304
F. Vanderveken et al.
with B 1 and B 2 the linear isotropic and anisotropic magnetoelastic coupling constants, respectively (Jm
−3 ). It is worth noting that the magnitude of the saturation
magnetization has no influence on the magnetoelastic energy or strain state, which are
rather determined by the orientation of the magnetization vector. The magnetization
orientation is defined by the vector
ζ =
⎡
⎣
ζ x
ζ y
ζ z
⎤
⎦ =
1
M s
⎡
⎣
M x
M y
M z
⎤
⎦ .
(12.58)
Substituting this into (12.57) leads to
E mel =B 1
ε xx
ζ
2
x −
1
3
+ ε yy
ζ
2
y −
1
3
+ ε zz
ζ
2
z −
1
3
+ 2B 2
ε xy ζ x ζ y + ε yz ζ y ζ z + ε zx ζ x ζ z
.
(12.59)
Based on this expression for the magnetoelastic energy density, it is possible to
calculate the magnetostrictive body force, which is given by
f mel = ∇ · ¯
σ mel = ∇ ·
dE mel
dε ij
,
(12.60)
leading to
f mel = 2B 1
⎡
⎢
⎢
⎢
⎢
⎣
ζ x
∂ζ x
∂x
ζ y
∂ζ y
∂ y
ζ z
∂ζ z
∂z
⎤
⎥
⎥
⎥
⎥
⎦
+ B 2
⎡
⎢
⎢
⎢
⎢
⎣
ζ x
∂ζ y
∂y
+
∂ζ z
∂z
+ ζ y
∂ζ x
∂ y
+ ζ z
∂ζ x
∂z
ζ y
∂ζ x
∂ x
+
∂ζ z
∂z
+ ζ x
∂ζ y
∂ x
+ ζ z
∂ζ y
∂z
ζ z
∂ζ x
∂ x
+
∂ζ y
∂ y
+ ζ x
∂ζ z
∂ x
+ ζ y
∂ζ z
∂ y
⎤
⎥
⎥
⎥
⎥
⎦
.
(12.61)
There are three important parameters that determine the strength of the magnetostrictive body force: the magnetoelastic coupling constants, the magnetization orientation,
and the gradient of the magnetization orientation.
The magnetostrictive body force affects the elastodynamics and thus needs to be
added to the elastodynamic equation (12.44) in magnetostrictive media. This allows
for the analytical description of the influence of the magnetization direction on the
elastodynamics and the properties of (magneto-)elastic waves.
Another important quantity is the magnetostrictive strain, which is the additional
strain originating from the magnetostrictive effect. For a material with cubic symmetry, the magnetostrictive strain is given by [24, 42, 43]
¯
ε mel =
3
2
⎡
⎢
⎣
λ 100
ζ
2
x −
1
3
λ 111 ζ x ζ y
λ 111 ζ x ζ z
λ 111 ζ y ζ x λ 100
ζ
2
y −
1
3
λ 111 ζ y ζ z
λ 111 ζ z ζ x
λ 111 ζ z ζ y
λ 100
ζ
2
z −
1
3
⎤
⎥
⎦ .
(12.62)
F. Vanderveken et al.
with B 1 and B 2 the linear isotropic and anisotropic magnetoelastic coupling constants, respectively (Jm
−3 ). It is worth noting that the magnitude of the saturation
magnetization has no influence on the magnetoelastic energy or strain state, which are
rather determined by the orientation of the magnetization vector. The magnetization
orientation is defined by the vector
ζ =
⎡
⎣
ζ x
ζ y
ζ z
⎤
⎦ =
1
M s
⎡
⎣
M x
M y
M z
⎤
⎦ .
(12.58)
Substituting this into (12.57) leads to
E mel =B 1
ε xx
ζ
2
x −
1
3
+ ε yy
ζ
2
y −
1
3
+ ε zz
ζ
2
z −
1
3
+ 2B 2
ε xy ζ x ζ y + ε yz ζ y ζ z + ε zx ζ x ζ z
.
(12.59)
Based on this expression for the magnetoelastic energy density, it is possible to
calculate the magnetostrictive body force, which is given by
f mel = ∇ · ¯
σ mel = ∇ ·
dE mel
dε ij
,
(12.60)
leading to
f mel = 2B 1
⎡
⎢
⎢
⎢
⎢
⎣
ζ x
∂ζ x
∂x
ζ y
∂ζ y
∂ y
ζ z
∂ζ z
∂z
⎤
⎥
⎥
⎥
⎥
⎦
+ B 2
⎡
⎢
⎢
⎢
⎢
⎣
ζ x
∂ζ y
∂y
+
∂ζ z
∂z
+ ζ y
∂ζ x
∂ y
+ ζ z
∂ζ x
∂z
ζ y
∂ζ x
∂ x
+
∂ζ z
∂z
+ ζ x
∂ζ y
∂ x
+ ζ z
∂ζ y
∂z
ζ z
∂ζ x
∂ x
+
∂ζ y
∂ y
+ ζ x
∂ζ z
∂ x
+ ζ y
∂ζ z
∂ y
⎤
⎥
⎥
⎥
⎥
⎦
.
(12.61)
There are three important parameters that determine the strength of the magnetostrictive body force: the magnetoelastic coupling constants, the magnetization orientation,
and the gradient of the magnetization orientation.
The magnetostrictive body force affects the elastodynamics and thus needs to be
added to the elastodynamic equation (12.44) in magnetostrictive media. This allows
for the analytical description of the influence of the magnetization direction on the
elastodynamics and the properties of (magneto-)elastic waves.
Another important quantity is the magnetostrictive strain, which is the additional
strain originating from the magnetostrictive effect. For a material with cubic symmetry, the magnetostrictive strain is given by [24, 42, 43]
¯
ε mel =
3
2
⎡
⎢
⎣
λ 100
ζ
2
x −
1
3
λ 111 ζ x ζ y
λ 111 ζ x ζ z
λ 111 ζ y ζ x λ 100
ζ
2
y −
1
3
λ 111 ζ y ζ z
λ 111 ζ z ζ x
λ 111 ζ z ζ y
λ 100
ζ
2
z −
1
3
⎤
⎥
⎦ .
(12.62)