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Average Theorems in Large Deformations
Appendix 1 Average Theorems in Large Deformations
In this appendix, the average theorems presented in Chapter 4 are extended to the case of large deformation processes. In the developed framework, all the operators are valid only in Cartesian coordinates and no curvilinear coordinates are considered. A1.1. Preliminaries A continuum body of volume V0 at time t0 is considered to be in the undeformed / reference configuration and occupies the space D0 , bounded by the surface ∂D0 with unit vector N (see Figure A1.1). At time t, the body has been moved and deformed. In the deformed / current configuration, it occupies the space D (volume Vt ), bounded by the surface ∂D with unit vector n. Any point P on this body can be described with the help of a fixed point O in two ways: (1) with the position vector X of the reference configuration and (2) the position vector x of the current configuration (Figure A1.1). Both X and x refer to Cartesian coordinate systems. In the following, simplified notation is utilized: {•}0 =
1 V0
{•}t =
1 Vt
D0
D
{•} dV ,
{•} dv ,
{•}0 = {•}t =
1 V0
1 Vt
{•} dS , ∂D0
{•} ds . ∂D
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Thermomechanical Behavior of Dissipative Composite Materials
N
n
∂D0
∂D D
D0
x
X
current configuration
reference configuration
Figure A1.1. Continuum body in reference and current configuration
In this formalism, the divergence theorem [2.11] and [2.10] for a tensor A of arbitrary order can be written in compact form as divA0 = A·N 0 and divAt = A·nt . For a second-order tensor A, the following properties hold: AT 0 = AT0 , , TA t = ATt ,
,
. .
AT AT
/ /
0
= AT0 ,
t
= ATt .
Moreover, the periodic tensor properties described in Section 4.1 hold also in the large deformation framework (either in the reference space D0 or in the current space D). A1.2. Hill’s Lemma and the Hill–Mandel theorem In this section, there is frequent use of the tensor operations and the definitions presented in Chapter 1. A1.2.1. Mechanical problem The following notion for the reference configuration is adopted:
Appendix 1
241
D EFINITION A1.1.– Kinematically admissible deformation gradient F is considered to refer to every symmetric second-order tensor that is related with the displacement vector u through the relation F = I + Gradu = Grad(u + X) . D EFINITION A1.2.– Statically admissible Piola stress Σ refers to every symmetric second-order tensor that (ignoring body forces) satisfies the equilibrium equation DivΣ = 0 . Using these definitions and the divergence theorem [2.11], the volume average for a kinematically admissible deformation gradient is given by (see the properties in section 1.1): F 0 = Grad(u + X)0 = Div ([u + X] ⊗ I)0 = [u + X] ⊗ N 0 ,
[A1.1]
and for a statically admissible Piola stress: Σ0 = GradX ˜· ΣT0 = Div (X ⊗ Σ)T0 − X ⊗ DivΣT0 = [Σ·N ] ⊗ X0 .
[A1.2]
The average mechanical work on the body, produced by a kinematically admissible strain F and a statically admissible stress Σ, is half of the quantity: Σ:F 0 = Σ:Grad(u + X)0 = Div ([u + X]·Σ)0 − (u + X)·DivΣ0 = [u + X]·[Σ·N ]0 .
[A1.3]
Based on the above definitions and properties, the Hill’s lemma is expressed in the following way.
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Thermomechanical Behavior of Dissipative Composite Materials
L EMMA A1.1.– Let F be a kinematically admissible deformation gradient and Σ be a statically admissible Piola stress. Then, it holds that Σ:F 0 − Σ0 : F 0 = [u + X − F 0 ·X] · [Σ·N − Σ0 ·N ]0 .
[A1.4]
Indeed, using the divergence theorem [2.11] and equations [A1.1], [A1.2] and [A1.3] we obtain [u + X − F 0 ·X] · [Σ·N − Σ0 ·N ]0 = [u + X]·[Σ·N ]0 − F 0 : [Σ·N ] ⊗ X0 − Σ0 : [u + X] ⊗ N 0 + [F 0 · Σ0 ]: X ⊗ N 0 ˜ = Σ:F 0 − F 0 : Σ0 − Σ0 : F 0 + F 0 : Σ0 = Σ:F 0 − Σ0 : F 0 . Additionally, the Hill–Mandel theorem can be stated in the following way. T HEOREM A1.1.– Let F be a kinematically admissible deformation gradient and Σ be a statically admissible Piola stress. Then, the three types of conditions: 1) u = [F 0 − I] ·X on the boundary ∂D0 ; 2) Σ·N = Σ0 ·N on the boundary ∂D0 ; 3) F = F 0 + Gradz, with z periodic and Σ·N anti-periodic; satisfy the energy equivalence Σ:F 0 = Σ0 : F 0 .
[A1.5]
For the first two conditions, the proof is a direct consequence of the Hill’s lemma [A1.4]. For the third condition, using the properties of tensor derivatives discussed in section 1.1, the divergence theorem [2.11], the
Appendix 1
243
definition A1.2 and the properties of the products between periodic and anti-periodic vectors described in Section 4.1, we obtain Σ:F 0 = Σ0 : F 0 + Σ:Gradz0 = Σ0 : F 0 + Σ:Gradz + z ·DivΣ0 = Σ0 : F 0 + Div (z ·Σ)0 = Σ0 : F 0 + z · [Σ·N ]0 = Σ0 : F 0 . Similar results can be obtained for the current configuration by considering the inverse of deformation gradient F −1 as the kinematically admissible tensor and the Cauchy stress σ as the statically admissible tensor. N OTE.– The Hill’s lemma and the Hill–Mandel theorem are valid for the ˙ products Σ: F˙ and Σ:F . The proof is analogous. A1.2.2. Thermal and other problems The thermal problem is treated in exactly the same way as in the small deformations case. The only difference is that ∇θ and q are substituted by ∇θ 0 and q 0 , respectively. With regard to electric or magnetic phenomena, appropriate forms of the Hill’s lemma and the Hill–Mandel theorem can be derived whether scalar or vector potentials are utilized [CHA 14]. A1.3. Useful identities – For the boundary conditions of the theorem A1.1 and admissible fields, the following hold: , 1) Σ·F T 0 = Σ0 · F T0 ; , 2) F ·ΣT 0 = F 0 · ΣT0 . The proof of this statement is as follows. For any type of boundary condition, the properties of tensor derivatives discussed in section 1.1, the
244
Thermomechanical Behavior of Dissipative Composite Materials
definitions A1.1, A1.2 and the divergence theorem [2.11] allow us to write ,
Σ·F T
0
1 0 = Σ· [Grad(u + X)]T
0
= Σ ˜· Grad(u + X) + DivΣ ⊗ [u + X]0 1 0 = [Div ([u + X] ⊗ Σ)]T = Div ([u + X] ⊗ Σ)T0 0
= [u + X] ⊗ ,
F ·ΣT
0
[Σ·N ]T0
= [Σ·N ] ⊗ [u + X]0 ,
= Grad(u + X) ˜· Σ0 = Grad(u + X) ˜· Σ + [u + X] ⊗ DivΣ0 = Div ([u + X] ⊗ Σ)0 = [u + X] ⊗ [Σ·N ]0 .
At this point, the proof is split in three parts: 1) For u = [F 0 − I]·X on ∂D0 , the use of equation [A1.2] yields ,
Σ·F T
0
= [Σ·N ] ⊗ [F 0 ·X]0 = [Σ·N ] ⊗ X0 · F T0 = Σ0 · F T0 ,
,
F ·ΣT
0
= [F 0 ·X] ⊗ [Σ·N ]0 = F 0 · X ⊗ [Σ·N ]0 = F 0 · [Σ·N ] ⊗ XT0 = F 0 · ΣT0 .
2) For Σ·N = Σ0 ·N on ∂D0 , the use of equation [A1.1] yields ,
Σ·F T
0
= [Σ0 ·N ] ⊗ [u + X]0 = Σ0 · N ⊗ [u + X]0 = Σ0 · [u + X] ⊗ N T0 = Σ0 · F T0 ,
,
F ·ΣT
0
= [u + X] ⊗ [Σ0 ·N ]0 = [u + X] ⊗ N 0 · ΣT0 = F 0 · ΣT0 .
Appendix 1
245
3) For F = F 0 + Gradz with z periodic and Σ·N anti-periodic, the use of the properties of the products between periodic and anti-periodic vectors described in section 4.1 yields 1 0 , Σ·F T 0 = Σ0 · F T0 + Σ· [Gradz]T
0
= Σ0 ·
F T0
= Σ0 ·
F T0
+ Σ ˜· Gradz + DivΣ ⊗ z0 1 0 = Σ0 · F T0 + [Div (z ⊗ Σ)]T 0
+ Div (z ⊗
Σ)T0
= Σ0 · F T0 + z ⊗ [Σ·N ]T0 = Σ0 · F T0 , ,
F ·ΣT
0
= F 0 · ΣT0 + Gradz ˜· Σ0 = F 0 · ΣT0 + Gradz ˜· Σ + z ⊗ DivΣ0 = F 0 · ΣT0 + Div (z ⊗ Σ)0 = F 0 · ΣT0 + z ⊗ [Σ·N ]0 = F 0 · ΣT0 .
– The conservation of mass [2.16] and the properties of Table 1.3 lead to the relations J0 =
Vt , V0
, −1 V0 J = , t Vt
, V0 ρt = ρ0 J −1 t = ρ0 0 . Vt
– For the boundary conditions of the theorem A1.1 and admissible fields, the following relation holds: σt =
V0 Σ0 · F T0 . Vt
Indeed, using the first identity in this section, the relation between Cauchy and Piola stress tensors, σ = J −1 Σ · F T , and the properties of Table 1.3, we
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Thermomechanical Behavior of Dissipative Composite Materials
obtain , V0 , V0 Σ·F T 0 = σt = J −1 Σ·F T t = Σ0 · F T0 . Vt Vt – In a “brick” shaped body, periodic fields in undeformed configuration remain periodic in the deformed configuration. Extended discussion about this point can be found in the literature [COS 05, BET 12]. A “brick” shaped body is defined by three linearly independent vectors r i , i = 1, 2, 3, mutually orthogonal. Under these conditions, the boundary of the body consists of six faces such that for each face F we can find a unique corresponding face Fh that can be viewed as the translation of F by one of the six vectors ±r i . For every point X ∈ F , a unique point X h ∈ Fh exists such that X − X h = ±r i . Under periodic motion, it holds that u(X, t) = [F 0 − I]·X + z(X, t), with z(X, t) = z(X h , t). So, we obtain u(X, t) − uh (X h , t) = [F 0 − I]· [X − X h ] = ±[F 0 − I]·r i , where the vector ±[F 0 − I]·r i is unique for the pair of homologous faces to which X and X h belong. Thus, the two faces translate by a constant vector in the deformed configuration. The main consequence of the last equation is that fields that are periodic and anti-periodic over the body preserve their character throughout the deformation process even when viewed as fields defined over the deformed configuration.
In this appendix, the average theorems presented in Chapter 4 are extended to the case of large deformation processes. In the developed framework, all the operators are valid only in Cartesian coordinates and no curvilinear coordinates are considered. A1.1. Preliminaries A continuum body of volume V0 at time t0 is considered to be in the undeformed / reference configuration and occupies the space D0 , bounded by the surface ∂D0 with unit vector N (see Figure A1.1). At time t, the body has been moved and deformed. In the deformed / current configuration, it occupies the space D (volume Vt ), bounded by the surface ∂D with unit vector n. Any point P on this body can be described with the help of a fixed point O in two ways: (1) with the position vector X of the reference configuration and (2) the position vector x of the current configuration (Figure A1.1). Both X and x refer to Cartesian coordinate systems. In the following, simplified notation is utilized: {•}0 =
1 V0
{•}t =
1 Vt
D0
D
{•} dV ,
{•} dv ,
{•}0 = {•}t =
1 V0
1 Vt
{•} dS , ∂D0
{•} ds . ∂D
240
Thermomechanical Behavior of Dissipative Composite Materials
N
n
∂D0
∂D D
D0
x
X
current configuration
reference configuration
Figure A1.1. Continuum body in reference and current configuration
In this formalism, the divergence theorem [2.11] and [2.10] for a tensor A of arbitrary order can be written in compact form as divA0 = A·N 0 and divAt = A·nt . For a second-order tensor A, the following properties hold: AT 0 = AT0 , , TA t = ATt ,
,
. .
AT AT
/ /
0
= AT0 ,
t
= ATt .
Moreover, the periodic tensor properties described in Section 4.1 hold also in the large deformation framework (either in the reference space D0 or in the current space D). A1.2. Hill’s Lemma and the Hill–Mandel theorem In this section, there is frequent use of the tensor operations and the definitions presented in Chapter 1. A1.2.1. Mechanical problem The following notion for the reference configuration is adopted:
Appendix 1
241
D EFINITION A1.1.– Kinematically admissible deformation gradient F is considered to refer to every symmetric second-order tensor that is related with the displacement vector u through the relation F = I + Gradu = Grad(u + X) . D EFINITION A1.2.– Statically admissible Piola stress Σ refers to every symmetric second-order tensor that (ignoring body forces) satisfies the equilibrium equation DivΣ = 0 . Using these definitions and the divergence theorem [2.11], the volume average for a kinematically admissible deformation gradient is given by (see the properties in section 1.1): F 0 = Grad(u + X)0 = Div ([u + X] ⊗ I)0 = [u + X] ⊗ N 0 ,
[A1.1]
and for a statically admissible Piola stress: Σ0 = GradX ˜· ΣT0 = Div (X ⊗ Σ)T0 − X ⊗ DivΣT0 = [Σ·N ] ⊗ X0 .
[A1.2]
The average mechanical work on the body, produced by a kinematically admissible strain F and a statically admissible stress Σ, is half of the quantity: Σ:F 0 = Σ:Grad(u + X)0 = Div ([u + X]·Σ)0 − (u + X)·DivΣ0 = [u + X]·[Σ·N ]0 .
[A1.3]
Based on the above definitions and properties, the Hill’s lemma is expressed in the following way.
242
Thermomechanical Behavior of Dissipative Composite Materials
L EMMA A1.1.– Let F be a kinematically admissible deformation gradient and Σ be a statically admissible Piola stress. Then, it holds that Σ:F 0 − Σ0 : F 0 = [u + X − F 0 ·X] · [Σ·N − Σ0 ·N ]0 .
[A1.4]
Indeed, using the divergence theorem [2.11] and equations [A1.1], [A1.2] and [A1.3] we obtain [u + X − F 0 ·X] · [Σ·N − Σ0 ·N ]0 = [u + X]·[Σ·N ]0 − F 0 : [Σ·N ] ⊗ X0 − Σ0 : [u + X] ⊗ N 0 + [F 0 · Σ0 ]: X ⊗ N 0 ˜ = Σ:F 0 − F 0 : Σ0 − Σ0 : F 0 + F 0 : Σ0 = Σ:F 0 − Σ0 : F 0 . Additionally, the Hill–Mandel theorem can be stated in the following way. T HEOREM A1.1.– Let F be a kinematically admissible deformation gradient and Σ be a statically admissible Piola stress. Then, the three types of conditions: 1) u = [F 0 − I] ·X on the boundary ∂D0 ; 2) Σ·N = Σ0 ·N on the boundary ∂D0 ; 3) F = F 0 + Gradz, with z periodic and Σ·N anti-periodic; satisfy the energy equivalence Σ:F 0 = Σ0 : F 0 .
[A1.5]
For the first two conditions, the proof is a direct consequence of the Hill’s lemma [A1.4]. For the third condition, using the properties of tensor derivatives discussed in section 1.1, the divergence theorem [2.11], the
Appendix 1
243
definition A1.2 and the properties of the products between periodic and anti-periodic vectors described in Section 4.1, we obtain Σ:F 0 = Σ0 : F 0 + Σ:Gradz0 = Σ0 : F 0 + Σ:Gradz + z ·DivΣ0 = Σ0 : F 0 + Div (z ·Σ)0 = Σ0 : F 0 + z · [Σ·N ]0 = Σ0 : F 0 . Similar results can be obtained for the current configuration by considering the inverse of deformation gradient F −1 as the kinematically admissible tensor and the Cauchy stress σ as the statically admissible tensor. N OTE.– The Hill’s lemma and the Hill–Mandel theorem are valid for the ˙ products Σ: F˙ and Σ:F . The proof is analogous. A1.2.2. Thermal and other problems The thermal problem is treated in exactly the same way as in the small deformations case. The only difference is that ∇θ and q are substituted by ∇θ 0 and q 0 , respectively. With regard to electric or magnetic phenomena, appropriate forms of the Hill’s lemma and the Hill–Mandel theorem can be derived whether scalar or vector potentials are utilized [CHA 14]. A1.3. Useful identities – For the boundary conditions of the theorem A1.1 and admissible fields, the following hold: , 1) Σ·F T 0 = Σ0 · F T0 ; , 2) F ·ΣT 0 = F 0 · ΣT0 . The proof of this statement is as follows. For any type of boundary condition, the properties of tensor derivatives discussed in section 1.1, the
244
Thermomechanical Behavior of Dissipative Composite Materials
definitions A1.1, A1.2 and the divergence theorem [2.11] allow us to write ,
Σ·F T
0
1 0 = Σ· [Grad(u + X)]T
0
= Σ ˜· Grad(u + X) + DivΣ ⊗ [u + X]0 1 0 = [Div ([u + X] ⊗ Σ)]T = Div ([u + X] ⊗ Σ)T0 0
= [u + X] ⊗ ,
F ·ΣT
0
[Σ·N ]T0
= [Σ·N ] ⊗ [u + X]0 ,
= Grad(u + X) ˜· Σ0 = Grad(u + X) ˜· Σ + [u + X] ⊗ DivΣ0 = Div ([u + X] ⊗ Σ)0 = [u + X] ⊗ [Σ·N ]0 .
At this point, the proof is split in three parts: 1) For u = [F 0 − I]·X on ∂D0 , the use of equation [A1.2] yields ,
Σ·F T
0
= [Σ·N ] ⊗ [F 0 ·X]0 = [Σ·N ] ⊗ X0 · F T0 = Σ0 · F T0 ,
,
F ·ΣT
0
= [F 0 ·X] ⊗ [Σ·N ]0 = F 0 · X ⊗ [Σ·N ]0 = F 0 · [Σ·N ] ⊗ XT0 = F 0 · ΣT0 .
2) For Σ·N = Σ0 ·N on ∂D0 , the use of equation [A1.1] yields ,
Σ·F T
0
= [Σ0 ·N ] ⊗ [u + X]0 = Σ0 · N ⊗ [u + X]0 = Σ0 · [u + X] ⊗ N T0 = Σ0 · F T0 ,
,
F ·ΣT
0
= [u + X] ⊗ [Σ0 ·N ]0 = [u + X] ⊗ N 0 · ΣT0 = F 0 · ΣT0 .
Appendix 1
245
3) For F = F 0 + Gradz with z periodic and Σ·N anti-periodic, the use of the properties of the products between periodic and anti-periodic vectors described in section 4.1 yields 1 0 , Σ·F T 0 = Σ0 · F T0 + Σ· [Gradz]T
0
= Σ0 ·
F T0
= Σ0 ·
F T0
+ Σ ˜· Gradz + DivΣ ⊗ z0 1 0 = Σ0 · F T0 + [Div (z ⊗ Σ)]T 0
+ Div (z ⊗
Σ)T0
= Σ0 · F T0 + z ⊗ [Σ·N ]T0 = Σ0 · F T0 , ,
F ·ΣT
0
= F 0 · ΣT0 + Gradz ˜· Σ0 = F 0 · ΣT0 + Gradz ˜· Σ + z ⊗ DivΣ0 = F 0 · ΣT0 + Div (z ⊗ Σ)0 = F 0 · ΣT0 + z ⊗ [Σ·N ]0 = F 0 · ΣT0 .
– The conservation of mass [2.16] and the properties of Table 1.3 lead to the relations J0 =
Vt , V0
, −1 V0 J = , t Vt
, V0 ρt = ρ0 J −1 t = ρ0 0 . Vt
– For the boundary conditions of the theorem A1.1 and admissible fields, the following relation holds: σt =
V0 Σ0 · F T0 . Vt
Indeed, using the first identity in this section, the relation between Cauchy and Piola stress tensors, σ = J −1 Σ · F T , and the properties of Table 1.3, we
246
Thermomechanical Behavior of Dissipative Composite Materials
obtain , V0 , V0 Σ·F T 0 = σt = J −1 Σ·F T t = Σ0 · F T0 . Vt Vt – In a “brick” shaped body, periodic fields in undeformed configuration remain periodic in the deformed configuration. Extended discussion about this point can be found in the literature [COS 05, BET 12]. A “brick” shaped body is defined by three linearly independent vectors r i , i = 1, 2, 3, mutually orthogonal. Under these conditions, the boundary of the body consists of six faces such that for each face F we can find a unique corresponding face Fh that can be viewed as the translation of F by one of the six vectors ±r i . For every point X ∈ F , a unique point X h ∈ Fh exists such that X − X h = ±r i . Under periodic motion, it holds that u(X, t) = [F 0 − I]·X + z(X, t), with z(X, t) = z(X h , t). So, we obtain u(X, t) − uh (X h , t) = [F 0 − I]· [X − X h ] = ±[F 0 − I]·r i , where the vector ±[F 0 − I]·r i is unique for the pair of homologous faces to which X and X h belong. Thus, the two faces translate by a constant vector in the deformed configuration. The main consequence of the last equation is that fields that are periodic and anti-periodic over the body preserve their character throughout the deformation process even when viewed as fields defined over the deformed configuration.