficulty to have in any case $\Psi$ a concave function of $T$. Experiments show that rigidity matrix $K$ depends on $T$. With this result, it is easy to have $\Psi$ a concave function of $T$, [18]. In this presentation, we keep the schematic expression for $\Psi$ and we note that we will assume the solid is not deformed when colliding, i.e., $\mathop{\mathrm{d}iv}\vec{u}=0$. In this situation, the schematic free energy is a concave function of $T$.

2.7 The Pseudo-potential of Dissipation

From experiments, it is known that the behaviour of shape memory alloys depends on time, i.e., the behaviour is dissipative. We define a pseudo-potential of dissipation with

[TABLE]

where $\lambda\geq 0$ represents the thermal conductivity and $k_{v}>0$, $c\geq 0$, $\upsilon\geq 0$ stand for collisions viscosities related to macroscopic and microscopic dissipative phenomena.

Remark 6**.**

A slightly more sophisticated pseudo-potential of dissipation is

[TABLE]

involving different viscosities for the phases $c_{i}\geq 0$, $\upsilon_{i}\geq 0$.

The pseudo-potential takes into account the mass balance relationship (i.e., Eq. (5)), $I_{0}$ being the indicator function of the origin of $\mathbb{R}$.

2.8 The Constitutive Laws

They are given by relationships (13), (14) and (12)

[TABLE]

giving

[TABLE]

and by relationship (15)

[TABLE]

where $P$ is the percussion reaction pressure due to the mass balance, and

[TABLE]

The internal energy within the small perturbation assumption is

[TABLE]

2.9 The Equations in a Collision

They result from the energy balance, the equations of motion, the constitutive laws and the initial situation, i.e., the situation before the collision.

The Energy Balance

We assume adiabatic evolution

[TABLE]

The energy balance results from relationships (6), (12) and (13)

[TABLE]

Note that the reactions, for instance $\vec{B}_{reca}^{fe}$, work, with work

[TABLE]

This a property of collisions: the reactions to perfect constraints work whereas they do not work in smooth evolutions, [16].

The Equations of Motion and the Mass Balance

The equations of motion are (1), (2), (3) and (4). For the sake of simplicity, we assume the voids volume fraction, [28], [17],

[TABLE]

is null and does not evolve in the collision, [28], [17]. Assuming no interpenetration before collision

[TABLE]

the mass balance relationship (5) gives

[TABLE]

and

[TABLE]

The Constitutive Laws

For the sake of simplicity, we assume the material is undeformed $\varepsilon(\vec{u})=0$ at collision time. Thus, using also (26), we have

[TABLE]

and by relationship (15)

[TABLE]

It results internal force $\vec{B}^{p}$ is

[TABLE]

and by letting

[TABLE]

with

[TABLE]

[TABLE]

The angular momentum equation of motion (2) is satisfied and there are four equations for the unknowns $\vec{U}^{+}$, $\vec{\beta}^{+}$, $\hat{P}$ and $T^{+}$: the equations of motion for $\vec{U}^{+}$ and $\vec{\beta}^{+}$, the mass balance and the equation of motion related to $\vec{\beta}^{+}$ for the percussion $\hat{P}$ and the energy balance for the temperature $T^{+}$.

2.10 The Mechanical Equations

As already said, there is an external surface percussion $\vec{G}^{p}$, for instance an hammer stroke on part $\Gamma_{1}$ of the boundary, the solid being fixed to a support on part $\Gamma_{0}$, with $\Gamma_{0},\Gamma_{1}$ a partition of boundary $\delta\Omega$. We assume the solid is at rest before collision and its temperature is uniform

[TABLE]

The equations $\vec{U}^{+}$, $\vec{\beta}^{+}$ and $\hat{P}$ are

[TABLE]

The boundary conditions are

[TABLE]

where percussion $\vec{G}^{p}$ is the given hammer percussion on part $\Gamma_{1}$. The solid is fixed on an immobile support on part $\Gamma_{0}$.. Quantities $\beta^{-}$ before collision are known.

Remark 7**.**

We may note that any term of the free energy which depends on $\beta_{1}+\beta_{2}+\beta_{3}$ does not intervene in the equation giving $\vec{\beta}^{+}$ because its derivatives with respect to the $\beta_{i}$ are absorbed by the reaction percussion pressure $P$. This is the case of quantity $CT^{+}\text{Log}T^{+}$.

2.11 The Thermal Equation

The equation for $T^{+}$ is using the mass balance ((26))

[TABLE]

with boundary condition

[TABLE]

assuming no external heat impulse on part $\Gamma_{1}$ of the boundary and $\underline{T}$ or $T^{+}$ is given on part $\Gamma_{0}$. Note that another boundary condition may be

[TABLE]

assuming the surface heat impulse is proportional to the temperature difference with the exterior. Temperature $T^{-}$ before collision is known.

Quantity

[TABLE]

is the dissipated work due to the microscopic motions producing the phase change. Because the thermal effects are mainly due to the macroscopic velocity discontinuities, we assume it is negligible compared to the dissipated work

[TABLE]

due to the macroscopic motion. We assume also that in the internal energy quadratic quantity

[TABLE]

is negigible compared to

[TABLE]

Thus the thermal equation becomes

[TABLE]

3 Closed Form Example

We assume the solid is struck by an hammer and that the dissipated work is known

[TABLE]

neglecting the dissipated work due to phase changes.

The main assumption is that the volume fractions and temperatures are homogeneous, i.e., their values do not depends on space variable $x$. The equations become non linear algebraic equations

[TABLE]

We may prove that system (43) has one and only one solution depending on the quantities before collision.

We investigate the situation where a mixture of the three phases can coexist after the collision

[TABLE]

The equations are

[TABLE]

with

[TABLE]

We choose initial state

[TABLE]

which is an equilibrium when

[TABLE]

that we assume.

We get from the first equation, subtracting line 1 from line 2 and line 1 from line 3,

[TABLE]

With the last equation, we have

[TABLE]

It results

[TABLE]

Note that if there is not dissipation, the temperature has to be equal to the phase change temperature. From the second equation, we get

[TABLE]

giving

[TABLE]

Function $\mathcal{T\rightarrow}T^{+}$ is increasing. This is the solution as long as

[TABLE]

or

[TABLE]

To satisfy these conditions, the dissipated work $\mathcal{T}$ has to verify

[TABLE]

In case there is dissipation, $c>0$, the three phases may coexist at temperatures different from $T_{0}$ whereas the temperature has to be equal to $T_{0}$ in case there is not dissipation, $c=0$. Of course, depending on the temperature before collision, the dissipated work $\mathcal{T}$ has to be not too small and not too large. The complete phase change occurs for a dissipated work large enough. For a very weak hammer stroke there is only an increase of temperature and no phase change.

4 The PDE System: Existence and Uniqueness of Solutions

In this section we introduce some convex sets in order to eliminate one of volume fractions of $\vec{\beta}^{+}$ and reformulate equation of motion for the microscopic motions (29)–(34) with (38) and with the associated boundary conditions. Then an existence and uniqueness of solutions of the problem theorem is proved.

4.1 The Problem

The state

[TABLE]

with $\mathop{\mathrm{g}rad}T^{-}=0$ and velocity

[TABLE]

before collision are given. The unknowns are

[TABLE]

the state and velocity after collision. The equation are

• •

the mass balance

[TABLE]

• •

the equation of motion for the macroscopic motion

[TABLE]

• •

the equation of motion for the microscopic motions

[TABLE]

• •

the energy balance

[TABLE]

4.2 Some Convex Sets

Let define convex set

[TABLE]

It is a plane triangle in $\mathbb{R}^{3}$, intersection of tetrahedron $C$ and plane $\gamma_{1}+\gamma_{2}+\gamma_{3}=1$.

Let define function

[TABLE]

We have

[TABLE]

and

[TABLE]

We have equality

[TABLE]

because the interior of convex set $C$ is not empty, [26].

The plane triangle in $\mathbb{R}^{2}$

[TABLE]

is a convex set. It is easy to prove

[TABLE]

4.3 The Equation of Motion for the Microscopic Motions

By using relationship (48), the equations (29)-(35) become

[TABLE]

An easy computation using relationship (49) shows that system (53) is equivalent to

[TABLE]

4.4 Another Equation of Motion for the Microscopic

Motions

By choosing the more sophisticated interfacial energy (21) and pseudo-potential of dissipation (24) with

[TABLE]

Let us note that phase one still intervenes in the physical properties of the alloy because there are no voids, see [14], [18]

[TABLE]

With these choices the equations giving the $\beta_{i}^{+}$ are

[TABLE]

By using relationship (49) we get

[TABLE]

4.5 The PDE System

We introduce notations to make precise the mathematical formulation of the problem and prove the existence and uniqueness theorem.

4.5.1 Notation

In order to give a precise formulation of our problem, let us denote by $\Omega$ a bounded, convex set in $\mathbb{R}^{3}$ with $C^{1}$ boundary $\Gamma$. Let $(\Gamma_{0},\Gamma_{1})$ be a partition of $\partial\Omega$ into two measurable sets such that both $\Gamma_{0}$ has positive surface measure and it’s Lipschitz. Finally, we introduce the Hilbert triplet $(V,H,V^{\prime})$ where

[TABLE]

and identify, as usual, $H$ (which stands either for the space $L^{2}(\Omega)$ or for $(L^{2}(\Omega))^{3}$) with its dual space $H^{\prime}$, so that $V\hookrightarrow H\hookrightarrow V^{\prime}$ with dense and continuous embeddings. Moreover, we denote by $\|\cdot\|_{X}$ the norm in some space $X$ and by $\langle\cdot,\cdot\rangle$ the duality pairing between $V$ and $V^{\prime}$ and by $(\cdot,\cdot)$ the scalar product in $H$. In the space $V$ we introduce the inner product

[TABLE]

We define the Hilbert space

[TABLE]

endowed with the usual norm. In addition, we introduce on $\mathbf{W}\times\mathbf{W}$ a bilinear symmetric continuous form $a(\cdot,\cdot)$ defined by

[TABLE]

Note here that (since $\Gamma_{0}$ has positive measure), thanks to Korn’s inequality (cf., e.g., [7], [12, p. 110]), there exists a positive constant $c$ such that

[TABLE]

Moreover, we introduce the space

[TABLE]

and we do not distinguish in the notation the spaces $W^{1,p}(\Omega)$ and $(W^{1,p}(\Omega))^{3}$ as well as the norms $\|\cdot\|_{1,p}$ and $\|\cdot\|_{p}$ which stand for the usual norms in $W^{1,p}(\Omega)$, $W^{1,p}(\Omega)^{3}$ and $L^{p}(\Omega$, $(L^{p}(\Omega))^{3}$, respectively.

We use equation (77) as equation of motion for the microscopic motions. Then, choosing for simplicity and without any loss of generality $\rho=k_{v}=c=\upsilon+k=\lambda=1$, $l_{a}=T_{0}$, and $\beta_{2}^{-}=\beta_{3}^{-}=0$, $\vec{U}^{-}=0$, we can rewrite our system, coupling (29), (34), (35), (38), and (77), in the new variables $(\vartheta,{\mathbf{u}},\chi_{2},\chi_{3})$, corresponding to the previous $(T^{+},\vec{U}^{+},\beta_{2},\beta_{3})$, as

[TABLE]

Notice that $\varepsilon_{ij}({\mathbf{u}}):=(\partial_{x_{j}}u_{i}+\partial_{x_{i}}u_{j})/2,$ $i,j=1,2,3,$ are the components of the standard linearized strain tensor $\varepsilon({\mathbf{u}})$ and $\mathbf{n}$ stands for the outward normal vector to $\Gamma.$ Concerning data, $f:\Omega\rightarrow\mathbb{R}$ represents a known source term (it was $f=-\Delta T^{-}+T^{-}+T_{0}\beta_{3}^{-}$ in the previous sections), $h_{i}:\Omega\rightarrow\mathbb{R}$, $i=2,3$ are given boundary data (it was $h_{i}=\upsilon\partial_{\mathbf{n}}\beta_{i}^{-}$ in previous sections and so we should have $h_{i}=0$ here, but we prefer to let it be more general, different from 0), $\Pi:\Gamma\rightarrow\mathbb{R}$ denotes an energy flux coming from the exterior of the system (it was $\Pi=-\partial_{\mathbf{n}}T^{-}$ in the previous sections), and $\mathbf{g}:\Gamma_{1}\rightarrow\mathbb{R}^{3}$ yields the external contact force applied to $\Gamma_{1}$ (it was $\mathbf{g}=\vec{G}^{p}$ in the previous sections). Moreover the maximal monotone graph $\partial I_{K},$ representing the subdifferential of the indicator function $I_{K}$ of the plane triangle $K$. Set $K$ is convex and contains the admissible phase proportions. We also notice that $I_{K}(\chi_{2},\chi_{3})=0$ if $[\chi_{2},\chi_{3}]\in K,$  $=+\infty$ otherwise. For definitions and basic properties of maximal monotone operators and subdifferentials of convex functions, we refer, for instance, to [4], [26].

Let us comment now on the fact that, with the help of the usually considered boundary conditions (92)–(93) (see, e.g., [11]), from (89) it turns out that (cf.  [12, Thm. 6.2, p. 168] and [29, Thm. 1.1, p. 437]), at almost any time $t,$ the velocity ${\mathbf{u}}$ can be completely determined in terms of the datum $\mathbf{g}$. Thus, we may introduce the operator $F_{\mathbf{g}}$, which maps $\mathbf{g}$ into $\varepsilon({\mathbf{u}}):\varepsilon({\mathbf{u}})$ where $\mathbf{u}$ stands for the related solution of (89), (92)–(93). Then, you consider the following system, denoted by $(SMA)$, whose unknowns are now the absolute temperature $\theta$ and the phase variables $\chi_{2},\chi_{3}$.

[TABLE]

Remark 8**.**

Let us note here that in this formulation of our system we consider just the case $\beta_{2}^{-}=\beta_{3}^{-}=0$, $\vec{U}^{-}=0$ for simplicity, however our existence and uniqueness results could be extended to the case these quantities are different from zero sufficiently regular. Moreover, we could handle the case where inclusion (77) is replaced by (66) in the same way due to the fact that the only difference would be having a positive definite matrix acting on $(\chi_{2},\chi_{3})$ instead of the identity matrix, that could be treated in a very similar manner.

In case $\vec{U}^{-}\neq 0$ with D($$\vec{U}^{-})\neq 0, we choose a pseudo-potential of dissipation satisfying

[TABLE]

implying that there is no collision if there is no hammer stroke, $\vec{G}^{p}=0$. Nevertheless the schematic simple pseudo-potential of dissipation (22) is adapted to account for what occurs for large hammer strokes, i.e., for large $D(\vec{U}^{+}+\vec{U}^{-})/2$.

4.6 The Existence and Uniqueness result

Let us first make precise the statement regarding the operator $F_{\mathbf{g}}$, whose proof can be found as result of [12, Thm. 6.2, p. 168], a slight modification of [29, Thm. 1.1, p. 437], and [7, Thm. 6.3-.6, p. 296] (cf. also [27, p. 260]).

Lemma 9**.**

(Higher integrability of the gradient) Given $\mathbf{g}\in L^{2}(\Gamma_{1})^{3})$, there exists a unique ${\mathbf{u}}\in\mathbf{W}$ weak solution of

[TABLE]

Moreover, there exists $\gamma>0$ such that the following statement holds: whenever $2<p<2+\gamma$, $\mathbf{g}\in(L^{2p/3}(\Gamma_{1}))^{3},$, and ${\mathbf{u}}$ is a weak solution of (104), then ${\mathbf{u}}\in(W^{1,p}(\Omega))^{3}$ and

[TABLE]

where $\gamma$ depends only on $\Omega$, $\Gamma_{0}$, and $C$ depends only on $p$ and these quantities. Hence, if we denote by $F_{\mathbf{g}}$ the operator

[TABLE]

where ${\mathbf{u}}$ denotes the unique weak solution of (104) corresponding to $\mathbf{g}$, then, there holds

[TABLE]

where $C$ deoends on $\Omega$, $\Gamma_{0}$, and $p$. Finally, if $\Omega$ is a $C^{2}$ domain, the closures of $\Gamma_{0}$ and $\Gamma_{1}$ do not intersect and $\mathbf{g}\in(L^{2}(\Gamma_{1}))^{3}$, then ${\mathbf{u}}\in(H^{2}(\Omega))^{3}$ and the following two estimates hold true

[TABLE]

for every $\mathbf{g}_{i}\in(L^{2}(\Gamma_{1}))^{3}$, $i=1,2$, where the constant $C_{2}$ depends on the problem data and also on the $\|{\mathbf{u}}\|_{(H^{2}(\Omega))^{3}}$.

Remark 10**.**

Let us note that we could actually generalize here the form of the elasticity bilinear form $a$ including an elasticity matrix which do not need to be exactly the identity matrix but it needs to be a symmetric matrix, positive definite with $L^{\infty}$ coefficients. Moreover we could include a non-zero volume percussion on the right hand side in (104). The regularity requirement we would need in order to apply the results of [29] is $L^{p}(\Omega)^{3}$. In fact, also the third part of the Lemma extends to the case of an anisotropic and inhomogeneous material, for which the elasticity tensors $\mathrm{R}_{e}$ is of the form $\mathrm{R}_{e}=(g_{ijkh})$, with functions

[TABLE]

satisfying the classical symmetry and ellipticity conditions (with the usual summation convention)

[TABLE]

Notice moreover that in the latter part of Lemma 9 the compatibility condition between $\Gamma_{0}$ and $\Gamma_{1}$ is necessary. Indeed, without the latter geometric condition, the elliptic regularity results ensuring the (crucial) $(H^{2}(\Omega))^{3}$-regularity of ${\mathbf{u}}$ may fail to hold, see [7, Chap. VI, Sec. 6.3]. In particular, the last estimate (106), which will be used in order to prove the stability estimate (110), is obtained by means of Young inequality and the standard stability estimate for ${\mathbf{u}}$

[TABLE]

which can be proved by testing the diferences of (104)’s by ${\mathbf{u}}_{1}-{\mathbf{u}}_{2}$.

Now we are in the position to state and prove our existence and uniqueness result

Theorem 11**.**

(Existence and Uniqueness) Assume that $T_{0}\in\mathbb{R}$, there exists $\gamma>0$ such that $\mathbf{g}\in(L^{2p/3}(\Gamma_{1}))^{3}$ for some $2<p<2+\gamma$, $\Pi\in W^{1-1/p,p}(\Gamma)$, $h_{i}\in W^{1/2,2}(\Gamma)$, $i=2,3$, $f\in L^{2}(\Omega)$. Then there exists a unique solution $(\theta,\chi_{2},\chi_{3})\in(H^{1}(\Omega)\cap W^{2,p/2}(\Omega))\times H^{2}(\Omega)\times H^{2}(\Omega)$ of system (SMA) such such that

[TABLE]

where $C$ depends only on the problem data and on $p$. Finally, if $\Omega$ is a $C^{2}$ domain, the closures of $\Gamma_{0}$ and $\Gamma_{1}$ do not intersect and $\mathbf{g}_{i}\in(L^{2}(\Gamma_{1}))^{3}$, $f_{i}\in H$, $(h_{i})_{1},\,(h_{i})_{2}\in L^{2}(\Gamma)$, $i=1,2$, then the following stability estimate holds true

[TABLE]

where $(\theta_{i},(\chi_{2})_{i},(\chi_{3})_{i})$ are two solutions corresponding to data $\mathbf{g}_{i}\in(L^{2}(\Gamma_{1}))^{3}$, $f_{i}\in H$, $(h_{i})_{1},\,(h_{i})_{2}\in L^{2}(\Gamma)$, $i=1,2$, respectively, and the constant $C$ depends only on the data of the problem.

Proof of Theorem 11.

The main idea of the proof of existence of solutions is to use a fixed point argument (Shauder theorem), that is, to prove that a suitable operator

[TABLE]

admits at least a fixed point. To do that we employ a standard Shauder theorem by proving that the operator $D$ is continuous and compact. First we fix $(\tilde{\chi}_{2},\tilde{\chi}_{3})\in(L^{2}(\Omega)\times L^{2}(\Omega))\cap K$ and we substitute $\tilde{\chi}_{3}$ in ($SMA_{1}$) in place of $\chi_{3}$ and we find (cf. [22, Ch. 2]) a unique $\tilde{\theta}\in V\cap W^{2,p/2}(\Omega)$, $\tilde{\theta}=D_{1}((\tilde{\chi}_{2},\tilde{\chi}_{3}))$ solution of ($SMA_{1}$) coupled with ($SMA_{4}$) and since $|\tilde{\chi}_{3}|\leq C$, then $\|\tilde{\theta}\|_{V\cap W^{2,p/2}(\Omega)}\leq S_{1}$ independently of $\tilde{\chi}_{3}$. Consider now the differential inclusion ($SMA_{2}$), ($SMA_{3}$) with $\tilde{\theta}$ in place of $\theta$. Then, by applying known regularity results (cf., e.g., [5]), we have that there exists a unique solution $(\chi_{2},\chi_{3})=D_{2}(\tilde{\theta})\in H^{2}(\Omega)\times H^{2}(\Omega)$ and

[TABLE]

where $S_{2}$ and $S_{3}$ depend only on the data of the problem, but not on the choice of $(\tilde{\chi}_{2},\tilde{\chi}_{3})$. From the above argument it follows that the operator $D::\,L^{2}(\Omega)\times L^{2}(\Omega)\rightarrow H^{2}(\Omega)\times H^{2}(\Omega)$ specified by $D=D_{2}\circ D_{1}$ is a compact operator in $H\times H$. It remains to prove that the operator is continuous, but this follows from the following estimate. Let $\theta_{i}=D_{1}((\tilde{\chi}_{2})_{i}),(\tilde{\chi}_{3})_{i})$, $i=1,2$, then, we easily get

[TABLE]

Moreover, by monotonicity arguments, letting $((\chi_{2})_{i},(\chi_{3})_{i})=D_{2}(\theta_{i})$, $i=1,2$, we get

[TABLE]

which implies the continuity of the operator $D$. Uniqueness of solutions follows by taking the differences of two equations ($SMA_{1}$) and ($SMA_{2}$) and testing the first by the differences of $\theta$’s and the second by differences of $(\chi_{2},\chi_{3})$’s, solutions associated to the same data. This, thanks to a cancellation and to monotonicity arguments leads to the estimate

[TABLE]

This concludes the proof of the first part of the Theorem. The stability estimate (110) follows from the following estimate. We take the differences of the two equations ($SMA_{1}$) and ($SMA_{2}$) corresponding to the two solutions $(\theta_{i},(\chi_{2})_{i},(\chi_{3})_{i})$, $i=1,2$, and test them by $\theta_{1}-\theta_{2}$ and $((\chi_{2})_{1}-(\chi_{2})_{2},(\chi_{3})_{1}-(\chi_{3})_{2})$, respectively. Then summing up the two resulting equations, we get

[TABLE]

Using now (106), we obtain exactly (110). This concludes the proof of Theorem 11.

5 Numerical Examples

In order to obtain numerical results for engineering applications, the proposed predictive theory has been also implemented in a computational framework.

The thermomechanical state $(\vec{U}^{+},\vec{\beta}^{+},T^{+})$ after the collision is obtained by solving Eqs. (29), (33) and (38) by means of the finite element method. The post-collision velocity $\vec{U}^{+}$ is obtained from Eq. (29). Based on this solution, the post-collision alloy composition $\vec{\beta}^{+}$ and temperature $T^{+}$ are obtained by solving the coupled Eqs. (33) and (38) through an iterative numerical scheme [23].

Moreover, the evolution of the thermomechanical state of the structure after the collision is also predicted. To reach this goal, the constitutive model for SMA smooth evolution presented in [17, 18], and enriched in [23, 24], has been employed. The model accounts for the typical shape-memory (i.e., thermal induced transformations) and pseudoelastic (i.e., stress-induced transformations) effects in SMAs. It is based on a set of solving equations analogous to Eqs. (29), (33) and (38), but formulated under smooth evolution assumptions. Hence, employed equations allow to compute structure displacements, as well as the evolution of alloy composition and of the temperature field, taking the post-collision state $(\vec{U}^{+},\vec{\beta}^{+},T^{+})$ as initial condition. The solution strategy for the post-collision evolution is based on an incremental algorithm which employs an explicit Euler time discretization (i.e., an updated-Lagrangian formulation) and a finite-element spatial discretization.

Model parameters are chosen referring to a Ni-Ti alloy: $\rho=6500$ kg/m3, $l_{a}=80$ MJ/(m3), $C=5.4$ MJ/(m3K), $\lambda=18$ Ws/(Km), and $T_{o}=332.75$ K [25]. The finite-element discretizations employ quadratic Lagrange basis functions and a mesh element size equal about to one cent of the structure maximum size.

In agreement with the theoretical framework previously described, the collision is assumed to be adiabatic. Addressing the thermal problem in the smooth evolution after the collision, a convective heat transfer is prescribed on the boundary, with convection coefficient equal to $100$ W/(m2K) and external temperature equal to $T_{ext}=0.9T_{o}$.

5.1 A Surface Percussion is Applied to a Solid

We present numerical results for a two-dimensional SMA solid subjected to an external percussion, [23]. As depicted in Fig. 1, the solid is squared in shape ($L_{o}=1$ mm wide), fixed on the bottom face to an immobile obstacle and free on the left and right sides.

Before the collision, the solid is at rest ($U^{-}=0$) and at uniform temperature $T^{-}=T_{ext}=0.9T_{o}$, such that the alloy mixture results (for temperatures below the transformation temperature $T_{o}$, the martensitic phase is at equilibrium): $\beta_{1}^{-}=\beta_{2}^{-}=0.5$ and $\beta_{3}^{-}=0$. A percussion $\vec{G}^{p}$ inclined of an angle $\alpha=60^{\circ}$ with respect to the horizontal direction is applied at the center of the top face on a segment with length $L_{o}/3$. (see Fig. 1).

5.1.1 Velocity and Volume Fractions after the Collision

As shown in Fig. 1, a non null velocity field $\vec{U}^{+}$ is obtained after the collision. The temperature $T^{+}$ mainly increases where the percussion is applied and on the fixed constraint where a percussion reaction originates (see Fig. 1). In these regions, the dissipated work is large, determining the increase of the temperature. In turn, the latter is coupled with the appearance of the austenite phase $\beta_{3}^{+}$ (dominant at large temperature) and the disappearance of martensites $\beta_{1}^{+}=\beta_{2}^{+}$ (see Fig. 1).

5.1.2 Evolution following the Collision: Position and Austenite

Volume Fraction Depending on Time

The evolution of the thermomechanical state of the structure after the collision is shown in Fig. 2 where the time-evolution of the austenite volume fraction $\beta_{3}$ in the current configuration of the solid is reported.

The predicted response is affected by both collision-induced and stress-induced transformation mechanisms. The former determines indeed a non-uniform alloy mixture at the beginning of the post-collision evolution (i.e., $\vec{\beta}^{+}$). On the other hand, stress-induced transformations are due to the deformation of the structure induced by the post-collision velocity field (i.e., $\vec{U}^{+}$). In particular, as shown in Fig. 2, the solid globally rotates and the collision-induced austenite progressively disappears with time due to stress-induced phase change. Accordingly, martensites appear, associated with material pseudoelastic response.

Since material properties after the collision depend on $\vec{\beta}^{+}$, the post-collision structural deformation (associated with a non-trivial $\vec{U}^{+}$) is affected by collision-induced transformation mechanisms. Therefore, collision-induced and stress-induced mechanisms are strongly coupled each other in determining the post-collision evolution of the structure, both in terms of deformation and alloy composition. Clearly, austenite-martensite phase change mechanisms are also coupled with the evolution of the temperature, whose initial condition after the collision is given by the non-uniform field $T^{+}$.

Acknowledgments

The financial support of the FP7-IDEAS-ERC-StG #256872 (EntroPhase) and of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d’Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l’ATtRattività dell’ateneo pavese” is gratefully acknowledged. The paper also benefited from the support of the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) for ER. The financial support of the State of Lower Saxony (Germany) in the framework of the Masterplan “Smart Biotecs” is acknowledged by MM.