Name: volume change under hydrostatic pressure
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 21 s
Channel: JoVE

Der Volumenmodul misst die Beständigkeit eines Materials gegen gleichmäßige Kompression. Sie ist definiert als die Proportionalitätskonstante zwischen einer Druckänderung und der daraus resultierenden relativen Volumenänderung.
Betrachten Sie einen isotropen Würfel mit Volumeneinheit. Unter normalen Spannungen verformt es sich zu einem rechteckigen Quader mit einem neuen Volumen.
Die Differenz zwischen diesem neuen Volumen und dem ursprünglichen wird als Erweiterung des Materials bezeichnet. Die Dilatation kann als Summe der Dehnungen in alle drei Richtungen ausgedrückt werden.
Bei einem Körper, der einem gleichmäßigen hydrostatischen Druck ausgesetzt ist, entspricht jede Spannungskomponente dem negativen hydrostatischen Druck.
Setzt man diese Werte in die Dilatationsgleichung ein, erhält man einen Ausdruck, der die Konstante einführt, die als Volumenmodul bekannt ist und in den gleichen Einheiten wie der Elastizitätsmodul ausgedrückt wird.
Stabile Materialien unter hydrostatischem Druck verringern ihr Volumen, wodurch die Dilatation negativ und der Volumenmodul positiv wird.
Ein ideales Material mit einer Poissonzahl von Null kann sich dehnen, ohne sich seitlich zusammenzuziehen. Umgekehrt bedeutet die Poissonzahl von 0,5 eine perfekte Inkompressibilität.

volumenmodul

Der Kompressionsmodul ist ein wissenschaftlicher Begriff, der den Widerstand eines Materials gegenüber gleichmäßiger Kompression beschreibt. Es ist die Proportionalitätskonstante, die eine Druckänderung mit der daraus resultierenden relativen Volumenänderung verknüpft.

<img alt="Equation1" src="/files/ftp_upload/15601/15601_Equation_1.svg" style="height: 44px;">

Dieses Konzept wird deutlicher, wenn man sich ein isotropes Materialelement als Würfel mit Einheitsvolumen vorstellt. Wenn dieser Würfel Normalspannungen ausgesetzt wird, verformt er sich und verändert seine Form in ein rechteckiges Parallelepiped mit einem anderen Volumen. Die Diskrepanz zwischen diesem neuen Band und dem Original wird als Materialausdehnung bezeichnet. Die Dilatation kann als kumulative Summe der Dehnungen in den drei Raumrichtungen berechnet werden. Wenn der Körper unter gleichmäßigem hydrostatischem Druck steht, entspricht jede Spannungskomponente dem Negativ dieses Drucks. Durch Einsetzen dieser Werte in die Dilatationsformel erhält man einen Ausdruck, der den Kompressionsmodul einführt.

<img alt="Equation2" src="/files/ftp_upload/15601/15601_Equation_2.svg" style="height: 48px;">

Dieser Modul hat die gleichen Einheiten wie der Elastizitätsmodul. Unter hydrostatischem Druck verkleinern stabile Materialien ihr Volumen, wodurch die Dilatation negativ und der Kompressionsmodul positiv wird. Ein ideales Material mit einem Poisson-Verhältnis von Null könnte sich ohne seitliche Kontraktion in eine Richtung dehnen. Andererseits wäre ein Material mit einer Poissonzahl von 0,5 vollkommen inkompressibel.

Volumenmodul

Der Volumenmodul misst die Beständigkeit eines Materials gegen gleichmäßige Kompression. Sie ist definiert als die Proportionalitätskonstante zwischen einer Änderung des Drucks ...

Education

Lehre

JoVE Core

Mechanical Engineering

Maschinenbau

Unit 1

Einheit 1

Stress and Strain - Axial Loading

Spannung und Dehnung – axiale Belastung

KEY TERMS AND CONCEPTS

SCHLÜSSELBEGRIFFE UND KONZEPTE

Bulk Modulus

volume change under hydrostatic pressure

Bulk modulus describes how much a material resists a uniform squeeze. It links a change in pressure to the relative change in volume. For high school students, it is easiest to think of it as a measure of how hard it is to compress a material evenly from all sides.

The idea becomes clearer when an isotropic material element is treated as a cube with unit volume. When normal stresses act on the cube, it deforms into a rectangular parallelepiped and its volume changes. The difference between the new volume and the original volume is called dilatation. Dilatation is found by adding the strains in the three spatial directions.

Under uniform hydrostatic pressure, each stress component is equal to the negative of that pressure. Substituting those stress values into the dilatation formula leads to the expression for bulk modulus. The bulk modulus has the same units as the modulus of elasticity. Stable materials usually lose volume under hydrostatic pressure, so dilatation is negative and the bulk modulus is positive.

The transcript also connects bulk behavior to Poisson's ratio. An ideal material with a Poisson's ratio of zero could stretch in one direction without a lateral contraction. A material with a Poisson's ratio of 0.5 would be perfectly incompressible.

Volume Change Under Hydrostatic Pressure

normal strain under axial loading

Axial Stress and Strain in a Rod

Axial stress and strain in a rod describe how a material changes length when a force is applied along its axis. The force may stretch the rod or compress it. In mechanics of materials, this deformation is called normal strain. Normal strain is found by dividing the change in length by the original length. This gives a unitless ratio that shows how much the material deforms under load. A load-deformation diagram can show elongation, but it cannot predict how another rod will deform. When a...

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stress-strain diagram - ductile materials

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stress-strain diagram - brittle materials

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hooke's law

Stress-Strain Limits in Materials

Stress-strain limits in materials explain how solids respond when force is applied. Stress is the applied force per area, and strain is the change in shape or size that results. In the elastic range, strain stays directly proportional to stress. The constant that connects them is called the modulus of elasticity, or Young's modulus. This linear behavior continues only up to the proportional limit. After that point, the stress-strain relationship becomes nonlinear. For ductile materials, this...

plastic behavior

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fatigue

Fatigue Failure in Repeated Loading

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deformation of member under multiple loadings

deformation-of-member-under-multiple-loadings

Calculating Rod Stretch from Mixed Loads

Calculating rod stretch from mixed loads starts with breaking the rod into parts. This step is needed when the rod is made of different materials or has changes in cross-section. Each part has its own internal force, cross-sectional area, length, and modulus of elasticity. These values are used to find the deformation of the full rod. For a member with a variable cross-section, the strain is not constant. Instead, it changes with position along the rod. The deformation of a small element is...

statically indeterminate problem solving

Axial Rod Reactions with Redundant Support

Axial rod reactions with a redundant support are found by combining equilibrium with deformation. The structure here uses two cylindrical rods, one steel and one brass, joined at point B and held by rigid supports at points A and C. The goal is to determine the reactions at A and C and the deflection at B. This setup is statically indeterminate because the supports create more unknown reactions than the equilibrium equations can solve. Statics alone is not enough to find every internal force...

thermal strain

Thermal Expansion of a Free Rod

Thermal strain describes how a structure changes length when temperature changes. It is important in real-world design because structures do not always stay at one temperature. Temperature shifts can affect how they behave and how long they last. A simple example is a homogeneous rod with a uniform cross-section resting freely on a flat horizontal surface. When the rod is heated, it elongates. The amount of elongation depends on the temperature change and the rod’s original length. This...

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poisson's ratio

Material Strain Under Axial Loading

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generalized hooke's law

Multiaxial Stress and Strain Relation

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bulk modulus

Bulk modulus describes how much a material resists a uniform squeeze. It links a change in pressure to the relative change in volume. For high school students, it is easiest to think of it as a measure of how hard it is to compress a material evenly from all sides. The idea becomes clearer when an isotropic material element is treated as a cube with unit volume. When normal stresses act on the cube, it deforms into a rectangular parallelepiped and its volume changes. The difference between the...

shearing strain

Shear Modulus and Angular Change

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relation between poisson's ratio, modulus of elasticity and modulus of rigidity

relation-between-poisson-s-ratio-modulus-elasticity-modulus

Axial Load Effects on Shear Strain

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saint-venant's principle

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residual stresses

Residual Stresses After Welding and Cooling

Residual stresses are stresses that remain in a structure after the original load or heat source is removed. They often form when different parts of a material undergo uneven plastic deformation, which means permanent shape change. A rod stretched past its yield point is one example. It will not return to its original length, and some stress stays inside even after the load is gone. These stresses can also develop during welding. Heat changes the material during the weld and can cause plastic...

key terms and concepts

Learn how hydrostatic pressure changes a material’s volume, how dilatation is calculated, and why bulk modulus stays positive for stable materials.

Volume Change Under Hydrostatic Pressure | JoVE K12

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mechanical engineering

Volume Change Under Hydrostatic Pressure

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